**Modern Methods (Without Determining the Contact Angle and Surface Tension) for Estimating the Surface Properties of Materials (Using Video and Computer Technology)**

A.O. Titov, I.I. Titova, M.O. Titov and O.P. Titov

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/61041

#### **Abstract**

In the study of surface phenomena including nanotechnology, the main and only instrumentally determined parameters are the surface tension and the contact angle. These indicators have been introduced over 200 years ago, and any new inventions in this area do not exist anymore. In line with this, we have developed a new method and device for determining the surface activity.

The basis of the method and the device makes use of video cameras to record the droplet size and changes in the surface layer of the known thickness of the liquid droplets from the impact of surfactant substances (surfactants). Committed changes are then processed by computer programs and calculated parameters, which can be characterized by the surfactant, the surface where the liquid is and the liquid itself. Determining the surface tension or contact angle is not necessary.

Exploring the possibility of estimating the surface properties of bulk and powdered materials, without determining the surface tension and contact angle, moving particles that are conventionally divided into six groups have been detected. The possibility of moving objects to glow is suggested by a possible mechanism of movement and glow through air oxidation of the organic compounds used as surfactants in the experiments. Score particle velocity indicates that they may move at a speed of 10 - 15 and 100 - 150 mm / sec. The results of the evaluation of the surface properties of the particulate material were obtained without measuring the contact angle and surface tension.

© 2015 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

**Keywords:** Surface tension, contact angle, the thickness of the liquid layer, the indi‐ cator grid, moving particles, , speed of movement, agglomerate, fast moving, glow, silica, nanoparticles

## **1. Introduction**

#### **1.1. Determination method of surface activity indicators that move liquid surfactants [1]**

Surface activity indicators when using this method and the device are:


A new method for determining the surface activity is carried out on the installation (see. Figure 1). [2]

**Sheme 1.** General scheme of installation. 1. A table with an adjustable horizontal; 2. Object - preparation (surface un‐ der study); 3. The camcorder recorded the size of the drop; 4. The camcorder recorded changes in the surface of the object - the drug; 5. Pipette; 6. Indicator mesh; 7. Lights

On the table with an adjustable level of the horizontal surface of the stack 1 of the plate material properties 2, the surface of which should be investigated (Scheme 1 indicates how an object is prepared). For the retention of the investigated liquid on a surface layer with 0.1-1 mm thickness, the material deposited on the circumference of a hydrophobic material depends if the liquid is polar or not, or if the liquid are solutions of various substances. The effect of which Modern Methods (Without Determining the Contact Angle and Surface Tension) for Estimating the Surface… http://dx.doi.org/10.5772/61041 181

**Figure 1.** Measurements: A - B range displaced liquid layer. A - C size scaling

**Keywords:** Surface tension, contact angle, the thickness of the liquid layer, the indi‐ cator grid, moving particles, , speed of movement, agglomerate, fast moving, glow,

**1.1. Determination method of surface activity indicators that move liquid surfactants [1]**

**1.** The amount of liquid which can move one kilogram of surfactant. The value of this parameter varies from tens up to hundreds of thousands of units. The indicator can be

**2.** The amount of fluid retained in the surface per unit time is calculated based on the first index and the surfactant supplements characteristic may be indicative of the characteris‐

**3.** The velocity of propagation of capillary and microwaves. This indicator complements the

A new method for determining the surface activity is carried out on the installation (see. Figure

**Sheme 1.** General scheme of installation. 1. A table with an adjustable horizontal; 2. Object - preparation (surface un‐ der study); 3. The camcorder recorded the size of the drop; 4. The camcorder recorded changes in the surface of the

On the table with an adjustable level of the horizontal surface of the stack 1 of the plate material properties 2, the surface of which should be investigated (Scheme 1 indicates how an object is prepared). For the retention of the investigated liquid on a surface layer with 0.1-1 mm thickness, the material deposited on the circumference of a hydrophobic material depends if the liquid is polar or not, or if the liquid are solutions of various substances. The effect of which

Surface activity indicators when using this method and the device are:

silica, nanoparticles

converted to units of energy - joules.

tics of the surface and the liquid.

object - the drug; 5. Pipette; 6. Indicator mesh; 7. Lights

**1. Introduction**

180 Surface Energy

first two.

1). [2]

should be further investigated. Then, the camera or film camera 4 is set so that the boundary line and the center of the bounding figures were clearly visible in the viewfinder and on the possibility of occupying the entire frame (Image sharpness adjustment). After setting up the sharpness, the line is set with a scale of 1 mm, and the camera fixed for subsequent scaling measurements. The line is set perpendicular to the optical axis of the lens fixing process chamber 4 exactly at the diameter of the circle. After that, the line is removed.

In limited hydrophilic or hydrophobic substance circles, the test liquid is measured in an amount necessary to produce the liquid layer thickness selected by the researcher.

Exactly over the center of the bounding shapes, such as circles, a calibrated weight drop was established at the diameter of the capillary pipette tip 5 so that the drop fell from it as accurately as possible at the center of the figure. The edge of the pipette tip was mounted at a height of 4-30 mm. Illuminator 7 display grid 6 is set so that the reflection from the surface of the liquid in the image grid locking chamber 4 was clearly visible.

The camera captured an image at the same time to determine the volume of a drop at the moment of separation from the capillary pipette including 3 cameras that captured a larger scale, drop, and surfactant solution or sample liquid that was introduced into the center of the circle. To fix the process of the moving fluid, the film footage was consistently studied to determine the distance from the center of the drop to the ground "wave motion" and in accordance with the scale that converted the results into units of length. Similarly, the droplet diameter at the time of separation from the capillary pipette was determined.

If it is necessary to define or map out the properties of the surfactant, one can use the "standard" surface, which are heat-resistant hydrophobic films or writing paper, or surface-modified paper, for example with gelatin.

When working with paper, it is applied to the circumference of the necessary internal diameter of the hydrophobic dye. The line width of the bounding figure is 5 - 6 mm. The damaged part of the paper with her bounding figure (object - a drug) is soaked in a solvent (for example water) for a certain time (for example 10 minutes) and placed on a table or laid on a planeparallel plate (thick glass). In this straightened paper, because it is removed from air extruded by a glass tube with rounded ends, (for example a diameter of the pipette 10 - 15 mm) or other device, (for example a roller or roller glossing pictures for unfolding and adhering wallpaper). In the area of the paper, you applied bounded by lines (circle, square). The test liquid is applied in an amount necessary to produce the layer thickness that is determined by experimental conditions. In the center of the pipette tip, the locking chamber is included and applied to the center of the bounding shape of the object - a drop of solution of the test drug surfactant.

## **2. Determination of the amount of the liquid conveyed surfactant**

To determine the amount of fluid displaced in the scan frame, the frame corresponded to the maximum radius (diameter) of the displaced fluid layer using standard computer programs to measure it. Figure 1 shows an example of the measurement. After finding the radius determined by

the amount of moving water, its density is given. For example, the volume of the displaced fluid was found to be 9.4 x 10-7 m3, the water depth was 5.0 x 10-4 m, and the weight of displaced fluid was 0.00094 kg. When the surfactant concentration is 5 kg/m3 , the diameter is 0.00229 m, and the droplet amount of the surfactant is equal to 3.15 x 10-8 kg, then, the specific liquid transfer amounts to 29,688.55 kg. One kg of surfactants is able to move the 29.7 tons of water. This is consistent with the work produced in J 7106.37.

Fig. 2 shows the results of a study based on the effect of these parameters on the amount of fluid displacement surfactants. Figure 3 shows the results of the determination of the perfect work in moving liquid surfactants.

From these results, it is evident that one kilogram of surfactant with a great deal of movement implements 110,000 joules, and it depends on the area of the movement. The greater the radius of the bounding area of a circle, the more work is done. The quantity of the displaced fluid is enormous given that 1 kg of the liquid surfactant mass moves in 370000 kg., and one of the surfactant molecules can move 3500000 water molecules [4]. It can be assumed that the distribution of the surfactant on the surface is liberated in the form of a film, which pushes their hydrophobic boundary forming a hydrophilic water wave motion (see Fig. 9, wave movement is marked **С**).

If it is necessary to define or map out the properties of the surfactant, one can use the "standard" surface, which are heat-resistant hydrophobic films or writing paper, or surface-modified

When working with paper, it is applied to the circumference of the necessary internal diameter of the hydrophobic dye. The line width of the bounding figure is 5 - 6 mm. The damaged part of the paper with her bounding figure (object - a drug) is soaked in a solvent (for example water) for a certain time (for example 10 minutes) and placed on a table or laid on a planeparallel plate (thick glass). In this straightened paper, because it is removed from air extruded by a glass tube with rounded ends, (for example a diameter of the pipette 10 - 15 mm) or other device, (for example a roller or roller glossing pictures for unfolding and adhering wallpaper). In the area of the paper, you applied bounded by lines (circle, square). The test liquid is applied in an amount necessary to produce the layer thickness that is determined by experimental conditions. In the center of the pipette tip, the locking chamber is included and applied to the center of the bounding shape of the object - a drop of solution of the test drug surfactant.

**2. Determination of the amount of the liquid conveyed surfactant**

displaced fluid was 0.00094 kg. When the surfactant concentration is 5 kg/m3

water. This is consistent with the work produced in J 7106.37.

work in moving liquid surfactants.

movement is marked **С**).

To determine the amount of fluid displaced in the scan frame, the frame corresponded to the maximum radius (diameter) of the displaced fluid layer using standard computer programs to measure it. Figure 1 shows an example of the measurement. After finding the radius

the amount of moving water, its density is given. For example, the volume of the displaced fluid was found to be 9.4 x 10-7 m3, the water depth was 5.0 x 10-4 m, and the weight of

0.00229 m, and the droplet amount of the surfactant is equal to 3.15 x 10-8 kg, then, the specific liquid transfer amounts to 29,688.55 kg. One kg of surfactants is able to move the 29.7 tons of

Fig. 2 shows the results of a study based on the effect of these parameters on the amount of fluid displacement surfactants. Figure 3 shows the results of the determination of the perfect

From these results, it is evident that one kilogram of surfactant with a great deal of movement implements 110,000 joules, and it depends on the area of the movement. The greater the radius of the bounding area of a circle, the more work is done. The quantity of the displaced fluid is enormous given that 1 kg of the liquid surfactant mass moves in 370000 kg., and one of the surfactant molecules can move 3500000 water molecules [4]. It can be assumed that the distribution of the surfactant on the surface is liberated in the form of a film, which pushes their hydrophobic boundary forming a hydrophilic water wave motion (see Fig. 9, wave

, the diameter is

paper, for example with gelatin.

182 Surface Energy

determined by

**Figure 2.** The change in the specific amount of water transferred and surfactants "balancer", depending on the content of the surfactant and the confining diameter of the circle (the layer thickness of 0.0005 m). Bounding circle radii (curves upward) 0.04; 0.05; 0.06; 0.07; 0.08 m. [1,3]

Surfactant content, g / cm3

**Figure 3.** The change in the specific operation when water is moving and a surfactant "equalizer", depending on the content of the surfactant and the confining diameter of the circle (layer thickness 0.005 m). Bounding circle radii (curves upward) 0.04; 0.05; 0.06; 0.07; 0.08 m [1].

Change in the diameter of the displaced liquid layer during the movement is shown in Figure 4. It can be seen that the diameter increases rapidly passing through the maximum point and then decreases slowly.

Increase in the diameter occurs almost in a straight line (see Figure 5). A reduction in the diameter is well described by a second degree polynomial shape and is very close to the

**Figure 4.** Change in the diameter of the displaced fluid layer

exponent (Fig. 6). From the above results, it can be seen that the performance can be considered to characterize a surfactant, wherein the range considerably exceeds the measurement range of the contact angle (0 to 180) and the range of values of the surface tension (20 to 480 [5]). This has been determined for we have studied a range of values from tens to hundreds of thousands of units [1].

**Figure 5.** Increase in the diameter of the displaced fluid layer

## **3. The determination of surface properties**

To study the surface properties of the liquid, the time was determined where the greatest range of movement in the number of staff from the moment of touching a drop of surfactant liquid

Modern Methods (Without Determining the Contact Angle and Surface Tension) for Estimating the Surface… http://dx.doi.org/10.5772/61041 185

**Figure 6.** Reducing the diameter of the displaced fluid layer

exponent (Fig. 6). From the above results, it can be seen that the performance can be considered to characterize a surfactant, wherein the range considerably exceeds the measurement range of the contact angle (0 to 180) and the range of values of the surface tension (20 to 480 [5]). This has been determined for we have studied a range of values from tens to hundreds of thousands

0,5 1 1,5 2 2,5 3 Time, sec

y = 13,679x - 6,0082

= 0,9736

R2

0,5 0,7 0,9 1,1

Time, sec

To study the surface properties of the liquid, the time was determined where the greatest range of movement in the number of staff from the moment of touching a drop of surfactant liquid

of units [1].

The diameter of the

184 Surface Energy

displaced layer, cm

0

**Figure 5.** Increase in the diameter of the displaced fluid layer

**3. The determination of surface properties**

2

4

displaced layer, cm

The diameter of the

6

8

10

**Figure 4.** Change in the diameter of the displaced fluid layer

to a frame moving with the largest radius was determined. To know the frame rate, the time between frames is multiplied by the number of frames to achieve the greatest range of fluid movement. Then, multiply the result with the time to reach the maximum displacement of the liquid, the number of displaced kilogram of surfactant liquid finding surface. The results are shown in Fig. 7.

**Figure 7.** The change of parameters depending on the surfactant concentration: 1. The specific amount of fluid dis‐ placement (kg liquid / kg of surfactant) 2. The specific amount of fluid displacement per unit time ((kg liquids / kg surfactant) \* sec)

From these results, it is seen that the amount of fluid displaced is changed, passed through a maximum, which is a concentration of 4 kg/m3 . Perhaps, this is due to the properties of surfactant solutions, which vary significantly with increasing concentration and the formation of micellar nanostructures.

## **4. Defining the properties of the fluid**

The liquid can be characterized by the methods given above. At the same time, we observed the regularity of occurrence and propagation of microwaves, which can also be used to characterize the surface properties of the fluid [6].

To determine the distribution ranges of microwaves, reflection display grids were used, which is clearly seen on the surface of the liquid in the video. The excitation of the surface of the liquid carried the water droplets drop at a height of 0.02 m on the surface of the water and poured in a circumscribing circle on the paper surface. The different amount introduced into the circle bounded the fluid to change the thickness of the layer from 0.0002 m to 0.001 m on the surface of the flooded paper.

Changes occurring in the interaction of water droplets to the surface layer of water were recorded with a video camera. Then, on a frame scan, selected images in which measurements were recorded changed. For this purpose, standard computer programs were used. Similarly, the effect of surfactant concentration on the velocity of the waves was included in the study. It has been shown that the water droplets in contact with the surface of the water formed microwaves first (see Fig. 8) and then moved over the water surface substantially at a constant speed (70 - 75)\*10-2 m/s (see Figure 10). With the passage of the microwave image, the grid display disappears. The resulting large waves then move on to the liquid surface at an average rate of 32\*10-2 m/s (see Figure 8). The increase in the thickness of the water layer in area C increases the distance of propagation of the microwaves. Considering the studies of American astronauts, it can be assumed that microwave oscillations are produced by the surface film having a thickness of 0.00015 m [7]. Their height (amplitude of oscillation) may not exceed the double thickness of the film but interaction of the drops of surfactant solution with the water also causes the appearance of these two types of waves. However, the larger wave, starting from the center of the dropping, and then moved to the edges of the bounding circle is already under the action "move fluid wave". (See fig. 9 mark C).

She seemed to be pushed in front of the wave, and the trailing edge of the wave becomes steeper. This wave slows down when approaching the edge of the bounding circle and its movement initially opens the surface of the liquid (see Fig. 9. mark A) and the "boundary layer" of liquid (see Fig. 9). In the notation, the line indicating the mesh is clearly visible, under the influence of a force field where the liquid surface is. Moreover, at low concentrations of surfactants and a strong interaction with the liquid lying below the surface of the "wave motion of the liquid", the "boundary layer" is rolled on the surface. It was great all the time and zone A was determined to have an observed free liquid. The introduction of surfactants somewhat reduces the velocity of the propagation of large capillary waves (Without the surfactant average speed is 32\*10-2m/s capillary wave and with surfactants, it is 20.5\*10-2 m/s). Moreover, the microwaves are observed on the surface of large waves because the indicator on the surface of the liquid keeper is not visible (Figures 8 and 9). A large wave of "moving fluid" slows down when approaching the edge of the circle to this limit: (13-15)\*10-2 m/s. The speed of microwaves remains almost constant at the range of 65 – 75\*10-2 m/s. This speed can be detected only two times during the passage of the microwaves on the investigated surface. Furthermore, microwaves quickly reach the circle limit.

From these results, it is seen that the amount of fluid displaced is changed, passed through a

surfactant solutions, which vary significantly with increasing concentration and the formation

The liquid can be characterized by the methods given above. At the same time, we observed the regularity of occurrence and propagation of microwaves, which can also be used to

To determine the distribution ranges of microwaves, reflection display grids were used, which is clearly seen on the surface of the liquid in the video. The excitation of the surface of the liquid carried the water droplets drop at a height of 0.02 m on the surface of the water and poured in a circumscribing circle on the paper surface. The different amount introduced into the circle bounded the fluid to change the thickness of the layer from 0.0002 m to 0.001 m on the surface

Changes occurring in the interaction of water droplets to the surface layer of water were recorded with a video camera. Then, on a frame scan, selected images in which measurements were recorded changed. For this purpose, standard computer programs were used. Similarly, the effect of surfactant concentration on the velocity of the waves was included in the study. It has been shown that the water droplets in contact with the surface of the water formed microwaves first (see Fig. 8) and then moved over the water surface substantially at a constant speed (70 - 75)\*10-2 m/s (see Figure 10). With the passage of the microwave image, the grid display disappears. The resulting large waves then move on to the liquid surface at an average rate of 32\*10-2 m/s (see Figure 8). The increase in the thickness of the water layer in area C increases the distance of propagation of the microwaves. Considering the studies of American astronauts, it can be assumed that microwave oscillations are produced by the surface film having a thickness of 0.00015 m [7]. Their height (amplitude of oscillation) may not exceed the double thickness of the film but interaction of the drops of surfactant solution with the water also causes the appearance of these two types of waves. However, the larger wave, starting from the center of the dropping, and then moved to the edges of the bounding circle is already

She seemed to be pushed in front of the wave, and the trailing edge of the wave becomes steeper. This wave slows down when approaching the edge of the bounding circle and its movement initially opens the surface of the liquid (see Fig. 9. mark A) and the "boundary layer" of liquid (see Fig. 9). In the notation, the line indicating the mesh is clearly visible, under the influence of a force field where the liquid surface is. Moreover, at low concentrations of surfactants and a strong interaction with the liquid lying below the surface of the "wave motion of the liquid", the "boundary layer" is rolled on the surface. It was great all the time and zone A was determined to have an observed free liquid. The introduction of surfactants somewhat reduces the velocity of the propagation of large capillary waves (Without the surfactant

. Perhaps, this is due to the properties of

maximum, which is a concentration of 4 kg/m3

**4. Defining the properties of the fluid**

characterize the surface properties of the fluid [6].

under the action "move fluid wave". (See fig. 9 mark C).

of micellar nanostructures.

186 Surface Energy

of the flooded paper.

**Figure 8.** Microwaves (area B) and capillary movement (area A) on the surface of the liquid. In the area of microwave indicator net disappears.

**Figure 9.** View of the water layer after exposure to drops of surfactant.

The effect of opening the boundary layer fluid wave motion can be used to estimate and visually observe the thickness of the boundary layers. Also, the properties of microwaves to increase the range of the spread with increasing thickness in the liquid layer can be used to estimate the thickness of the layer of water associated with the surface on which the liquid is present and to evaluate the interaction energy of the liquid with the surface [6]. In the method of the moving fluid, a surfactant is proposed for use due to its characteristics and surface properties in surfactant identification, including metrology and nanomaterials.

**Figure 10.** The changing speed of microwaves on the surface from the impact of a drop of water.

**Figure 11.** The change in the speed of the capillary wave action due to the drops of surfactant. The water depth is 0.0005 m.

Studies have shown that in a limited space, microwaves do not change their speed, while the larger capillary waves significantly reduce its speed when approaching the limiting barrier. This can serve as a basis for concluding that occurs due to fluctuations in the microwave surface of the liquid film. A wave of migration has a pulsating speed like bumping and the surfacing of the boundary layer, slowing down and crashing "surfacing" acceleration.

Microwaves can be used to determine the thickness of the boundary layers created on the surface, study the increasing the thickness of the layers of water, and expose small water droplets on the surface dropping them from a small height.

After recording a video, defined images, in which the measurements of the amount of liquid transfer and the range of wave propagation, which are the characteristics of the surfactant properties of liquids and surfaces, can be seen frame by frame. At the same time, the images can determine the thin, visible layer of liquid. Figure 12 shows a view of the exposed surface of the surfactant layer of liquid. Figure 12 shows three clearly visible that are staggered. Zone A - is the surface (substrate) excepted from the liquid, zone B – is the tampering surfactant interfa‐ cial layer of liquid, and zone C – is free (not bound to the substrate) and fluid displacement.

increase the range of the spread with increasing thickness in the liquid layer can be used to estimate the thickness of the layer of water associated with the surface on which the liquid is present and to evaluate the interaction energy of the liquid with the surface [6]. In the method of the moving fluid, a surfactant is proposed for use due to its characteristics and surface

properties in surfactant identification, including metrology and nanomaterials.

**Figure 10.** The changing speed of microwaves on the surface from the impact of a drop of water.

0.0005 m.

188 Surface Energy

**Figure 11.** The change in the speed of the capillary wave action due to the drops of surfactant. The water depth is

Studies have shown that in a limited space, microwaves do not change their speed, while the larger capillary waves significantly reduce its speed when approaching the limiting barrier. This can serve as a basis for concluding that occurs due to fluctuations in the microwave surface of the liquid film. A wave of migration has a pulsating speed like bumping and the surfacing

Microwaves can be used to determine the thickness of the boundary layers created on the surface, study the increasing the thickness of the layers of water, and expose small water

of the boundary layer, slowing down and crashing "surfacing" acceleration.

droplets on the surface dropping them from a small height.

**Figure 12.** Three zones. Surfactant - "equalizer" with a concentration of 3 kg/m3. The diameter of the circle bounding 0.144 m: A - surface freed from water; B - surface boundary layer; C - moving layer of fluid

**Figure 13.** Izmenenie microwaves spread on the water surface depending on the thickness of the substrate - the paper

Since there is liquid on the surface layer bordering the air and the volume of liquid, it is possible to conclude that the thickness of the liquid on any surface exceed the total thickness of the surface layers and that the interfacial layer of the rest of the liquid will be "free". US astronauts conducting experiments in space determined that a double layer of water in the air interface when there is no free liquid is equal to 300 microns (0.0003 m). Consequently, a single layer is equal to 0.00015 meters, and the thickness of the liquid layer above the value property of this layer should undergo changes. We believe that the easiest way to define these changes is that it can serve as a distance propagation of microwaves on the surface of the liquid layer of known thickness. In this paper, we change the thickness of the liquid layer from 0.0002 to 0.0006 m.

**Figure 14.** The dependence of the radial propagation of the microwaves on the surface of its layer thickness.

The determination of microwave range is carried out by the disappearance of the indicator grid lines on the surface of the water (see. Fig. 15). The results of the measurements are shown in Figure 13, which illustrates the sharp increase in the area of microwave propagation beginning with 0.0003 mm. Perhaps, this is the thickness of the water layer in which the surface layer and the boundary layers start to move away from each other. Then, when the surface layer has a thickness of 0.15 mm, the boundary layer will have a thickness of 0.3 minus 0.15 = 0.15 mm. Approximately, the same value is obtained when considering the radial distribution of microwaves (see Fig. 4). Direct drawn through the experimental points intersects at the horizontal axis at 0.12 mm. In other words, given the zero value of the radius of the microwaves on the propagation of the liquid layer with 0.12 mm thickness, the liquid layer is bonded to a substrate.

**Figure 15.** The type of surface water layer with a thickness of 0.4 mm. The distribution of microwaves on the surface is observed by the disappearance of the grid display.

Thus, the method for moving the liquid surfactant can be used to characterize the boundary layer thickness.

## **5. Contactless displacement liquid surfactants**

Since there is liquid on the surface layer bordering the air and the volume of liquid, it is possible to conclude that the thickness of the liquid on any surface exceed the total thickness of the surface layers and that the interfacial layer of the rest of the liquid will be "free". US astronauts conducting experiments in space determined that a double layer of water in the air interface when there is no free liquid is equal to 300 microns (0.0003 m). Consequently, a single layer is equal to 0.00015 meters, and the thickness of the liquid layer above the value property of this layer should undergo changes. We believe that the easiest way to define these changes is that it can serve as a distance propagation of microwaves on the surface of the liquid layer of known thickness. In this paper, we change the thickness of the liquid layer from 0.0002 to 0.0006 m.

**Figure 14.** The dependence of the radial propagation of the microwaves on the surface of its layer thickness.

substrate.

190 Surface Energy

The determination of microwave range is carried out by the disappearance of the indicator grid lines on the surface of the water (see. Fig. 15). The results of the measurements are shown in Figure 13, which illustrates the sharp increase in the area of microwave propagation beginning with 0.0003 mm. Perhaps, this is the thickness of the water layer in which the surface layer and the boundary layers start to move away from each other. Then, when the surface layer has a thickness of 0.15 mm, the boundary layer will have a thickness of 0.3 minus 0.15 = 0.15 mm. Approximately, the same value is obtained when considering the radial distribution of microwaves (see Fig. 4). Direct drawn through the experimental points intersects at the horizontal axis at 0.12 mm. In other words, given the zero value of the radius of the microwaves on the propagation of the liquid layer with 0.12 mm thickness, the liquid layer is bonded to a In the study of fluid displacement, it was observed that microwaves begin to form when the surface of the water droplet is at a small distance in drip surfactants. Consequently, using surfactant with high volatility can cause a slide on the surface layer of water from the gas phase surfactant [8].

For the experiment, in the method described above, the only investigated material were a small square and a thickness of 1 mm. They were placed in the center of the bounding shape (a circle) and the amount of water is taken. Having regard to the volume of the test material, a layer of water with a predetermined thickness was generated on it. The objects of the study used were duralumin discs, silicon hereinafter - the "silicon", and LiNbO3 of the "Lithium". The capillary pipette was placed at different heights (from 1.5 to 4mm) from the surface being studied, on which the fluid (water) was located. Also, there was a change the thickness of the liquid layer from 0.3 to 0.6 mm. The measurement results are shown in Table 1.

Observations have shown that a decrease in the height of the capillary over the studied surface purifying it from the liquid is more intense. The surface on which the liquid was, is cleared of the water layer quickly. A thickness of 0.3 mm in the water displaces the surfactant on the surface being studied for 6-7 seconds. Table 1 with a reduced thickness of the liquid cleansing surface is faster. The greatest speed of moving liquid is observed on silicon, and the smallest on the paper. One can assume that the communication between the water surface and the paper is higher than the surface of lithium and silicon wafers. At the same time, the thickness of the water layer, which can still be displaced under the action of the surfactant, is less in paper plates than to silicon and lithium. The paper limited the thickness of the water layer at 0.4 mm, silicon between 0.5 to 0.6 mm, and lithium. This effect is used to evaluate the thickness of the boundary layer of water on the surface of the materials investigated [9]. However, there is a contradiction. Lower speed happens in thin layers of water, which may be due to surface roughness. The average value of surface roughness in micron samples obtained on the instrument "Profilers - 296" is : paper - 4.55 ; Li - 1.16 ; and silicon - 0.75. The physical nature of this indicator is that the larger the value, the greater the difference between the highest and the lowest point on the surface (vertical drop), therefore, the greater roughness. Consequently, the roughest paper will then be lithium and silicon. In the reverse speed of the fluid buildup, with a correlation coefficient of - 0.999 and the same parameters (height above the surface of the capillary pipette and the thickness of 3.5 mm to 0.3 mm) fluid velocity of the liquid will move: paper - 2.62; Li - 16.49 ; and silicon - 18.9 mm3 /sec. At first glance, these results confirm the conclusion about the effect of roughness on the velocity of the fluid. However, this aspect requires a more detailed study. For example, if one adopts the hypothesis about the impact on the speed of the fluid, the strength of the molecules of the liquid from the surface under study, it turns out that the paper bond strength liquid is higher than the surface of silicon wafers and lithium. Therefore, the rate of fluid movement across the paper surface is less than the surface of the silicon wafer and lithium. But, as we have noted above, the movement and penetration of the liquid occurs on the investigated surface with a liquid layer of different thicknesses. For the paper, the value of the minimum thickness of the fluid at which there is a breakthrough of the liquid layer is 0.4 mm to 0.6 mm, and lithium-silicon is 0.5mm [9]. This means, lithium and silicon binds more water than paper. According to fluid handling associated with more water and logically, paper - the stronger of the coupling, the smaller will move liquid per unit of time (for more details see below).



