Page 388 - Modelling in Transport Phenomena A Conceptual Approach

368 CHAPTER 9. STEADY MTcRoscopIc BALANCES WITH GEN.

9.3.1.2 Nusselt number for a thermally developed flow

Substitution of J3q. (9.3-1) into Eq. (9.3-9) gives
ZpGp(v,) [1- (31 ,“ 8”, (9.3-39)

= - - (YE)
It should always be kept in mind that the purpose of solving the above equation for
temperature distribution is to obtain a correlation to use in the design of heat trans-
fer equipment, such as, heat exchangers and evaporators. As shown in Chapter 4,
heat transfer correlations are expressed in terms of the Nusselt number. Therefore,
Eq. (9.3-39) will be solved for a thermally developed flow for two different types of
boundary conditions, i.e., constant wall heat flux and constant wall temperature,
to determine the Nusselt number.

Constant wall heat flux
In the case of a constant wall heat flux, as shown in Example 9.7, the temperature
gradient in the axial direction is constant and expressed in the form

(9.3-40)

Since we are interested in the determination of the Nusselt number, it is appropriate
to express aT/az in terms of the Nusselt number. Note that the Nusselt number
is given by
(9.3-41)

Therefore, Eq. (9.3-40) reduces to
aT
-- Nu(Tw -Tb)k (9.3-42)
-
P CPR2 (VZ)
Substitution of Eq. (9.3-42) into Eq. (9.3-39) yields

(9.3-43)

In terms of the dimensionless variables

(9.3-44)

(9.3-45)
J3q. (9.3-43) takes the form

(9.3-46)