# 4.12: Untitled Page 67

- Page ID
- 18200

## Chapter 4

*M*

*c*

*A*

**v n**

*A A*

*dA *

(4‐67)

*Aexit*

On the basis of the discussion in Sec. 4.2, we assume that the *diffusive flux * is negligible (

) so that the above result takes the form

*M*

*c*

*A*

**v n**

*A*

*dA *

(4‐68)

*Aexit*

It is possible that both *c * and **v** vary across the exit and a detailed evaluation of *A*

the area integral is required in order to determine the molar flow rate of species *A*. In general this is not the case; however, it is very important to be aware of this possibility.

4.5.1 *Average concentrations*

In Sec. 3.2.1 we defined a volume average density and we use the same definition here for the

e

volum average concentration given by

(4‐69)

At entrances and exits, we often work with the “bulk concentr tion”

a

or “cup

mixed concentration” that was defined earlier in Sec. 3.2.1. For the concentration, *cA *, we repeat the definition according to *cA *

**v n** *dA*

1

*A*

*c *

*exit*

*c ***v**

*A b*

*A*

**n** *dA *

(4‐70)

*Q*

*exit*

**v ** *dA*

*A exit*

**n**

*A exit*

In terms of the *bulk concentration* the molar flow rate given by Eq. 4‐68 can be expressed as

(4‐71)

in which it is understood that

and *c *

*A b * represent the molar flow

rate and

bulk concentration at the exit.

In addition to the bulk or cup‐mixed

concentration, one may encounter the area average concentration denoted by

*c *

*A *

and defined at

an exit according to

116

1

*c *

*A*

*c dA*

(4‐72)

*A*

*A*

*exit Aexit*

If the concentration is constant over *Aexit *, the area average concentration is equal to this constant value, i.e.

(4‐73)

We often refer to this condition as a “flat” concentration profile, and for this case e

w have

(4‐74)

Under these circumstances the molar flow rate takes the form

(4‐75)

The conditions for which *cA * can be treated as a constant over an exit or an ntrance

e

are likely to occur in many practical applications.

When the flow is turbulent, there are rapid velocity fluctuations about the mean or time‐averaged velocity. The velocity fluctuations tend to create uniform velocity profiles and they play a crucial role in the transport of mass *orthogonal* to the direction of the mean flow. The contribution of turbulent fluctuations to mass transport *parallel* to the direction of the mean flow can normally be neglected and we will do so in our treatment of macroscopic mass balances. In a subsequent courses on fluid mechanics and mass transfer, the influence of turbulence will be examined more carefully. In our treatment, we will make use of the reasonable approximation that the turbulent velocity profile is *flat* and this means that **v ** **n ** is constant over *A exit *. Both turbulent and laminar velocity profiles are illustrated in Figure 4‐7 and there we see that the velocity for turbulent flow s

i nearly constant o

r

ve a major portion of the flow field. If we

make the “flat velocity profile” assumption, we can express Eq.

4‐70 as

117