Page 446 - Bird R.B. Transport phenomena

428 Chapter 14 Interphase Transport in Nonisothermal Systems

§14.2 ANALYTICAL CALCULATIONS OF HEAT TRANSFER
COEFFICIENTS FOR FORCED CONVECTION
THROUGH TUBES AND SLITS
Recall from Chapter 6, where we defined and discussed friction factors, that for some
very simple laminar flow systems we could obtain analytical formulas for the (dimen-
sionless) friction factor as a function of the (dimensionless) Reynolds number. We would
like to do the same for the heat transfer coefficient, h, which, however, is not dimension-
less. Nonetheless we can construct with it a dimensionless quantity, Nu = hD/k, the
Nusselt number, using the fluid thermal conductivity к and a characteristic length D that
must be specified for each flow system. Two other related dimensionless groups are
commonly used: the Stanton number, St = Nu/RePr, and the Chilton-Colburn j-factor for
1/3
heat transfer, j H = Nu/RePr . Each of these dimensionless groups may be "decorated"
with subscript 1, a, In, or m, corresponding to the subscript on the Nusselt number.
By way of illustration, let us return to §10.8 where we discussed the heating of a
fluid in laminar flow in a tube, with all the fluid properties being considered constant.
From Eq. 10.8-33 and Eq. 10.8-31 we can get the difference between the wall temperature
and the bulk temperature:

T 1 nr I л у j _ *• J- If '*•
о - Т - \Ц + T^-JI -
ь
11 /aJD\
(14.2-1)
in which R and D are the radius and diameter of the tube. Solving for the wall flux we
get
Л О I 1,\
о - T ) (14.2-2)
b
Then making use of the definition of the local heat transfer coefficient /z —namely, that
loc
q 0 = /z (T — T^)—we find that
loc
0
|
*юс = Л Ш or Nu = ^ = ? (14.2-3)
loc
11 \D/ /c 11
This result is the entry in Eq. (L) of Table 14.2-1—namely, for the laminar flow of a con-
stant-property fluid with a constant wall heat flux, for very large z. The other entries in
Table 14.2-1 and 2 may be obtained in a similar way. 1 Some Nusselt numbers for New-
tonian fluids with constant physical properties are shown in Fig. 14.2-1. 2

1
These tables are taken from R. B. Bird, R. C. Armstrong, and O. Hassager, Dynamics of Polymeric
Liquids, Vol. 1, Fluid Mechanics, 1st edition, Wiley, New York, (1987), pp. 212-213. They are based, in turn,
on W. J. Beek and R. Eggink, De Ingenieur, 74, (35) Ch. 81-Ch. 89 (1962) and J. M. Valstar and W. J. Beek,
De Ingenieur, 75, (1), Ch. 1-Ch. 7 (1963).
2 The correspondence between the entries of Tables 14.2-1 and 2 and problems in this book is as
follows (O = circular tube, || = plane slit):
Eq. (C) Problem 12D.4 O; 12D.5 || Laminar Newtonian
Eq. (F) Problem 12D.3 O; 12D.5 || Laminar Newtonian
Eq. (G) Problem 10B.9(a) O; 10B.9(b) || Plug flow
Eq. (I) Problem 12D.7 O; 12D.6 || Laminar Newtonian
Eq. (K) Problem 10D.2 О Laminar non-Newtonian
Eq. (L) Problem 12D.6 || Laminar Newtonian
Equations analogous to Eqs. (K) in Tables 14.2-1 and 2 are given for turbulent flow in Eqs. 13.4-19 and
13C.1-1.