**8. Packed bed height**

Differential form of contact is a usual method to contact phases in different separation processes. High surface area for contacting phases, less pressure drop, simplicity and low cost are the factors that cause this type of mass exchanger to be more useful. Height and number of transfer units are of important parameters that need to be calculated in packed bed contactors. During contact between phases along the height of the tower, concentration of separating component in two phases changes continuously and this type of mass transfer changes the rate of two streams as well. Sometimes transport of mass is occurred in the case of transfer into stagnant components e. g. absorption of ammonia from air with water, and sometimes it occurred as equimolar countertransfer e. g. in distillation process.

Fig. 19. A schematic of packed bed tower (left) and an element of tower in z-direction (right)

Mass Transfer - The Skeleton of Purification Processes 803

In the case when transfer of component through stagnant components takes place and for low mass transfer rate, the K coefficients should be used to define flux relation and equation

A yA A A A

(1 y ) (1 y ) (1 y ) 1 y ln

Indeed when component A transfers through stagnant components, e.g. absorbing ammonia from air using water, flow rate L cannot be assumed as a constant and it can be formulated

> s A

<sup>L</sup> <sup>L</sup> 1 y

where Ls is part of the light phase flow rate that is nontransferable or stagnant. Replacing new values for NA and L from equations (36) and (38), respectively, into equation (29), other

When the rate of mass transfer is high, using F coefficient as mass transfer coefficient is usual and other form of definition of NA is needed. The governing equations in this case

In the present chapter it tried to figure out a mass transfer sense related to more useful purification processes in chemical industries. A common sense in different processes was obtained based on classifying all separation processes into two broad categories of absorption-like and desorption-like processes. This classification can help the reader to get a similar sense when see separation processes individually. Referring denser phase as a heavy phase in comparison with the second phase causes the processing calculations to be simplified and categorized easily. The basic concepts of mass transfer were used to describe what happened in the interface between two phases and a brief discussion on the phase

Also it tried to bring a good summary on significant processing calculations in different contact forms of phases and finally the calculations on two mass exchangers of tray towers

Geankoplis, C. J.; (2003). *Transport Processes And Separation Process Principles*, New Jersey,

Henley, E. J.; & Seader, J. D. (1981). *Equilibrium-Stage Separation Operations In Chemical* 

− −− − = <sup>−</sup>

A \*M

relations are produced for height of packed bed and number of transfer units.

have been presented in ref. (Hines & Maddox, 1985).

equilibria was used to generate equilibrium curves.

and packed-bed columns were presented briefly.

*Engineering*, John Wiley & Sons.

Prentice Hall PTR.

'

K

\* \* y

A \*M

\* A A

1 y

−

A \* A

N K (y y ) (y y ) (1 y ) = −= − <sup>−</sup> (36)

(37)

<sup>=</sup> <sup>−</sup> (38)

(31) should be replaced with

where

as

**9. Conclusion** 

**10. References** 

A schematic of packed bed tower is shown in Fig. 19. The two phases of L and H flow inside the tower in countercurrent form. L and H show the flow rates as kmol/(time)(tower cross sectional area). The mass balance equation on one element, dZ from the tower as shown in RHS of Fig. 19 can be simplified as

$$\mathbf{d(L\mathbf{y}\_A) = d(H\mathbf{x}\_A)}\tag{28}$$

The mass balance equation on phase-L can be written as well

$$-\mathbf{N\_{A}dS} = \mathbf{d(Ly\_{A})} \tag{29}$$

where dS is the total surface area that mass can be transferred from it, in dZ element, per tower cross sectional area.

In packed beds dS can be calculated from its relation with packing specific area, a and dZ as

$$\mathbf{dS} = \mathbf{a} \,\mathrm{d}\mathbf{Z} \tag{30}$$

The packing specific area is defined as total surface area of packing per volume of tower. A proper relation can be written for flux of component A e.g. in the case of equimolar countertransfer in low mass transfer rate based on overall mass transfer coefficient in light phase the following relation can be written

$$\mathbf{N\_A} = \mathbf{K\_y'}(\mathbf{y\_A} - \mathbf{y\_A'}) \tag{31}$$

The flow rate L is assumed to be constant in the case of equimolar countertransfer. Replacing dS from equation (30) and NA from equation (31) into equation (29) and considering L as a constant

$$-\mathbf{K}\_{\mathbf{y}}(\mathbf{y}\_{\text{A}} - \mathbf{y}\_{\text{A}}^{\bullet})\,\text{a}\,\text{d}\mathbf{Z} = \text{L}\,\text{dy}\_{\text{A}}\tag{32}$$