**Table 1.** The influence of parameters on the movement of water surfactants from the gas phase.

During the experiment, it was observed that erodible ring structure appear at the surface layer of the water, which later disappears. (see Figs. 16 and 17)

**Figure 16.** Position of the edges forming a ring structure.

is higher than the surface of lithium and silicon wafers. At the same time, the thickness of the water layer, which can still be displaced under the action of the surfactant, is less in paper plates than to silicon and lithium. The paper limited the thickness of the water layer at 0.4 mm, silicon between 0.5 to 0.6 mm, and lithium. This effect is used to evaluate the thickness of the boundary layer of water on the surface of the materials investigated [9]. However, there is a contradiction. Lower speed happens in thin layers of water, which may be due to surface roughness. The average value of surface roughness in micron samples obtained on the instrument "Profilers - 296" is : paper - 4.55 ; Li - 1.16 ; and silicon - 0.75. The physical nature of this indicator is that the larger the value, the greater the difference between the highest and the lowest point on the surface (vertical drop), therefore, the greater roughness. Consequently, the roughest paper will then be lithium and silicon. In the reverse speed of the fluid buildup, with a correlation coefficient of - 0.999 and the same parameters (height above the surface of the capillary pipette and the thickness of 3.5 mm to 0.3 mm) fluid velocity of the liquid will

the conclusion about the effect of roughness on the velocity of the fluid. However, this aspect requires a more detailed study. For example, if one adopts the hypothesis about the impact on the speed of the fluid, the strength of the molecules of the liquid from the surface under study, it turns out that the paper bond strength liquid is higher than the surface of silicon wafers and lithium. Therefore, the rate of fluid movement across the paper surface is less than the surface of the silicon wafer and lithium. But, as we have noted above, the movement and penetration of the liquid occurs on the investigated surface with a liquid layer of different thicknesses. For the paper, the value of the minimum thickness of the fluid at which there is a breakthrough of the liquid layer is 0.4 mm to 0.6 mm, and lithium-silicon is 0.5mm [9]. This means, lithium and silicon binds more water than paper. According to fluid handling associated with more water and logically, paper - the stronger of the coupling, the smaller will move liquid per unit of time

> **The diameter of the layer of fluid displaced by 20 frames (4 seconds), mm**

3,5 0,3 1,4 16,7 65,99 5,4 16,49 12,22 3,5 0,4 6,8 17,05 91,32 10,8 22,83 8,45 2,0 0,5 0,6 13,12 67,61 4,6 16,90 14,69

3,5 0,3 2,0 17,91 75,61 6,0 18,90 12,60 2,5 0,5 0,8 17,05 114,15 4,8 28,53 23,78 2,0 0,5 0,8 17,52 120,55 4,8 30,13 25,11

3,5 0,3 8,2 6,67 10,51 12,2 2,62 0,86 3,0 0,3 5,0 9,74 22,36 9,0 5,59 2,48

**The volume of the displaced fluid mm3**

**Time from onset of exposure, sec**

**The speed of movement through fluid 20 frames (4 seconds), mm3**

**/s**

**The speed of movement from the start of exposure, mm3 /s**

/sec. At first glance, these results confirm

move: paper - 2.62; Li - 16.49 ; and silicon - 18.9 mm3

(for more details see below).

**The height above the surface of the object, mm**

**The water depth, mm**

**Breakthrough time the total thickness of the water layer, sec**

**Object**

192 Surface Energy

lithium

silicon

paper

**Figure 17.** Position of the edges of the ring structure formed from structure.

In Figure 16, arrows 1 and 2 indicate that the position of the edges formed a ring structure. Arrow 3, on the other hand, shows the emergence of a new ring structure. In Figure 17, arrows 4 and 5 show the position of the edges of the ring structure formed, which is marked by the arrow 3 in Figure 16. Arrow 6, then, shows the emergence of a new ring structure. There are several explanation of this effect:


The first explanation is the most acceptable. In previous studies on the movement of a drop of liquid surfactant, the free liquid displacement is observed on the surface of the water bound to the gelatin layer and then destroys the boundary layer. Below are the data from this work.

The movement to the peripheral areas of liquid drug is slow and comes in two stages. The first stage is under the influence of highly concentrated surfactant solution (80 kg / m3) "sulphonol" located above the moving fluid bed bound. After a while, the fluid begins to move associated with the gelatin layer of water, freeing the surface and causing it to become dull. (see Figs. 18-20)

The time taken for the movement of fluid may be determined by the number of frames from the starting points. Since the beginning of the movement, it is necessary to frame up to № 10. The movement of "free" liquid ends at frame 43 (Fig. 18). At a frequency of shooting 24 frames per second on the move, 1.375 sec have been spent. Without noticeable changes in the layer, "bound" water was up to the frame number 60 in 0.7 seconds. Further movement of the associated layer to frame 215 occurred within 6.45 sec.

**Figure 18.** View of the water surface after movement of the "free" liquid on the surface of the water associated with gelatin frame 43. A - bound fluid layer; B - bound (shaft) move "free" liquid.

**Figure 19.** The start of the movement associated with the liquid layer on the surface of the gelatin block 81. A - bound fluid layer; B - bound (shaft) move "free" liquid; C - free gelatin surface.

Bound liquid layer on the surface of the gelatin is not destroyed by the surfactant concentration of 5 g / cm3 . Movement of the "free" liquid only occurred on the surface of the "bound".

**1.** This stratified destruction layer are associated with the studied surface water, thus, water layers in the associated liquid layer have varying degrees of communication between

**2.** The impact of the electric potential between the capillary pipette and the surface is being

The first explanation is the most acceptable. In previous studies on the movement of a drop of liquid surfactant, the free liquid displacement is observed on the surface of the water bound to the gelatin layer and then destroys the boundary layer. Below are the data from this work.

The movement to the peripheral areas of liquid drug is slow and comes in two stages. The first stage is under the influence of highly concentrated surfactant solution (80 kg / m3) "sulphonol" located above the moving fluid bed bound. After a while, the fluid begins to move associated with the gelatin layer of water, freeing the surface and causing it to become dull. (see Figs.

The time taken for the movement of fluid may be determined by the number of frames from the starting points. Since the beginning of the movement, it is necessary to frame up to № 10. The movement of "free" liquid ends at frame 43 (Fig. 18). At a frequency of shooting 24 frames per second on the move, 1.375 sec have been spent. Without noticeable changes in the layer, "bound" water was up to the frame number 60 in 0.7 seconds. Further movement of the

**Figure 18.** View of the water surface after movement of the "free" liquid on the surface of the water associated with

**Figure 19.** The start of the movement associated with the liquid layer on the surface of the gelatin block 81. A - bound

associated layer to frame 215 occurred within 6.45 sec.

gelatin frame 43. A - bound fluid layer; B - bound (shaft) move "free" liquid.

fluid layer; B - bound (shaft) move "free" liquid; C - free gelatin surface.

**3.** This "microdroplets" evaporating surfactants spreads over the surface of water.

them.

194 Surface Energy

studied.

18-20)

**Figure 20.** The type of surface bound gelatin layer of water moving frame 215. A - bound fluid layer; B - bound (shaft) move "bound" liquid; C - matt exempt from bound water surface layer of gelatin.

The surface of the paper layer of bound water is destroyed under the action of a surfactant and a small concentration of 5 kg/m3 .

It is possible that ring structures will form in a contactless displacement fluid given a scenario similar to the one above. The observed circular formations change their dimensions slowly. Therefore, we assume that this is the most likely scenario of the destruction process of boundary layers.

The second assumption about the nature of electricity does not exclude, but rather comple‐ ments the first scenario. More so, only after grounding the tripod holding capillary dropper measurement results in our experiments stabilized.

The third assumption, in principle, is unlikely, as it implies a rapid condensation of the evaporated molecules in microdroplets. Evaporation or not molecular, and cluster. We are assuming that the surfactant molecules at the water surface form a monomolecular film that is moved by the water. The highest measured diameters of the annular formations (see Fig. 16 and 17) are calculated using the table data area and the height of surfactant molecules on their surface. Measurements and calculations showed that the diameter of the "microdroplets" with the observed size of the circular formations may be within 0.00020245 m. 2,82743E + 13 contains more than molecules. The droplet size is large enough. Therefore, it can be assumed that the surfactant molecules at the surface either does not fit into a continuous film upon evaporation or is the molecules and the clusters containing few molecules are detached from the total weight. The first assumption is most likely because the calculations carried out the work of Karbainova, A.N. [4] It was shown that one molecule surfactant can move 3.5 million molecules of water. If you use this value to calculate the number of surfactant molecules needed to move the volumes of liquid, which are shown in Table 1, it turns out that the greatest number of molecules that moved will be equal to 1,15233E+15 and the smallest is 1,00465E+14. A number of the above-mentioned droplet size must be delivered to the first surface 41 in the second case 4. Naturally, the number of droplets is significantly larger and significantly smaller in droplet size, possibly approaching nanoscale.

The measurement of the contact angle (wetting angle) showed that the contact angle of water droplets on lithium (45o) is lower than silicon (55 °)(See drawings on the left and right). That is, lithium is more hydrophilic than silicon and accordingly, the rate of water movement on lithium is less than silicon. Water molecules bind more strongly to the surface of lithium, so it is necessary to expend more energy for their movement and other things being equal, the amount of water transferred will be smaller. For example, when the height of the capillary is 2.0 mm, the volume rate of water transferred to lithium is 67.61 mm3 , 16.9 mm3 /sec and silicon is at 120.55 mm3 , 30.13 mm3 /sec)(Tab. 1). When the value of the difference in contact angle is 10o between silicon and lithium, the volume rate of water movement between the surfaces is practically twice of that of the silicon. Consequently, new hydrophilic indicators give a more differentiated picture.

The change in the surface properties of the skin treatment processes was also investigated. A sample size of 4x4 cm was glued to a sheet of paper, coated thereon the limit line as a circle, and treated sequentially with water, aqueous acids and salt, an aqueous solution of a chromi‐ um tanning agent. After completing these processes, the speed of the movement of the water layer with thickness of 0.2 mm on the surface of the samples is determined. After exposure to water, the travel speed was found to be - 2.05 mm/sec for samples after exposure to acid and salt; 3.37 mm/sec after exposure to chrome tanning agent; and 6.86 mm/sec. The results showed that during the treatment, there is an increase in the velocity of the fluid, i.e. water-repellency at the surface of the skin as a consequence of reducing the surface water connection and therefore increases the speed of movement. This corresponds to the theoretical concepts of the science of skin.

Thus, studies have shown the possibility of estimating the properties of different surfaces by the use of non-contact displacement fluid. It is shown that the movement of the liquid surfactant influences the surface roughness and the strength of molecular interaction with the surface of the liquid.

The study of fluid motion on the surface of duralumin has shown that there are several features that can be seen only on light reflecting surfaces. Studies were therefore conducted as set forth above.

The process of moving the non-contact liquid surfactant consists of several stages. At the beginning of travel, a surfactant element removes water from the surface layers of the molecules that are fixed in this layer due to the interaction with the molecules of air between them. During the experiment, it was observed that erodible ring structures appear on the surface layer of water, which disappears over time (see Fig. 21).

In our opinion, this layering destruction of water is bound to the surface under the study. Therefore, the layers of bound water in the liquid layer have different degrees of communi‐ cation between them. Furthermore, the water layer is structured on one side with air on the other side of the surface where the liquid is located. With the structure status stored in the interaction of the surface of the water with a surfactant, which is embedded in the layer structure and the stability limit is exceeded, new created surfactant molecules embedded structure begins its destruction and the movement of the liquid layer. This altered state Modern Methods (Without Determining the Contact Angle and Surface Tension) for Estimating the Surface… http://dx.doi.org/10.5772/61041 197

**Figure 21.** The formation of the ring structures (arrows) in the water layer of 0.4 mm thickness on a paper surface.

structures, with embedded surfactant molecules, are maintained long enough, which, accord‐ ing to our measurements, is more than 20 seconds.

The measurement of the contact angle (wetting angle) showed that the contact angle of water droplets on lithium (45o) is lower than silicon (55 °)(See drawings on the left and right). That is, lithium is more hydrophilic than silicon and accordingly, the rate of water movement on lithium is less than silicon. Water molecules bind more strongly to the surface of lithium, so it is necessary to expend more energy for their movement and other things being equal, the amount of water transferred will be smaller. For example, when the height of the capillary is

10o between silicon and lithium, the volume rate of water movement between the surfaces is practically twice of that of the silicon. Consequently, new hydrophilic indicators give a more

The change in the surface properties of the skin treatment processes was also investigated. A sample size of 4x4 cm was glued to a sheet of paper, coated thereon the limit line as a circle, and treated sequentially with water, aqueous acids and salt, an aqueous solution of a chromi‐ um tanning agent. After completing these processes, the speed of the movement of the water layer with thickness of 0.2 mm on the surface of the samples is determined. After exposure to water, the travel speed was found to be - 2.05 mm/sec for samples after exposure to acid and salt; 3.37 mm/sec after exposure to chrome tanning agent; and 6.86 mm/sec. The results showed that during the treatment, there is an increase in the velocity of the fluid, i.e. water-repellency at the surface of the skin as a consequence of reducing the surface water connection and therefore increases the speed of movement. This corresponds to the theoretical concepts of the

Thus, studies have shown the possibility of estimating the properties of different surfaces by the use of non-contact displacement fluid. It is shown that the movement of the liquid surfactant influences the surface roughness and the strength of molecular interaction with the

The study of fluid motion on the surface of duralumin has shown that there are several features that can be seen only on light reflecting surfaces. Studies were therefore conducted as set forth

The process of moving the non-contact liquid surfactant consists of several stages. At the beginning of travel, a surfactant element removes water from the surface layers of the molecules that are fixed in this layer due to the interaction with the molecules of air between them. During the experiment, it was observed that erodible ring structures appear on the

In our opinion, this layering destruction of water is bound to the surface under the study. Therefore, the layers of bound water in the liquid layer have different degrees of communi‐ cation between them. Furthermore, the water layer is structured on one side with air on the other side of the surface where the liquid is located. With the structure status stored in the interaction of the surface of the water with a surfactant, which is embedded in the layer structure and the stability limit is exceeded, new created surfactant molecules embedded structure begins its destruction and the movement of the liquid layer. This altered state

surface layer of water, which disappears over time (see Fig. 21).

, 16.9 mm3

/sec)(Tab. 1). When the value of the difference in contact angle is

/sec and silicon

2.0 mm, the volume rate of water transferred to lithium is 67.61 mm3

, 30.13 mm3

is at 120.55 mm3

196 Surface Energy

science of skin.

surface of the liquid.

above.

differentiated picture.

It is possible that the formation of dark spots before the breakthrough of the liquid layer on the duralumin becomes a specific interest (see Fig. 3). We assume that this is due to a decrease in the thickness of water for up to 3 - 5 nm or less than the wavelength of light [10]. And the moving layer of water can "roll" in the study water to the surface film thickness of the dark spots. Figure 14 shows the formation of a water film (marked by an arrow) which is then collected in the droplet.

It is possible that the formation of a dark film was due to depressions in the surface of duralumin. Then, this effect can be used to assess mechanical defects on surfaces with almost nanometer thickness. The figure also shows the optical transition to deepen or to a defect or when not to manifest itself. At any rate, the reflection capillary is not changed. To detect this defect by optical methods, a special instrument base and an increase in size is required.

**Figure 23.** The sequence of frames showing the formation of a thin film of water at dark moving surfactant-ohm, which is later collected in a drop (marked by an arrow).

The method of moving the liquid surfactant was used to assess the particulate materials, including water-soluble measurement results that are shown in Table 2. For studies, particulate materials are mixed with water and were made into a pattern in which the material surface is leveled. The template was placed in the center of the bounding figure and above the layer of water creating a thickness of 0.2 mm. Capillary with isobutyl alcohol surfactants are over the patterns. The camcorder was then turned on. It was then seen that the capillary was fed vertically to the surface of the template.


**Table 2.** The speed of the liquid (water) on the surface of particulate materials.

The table shows that the velocity of the liquid material on the surface can serve to characterize the extent of water interaction with the material. For example, the reaction of water with the surface of the copper oxide is insignificant as the moving speed of 85.1 mm / sec is much greater than on the surfaces of other materials - sand, iron oxide, lime, and others, but is very close to the speed of movement over the surface of table salt. Therefore, their surface is more hydro‐ phobic [10, 11].

## **6. Moving particles [12]**

Studies of particulate materials by fluid displacement are possible to detect moving objects using silicon oxide.

We explored the possibility of estimating the surface properties of powder and granular materials, without determining the surface tension and wetting angle [4]. We found that with the interaction of the surfactant with the liquid layer above the surface of the sand amd after bringing the surfactant in the sand into contactthe camcorder recorded self-propelled objects of different types.

Conventionally, these are the six types:


**Figure 23.** The sequence of frames showing the formation of a thin film of water at dark moving surfactant-ohm, which

The method of moving the liquid surfactant was used to assess the particulate materials, including water-soluble measurement results that are shown in Table 2. For studies, particulate materials are mixed with water and were made into a pattern in which the material surface is leveled. The template was placed in the center of the bounding figure and above the layer of water creating a thickness of 0.2 mm. Capillary with isobutyl alcohol surfactants are over the patterns. The camcorder was then turned on. It was then seen that the capillary was fed

The table shows that the velocity of the liquid material on the surface can serve to characterize the extent of water interaction with the material. For example, the reaction of water with the surface of the copper oxide is insignificant as the moving speed of 85.1 mm / sec is much greater than on the surfaces of other materials - sand, iron oxide, lime, and others, but is very close to the speed of movement over the surface of table salt. Therefore, their surface is more hydro‐

Studies of particulate materials by fluid displacement are possible to detect moving objects

We explored the possibility of estimating the surface properties of powder and granular materials, without determining the surface tension and wetting angle [4]. We found that with the interaction of the surfactant with the liquid layer above the surface of the sand amd after bringing the surfactant in the sand into contactthe camcorder recorded self-propelled objects

**Disperse powder and granular water-soluble**

SiO2 Fe2O3 CaOH2 ZnO MgO CuO NaCl NaHCO3 58,50 7,64 10,69 12,06 17,29 85,1 74,18 57,0

is later collected in a drop (marked by an arrow).

vertically to the surface of the template.

**Table 2.** The speed of the liquid (water) on the surface of particulate materials.

Fluid movement speed, mm / sec

198 Surface Energy

phobic [10, 11].

using silicon oxide.

of different types.

**6. Moving particles [12]**

**Figure 24.** The first type of moving objects. Large objects resembling a UFO. Getting traffic frame 698, continued frames 708 - 718. The arrows marked the position of the object. The speed of movement of the object is 10-15 mm/sec. Its size is larger than 5 mm. (moving from right to left, top) In the shade (to 698), a bright object is seen on a dark background. And on a light background is a dark object (to 708 and 718). This can be interpreted as the illumination object. Similar changes were observed for the other types. See Fig. 29. Frames 178, 189, 198, 276, 292, 328.

From Figure 24, the movement of the object causes display grid lines curves to be reflected from the surface of the water. This indicates that the object is moving on the water surface. But, there are objects moving in the water beneath the surface. (see Fig. 25).

**Figure 25.** Underwater object. Changes in the display grid lines (moving from right to left, top). Movement is seen be‐ neath the surface. The speed of movement of the object is 10-15 mm/sec. Its size is approximately 4 mm.

From figure 25, the object moves from left to right and slightly upwards. Speed estimation is approximately 10 - 15 mm/sec. It is sufficient, in principle, to quickly notice the object with an unaided eye movement.

Given that the thickness of the surface layer of water on the air interface is 0.15 mm [7], it can be assumed that the object moved at least at this depth. It follows that the movement of the objects does not only affect surface forces. In principle, there were hopes to explain the observed motion of the objects.

Large objects, in the form of agglomerates, obviously also moves under the water surface (see Fig. 26) as reflected by the changes in the surface mesh of the indicator observed.

**Figure 26.** The third type of moving objects - shapeless aggregates. The speed of movement of the object is about 35 - 40 mm / sec (the trajectory 3 in Fig. 27).

For the experiment, the recorded moving agglomerates were characterized by the emergence of many objects. Fig. 27 shows the trajectories of some objects.

**Figure 27.** The trajectories of the objects

From the observation of the movement of objects, it was noted that in the beginning, a moving black circular object in shots 17 - 39 (path 4) has initiated. Then, it continues to become an agglomerate (path 3) and, following the black object, both come to a single point (see Fig. 26) (Frame 34 - 84). At this point, there is intense movement of the objects in a small volume. From this point, after a while, the new black object starts to move quickly (frames 85 - 100 trajectory 5). Along with the sinter, a noticeable movement was produced but in reverse, which was showed by the dark object in frames 64 - 74 (milestone trajectory 6). Another dark object (path 7 shots 69 - 78) was seen moving together with other large agglomerates. Initially, the agglom‐ erate is not noticeable and suddenly appears near this point object and continues to move together. At the bottom, starting with frame 58, two objects are substantially on the same trajectory at first and then a large object is shown(1 and 2 of the trajectory).

Moving objects change their size and position. It can be assumed that this is due to the addition and elimination of particles that make up the objects or the rearrangement of it. Before the experiment, the sand used in the experiment was sifted through a sieve with 1 mm openings. The observed objects have the same parameters, significantly exceeding this size. Consequent‐ ly, the objects, themselves, are formed during the experiment.

**Figure 26.** The third type of moving objects - shapeless aggregates. The speed of movement of the object is about 35 -

For the experiment, the recorded moving agglomerates were characterized by the emergence

From the observation of the movement of objects, it was noted that in the beginning, a moving black circular object in shots 17 - 39 (path 4) has initiated. Then, it continues to become an

of many objects. Fig. 27 shows the trajectories of some objects.

40 mm / sec (the trajectory 3 in Fig. 27).

200 Surface Energy

**Figure 27.** The trajectories of the objects

**Figure 28.** The trajectories of the cyclic objects. The direction of movement is indicated by the arrows.

In our point of view, the time of passage of the objects in the shadow of the device unit is important. In general, the moving objects appear darker than the surrounding background (there is light merging with the general background). However, objects remain visible and lighter than the shade behind the shadow of the devices. In our opinion, this indicates that the objects glow. Indeed, the cyclically moving objects in the original frames show that the objects marked by the arrows in Figure 28 had a yellow gold color (like light).

**Figure 29.** The successive frames of the cyclic movement of objects (path 1 frames 168-207. Trajectory 2 frames 215-328 Fig. 28)

It is believed that the water in the seas and lakes by chemiluminescence illuminates during the oxidation of organic substances with oxygen. Moreover, the luminescence observed depends on the ultrasound and the purity of the water that can be suppressed. The purer the water, the less intense is the glow [13]. Possibly, the ultrasound destroys the self-organizing structure of the particles, leading to a decrease in the luminescence. In purer water, the smaller particles can be united and pushed downward for the oxidation of organic substances. With regard to our experience, these facts suggest that the association of the particles into larger agglomerates causes the acceleration of the oxidation reaction. Any association and creation of complex structures in itself contribute to the acceleration of electrons, which causes the glow and helps create movement. At the same time, it can be assumed that the particles are oxidation catalysts and as a result, have a visible glow.

The fourth type of moving objects performs cyclic movements (Fig. 29). On the frames 281, 292, and 328, large arrows labeled objects rod type 5. In frames 178, 189, 198, 204, 273, 276, 281, 292, and 328, moving objects are in the shade and have a bright appearance that may shine.

We also recorded fast-moving underwater objects (see Fig. 30). Their speed is 100 - 150 mm/ sec. A total of four frames recorded the motion of this particle.

It is possible that there are particles moving with an even higher speed. But they could not be fixed, due to the technical capabilities of the equipment.

**Figure 30.** A fast-moving underwater object.

These results show that research in this area are the prospects in the creation of sufficiently large, even visible to the naked eye, objects moving independently of nanoparticles in water. It is then necessary to understand the principles that unite and move the newly created objects.

## **Author details**

In our point of view, the time of passage of the objects in the shadow of the device unit is important. In general, the moving objects appear darker than the surrounding background (there is light merging with the general background). However, objects remain visible and lighter than the shade behind the shadow of the devices. In our opinion, this indicates that the objects glow. Indeed, the cyclically moving objects in the original frames show that the objects

**Figure 29.** The successive frames of the cyclic movement of objects (path 1 frames 168-207. Trajectory 2 frames 215-328

It is believed that the water in the seas and lakes by chemiluminescence illuminates during the oxidation of organic substances with oxygen. Moreover, the luminescence observed depends on the ultrasound and the purity of the water that can be suppressed. The purer the water, the less intense is the glow [13]. Possibly, the ultrasound destroys the self-organizing structure of the particles, leading to a decrease in the luminescence. In purer water, the smaller particles can be united and pushed downward for the oxidation of organic substances. With regard to our experience, these facts suggest that the association of the particles into larger agglomerates

Fig. 28)

202 Surface Energy

marked by the arrows in Figure 28 had a yellow gold color (like light).

A.O. Titov\* , I.I. Titova, M.O. Titov and O.P. Titov

\*Address all correspondence to: fibrilla45@mail.ru

East-Siberian State University of Technology and Management, Russia

## **References**

[1] Patent RUSSIA №2362141Sposob determine the amount of fluid moved surfactant http://www.findpatent.ru/patent/236/2362141.


**A General-Purpose Multiphase/Multispecies Model to Predict the Spread, Percutaneous Hazard, and Contact Dynamics for Nonporous and Porous Substrates and Membranes**

Navaz Homayun, Zand Ali, Gat Amir and Atkinson Theresa

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/60807

#### **Abstract**

[2] Patent number 2362979 RUSSIA device for determining the distance of propagation

[3] Moving liquid surfactant as a way of assessing the properties of surfactant, liquids, and surfaces. Titov, AO; Titov, O. P.; Titov, M. O.; Karbainov, AN Journal of Physics: Conference Series, Volume 291, Issue 1, article id. 012 011, 7 pp. (2011). http://iops‐

[6] Titov II, Titov SA, Titov OP structure of the layer of liquid (water). Nanotechnics 3

[7] Experiments with soap film without soap. The journal "Science and Life" № 1, 2004.

[8] RUSSIAN Patent number 2,510,011 Method of determining the amount of liquid con‐

[9] Patent number 2510495 RUSSIA method for determining the thickness of the boun‐

[10] http://gisap.eu/ru/node/8855 EVALUATION surface properties of materials move‐ ment of fluid surfactant International Academy of Science and Higher Education

[11] Patent number 2527702 RUSSIA method for determining the properties of particulate materials in contact with water and surfactantshttp://www.findpatent.ru/patent/

[12] Patent number 2524556 RUSSIA visualization method of self-organization and move‐

ment of objects http://www.findpatent.ru/patent/252/2524556.html

[13] Baikal water glow http://baikaler.ru/news/2012/05/31/676/

[4] http://www.nanometer.ru/2010/11/05/internet\_olimpiada\_220718.html

[5] https://ru.wikipedia.org/wiki/Поверхностное\_натяжение

of microwaves on the surface of the liquid layer.

cience.iop.org/1742-6596/291/1/012011/

http://www.nkj.ru/archive/articles/5029/

veyed surfactant in the gas phase.

dary layer of water.

(IASHE, London, UK

252/2527702.html

(35)2013

204 Surface Energy

A computational model to solve the coupled transport equations with chemical reaction and phase change for a liquid sessile droplet or the contact and spread of a sessile droplet between two approaching porous or non-porous surfaces, is developed. The model is general therefore it can be applied to toxic chemicals (contact hazard), drug delivery through porous organs and membranes, combus‐ tion processes within porous material, and liquid movements in the ground. The equation of motion and the spread of the incompressible liquid available on the primary surface for transfer into the contacting surface while reacting with other chemicals (or water) and/or the solid substrate are solved in a finite difference domain with adaptive meshing. The comparison with experimental data demon‐ strated the model is robust and accurate. The impact of the initial velocity on the spread topology and mass transfer into the pores is also addressed.

**Keywords:** Liquid spread in porous materials, gaseous spread in porous mate‐ rial, evaporation in porous material, phase change in porous materials, multiphase process in porous materials, multi-species process in porous materials

© 2015 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

## **1. Introduction**

Many hazardous chemicals are dispersed in their liquid phase as droplets and reside on the surface of the liquid shortly after dispersion. In this state they are referred to as sessile droplets. Many coupled processes commence after this stage, resulting in different forms of threats, depending on the surface and physical properties of the liquid droplet. For nonporous surfaces, if the vapor pressure of the liquid is high, the threat of exposure is through the respiratory system, and if the vapor pressure is low, a percutaneous or direct contact hazard can be expected. Furthermore, should liquid droplets enter a chemical reaction with the surface, the products or the remaining reactants can reduce or augment the threat. In general, the following processes can be expected on a nonporous surface:


For porous surfaces the following processes can be simultaneously present:


Although nonporous and porous surfaces share some processes, there are some differences in their outcome. Furthermore, in case of temperature variation, phase change can occur, thereby necessitating the inclusion of the energy equation in exposure risk models.

It is evident that the processes such as capillary transport (for porous surfaces), surface evaporation, in-pore or secondary evaporation, adsorption, and chemical reaction are initiated simultaneously as soon as a liquid droplet contacts a surface. In the case of a contact, a liquid droplet can be transported to the secondary (or contacting surface). This can occur when the droplet is sandwiched between the two surfaces, with the droplet forming a liquid bridge, or long after the droplet has disappeared from the surface. Therefore, an additional coupled process, which is mainly physical (liquid bridge spread), can be added to the above processes. The end result of all these processes determines the available chemical for atmospheric transport in the form of vapor or percutaneous hazard on surfaces available for transfer to a secondary surface upon contact.

**1. Introduction**

206 Surface Energy

**a.** Surface evaporation

**a.** Surface evaporation

**b.** Capillary transport

postcontact phase

processes can be expected on a nonporous surface:

droplet on the nonporous or primary surface

**d.** Chemical reaction and adsorption inside the pores

**b.** Chemical reaction or adsorption at the contact surface only

**d.** Spread of the liquid bridge as the gap between the two surfaces closes. For porous surfaces the following processes can be simultaneously present:

**c.** Evaporation through the pores referred to as secondary evaporation

**f.** Spread of the liquid bridge if the conditions for its formation are met

hazard can exist even though the liquid droplet cannot be seen.

necessitating the inclusion of the energy equation in exposure risk models.

