# 8.15: Untitled Page 204

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## Chapter 8

function of time; however, without the knowledge of how *M *( *t*)

changes with

time we can only determine *x * as a function of *M*

*A*

. This represents a classic

situation in many batch processes where one can only determine the *changes that* *take* *place between one state and another*. In this analysis the state of the system is characterized by *x * and *M*

*A*

.

Returning to Eq. 8‐76 we divide by *x * *M * and multiply by *dt * in order to *A*

obtain

*dM*

*d x*

*A*

1 *eff *

*x *

(8‐79)

*M*

*A*

Since

will generally depend on the temperature, and the temperature at *eff*

which the solution boils will depend on *x * , we need to determine how

*A*

*eff*

depends upon *x * before the variables in Eq. 8‐79 can be completely separated.

*A*

Here we will avoid this complication and treat

as a constant so that Eq. 8‐79

*eff*

can be integrated leading to

*x *

*M *( *t*)

*A*

*d*

*d*

1

(8‐80)

*eff *

o

o

*x*

*A*

*M*

Evaluation of the integrals allows one to obtain a solution for *x * given by *A*

1

*eff*

*M *( *t*)

o

*x * *x *

(8‐81)

*A*

*A*

o

*M *

Once again, we must remember that

is a *process equilibrium relation* that will

*eff*

generally depend on the temperature which will change during the course of a batch distillation. Nevertheless, we can use Eq. 8‐81 to provide a *qualitative* *indication* of how the mole fraction of the liquid phase changes during the course of a batch distillation process.

When

is greater than one (

1 ) we can see from Eq. 8‐75 that the

*eff*

*eff*

vapor phase is *richer* in species *A* than the liquid, and Eq. 8‐81 predicts a *decreasing* value of *x * as the number of moles of liquid decreases. For the case *A*

where

takes on a variety of values, we have indicated the normalized mole *eff*

*Transient Material Balances *

381

fraction

o

*x * */ x * as a function of

o

*M *( *t*) */ M*

*A*

*A*

in Figure 8‐12. There we can see

that a significant separation takes place when

is either large or small

*eff*

compared to one. The results presented in Figure 8‐12 are certainly quite plausible; however, one must keep in mind that they are based on the *process* *equilibrium relation* represented by Eq. 8‐75. Whenever one is confronted with an assumption of uncertain validity, experiments should be performed, or a more comprehensive theory should be developed, or both.

*Figure 8‐12*. Composition of a binary system in a batch still **8.6 Problems **

*Section* 8.1

8‐1. A tank containing 200 gallons of saturated salt solution (3 lbm of salt per gallon) is to be diluted by the addition of brine containing 1 lbm of salt per gallon.

If this solution enters the tank at a rate of 4 gallons per minute and the mixture

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