By separating the variables and integration the following equation is obtained

$$\mathbf{Z} = \oint\_{0} \mathbf{d} \mathbf{Z} = \int\_{\mathbf{y}\_{\text{A1}}}^{\mathbf{y}\_{\text{A2}}} \frac{\mathbf{L}}{\mathbf{K}\_{\text{y}}^{\ast} \mathbf{a}} \frac{\mathbf{d} \mathbf{y}\_{\text{A}}}{(\mathbf{y}\_{\text{A}}^{\bullet} - \mathbf{y}\_{\text{A}})} \tag{33}$$

If ' H L/K a OL y <sup>=</sup> and is defined as height of the transfer units for the overall mass transfer coefficient and is assumed to be constant, the following equation is obtained from equation (33)

$$Z = \mathbf{H}\_{\text{OL}} \int\_{\mathbf{y}\_{\text{Al}}}^{\mathbf{y}\_{\text{A}2}} \frac{\text{dy}\_{\text{A}}}{\text{y}\_{\text{A}}^{\bullet} - \text{y}\_{\text{A}}} \tag{34}$$

The value of integral is defined as the number of transfer units, NOL as

$$\mathbf{N\_{OL}} = \int\_{\mathbf{y\_{Al}}}^{\mathbf{y\_{A2}}} \frac{\mathbf{dy\_{A}}}{\mathbf{y\_{A}^\* - \mathbf{y\_{A}}}} \tag{35}$$

In the case when transfer of component through stagnant components takes place and for low mass transfer rate, the K coefficients should be used to define flux relation and equation (31) should be replaced with

$$\mathbf{N\_A} = \mathbf{K\_y}(\mathbf{y\_A} - \mathbf{y\_A^\*}) = \frac{\mathbf{K\_y^\prime}}{(\mathbf{l} - \mathbf{y\_A}) \mathbf{\ast\_M}}(\mathbf{y\_A} - \mathbf{y\_A^\*}) \tag{36}$$

where

802 Mass Transfer - Advanced Aspects

A schematic of packed bed tower is shown in Fig. 19. The two phases of L and H flow inside the tower in countercurrent form. L and H show the flow rates as kmol/(time)(tower cross sectional area). The mass balance equation on one element, dZ from the tower as shown in

where dS is the total surface area that mass can be transferred from it, in dZ element, per

In packed beds dS can be calculated from its relation with packing specific area, a and dZ as

The packing specific area is defined as total surface area of packing per volume of tower. A proper relation can be written for flux of component A e.g. in the case of equimolar countertransfer in low mass transfer rate based on overall mass transfer coefficient in light

The flow rate L is assumed to be constant in the case of equimolar countertransfer. Replacing dS from equation (30) and NA from equation (31) into equation (29) and

By separating the variables and integration the following equation is obtained

Z y

A2

L dy Z dZ

A1

If ' H L/K a OL y <sup>=</sup> and is defined as height of the transfer units for the overall mass transfer coefficient and is assumed to be constant, the following equation is obtained from equation

A2

y

A1

OL \*

y dy Z H

A2

y

OL \*

y dy <sup>N</sup>

A1

The value of integral is defined as the number of transfer units, NOL as

d(Ly ) d(Hx ) A A = (28)

−N dS d(Ly ) A A = (29)

dS a dZ = (30)

' \* N K (y y ) A yA A = − (31)

' \* −− = K (y y )a dZ Ldy yA A <sup>A</sup> (32)

K a (y y ) = = <sup>−</sup> ∫ ∫ (33)

y y <sup>=</sup> <sup>−</sup> ∫ (34)

y y <sup>=</sup> <sup>−</sup> ∫ (35)

A

' \* y AA 0 y

A

A A

A

A A

RHS of Fig. 19 can be simplified as

tower cross sectional area.

considering L as a constant

(33)

phase the following relation can be written

The mass balance equation on phase-L can be written as well

$$(\mathbf{l} - \mathbf{y}\_{\mathcal{A}})\_{\ast \mathcal{M}} = \frac{(\mathbf{l} - \mathbf{y}\_{\mathcal{A}}) - (\mathbf{l} - \mathbf{y}\_{\mathcal{A}}^{\star})}{\ln \frac{\mathbf{l} - \mathbf{y}\_{\mathcal{A}}}{\mathbf{l} - \mathbf{y}\_{\mathcal{A}}^{\star}}} \tag{37}$$