Many hazardous chemicals are dispersed in their liquid phase as droplets and reside on the surface of the liquid shortly after dispersion. In this state they are referred to as sessile droplets. Many coupled processes commence after this stage, resulting in different forms of threats, depending on the surface and physical properties of the liquid droplet. For nonporous surfaces, if the vapor pressure of the liquid is high, the threat of exposure is through the respiratory system, and if the vapor pressure is low, a percutaneous or direct contact hazard can be expected. Furthermore, should liquid droplets enter a chemical reaction with the surface, the products or the remaining reactants can reduce or augment the threat. In general, the following

**c.** Formation of liquid bridge when a secondary surface come into contact with the sessile

**e.** Possible formation of liquid bridge when a secondary surface comes into contact with any possible sessile droplet left due to capillary transport on the primary substrate in the

**g.** Transfer of chemicals from the primary surface to the contacting surface occurs when the droplet on the primary substrate is observable or nonobservable. Therefore, contact

Although nonporous and porous surfaces share some processes, there are some differences in their outcome. Furthermore, in case of temperature variation, phase change can occur, thereby

It is evident that the processes such as capillary transport (for porous surfaces), surface evaporation, in-pore or secondary evaporation, adsorption, and chemical reaction are initiated simultaneously as soon as a liquid droplet contacts a surface. In the case of a contact, a liquid droplet can be transported to the secondary (or contacting surface). This can occur when the droplet is sandwiched between the two surfaces, with the droplet forming a liquid bridge, or When a wetting fluid is introduced into a porous medium, in the absence of external forces, the distribution of the liquid in pores is influenced by pressure differential, capillary, gravita‐ tional, and viscous forces. The main driver for the spread of this wetting fluid is capillary pressure. As the porous medium imbibes the liquid, the liquid flow rate decreases due to the domination of viscous forces, causing a reduction in the flow capillary number. The spread dynamics are influenced by the porous medium thickness, where two- and three-dimensional flows occur. For thin porous media, the liquid flows in a radial direction and can be modeled by the method of "common-lines" [1, 2]. In these solutions, two parameters need to be experimentally determined. In thick porous media, the flow becomes three-dimensional, although a cylindrical symmetry has been assumed in the past [3, 4] The experimental results of Denesuk et al. [5] verified this three-dimensional nature of the flow. Holman et al. [6] studied the surface spread in a porous medium as a function of permeability and have demonstrated that the surface spread is the dominant mechanism as compared to droplet penetration dynamics. In the above studies, a well-defined interface between the wetted and nonwetted volumes existed. It is known that spread dynamics becomes more complex when the saturation is less than unity, as demonstrated via MRI experiments by Mantle et al. [7]

When a droplet spreads into a porous substrate, saturation gets distributed between 0 and 1, necessitating a multiphase flow approach. Therefore, the governing equations need to account for the presence of all the phases [8, 9]. In these equations, two additional transport parameters are present: a relative permeability and a capillary pressure function that includes the interaction parameters between phases as shown by Chen et al. [10]. The relative permeability can be modeled using a power function as indicated by Brooks and Corey, [11] or van Genuchten equations, [12] in which each equation has a large number of parameters expressing the relative permeability. The capillary pressure is often modeled using the Leverett [13]and Udell [14] *J*-function.

In solving the droplet spread on or into a porous substrate, the shape of the droplet on the porous substrate surface (droplet free volume) and the shape of the wetted porous substrate imprint must be determined. Starov et al. [2, 15] have solved the governing equations to obtain the shape of a droplet free volume that spreads into a thin porous medium where he found a large variation of the droplet base radius. Using full numerical calculations for a threedimensional porous medium, Alleborn et al. [4] have come to a similar conclusion. In respect to the imprint shape, the spread of the droplet is usually modeled by clearly separating the fully saturated and non-saturated regions. However, the spread dynamic is more complex due to the existence of partially saturated regions and varying transport parameters within the pores as indicated by Lenormand et al. [16]

The change of the flow mechanism, and the formation of partially saturated regions, requires solving the droplet spread into porous medium as a multiphase problem for which the multiphase flow parameters have to be determined. The multiphase parameters depend on the type of process and the predominance of gravity, viscous, or capillary forces. [16] Valava‐ nides and Payatakes [17] summarized these influences on the multiphase parameters, where, in addition to phase content or phase saturation and emerging forces, solid–fluid contact angles, ratios of phase viscosities and flow rates, and the flow history (e.g., drainage or imbibition) altered the behavior of multiphase parameters. Discrete pore network models provide an alternative approach to elucidate the transfer phenomena and to evaluate transport parameters. In network models, an actual porous medium is represented as a network of pores that are connected by throats. [18] The phase progression is set using a potential threshold and the flow pattern is determined thereafter, as outlined by Prat. [19] From the phase distribution, the multiphase parameters are calculated (see, e.g., Constantinides and Payatakes [20]).

Further studies using a discrete pore network split the flow into primary and secondary regimes. The primary flow occurs when the volume of the sessile drop is greater than zero and the secondary flow happens after the sessile droplet volume becomes zero [21, 22, 23, 24]. In the secondary flow regime, the flow front at the surface becomes an irregular flow front, forming a heterogeneous porous medium at the surface. [25]

In spite of all previous progress in the field of porous media flow, there was still a need for a production code (or model) that included all the physiochemical processes in order to calculate the amount of a hazardous liquid droplet that poses a threat. The first problem encountered, in a realistic scenario, is the evaporation of a sessile droplet that is deposited onto an imper‐ meable surface and exposed to outside wind conditions. This is called "convective evapora‐ tion" and its driving force is the outdoor wind velocity. As mentioned previously, a convective evaporation (laminar or turbulent) model needs to be developed in the absence of all other processes, i.e., for non-permeable non-reacting surfaces. An evaporation model for a sessile droplet on a non-porous surface with stagnant air above it is discussed by Popov [26]. Hu et al. [27] used finite element method to mesh the domain and then solved the mass conservation equations for the vapor concentration to find the evaporation rate. The above models assume no motion of air above the sessile droplet eliminating the convective effects. Baines et al. [28] developed a very simple convective evaporation model that did not compare with experi‐ mental results very well and in some cases overpredicted the evaporation rate by 90%.

A comprehensive literature review of the existing models is presented by Winter et al. [29] All of the previous studies focus on capturing the physics of evaporation and do not address the role of turbulence on evaporation. The effect of turbulence on evaporation of droplets that are moving in the core flow has been addressed by Navaz et al. [30] and Ward et al. [31] for spray combustion problems where high density and temperature gradients will reduce the evapo‐ ration time to the scale of turbulence fluctuations. However, for smaller or more moderate temperature gradients, the turbulence time scale will be smaller than the evaporation time scale. Therefore, a new approach should be developed to examine the effects of turbulence on evaporation of sessile drops. There are numerous works that examine atmospheric turbulence. The effect of the free stream Reynolds number on shear stress and friction velocity at a wall under outdoor conditions is discussed by Metzger and Klewicki. [32] Their study was performed on the Great Salt Lake Desert and spans over *Re*=2000 – 5e6 or *Re*=2000 - 5× 106 . There is also a dataset for friction velocity measured over a period of several months compiled from the outdoor wind data of Klinger et al. [33]. It has been mentioned by Weber [34] that the main scaling parameter in similarity theory of atmospheric boundary layer is friction velocity. He concludes that there could be a significant difference in friction velocity depending on the type of air flow. Several studies have focused on the varying turbulence intensity in wind-driven flows under outdoor conditions [35, 36]. Their approach relies on the validity of some wall function that relates the friction velocity to the mean free stream velocity and height. Upon knowing the friction velocity, the shear stress and turbulence intensity are correlated. For the evaporation problem, the previous studies have correlated the free stream velocity (or Reynolds number) to the friction velocity. However, they have not addressed conditions where turbulence intensity can change while the mean free stream velocity stays constant, or the turbulence intensity can be altered without changing the free stream velocity (or mean velocity in a channel flow).

multiphase flow parameters have to be determined. The multiphase parameters depend on the type of process and the predominance of gravity, viscous, or capillary forces. [16] Valava‐ nides and Payatakes [17] summarized these influences on the multiphase parameters, where, in addition to phase content or phase saturation and emerging forces, solid–fluid contact angles, ratios of phase viscosities and flow rates, and the flow history (e.g., drainage or imbibition) altered the behavior of multiphase parameters. Discrete pore network models provide an alternative approach to elucidate the transfer phenomena and to evaluate transport parameters. In network models, an actual porous medium is represented as a network of pores that are connected by throats. [18] The phase progression is set using a potential threshold and the flow pattern is determined thereafter, as outlined by Prat. [19] From the phase distribution, the multiphase parameters are calculated (see, e.g., Constantinides and Payatakes [20]).

Further studies using a discrete pore network split the flow into primary and secondary regimes. The primary flow occurs when the volume of the sessile drop is greater than zero and the secondary flow happens after the sessile droplet volume becomes zero [21, 22, 23, 24]. In the secondary flow regime, the flow front at the surface becomes an irregular flow front,

In spite of all previous progress in the field of porous media flow, there was still a need for a production code (or model) that included all the physiochemical processes in order to calculate the amount of a hazardous liquid droplet that poses a threat. The first problem encountered, in a realistic scenario, is the evaporation of a sessile droplet that is deposited onto an imper‐ meable surface and exposed to outside wind conditions. This is called "convective evapora‐ tion" and its driving force is the outdoor wind velocity. As mentioned previously, a convective evaporation (laminar or turbulent) model needs to be developed in the absence of all other processes, i.e., for non-permeable non-reacting surfaces. An evaporation model for a sessile droplet on a non-porous surface with stagnant air above it is discussed by Popov [26]. Hu et al. [27] used finite element method to mesh the domain and then solved the mass conservation equations for the vapor concentration to find the evaporation rate. The above models assume no motion of air above the sessile droplet eliminating the convective effects. Baines et al. [28] developed a very simple convective evaporation model that did not compare with experi‐ mental results very well and in some cases overpredicted the evaporation rate by 90%.

A comprehensive literature review of the existing models is presented by Winter et al. [29] All of the previous studies focus on capturing the physics of evaporation and do not address the role of turbulence on evaporation. The effect of turbulence on evaporation of droplets that are moving in the core flow has been addressed by Navaz et al. [30] and Ward et al. [31] for spray combustion problems where high density and temperature gradients will reduce the evapo‐ ration time to the scale of turbulence fluctuations. However, for smaller or more moderate temperature gradients, the turbulence time scale will be smaller than the evaporation time scale. Therefore, a new approach should be developed to examine the effects of turbulence on evaporation of sessile drops. There are numerous works that examine atmospheric turbulence. The effect of the free stream Reynolds number on shear stress and friction velocity at a wall under outdoor conditions is discussed by Metzger and Klewicki. [32] Their study was

or *Re*=2000 - 5× 106

. There

performed on the Great Salt Lake Desert and spans over *Re*=2000 – 5e6

forming a heterogeneous porous medium at the surface. [25]

208 Surface Energy

Once a droplet spreads into a porous substrate, secondary evaporation (from the porous substrate) occurs. This mass transport and vapor diffusion is hindered by granules of the porous medium, and the vapor may face liquid parcels of the agent that act as additional obstructions. These obstructions greatly alter the transport properties of the vapor inside the pores and the transport rate toward the surface. Basically, the transport properties for the vapor phase become a function of porosity and saturation. These properties are yet to be determined.

As mentioned earlier, capillary transport of sessile droplet of a chemical agent on a porous substrate can be further complicated by the presence of chemical reaction. The chemical reaction can occur between the chemical agent and the constituents of a porous substrate or the chemical agent and pre-existing chemicals inside the pores, such as water or other chemicals. It is known that some chemical agents undergo hydrolysis reaction when they encounter moisture in the soil or sand, as it will be shown later. A simple set of one-dimensional mass and energy equations for a medium with no motion (conductive only) with chemical reaction was solved by Sadig [37, 38] who also demonstrated how the local properties are affected by chemical reaction, as the composition of constituents vary in time. Xu et al., [39] Lichtner, [40] and Steefel [41] have solved a very simple reactive flow with diffusion in geological settings dealing with ion transport. To have a general-purpose model, it is required that the conservation equations for all the phases (solid, liquid, and gaseous) and existing constituents, in forms of reactants and products with a chemical reaction, be solved simulta‐ neously in time. The variation of species concentration in any of the existing phases will alter the local properties of liquid, gas, or solid mixture. Furthermore, the production or destruction of any solid phase will also change the local porosity of the medium.

A low-volatile chemical in its liquid form poses a threat upon contact with skin or other materials (vehicles, etc.). Mass transfer to the secondary surface occurs during this process. However, the contact may occur before a sessile droplet completely sinks in the primary substrate, if porous, or after a sessile droplet is not observable on the surface. In the first case, a liquid bridge is formed between the two approaching surfaces and may spread depending on the surface properties. Furthermore, if the primary surface or substrate is non-porous, a liquid bridge is definitely formed and spreads as the distance between the approaching surfaces decreases. The forces applied by liquid bridges connected to static supporting surfaces were studied extensively by Butt et al. [42] However, in many applications, it is common that at least one of the bodies is moving, and thus, the motion of the solid body may be influenced by the forces associated with the liquid bridge [43, 44, 45]. Pitois et al. [46] studied the forces applied by a liquid bridge connecting two spheres moving at a constant speed relative to each other. Similarly, Meurisse and Querry [47] studied the effects of liquid bridges connecting two parallel flat plates, moving perpendicularly to the plane of the plates, at a constant speed, or at a constant force applied on the liquid. Both Pitois et al. [46] and Meurisse and Querry [47] observed a rapid change from an attractive force due to capillary effects to a repulsive force due to viscous effects for the case of surfaces approaching each other at a constant speed. De Souza et al. [48] studied the effect of contact angle hysteresis on the capillary forces in the absence of significant viscous forces and obtained good agreement between the experimental data and the existing models. The capillary force measurement of liquid bridges was examined and shown to be a reliable method for estimating advancing and receding wetting angles.

The absorption of chemical into the skin is of primary interest due to threat from contact with hazardous materials. Experimental studies of Ngo, O'Malley, and Maibach [49] have attempt‐ ed to quantify mass absorption into the skin. This is a complex problem because skin chemistry and perspiration rate may vary between individuals affecting the absorption rate. Although some simplified models based on linear regression exist to estimate the absorption rate, a more comprehensive and scientifically sound approach provides the solution of coupled equations for multi-species systems, as discussed hereinafter.

However, there was still a need for a model that can predict the available threat from a liquid chemical toxicant release. This issue has been addressed by Kilpatrick et al. [50], Savage [51, 52], and Munro et al. [53], and prompted the Department of Defense (DoD), Edgewood Chemical and Biological Center (ECBC), and Defense Threat Reduction Agency (DTRA) to initiate a comprehensive research project called Agent Fate Program that was geared toward chemical agents of warfare. This project led to the development of the first-generation production code COMCAD (COmputational Modeling of Chemical Agent Dispersion) by Navaz et al. [54, 55], and the next generation for contact hazard MOCHA (Modeling Of Contact HAzard) codes. [56] Since the model is general, it can not only be applied to chemical agents of the warfare, but also any other liquid that can be disseminated into the environmental substrates in a sessile droplet form (e.g., pesticides, toxic industrial chemicals (TICs), and toxic industrial materials (TIMs)). This chapter summarizes the result of this research and its application to a variety of problems.

#### **1.1. Mathematical model**

The species mass and momentum, and energy equations for a multicomponent system in a porous medium with all phases being active is rather complex and are given in Navaz et al. [57], Navaz et al., [55] Vafai, [9] Navaz et al., [58] and Kuo. [59] The continuity equation for a multicomponent system for all phases includes any loss of mass due to surface and secondary evaporation, and cross-over mass due to chemical reaction, phase change, and adsorption. The momentum equations in porous media are considered for liquid and gaseous phases. Local thermal equilibrium is considered for the energy equation. That is to say that the local temperatures of all phases are equal (*Ts* =*T*<sup>ℓ</sup> =*Tg*). Note that the solid phase is stagnant, but the porosity will be changing in time if any solid constituent is formed due to phase change or chemical reaction. The continuity equation for the porosity is eliminated, but the porosity is updated in each time step (lagged by one time step). Although for an open system under the usual atmospheric conditions the gaseous phase pressure normally stays at the environmental conditions, the pressure term is included in the equations to extend the applicability of the model for the future applications, in which the pressure may change (closed systems or existence of explosions). All the conservation equations are solved explicitly on a finite difference mesh as will be described later. The sub-models describing the source terms (e.g., evaporation), chemical kinetics, and adsorption models are described in separate sections. We start with explaining the evaporation rate term.

#### **Mass Conservation for Multiple Solid System (Substrate – Solid**" *k* ")

$$\frac{\partial \rho\_{sk} \rho\_s}{\partial t} = -\dot{\alpha}\_{sk}^{\text{Rauction}} - \dot{\alpha}\_{sk}^{\text{Adjorption}} \tag{1}$$

#### **Mass Conservation for Multiple Solid System (Liquid**"*i*")

surfaces decreases. The forces applied by liquid bridges connected to static supporting surfaces were studied extensively by Butt et al. [42] However, in many applications, it is common that at least one of the bodies is moving, and thus, the motion of the solid body may be influenced by the forces associated with the liquid bridge [43, 44, 45]. Pitois et al. [46] studied the forces applied by a liquid bridge connecting two spheres moving at a constant speed relative to each other. Similarly, Meurisse and Querry [47] studied the effects of liquid bridges connecting two parallel flat plates, moving perpendicularly to the plane of the plates, at a constant speed, or at a constant force applied on the liquid. Both Pitois et al. [46] and Meurisse and Querry [47] observed a rapid change from an attractive force due to capillary effects to a repulsive force due to viscous effects for the case of surfaces approaching each other at a constant speed. De Souza et al. [48] studied the effect of contact angle hysteresis on the capillary forces in the absence of significant viscous forces and obtained good agreement between the experimental data and the existing models. The capillary force measurement of liquid bridges was examined and shown to be a reliable method for estimating advancing and receding wetting angles.

The absorption of chemical into the skin is of primary interest due to threat from contact with hazardous materials. Experimental studies of Ngo, O'Malley, and Maibach [49] have attempt‐ ed to quantify mass absorption into the skin. This is a complex problem because skin chemistry and perspiration rate may vary between individuals affecting the absorption rate. Although some simplified models based on linear regression exist to estimate the absorption rate, a more comprehensive and scientifically sound approach provides the solution of coupled equations

However, there was still a need for a model that can predict the available threat from a liquid chemical toxicant release. This issue has been addressed by Kilpatrick et al. [50], Savage [51, 52], and Munro et al. [53], and prompted the Department of Defense (DoD), Edgewood Chemical and Biological Center (ECBC), and Defense Threat Reduction Agency (DTRA) to initiate a comprehensive research project called Agent Fate Program that was geared toward chemical agents of warfare. This project led to the development of the first-generation production code COMCAD (COmputational Modeling of Chemical Agent Dispersion) by Navaz et al. [54, 55], and the next generation for contact hazard MOCHA (Modeling Of Contact HAzard) codes. [56] Since the model is general, it can not only be applied to chemical agents of the warfare, but also any other liquid that can be disseminated into the environmental substrates in a sessile droplet form (e.g., pesticides, toxic industrial chemicals (TICs), and toxic industrial materials (TIMs)). This chapter summarizes the result of this research and its

The species mass and momentum, and energy equations for a multicomponent system in a porous medium with all phases being active is rather complex and are given in Navaz et al. [57], Navaz et al., [55] Vafai, [9] Navaz et al., [58] and Kuo. [59] The continuity equation for a multicomponent system for all phases includes any loss of mass due to surface and secondary evaporation, and cross-over mass due to chemical reaction, phase change, and adsorption. The momentum equations in porous media are considered for liquid and gaseous phases. Local

for multi-species systems, as discussed hereinafter.

application to a variety of problems.

**1.1. Mathematical model**

210 Surface Energy

$$\frac{\partial \left( \boldsymbol{\varrho} \boldsymbol{\rho}\_{\boldsymbol{\iota}\boldsymbol{\varepsilon}} \boldsymbol{s}\_{\boldsymbol{\iota}\boldsymbol{\iota}} \right)}{\partial t} + \nabla \left( \boldsymbol{\varrho} \boldsymbol{\rho}\_{\boldsymbol{\iota}\boldsymbol{\varepsilon}} \boldsymbol{s}\_{\boldsymbol{\iota}\boldsymbol{\iota}} \vec{V}\_{\boldsymbol{\iota}\boldsymbol{\iota}} \right) = \left( -\dot{\boldsymbol{\rho}}\_{\boldsymbol{\iota}\boldsymbol{\iota}}^{\text{Soundary Enaporation}} - \dot{\boldsymbol{\rho}}\_{\boldsymbol{\iota}\boldsymbol{\iota}}^{\text{Surnă ex Equation}} - \dot{\boldsymbol{\alpha}}\_{\boldsymbol{\iota}\boldsymbol{\iota}}^{\text{Racanton}} - \dot{\boldsymbol{\alpha}}\_{\boldsymbol{\iota}\boldsymbol{\iota}}^{\text{Absorption}} \right) \tag{2}$$

$$\boldsymbol{\rho}\_{\boldsymbol{\ell}-\text{mixture}} = \sum\_{l=1}^{N(\text{liquid})} \boldsymbol{\rho}\_{\boldsymbol{\ell}l} \mathbf{C}\_{\ell l} \qquad \mathbf{C}\_{\ell l} = \text{Mass} \quad \text{Fraction} \\ \boldsymbol{\eta} = \frac{\boldsymbol{\rho}\_{\boldsymbol{\ell}l} \mathbf{s}\_{\ell l}}{\sum\_{l=1}^{N} \boldsymbol{\rho}\_{\boldsymbol{\ell}l} \mathbf{s}\_{\ell l}} \text{ , } \mathbf{s}\_{\ell} = \sum\_{l=1}^{N(\text{liquid})} \mathbf{s}\_{\ell l} \tag{3}$$

$$\vec{V}\_{\ell i} = -\frac{\mathbf{K}\mathbf{k}\_{\ell i}}{\mu\_{\ell}} (\nabla P\_{\ell i} - \rho\_{\ell i}\mathbf{g}\mathbf{s}\_{\ell i}) \tag{4}$$

#### *j* **Mass Conservation for Multiple Gaseous System (Gas" ")**

$$\frac{\partial \left(\boldsymbol{\varrho}\boldsymbol{\rho}\_{\mathcal{S}}\mathbf{s}\_{\mathcal{S}}\right)}{\partial t} + \nabla \left(\boldsymbol{\varrho}\boldsymbol{\rho}\_{\mathcal{S}\mathcal{S}}\mathbf{s}\_{\mathcal{S}}\vec{V}\_{\mathcal{S}^{\complement}}\right) = \nabla \bullet \left(\boldsymbol{\rho}\_{\mathcal{S}}\mathbf{D}\_{j-\text{mix}}\nabla \mathbf{C}\_{j}\right) + \dot{\boldsymbol{\rho}}\_{\ell,j-\text{j}}^{\text{Sominary\text{ }Exponential}} - \dot{\boldsymbol{\alpha}}\_{\mathcal{S}^{\complement}}^{\text{Racation}} - \dot{\boldsymbol{\alpha}}\_{\mathcal{S}^{\complement}}^{\text{Adsorption}} \tag{5}$$

$$\vec{V}\_{\text{gj}} = -\frac{Kk\_{\text{gj}}}{\mu\_{\text{g}}} \left(\nabla P\_{\text{gj}} - \rho\_{\text{gj}} \text{gs}\_{\text{g}}\right) \cdot D\_{\text{(\rightarrow mixture)}} \nabla C\_{\text{gj}} \tag{6}$$

Relative Permeability to constituent of the gas phase *gj k* = *jth*

$$
\mu\_{\text{g}}\text{(gas\\_mixture\\_viscosity)} = \sum\_{j=1}^{M(\text{Gas})} \mu\_{\text{g}}\text{C}\_{\text{g}\prime} \tag{7}
$$

Density of constituent of the gas phase Velocity component of gas phase constituent *gj gj jth V jth* r = <sup>=</sup> <sup>r</sup>

where

$$s\_g = 1 - \sum\_{i}^{N(\text{Lipids})} s\_{\ell i} \quad \text{and} \qquad \rho\_g = \sum\_{j=1}^{M(\text{Gauss})} \rho\_{gj} \tag{8}$$

*D Effective Diffusion Coefficient of gas species "j" into the mixture j mix* - =

$$\mathbf{C}\_{\circ} = \text{Mass} \cdot \text{Fraction} = \frac{\rho\_{\text{gj}}}{\rho\_{\text{g}}} \tag{9}$$

$$P\_{\langle\rangle} = P - P\_{\langle\rangle} \text{ (Capillary Pressure)} + \rho\_{\langle\rangle}gh^\* \tag{10}$$

\* Local height of the droplet as a function of time (hydrostatic pressure) *h* =

Inter-Phase Chemical Species production or destruction w = &

Saturation permeability *K* =

( ) Relative permeability is a f *k ks* = <sup>l</sup> l <sup>2</sup> unction of the local saturation *s* = l

( ) , , , Local porosity is a function of local density of the solid phase *sk* j jr *xyz* =

$$k\_{\rm gj} = \mathbf{1} + s\_\ell^2 \left(\mathbf{2}s\_\ell - \mathbf{3}\right) \tag{11}$$

#### **Energy Equation**

$$\begin{split} & \frac{\partial}{\partial t} \Big[ \rho\_s \rho\_s H\_s + \rho\_t \rho\_t H\_t + \rho\_g \left( \rho\_g H\_g - P \right) \Big] + \\ & + \nabla \bullet \dot{\mathbf{q}} + \nabla \bullet \left( \rho\_s \rho\_t H\_t \vec{V}\_\ell + \rho\_g \rho\_g H\_g \vec{V}\_g \right) + \nabla \bullet \left( -\bar{\ddot{\mathbf{r}}}\_{\text{mot}} \vec{V}\_{\text{n\ell}} - \bar{\ddot{\mathbf{r}}}\_{\text{mug}} \vec{V}\_{\text{ng}} \right) \\ & \rho \Big[ -\nu\_l \varrho - \nu\_g \varrho \Big] + \sum\_{i=1}^{N\_{\text{ads}}} \dot{\rho}\_{s-\ell i} \Delta h\_{s-\ell i} + \sum\_{i=1}^{N\_{\text{ads}}} \dot{\rho}\_{\ell-g\ell} \Delta h\_{\ell-g\ell} + \sum\_{i=1}^{N\_{\text{ads}}} \dot{\rho}\_{s-g\ell} \Delta h\_{s-g\ell} = 0 \end{split} \tag{12}$$

A General-Purpose Multiphase/Multispecies Model to Predict the Spread, Percutaneous Hazard, and... http://dx.doi.org/10.5772/60807 213

where

( )

*M Gases*

*j*

=

(gas mixture viscosity) ,

Density of constituent of the gas phase

( ) ( )

*D Effective Diffusion Coefficient of gas species "j" into the mixture j mix* - =

*gj*

*C Mass Fraction*

( )

( ) ( )

*HV HV V V*

*gg g g mn n mng ng*

**q** t t (12)


11 1

== =

*g si si gi gi s gi s gi ii i*

r

&&&

*liquids solids solids*

*N N N*

ll l l l l

r r rr v v v v &

*gg h h h*

é ù -- + D + D + D =

\* (Capillary Pressure) *<sup>i</sup> ci <sup>i</sup> P PP* = - +

\* Local height of the droplet as a function of time (hydrostatic pressure) Inter-Phase Chemical Species production or destruction

, , , Local porosity is a function of local density of the solid phase *sk*

*g*

r*gh* l l (10)

<sup>2</sup> unction of the local saturation

*s*

l

0

=

( ) <sup>2</sup> 1 23 *gj k ss* =+ - l l (11)

r

r

r

*N Liquids M Gases g i g gj i j*

*jth V jth*

1 and

*j*

( )

( )

*ss s g gg*

ll l

 rj

rj

*H H HP*

é ù + + -+ ë û

 r  j r

> rj

l ll l l

+Ñ · + Ñ · + + Ñ · - -

ë û ååå

*h*

= = = = <sup>l</sup>

*K k ks*

l

=

j jr*xyz*

**Energy Equation**

*t*

ru

rj

 u

¶

¶

w

&

Saturation permeability Relative permeability is a f

*s s*

Velocity component of gas phase constituent

r

*g gj gj*

m

*gj gj*

r = <sup>=</sup> <sup>r</sup>

where

212 Surface Energy

1

 m*C*

1

 r = <sup>=</sup> - = å å <sup>l</sup> (8)

= = (9)

<sup>=</sup> å (7)

$$\begin{aligned} \boldsymbol{\rho}\_s &= (1 - \boldsymbol{\rho}), & \boldsymbol{\rho}\_t &= \boldsymbol{s}\_t \boldsymbol{\rho}, & \boldsymbol{\rho}\_g &= (1 - \boldsymbol{s}\_t) \boldsymbol{\rho}\_t & V^2 = \boldsymbol{\eta} u^2 + \boldsymbol{\upsilon}^2 + \boldsymbol{w}^2 \end{aligned} \tag{13}$$

$$\begin{aligned} \dot{\boldsymbol{\rho}} &= \text{Heat} \quad \text{flux} = -\boldsymbol{k}\_{\circ \boldsymbol{\rho}} \nabla T \quad \text{at} &= \text{Latent heat of phase change} \\ \dot{\boldsymbol{\rho}} &= \text{rate of phase Change} & H = \text{Total enthalpy} = \boldsymbol{c}\_p T + \frac{V^2}{2} \\ \boldsymbol{c}\_p &= \text{Specific heat} & \bar{\boldsymbol{\tau}} = \text{Shear stress tensor given by:} \\ \boldsymbol{\tau} &= \boldsymbol{w} \cdot \boldsymbol{\omega} \begin{bmatrix} \gamma\_2 \boldsymbol{\partial}u & 2 \left( \hat{\boldsymbol{\varepsilon}} u\_+ \right) \boldsymbol{\partial}v & \boldsymbol{\varepsilon} u\_- \end{bmatrix} \end{aligned} \tag{14}$$

$$\begin{aligned} \pi\_{xx} &= \mu \left| 2 \frac{\partial u}{\partial x} - \frac{2}{3} \left( \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} \right) \right|, & \qquad \pi\_{xy} &= \pi\_{yx} = \mu \left| \frac{\partial u}{\partial y} + \frac{\partial v}{\partial x} \right|, \\ \pi\_{xz} &= \pi\_{zx} = \mu \left[ \frac{\partial w}{\partial x} + \frac{\partial u}{\partial z} \right] \end{aligned} \qquad \qquad \begin{aligned} \pi\_{xy} &= \pi\_{yx} = \mu \left[ \frac{\partial u}{\partial y} + \frac{\partial v}{\partial x} \right], \\ \pi\_{yy} &= \mu \left[ 2 \frac{\partial v}{\partial y} - \frac{2}{3} \left( \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} \right) \right], \\ \pi\_{zz} &= \mu \left[ 2 \frac{\partial w}{\partial z} - \frac{2}{3} \left( \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} \right) \right] \end{aligned} \tag{14}$$

#### **1.2. Evaporation models**

When surface evaporation of a sessile droplet on a non-porous substrate occurs, one or both of the following mechanisms determines the topology of the sessile droplet. Either surface evaporation takes place with the droplet's base area (radius) remaining constant while the height of the droplet decreases, or the surface evaporation causes the shrinking of a droplet while the ratio *h/r* remains constant. It has been observed that for the evaporation of nerve agents (HD, VX, and GD) the process starts with the first mechanism and when the contact angle reaches about 12°, the second mechanism is initiated.

