Page 329 - Bird R.B. Transport phenomena

§10.8 Forced Convection 313

term containing the viscosity is the viscous heating, which we shall neglect in this dis-
cussion. The last term in the first bracket, corresponding to heat conduction in the axial
direction, will be omitted, since we know from experience that it is usually small in com-
parison with the heat convection in the axial direction. Therefore, the equation that we
want to solve here is

(10.8-12)

This partial differential equation, when solved, describes the temperature in the fluid as
a function of r and z. The boundary conditions are

B.C. 1: atr = 0, T = finite (10.8-13)
B.C. 2: at r = R, kj^ = q 0 (constant) (10.8-14)

B.C.3: at z = 0, T = T, (10.8-15)
We now put the problem statement into dimensionless form. The choice of the dimen-
sionless quantities is arbitrary. We choose

0 = = = ?L
T¥7T ^ i £ -~ ~T- dO.8-16,17,18)
2
VoR/k R PC pv z>maxR /k
Generally one tries to select dimensionless quantities so as to minimize the number of
parameters in the final problem formulation. In this problem the choice of £ = r/R is a
natural one, because of the appearance of r/R in the differential equation. The choice for
the dimensionless temperature is suggested by the second and third boundary condi-
tions. Having specified these two dimensionless variables, the choice of dimensionless
axial coordinate follows naturally.
The resulting problem statement, in dimensionless form, is now

1 д Ы 8 , (10.8-19)

with boundary conditions
B.C. 1: at f = 0, 0 = finite (10.8-20)
B.C. 2: a t f = l , ^ = 1 (10.8-21)

B.C. 3: at I = 0, 0 = 0 (10.8-22)
The partial differential equation in Eq. 10.8-19 has been solved for these boundary condi-
3
tions, but in this section we do not give the complete solution.
It is, however, instructive to obtain the asymptotic solution to Eq. 10.8-19 for large f.
After the fluid is sufficiently far downstream from the beginning of the heated section,
one expects that the constant heat flux through the wall will result in a rise of the fluid
temperature that is linear in f. One further expects that the shape of the temperature pro-
files as a function of £ will ultimately not undergo further change with increasing £ (see
Fig. 10.8-3). Hence a solution of the following form seems reasonable for large £:

(10.8-23)
0
in which C o is a constant to be determined presently.

3
R. Siegel, E. M. Sparrow, and T. M. Hallman, AppL Sci. Research, A7, 386-392 (1958). See Example
12.2-1 for the complete solution and Example 12.2-2 for the asymptotic solution for small £.