Page 454 - Bird R.B. Transport phenomena

434 Chapter 14 Interphase Transport in Nonisothermal Systems

The dimensionless temperature gradient appearing in Eq. 14.3-3 could, in principle,
be evaluated by differentiating the expression for T obtained by solving Eqs. 11.5-7, 8,
and 9 with the boundary conditions

at z = 0, v = b forO (14.3-4)
z
at f = \, v = 0 for z > 0 (14.3-5)
at r = 0 and z = 0 op = о (14.3-6)
at z = 0, f =1 forO < r <^ (14.3-7)
at f = \ f = 0 forO < z < L/D (14.3-8)
2
where v = v/{v \ and Ф = (9> - WJ/p(v )\. As in §6.2, we have neglected the d /dz 2
z
z
terms of the equations of change on the basis of order-of-magnitude reasoning similar to
that in §4.4. With those terms suppressed, upstream transport of heat and momentum
are excluded, so that the solutions upstream of plane 2 in Fig. 14.3-1 do not depend on
L/D.
From Eqs. 11.5-7, 8, and 9 and these boundary conditions, we conclude that the di-
mensionless instantaneous temperature distribution must be of the following form:
f = f (г, 0, z, t; Re, Pr, Br) for 0 < z < L/D (14.3-9)
Substitution of this relation into Eq. 14.3-3 leads to the conclusion that Nuj(0 = Nu^Re,
Pr, Br, L/D, 0. When time-averaged over an interval long enough to include all the tur-
bulent disturbances, this becomes
Nu = Nil,(Re, Pr, Br, L/D) (14.3-10)
1
A similar relation is valid when the flow at plane 1 is fully developed.
If, as is often the case, the viscous dissipation heating is small, the Brinkman number
can be omitted. Then Eq. 14.3-10 simplifies to

Nil, = Nu^Re, Pr, L/D) (14.3-11)
Therefore, dimensional analysis tells us that, for forced-convection heat transfer in circu-
lar tubes with constant wall temperature, experimental values of the heat transfer coeffi-
cient hi can be correlated by giving Кщ as a function of the Reynolds number, the
Prandtl number, and the geometric ratio L/D. This should be compared with the similar,
but simpler, situation with the friction factor (Eqs. 6.2-9 and 10).
The same reasoning leads us to similar expressions for the other heat transfer coeffi-
cients we have defined. It can be shown (see Problem 14.B-4) that

Nu, = Nu (Re, Pr, L/D) (14.3-12)
fl
Nu = Nu (Re, Pr, L/D) (14.3-13)
ln ln
Nu = Nu (Re, Pr, L/D) (14.3-14)
loc loc
in which Nu fl = h D/k, Nu ln = h D/k, and Nu ]oc = h D/k. That is, to each of the heat
]n
a
]oc
transfer coefficients, there is a corresponding Nusselt number. These Nusselt numbers
are, of course, interrelated (see Problem 14.B-5). These general functional forms for the
Nusselt numbers have a firm scientific basis, since they involve only the dimensional
analysis of the equations of change and boundary conditions.
Thus far we have assumed that the physical properties are constants over the tem-
perature range encountered in the flow system. At the end of §14.2 we indicated that
evaluating the physical properties at the film temperature is a suitable empiricism. How-
ever, for very large temperature differences, the viscosity variations may result in such a
large distortion of the velocity profiles that it is necessary to account for this by introduc-
ing an additional dimensionless group, /!,&//% where /л is the viscosity at the arithmetic
ь