Irrespective of the evaporation mechanism (*r*=*const*. or *λr*= *h/r* =*const*.), the volume and surface area of a sessile droplet in a form of a spherical cap is given by:

$$\begin{aligned} \forall &= \frac{1}{3}\pi h^2 (\Im R\_s - h) = \frac{1}{6}\pi h (\Im r^2 + h^2) \\ A\_s &= 2\pi Rh \end{aligned} \tag{15}$$

The dynamics and topology of this model is shown in Figure 1. where the evaporation is initiated with the first mechanism and then is switched to the *h/r=const.* case.

Having the mass rate of evaporation (*m*˙ ) equal to the negative change of the mass (*m*) left on the surface at time (*t*) and it can be written as:

$$\dot{m} = -\frac{d}{dt}m = -\frac{d}{dt}(\rho\_\epsilon \ V) = -\rho\_\epsilon \frac{\pi}{2} (r^2 + h^2) \frac{dh}{dt} \tag{16}$$

where *( ρ*<sup>ℓ</sup> *)* is the liquid density. From Equation (16) the height of the droplet is calculated as:

$$\frac{dh}{dt} = \frac{-\dot{m}}{\rho\_\epsilon \frac{\pi}{2} \left(r^2 + h^2\right)}\tag{17}$$

**Figure 1.** Schematic for the Topology and Dynamics of HD Evaporation on a Non-Permeable Surface.

The instantaneous droplet height can be obtained by integrating Equation (17), provided that the forcing function *m*˙ is known. There are numerous expressions in the literature for the evaporation rate *m*˙ given by researchers in spray combustion field [58, 59, 60, 61] for spherical droplets following a gas trajectory in a rocket engine. The current model follows the same process with a few exceptions. Toxic chemicals are generally tested in small wind tunnels mainly for better control of contamination. The flow in these wind tunnels is basically a channel flow and a boundary layer plate cannot be installed (due to size constraints) to create a scalable model to the open air scenarios. To overcome this issue, the Reynolds number in Equation (18) was based on the friction velocity instead of the free stream velocity. Embedded in the friction velocity is the impact of turbulence on evaporation. This is discussed by Navaz et al. [57] and is the focus of Section 1.3. Further modification was necessary to define a new transfer *cp*(*Tg* −*T*boiling point)

number, B. The transfer number in combustion is defined as: *h fg* where *hfg* is the latent heat of evaporation. Although this definition will work in a combustion chamber, it does not apply to the evaporation of toxic chemicals under environmental conditions that mainly involve the vapor pressure of the sessile droplet and convective mass transfer. It is assumed that the evaporation process takes place isothermally and any minimal change of temperature at the interface and its conduction through the sessile droplet can be ignored. Two different length scales are defined depending on the droplet topology. [57] The following surface evaporation model, Equation (18), provides the source term (*ρ*˙ <sup>ℓ</sup> *surface evaporation*) in Equations (2) and (5) for sessile droplets, which is derived, calibrated, and used for this study.

( ) ( ) 2 2 2

 r

where *( ρ*<sup>ℓ</sup> *)* is the liquid density. From Equation (16) the height of the droplet is calculated as:

&

( ) 2 2

p

l l & (16)

(17)

*d d dh m m V rh dt dt dt*

2

+ <sup>l</sup>

*dh m dt r h* p


r

**Figure 1.** Schematic for the Topology and Dynamics of HD Evaporation on a Non-Permeable Surface.

number, B. The transfer number in combustion is defined as:

The instantaneous droplet height can be obtained by integrating Equation (17), provided that the forcing function *m*˙ is known. There are numerous expressions in the literature for the evaporation rate *m*˙ given by researchers in spray combustion field [58, 59, 60, 61] for spherical droplets following a gas trajectory in a rocket engine. The current model follows the same process with a few exceptions. Toxic chemicals are generally tested in small wind tunnels mainly for better control of contamination. The flow in these wind tunnels is basically a channel flow and a boundary layer plate cannot be installed (due to size constraints) to create a scalable model to the open air scenarios. To overcome this issue, the Reynolds number in Equation (18) was based on the friction velocity instead of the free stream velocity. Embedded in the friction velocity is the impact of turbulence on evaporation. This is discussed by Navaz et al. [57] and is the focus of Section 1.3. Further modification was necessary to define a new transfer

latent heat of evaporation. Although this definition will work in a combustion chamber, it does not apply to the evaporation of toxic chemicals under environmental conditions that mainly involve the vapor pressure of the sessile droplet and convective mass transfer. It is assumed that the evaporation process takes place isothermally and any minimal change of temperature at the interface and its conduction through the sessile droplet can be ignored. Two different length scales are defined depending on the droplet topology. [57] The following surface

*cp*(*Tg* −*T*boiling point) *h fg*

where *hfg* is the

=- =- =- + r

214 Surface Energy

$$\begin{aligned} \stackrel{\circ}{m}\_{\text{surface}} \quad & \text{evaporation} = 2\pi L\_c \frac{\mu}{\text{Pr}} \Big( \text{C}\_f + \text{C}\_1 \stackrel{\circ}{\text{Re}}^m \stackrel{\circ}{\text{Pr}}^m \eta \Big) \ell n \Big[ 1 + B \Big] \\\ \stackrel{L}{L}\_c = \text{Length scale} \begin{cases} \text{Mechanical} \, 1 \text{: } = & 2\pi R\_s \stackrel{\circ}{\text{R}} \\ \text{Mechanical} \, 2 \text{: } = & 2\pi R\_s \Big[ 1 - \text{sin}\left(\frac{\pi}{2} - \theta\_a\right) \Big] \end{cases} \end{aligned}$$

Re Reynolds number based on the radius of curvature and friction velocity Gas mixture density *g* r = =

$$B = \text{Transfer Number} = \left(\frac{y}{1-y}\right)^{\mathcal{E}}, \quad y = \frac{P\_{\nu j}}{P} \tag{18}$$

Vapor ressure of species " where: is the mass of liquid species " t each node , , , , Model constants to be determined experimentally <sup>1</sup> *P p j" <sup>j</sup> m j" a C Cc c f mn* u e = =

The evaporation model for flow through the pores is also based on the methodology discussed by Navaz et al. [55, 58]. A numerical investigation of this process on hot surfaces has been published by Nikolopoulos et al. [62] However, in the context of finite difference or finite volume framework, a new model needs to be developed. It is assumed that the liquid phase at each node is spherical and the instantaneous length scale is found by knowing the mass and density of each liquid species at this node. The evaporation rates in the species continuity equation (Eq. 2) are obtained as indicated by Equation (19).

$$\begin{aligned} \text{(\\_Secondary evaporation)} &= 4\pi \rho \frac{D}{\text{g}} \text{D}\_{\text{j}-\text{Mix}} \, ^L \text{c} \left( \text{C}\_1 + \text{C}\_2 \, \text{Re}^m \text{Sc}^n \right) \ell n \left[ 1 + \frac{\left( \frac{P\_{\text{v}j}}{P} \right)^{\mathcal{E}}}{1 - \left( \frac{P\_{\text{v}j}}{P} \right)^{\mathcal{E}}} \right] \\\\ \text{Secondary Equation} \end{aligned}$$

Secondary Evaporation / *<sup>m</sup> cell* r = " & &

Local Schmidt Number Gas mixture density Effective Diffusion Coefficient of species " into the mixture Vapor Pressure of species " Gas mixture pressure *<sup>j</sup> Sc Dj Mix g D j" j Mix P j" <sup>j</sup> P* n r u = = - = <sup>=</sup> - = = (19) 1 3 *<sup>m</sup> <sup>j</sup> <sup>L</sup>* l

$$\begin{aligned} L\_{\mathcal{L}} &= \text{Characteristic Length} = \left(\frac{m\_{\ell\dot{f}}}{\rho\_{\ell\dot{f}}}\right)^{\sqrt{3}}\\ \text{Re} &= \text{Reynolds Number} = \frac{L\_{\mathcal{L}}\sqrt{\nu\_{\ell}^{2} + \nu\_{\ell}^{2} + \overline{w}\_{\ell}^{2}}}{\nu\_{\dot{f}}} \end{aligned}$$

where: is the mass of liquid species " t each node , , , , , Model constants to be determined experimenta <sup>123</sup> *<sup>m</sup> j" a <sup>j</sup> C C C mn* e = l

lly

#### **1.3. Scalability of surface evaporation model**

As mentioned earlier, the Reynolds number in the evaporation model is based on friction velocity. This is due to the fact that a similar solution can be found for all cases involving the evaporation of a sessile droplet. Furthermore, the turbulence also speeds up the evaporation and it was postulated that the turbulence effects will be embedded in the friction velocity.

The model from Equation (18) suggests that the evaporation is influenced by geometry (because of the exposed surface to convective motion), convective transfer (turbulence and free stream velocity), and driving force (temperature and/or mole fraction). Based on analysis performed by similitudes, non-dimensional groups that lump all these effects and can influence the evaporation rate were derived. It was further postulated that the characteristic velocity representing the convective evaporation would be the friction velocity and not the traditional mean flow velocity. Based on these analyses, a geometrical factor of *V*/*r*<sup>2</sup> defining the length scale was defined. Basically, a sessile droplet will "see" the friction velocity rather than the mean velocity of the flow. Furthermore, the turbulence effect that has an impact on evaporation rate is embedded in the friction velocity. Therefore, expressing the Reynolds number, which is the driver for convective evaporation in terms of the friction velocity, seems like a plausible proposition. Based on the similitude analysis that was explained earlier, a nondimensional time scale is defined as:

A General-Purpose Multiphase/Multispecies Model to Predict the Spread, Percutaneous Hazard, and... http://dx.doi.org/10.5772/60807 217

$$
\hbar \mathfrak{u}^\* \left( V \left/ r^2 \right)^{-1} \ell n \left( 1 + B \right) = \mathfrak{t} \mathfrak{u}^\* \left( r^2 \slash V \right) \ell n \left( 1 + B \right) \tag{20}
$$

The model was used to calculate the evaporation rate and the normalized amount of mass left as a function of the non-dimensional time shown in Equation (20) for a variety of droplet initial volume, friction velocity, and initial contact angle (or base radius). The HD properties were used for this study. Figure 1.3.1 shows the collective effects of this study and it is observed that regardless of the droplet initial topology, air velocity, and turbulence intensity, a similar solution exists. This verifies the validity of our earlier postulate about the friction velocity and its ability to scale wind tunnel data to open air evaporation results. An identical experimental study was conducted with HD and the same similar solutions were calculated and plotted analogous to Figure 1.3.1 to create Figure 2.. It is observed that the experimental results verify the results obtained by the mathematical model.

Local Schmidt Number

= =

*<sup>j</sup> Sc*

Gas mixture density

*g*

=

=

<sup>=</sup> -

*c*

r

216 Surface Energy

*P*

u

=

Gas mixture pressure

Vapor Pressure of species "

Characteristic Length

e

=

*<sup>m</sup> <sup>j</sup> <sup>L</sup>*

ç ÷ = = ç ÷

Re Reynolds Number

= =

123

dimensional time scale is defined as:

*C C C mn*

l

**1.3. Scalability of surface evaporation model**

*P j" <sup>j</sup>*

*D j" j Mix*

Effective Diffusion Coefficient of species " into the mixture

1 3

22 2

ll l

, , , , , Model constants to be determined experimenta

As mentioned earlier, the Reynolds number in the evaporation model is based on friction velocity. This is due to the fact that a similar solution can be found for all cases involving the evaporation of a sessile droplet. Furthermore, the turbulence also speeds up the evaporation and it was postulated that the turbulence effects will be embedded in the friction velocity.

The model from Equation (18) suggests that the evaporation is influenced by geometry (because of the exposed surface to convective motion), convective transfer (turbulence and free stream velocity), and driving force (temperature and/or mole fraction). Based on analysis performed by similitudes, non-dimensional groups that lump all these effects and can influence the evaporation rate were derived. It was further postulated that the characteristic velocity representing the convective evaporation would be the friction velocity and not the

the length scale was defined. Basically, a sessile droplet will "see" the friction velocity rather than the mean velocity of the flow. Furthermore, the turbulence effect that has an impact on evaporation rate is embedded in the friction velocity. Therefore, expressing the Reynolds number, which is the driver for convective evaporation in terms of the friction velocity, seems like a plausible proposition. Based on the similitude analysis that was explained earlier, a non-

traditional mean flow velocity. Based on these analyses, a geometrical factor of *V*/*r*<sup>2</sup>

*j*

r

ç ÷ è ø

l

æ ö

l

*Lu w*

n

*j*

u

+ +

(19)

lly

defining

*Dj Mix*


where: is the mass of liquid species " t each node

*<sup>m</sup> j" a <sup>j</sup>*

*c*

n

**Figure 2.** Similar solutions in the form of the percentage of mass left as a function of dimensionless time that is reduced with respect to geometrical, convective, and driving force parameters.

The significance of this result is in identifying the similar solution that lumps the effects of free stream momentum and turbulence (*u\** ), the geometrical factor (*r2 /V*), and heat or mass transfer (transfer number) into a single scalable variable. The variation in the initial contact angle makes a very small difference in the curve. Therefore, if experiments are conducted in different wind tunnels with different droplet sizes and different transfer numbers (liquids), the same evaporation rate curve should be obtained for all wind tunnels as long as the results are

**Figure 3.** Similar solutions for the available experimental data in the form of the percentage of mass left as a function of dimensionless time.

expressed in the non-dimensional time scales. In doing so, the friction velocity should be measured in each experiment. This is an important conclusion and should be directly tested to verify the results of Figures 2 and 3.

It is evident that the free stream velocity and turbulence intensity should be treated as independent variables, i.e., one can be altered while the other remains constant. A series of boundary layer measurement made at Caltech's 6'× 6 ' wind tunnel altered the turbulence intensity while the mean velocity was maintained constant. The turbulence intensity was altered by the number of bungee cords in the wind tunnel. A schematic of the wind tunnel and the cords for changing the turbulence level is shown in Figure 4. The wall shear stress was measured by the oil film technique in which a drop of olive oil is placed on a surface under a monochromatic light source. The combined reflection from oil and the glass surface create an interference image pattern as shown in Figure 5. The fringe spacing growth rate can be correlated to the wall shear stress as *τ<sup>w</sup>* <sup>=</sup> <sup>2</sup>*nμ λwave dw dt* where *<sup>n</sup>* is the index of refraction for oil, *<sup>μ</sup>* is the dynamic viscosity, and *λ*wave is the light wavelength.

The turbulence intensity was measured by hot wire anemometry. Particle image velocimetry (PIV) can also be used to find the components of velocity fluctuations that will lead into the turbulence intensity calculations. Table 1 shows the wall shear stress as a function of the free stream velocity and turbulence intensity. The friction velocity (*u* \* = *τ<sup>w</sup>* / *ρ*) is calculated by

finding the wall shear stress through interpolation from Table 1. The density can be obtained from the pressure and temperature data. stream velocity and turbulence intensity. The friction velocity (*u w* \* ) is calculated by finding the wall shear stress through interpolation from Table 1.3.1. The density can be obtained

Figure 1.3.3 Schematic of Caltech's 6ʹ x 6 ʹ Wind Tunnel **Figure 4.** Schematic of Caltech's 6' x 6 ' Wind Tunnel

from the pressure and temperature data.

expressed in the non-dimensional time scales. In doing so, the friction velocity should be measured in each experiment. This is an important conclusion and should be directly tested

**Figure 3.** Similar solutions for the available experimental data in the form of the percentage of mass left as a function of

It is evident that the free stream velocity and turbulence intensity should be treated as independent variables, i.e., one can be altered while the other remains constant. A series of boundary layer measurement made at Caltech's 6'× 6 ' wind tunnel altered the turbulence intensity while the mean velocity was maintained constant. The turbulence intensity was altered by the number of bungee cords in the wind tunnel. A schematic of the wind tunnel and the cords for changing the turbulence level is shown in Figure 4. The wall shear stress was measured by the oil film technique in which a drop of olive oil is placed on a surface under a monochromatic light source. The combined reflection from oil and the glass surface create an interference image pattern as shown in Figure 5. The fringe spacing growth rate can be

*λwave*

*dw*

The turbulence intensity was measured by hot wire anemometry. Particle image velocimetry (PIV) can also be used to find the components of velocity fluctuations that will lead into the turbulence intensity calculations. Table 1 shows the wall shear stress as a function of the free

*dt* where *<sup>n</sup>* is the index of refraction for oil, *<sup>μ</sup>*

= *τ<sup>w</sup>* / *ρ*) is calculated by

to verify the results of Figures 2 and 3.

dimensionless time.

218 Surface Energy

correlated to the wall shear stress as *τ<sup>w</sup>* <sup>=</sup> <sup>2</sup>*nμ*

is the dynamic viscosity, and *λ*wave is the light wavelength.

stream velocity and turbulence intensity. The friction velocity (*u* \*

varying droplet volume, mean air speed, and temperature conditions. The turbulence intensity **Figure 5.** Fringe Spacing Growth Rate for Oil

was estimated using the Edgewood Chemical and Biological Center (ECBC) wind tunnel data for the wall shear stress at the velocities that are mentioned hereinto. The wind tunnel tests were conducted for four different free stream velocities, *u* (m/s)={0.26, 1.77, 3.00, 3.66}, three initial droplet volumes, *V(L)=*{1, 6, 9}, and five different temperatures, *T*( o C)={15, 25, 35, 50, 55}. A numerical procedure that solves the governing Equation (1.2.3) along with Equation (1.2.4) was developed. The scheme is based on the Runge–Kutta fourth order integration algorithm. For the sake of brevity, the model predictions are compared with the experimental data for nine of approximately 60 cases in Figure 1.3.5. An excellent agreement is seen for all droplet sizes, air velocities, and temperatures. Detailed comparisons can be found in Navaz et al.54, 55 In a group of wind tunnel experiments, HD droplet evaporation rates were measured for varying droplet volume, mean air speed, and temperature conditions. The turbulence intensity was estimated using the Edgewood Chemical and Biological Center (ECBC) wind tunnel data for the wall shear stress at the velocities that are mentioned hereinto. The wind tunnel tests were conducted for four different free stream velocities, *u∞*(m/s)={0.26, 1.77, 3.00, 3.66}, three initial droplet volumes, V(μL)={1, 6, 9}, and five different temperatures, T(°C)={15, 25, 35, 50, 55}. A numerical procedure that solves the governing Equation (17) along with Equation (18) was developed. The scheme is based on the Runge–Kutta fourth order integration algorithm. For the sake of brevity, the model predictions are compared with the experimental data for

The analysis revealed that there is a difference in how each parameter influences the

evaporation rate. Having the analytical formulation validated, the model can be used for

19

nine of approximately 60 cases in Figure 6. An excellent agreement is seen for all droplet sizes, air velocities, and temperatures. Detailed comparisons can be found in Navaz et al. [54, 55]

The analysis revealed that there is a difference in how each parameter influences the evapo‐ ration rate. Having the analytical formulation validated, the model can be used for parametric studies. To do this, the evaporation rate had to be calculated as the mean flow velocity and turbulence intensities were changed independently. A small (V0=1 μL) and a large (V0=9 μL) droplet were exposed to a constant mean boundary layer velocity as the turbulence intensity varied from 0 (laminar) to 6%. Then the mean velocity changed and the same calculations were repeated. Figure 7. (a and b) depict the results of this study and emphasizes the fact that that turbulence intensity is a driving force in convective cooling and the most viable method to account for its contribution is through friction velocity. A similar analysis was performed to examine the significance of the free stream velocity on the evaporation by considering the smallest and largest droplets as shown in Figure 7. (c and d). In this case, a free stream turbulence intensity of 2% with a constant prescribed free stream temperature of 35°C showed that if the free stream velocity is increased by a factor of about 14, the evaporation rate will decrease by a factor of about 4. By the same token, each free stream velocity can be correlated to the evaporation time.

In the last parametric study, the free stream velocity and turbulence intensity were maintained constant for the two droplet sizes and the temperature was varied. It appeared the air tem‐ perature (T) had the most significant effect on evaporation as seen from Figure 7. (e and f). For a 14% increase in absolute temperature, the evaporation rate is increased by a factor of 24 due to the vapor pressure being a strong function of temperature (the vapor pressure of HD at 30°C is about 0.25 mmHg).

There are two uncertainties that are encountered in the process of model validation with outdoor data: pertinent free stream velocity and turbulence intensity of the atmospheric flow. In the free atmospheric air flow, the wind velocity varies as a function of height (or vertical direction) and time, i.e., *V*(*y, t*). Furthermore, two velocities can be used as the 'free stream velocity' in the prediction of the droplet evaporation rate, one being the instantaneous wind velocity at the specific height *V*(*y*spec*,t*), and the other the height (*y*) averaged velocity *V*¯(*t*). In the outdoor experiments, the instantaneous velocity data were taken at seven or eight vertical locations up to approximately *H*=*5* m high.

The instantaneous velocity profile was used to find the transient average velocity, which is given by:

$$
\overline{V}(t) = \frac{1}{H} \int\_0^H V(y, t) dy\tag{21}
$$

The same procedure was performed for the temperature profile and the graphical represen‐ tations of these averaged values are embedded in all figures related to the outdoor data.

A General-Purpose Multiphase/Multispecies Model to Predict the Spread, Percutaneous Hazard, and... http://dx.doi.org/10.5772/60807 221

nine of approximately 60 cases in Figure 6. An excellent agreement is seen for all droplet sizes, air velocities, and temperatures. Detailed comparisons can be found in Navaz et al. [54, 55]

The analysis revealed that there is a difference in how each parameter influences the evapo‐ ration rate. Having the analytical formulation validated, the model can be used for parametric studies. To do this, the evaporation rate had to be calculated as the mean flow velocity and turbulence intensities were changed independently. A small (V0=1 μL) and a large (V0=9 μL) droplet were exposed to a constant mean boundary layer velocity as the turbulence intensity varied from 0 (laminar) to 6%. Then the mean velocity changed and the same calculations were repeated. Figure 7. (a and b) depict the results of this study and emphasizes the fact that that turbulence intensity is a driving force in convective cooling and the most viable method to account for its contribution is through friction velocity. A similar analysis was performed to examine the significance of the free stream velocity on the evaporation by considering the smallest and largest droplets as shown in Figure 7. (c and d). In this case, a free stream turbulence intensity of 2% with a constant prescribed free stream temperature of 35°C showed that if the free stream velocity is increased by a factor of about 14, the evaporation rate will decrease by a factor of about 4. By the same token, each free stream velocity can be correlated

In the last parametric study, the free stream velocity and turbulence intensity were maintained constant for the two droplet sizes and the temperature was varied. It appeared the air tem‐ perature (T) had the most significant effect on evaporation as seen from Figure 7. (e and f). For a 14% increase in absolute temperature, the evaporation rate is increased by a factor of 24 due to the vapor pressure being a strong function of temperature (the vapor pressure of HD at 30°C

There are two uncertainties that are encountered in the process of model validation with outdoor data: pertinent free stream velocity and turbulence intensity of the atmospheric flow. In the free atmospheric air flow, the wind velocity varies as a function of height (or vertical direction) and time, i.e., *V*(*y, t*). Furthermore, two velocities can be used as the 'free stream velocity' in the prediction of the droplet evaporation rate, one being the instantaneous wind velocity at the specific height *V*(*y*spec*,t*), and the other the height (*y*) averaged velocity *V*¯(*t*). In the outdoor experiments, the instantaneous velocity data were taken at seven or eight vertical

The instantaneous velocity profile was used to find the transient average velocity, which is

The same procedure was performed for the temperature profile and the graphical represen‐ tations of these averaged values are embedded in all figures related to the outdoor data.

*V t V y t dy <sup>H</sup>*<sup>=</sup> ò (21)

0 <sup>1</sup> () ( ,) *H*

to the evaporation time.

220 Surface Energy

is about 0.25 mmHg).

given by:

locations up to approximately *H*=*5* m high.

Figure 1.3.5 Surface evaporation model prediction compared with wind tunnel experimental data for the following HD droplet sizes, wind speeds, and temperatures: (a) 1 µL, 1.77 m/s, 15<sup>o</sup> C, (b) 1 µL, 0.26 m/s, 35<sup>o</sup> C, (c) 1 µL, 1.77 m/s, 35<sup>o</sup> C, (d) 1 µL, 3.66 m/s, 50<sup>o</sup> C, (e) 6 µL, 1.77 m/s, 15<sup>o</sup> C, (f) 6 µL, 3.00 m/s, 35<sup>o</sup> C, (g) 9 µL, 3.00 m/s, **Figure 6.** Surface evaporation model prediction compared with wind tunnel experimental data for the following HD droplet sizes, wind speeds, and temperatures: (a) 1 μL, 1.77 m/s, 15°C, (b) 1 μL, 0.26 m/s, 35°C, (c) 1 μL, 1.77 m/s, 35°C, (d) 1 μL, 3.66 m/s, 50°C, (e) 6 μL, 1.77 m/s, 15°C, (f) 6 μL, 3.00 m/s, 35°C, (g) 9 μL, 3.00 m/s, 15°C, (h) 9 μL, 1.77 m/s, 35°C, and (i) 9 μL, 3.00 m/s, 55°C.

The second uncertainty encountered in model application is that the free stream turbulence intensity levels could be a function of time. In the atmospheric flow, one expects the existence of smaller turbulence intensity. In this work, the turbulence intensity for the outdoor experi‐ ments was assumed to be about 2% based on the "averaged" value of velocity measurements. Note that the turbulence intensity is an independent variable and needs to be extracted from the actual velocity measurement with high sampling rate. 21 15o C, (h) 9 µL, 1.77 m/s, 35<sup>o</sup> C, and (i) 9 µL, 3.00 m/s, 55<sup>o</sup> C.

Numerous model validations with open air test data are compiled by Navaz et al. [54, 55]. Two of these cases are presented here and an excellent agreement between outdoor data and model prediction are observed and summarized in Figure 8.(*a* and *b*).


0.0765 0.4096 0.4450 0.5115

**Table 1.** Wall Shear Stress Data as a Function of Turbulence Intensity and Free Stream Velocity 15

Figure 1.3.6 Performing parametric study to explore the effect of turbulence intensity (*a* and *b*), free stream velocity (*c* and *d*), and air temperature (*e* and *f*) on evaporation. **Figure 7.** Performing parametric study to explore the effect of turbulence intensity (a and b), free stream velocity (c and d), and air temperature (e and f) on evaporation.

The instantaneous velocity profile was used to find the transient average velocity, which

#### **1.4. Porous media models** is given by:

The porous medium consists of solid inclusions with void spaces in between. The porosity is the ratio of the void volume to the total volume. It is an averaged property of a porous medium for "relatively" homogeneous substrates. For heterogeneous substrates, the porosity becomes a local value expressed as *ϕ =ϕ*(*x,y,z*). Furthermore, in the presence of chemical reaction, the *H tyV dy <sup>H</sup> tV* 0 <sup>1</sup> *),()(* (1.2.7) The same procedure was performed for the temperature profile and the graphical representations of these averaged values are embedded in all figures related to the outdoor data.

The second uncertainty encountered in model application is that the free stream turbulence intensity levels could be a function of time. In the atmospheric flow, one expects the existence of smaller turbulence intensity. In this work, the turbulence intensity for the outdoor

experiments was assumed to be about 2% based on the "averaged" value of velocity measurements. Note that the turbulence intensity is an independent variable and needs to be

extracted from the actual velocity measurement with high sampling rate.

Numerous model validations with open air test data are compiled by Navaz et al.54, 55. Two of these cases are presented here and an excellent agreement between outdoor data and A General-Purpose Multiphase/Multispecies Model to Predict the Spread, Percutaneous Hazard, and... http://dx.doi.org/10.5772/60807 223

model prediction are observed and summarized in Figure 1.3.7(*a* and *b*).

with the measurements for 1 L average HD drop size. The embedded figures represent the instantaneous wind velocity and temperature. **Figure 8.** Model predictions for real-time wind velocity and temperature data as compared with the measurements for 1 μL average HD drop size. The embedded figures represent the instantaneous wind velocity and temperature.

local porosity changes in time as any of the reaction products may appear in solid phase making the porosity a function of time (or local solid phase density), i.e.. *ϕ =ϕ*(*x,y,z,t*) *=ϕ(x,y,z,ρsk*)*.* For the time-dependent porosity, the gradient of the porosity should be included in the governing equations. This will make the equations more complex. However, since the reactions are generally slow, the porosity distribution and change in time is lagged by one time step (taken from the previous time step). This makes the algorithm less complicated and still very accurate. *1.4 Porous Media Models*  The porous medium consists of solid inclusions with void spaces in between. The porosity is the ratio of the void volume to the total volume. It is an averaged property of a porous medium for "relatively" homogeneous substrates. For heterogeneous substrates, the porosity becomes a local value expressed as  *=*(*x,y,z*). Furthermore, in the presence of chemical reaction, the local

Another property of the porous medium is the saturation permeability (*K* in Equation 10) that needs to be known. It is a momentum transport property of the porous medium. It is basically a measure of how "easily" fluid flows through fully saturated porous material. Therefore, this permeability is referred to as single-phase permeability, saturation permeability, or simply permeability. An experiment can be performed with any nonvolatile liquid including water to measure this property. [54, 63]. porosity changes in time as any of the reaction products may appear in solid phase making the porosity a function of time (or local solid phase density), i.e..  *=*(*x,y,z,t*) *=(x,y,z,sk*)*.* For the time-dependent porosity, the gradient of the porosity should be included in the governing equations. This will make the equations more complex. However, since the reactions are generally slow, the porosity distribution and change in time is lagged by one time step (taken from the previous time step). This makes the algorithm less complicated and still very accurate. Another property of the porous medium is the saturation permeability (*K* in Equation

We have measured the saturation permeability for different porosities using glass beads. This was a necessary step for nonhomogeneous porous media. Saturation permeability is a function of porosity and changes locally as the porosity will change, especially due to those chemical reactions that produce an additional solid phase as a product. This relationship is: 1.1.10) that needs to be known. It is a momentum transport property of the porous medium. It is basically a measure of how "easily" fluid flows through fully saturated porous material. Therefore, this permeability is referred to as single-phase permeability, saturation permeability,

**1.4. Porous media models**

d), and air temperature (e and f) on evaporation.

