# 8.15: Untitled Page 204

## 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

   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  382