# 8.10: Untitled Page 199

## Chapter 8

so that Eq. 8‐40 takes the form

d 

D

V

  Q

D 2

2

dt





rate at which

rate of accumulation

cellular material leaves

of cellular material

the chemostat

in the chemostat

(8‐43)

  Q

r V

D 1 1

D





rate at which

rate of production

cellular material enters

of cellular material

the chemostat

in the chemostat

It is the term on the right hand side of this result that is important to us since it represents the mass rate of production of cellular material in the chemostat. Rather than work directly with this quantity, there is a tradition of using the rate of production of cells to describe the behavior of the chemostat. We define the average mass of a cell in the chemostat by

mass of cellular material

average mass

per unit volume

m

 

 

(8‐44)

cell

of a cell

number of cells

per unit volume

and we represent the number of cells per unit volume by

number of cells

n  

(8‐45)

per unit volume

This allows us to express the mass of cellular material per unit volume according to

    nm

(8‐46)

D

cell

Given these definitions, we can divide Eq. 8‐43 by the constant, m

, to obtain a

cell

macroscopic balance for the number density of cells that takes the form dn

V

  nQ   nQ

r m

V

(8‐47)

2

2

1 1

D cell

dt

Here we have assumed that the average mass of a cell in the chemostat is independent of time, and this may not be correct for transient processes. In addition, Eq. 8‐46 is based on the assumption that all of species D is contained within the cells. This is consistent with the cellular processes illustrated in

Transient Material Balances

373

Figure 8‐9; however, that illustration does not take into account the process of cell death (Bailey and Ollis, 1986). Because of cell death, Eq. 8‐46 represents an over-estimate of the number of cells per unit volume.

Traditionally, one assumes that the volumetric flow rates entering and leaving the chemostat are equal so that Eq. 8‐47 simplifies to dn

  nQ V   nQ V

  r m

(8‐48)

2 

1 

D

cel

dt

l

This represents a governing differential equation for cells per unit volume,  n ; however, it is the cell concentration at the exit,  n , that we wish to predict, and 2

this prediction is usually based on the assumption of a perfectly mixed system as described in Sec. 8.1. This assumption leads to  n   n and it allows us to 2

express Eq. 8‐48 in the form

dn

  n  Q V    nQ V

r m

(8‐49)

1 

D cell

dt













outflow

inflow

production

accumulation

In previous sections of this chapter the quantity, V / Q , was identified as the mean residence time and denoted by  . However, in the biochemical engineering literature, the tradition is to identify Q / V as the dilution rate and denote it by D.

Following this tradition we express Eq. 8‐49 in the form

dn

  n   nD   r m

(8‐50)

1 

D

ce

dt

ll

where the term on the right hand side should be interpreted as

number of cells

r m

  produced per unit

(8‐51)

D

cell