Page 382 - Modelling in Transport Phenomena A Conceptual Approach

362 CHAPTER 9. STEADY MICROSCOPIC BALANCES WITH GEN.
Convection in - (9.3-8)
Substitution of Eqs. (9.3-2) and (9.3-3) into Eq. (9.3-7) yields
d2T

z-direction
r-direction r-direction Conduction in
Note that in the z-direction energy is transported both by convection and conduc-
tion. As stated by Eq. (2.4-8), conduction can be considered negligible with respect
to convection when PeH >> 1. Under these circumstances, Eq. (9.3-8) reduces to

(9.3-9)

As an engineer, we axe interested in the variation of the bulk fluid temperature,
Tb, rather than the local temperature, T. For forced convection heat transfer in a
circular pipe of radius R, the bulk fluid temperature defined by Eq. (4.1-2) takes
the form
I"" Jd" v,T r drd6
Tb = (9.3-10)
1" v, r drd0
Note that while the fluid temperature, T, depends on both the radial and the axial
coordinates, the bulk temperature, Tb, depends only on the axial direction.
To determine the governing equation for the bulk temperature, it is necessary
to integrate Eq. (9.3-9) over the cross-sectional area of the pipe, i.e.,

Since v, # v,(z), the integral on the left-side of Eq. (9.3-11) can be rearranged as
1'"Jd" - drd6 = 1'" 1" r drd6
E
v,
r

= 2 (rlRv,Trdrd6) (9.3-12)
dz

Substitution of Eq. (9.3-10) into Eq. (9.3-12) yields

(9.3-13)