Page 330 - Bird R.B. Transport phenomena

314 Chapter 10 Shell Energy Balances and Temperature Distributions in Solids and Laminar Flow

Tube wall
z = 0
Region of
small z

Slope at r = R
same for all z > 0

Shape of profiles
Region of is same—they
are displaced
large z upward with
increasing z

Fig. 10.8-3. Sketch showing how one expects the temperature
T(r, z) to look for the system shown in Fig. 10.8-2 when the
fluid is heated by means of a heating coil wrapped uniformly
around the tube (corresponding to q positive).
0

The function in Eq. 10.8-23 is clearly not the complete solution to the problem; it
does allow the partial differential equation and boundary conditions 1 and 2 to be satis-
fied, but clearly does not satisfy boundary condition 3. Hence we replace the latter by an
integral condition (see Fig. 10.8-4),
f2n fR
A
Condition 4: 2irRzq 0 = рСЛТ - T )v r dr dd (10.8-24)
z
x
Jo Jo
or, in dimensionless form,
= Г (10.8-25)
Jo
This condition states that the energy entering through the walls over a distance £ is the
same as the difference between the energy leaving through the cross section at £ and that
entering at £ = 0.
Substitution of the postulated function of Eq. 10.8-23 into Eq. 10.8-19 leads to the fol-
lowing ordinary differential equation for ^ (see Eq. C.l-11):

(10.8-26)