0

 *H tyV dy <sup>H</sup>*

is given by:

*tV*

222 Surface Energy

The porous medium consists of solid inclusions with void spaces in between. The porosity is the ratio of the void volume to the total volume. It is an averaged property of a porous medium for "relatively" homogeneous substrates. For heterogeneous substrates, the porosity becomes a local value expressed as *ϕ =ϕ*(*x,y,z*). Furthermore, in the presence of chemical reaction, the

The same procedure was performed for the temperature profile and the graphical representations

The second uncertainty encountered in model application is that the free stream turbulence intensity levels could be a function of time. In the atmospheric flow, one expects the existence of smaller turbulence intensity. In this work, the turbulence intensity for the outdoor

<sup>1</sup> *),()(* (1.2.7)

Figure 1.3.6 Performing parametric study to explore the effect of turbulence intensity (*a* and

**Figure 7.** Performing parametric study to explore the effect of turbulence intensity (a and b), free stream velocity (c and

*b*), free stream velocity (*c* and *d*), and air temperature (*e* and *f*) on evaporation.

The instantaneous velocity profile was used to find the transient average velocity, which

(*d*) (*e*) (*f*)

of these averaged values are embedded in all figures related to the outdoor data.

experiments was assumed to be about 2% based on the "averaged" value of velocity measurements. Note that the turbulence intensity is an independent variable and needs to be

extracted from the actual velocity measurement with high sampling rate.

**Free Stream Velocity (m/s) Turbulence Intensity**

**Table 1.** Wall Shear Stress Data as a Function of Turbulence Intensity and Free Stream Velocity 15

0% **0.3–0.4% 2.6%** 4.1–5.4% **τw (Shear Stress at the wall, Pascal)**

 0.0 0.0 0.0 0.0 0.0037 0.0249 0.0251 0.0253 0.0147 0.0626 0.0663 0.0723 0.0466 0.2048 0.2102 0.2483 0.0765 0.4096 0.4450 0.5115

0.0765 0.4096 0.4450 0.5115

(*a*) (*b*) (*c*)

$$K = \exp\left(\log\_{10}\varphi(\mathbf{x}, y, z, t) - 2\mathbf{2}.67\right) \tag{22}$$

23

where *K* is the saturation permeability in m2 or ft2 and *ϕ* is the instantaneous local porosity (0.3, 0.5, etc.). This equation is general and valid for porosities up to 60% or 0.60. This procedure is detailed by Zand et al. [63]

Contrary to the single-phase (fully saturated) flow, multiphase flow can occur when a fluid phase prefers a specific path, thereby leaving some of the porous medium regions filled with the originally present phase, e.g., vapor–air mixture. The measure of how "easily" a phase flows through the porous medium in the multiphase flow is referred to as phase permeability. The relative permeability for each constituent in liquid and gaseous phase (*k*ℓ*<sup>i</sup>* , *kgj*) is defined as a ratio of the phase permeability to the saturation permeability and 0≤*k*ℓ*<sup>i</sup> and kgj* ≤1 (zero for immobile phase and one for fully saturated flow). We eliminate the index "*i*" and "*j*" for simplicity, keeping in mind that these properties belong to each constituent in the liquid and/ or gas phase. Hence, it is evident that the relative permeability is a function of the phase content or saturation, i.e.,*k*<sup>ℓ</sup> =*k*ℓ(*s*ℓ) and *kg* =*kg*(*sg*).

A hybrid approach (combined modeling and experimental studies) was used to find expres‐ sions for relative permeability. [54, 55] This hybrid approach needs to be performed separately for agents and the substrates of interest.

$$\begin{aligned} k\_{\ell} &= s\_{\ell}^{k}, \quad k = \mathbf{2} \\ k\_{\mathcal{g}} &= \mathbf{1} + s\_{\ell}^{2} \left( \mathbf{2}s\_{\ell} - \mathbf{3} \right) \end{aligned} \tag{23}$$

The capillary pressure function (see Equation 10) is not known for sessile droplets. Therefore, it is postulated that it has the following form:

$$\begin{aligned} p\_{\boldsymbol{cl}} &= f\_{\boldsymbol{i}} \big( \* \Big) \frac{\sigma\_{\boldsymbol{i}} \cos \theta\_{\boldsymbol{i}}}{\sqrt{\mathbb{K}/\rho}} J(s\_{\boldsymbol{\iota}}), \\ \text{where} & \quad J(\boldsymbol{s}\_{\boldsymbol{\iota}}) = 1.417 \{1 - \boldsymbol{s}\_{\boldsymbol{\iota}}\} - 2.120 \{1 - \boldsymbol{s}\_{\boldsymbol{\iota}}\}^2 + 1.263 \{1 - \boldsymbol{s}\_{\boldsymbol{\iota}}\}^3 \end{aligned} \tag{24}$$

where *σ<sup>i</sup>* is the surface tension, *θ<sup>i</sup>* is the contact angle inside the pores, *K* is the saturation permeability, *s*ℓ*<sup>i</sup>* is the saturation for each constituent, and *f <sup>i</sup>* (\*) is a function to be determined. This form is based on the original equation proposed by Leverett [13] and Udell [14]. In Equation (24), a geometrical scale is defined as *K* / *φ*, which is not necessarily the correct value as capillary pressure can be scaled with a different geometrical scale. Therefore, the capillary pressure is corrected for a potential error in geometrical scale using the function *f <sup>i</sup>* (\*).

Initially, the Buckingham theorem was used to identify the basic non-dimensional groups and the computational model developed was utilized to combine these basic **Π** groups into a more meaningful similar solution. A value of *fi (\*)=*1 was assumed and numerical simulations were carried out over a wide range of initial droplet volume and wetted area (maintaining a constant initial contact angle), porous medium saturation permeability, liquid viscosity, and surface tension. For a constant initial contact angle and a specific substrate, a similar solution can be obtained with the following **Π** groups:

$$\begin{aligned} \mathbf{II\_1} &= \text{function}(\mathbf{II\_2}) \quad \text{for a constant contact angle} \\ \mathbf{II\_1} &= \frac{t \sigma K^{0.5}}{\mu \left( \frac{V\_{\text{on the surface}}}{r^2} \right)^2} \end{aligned} \qquad \qquad \begin{aligned} \mathbf{II\_2} &= \frac{V\_{\text{on the surface}}}{r^3} \end{aligned} \tag{25}$$

It should be noted again that the properties must be calculated for each constituent in the liquid phase marked by subscript "*i*" earlier. In Equation (25) *r* and *V* represent the wetted radius and the volume of the droplet left on the surface, respectively, and *t* is the time taken by a droplet to disappear from the surface. Figure 9.(*a*) shows a family of similar solution for these non-dimensional groups. Although this curve proves that the proposed non-dimensional groups are correct scalable quantities, it becomes a singular function when *V*on the surface→0. Note that the properties relate to each liquid constituent on the surface only.

The above functions were used in an alternate method to eliminate this singularity. Again, the value *f(\*)*=1 was kept, and the porosity and contact angle were changed to generate a family of curves with returning the instantaneous volume and wetted radius to their initial values, i.e., *VInitial* and *rInitial*. Figure 9.(*b*) is a collection of modified **Π<sup>1</sup>** group as a function of contact angle for substrate porosity ranging from 0.1 to 0.9. The following procedure can be used to find the unknown *f*(\*). The details of this method and all the experimental results and valida‐ tions are compiled and discussed by Navaz et al. [64]. The process of finding *f*(\*) is as follows:


Contrary to the single-phase (fully saturated) flow, multiphase flow can occur when a fluid phase prefers a specific path, thereby leaving some of the porous medium regions filled with the originally present phase, e.g., vapor–air mixture. The measure of how "easily" a phase flows through the porous medium in the multiphase flow is referred to as phase permeability.

as a ratio of the phase permeability to the saturation permeability and 0≤*k*ℓ*<sup>i</sup> and kgj* ≤1 (zero for immobile phase and one for fully saturated flow). We eliminate the index "*i*" and "*j*" for simplicity, keeping in mind that these properties belong to each constituent in the liquid and/ or gas phase. Hence, it is evident that the relative permeability is a function of the phase content

A hybrid approach (combined modeling and experimental studies) was used to find expres‐ sions for relative permeability. [54, 55] This hybrid approach needs to be performed separately

> ( ) <sup>2</sup> , 2 1 23

The capillary pressure function (see Equation 10) is not known for sessile droplets. Therefore,

() ( ) ( ) ( ) 2 3

= -- - + -

*J s s s s*

*iii i*

is the contact angle inside the pores, *K* is the saturation

*(\*)=*1 was assumed and numerical simulations were

(\*) is a function to be determined.

(\*).

l ll l

This form is based on the original equation proposed by Leverett [13] and Udell [14]. In Equation (24), a geometrical scale is defined as *K* / *φ*, which is not necessarily the correct value as capillary pressure can be scaled with a different geometrical scale. Therefore, the capillary

Initially, the Buckingham theorem was used to identify the basic non-dimensional groups and the computational model developed was utilized to combine these basic **Π** groups into a more

carried out over a wide range of initial droplet volume and wetted area (maintaining a constant initial contact angle), porous medium saturation permeability, liquid viscosity, and surface tension. For a constant initial contact angle and a specific substrate, a similar solution can be

l l

, *kgj*) is defined

(23)

(24)

The relative permeability for each constituent in liquid and gaseous phase (*k*ℓ*<sup>i</sup>*

*k*

*ks k k ss* = = =+ -

where 1.417 1 2.120 1 1.263 1

pressure is corrected for a potential error in geometrical scale using the function *f <sup>i</sup>*

is the saturation for each constituent, and *f <sup>i</sup>*

l l

*g*

l

or saturation, i.e.,*k*<sup>ℓ</sup> =*k*ℓ(*s*ℓ) and *kg* =*kg*(*sg*).

for agents and the substrates of interest.

it is postulated that it has the following form:

*p f Js <sup>K</sup>*

<sup>=</sup>

is the surface tension, *θ<sup>i</sup>*

meaningful similar solution. A value of *fi*

obtained with the following **Π** groups:

where *σ<sup>i</sup>*

224 Surface Energy

permeability, *s*ℓ*<sup>i</sup>*

s

( ) ( )

*i i ci i i*

cos \* ,

 q

j

$$\frac{1}{f\_i(^{\ast})\_{\text{Unbon}}} = \frac{\text{nondimensional measured time for the droplet to disappear from the surface}}{\text{nondimensional reference time ( $\mathbf{I}\_1$  Read from Figure 9b)}}\tag{26}$$

**•** Using the calculated value from step 2 should yield the correct *f(\*)* value that can be used in Equation (24) for simulating any other scenario.

Table 2. shows the results of this study for five different liquids (including HD and VX, known as Mustard and nerve agents, respectively). More validations for this method is compiled by Navaz et al. [54]

Another transport property that needs to be known is the effective diffusion coefficient for the gas phase as indicated in Equations (5) and (6). The evaporation of a liquid inside a porous material generates a vapor phase concentration profile throughout the porous substrate, and the transport is described by a specie diffusion equation. This equation requires the diffusion coefficient as a transport parameter. In a porous medium, the molecular diffusion coefficient needs to be reduced as the vapor transport is affected by a presence of solid and liquid phases. \* 1

*<sup>i</sup> Unknown f*

Measure the initial contact angle, and based on the value of porosity, find the modified

Using the calculated value from step 2 should yield the correct *f(\*)* value that can be used

nondimensi onal measured time for the droplet to disppear from the surface

1

(1.4.5)

nondimensi onal reference time ( Read from Figure 1.4.1b)

**1** group representing *tref* (reference time).

Determine the function *f(\*)* based on the following equation:

in Equation (1.4.3) for simulating any other scenario.

Figure 1.4.1 Nondimensional groups and the similar solutions for the groups (*a*), and the nondimensional reference time as a function of the initial contact angle and porosity (*b*). **Figure 9.** Nondimensional groups and the similar solutions for the **Π** groups (a), and the nondimensional reference time as a function of the initial contact angle and porosity (b).


Glycerin 12.30 11.95 VX 1.50 1.45 **Table 2.** Measured and calculated time for the disappearance of a droplet from the surface of a porous material

This results in a definition of an effective diffusion coefficient, which is a function of porosity and saturation, as shown below: HD 0.90 0.87

$$D\_{\rm eff} = f\left(D\_{\rm molecular}, \mathfrak{g}, \mathfrak{s}\_{\ell}\right) \tag{27}$$

The molecular diffusion coefficient as a function of temperature and molecular structure was taken from Treyball. [65] The function that describes how porosity and saturation attenuate this quantity was derived experimentally and is detailed by Navaz et al. [54, 55]

The effective diffusion coefficient, as given by Equation (27), is measured on a porous substrate bed where the boundary concentrations are given and transport takes place in the principal direction *L*. With reference to Figure 10., if the vapor is traveling from the lower surface to the upper surface (or vice versa), a one-dimensional diffusion problem governs this transport. The measurement technique is based on the gas chromatography and mass spectroscopy (GC/MS) substrate bed where the boundary concentrations are given and transport takes place in the

The effective diffusion coefficient, as given by Equation (1.4.6), is measured on a porous

The molecular diffusion coefficient as a function of temperature and molecular structure was taken from Treyball.65 The function that describes how porosity and saturation attenuate this

Another transport property that needs to be known is the effective diffusion coefficient for

(1.4.6)

the gas phase as indicated in Equations (1.1.5) and (1.1.6). The evaporation of a liquid inside a porous material generates a vapor phase concentration profile throughout the porous substrate, and the transport is described by a specie diffusion equation. This equation requires the diffusion coefficient as a transport parameter. In a porous medium, the molecular diffusion coefficient needs to be reduced as the vapor transport is affected by a presence of solid and liquid phases. This results in a definition of an effective diffusion coefficient, which is a function of porosity

method when the steady state of one-dimensional diffusion is achieved. At steady state, a constant evaporation rate is achieved, and therefore a constant slope mass loss or zero derivative of mass loss in time (*m*˙ =*dm* / *dt*) should be observed. Then the equation in Figure (23) is used to find *Deff*. transport. The measurement technique is based on the gas chromatography and mass spectroscopy (GC/MS) method when the steady state of one-dimensional diffusion is achieved. At steady state, a constant evaporation rate is achieved, and therefore a constant slope mass loss or zero derivative of mass loss in time (*m dm*/ *dt* ) should be observed. Then the equation in

quantity was derived experimentally and is detailed by Navaz et al.54, 55

Figure 1.4.2 Experimental concept for the effective diffusivity measurements. **Figure 10.** Experimental concept for the effective diffusivity measurements.

After the data reduction, the effective diffusion coefficient (Eq. 1.4.7) is obtained. After the data reduction, the effective diffusion coefficient (Eq. 28) is obtained.

$$D\_{j-\text{aux}} = \left(-0.5855\left(1 - s\_{\ell i}\right)^3 + 0.4591\left(1 - s\_{\ell i}\right)^2 + 0.1264\left(1 - s\_{\ell i}\right)\right) \rho\left(\mathbf{x}, y, z, t\right) D\_M \tag{28}$$

27

65 Treyball, R. E. (1980), *Mass-Transfer Operation*, McGraw Hill. where *DM* is the molecular diffusion coefficient.

and saturation, as shown below:

*D f D , ,s eff molecular*

Figure (1.4.2) is used to find *Deff*.

This results in a definition of an effective diffusion coefficient, which is a function of porosity

(*<sup>a</sup>*) (*b*)

Figure 1.4.1 Nondimensional groups and the similar solutions for the groups (*a*), and the

**Figure 9.** Nondimensional groups and the similar solutions for the **Π** groups (a), and the nondimensional reference

**Substance Experiment (time, s) – Extracted from Video Model (time, s)**

1-2 Propandiol 1.45 1.50 Castor Oil 27.50 28.30 Glycerin 12.30 11.95 VX 1.50 1.45 HD 0.90 0.87

nondimensional reference time as a function of the initial contact angle and

Table 1.4.1 shows the results of this study for five different liquids (including HD and VX, known as Mustard and nerve agents, respectively). More validations for this method is compiled

Table 1.4.1 Measured and calculated time for the disappearance of a droplet from the surface of

**Substance Experiment (time, s) – Extracted from Video Model (time, s)** 

1-2 Propandiol 1.45 1.50 Castor Oil 27.50 28.30 Glycerin 12.30 11.95 VX 1.50 1.45 HD 0.90 0.87

**Table 2.** Measured and calculated time for the disappearance of a droplet from the surface of a porous material

Measure the initial contact angle, and based on the value of porosity, find the modified

Using the calculated value from step 2 should yield the correct *f(\*)* value that can be used

nondimensi onal measured time for the droplet to disppear from the surface

1

(1.4.5)

nondimensi onal reference time ( Read from Figure 1.4.1b)

**1** group representing *tref* (reference time).

\* 1

226 Surface Energy

*<sup>i</sup> Unknown f*

Determine the function *f(\*)* based on the following equation:

in Equation (1.4.3) for simulating any other scenario.

The molecular diffusion coefficient as a function of temperature and molecular structure was taken from Treyball. [65] The function that describes how porosity and saturation attenuate

The effective diffusion coefficient, as given by Equation (27), is measured on a porous substrate bed where the boundary concentrations are given and transport takes place in the principal direction *L*. With reference to Figure 10., if the vapor is traveling from the lower surface to the upper surface (or vice versa), a one-dimensional diffusion problem governs this transport. The measurement technique is based on the gas chromatography and mass spectroscopy (GC/MS)

j

<sup>l</sup> ) (27)

*D fD s eff* = ( *molecular* , ,

this quantity was derived experimentally and is detailed by Navaz et al. [54, 55]

and saturation, as shown below:

by Navaz et al.54

porosity (*b*).

time as a function of the initial contact angle and porosity (b).

a porous material

#### **1.5. Contact dynamics and governing equations for the liquid bridge**

Many parameters affect the amount or the concentration of transferred liquid into a secondary surface coming into contact with a sessile droplet on a substrate. The approach velocity between two surfaces affects the amount of mass transfer due to the resulting footprint after the contact. The mass transfer between the two surfaces, after the contact, occurs through the area of this footprint. There are basically two pathways by which a contact transfer can occur. These are schematically shown in Figures 29 and 30.

In this document, the bottom and top surfaces are referred to as the primary and contacting or secondary surfaces, respectively. The model is purely driven by the physiochemical properties of the medium, solids, liquids, and gases. When a droplet is placed on a primary surface (Figures 11a and 12a), capillary transport, surface evaporation (due to wind and temperature), and chemical reaction with the surface or other preexisting chemicals in the pores (such as moisture) are simultaneously initiated and the chemical available for transfer

**Figure 11.** Pathway 1: (a) Spread into the bottom surface, (b) contact dynamics, and (c) contact transfer Figure 1.5.1 Pathway 1: (a) Spread into the bottom surface, (b) contact dynamics, and (c)

*<sup>j</sup> mix <sup>i</sup> <sup>i</sup> <sup>i</sup> DM D* 0.5855 1 *s* 0.45911 *s* 0.1264 1 *s x*, *y*,*z*,*t* <sup>3</sup> <sup>2</sup>

*1.5 Contact Dynamics and Governing Equations for the* 

secondary surface coming into contact with a sessile droplet on a substrate. The approach

area of this footprint. There are basically two pathways by which a contact transfer can occur.

Many parameters affect the amount or the concentration of transferred liquid into a

(1.4.7)

These are schematically shown in Figures 1.5.1 and 1.5.2.

contact transfer

contact transfer

where *DM* is the molecular diffusion coefficient.

*Liquid Bridge* 

Figure 1.5.2 Pathway 2: (a) Spread into the bottom surface, (b) contact dynamics, and (c) **Figure 12.** Pathway 2: (a) Spread into the bottom surface, (b) contact dynamics, and (c) contact transfer

through contact is the end product or resultant of all these processes. We refer to this phe‐ nomenon as the precontact process. After this point, two possible scenarios could occur: creation of a liquid bridge between the two surfaces and its spread as the two surfaces get closer (Figure 11.b), or complete disappearance of the liquid droplet from the surface of the substrate (Figure 12.b). The last stage or post-contact period is when both surfaces come together and capillary transport occurs at the interface. The mass transfer to the contacting surface always occurs regardless of which pathway is taken. It should be noted when the upper surface comes into contact with or without the formation of the liquid bridge, chemical reaction, and evaporation can also be initiated in the upper surface. In this document, the bottom and top surfaces are referred to as the primary and contacting or secondary surfaces, respectively. The model is purely driven by the physiochemical properties of the medium, solids, liquids, and gases. When a droplet is placed on a primary surface (Figures 1.5.1a and 1.5.2a), capillary transport, surface evaporation (due to wind and temperature), and chemical reaction with the surface or other preexisting chemicals in the pores (such as moisture) are simultaneously initiated and the chemical available for transfer through

The upper surface moves according to Newton's second law assuming constant acceleration. The *y*-coordinate of the lower boundary of the upper surface (in motion) can be calculated by the equation of motion:

$$\begin{aligned} \text{Equation of Motion:} \quad y &= \frac{F}{2m}t^2 + V\_o t + y\_o \\ y &= \text{Coordinate system (Vertical)} \quad V\_o = \text{Initial Velocity} \\ y\_o &= \text{Initial space between the two surfaces} \\ F &= \text{Force exerted on the upper surface to move it down,} \\ m &= \text{mass of the upper surface} \end{aligned} \tag{29}$$

The liquid bridge has some free surface. It has been reported by Coblas et al. [66] that the shape of the free surface and its evolution varies as a function of inertial effects. For hydrophilic liquid bridges formed upon contact with small droplets (that is usually the case in chemical agent contamination and spray of pesticides on common surfaces), the liquid bridge is influenced by capillary flow and it generally assumes the shape of a hyperbloid when the velocity of approach is not significantly high (Longley et al.) [67]. The shape of the free surface of the liquid bridge (bulging in or out) has a very minimal effect on the amount of mass transfer into the contacting surfaces. Rather, the footprint area of the liquid bridge defines the surface available for the mass transfer. In the current work, this footprint is assumed to be in the shape of a circle, although the algorithm developed here may be modified to explore other shapes.

The contact angle of this hyperboloid and the surfaces can be found through experiments, although they will not affect the overall transfer of the mass. We have taken this angle to be π/8. The mass transfer into the porous media is calculated in a time step. Then the remaining mass of the liquid bridge is updated and its new volume is calculated. The separating distance is known from the equation of motion. By knowing the volume and the height of the liquid bridge, the contacting surface area (assumed to be circular) is calculated. The radius increase in time will determine the spread rate. Figure 13. shows that the arc FMG revolves around the *C–C'* axis to produce a hyperboloid. We need to find the volume of this body of revolution analytically. If point *A* is the centroid of the surface engulfed by the arc *FMG*, the volume of the partial torus shaped resulting from this revolution according to Pappus centroid theorem is:

Volume of the partial torus = *<sup>V</sup> pt* =2*πAO*¯ *A* where *A* is the area of the arc. Then the volume of the liquid between the two surfaces will be:

through contact is the end product or resultant of all these processes. We refer to this phe‐ nomenon as the precontact process. After this point, two possible scenarios could occur: creation of a liquid bridge between the two surfaces and its spread as the two surfaces get closer (Figure 11.b), or complete disappearance of the liquid droplet from the surface of the substrate (Figure 12.b). The last stage or post-contact period is when both surfaces come together and capillary transport occurs at the interface. The mass transfer to the contacting surface always occurs regardless of which pathway is taken. It should be noted when the upper surface comes into contact with or without the formation of the liquid bridge, chemical

In this document, the bottom and top surfaces are referred to as the primary and contacting or secondary surfaces, respectively. The model is purely driven by the physiochemical properties of the medium, solids, liquids, and gases. When a droplet is placed on a primary surface (Figures 1.5.1a and 1.5.2a), capillary transport, surface evaporation (due to wind and temperature), and chemical reaction with the surface or other preexisting chemicals in the pores (such as moisture) are simultaneously initiated and the chemical available for transfer through

Figure 1.5.2 Pathway 2: (a) Spread into the bottom surface, (b) contact dynamics, and (c)

**Figure 12.** Pathway 2: (a) Spread into the bottom surface, (b) contact dynamics, and (c) contact transfer

**Figure 11.** Pathway 1: (a) Spread into the bottom surface, (b) contact dynamics, and (c) contact transfer

Figure 1.5.1 Pathway 1: (a) Spread into the bottom surface, (b) contact dynamics, and (c)

*<sup>j</sup> mix <sup>i</sup> <sup>i</sup> <sup>i</sup> DM D* 0.5855 1 *s* 0.45911 *s* 0.1264 1 *s x*, *y*,*z*,*t* <sup>3</sup> <sup>2</sup>

*1.5 Contact Dynamics and Governing Equations for the* 

secondary surface coming into contact with a sessile droplet on a substrate. The approach velocity between two surfaces affects the amount of mass transfer due to the resulting footprint after the contact. The mass transfer between the two surfaces after the contact occurs through the area of this footprint. There are basically two pathways by which a contact transfer can occur.

Many parameters affect the amount or the concentration of transferred liquid into a

(1.4.7)

These are schematically shown in Figures 1.5.1 and 1.5.2.

contact transfer

contact transfer

where *DM* is the molecular diffusion coefficient.

*Liquid Bridge* 

228 Surface Energy

The upper surface moves according to Newton's second law assuming constant acceleration. The *y*-coordinate of the lower boundary of the upper surface (in motion) can be calculated by

> *<sup>F</sup> y t Vt y m*

= ++

Coordinate system (Vertical) Initial Velocity

*o o o*

(29)

Force exerted on the upper surface to move it down,

Initial space between the two surfaces

reaction, and evaporation can also be initiated in the upper surface.

<sup>2</sup> Equation of Motion: 2

= the upper surface

*y V*

= =

mass of

*o*

= =

*y F m*

the equation of motion:

$$V\_{L\!\!\!\!\!\/]} = \pi r^2 \left( t \right) h\left( t \right) - V\_{pt} \tag{30}$$

where *t* represents the time. Note that both the footprint radii and the liquid bridge height both are functions of time and are calculated in each time step. If*θ* ^ <sup>=</sup>*GBF* , then from basic trigonometry:

$$\begin{aligned} AB &= \frac{4R(t)\sin^3\frac{\theta}{2}}{3(\theta - \sin\theta)}\\ R(t) &= \frac{h(t)}{2\cos\beta} \\ V\_{\perp\downarrow} &= \pi r^2(t)h(t) - 2\pi \left[\frac{R^2(t)}{2}(\theta - \sin\theta)\right] \left[r(t) + \frac{h(t)}{2}\tan\beta - AB\right] \end{aligned} \tag{31}$$

**Figure 13.** Schematic of the hyperboloid geometry

The equation correlating instantaneous *r*(*t*) and *h*(*t*) with the liquid volume ∀*Liquid Bridge* (*t*) can be obtained as:

$$\begin{aligned} r^2(t) - \frac{h(t)G(\beta)}{4\cos^2\beta}r(t) - \frac{h^2(t)G(\beta)\tan\beta}{8\cos^2\beta} + \frac{h^2(t)}{6} - \frac{\forall\_{\text{Lipid}\text{ }B\text{ridge}}}{\pi h(t)} &= 0\\ G(\beta) = \pi - 2\beta - \sin\left(\pi - 2\beta\right) \end{aligned} \tag{32}$$

*β* is the contact angle between the surface and the liquid bridge. The droplet base radius (footprint), *r*(*t*), is calculated from the above equation. Both domains, or media, are remeshed and adapted to the new footprint of the liquid bridge. The remeshing is done after each time step.

As the gap between the two surfaces is closing, the height will reduce. At the same time, the liquid is absorbed into the porous media. Therefore, *h(t)* and ∀*Liquid Bridge* (*t*) are updated in each time step. Then the new footprint radius is obtained by Equation (32). It should be noted that when the height of the liquid bridge becomes less than a minimum threshold, the spread is stopped. This minimum threshold is a function of surface roughness and viscosity of the fluid. However, it is quite possible to assume a value of 10–100 μm for this minimum height. At this point the radius of the footprint remains constant and the absorption is driven solely by capillary pressure causing a reduction in *h*(*t*)*.* This value of *h*(*t*) can be calculated by solving the above equation again with the "final" liquid footprint radius remaining a constant. This equation is given as:

$$\left(\frac{\pi}{6} - \frac{\pi G\left(\beta\right)\tan\beta}{8\cos^2\beta}\right)h^3\left(t\right) - \frac{\pi r\left(t\right)G\left(\beta\right)}{4\cos^2\beta}h^2\left(t\right) + \pi r^2\left(t\right)h\left(t\right) - \forall\_{\text{Lipid. Bridge}} = 0\tag{33}$$

This will be a cubic equation in *h*(*t*)*.* As the absorption into the porous media occurs, the volume of the very thin liquid bridge is updated and the new *h*(*t*) is calculated. In our model, this switching occurs when *h*(*t*) < 0.08 mm. This value is a function of surface roughness and can be specified by the user.

