# 8.10: Untitled Page 199

- Page ID
- 18332

## 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

*n* *m *

(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 *d* *n*

*V*

*n* *Q * *n* *Q*

*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

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 *d* *n*

*n* *Q V * *n* *Q 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

*d* *n*

*n* *Q V * *n* *Q 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

*d* *n*

*n* *n* *D * *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*