Page 387 - Modelling in Transport Phenomena A Conceptual Approach

9.3. HEAT TRANSFER WITH CONVECTION 367

Solution

For a thermally developed flow, the ratio given in Eq. (9.3-36) depends only on the
radial coordinate T, ie.,
T - Tb
-= f (r)
Tw - Tb
Differentiation of Eq. (1) with respect to T gives

which is valid at all points within the flow field. Evaluation of Eq. (2) at the surface
of the pipe yields
= (Tw - Tb) ;i;; (3)
E Ir=R df /r=R
On the other hand, the heat flm at the wall is expressed as

Substitution of Eq. (3) into Eq. (4) gives

Example 9.7 For a thermally developed flow, show that the temperature gradient
in the axial direction, aT/8z, remains constant for a constant wall heat flux.
Solution
The heat flux at the wall is given by
qrlr=R = h (T! - Tb) = constant (1)

Since h is constant for a thermally developed flow, Eq. (1) implies that
Tw - Tb = constant (2)
or,
dTw dTb
-=-.
dz dz (3)
Therefore, Eq. (9.3-38) simplifies to

dT dTb dTw
-=-=- (4)
dz dz dz
Since dTbldz is constant according to Eq. (9.3-31), dT/dz also remains constant,
a.e.,
-- - dTb - dz mep (5)
-- -- - -- ?rDqw - constant
dTw
az dz