**2.1. Results**

mocouples were inserted but at lower depth, 2.2 cm from the surface, to measure the bottom

**Figure 2.** Details of (a) Wind tunnel: 1. smoothing grid; 2. observation port; 3.thermocouple socket; 4. joint, (b) Glass

tray (clear) and flat plate (gray): 1. the leading edge; 2. thermocouple socket; 3. thermocouples.

temperature distribution.

168 Wind Tunnel Designs and Their Diverse Engineering Applications

A bed of the desert sand was subjected to forced convective drying at 84 o C. The wet-bulb temperature of the air was 35.5 o C. The resulting surface temperature vs. time at different distances from the leading is shown in Figure 3, which demonstrates clearly the stages of the drying process; i.e. the pre-constant rate period, the constant rate period, and the falling rate period.

The temperature of the bed rose from ambient temperature 23º C to a steady value at time = 70 minute. This initial period, termed the pre-constant rate period, is usually short. The sur‐ face temperatures remained constant for a period of 140 minutes, indicating the constant rate period. The surface temperature at the distance x=1cm, from the leading edge was 36 o C. This temperature was greater than that at x=50 cm and at x= 100 cm by 1 o C and 2 o C respec‐ tively. The surface temperatures were close to the wet-bulb temperature.

During the constant rate period, the surface of the solid is so wet that a continuous film of water exists on the drying surface. This water is entirely unbound water and exerts a vapour pressure equal to that of pure water at the same temperature. The rate of moisture move‐ ment within the solid is sufficient to maintain a saturated condition at the surface.

**Figure 3.** Temperature distribution profile of the surface for the desert sand bed. Figure 3

At a specific point, *t* = 240 minutes, the surface temperature at all positions rose gradually, indicating the end of the constant-rate period and the beginning of the falling-rate period. In the falling-rate period, there was insufficient water on the surface to maintain a continuous film. The entire surface was no longer wetted and dry patches began to form. The surface temperature continued to rise for a longer time until it approximated to the air temperature. A thin dried layer appeared on the entire surface.

The temperatures at the surface and the bottom at different distances from the leading edge are shown in Figure 4. The profiles show that when the surface became dry, the bottom re‐ mained wet at a constant temperature for a longer time than that for the surface. During the falling-rate periods a receding evaporation front divided the system into a hotter, dry zone near the surface and a wet zone towards the bottom of the sample [22]. The evaporation plane receded from the surface toward the bottom. The temperatures then rose quickly when the dry zone extended throughout the bed.

At a specific point, *t* = 240 minutes, the surface temperature at all positions rose gradually, indicating the end of the constant-rate period and the beginning of the falling-rate period. In the falling-rate period, there was insufficient water on the surface to maintain a continuous film. The entire surface was no longer wetted and dry patches began to form. The surface temperature continued to rise for a longer time until it approximated to the air temperature.

**0 100 200 300 400 500 600 700**

**++ 1 cm \*\* 50 cm xx 100 cm**

**Time / minutes**

The temperatures at the surface and the bottom at different distances from the leading edge are shown in Figure 4. The profiles show that when the surface became dry, the bottom re‐ mained wet at a constant temperature for a longer time than that for the surface. During the falling-rate periods a receding evaporation front divided the system into a hotter, dry zone near the surface and a wet zone towards the bottom of the sample [22]. The evaporation plane receded from the surface toward the bottom. The temperatures then rose quickly

A thin dried layer appeared on the entire surface.

**Figure 3.** Temperature distribution profile of the surface for the desert sand bed. Figure 3

**0**

**10**

**20**

**30**

**40**

**50**

**Temprature / oC**

**60**

**70**

**80**

**90**

170 Wind Tunnel Designs and Their Diverse Engineering Applications

when the dry zone extended throughout the bed.

Figure 4 **Figure 4.** Temperatures of surface and bottom for the desert sand bed at distance of 1 cm, 50 cm and 100 cm from the leading edge, respectively, from the top.

A bed of the beach sand was dried at an air temperature of 83 o C. The wet bulb temperature was 35 o C. The temperature distribution profile at different distances from the leading edge is shown in Figure 5. The stages of the drying process can easily be recognized from the tem‐ perature profile. However, the surface temperature took longer to approach to the air tem‐ perature.

Figure 5 **Figure 5.** Temperature distribution profile of the surface for the beach sand bed).

Figure 6 shows the surface and the bottom temperatures at distances from the leading edge of 1 cm, 50 cm 100 cm, respectively. It can be seen that the surface and bottom temperatures increased rapidly at some times and decreased others. Also, the temperature profile did not increase gradually like that of the desert sand. This can be attributed to the nature of the beach sand, which comprises different types of small shells of various shapes, and contain tiny hollows. The trapped water in these hollows forms small bubbles which can explode with increasing temperature. Therefore the temperature of the bed changes suddenly at such times. The temperature at the bottom of the bed indicated it remained wet during the falling-rate period.