Indeed when component A transfers through stagnant components, e.g. absorbing ammonia from air using water, flow rate L cannot be assumed as a constant and it can be formulated as

$$\mathbf{L} = \frac{\mathbf{L}\_{\mathbf{s}}}{1 - \mathbf{y}\_{\mathbf{A}}} \tag{38}$$

where Ls is part of the light phase flow rate that is nontransferable or stagnant. Replacing new values for NA and L from equations (36) and (38), respectively, into equation (29), other relations are produced for height of packed bed and number of transfer units.

When the rate of mass transfer is high, using F coefficient as mass transfer coefficient is usual and other form of definition of NA is needed. The governing equations in this case have been presented in ref. (Hines & Maddox, 1985).

### **9. Conclusion**

In the present chapter it tried to figure out a mass transfer sense related to more useful purification processes in chemical industries. A common sense in different processes was obtained based on classifying all separation processes into two broad categories of absorption-like and desorption-like processes. This classification can help the reader to get a similar sense when see separation processes individually. Referring denser phase as a heavy phase in comparison with the second phase causes the processing calculations to be simplified and categorized easily. The basic concepts of mass transfer were used to describe what happened in the interface between two phases and a brief discussion on the phase equilibria was used to generate equilibrium curves.

Also it tried to bring a good summary on significant processing calculations in different contact forms of phases and finally the calculations on two mass exchangers of tray towers and packed-bed columns were presented briefly.

### **10. References**

Geankoplis, C. J.; (2003). *Transport Processes And Separation Process Principles*, New Jersey, Prentice Hall PTR.

Henley, E. J.; & Seader, J. D. (1981). *Equilibrium-Stage Separation Operations In Chemical Engineering*, John Wiley & Sons.

**35** 

*South Korea* 

**Microdroplets for the Study of Mass Transfer** 

As the development in the microfabrication technology in the last two decades has allowed the easy fabrication of microchannels with low cost, many studies have been conducted on the transport of fluid and the realization of various functions using fluids (Whitesides, 2006; Dittrich et al., 2006). The realization of the microchannel-based fluidic system and the relevant study are called microfluidics. In the scale in which microfluidics is concerned, the surface force is dominant over the body force, since the surface-to-volume ratio is large. The dominant influence of the surface force allows the production as well as the movement and control of micro-sized droplets in a microchannel. This study area is called droplet-based microfluidics (Beebe et al., 2002; Kim, 2004; Stone et al., 2004). Droplet-based microfluidics is expected to enable chemical and biological applications such as particle synthesis (Frenz et al., 2008), microextraction (Mary et al., 2008), and protein crystallization (Zheng et al., 2003). In a general microfluidics system, the Reynolds number (Re) is very small as 0.1-10, and thus the fluid forms a laminar flow. Such a laminar flow makes it difficult for two different fluids to be mixed with each other. However, if droplets are used, different fluids can be mixed with each other, because an internal circulation flow takes place in the droplets (Tice

This chapter describes the microextraction based on the droplet-based microfluidics. Firstly, we will explain the electrohydrodynamic droplet generation and control technology in the aqueous two-phase system (ATPS) that we employed for the study, and the application of the generated droplets to microextraction. In particular, we were able to control the rate of extraction, which was impossible in the previous extraction methods, and analyzed the microextraction behaviour by simulating the phenomena based on a simple dissolving

The technology that is firstly required in droplet-based microfluidics is the method to generate droplets in a microchannel. Droplet generation is related with capillary number (Ca), which is the ratio of viscous force to interfacial tension (Squires & Quake, 2005). In a macroscopic system, droplets can be easily generated by vigorously shaking immiscible fluids, but the size distribution is very wide. In a microfluidics system, on the contrary, droplets are generated by various controllable methods so that the size distribution can be limited. Microfluidic methods for forming droplets can be either passive or active. Most methods are passive, relying on the flow field to deform the interface and promote the

**1. Introduction** 

et al., 2003).

model.

**1.1 Droplet-based microfluidics** 

Young Hoon Choi, Young Soo Song and Do Hyun Kim *Departmenet of Chemical and Biomolecular Engineering, KAIST* 


