# 4.13: Untitled Page 68

- Page ID
- 18201

## Chapter 4

*c ***v **

**n**

*A*

*dA*

*c*

*AdA*

*A*

**v **

*exit*

*A exit*

**n**

*c *

*A b*

**v **

*dA*

*dA*

**n**

**v n**

*A exit*

*A exit*

(4‐76)

1

*c*

*AdA*

*cA*

*A exit Aexit*

For this case, the molar flow rate at the exit takes the form *M*

(4‐77)

*A*

*cA*

*e*

*Q xit , *

*flat velocity profile*

To summarize, we note that Eq. 4‐71 is an *exact* representation of the molar flow rate in terms of the *bulk* *concentration* and the volumetric flow rate. When the *Figure 4‐7*. Laminar and turbulent velocity profiles for flow in a tube concentration profile can be *approximated* as flat, the molar flow rate can be represented in terms of the *constant concentration* and the volumetric flow rate as indicated by Eq. 4‐75. When the velocity profile can be *approximated* as flat, the molar flow rate can be represented in terms of the *area average concentration* and the volumetric flow rate as indicated by Eq. 4‐77. If one is working with the species mass balance given by Eq. 4‐7, the development represented by Eqs. 4‐64

through 4‐77 can be applied simply by replacing *c * with

*A*

*A *.

**4.6 Alternate Flow Rates **

There are a number of relations between species flow rates and total flow rates that are routinely used in solving macroscopic mass or mole balance problems *provided* that either the velocity profile is flat or the concentration profile is flat. For example, we can always write Eq. 4‐68 in the form

118

*c*

*A*

*M*

**v ****n**

**v ****n**

**v ****n**

*A*

*c*

*A dA*

*c*

*dA*

*x*

*Ac dA* (4‐78)

*c*

*A exit*

*A exit*

*A exit*

If either

or *xA * is constant over the area of the exit, we can express this result as

(4‐79)

where

is the total molar flow rate defined by

(4‐80)

If the individual molar flow rates are known and one desires to determine the area averaged mole fraction at an entrance or an exit, it is given by (4‐81)

provided that either

or *x * is constant over the area of the entrance or the *A*

exit. It will be left as an exercise for the student to show that similar relations exist between mass fractions and mass flow rates. For example, a form analogous to Eq. 4‐79 is given by

(4‐82)

and the mass fraction at an entrance or an exit can be expressed as (4‐83)

One *must* keep in mind that the results given by Eqs. 4‐79 through 4‐83 are only valid when either the concentration (density) is constant or the molar (mass) flux is constant over the entrance or exit.

When neither of these simplifications is valid, we express Eq. 4‐78 as *M*

*A*

*x*

**v n**

*A c*

*dA*

*xA b M *

(4‐84)

*Aexit*

119