A bed of the beach sand was dried at an air temperature of 83 o

172 Wind Tunnel Designs and Their Diverse Engineering Applications

C. The temperature distribution profile at different distances from the leading edge

**0 100 200 300 400 500 600 700**

**++ 1 cm \*\* 50 cm xx 100 cm**

**Time / minutes**

Figure 6 shows the surface and the bottom temperatures at distances from the leading edge of 1 cm, 50 cm 100 cm, respectively. It can be seen that the surface and bottom temperatures increased rapidly at some times and decreased others. Also, the temperature profile did not increase gradually like that of the desert sand. This can be attributed to the nature of the beach sand, which comprises different types of small shells of various shapes, and contain tiny hollows. The trapped water in these hollows forms small bubbles which can explode with increasing temperature. Therefore the temperature of the bed changes suddenly at such times. The temperature at the bottom of the bed indicated it remained wet during the

Figure 5 **Figure 5.** Temperature distribution profile of the surface for the beach sand bed).

is shown in Figure 5. The stages of the drying process can easily be recognized from the tem‐ perature profile. However, the surface temperature took longer to approach to the air tem‐

was 35 o

perature.

**0**

falling-rate period.

**10**

**20**

**30**

**40**

**50**

**Temprature / oC**

**60**

**70**

**80**

**90**

C. The wet bulb temperature

**Figure 6.** Temperatures of surface and bottom for the beach sand bed at distance of 1 cm, 50 cm and 100 cm from Figure 6 the leading edge, respectively, from the top.

A similar investigation was carried out for a bed of the glass beads at an air temperature of 84 o C. The wet bulb temperature was 37 o C. Figure 7 shows temperature versus time at dif‐ ferent distances from the leading edge of the bed surface. The temperature distribution pro‐ file again illustrates clearly the stages of the drying process. The results for the surface and bottom temperatures for the glass beads were almost similar to those for the desert sand as shown in Figure 4.

Figure 7 **Figure 7.** Temperature distribution profile of the surface for the glass-bead bed.

#### **2.2. Discussion**

A thin film adjacent to the surface always exists when a forced flow passes over a flat plate and forms what is called a hydrodynamic boundary layer. The influence of the surface tem‐ perature reaches deeper into the fluid, thus causing the formation of a thermal boundary layer. It is well known, the thickness of the thermal boundary layer increases with increas‐ ing distance from the leading edge. This layer is affected by the geometry of the system, roughness of the surface and the fluid properties.

For the case in which the heated section is preceded by an unheated straight length, the local Nusselt number (*Nu*x) is represented in [23-25] as:

A similar investigation was carried out for a bed of the glass beads at an air temperature of

ferent distances from the leading edge of the bed surface. The temperature distribution pro‐ file again illustrates clearly the stages of the drying process. The results for the surface and bottom temperatures for the glass beads were almost similar to those for the desert sand as

**0 100 200 300 400 500 600 700**

**++ 1 cm**

**\*\* 50 cm**

**xx 100 cm**

**Time/ minutes**

A thin film adjacent to the surface always exists when a forced flow passes over a flat plate and forms what is called a hydrodynamic boundary layer. The influence of the surface tem‐ perature reaches deeper into the fluid, thus causing the formation of a thermal boundary layer. It is well known, the thickness of the thermal boundary layer increases with increas‐ ing distance from the leading edge. This layer is affected by the geometry of the system,

Figure 7 **Figure 7.** Temperature distribution profile of the surface for the glass-bead bed.

roughness of the surface and the fluid properties.

C. Figure 7 shows temperature versus time at dif‐

84 o

shown in Figure 4.

**0**

**2.2. Discussion**

**10**

**20**

**30**

**40**

**50**

**Temprature/ oC**

**60**

**70**

**80**

**90**

C. The wet bulb temperature was 37 o

174 Wind Tunnel Designs and Their Diverse Engineering Applications

$$\mathrm{Nu}\_{\mathrm{x}} = \frac{\mathrm{h}\,\mathrm{x}}{\mathrm{k}} = \frac{0.323 \,\mathrm{(Re\_{x})}^{0.5} \mathrm{(Pr)}^{1/3}}{[1 - (\mathrm{x\_{o}}/\mathrm{x})^{3/4}]^{1/3}} \tag{1}$$

where *Rex* is Reynolds number with respect to length and *x* is the length of the flat plate in (m).

Figure 8 shows a plot of variation in the local heat transfer coefficient versus the distance from the leading edge. The plot indicates that the values of the coefficient decreased signifi‐ cantly when *x* increased from the leading edge, and then it remains virtually constant for large *x* values. A plot of variation of the mass transfer coefficient could be expected to be almost similar to that in Figure 8 because of the similarity in the transport coefficient equa‐ tions. This result with the concept of the boundary layer thickness demonstrates that resist‐ ance to heat and mass transfer to, or from, the bed increases with increasing the distance from the leading edge.