#### **1.6. Chemistry model**

The equation correlating instantaneous *r*(*t*) and *h*(*t*) with the liquid volume ∀*Liquid Bridge* (*t*) can

2 2

*Liquid Bridge htG h tG h t*


 b

 bb

tan

*β* is the contact angle between the surface and the liquid bridge. The droplet base radius (footprint), *r*(*t*), is calculated from the above equation. Both domains, or media, are remeshed and adapted to the new footprint of the liquid bridge. The remeshing is done after each time

As the gap between the two surfaces is closing, the height will reduce. At the same time, the liquid is absorbed into the porous media. Therefore, *h(t)* and ∀*Liquid Bridge* (*t*) are updated in each time step. Then the new footprint radius is obtained by Equation (32). It should be noted that when the height of the liquid bridge becomes less than a minimum threshold, the spread is stopped. This minimum threshold is a function of surface roughness and viscosity of the fluid. However, it is quite possible to assume a value of 10–100 μm for this minimum height. At this point the radius of the footprint remains constant and the absorption is driven solely by capillary pressure causing a reduction in *h*(*t*)*.* This value of *h*(*t*) can be calculated by solving the above equation again with the "final" liquid footprint radius remaining a constant. This

( )

*h t*

p

"

0

0

(32)

(33)

( ) () ( ) ( ) () ( ) ( )

2 2

2

( ) ( ) () ( ) () () () <sup>3</sup> 2 2

 p

6 8cos 4cos *Liquid Bridge*

 b

 b

ç ÷ - - + -" =

2 2

p

*h t h t r tht*

4cos 8cos 6

( ) ( )

=- - -

 b

b

b

*r t r t*

**Figure 13.** Schematic of the hyperboloid geometry

tan

b

 b

pb

æ ö

è ø

*G rtG*

2 sin 2

 p b

be obtained as:

230 Surface Energy

step.

*G*

equation is given as:

p bp

The chemistry model is based on the Joint Army–Navy–NASA–Air Force (JANNAF) standard methodology. This methodology has three components: a reaction rate processor, species production destruction term calculator, and NASA thermodynamic file (provides specific heat, enthalpy, entropy, and Gibbs free energy) as a function of temperature. [68, 69, 70] This is shown in Equation (34). The outcome of this section will provide the term *ω*˙ Reaction in the governing equations.

$$\begin{aligned} \frac{\overline{C}\_p}{\overline{R}} &= \frac{a\_1}{T^2} + \frac{a\_2}{T} + a\_3 + a\_4 T + a\_5 T^2 + a\_6 T^3 + a\_7 T^4 \\ \frac{\overline{h}}{\overline{R}T} &= -\frac{a\_1}{T^2} + a\_2 \frac{\ell nT}{T} + a\_3 + \frac{a\_4 T}{2} + \frac{a\_5 T^2}{3} + \frac{a\_6 T^3}{4} + \frac{a\_7 T^4}{5} + \frac{a\_8}{T} \\ \frac{\overline{s}}{\overline{R}} &= -\frac{a\_1}{2T^2} - \frac{a\_2}{T} + a\_3 \ell nT + a\_4 T + \frac{a\_5 T^2}{2} + \frac{a\_6 T^3}{3} + \frac{a\_7 T^4}{4} + a\_9 \\ \frac{\overline{g}}{\overline{R}} &= \frac{\overline{h}}{\overline{R}T} - \frac{\overline{s}}{\overline{R}} \end{aligned} \tag{34}$$

#### *1.6.1. Bidirectional reactions*

A general chemical reaction can be written in terms of its stoichiometric coefficients *νij* and *νij* ' as:

$$\sum\_{i=1}^{\text{NSP}} \nu\_{ij} \overline{\mathbf{M}}\_i \Leftrightarrow \sum\_{i=1}^{\text{NSP}} \nu\_{ij}^{\cdot} \overline{\mathbf{M}}\_i \tag{35}$$

where *M*¯ *i* represents the *i-*th chemical species name (*i=*1, 2,...., NSP) and *j* represents the *j-*th reaction (*j=*1, 2,..., *L*). These reactions proceed (forward or reverse) according to the law of mass action, which states: "The rate at which an elementary reaction proceeds is proportional to the product of the molar concentrations of the reactant each raised to a power equal to its stoi‐ chiometric coefficient in the reaction equation."

Let *M*¯ denote the molar concentration of species *<sup>i</sup>*. The forward (left to right) reaction rate, *Ratej* (*L* →*R*), for reaction *j* can be written as:

$$\text{Rate}\_{/}\left(L \to R\right) = k\_{/\_{/}} \prod\_{i=1}^{\text{NSP}} \left[\overline{M}\_{i}\right]^{\nu\_{q}}\tag{36}$$

where *<sup>k</sup> <sup>f</sup> <sup>j</sup>* is the forward reaction rate constant. For species *i*, *νij*moles on the left side of the reaction becomes *νij* ' moles on the right side of the reaction. Consequently, the forward reaction for reaction *j* yields a time rate of change in the molar concentration of species *i* as follows:

$$d\left(Forward\right) = \frac{d\left[\overline{M}\_i\right]\_j}{dt} = \left(\dot{\nu\_{\dot{\eta}}} - \nu\_{\dot{\eta}}\right) k\_{f\_{\ne}} \prod\_{i=1}^{NSP} \left[\overline{M}\_i\right]^{\nu\_{\eta}} \tag{37}$$

Similarly, the reverse reaction for reaction *j* yields:

$$\left(\text{Re}\,\text{vers}se\right) = \frac{d\left\lfloor \bar{M}\_{i} \right\rfloor\_{j}}{dt} = \left(\nu\_{ij} - \nu\_{ij}^{\cdot}\right) k\_{b\_{j}} \prod\_{l=1}^{\text{NSP}} \left\lfloor \bar{M}\_{l} \right\rfloor^{\nu\_{q}} \tag{38}$$

where *kbj* is the backward reaction rate constant. Thus, the net rate of change in the molar concentration of species *i* for reaction *j* (denoted by *Xij*) is as follows:

$$X\_{\boldsymbol{\eta}} = \frac{d\left[\overline{\boldsymbol{M}}\_{\boldsymbol{l}}\right]\_{\boldsymbol{l}}}{dt} = \left(\boldsymbol{\nu}\_{\boldsymbol{\eta}}^{\cdot} - \boldsymbol{\nu}\_{\boldsymbol{\eta}}\right) \left[\boldsymbol{k}\_{\boldsymbol{l}\_{\boldsymbol{\eta}}}\prod\_{i=1}^{\text{NSP}} \left[\overline{\boldsymbol{M}}\_{\boldsymbol{l}}\right]^{\boldsymbol{\nu}\_{\boldsymbol{\eta}}} - \boldsymbol{k}\_{\boldsymbol{b}\_{\boldsymbol{l}\_{\boldsymbol{\eta}}}}\prod\_{i=1}^{\text{NSP}} \left[\overline{\boldsymbol{M}}\_{\boldsymbol{l}\_{\boldsymbol{\eta}}}\right]^{\boldsymbol{\nu}\_{\boldsymbol{\eta}}}\right] \tag{39}$$

The "species production rate" is the time rate of change for the species density. For reaction *j* the net species production rate for species *i* is *MwiXij* where *Mwi* is the molecular weight of species *i*. Summing over all reactions gives the net species production rate *ω<sup>i</sup>* for the reaction set

$$\alpha \rho\_i = \mathbf{M}\_{\rm ui} \sum\_{j=1}^{L} \mathbf{X}\_{ij} \tag{40}$$

The molar concentration of species *i* can be written as *ρ<sup>i</sup> Mwi* , which is the species *i* mass density divided by the species *i* molecular weight.

$$\left[\left.\widetilde{M}\_{i}\right]\right] = \frac{\rho\_{i}}{M\_{wi}} = \frac{\rho \mathbf{C}\_{i}}{M\_{wi}}\tag{41}$$

It follows that in terms of species *i* mass fraction Equation (39) becomes:

A General-Purpose Multiphase/Multispecies Model to Predict the Spread, Percutaneous Hazard, and... http://dx.doi.org/10.5772/60807 233

$$\begin{aligned} \mathbf{X}\_{\boldsymbol{\eta}} &= \left(\boldsymbol{\nu}\_{\boldsymbol{\eta}} - \boldsymbol{\nu}\_{\boldsymbol{\eta}}\right) \left[\mathbf{k}\_{f\_{\boldsymbol{\eta}}} \prod\_{i=1}^{\text{NSP}} \left(\frac{\rho \mathbf{C}\_{i}}{\mathbf{M}\_{\text{val}}}\right)^{\boldsymbol{\nu}\_{\boldsymbol{\eta}}} - \mathbf{k}\_{b\_{j}} \prod\_{i=1}^{\text{NSP}} \left(\frac{\rho \mathbf{C}\_{i}}{\mathbf{M}\_{\text{uv}}}\right)^{\boldsymbol{\nu}\_{\boldsymbol{\eta}}}\right] \qquad \text{or} \\ \mathbf{X}\_{\boldsymbol{\eta}} &= \left(\boldsymbol{\nu}\_{\boldsymbol{\eta}} - \boldsymbol{\nu}\_{\boldsymbol{\eta}}\right) \left[\mathbf{K}\_{i} \prod\_{l=1}^{\text{NSP}} \left(\frac{\rho \mathbf{C}\_{i}}{\mathbf{M}\_{\text{uu}}}\right)^{\boldsymbol{\nu}\_{\boldsymbol{\eta}}} - \prod\_{i=1}^{\text{NSP}} \left(\frac{\rho \mathbf{C}\_{i}}{\mathbf{M}\_{\text{uu}}}\right)^{\boldsymbol{\nu}\_{\boldsymbol{\eta}}}\right] \mathbf{k}\_{f} \\ \mathbf{K}\_{j} &= \frac{\mathbf{k}\_{f\_{j}}}{\mathbf{k}\_{b\_{j}}} \qquad \text{and} \qquad \mathbf{k}\_{j} = \mathbf{k}\_{b\_{j}} \end{aligned} \tag{42}$$

The reaction rate *kj* is from right to left (reverse) in the above equation and is often represented by the Arrhenius form:

$$k\_{\rangle} = a\_{\rangle} T^{-n\_{\rangle}} \exp\left(\frac{-b\_{\rangle}}{\overline{R}T}\right) \tag{43}$$

where *aj* is the preexponential coefficient, *nj* is the temperature dependence of the preexpo‐ nential factor, and *bj* is the activation energy. The ratio of forward to backward rate, *Kj* in Equation (42) is related to the equilibrium constant, *K*eql,by the following equation:

$$K\_{\rangle} = \frac{k\_{f\_{\rangle}}}{k\_{b\_{\rangle}}} = K\_{\alpha \eta l} \left(\overline{R}T\right)^{l\_{\rangle}} \tag{44}$$

where

where *<sup>k</sup> <sup>f</sup> <sup>j</sup>*

232 Surface Energy

where *kbj*

set

reaction becomes *νij*

'

Similarly, the reverse reaction for reaction *j* yields:

Re

*d M*

*dt*

The molar concentration of species *i* can be written as *ρ<sup>i</sup>*

divided by the species *i* molecular weight.

is the forward reaction rate constant. For species *i*, *νij*moles on the left side of the

for reaction *j* yields a time rate of change in the molar concentration of species *i* as follows:

n n

*NSP <sup>i</sup> <sup>j</sup>*

'

is the backward reaction rate constant. Thus, the net rate of change in the molar

( ) '

*j j*

*i i*

= =

n

1 1

ëû ëû ë û

*Mwi*

*NSP <sup>i</sup> <sup>j</sup>*

( ) ( ) '

*d M*

concentration of species *i* for reaction *j* (denoted by *Xij*) is as follows:

'

n n

*dt*

é ù

*Forward k M dt*

( ) ( ) '

*verse k M*

*NSP NSP <sup>i</sup> <sup>j</sup> ij ij ij f i b i*

é ù é ù ë û = = - - ê ú éù éù

The "species production rate" is the time rate of change for the species density. For reaction *j* the net species production rate for species *i* is *MwiXij* where *Mwi* is the molecular weight of

> 1 *L i wi ij j*

> > *i i*

 r

*<sup>C</sup> <sup>M</sup> M M* r

*wi wi*

 *M X*=

*X kMkM*

species *i*. Summing over all reactions gives the net species production rate *ω<sup>i</sup>*

w

*i*

It follows that in terms of species *i* mass fraction Equation (39) becomes:

n n

*d M*

é ù

moles on the right side of the reaction. Consequently, the forward reaction

1

1

=

*j*

*ij ij b i i*

=

*j*

*ij ij f i i*

*ij*

*ij*

n

ë û = = - é ù Õë û (38)

*ij ij*

 n

<sup>=</sup> å (40)

é ù = = ë û (41)

, which is the species *i* mass density

Õ Õ (39)

for the reaction

n

ë û = = - é ù Õë û (37)

$$\mathcal{A}\_{\rangle} = \sum\_{i=1}^{NSP} \left( \stackrel{\cdot}{\nu\_{ij}} - \nu\_{ij} \right) \tag{45}$$

The quantity |*λ<sup>j</sup>* | + 1 is known as the order of the *j-*th reaction and the equilibrium constant is:

$$\begin{aligned} K\_{\text{eq}} &= \exp\left(-\frac{\Delta F}{\overline{R}T}\right) & \Delta F &= \sum\_{i=1}^{\text{NSP}} f\_i \nu\_{ij} - \sum\_{i=1}^{\text{NSP}} f\_i \nu\_{ij} \\ f\_i &= \text{Gibbs} \text{ Free Energy} = \text{Chemical Potential} = \overline{h}\_i - T\overline{S}\_i \\ \overline{h}\_i &= \text{Molar specific enthalpy} \\ \overline{S}\_i &= \text{Molar specific entropy} \end{aligned} \tag{46}$$

and

$$\begin{aligned} \mathbf{X}\_{\boldsymbol{\eta}} &= \left(\boldsymbol{\nu}\_{\boldsymbol{\eta}} - \boldsymbol{\nu}\_{\boldsymbol{\eta}}\right) \Big[ \mathbf{K}\_{\boldsymbol{\eta}} \prod\_{l=1}^{\text{NSP}} \overline{\mathbf{C}}\_{i}^{\nu\_{\boldsymbol{\eta}}} - \rho^{\lambda\_{\boldsymbol{\eta}}} \prod\_{l=1}^{\text{NSP}} \overline{\mathbf{C}}\_{i}^{\nu\_{\boldsymbol{\eta}}} \Big] k\_{j} \rho^{\sum\_{i=1}^{\text{NSP}} \nu\_{\boldsymbol{\eta}}} \\ \overline{\mathbf{C}}\_{i} &= \frac{\mathbf{C}\_{i}}{M\_{\boldsymbol{\eta}\boldsymbol{\eta}}} \Big( \text{Molar mass fraction} \Big) \end{aligned} \tag{47}$$

#### *1.6.2. Third body dissociation recombination reactions*

Reactions involving a third body have a distinct reaction rate for each particular third body. Benson and Fueno [71] have shown theoretically that the temperature dependence of recom‐ bination rates is approximately independent of the third body. Available experimental recombination rate data also indicates that the temperature dependence of recombination rates is independent of the third body within the experimental accuracy of the measurements. Assuming that the temperature dependence of recombination rates is independent of the third body, the recombination rate associated with the *k-*th species (third body) can be represented as:

$$k\_{kj} = a\_{kj} T^{-n\_j} \exp\left(\frac{-b\_j}{\overline{R}T}\right) \tag{48}$$

where only the constant *akj* is different for different species (third bodies). Assuming that the reference species (third body) whose rate is specified in the program input has index *k = k*ref*,* we may write:

$$\begin{aligned} k\_{kj} &= \frac{a\_{kj}}{a\_{k\_{nj}}} a\_{k\_{nj}} T^{-n\_j} \exp\left(\frac{-b\_j}{\overline{R}T}\right) = m\_{kj} k\_{ref(j)} \quad \text{for } k^{th} \text{ species} \\ m\_{kj} &= \frac{a\_{kj}}{a\_{k\_{nj}}} \end{aligned} \tag{49}$$

For the *k-*th species (third body) the stoichiometric coefficient is 1 on both sides of reaction *j* giving:

$$\begin{split} \mathbf{X}\_{ij} \text{(for } 3^{rd} \text{ body } k) &= \left(\boldsymbol{\nu}\_{ij} - \boldsymbol{\nu}\_{ij}\right) \left[ \mathbf{K}\_{j} \frac{\rho \mathbf{C}\_{k}}{\mathbf{M}\_{\text{uk}}} \prod\_{l=1}^{\text{NSP}} \left(\frac{\rho \mathbf{C}\_{i}}{\mathbf{M}\_{\text{uk}}}\right)^{\boldsymbol{\nu}\_{q}} - \frac{\rho \mathbf{C}\_{k}}{\mathbf{M}\_{\text{uk}}} \prod\_{l=1}^{\text{NSP}} \left(\frac{\rho \mathbf{C}\_{i}}{\mathbf{M}\_{\text{uk}}}\right)^{\boldsymbol{\nu}\_{q}}\right] \\ &= \left(\boldsymbol{\nu}\_{ij} - \boldsymbol{\nu}\_{ij}\right) \left[ \mathbf{K}\_{j} \prod\_{l=1}^{\text{NSP}} \left(\frac{\rho \mathbf{C}\_{l}}{\mathbf{M}\_{\text{uk}}}\right)^{\boldsymbol{\nu}\_{q}} - \prod\_{l=1}^{\text{NSP}} \left(\frac{\rho \mathbf{C}\_{i}}{\mathbf{M}\_{\text{uk}}}\right)^{\boldsymbol{\nu}\_{q}} \right] \boldsymbol{m}\_{li} \boldsymbol{k}\_{nq\{f\}} \end{split} \tag{50}$$

Summing over all third bodies *k* we get the total *Xij*:

$$\begin{aligned} \mathbf{X}\_{\boldsymbol{\eta}} &= \sum\_{k=1}^{\text{NSP}} \mathbf{X}\_{\boldsymbol{\eta}} \left( \text{for } \mathbf{3}^{\text{rd}} \text{ body } k \right) = \left( \boldsymbol{\nu}\_{\boldsymbol{\eta}} - \boldsymbol{\nu}\_{\boldsymbol{\eta}} \right) \left[ \mathbf{K}\_{j} \prod\_{l=1}^{\text{NSP}} \left( \frac{\rho \mathbf{C}\_{i}}{\mathbf{M}\_{\text{ul}}} \right)^{\text{V}\_{\boldsymbol{\eta}}} - \prod\_{l=1}^{\text{NSP}} \left( \frac{\rho \mathbf{C}\_{i}}{\mathbf{M}\_{\text{ul}}} \right)^{\text{V}\_{\boldsymbol{\eta}}} \right] \mathbf{M}\_{j} \mathbf{k}\_{n\boldsymbol{\eta}\left(\boldsymbol{\ell}\right)} \\\ \mathbf{M}\_{j} &= \rho \sum\_{k=1}^{\text{NSP}} m\_{k\boldsymbol{\eta}} \left( \frac{\mathbf{C}\_{k}}{\mathbf{M}\_{\text{ul}}} \right) = \rho \sum\_{k=1}^{\text{NSP}} m\_{k\boldsymbol{\eta}} \overline{\mathbf{C}\_{k}} = \rho \sum\_{k=1}^{\text{NSP}} m\_{k\boldsymbol{\eta}} \left[ \overline{\mathbf{M}}\_{k} \right] \end{aligned} \tag{51}$$

This is the standard production rate modified by the factor *Mj* , which is an effective third body molar concentration. In terms of *C*¯ *i* we may rewrite Eq. (51) as:

$$\begin{aligned} \mathbf{X}\_{\boldsymbol{\dot{\boldsymbol{\eta}}}} &= \sum\_{k=1}^{\text{NSP}} \mathbf{X}\_{\boldsymbol{\dot{\boldsymbol{\eta}}}} \Big( \text{for } \mathbf{3}^{\text{rd}} \text{ body } k \Big) = \left( \mathbf{\dot{\nu}}\_{\boldsymbol{\dot{\boldsymbol{\eta}}}} - \boldsymbol{\nu}\_{\boldsymbol{\dot{\boldsymbol{\eta}}}} \right) \Big[ \mathbf{K}\_{\boldsymbol{\boldsymbol{\eta}}} \prod\_{l=1}^{\text{NSP}} \overline{\mathbf{C}}\_{\boldsymbol{\imath}}^{\text{w}\_{\boldsymbol{\eta}}} - \boldsymbol{\rho}^{\text{t}\_{l}} \prod\_{l=1}^{\text{NSP}} \overline{\mathbf{C}}\_{\boldsymbol{\imath}}^{\text{v}\_{\boldsymbol{\eta}}} \Big] \mathbf{M}\_{\boldsymbol{\eta}} k\_{\boldsymbol{\rho}} \rho^{\sum\_{\text{sie}}^{\text{ssp}} \mathbf{v}\_{\boldsymbol{\eta}}} \\\\ \mathbf{M}\_{\boldsymbol{\dot{\cdot}}} &= \begin{cases} \rho \sum\_{l=1}^{\text{NSP}} m\_{\boldsymbol{\eta}} \overline{\mathbf{C}}\_{\boldsymbol{\dot{\boldsymbol{\eta}}}} & \text{for reactions involving } \mathbf{3}^{\text{rd}} \text{ bodies} \\ 1 & \text{for all other reactions} \end{cases} \end{aligned} \tag{52}$$

#### *1.6.3. Unidirectional reactions*

and

234 Surface Energy

as:

we may write:

giving:

'

n n ( )

*i*

n n

*wi*

*i*

*<sup>C</sup> <sup>C</sup> M*

*1.6.2. Third body dissociation recombination reactions*

=

( )

Molar mass fraction

*ij ij ij j i i j i i*

*X KC Ck*

<sup>1</sup> ' 1 1

r

Reactions involving a third body have a distinct reaction rate for each particular third body. Benson and Fueno [71] have shown theoretically that the temperature dependence of recom‐ bination rates is approximately independent of the third body. Available experimental recombination rate data also indicates that the temperature dependence of recombination rates is independent of the third body within the experimental accuracy of the measurements. Assuming that the temperature dependence of recombination rates is independent of the third body, the recombination rate associated with the *k-*th species (third body) can be represented

> exp *<sup>j</sup> n j*

where only the constant *akj* is different for different species (third bodies). Assuming that the reference species (third body) whose rate is specified in the program input has index *k = k*ref*,*

For the *k-*th species (third body) the stoichiometric coefficient is 1 on both sides of reaction *j*

'

 n

*NSP NSP rd ki ki*

rr

*ij ij*

 r

*CC CC X kK*

*i i ij ij j kj ref j i i wi wi*

*C C <sup>K</sup> m k M M*


*kj n j th*

*b*

*RT*

( ) exp for species *<sup>j</sup>*

1 1

èø èø ë û

*MM MM*

= =

Õ Õ

n

é ù æö æö ê ú <sup>=</sup> - -= ç÷ ç÷

*wk i i wi wk wi*

ç ÷ è ø

*kj kj*

*kj k kj ref j*

*k aT mk k*

ç ÷ è ø

*a b*


*a RT*

*ref ref*

*ref*

*k*

( ) ( )

*ij ij ij j*

r

( ) ( )

èø èø ë û

n

1 1

= =

*NSP NSP*

Õ Õ

é ù æö æö ê ú - - ç÷ ç÷

'

n n

*a*

*k kj*

*a*

*kj*

=

*m*

for 3 body

*k aT*

nl

*NSP NSP*

Õ Õ

= = é ù <sup>å</sup> =- - ê ú ë û

'

 r<sup>=</sup>

*ij j ij ij i*

 n *NSP*

n

(47)

(48)

(49)

(50)

'

 n

*ij ij*

rr

For unidirectional reactions the reaction only proceeds from left to right and has the assumed form

$$
\beta \overleftrightarrow{\mathcal{B}} \rightarrow \alpha\_1 \overleftrightarrow{A}\_1 + \alpha\_2 \overleftrightarrow{A}\_2 + ... + \alpha\_n \overleftrightarrow{A}\_n \tag{53}
$$

where the chemical species *B*¯ does not occur on the right side of the equation. Let *B*¯ have species index *K-UNI*. A unidirectional reaction is obtained by setting the backward reaction rate constant to zero, so that for reaction *j*:

$$\begin{aligned} \mathbf{X}\_{\boldsymbol{\beta}} &= \left(\boldsymbol{\nu}\_{\boldsymbol{\beta}} - \boldsymbol{\nu}\_{\boldsymbol{\beta}}\right) \left(\frac{\rho \mathbf{C}\_{\boldsymbol{K}-\text{IDN}}}{M\_{\boldsymbol{w}\_{\text{-
\text{IDN}}}}}\right) k\_{\boldsymbol{\beta}}\\ \text{So that:}\\ \frac{d\left[\overline{\boldsymbol{B}}\right]}{dt} &= -\beta \rho \overline{\mathbf{C}}\_{\boldsymbol{K}-\text{IDN}} k\_{\boldsymbol{\beta}} \quad \text{and} \qquad \frac{d\left[\overline{\boldsymbol{A}}\right]}{dt} = -\alpha\_{i} \rho \overline{\mathbf{C}}\_{\boldsymbol{K}-\text{IDN}} k\_{\boldsymbol{\beta}} \end{aligned} \tag{54}$$

#### **2. Numerical scheme and algorithm**

The governing equations are transformed into the computational domain by rewriting them into a vector form with **U** being the conserved variable vector, **E**, **F**, and **H** are the inviscid flux vectors, and **Ev**, Fv, and Hv the viscous fluxes in each three directions, respectively. Vector G contains all source terms.

$$\frac{\partial \mathbf{U}}{\partial \mathbf{t}} + \frac{\partial \mathbf{E}}{\partial \mathbf{x}} + \frac{\partial \mathbf{E}\_{\mathbf{v}}}{\partial \mathbf{x}} + \frac{\partial \mathbf{F}}{\partial \mathbf{y}} + \frac{\partial \mathbf{F}\_{\mathbf{v}}}{\partial \mathbf{y}} + \frac{\partial \mathbf{H}}{\partial \mathbf{z}} + \frac{\partial \mathbf{H}\_{\mathbf{v}}}{\partial \mathbf{z}} + \mathbf{G} = \mathbf{0} \tag{55}$$

The following transformation is used to cast the conservation equations in the computational domain of *ξ*, *η*, and *ζ* that correspond to *x, y,* and *z*.

$$\begin{aligned} \xi &= \xi \left( \mathbf{x}, \mathbf{y}, z \right) \\ \eta &= \eta \left( \mathbf{x}, \mathbf{y}, z \right) \\ \zeta &= \zeta \left( \mathbf{x}, \mathbf{y}, z \right) \end{aligned} \tag{56}$$

According to the chain rule of differentiation:

$$\begin{aligned} \frac{\partial}{\partial \mathbf{x}} &= \xi\_x \frac{\partial}{\partial \xi} + \eta\_x \frac{\partial}{\partial \eta} + \zeta\_x \frac{\partial}{\partial \zeta} \\ \frac{\partial}{\partial y} &= \xi\_y \frac{\partial}{\partial \xi} + \eta\_y \frac{\partial}{\partial \eta} + \zeta\_y \frac{\partial}{\partial \zeta} \\ \frac{\partial}{\partial z} &= \xi\_z \frac{\partial}{\partial \xi} + \eta\_z \frac{\partial}{\partial \eta} + \zeta\_z \frac{\partial}{\partial \zeta} \end{aligned} \tag{57}$$

where

$$\begin{aligned} \xi\_x &= \frac{y\_\eta z\_\zeta - y\_\zeta z\_\eta}{|\mathbf{n}|}, \qquad \xi\_y = -\frac{x\_\eta z\_\zeta - x\_\zeta z\_\eta}{|\mathbf{n}|}, \qquad \xi\_z = -\frac{x\_\eta y\_\zeta - x\_\zeta y\_\eta}{|\mathbf{n}|}\\ \eta\_x &= -\frac{y\_\zeta z\_\zeta - y\_\zeta z\_\zeta}{|\mathbf{n}|}, \qquad \eta\_y = \frac{x\_\zeta z\_\zeta - x\_\zeta z\_\zeta}{|\mathbf{n}|}, \qquad \eta\_z = -\frac{x\_\xi y\_\zeta - x\_\zeta y\_\zeta}{|\mathbf{n}|}\\ \zeta\_x &= \frac{y\_\zeta z\_\eta - y\_\eta z\_\zeta}{|\mathbf{n}|}, \qquad \zeta\_y = -\frac{x\_\xi z\_\eta - x\_\eta z\_\zeta}{|\mathbf{n}|}, \qquad \zeta\_z = \frac{x\_\xi y\_\eta - x\_\eta y\_\zeta}{|\mathbf{n}|} \end{aligned} \tag{58}$$

With the norm of:

$$\left|\mathbf{n}\right| = \mathbf{x}\_{\boldsymbol{\xi}} \mathbf{y}\_{\eta} \mathbf{z}\_{\boldsymbol{\xi}} - \mathbf{x}\_{\boldsymbol{\xi}} \mathbf{y}\_{\boldsymbol{\xi}} \mathbf{z}\_{\eta} + \mathbf{x}\_{\eta} \mathbf{y}\_{\boldsymbol{\xi}} \mathbf{z}\_{\boldsymbol{\xi}} - \mathbf{x}\_{\eta} \mathbf{y}\_{\boldsymbol{\xi}} \mathbf{z}\_{\boldsymbol{\xi}} + \mathbf{x}\_{\boldsymbol{\xi}} \mathbf{y}\_{\boldsymbol{\xi}} \mathbf{z}\_{\eta} - \mathbf{x}\_{\boldsymbol{\xi}} \mathbf{y}\_{\eta} \mathbf{z}\_{\boldsymbol{\xi}} \tag{59}$$

Equation (55) will now be transformed into:

( ) ( ) ( ) ( ) ( ) xyz xyz xyz x yz x yz x y = 0 1 1 with t 11 1 , , 11 1 , , *J J JJ J JJ J* ¶¶¶ ¶ ¶ ¶ ¶x ¶h z ¶x ¶h z xxx hhh z z z x xx h hh z z z ¶ ¶ ++++ + + + = = ¶ ¶ = ++ = ++ = ++ = ++ = ++ = ++ % % %% % %% % % % % %% % %%% **vv v v v v vv v v v v v U E FH E FH G U U, G G E E F HF E F HH E F H E E F H F E F H H Ev F** ( <sup>z</sup> ) **Hv** (60)

The Jacobian of the transformation is:

vectors, and **Ev**, Fv, and Hv the viscous fluxes in each three directions, respectively. Vector G

The following transformation is used to cast the conservation equations in the computational

( ) ( ) ( )

*xx x*

 xhz

 xhz

 xhz

xhz

¶¶¶ ¶ =++ ¶¶¶ ¶ ¶¶¶ ¶ =++ ¶¶¶ ¶ ¶¶¶ ¶ =++ ¶¶ ¶ ¶

*yy y*

xhz

*zz z*

*yz yz xz xz xy xy*

**n nn**

*yz yz xz xz xy xy*

**nn n**

 zx

> hx

 zh

 x

 z

> z xh

**n** =-+-+- (59)

hz

xz

 hx

 z hx

xh

 zh

> zx

xhz

, ,


hz

*xy z*

*xyz*

hhh

*xy z*

 x zh

, ,

xz


, ,

 h z x h xz


*yz yz xz xz xy xy*

**n nn**

*xyz xyz xyz xyz xyz xyz*

xh

, , , , , ,

*xyz xyz xyz* ¶ ¶

¶ ¶

**v v UEE FF H Hv <sup>G</sup>** (55)

t *xxyyz z*

 ¶

++ ++ ++ +

 ¶¶

 ¶

x x

= = =

h h

z z

*x*

*y*

*z*

 ¶ = 0

(56)

(57)

(58)

contains all source terms.

