# 5.2: Untitled Page 100

- Page ID
- 18233

## Chapter 5

study of the mass transfer of species *A*, and this is something that will take place in subsequent courses in the chemical engineering curriculum.

In a problem of this type, we wish to know how the concentrations in the *exit* *streams* are related to the concentrations in the *inlet streams*, and this leads to the use of the control volume illustrated in Figure 5‐6. The macroscopic mole balance for species *A* takes the form

Species *A*:

*c*

*dA * 0

(5‐51)

**v n**

*A*

*A* e

in which **v ** **n ** is used in place of **v ** **n ** with the idea that diffusive effects are *A*

negligible at the *entrances and exits* of the system. Since the process equilibrium *Figure 5‐6*. Control volume for mixer‐settler

relation given by Eq. 5‐50 is expressed in terms of mole fractions, it is convenient to express Eq. 5‐51 in the form (see Sec. 4.6)

Species *A*:

*x c * *dA * 0

(5‐52)

**v n**

*A*

*A* e

When this result is applied to the control volume illustrated in Figure 5‐6, we obtain

Species *A*:

( *x *) *M *

( *y *) *M *

( *y *) *M *

*x*

*M*

*A * 1

1

*A * 2

2

*A * 3

3

(

)

*A * 4

4

0

(5‐53)

*Two‐Phase Systems* & *Equilibrium Stages* 181

Here we have used *x * to represent the mole fraction of species *A* in the aqueous *A*

phase and *y * to represent the mole fraction of species *A* in the organic phase. In *A*

addition, we have used *M*

to represent the total molar flow rate in each of the

four streams. At the two exits of the settler, we assume that the organic and aqueous streams are in equilibrium, thus the mixer‐settler is considered to be an *equilibrium stage* and Eq. 5‐50 is applicable. When changes in the mole fraction of species *A* are sufficiently small, we can use the approximations given by *M*

*M*

*M* *, *

*M*

*M*

*M*

(5‐54)

1

4

2

3

Here we note that the change in the molar flow rates can be estimated as *M*

( *x *) *M* *, *

*M*

( *y *) *M*

(5‐55)

*A*

*A*

in which ( *x *) and ( *y *) represent the changes in the mole fractions that occur *A*

*A*

between the inlet and outlet streams. From these estimate we conclude that *M*

*M* *, *

*M*

*M*

(5‐56)

whenever the change in the mole fractions are constrained by

( *x *) 1 *, *

( *y *) 1

(5‐57)

*A*

*A*

If we impose these constraints on the process illustrated in Figure 5‐6, we can use the approximation given by Eq. 5‐54 in order to express Eq. 5‐53 as ( *y *)

( *y *) ( *x *) ( *x *) *M* *M*

(5‐58)

*A * 3

*A * 2

*A * 1

*A * 4

We now make use of the process equilibrium relation given by Eq. 5‐50 to eliminate ( *x *) leading to

*A * 4

( *y *)

( *y *)

( *y *) ( *x *) *M* *M*

*M*

*M*

(5‐59)

*A*

*A*

*A*

*A * 3

3

2

1

*Keq,A*

Here it is convenient to arrange the macroscopic mole balance for species *A* in the form

*M* *M*

*M M*

( *y *) 1

( *y *) *K*

( *x *)

(5‐60)

*A * 3

*A * 2

*eq,A*

*A * 1

*K*

*K*

*eq,A*

*eq,A*

which suggests that we define an *absorption factor* according to