# 8.2: Untitled Page 191

- Page ID
- 18324

## volume average

*area average *

*concentration in*

*concentration*

* the tank*

*in the effluent*

This allows us to write Eq. 8‐10 in terms of the *single unknown*, *c * , in order to *A*

obtain

*d* *c *

*A*

*V*

*c * *Q * *c * *Q *

(8‐12)

*A*

*A * 1

*dt*

One must be very careful to understand that Eq. 8‐11 is an *approximation* based on the assumption that the difference between *c * and *c * is *small enough* so *A * 2

*A*

that it can be neglected (Whitaker, 1988).

It is convenient to divide Eq. 8‐12 by the volumetric flow rate and express the result as

*d* *c *

*A*

*c * *c *

(8‐13)

*A*

*A * 1

*dt*

Here represents the average residence time that is defined explicitly by

*average *

*V*

*residence*

(8‐14)

*Q*

* time *

In general, we are interested in processes for which the inlet concentration,

*c * , is a function of time, and a classic example is illustrated in Figure 8‐2.

*A * 1

There we have indicated that *c * undergoes a sudden change from o *c * to 1

*c *,

*A * 1

*A*

*A*

and we want to determine how the concentration in the tank, *c * , changes *A*

because of this change in the inlet concentration. The initial value problem associated with the sudden change in the inlet concentration is given by *d* *c *

*A*

1

*c * *c , *

*t * 0

*A*

*A*

(8‐15a).

*dt*

IC.

o

*c * *c , *

*t * 0

*A*

*A*

(8‐15b)

Equations 8‐15 are consistent with a situation in which the inlet concentration is originally fixed at o

*c * and then *instantaneously* switched from o

*c * to 1

*c * at *t * 0 .

*A*

*A*

*A*

358