Page 381 - Modelling in Transport Phenomena A Conceptual Approach

9.3. HEAT TRANSFER WTH CONVECTION 361

i
E

Figure 9.9 Forced convection heat transfer in a pipe.

In general, T = T(r, z) and from Table C.5 in Appendix C, the non-zero energy
flux components are
aT
e,=-k- (9.3-2)
dr
aT
e, = -k- +(pcpT)v, (9.3-3)
dz
Since there is no generation of energy, Eq. (9.21) simplifies to

(Rate of energy in) - (Rate of energy out) = 0 (9.3-4)

For a cylindrical differential volume element of thickness AT and length Az, as
shown in Figure 9.9, Eq. (9.3-4) is expressed as

( e,.[, 27rr Az + e,/, 27rr AT) - 247- + AT) Az + e,I,+px 27rr AT] = 0
(9.3-5)
Dividing Eq. (9.3-5) by 27rArA.z and taking the limit as Ar --f 0 and AB ---t 0
gives

(9.3-6)

(9.3-7)