**Figure 8.** Local heat transfer coefficient vs. distance from the leading edge.

A model proposed in a previous paper [26] was modified to find a method for prediction of surface temperature distribution. The equation of energy can be represented as:

$$\text{Q cp } \frac{\partial \text{ T}}{\partial \text{ t}} = \nabla(\text{k} \,\text{VT}) + \nabla(\frac{\text{D}\_{\text{v}} \text{M}}{\text{RT}} \nabla \text{P}\_{\text{v}}) \Delta \text{hv} \tag{2}$$

At time zero, the whole body has a uniform initial temperature of *T*o, and the initial condi‐ tions are:

$$\mathbf{T\_O = T \big|\_{\mathbf{y} = 0} = T \big|\_{\mathbf{y} = \mathbf{h}}} \tag{3}$$

At the external surface, i.e. *y*=0, the boundary conditions can be written on the basis of Fig‐ ure 9 as:

$$-\mathbf{k}\,\frac{\partial}{\partial\mathbf{y}}\mathbf{\hat{u}}=\mathbf{h}\,(\mathbf{T\_a}-\mathbf{T\_{sx}})+\mathbf{\hat{m}}\,\Delta\mathbf{h}\mathbf{v}\tag{4}$$

where Tsx is the surface temperature at distance *x* from the leading edge.

The drying rate, m• varies with the time and can be defined as,

$$\stackrel{\bullet}{\rm m} = \frac{\rm M \, K\_{\rm G}^{\prime}}{\rm RT} (\rm P\_{\rm Sr} - \rm P\_{\rm V}) \tag{5}$$

where *K* ′ *<sup>G</sup>* is an overall mass transfer coefficient, defined in [27] as:

$$\frac{1}{\mathbf{K}\_{\mathbf{G}}^{\prime}} = \frac{1}{\mathbf{k}\_{\mathbf{c}}^{\prime}} + \frac{\mathbf{z}^{\prime}}{\mathbf{D}\_{\mathbf{eff}}} \tag{6}$$

where k<sup>c</sup> ' is the local mass transfer coefficient (m s-1), z" is the distance from the plate surface to the receding evaporation front in (m).

The boundary condition at *y* = h is:

$$-\mathbf{k} \,\, \frac{\partial \mathbf{T}}{\partial \mathbf{y}} = \mathbf{0} \tag{7}$$

Equation (2), with such boundary conditions, was solved by a finite difference method (a modified form the so-called explicit method). Therefore, the temperature distribution on the surface (i.e. *y* = 0) was calculated by using the model at different local points, *x.* Table 1 shows the physical properties of the sample materials.

**Figure 9.** Heat and mass transfer process for the wet material. NA: mass transfer flux and q: heat transfer flux.)

A model proposed in a previous paper [26] was modified to find a method for prediction of

At time zero, the whole body has a uniform initial temperature of *T*o, and the initial condi‐

At the external surface, i.e. *y*=0, the boundary conditions can be written on the basis of Fig‐

=h (T<sup>a</sup> <sup>−</sup>*Tsx*) + m•

( ) M K m PP G sr v R T

> 1 1 z" K kD <sup>G</sup> <sup>c</sup> eff

> > <sup>T</sup> k 0 y ¶

Equation (2), with such boundary conditions, was solved by a finite difference method (a modified form the so-called explicit method). Therefore, the temperature distribution on the surface (i.e. *y* = 0) was calculated by using the model at different local points, *x.* Table 1

¶

is the local mass transfer coefficient (m s-1), z" is the distance from the plate surface

v

v

=Ñ Ñ +Ñ Ñ (2)

TT T <sup>o</sup> y yh = = = = (3)

· ¢ <sup>=</sup> - (5)

= + ¢ ¢ (6)


Δhv (4)

surface temperature distribution. The equation of energy can be represented as:

¶

176 Wind Tunnel Designs and Their Diverse Engineering Applications

¶

−k <sup>∂</sup> <sup>T</sup> ∂ y

where Tsx is the surface temperature at distance *x* from the leading edge.

varies with the time and can be defined as,

*<sup>G</sup>* is an overall mass transfer coefficient, defined in [27] as:

tions are:

ure 9 as:

The drying rate, m•

where *K* ′

where k<sup>c</sup> '

to the receding evaporation front in (m).

shows the physical properties of the sample materials.

The boundary condition at *y* = h is:

<sup>T</sup> D M ρ cp (k T) ( P )Δhv <sup>t</sup> R T

0

Figure 10 shows the experimental and the predicted surface temperature distributions for the desert sand along the bed at various times. The predicted temperatures were in good agreement with the experimental results. From the graph, it can be seen that there is a signif‐ icant difference in the surface temperature between 0.1 m and 1 m. At time=195 minutes; i.e. during the constant-rate period, the difference in temperature was 2 o C.

