# 5.13: Untitled Page 111

- Page ID
- 18244

## Chapter 5

### 0 010

*. *

( *y *)

0 *. * 00064

(3)

*A * 1

15 *. * 67

In addition to calculating the exit mole fraction of acetone in the air stream, one may want to determine the mole fraction of acetone in the water stream leaving the cascade. This is obtained from Eq. 5‐93 which provides

( *y *) ( *y *)

0 *. * 010 0 00064

*. *

*A * 8

*A * 1

( *x *)

0 *. * 0031

*A * 7

*M*

*M*

(4)

*(* 90 kmol/h *) *(30 kmol/h)

Some times it is possible to change the operating characteristics of a cascade to achieve a desired result and this situation is considered in the following example.

Type III: Given the *inlet* mole fractions, ( *x *) and ( *y *)

, the

*A * o

*A N*1

system parameters *K*

, *M*

, the number of stages, *N*, and a

*eq,A*

*desired* value of ( *y *) , we would like to determine the molar flow *A * 1

rate of the ‐phase, *M*

.

In this case we consider an *existing unit* in which there are 6 stages. The *inlet* mole fractions are given by ( *x *) 0 and ( *y *) 0 *. * 010 , and the *A * o

*A * 7

specified parameters associated with the system are *K*

2 *. * 53 and

*eq,A*

*M*

30 kmol/h . The desired value of the mole fraction of acetone in the

exit air stream is ( *y *) 0 0005

*. *

. We begin the analysis with Eq. 5‐91 that

*A * 1

leads to

( *y *)

*A * 7

0 010

*. *

2

3

4

5

6

1 *A * *A * *A * *A * *A * *A*

20 *. * 0

(5)

( *y *)

0 *. * 0005

*A * 1

Since *M*

is an adjustable parameter, we need only solve this *implicit* *equation* for the absorption coefficient in order to determine the molar flow rate. Use of one of the iterative methods described in Appendix B leads to *A * 1 *. * 342

(6)

and from the definition of the absorption coefficient given by Eq. 5‐72 we determine the molar flow rate of the ‐phase to be

*M*

*A M* *K*

101 9

*. * kmol/h

(7)

*eq,A*

*Two‐Phase Systems* & *Equilibrium Stages *

203

Adjusting a molar flow rate to achieve a specific separation is a convenient operational technique provided that the auxiliary equipment required to hange

c

the flow rate is readily available.

5.6.2 *Sequential analysis‐graphical*

The algebraic solution to the cascade of equilibrium stages illustrated in Figure 5‐10 can also be represented in terms of a graph. In this case we consider the cascade illustrated in Figure 5‐14 in which the index *n * is bounded according *Figure 5‐14*. Cascade of equilibrium stages

to 1 *n * *N *. The macroscopic mole balance for species *A* associated with this control volume can be expressed as

( *x *) *M*

( *y *) *M*

( *x *) *M*

( *y *)

*M*

(5‐94)

*A n*

*A * 1

*A * o

*A n*1

*molar flow of species A*

*molar flow of species A*

*out of the control volume*

*into the control volume*

In this case we want to develop an expression for ( *y *) *A n* thus we arrange

1

Eq. 5‐94 in the form

( *y *)

( *x *)

*M*

*M*

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

*, *

*n * 1 *, * 2 *,...,N * (5‐95)

*A n*1

*A n *

*A * 1

*A * o

This is sometime called the “operating line” and it can be used in conjunction with the “equilibrium line”

( *y *)

*K*

( *x *) *, *

*n * 1 *, * 2 *,...,N *

(5‐96)

*A n*

*eq,A*

*A n*

to provide a graphical representation of the solution developed using the sequential analysis. To be precise, we note that Eqs. 5‐95 and 5‐96 represent a series of *operating points* and *equilibrium points*; however, the construction of *operating lines* and *equilibrium lines* is a useful graphical tool.

204