Page 389 - Modelling in Transport Phenomena A Conceptual Approach

9.3. HEAT TRANSFER WITH CONVECTION 369

It is important to note that 8 depends only on (or, T).
The boundary conditions associated with Eq. (9.3-46) are

(9.3-47)

at <=l 8=1 (9.3-48)
Integration of Eq. (9.3-46) with respect to 5 gives
)
t-@= ( E2 - f Nu +Cl (9.3-49)
de

where C1 is an integration constant. Application of Eq. (9.3-47) indicates that
CI = 0. Integration of Eq. (9.3-49) once more with respect to < and the use of the
boundary condition given by Eq. (9.3-48) gives

(9.3-50)
On the other hand, the bulk temperature in dimensionless form can be expressed
as
L1(l -t2)Wt
ob=-- Tb - Tb -o= (9.3-51)

Tw - Tb i1(l-F2)EdE
Substitution of Eq. (9.3-50) into Q. (9.3-51) and integration gives the Nusselt
El (9.3-52)
number as
NU= -

Constant wall temperature

When the wall temperature is constant, Eq. (9.3-38) indicates that
aT
(9.3-53)

The variation of Tb as a function of the axial position can be obtained from Eq.
(9.3-21) as
dTb (9.3-54)
-=- nD(h)z (Tw - T&,) exp [- (-) nD(h), Z]
dz mcp m cp
" /
(Tw -Tb)
Since the heat transfer coefficient is constant for a thermally developed flow, Q.
(9.3-54) becomes
dTb TD h (T, - Tb) - 4 h (T, - Tb) (9.3-55)
-
-- -
dz m cp D(vz ) P CP