236 Surface Energy

where

With the norm of:

¶

 ¶¶

domain of *ξ*, *η*, and *ζ* that correspond to *x, y,* and *z*.

According to the chain rule of differentiation:

hz

xz

xh

 zh

xx

 hx

zz

x hz

Equation (55) will now be transformed into:

 zx

¶¶

$$J = \xi\_x \left(\eta\_y \zeta\_z - \eta\_z \zeta\_y\right) - \xi\_y \left(\eta\_x \zeta\_z - \zeta\_x \eta\_z\right) + \xi\_z \left(\eta\_x \zeta\_y - \zeta\_x \eta\_y\right) \tag{61}$$

The explicit form of all of the flux vectors are described by Navaz et al. [55,56]. The above sets of equations are solved by finite difference method on a structured mesh. The explicit Runge– Kutta integration algorithm was used to find the distribution of all variables (gas, liquid, and solid concentrations, capillary pressure, velocities, etc.) in time. The program can accept externally generated mesh by commercial software such as GRIDGEN™ [72] or can generate a grid internally for one or two sessile droplet(s) residing on a porous on nonporous substrate. In the presence of a liquid bridge and its spread, a built-in and automated adaptive mesh generator is embedded in the computer model. The mesh or grid point distribution is based on Coon's Patch [73]. First, the mesh for the substrate is generated and then the mesh defining the droplet(s) is overlaid on the substrate geometry. The mesh points defining the droplets are collapsed on the surface of the substrate except in the area(s) that droplet(s) reside on the surface. Figure 14 shows the top view of the adaptive mesh when liquid bridge expands during the contact process. Figures 11 and 12 shows snapshots of the absorption process during the contact. It displays the precontact and postcontact configurations.

**Figure 14.** Top view of the adaptive mesh when liquid bridge still expanding, until the conditions for Equation (33) are reached.

The calculations start with the specified initial conditions and then the continuity and mo‐ mentum equations are numerically integrated in time to find the distribution of liquid inside the pores. The saturation is equal to unity (*s*<sup>ℓ</sup> =1) at the interface between the droplet or liquid bridge with the porous media. The hydrostatic pressure is also added to the capillary pressure for more accurate calculations. That is, *P*<sup>ℓ</sup> <sup>=</sup> *<sup>P</sup>* <sup>−</sup>*Pci* <sup>+</sup> *<sup>ρ</sup>*ℓ*gh* \* where *h\** is the local height of the droplet above the surface. Mass is being transported into porous medium according to (*ρ*ℓ*υ*˜*φ*)/ *J;* where *J* is the Jacobian for the transformation and *υ*˜ is the contra-variant vertical velocity given by: *υ*˜ <sup>=</sup>*ηxu*<sup>ℓ</sup> <sup>+</sup> *<sup>η</sup>yυ*<sup>ℓ</sup> <sup>+</sup> *<sup>η</sup>zw*ℓwith *u*ℓ, *<sup>υ</sup>*ℓ, *<sup>w</sup>*ℓ being the three components of the velocity and *ηx*, *ηy*, *η<sup>z</sup>* being the metrics of the transformation. The mass transfer is calculated in each time step and the instantaneous remaining mass yields the liquid bridge volume. The remaining mass divided by the liquid density will yield the instantaneous volume of the liquid bridge. Since the distance gap or the height of the liquid bridge is known from the equation of motion (Eq. 29), the temporal (or instantaneous) base or footprint radius can be found from Equa‐ tion(32). When the distance between surfaces become less than a threshold, the spread is stopped, that is, the base radius stays constant at its last value and the capillary transport into both surfaces continues, causing a reduction in the height of the liquid bridge. At this point the instantaneous or temporal height of the liquid bridge can be calculated from Equation (33). This process continues until the amount of liquid between the two surfaces becomes zero.

## **3. Test cases**

The computational model developed is called modeling of chemical hazard—dispersion and contact (MOCHA-DC). Test cases are designed to validate the physiochemical phenomena that are formulated in this chapter. More details are available in Navaz et al. [55, 56]. A test case describing the spread of VX nerve agent into sand is described. The capillary pressure, which is the main driver for the spread and transport of liquid into porous media, was obtained by the method outlined in Section 1.4.

#### **3.1. Capillary transport and secondary evaporation**

The topology of the penetration of a 6 μL VX droplet into the UK sand (*ϕ* = 0.44) after about 120 minutes is shown in Figure 15. VX has a very low vapor pressure at room temperature and so evaporation could be ignored. The shape of the liquid front is obtained experimentally at ECBC and is compared to the MOCHA-DC code prediction (saturation contours) in Figure 15. A comparison for the prediction of the depth of spread in time is observed from Figure 16. as compared with experimental data.

A similar validation was performed for the penetration of 1,2-propandiol, glycerin, and castor oil into a ceramic tile (*ϕ* = 0.26), coarse sand (*ϕ* = 0.31), and glass beads (*ϕ* = 0.28). The penetration was recorded after 2, 3, 5, 10, 20, 30, 60 minutes, for up to a week (for castor oil). The model prediction was excellent and they are detailed by Navaz et al. [55]. A sample case for glycerin spread into sand is shown in Figure 17.

experimentally at ECBC and is compared to the MOCHA-DC code prediction (saturation contours) in Figure 3.1.1. A comparison for the prediction of the depth of spread in time is A General-Purpose Multiphase/Multispecies Model to Predict the Spread, Percutaneous Hazard, and... http://dx.doi.org/10.5772/60807 239

= 0.44) after

or the height of the liquid bridge is known from the equation of motion (Eq. 1.5.1), the temporal (or instantaneous) base or footprint radius can be found from Equation(1.5.4). When the distance between surfaces become less than a threshold, the spread is stopped, that is, the base radius stays constant at its last value and the capillary transport into both surfaces continues, causing a reduction in the height of the liquid bridge. At this point the instantaneous or temporal height of the liquid bridge can be calculated from Equation (1.5.5). This process continues until the

The computational model developed is called modeling of chemical hazard—dispersion and contact (MOCHA-DC). Test cases are designed to validate the physiochemical phenomena that are formulated in this chapter. More details are available in Navaz et al.55, 56. A test case describing the spread of VX nerve agent into sand is described. The capillary pressure, which is the main driver for the spread and transport of liquid into porous media, was obtained by the

amount of liquid between the two surfaces becomes zero.

*3.1 Capillary Transport and Secondary Evaporation* 

observed from Figure 3.1.2 as compared with experimental data.

The topology of the penetration of a 6 L VX droplet into the UK sand (

about 120 minutes is shown in Figure 3.1.1. VX has a very low vapor pressure at room temperature and so evaporation could be ignored. The shape of the liquid front is obtained

*3 Test Cases* 

method outlined in Section 1.4.

Figure 3.1.1 Comparison of COMCAD prediction and experiment for VX on UK sand. **Figure 15.** Comparison of COMCAD prediction and experiment for VX on UK sand.

The calculations start with the specified initial conditions and then the continuity and mo‐ mentum equations are numerically integrated in time to find the distribution of liquid inside the pores. The saturation is equal to unity (*s*<sup>ℓ</sup> =1) at the interface between the droplet or liquid bridge with the porous media. The hydrostatic pressure is also added to the capillary pressure

droplet above the surface. Mass is being transported into porous medium according to (*ρ*ℓ*υ*˜*φ*)/ *J;* where *J* is the Jacobian for the transformation and *υ*˜ is the contra-variant vertical velocity given by: *υ*˜ <sup>=</sup>*ηxu*<sup>ℓ</sup> <sup>+</sup> *<sup>η</sup>yυ*<sup>ℓ</sup> <sup>+</sup> *<sup>η</sup>zw*ℓwith *u*ℓ, *<sup>υ</sup>*ℓ, *<sup>w</sup>*ℓ being the three components of the velocity and *ηx*, *ηy*, *η<sup>z</sup>* being the metrics of the transformation. The mass transfer is calculated in each time step and the instantaneous remaining mass yields the liquid bridge volume. The remaining mass divided by the liquid density will yield the instantaneous volume of the liquid bridge. Since the distance gap or the height of the liquid bridge is known from the equation of motion (Eq. 29), the temporal (or instantaneous) base or footprint radius can be found from Equa‐ tion(32). When the distance between surfaces become less than a threshold, the spread is stopped, that is, the base radius stays constant at its last value and the capillary transport into both surfaces continues, causing a reduction in the height of the liquid bridge. At this point the instantaneous or temporal height of the liquid bridge can be calculated from Equation (33). This process continues until the amount of liquid between the two surfaces becomes zero.

The computational model developed is called modeling of chemical hazard—dispersion and contact (MOCHA-DC). Test cases are designed to validate the physiochemical phenomena that are formulated in this chapter. More details are available in Navaz et al. [55, 56]. A test case describing the spread of VX nerve agent into sand is described. The capillary pressure, which is the main driver for the spread and transport of liquid into porous media, was obtained by

The topology of the penetration of a 6 μL VX droplet into the UK sand (*ϕ* = 0.44) after about 120 minutes is shown in Figure 15. VX has a very low vapor pressure at room temperature and so evaporation could be ignored. The shape of the liquid front is obtained experimentally at ECBC and is compared to the MOCHA-DC code prediction (saturation contours) in Figure 15. A comparison for the prediction of the depth of spread in time is observed from Figure

A similar validation was performed for the penetration of 1,2-propandiol, glycerin, and castor oil into a ceramic tile (*ϕ* = 0.26), coarse sand (*ϕ* = 0.31), and glass beads (*ϕ* = 0.28). The penetration was recorded after 2, 3, 5, 10, 20, 30, 60 minutes, for up to a week (for castor oil). The model prediction was excellent and they are detailed by Navaz et al. [55]. A sample case for glycerin

where *h\** is the local height of the

for more accurate calculations. That is, *P*<sup>ℓ</sup> <sup>=</sup> *<sup>P</sup>* <sup>−</sup>*Pci* <sup>+</sup> *<sup>ρ</sup>*ℓ*gh* \*

**3. Test cases**

238 Surface Energy

the method outlined in Section 1.4.

16. as compared with experimental data.

spread into sand is shown in Figure 17.

**3.1. Capillary transport and secondary evaporation**

39

**Figure 16.** Progression of depth in time for VX spread/transport through UK sand as predicted by the model and ex‐ periment

The secondary evaporation of several nerve agents are measured by Navaz et al.*<sup>55</sup>*. The validation of the model was performed using data from Reis et al. [74] and Mantle et al. [7] They studied the evaporation of water inside porous media using magnetic resonance imaging (MRI). Figure 18. shows an excellent comparison for the evaporation of water in sand and 120 and 400-μm diameter glass beads.

castor oil into a ceramic tile (

400-m diameter glass beads.

A similar validation was performed for the penetration of 1,2-propandiol, glycerin, and

0 2 4 6 8 10 12 14 16

Time1/2 (min1/2)

penetration was recorded after 2, 3, 5, 10, 20, 30, 60 minutes, for up to a week (for castor oil). The model prediction was excellent and they are detailed by Navaz et al.55. A sample case for

= 0.31), and glass beads (

= 0.28). The

Figure 3.1.2 Progression of depth in time for VX spread/transport through UK sand as

= 0.26), coarse sand (

predicted by the model and experiment

0

0.5

1

1.5

2

2.5

Depth (mm)

3

3.5

4

4.5

Experimental

Power law fit - Experimental Numerical prediction

Figure 3.1.4 shows an excellent comparison for the evaporation of water in sand and 120- and Figure 3.1.4 shows an excellent comparison for the evaporation of water in sand and 120- and Figure 3.1.3 Penetration of glycerin into the coarse sand after 3 minutes (depth ~ 6 mm) and **Figure 17.** Penetration of glycerin into the coarse sand after 3 minutes (depth ~ 6 mm) and wetted pad from the experi‐ mental study

Data collected in collaboration with Kiple Acquisition Science Technology Logistics & Engineering Inc. (KASTLE), B.O.I.S.-Filtry, Brno, Czech Republic and VOP-026 Sternberk, Brno, Czech Republic, was used to validate the model. The data was taken in open air and the model was run under the same conditions (wind speed and temperature) for EDB (ethylene dibromide). An excellent agreement between the model prediction and data exists, as can be seen from Figure 19. Data collected in collaboration with Kiple Acquisition Science Technology Logistics & Engineering Inc. (KASTLE), B.O.I.S.-Filtry, Brno, Czech Republic and VOP-026 Sternberk, Brno, Czech Republic, was used to validate the model. The data was taken in open air and the model was run under the same conditions (wind speed and temperature) for EDB (ethylene dibromide). An excellent agreement between the model prediction and data exists, as can be seen from Figure 3.1.5. 400-m diameter glass beads. Data collected in collaboration with Kiple Acquisition Science Technology Logistics & Engineering Inc. (KASTLE), B.O.I.S.-Filtry, Brno, Czech Republic and VOP-026 Sternberk, Brno, Czech Republic, was used to validate the model. The data was taken in open air and the model was run under the same conditions (wind speed and temperature) for EDB (ethylene dibromide). An excellent agreement between the model prediction and data exists, as can be seen wetted pad from the experimental study The secondary evaporation of several nerve agents are measured by Navaz et al.*<sup>55</sup>*. The validation of the model was performed using data from Reis et al.74 and Mantle et al.7 They studied the evaporation of water inside porous media using magnetic resonance imaging (MRI).

Figure 3.1.4 Model validation for the secondary evaporation of water inside sand and glass beads compared with MRI data **Figure 18.** Model validation for the secondary evaporation of water inside sand and glass beads compared with MRI data Figure 3.1.4 Model validation for the secondary evaporation of water inside sand and glass beads compared with MRI data

evaporation. Figure 3.1.5 Model/experiment comparisons with Czech Data for *6 µL* Ethylene Dibromide (EDB) droplet spread inside porous media (sand and glass beads) with secondary evaporation. **Figure 19.** Model/experiment comparisons with Czech Data for 6 μL Ethylene Dibromide (EDB) droplet spread inside porous media (sand and glass beads) with secondary evaporation.

(EDB) droplet spread inside porous media (sand and glass beads) with secondary

41

41

A General-Purpose Multiphase/Multispecies Model to Predict the Spread, Percutaneous Hazard, and... http://dx.doi.org/10.5772/60807 241

Figure 3.1.6 Model/experiment comparisons with Czech Data for *6 µL* Malathion (MAL) droplet spread inside porous media with secondary evaporation. **Figure 20.** Model/experiment comparisons with Czech Data for 6 μL Malathion (MAL) droplet spread inside porous media with secondary evaporation.

Wind tunnel test were performed for the evaporation of Malathion in dry concrete and glass beads. There was no (or minor) evaporation of Malathion inside dry concrete or glass beads due to the low vapor pressure of the species, as seen from Figure 3.1.6. However, it will be shown that in the presence of moisture or water this picture will change. Wind tunnel test were performed for the evaporation of Malathion in dry concrete and glass beads. There was no (or minor) evaporation of Malathion inside dry concrete or glass beads due to the low vapor pressure of the species, as seen from Figure 20. However, it will be shown that in the presence of moisture or water this picture will change.

#### *3.2 Chemical Reaction in Porous Media*  **3.2. Chemical reaction in porous media**

Data collected in collaboration with Kiple Acquisition Science Technology Logistics & Engineering Inc. (KASTLE), B.O.I.S.-Filtry, Brno, Czech Republic and VOP-026 Sternberk, Brno, Czech Republic, was used to validate the model. The data was taken in open air and the model was run under the same conditions (wind speed and temperature) for EDB (ethylene dibromide). An excellent agreement between the model prediction and data exists, as can be

**Figure 17.** Penetration of glycerin into the coarse sand after 3 minutes (depth ~ 6 mm) and wetted pad from the experi‐

The secondary evaporation of several nerve agents are measured by Navaz et al.*<sup>55</sup>*. The

studied the evaporation of water inside porous media using magnetic resonance imaging (MRI).

74 Reis, N., and Griffiths, R. (2003), "Investigation of the evaporation of the embedded liquid droplets from the

Figure 3.1.3 Penetration of glycerin into the coarse sand after 3 minutes (depth ~ 6 mm) and

validation of the model was performed using data from Reis et al.74 and Mantle et al.7

Figure 3.1.2 Progression of depth in time for VX spread/transport through UK sand as

= 0.26), coarse sand (

A similar validation was performed for the penetration of 1,2-propandiol, glycerin, and

0 2 4 6 8 10 12 14 16

Time1/2 (min1/2)

penetration was recorded after 2, 3, 5, 10, 20, 30, 60 minutes, for up to a week (for castor oil). The model prediction was excellent and they are detailed by Navaz et al.55. A sample case for

= 0.31), and glass beads (

= 0.28). The

They

predicted by the model and experiment

0

0.5

1

1.5

2

2.5

Depth (mm)

3

3.5

4

4.5

Experimental

Power law fit - Experimental Numerical prediction

glycerin spread into sand is shown in Figure 3.1.3.

Figure 3.1.4 Model validation for the secondary evaporation of water inside sand and glass

**Figure 18.** Model validation for the secondary evaporation of water inside sand and glass beads compared with MRI

Figure 3.1.4 Model validation for the secondary evaporation of water inside sand and glass

Figure 3.1.5 Model/experiment comparisons with Czech Data for *6 µL* Ethylene Dibromide

Figure 3.1.5 Model/experiment comparisons with Czech Data for *6 µL* Ethylene Dibromide

**Figure 19.** Model/experiment comparisons with Czech Data for 6 μL Ethylene Dibromide (EDB) droplet spread inside

(EDB) droplet spread inside porous media (sand and glass beads) with secondary

(EDB) droplet spread inside porous media (sand and glass beads) with secondary

beads compared with MRI data

porous surfaces using magnetic resonance imaging," *Int J Heat Mass Transfer*

wetted pad from the experimental study

beads compared with MRI data

evaporation.

porous media (sand and glass beads) with secondary evaporation.

evaporation.

Figure 3.1.4 shows an excellent comparison for the evaporation of water in sand and 120- and

Figure 3.1.4 shows an excellent comparison for the evaporation of water in sand and 120- and

Data collected in collaboration with Kiple Acquisition Science Technology Logistics & Engineering Inc. (KASTLE), B.O.I.S.-Filtry, Brno, Czech Republic and VOP-026 Sternberk, Brno, Czech Republic, was used to validate the model. The data was taken in open air and the model was run under the same conditions (wind speed and temperature) for EDB (ethylene dibromide). An excellent agreement between the model prediction and data exists, as can be seen

Data collected in collaboration with Kiple Acquisition Science Technology Logistics & Engineering Inc. (KASTLE), B.O.I.S.-Filtry, Brno, Czech Republic and VOP-026 Sternberk, Brno, Czech Republic, was used to validate the model. The data was taken in open air and the model was run under the same conditions (wind speed and temperature) for EDB (ethylene dibromide). An excellent agreement between the model prediction and data exists, as can be seen

seen from Figure 19.

mental study

castor oil into a ceramic tile (

240 Surface Energy

from Figure 3.1.5.

data

from Figure 3.1.5.

400-m diameter glass beads.

400-m diameter glass beads.

In this section series of validation, test cases are presented that involve chemical In this section series of validation, test cases are presented that involve chemical reaction(s) in different phases.

reaction(s) in different phases. We chose to simulate two cases for a liquid interacting with a solid. The first case was a 50 µL drop of sulfuric acid placed on a brick of solid sodium sulfide. The second case was a 50 µL drop of sulfuric acid on a mixture of 75% sand and 25% sodium sulfide. In both cases the samples were open to the air to allow the product gas to escape. Mass loss was monitored and rates of product formation/reactant degradation inferred. Earlier experiments documented that during the time scale for these reactions the sulfuric acid did not react with the sand (100% sand case) and adsorption of atmospheric water was insignificant. Therefore, the mass change was due to the reaction alone. The reaction stoichiometric equation is given below. The model simulation We chose to simulate two cases for a liquid interacting with a solid. The first case was a 50 μL drop of sulfuric acid placed on a brick of solid sodium sulfide. The second case was a 50 μL drop of sulfuric acid on a mixture of 75% sand and 25% sodium sulfide. In both cases the samples were open to the air to allow the product gas to escape. Mass loss was monitored and rates of product formation/reactant degradation inferred. Earlier experiments documented that during the time scale for these reactions the sulfuric acid did not react with the sand (100% sand case) and adsorption of atmospheric water was insignificant. Therefore, the mass change was due to the reaction alone. The reaction stoichiometric equation is given below. The model simulation of the liquid on the solid brick of sodium sulfide and the sand/sodium sulfide mixture are given in Figures 21a and b. Both results are validated with experimental data.

of the liquid on the solid brick of sodium sulfide and the sand/sodium sulfide mixture are given Na2S (s) + H2SO4 (aq) → Na2SO4 (s) + H2S (g)

41

41

in Figures 3.2.1a and 3.2.1b. Both results are validated with experimental data. (sodium sulfide + sulfuric acid → sodium sulfate + hydrogen sulfide)

Na2S (s) + H2SO4 (aq) → Na2SO4 (s) + H2S (g) (sodium sulfide + sulfuric acid → sodium sulfate + hydrogen sulfide) In the next test case, glass beads were chosen as the porous substrate and droplets of cyclo‐ hexanol and phosphoric acid were deposited about 1 mm apart. The reaction started when the two droplets diffused together due to capillary forces, and the reaction proceeded thereafter. Figure 22. shows the contours of remaining reactants and the amount of product (cyclohexene) as predicted by the MOCHA-DC code from the initiation of the process (diffusion and

acid reaction, and (*b*) 3:1 mixture of sand: sodium sulfate and 50 L sulfuric acid droplet **Figure 21.** Model/ experiment comparison for (a) 100% sodium sulfate and 50 μL sulfuric acid reaction, and (b) 3:1 mixture of sand: sodium sulfate and 50 μL sulfuric acid droplet

Figure 3.2.1 Model/ experiment comparison for (*a*) 100% sodium sulfate and 50 L sulfuric

reaction). The overall destruction of cyclohexanol is similar to experimental data and is described in Figure 22.. In the next test case, glass beads were chosen as the porous substrate and droplets of cyclohexanol and phosphoric acid were deposited about 1 mm apart. The reaction started when

The reaction of chemical warfare agents with water is of special interest in this section. This is called a hydrolysis reaction. VX and Malathion are among the chemicals that react with water. Water can be present as moisture in sand, concrete, and soil, or can be added to a preexisting chemical agent in a dry porous medium as rain. The model is capable of simulating all possible scenarios. The hydrolysis of VX proceeds as multiple chemical reaction processes that result in several breakdown products. The evolution of these products and the spread of a droplet therefore provide a reasonable validation of the interaction of the reaction and diffusion functions in the code. A simulation of a 6 μL droplet of *O*-ethyl-*s*-[2-*N*,*N*-(diisopropylami‐ no)ethyl] methylphosphonothioate (VX) spread into sand with porosity of 35%. The hydrolysis reaction rates were defined using those for VX in moist sand given in Brevett et al. [75, 76] Brevett et al.'s [75, 76] work indicates that VX degrades into several products in the presence of water. All of these reactions were taken as first order in VX, due to the reported 25-32 fold molar excess of water present in the experiments. The droplet size was taken from the same study. The porous field in the model extended beyond the region of droplet spread. All model boundaries were set as impermeable, as the experimental study involved degradation under sealed conditions. The quantity of VX and the appearance of breakdown products indicated by the models throughout the simulation were consistent with data from the experimental study. VX spread to a radius of 7 mm in the model, similar to the spread reported by Brevett et al. [75, 76] and Wagner et al. [77] The match between the amount of VXH+ in the experiment and in the model during the degradation process results from the model's ability to handle multiple reactions. The rate of production/destruction of each chemical species is a local event, calculated at each node. Species concentration and, therefore, the reaction rate vary across the saturation region within the sand, yielding a complex mixture of breakdown products. If the the two droplets diffused together due to capillary forces, and the reaction proceeded thereafter. Figure 3.2.2 shows the contours of remaining reactants and the amount of product (cyclohexene) as predicted by the MOCHA-DC code from the initiation of the process (diffusion and reaction). The overall destruction of cyclohexanol is similar to experimental data and is described in Figure 3.2.3. 

43

A General-Purpose Multiphase/Multispecies Model to Predict the Spread, Percutaneous Hazard, and... http://dx.doi.org/10.5772/60807 243

Figure 3.2.2 Development of reaction zone between cyclohexanol and phosphoric acid Modeled by the MOCHA-DC Code (from top left to lower right) t=0, t=12 s, t=144 s, t=3600 s. **Figure 22.** Development of reaction zone between cyclohexanol and phosphoric acid Modeled by the MOCHA-DC Code (from top left to lower right) t=0, t=12 s, t=144 s, t=3600 s.

fluid motion predicted in the model was inappropriate it would lead to unrealistic local concentrations, which would cause the local rates of production/destruction (based on local concentrations in the reaction model) to be inappropriate. The agreement between the model and the experimental data, therefore, demonstrates robustness in both the fluid motion and reaction modeling methods. The result of this study is shown in Figure 24.

## **3.3. Contact hazard**

43

reaction). The overall destruction of cyclohexanol is similar to experimental data and is

**Figure 21.** Model/ experiment comparison for (a) 100% sodium sulfate and 50 μL sulfuric acid reaction, and (b) 3:1

In the next test case, glass beads were chosen as the porous substrate and droplets of cyclohexanol and phosphoric acid were deposited about 1 mm apart. The reaction started when the two droplets diffused together due to capillary forces, and the reaction proceeded thereafter. Figure 3.2.2 shows the contours of remaining reactants and the amount of product (cyclohexene) as predicted by the MOCHA-DC code from the initiation of the process (diffusion and reaction). The overall destruction of cyclohexanol is similar to experimental data and is described in Figure

*(a)* (*b*)

acid reaction, and (*b*) 3:1 mixture of sand: sodium sulfate and 50 L sulfuric acid

Figure 3.2.1 Model/ experiment comparison for (*a*) 100% sodium sulfate and 50 L sulfuric

The reaction of chemical warfare agents with water is of special interest in this section. This is called a hydrolysis reaction. VX and Malathion are among the chemicals that react with water. Water can be present as moisture in sand, concrete, and soil, or can be added to a preexisting chemical agent in a dry porous medium as rain. The model is capable of simulating all possible scenarios. The hydrolysis of VX proceeds as multiple chemical reaction processes that result in several breakdown products. The evolution of these products and the spread of a droplet therefore provide a reasonable validation of the interaction of the reaction and diffusion functions in the code. A simulation of a 6 μL droplet of *O*-ethyl-*s*-[2-*N*,*N*-(diisopropylami‐ no)ethyl] methylphosphonothioate (VX) spread into sand with porosity of 35%. The hydrolysis reaction rates were defined using those for VX in moist sand given in Brevett et al. [75, 76] Brevett et al.'s [75, 76] work indicates that VX degrades into several products in the presence of water. All of these reactions were taken as first order in VX, due to the reported 25-32 fold molar excess of water present in the experiments. The droplet size was taken from the same study. The porous field in the model extended beyond the region of droplet spread. All model boundaries were set as impermeable, as the experimental study involved degradation under sealed conditions. The quantity of VX and the appearance of breakdown products indicated by the models throughout the simulation were consistent with data from the experimental study. VX spread to a radius of 7 mm in the model, similar to the spread reported by Brevett et al. [75, 76] and Wagner et al. [77] The match between the amount of VXH+ in the experiment and in the model during the degradation process results from the model's ability to handle multiple reactions. The rate of production/destruction of each chemical species is a local event, calculated at each node. Species concentration and, therefore, the reaction rate vary across the saturation region within the sand, yielding a complex mixture of breakdown products. If the

described in Figure 22..

3.2.3.

