# 8.9: Untitled Page 198

- Page ID
- 18331

## Chapter 8

Here *V*( *t*) represents the control volume illustrated in Figure 8‐10 and (

*A t*)

represents the surface area at which the speed of displacement is **w ** **n **.

*Figure 8‐10*. Control volume for chemostat

In order to develop the macroscopic balance for the *total density* *of cellular* *material*, we simply add Eqs. 8‐36 to obtain

371

*d*

*dV *

(**v ** **w**) **n** *dA *

(**v ** **w**)

*dt *

*dA*

*D*

*F F*

**n**

*G*

*G*

*V *( *t*)

*A*( *t*)

*A*( *t*)

(8‐37)

(**v ** **w**) **n** *dA * etc.

*r dV*

*H*

*H*

*D*

*A*( *t*)

*V*( *t*)

Here we need to be very clear that represents the *total density* of the cellular *D*

material and that this density is defined by

etc.

(8‐38)

*D*

*F*

*G*

*H*

In addition, we need to be very clear that *r * represents the *total mass rate of* *D*

*production* of cellular material, and that this mass rate of production is defined by *r*

*r * *r * *r * etc.

(8‐39)

*D*

*F*

*G*

*H*

There are other molecular species in the system illustrated in Figure 8‐9; however, we are interested in the rate of growth of cellular material, thus *r * is *D*

the quantity we wish to predict.

Returning to Eq. 8‐37, we note that terms such as (**v ** **w**) **n ** are *negligible* *G*

everywhere except at the *entrance* where cellular material may enter the chemostat, and at the *exit* where the product leaves the system. It is reasonable to assume that all the species associated with the cellular material move with the same velocity at the entrance and exit, and this allows us to express Eq. 8‐37 as *d*

*dV *

**v ** **n ** *dA *

(8‐40)

*dt *

*r dV*

*D*

*D D*

*D*

*V *( *t*)

*A*

*V *( *t*)

e

where *A* e represents the area of the entrance and the exit. It is important to understand that this result is based on the plausible assumption that all the velocities of the species remaining in the cell are the same

**v**

**v**

**v**

etc.

(8‐41)

*F*

*G*

*H*

and we have identified this *common velocity* by **v **. For the typical chemostat, it *D*

is reasonable to ignore variations in the control volume and to assume that the velocities at the entrance and exit are constrained by

**v ** **n**

**v**

**n ** *, *

at *A *

(8‐42)

*D*

H2O

e