# 8.12: Untitled Page 201

## Chapter 8

The quantity,  , is referred to as the specific growth rate, and if  is known one can use this result to predict  n as a function of time.

For many practical applications, there are no cells entering the chemostat, thus  n is zero and we are dealing with what is called a sterile feed. For a sterile 1

feed, the cell concentration is determined by the following governing equation and initial condition

dn

  nD    n

(8‐54a)

dt

IC.

n  n ,

t  0

(8‐54b)

o

Here n represents the initial concentration of cells in the chemostat and this is o

usually referred to as the inoculum. If we treat the specific growth rate,  , as a constant, the solution of the initial value problem for  n is straightforward and is left as an exercise for the student.

The steady‐state form of Eq. 8‐54a is given by

D   n  0

(8‐55)

and this indicates that the steady state can only exist when  n  0 or when D   . The first of these conditions is of no interest, while the second suggests that a steady‐state chemostat might be rather rare since adjusting the dilution rate, D Q V , to be exactly equal to the specific growth rate might be very difficult if the specific growth rate were constant. However, a little thought indicates that the specific growth rate,  , must depend on the concentration of the nutrients entering the chemostat, thus  can be controlled by adjusting the input conditions.

When the substrate B is present in excess, the rate of cell growth can be expressed in terms of the concentration of species A in the extracellular fluid,

c  , a reference concentration K , and other parameters according to A

A

  F  c , K , other parameters

(8‐56)

A

A

If a specific growth rate has the following characteristics

 0

c  

0

A

  

(8‐57)

  

 max

cA

KA

Transient Material Balances

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it could be modeled by what is known as Monod’s equation (Monod 1942)

 

max cA

Monod’s equation:

 

(8‐58)

K

  

A

cA

The parameter K is sometimes referred to as the “half saturation” since it A

represents the concentration at which the growth rate is half the maximum growth rate,  max . It should be clear that there are many other functional representations that would satisfy Eq. 8‐57; however, the form chosen by Monod has been used with reasonable success to correlate macroscopic experimental data (Monod, 1949).

8.5 Batch Distillation

Distillation is a common method of separating the components of a solution.

The degree of separation that can be achieved depends on the vapor‐liquid equilibrium relation and the manner in which the distillation takes place. Salt and water are easily separated in solar ponds in a process that is analogous to batch distillation. In that case the separation is essentially perfect since a negligible amount of salt is present in the vapor phase leaving the pond.

In this section we wish to analyze the unit illustrated in Figure 8‐11 which is sometimes referred to as a simple still. The process under consideration is obviously a transient one in which the unit is initially charged with M moles of o

a binary mixture containing species A and B. The initial mole fraction of species A is designated by o

x , and we will assume that the mole fraction of species A is A

small enough so that the ideal solution behavior discussed in Chapter 5 (see Eq. 5‐30) provides an equilibrium relation of the form

y

 

x

(8‐59)

A

AB A

Here y represents the mole fraction in the vapor phase and x represents the A

A

mole fraction in the liquid phase. In our analysis, we would like to predict the composition of the liquid during the course of the distillation process.   376