242 Surface Energy

droplet

mixture of sand: sodium sulfate and 50 μL sulfuric acid droplet

In this section, some test cases are presented to illustrate the model prediction for a sessile droplet spread dynamics between two moving nonporous or porous surfaces. The presented test cases will examine the gas and liquid mass transfer and the effect of chemical reactions during the droplet spread process. Figure 3.2.3 Comparison of destruction of cyclohexanol to experimental data

In the first case, the spread of de-ionized water between a nonporous and porous glass plates as a function of velocity of approach was examined. A linear stage actuator (Thorlabs™ LNR50SEK1) controls the distance between the two parallel plates and a load cell (Interface™ MB-LBF5) is designated for measuring the force acting on the solid body. Two optical windows (Newport™ 20BW40-30) are used as the upper and lower plates as shown in Figure 25.. A The reaction of chemical warfare agents with water is of special interest in this section. This is called a hydrolysis reaction. VX and Malathion are among the chemicals that react with water. Water can be present as moisture in sand, concrete, and soil, or can be added to a

**Figure 23.** Comparison of destruction of cyclohexanol to experimental data

**Figure 24.** A 6 μL VX droplet undergoes hydrolysis in damp sand at 50°C. Both agent and breakdown product amounts present in the sand, as measured in Brevett et al. [75, 76], are characterized by the model

feedback control loop was created, connecting the load-cell force input, the actuator position, and speed inputs and outputting the upper plate acceleration and speed. This feedback enabled to simulate a freely moving solid body with a given mass, *m*1, reacting to external forces and the force applied by the liquid bridge connected to a solid body with mass *m*2→*∞*. The plates were cleaned before each experiment using acetone, isopropanol, and nitrogen gas. The advancing and receding wetting angles were estimated from quasi-static force measure‐ ments. The experiment utilized a 200 μL deionized (DI) water as a sessile droplet deposited onto a fused silica optical window (an impermeable surface), and a porous glass (VykorTM 7930) was used as the permeable contacting surface. The physical properties of water are *μ* =0.001*Pa*.*s*, *ρ* =1000*kg* / *m*<sup>3</sup> , *σ* =0.072*N* / *m*. The porous glass had a porosity of 28% with permeability of 2.08 × 10–19 m2 /s (values provided by manufacturer). The surfaces were brought into contact with constant relative speeds of 0.5, 1.5, 2.0, and 2.5 μm/s, and the liquid bridge radius was measured in the experiment and compared to that calculated by the model (Figure 26). In all cases, there was a rapid initial change in geometry upon contact of the liquid with both plates, and a transition from sessile droplet geometry to the hyperboloid geometry of the liquid bridge. For smaller approach velocities, the absorption into the porous medium proceeded faster than the radial spreading throughout the liquid penetration process, causing a steady reduction in the footprint radius in time. However, for faster approach speeds (2 μm/ s and above) the radius of the liquid bridge was only reducing in the initial stages due to rapid penetration into the porous medium. After some penetration of the liquid into the porous medium, and thus increasing the viscous resistance of the flow, the liquid began to spread radially and the radius of the liquid bridge increased with time. This finding demonstrated that the depth of liquid penetration and the imprint radius of droplet into the porous medium is determined by the speed of the approaching surfaces. We have also modeled two additional approach velocities of 1.0 and 3.0 μm/s to demonstrate the consistency of the results (Figure 26.). The model and experimental results indicated that the spreading increases as the approach velocity increases. This causes a larger footprint to be issued by the process.

**Figure 23.** Comparison of destruction of cyclohexanol to experimental data

244 Surface Energy

**Figure 24.** A 6 μL VX droplet undergoes hydrolysis in damp sand at 50°C. Both agent and breakdown product

amounts present in the sand, as measured in Brevett et al. [75, 76], are characterized by the model

An experiment conducted to measure the amount of mass transfer for two surfaces coming into contact with a liquid droplet in between. A 20 μL glycerin droplet was deposited on a porous media composed of play sand (porosity = 35%). Then a cloth at an initial distance of 11 cm with the speed of 35 cm/s was brought into contact with the glycerin. The amount of mass absorbed during the contact process into the cloth was measures and is compared to the model prediction. These tests were conducted with three additional chemicals and the results were compared to the model's response. These conditions are shown in Figure 27.a and the agreement between the model and experimental data is seen to be excellent.

Another experiment was conducted with the same setup with the exception of varying the time of the contact. The purpose of this experiment is to demonstrate that even after the disappearance of the sessile droplet from the surface, mass transfer to a secondary porous material will take place. That is to say, even though a hazardous liquid substance may not be visible on a surface of a porous material, it can still pose a threat through contact. However, kitchen tile (porosity of 24%) with a lower porosity was selected for this experiment to have more control over the disappearance time of the sessile droplet from the surface. A wafer,

Figure 3.3.1 The experimental setup: two fused-silica plates connected by a DI water liquid **Figure 25.** The experimental setup: two fused-silica plates connected by a DI water liquid bridge. (a) Plates in fixture and (*a*\*) a closeup view of the liquid bridge (*b*) shows the porous glass used in the experiment

bridge. (a) Plates in fixture and (*a*\*) a closeup view of the liquid bridge (*b*) shows

fabricated in our laboratory from filter paper pulp, with a porosity of 60%, served as the secondary surface. Five separate experiments (three repetitions for each experiment) were conducted with a contact time to be, respectively, 1, 10, 20, 30, and 40 minutes after the droplet is placed on the kitchen tile. Every experiment was repeated three times at an initial gap of 2.5 cm between the two surfaces and a downward force of 1 N. The two surfaces were brought into contact. The amount of glycerin transferred into the secondary surface (wafer) was measured and compared favorably with the model predictions (Figure 27.b). These compari‐ sons indicate that the model was fairly accurate in predicting the amount of mass being absorbed into a contacting porous surface. the porous glass used in the experiment

In order to provide data for cases where contact between two surfaces results in a chemical reaction in one or both of the contacting surfaces, experiments were conducted measuring the rate of formation of product where one reactant resides on one surface and the other on the second, contacting, surface. Since there are many permutations for these types of experiments, we began our investigation for a case where the primary surface is nonporous, a glass slide, and the contacting surface is porous, an adsorbent pad. The reaction needed to be fast enough so that the product formation could be measured accurately and in a relatively short time span to avoid complications associated with evaporation of chemicals. The reaction chosen was addition of bromine to alkenes, since the progress of this reaction could be monitored both Figure 3.3.2 200 L of water on glass contacted by porous glass. The approach velocities of 0.5, 1.5, 2.0, and 2.5 m/s are compared with the model. Two additional velocities

the same "speed"

of 1.0 and 3.0 m/s are also modeled. For lower velocities the absorption into porous glass is at a faster pace than the spread speed, causing a continuous reduction in radius in time. As the approach velocity increases, the spread will proceed at a faster pace than the absorption, causing a continuous increase of radius in time. It appears that at velocity of 1.5 m/s both processes take place at

An experiment conducted to measure the amount of mass transfer for two surfaces coming into contact with a liquid droplet in between. A 20 µL glycerin droplet was deposited on

47

cm with the speed of 35 cm/s was brought into contact with the glycerin. The amount of mass

**Figure 26.** 200 μL of water on glass contacted by porous glass. The approach velocities of 0.5, 1.5, 2.0, and 2.5 μm/s are compared with the model. Two additional velocities of 1.0 and 3.0 μm/s are also modeled. For lower velocities the ab‐ sorption into porous glass is at a faster pace than the spread speed, causing a continuous reduction in radius in time. As the approach velocity increases, the spread will proceed at a faster pace than the absorption, causing a continuous increase of radius in time. It appears that at velocity of 1.5 μm/s both processes take place at the same "speed" cm between the two surfaces and a downward force of 1 N. The two surfaces were brought into contact. The amount of glycerin transferred into the secondary surface (wafer) was measured and compared favorably with the model predictions (Figure 3.3.3b). These comparisons indicate that the model was fairly accurate in predicting the amount of mass being absorbed into a contacting

porous surface.

is placed on the kitchen tile. Every experiment was repeated three times at an initial gap of 2.5

fabricated in our laboratory from filter paper pulp, with a porosity of 60%, served as the secondary surface. Five separate experiments (three repetitions for each experiment) were conducted with a contact time to be, respectively, 1, 10, 20, 30, and 40 minutes after the droplet is placed on the kitchen tile. Every experiment was repeated three times at an initial gap of 2.5 cm between the two surfaces and a downward force of 1 N. The two surfaces were brought into contact. The amount of glycerin transferred into the secondary surface (wafer) was measured and compared favorably with the model predictions (Figure 27.b). These compari‐ sons indicate that the model was fairly accurate in predicting the amount of mass being

**Figure 25.** The experimental setup: two fused-silica plates connected by a DI water liquid bridge. (a) Plates in fixture

the porous glass used in the experiment

and (*a*\*) a closeup view of the liquid bridge (*b*) shows the porous glass used in the experiment

(a\*)

246 Surface Energy

Figure 3.3.1 The experimental setup: two fused-silica plates connected by a DI water liquid

bridge. (a) Plates in fixture and (*a*\*) a closeup view of the liquid bridge (*b*) shows

In order to provide data for cases where contact between two surfaces results in a chemical reaction in one or both of the contacting surfaces, experiments were conducted measuring the rate of formation of product where one reactant resides on one surface and the other on the second, contacting, surface. Since there are many permutations for these types of experiments, we began our investigation for a case where the primary surface is nonporous, a glass slide, and the contacting surface is porous, an adsorbent pad. The reaction needed to be fast enough so that the product formation could be measured accurately and in a relatively short time span to avoid complications associated with evaporation of chemicals. The reaction chosen was addition of bromine to alkenes, since the progress of this reaction could be monitored both

Figure 3.3.2 200 L of water on glass contacted by porous glass. The approach velocities of

An experiment conducted to measure the amount of mass transfer for two surfaces coming into contact with a liquid droplet in between. A 20 µL glycerin droplet was deposited on

0.5, 1.5, 2.0, and 2.5 m/s are compared with the model. Two additional velocities

of 1.0 and 3.0 m/s are also modeled. For lower velocities the absorption into porous glass is at a faster pace than the spread speed, causing a continuous reduction in radius in time. As the approach velocity increases, the spread will proceed at a faster pace than the absorption, causing a continuous increase of radius in time. It appears that at velocity of 1.5 m/s both processes take place at

absorbed into a contacting porous surface.

the same "speed"

comparison of the model prediction with experimental data for each case, and (*b*) mass transfer of glycerin, that was initially deposited on sand, to a contacting cloth, where the amount transferred was a function of the time elapsed between deposition and the contact, the figure also shows the model prediction for each **Figure 27.** (*a*) Mass transferred to contacting surface as a function of the type of chemical and comparison of the model prediction with experimental data for each case, and (*b*) mass transfer of glycerin, that was initially deposited on sand, to a contacting cloth, where the amount transferred was a function of the time elapsed between deposition and the contact, the figure also shows the model prediction for each case.

Figure 3.3.3 (*a*) Mass transferred to contacting surface as a function of the type of chemical and

quantitatively and qualitatively. A 20 μL of styrene solution was placed on the surface of a glass slide and an adsorbent pad saturated with 1 mL of bromine in carbon tetrachloride solution was placed on top of the slide. The contact was stopped at various time intervals. The reaction was quenched by addition of excess cyclohexene to the pad, after removal from the slide, and adsorbent pad and the glass slide were each extracted with 2 mL of dichloromethane. The solutions were then analyzed by GC/MS and GC/FID. No traces of styrene or the addition case. In order to provide data for cases where contact between two surfaces results in a chemical reaction in one or both of the contacting surfaces, experiments were conducted measuring the rate of formation of product where one reactant resides on one surface and the

47

product were observed on the glass slide even for the smallest time step, indicating complete adsorption into the pad. Figure 28. shows the model/experiment comparison of the product form according to: adsorbent pad and the glass slide were each extracted with 2 mL of dichloromethane. The solutions were then analyzed by GC/MS and GC/FID. No traces of styrene or the addition product were observed on the glass slide even for the smallest time step, indicating complete adsorption into the pad. Figure 3.3.4 shows the model/experiment comparison of the product

quenched by addition of excess cyclohexene to the pad, after removal from the slide, and

other on the second, contacting, surface. Since there are many permutations for these types of experiments, we began our investigation for a case where the primary surface is nonporous, a glass slide, and the contacting surface is porous, an adsorbent pad. The reaction needed to be fast enough so that the product formation could be measured accurately and in a relatively short time span to avoid complications associated with evaporation of chemicals. The reaction chosen was addition of bromine to alkenes, since the progress of this reaction could be monitored both quantitatively and qualitatively. A 20 L of styrene solution was placed on the surface of a glass

The styrene–bromine reaction is as follows: C8H8(L)+Br2(L) = C8H8Br2(L) form according to:

The styrene–bromine reaction is as follows: C8H8(L)+Br2(L) = C8H8Br2(L)

Figure 3.3.4 Product formation in time for the styrene and bromine reaction from the experiment and as predicted by MOCHA-DC and the model showing the styrene droplet just prior to contact with the bromine saturated upper layer **Figure 28.** Product formation in time for the styrene and bromine reaction from the experiment and as predicted by MOCHA-DC and the model showing the styrene droplet just prior to contact with the bromine saturated upper layer

The second experiment is similar to the previous experiment but the glass substrate is replaced with permeable sand. A 60 L styrene droplet was placed on sand and after completely disappearing from the surface was contacted by filtered paper saturated by bromine. The mass exchange occurs at the interface such that the reaction takes place in both media. The results are shown in Table 3.3.1. The comparison is quite satisfactory and the model accurately predicts the mass transfer between the two contacting surfaces and the amount of product in both zones. The second experiment is similar to the previous experiment but the glass substrate is replaced with permeable sand. A 60 μL styrene droplet was placed on sand and after completely disappearing from the surface was contacted by filtered paper saturated by bromine. The mass exchange occurs at the interface such that the reaction takes place in both media. The results are shown in Table 3. The comparison is quite satisfactory and the model accurately predicts the mass transfer between the two contacting surfaces and the amount of product in both zones.


49

**Table 3.** Data and model reactant and product mass comparison in two porous media

In another test case, a 6 μL VX droplet is situated on moist sand (5% saturation). The reaction occurs until a secondary moist porous medium (cloth) is brought into contact with the first surface after about 0.5 hour. The remaining VX at the surface of the sand starts to react with water content in or from the contacting surface. The reaction in both zones continues for about 3 hours. Figure 29. shows the amount of VX left and all products during this process in real time. We do not have any experimental data to compare with these results, however; although there are no data available for this case, it is a very challenging case to run and it demonstrates the robustness and generality of the code.

**Figure 29.** VX on wet sand and contact with another wet porous surface

49

product were observed on the glass slide even for the smallest time step, indicating complete adsorption into the pad. Figure 28. shows the model/experiment comparison of the product

other on the second, contacting, surface. Since there are many permutations for these types of experiments, we began our investigation for a case where the primary surface is nonporous, a glass slide, and the contacting surface is porous, an adsorbent pad. The reaction needed to be fast enough so that the product formation could be measured accurately and in a relatively short time span to avoid complications associated with evaporation of chemicals. The reaction chosen was addition of bromine to alkenes, since the progress of this reaction could be monitored both quantitatively and qualitatively. A 20 L of styrene solution was placed on the surface of a glass slide and an adsorbent pad saturated with 1 mL of bromine in carbon tetrachloride solution was placed on top of the slide. The contact was stopped at various time intervals. The reaction was quenched by addition of excess cyclohexene to the pad, after removal from the slide, and adsorbent pad and the glass slide were each extracted with 2 mL of dichloromethane. The solutions were then analyzed by GC/MS and GC/FID. No traces of styrene or the addition product were observed on the glass slide even for the smallest time step, indicating complete adsorption into the pad. Figure 3.3.4 shows the model/experiment comparison of the product

The styrene–bromine reaction is as follows: C8H8(L)+Br2(L) = C8H8Br2(L)

The styrene–bromine reaction is as follows: C8H8(L)+Br2(L) = C8H8Br2(L)

Figure 3.3.4 Product formation in time for the styrene and bromine reaction from the

**Figure 28.** Product formation in time for the styrene and bromine reaction from the experiment and as predicted by MOCHA-DC and the model showing the styrene droplet just prior to contact with the bromine saturated upper layer

The second experiment is similar to the previous experiment but the glass substrate is replaced with permeable sand. A 60 μL styrene droplet was placed on sand and after completely disappearing from the surface was contacted by filtered paper saturated by bromine. The mass exchange occurs at the interface such that the reaction takes place in both media. The results are shown in Table 3. The comparison is quite satisfactory and the model accurately predicts the mass transfer between the two contacting surfaces and the amount of product in both zones.

with permeable sand. A 60 L styrene droplet was placed on sand and after completely disappearing from the surface was contacted by filtered paper saturated by bromine. The mass exchange occurs at the interface such that the reaction takes place in both media. The results are shown in Table 3.3.1. The comparison is quite satisfactory and the model accurately predicts the mass transfer between the two contacting surfaces and the amount of product in both zones.

Table 3.3.1 Data and model reactant and product mass comparison in two porous media

In another test case, a 6 μL VX droplet is situated on moist sand (5% saturation). The reaction occurs until a secondary moist porous medium (cloth) is brought into contact with the first surface after about 0.5 hour. The remaining VX at the surface of the sand starts to react with water content in or from the contacting surface. The reaction in both zones continues for about 3 hours. Figure 29. shows the amount of VX left and all products during this process in real time. We do not have any experimental data to compare with these results, however; although

MOCHA-DC 44.50 7.12 4.90 1.75 Data 43.74 7.40 5.13 1.56

**Table 3.** Data and model reactant and product mass comparison in two porous media

**Styrene Sand Styrene Pad Product Sand Product Pad** 

**Styrene Sand (mg) Styrene Pad (mg) Product Sand (mg) Product Pad (mg)**

droplet just prior to contact with the bromine saturated upper layer

The second experiment is similar to the previous experiment but the glass substrate is replaced

experiment and as predicted by MOCHA-DC and the model showing the styrene

form according to:

248 Surface Energy

form according to:

In the following test case, liquid and vapor transport through a dual layer fabric is examined when each of the layers has a different permeability. The layer with small permeability (pores of 0.1 μm) is defined hereafter as the outer later and the layer with the larger permeability (pores of 5 μm) is defined as the inner later. The first experiment started with a 40 μL de-ionized water droplet on the inner-layer. Figure 30.a shows the saturation distribution and its migra‐ tion to the outer layer. In the second experiment the droplet was placed on the outer layer and it did not penetrate to the inner layer as seen in Figure 30.b.

Computer simulations were performed with the same conditions. The evaporation module was activated to monitor the vapor concentration as well as liquid saturation. The MOCHA-DC prediction is shown in Figure 31. for the case of a droplet deposited on the low-porosity outer layer. The model indicates the liquid spreads laterally, instead of continuing to spread vertically, when it reaches the interface with the high-porosity inner layer. Due to the smaller pores, the capillary pressure at the outer layer is greater than that of the inner layer. It is seen that this spread pattern virtually eliminates the possibility of liquid passing through the higher porosity inner layer. Since the double layer material is thin, the aspect ratio of vertical direction in these plots is taken to be eight times greater than other directions to make the visuals clearer.

On the other hand, when the droplet is placed on the inner layer as seen from Figures 30a and 32, its preferred direction for movement is vertical when it reaches the interface between the layers, rather than spreading horizontally. In the top image of Figure 32., the droplet is initially placed on the inner layer and gradually is transported vertically, due to capillary pressure, toward the outer layer. The liquid keeps moving upward toward the outer layer saturating the upper portion of the fabric. The fluid therefore travels through both the layers. This behavior would facilitate the transport of fluid, such as sweat, through a garment constructed of such a material. As inner layer fluids reach the outer layer, they might evaporate in the surrounding environment. Conversely, when fluids contact the outer layer they do not reach the inner layer surface, thus preventing contamination of the inner surface. This result suggests it may be possible to eliminate the likelihood of hazardous liquid contact with skin if such a fabric were used for protective clothing.

**Figure 30.** Sessile droplet positioned on the inner layer (a) and the outer layer (b) on a double-layered fabric. The outer layer pore size is order of magnitude smaller compared with the inner layer. (Adapted from Gat et al. 2013.)

The vapor phase also followed the same pattern. Although a small amount of vapor was predicted at the inner layer, it was significantly smaller than the amount of vapor present in the outer layer. Figure 33. shows the distribution of mole fraction of water vapor inside both layers. The vapor inside the bottom layer is one to two orders of magnitude less than the upper layer, implying that the capillary pressure difference between the two layers can be utilized as a tool to prevent volatile chemicals from reaching skin.

Given that the pore size influences capillary pressure, a range of porous glasses were coupled to study the transport behavior. Figure 34. demonstrates the experimental and model predic‐ tion for all combinations of the two pore sizes for the evaporation of deionized water placed on the bottom surface of the dual layer.

A General-Purpose Multiphase/Multispecies Model to Predict the Spread, Percutaneous Hazard, and... http://dx.doi.org/10.5772/60807 251

porosity inner layer. Since the double layer material is thin, the aspect ratio of vertical direction in these plots is taken to be eight times greater than other directions to make the visuals clearer. On the other hand, when the droplet is placed on the inner layer as seen from Figures 30a and 32, its preferred direction for movement is vertical when it reaches the interface between the layers, rather than spreading horizontally. In the top image of Figure 32., the droplet is initially placed on the inner layer and gradually is transported vertically, due to capillary pressure, toward the outer layer. The liquid keeps moving upward toward the outer layer saturating the upper portion of the fabric. The fluid therefore travels through both the layers. This behavior would facilitate the transport of fluid, such as sweat, through a garment constructed of such a material. As inner layer fluids reach the outer layer, they might evaporate in the surrounding environment. Conversely, when fluids contact the outer layer they do not reach the inner layer surface, thus preventing contamination of the inner surface. This result suggests it may be possible to eliminate the likelihood of hazardous liquid contact with skin if such a

**Figure 30.** Sessile droplet positioned on the inner layer (a) and the outer layer (b) on a double-layered fabric. The outer

The vapor phase also followed the same pattern. Although a small amount of vapor was predicted at the inner layer, it was significantly smaller than the amount of vapor present in the outer layer. Figure 33. shows the distribution of mole fraction of water vapor inside both layers. The vapor inside the bottom layer is one to two orders of magnitude less than the upper layer, implying that the capillary pressure difference between the two layers can be utilized

Given that the pore size influences capillary pressure, a range of porous glasses were coupled to study the transport behavior. Figure 34. demonstrates the experimental and model predic‐ tion for all combinations of the two pore sizes for the evaporation of deionized water placed

layer pore size is order of magnitude smaller compared with the inner layer. (Adapted from Gat et al. 2013.)

as a tool to prevent volatile chemicals from reaching skin.

on the bottom surface of the dual layer.

fabric were used for protective clothing.

250 Surface Energy

**Figure 31.** Liquid transport and spread from the upper layer to the lower layer in the model. This time series shows the saturation contours in the center plane of the domain as the droplet is absorbed and moves and spreads laterally, but not into the lower layer of the dual layer fabric. The saturation within the lower layer remains zero.

**Figure 32.** Liquid transport and spread from the lower layer to the upper layer as progressed in time. The figures show the saturation contours at the center plane of the domain. The top picture is the initial condition and as the droplet is absorbed it moves toward the upper layer rather immediately and then spread laterally.

**Figure 33.** Image of the vapor transport and spread in the upper and lower layers when a droplet is placed on the upper surface. The vapor phase demonstrates a behavior similar to that of the liquid phase. The droplet is 40 μL water. The evaporation occurs somewhat faster at the edges of the wetted region. The mole fraction of the vapor phase is shown.

**Figure 34.** Evaporation vs. time for 200 μL ethanol droplet, positioned on the inner layer, into various combinations of nylon membranes (Scientific Tisch, TM) with 5 and 0.1 μm average pore size. Both model and experiment predict smaller evaporation time for the smaller inner pore size. For all cases the droplet was placed on the lower surface.

When the outer (upper) layer's pore size is an order of magnitude smaller than the inner (lower) layer's, the time for evaporation of a droplet on the lower surface was reduced by about ≈30−40%. This suggests that such a combination can be used to increase the liquid evaporation rate from the inner side of a protective fabric so as to enhance the pass through of perspiration. For the smallest outer layer pore size, the model predicts somewhat slower evaporation. This could be due to the mesh size, which may need to be refined for faster processes. However, since the results are fairly close to the experimental values, we decided not to repeat this calculation and emphasize the fact that there are a number of values to the dual layer fabric design that make it suitable for a variety of applications.

## **4. Nomenclature**

**4.1. General** *As Surface area a Pre-exponential coefficient b Activation energy C Mass fraction cp Specific heat at constant pressure D Diffusion coefficient* **E, F, H***Flux vectors in x, y, and z directions F Force f Gibb's Free Energy f(\*) Unknown quantity for capillary pressure* **G***Source term vector g Gravitational acceleration h Height of apex of a sessile droplet h* ¯*Molar enthalpy h\* Local height of the droplet H Enthalpy hfg Latent heat of evaporation H Maximum vertical distance for boundary layer measurements J Jacobian for the transformation K Saturated permeability for isotropic medium*

**Figure 34.** Evaporation vs. time for 200 μL ethanol droplet, positioned on the inner layer, into various combinations of nylon membranes (Scientific Tisch, TM) with 5 and 0.1 μm average pore size. Both model and experiment predict smaller evaporation time for the smaller inner pore size. For all cases the droplet was placed on the lower surface.

**Figure 33.** Image of the vapor transport and spread in the upper and lower layers when a droplet is placed on the upper surface. The vapor phase demonstrates a behavior similar to that of the liquid phase. The droplet is 40 μL water. The evaporation occurs somewhat faster at the edges of the wetted region. The mole fraction of the vapor phase is

shown.

252 Surface Energy

*Keql Equilibrium reaction rate k Reaction rate keff Effective conductivity kg*, *k*ℓ*Relative permeability in gas or liquid phases M Molecular weight Molar concentration m Mass n Index of refraction P Pressure Pv Vapor pressure* **q˙***Heat flux r Wetted area radius R or Rs Radius of curvature Re Reynolds number,u* <sup>∗</sup>*R* / *ν S Entropy s Saturation, fraction of the pore volume occupied by a phase (or state) Sc Schmidt number t Time T Temperature* **U***Conserved variables vector u\* Friction velocity V Velocity v and v' Stoichiometric coefficients υ*˜*Contra-variant vertical velocity u*, *υ*, *wVelocity components x, y, z Cartesian coordinate system X Net rate of change in molar concentration* **Greek Symbols** *α Thermal diffusivity*

*β Angle defined in*Figure 12. *ϕ Porosity λ* or *λr h/r λ<sup>j</sup>* or *λi<sup>j</sup> Reaction order μ Viscosity ρ Density ρ*˙*Rate of phase change σ Surface tension τ Shear stress* ∀*Volume ω Species production/destruction rate Contact angle a Contact angle at which the evaporation topology changes Contact angle inside the pores ξ, η, ζ Computational domain coordinates Subscripts A Air b Backward c Capillary eql Equilibrium f Forward fg Fluid to gas (evaporation) g Related to the gaseous phase i Liquid phase constituent j Gas phase constituent k Solid phase constituent* ℓ*, liq Related to the liquid phase m*, *n*, ℓ*Refer to xyz for shear stress tensor M Related to mass transfer*

*i*

*Keql Equilibrium reaction rate*

*kg*, *k*ℓ*Relative permeability in gas or liquid phases*

*keff Effective conductivity*

*M Molecular weight Molar concentration*

*n Index of refraction*

*Pv Vapor pressure*

*r Wetted area radius*

*Sc Schmidt number*

 *Friction velocity*

*T Temperature*

*R or Rs Radius of curvature*

*Re Reynolds number,u* <sup>∗</sup>*R* / *ν*

**U***Conserved variables vector*

*v and v' Stoichiometric coefficients υ*˜*Contra-variant vertical velocity*

*x, y, z Cartesian coordinate system*

*X Net rate of change in molar concentration*

*u*, *υ*, *wVelocity components*

**Greek Symbols**

*α Thermal diffusivity*

*s Saturation, fraction of the pore volume occupied by a phase (or state)*

*m Mass*

*P Pressure*

**q˙***Heat flux*

*S Entropy*

*t Time*

*u\**

*V Velocity*

*k Reaction rate*

254 Surface Energy

*n Temperature dependence o Initial pt Partial torus r Relative s Surface of the droplet, or related to the solid phase s* −ℓ*Solid to liquid phase change* ℓ− *gLiquid to gas phase change s* − *gSolid to gas phase change T Related to heat transfer vap Vapor phase* ∀*Volume w At the wall ∞Far field – Free stream Superscripts ~ Vector UNI Unidirectional per unit time*

*mix Mixture of liquid and gas phase*

## **Acknowledgements**

This project was collectively supported by The Air Force Research Laboratory, Human Effectiveness Directorate, Biosciences and Protection Division, Wright-Patterson AFB, US Army's Edgewood Chemical and Biological Center under contract, and The Defense Threat Reduction Agency (DTRA) under the contract HDTRA1-10-C-0064. The authors wish to thank all the above agencies and our CORs Drs. William Ginely, Sari Paikoff, and Brian Pate for their support and guiding this effort. Special thank also goes to Mr. Joseph B. Kiple of KASTLE Corporation for his involvement with the open air testing and sharing his experiences and vision with the entire team and Dr. D'Onofrio of ECBC for providing some of test data. We also appreciate Dr. Miroslav Skoumal and his team in Czech Republic for participating and sharing their state-of-the art open air testing facilities and their laboratories. A final word of thanks goes to the many Kettering University students who helped with the experimental and numerical studies.

## **Author details**

*mix Mixture of liquid and gas phase*

*s Surface of the droplet, or related to the solid phase*

*n Temperature dependence*

*s* −ℓ*Solid to liquid phase change*

ℓ− *gLiquid to gas phase change*

*s* − *gSolid to gas phase change*

*T Related to heat transfer*

*∞Far field – Free stream*

*UNI Unidirectional*

numerical studies.

**Acknowledgements**

This project was collectively supported by The Air Force Research Laboratory, Human Effectiveness Directorate, Biosciences and Protection Division, Wright-Patterson AFB, US Army's Edgewood Chemical and Biological Center under contract, and The Defense Threat Reduction Agency (DTRA) under the contract HDTRA1-10-C-0064. The authors wish to thank all the above agencies and our CORs Drs. William Ginely, Sari Paikoff, and Brian Pate for their support and guiding this effort. Special thank also goes to Mr. Joseph B. Kiple of KASTLE Corporation for his involvement with the open air testing and sharing his experiences and vision with the entire team and Dr. D'Onofrio of ECBC for providing some of test data. We also appreciate Dr. Miroslav Skoumal and his team in Czech Republic for participating and sharing their state-of-the art open air testing facilities and their laboratories. A final word of thanks goes to the many Kettering University students who helped with the experimental and

*vap Vapor phase*

∀*Volume*

*w At the wall*

*Superscripts*

*per unit time*

*~ Vector*

*o Initial*

256 Surface Energy

*r Relative*

*pt Partial torus*

Navaz Homayun1\*, Zand Ali1 , Gat Amir2 and Atkinson Theresa1

\*Address all correspondence to: hnavaz@kettering.edu

1 Kettering University, Flint, USA

2 Technion – Israel Institute of Technology, Haifa, Israel

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**Section 3**