During the falling-rate period, the difference in the surface temperature between 0.1 m and 1 m increased. At 510 minutes, the difference is 11 oC. At the bottom of the sand bed, the dif‐ ference in temperature along the bed also can be seen clearly (Figure 4).

**Figure 10.** Experimental and predicted surface temperatures of the desert sand bed; experimental results (symbols); predicted results (solid lines).

For the entire time of the experiment, the surface and the bottom temperatures decreased gradually with increasing distance from the leading edge. This caused by the resistance to heat transfer process which increased with increasing thickness of thermal boundary layer. In contrast, near the leading edge, the resistance to heat transfer diminishes, since the thick‐ ness of the thermal boundary layer in the vicinity of the surface thins. Therefore, the rate of heat transfer to the body increased, thereby raising the temperature of surface. Afterwards, heat transfer by conduction across the solid particles raises the temperature of the bed, and the portion closest to the leading edge dries faster than that at a greater distance.

Figure 11 shows both predicted and experimental results for the bed of beach sand. This Fig‐ ure shows that the computed temperature distribution was in general agreement with ex‐ perimental results. However, unsatisfactory results can be seen at times greater than 400 minutes, i.e. during the falling-rate period. This is due to the nature of the beach sand as dis‐ cussed before. The surface temperature of the beach sand sample decreased gradually with increasing distance from the leading edge. A significant difference in temperature between 0.1 m and 1 m can be seen clearly for both constant and falling-rate periods. During the con‐ stant-rate period, the difference in temperature was 2 o C, whereas in the falling-rate period the difference reached to 8 o C.

**Figure 11.** Experimental and predicted surface temperatures of the beach sand bed; experimental results (symbols); predicted results (solid lines).


**Table 1.** Physical properties for desert sand and beach sand.

For the entire time of the experiment, the surface and the bottom temperatures decreased gradually with increasing distance from the leading edge. This caused by the resistance to heat transfer process which increased with increasing thickness of thermal boundary layer. In contrast, near the leading edge, the resistance to heat transfer diminishes, since the thick‐ ness of the thermal boundary layer in the vicinity of the surface thins. Therefore, the rate of heat transfer to the body increased, thereby raising the temperature of surface. Afterwards, heat transfer by conduction across the solid particles raises the temperature of the bed, and

Figure 11 shows both predicted and experimental results for the bed of beach sand. This Fig‐ ure shows that the computed temperature distribution was in general agreement with ex‐ perimental results. However, unsatisfactory results can be seen at times greater than 400 minutes, i.e. during the falling-rate period. This is due to the nature of the beach sand as dis‐ cussed before. The surface temperature of the beach sand sample decreased gradually with increasing distance from the leading edge. A significant difference in temperature between 0.1 m and 1 m can be seen clearly for both constant and falling-rate periods. During the con‐

**Figure 11.** Experimental and predicted surface temperatures of the beach sand bed; experimental results (symbols);

C, whereas in the falling-rate period

the portion closest to the leading edge dries faster than that at a greater distance.

stant-rate period, the difference in temperature was 2 o

178 Wind Tunnel Designs and Their Diverse Engineering Applications

C.

the difference reached to 8 o

predicted results (solid lines).

We have found that the temperature distribution profiles determined for the flat beds of desert sand, beach sand and glass beads identified clearly the stages of drying. The tempera‐ ture profiles in general were almost similar. However, the beach sand profile showed irreg‐ ularity in temperature, due to the nature of the beach sand which contains a lot of shells with various shapes.

The temperature profiles also showed that when the whole surface of the bed became dry, the whole bottom of the bed remained wet. During the falling-rate periods, a receding evap‐ oration front divided the system into a hotter, dry zone near the surface and a wet zone to‐ wards the bottom of the sample.

The predicted transport coefficients have very large values close to the leading edge, where the thickness of the boundary layer approaches zero. In contrast, the values of the coeffi‐ cients decrease progressively with increased distance from the leading edge, where the boundary layer thickens. Hence the resistance to heat and mass transfer to, or from, the sur‐ face also increases. These variations in thickness and resistance have a significant effect on the temperature distribution along the bed and the drying rate.

A mathematical model has been modified to predict temperature distributions along the bed at various times. The model was compared with the experimental results for various beds and good agreement was obtained. We found that surface and bottom temperatures de‐ creased gradually with increasing distance from the leading edge, and the difference in tem‐ perature became clearer during the falling-rate period. The difference in the surface temperature was 11 o C for the case of desert sand, and was 8 o C for the case of beach sand. We concluded that the portion close to the leading edge dried faster than that at larger dis‐ tance, since the resistance to heat and mass transfer diminishes at that position.
