Page 400 - Bird R.B. Transport phenomena

382 Chapter 12 Temperature Distributions with More Than One Independent Variable

In Eq. 12.2-3, Ф, is the dissipation function given in Eq. 3.3-3. To get the temperature
с
profiles for forced convection, a two-step procedure is used: first Eqs. 12.2-1 and 2 are
solved to obtain the velocity distribution v(r, t); then the expression for v is substi-
tuted into Eq. 12.2-3, which may in turn be solved to get the temperature distribution
T(r, t).
Many analytical solutions of Eqs. 12.2-1 to 3 are available for commonly encoun-
1 7
tered situations. " . One of the oldest forced-convection problems is the Graetz-Nusselt
8
problem, describing the temperature profiles in tube flow where the wall temperature
undergoes a sudden step change at some position along the tube (see Problems
12D.2, 3, and 4). Analogous solutions have been obtained for arbitrary variations of
wall temperature and wall flux. 9 The Graetz-Nusselt problem has also been extended
to non-Newtonian fluids. 10 Solutions have also been developed for a large class of
laminar heat exchanger problems, 11 in which the wall boundary condition is provided
by the continuity of heat flux across the surfaces separating the two streams. A fur-
ther problem of interest is duct flow with significant viscous heating effects (the
2
Brinkman problem^ ).
In this section we extend the discussion of the problem treated in §10.8—namely, the
determination of temperature profiles for laminar flow of an incompressible fluid in a
circular tube. In that section we set up the problem and found the asymptotic solution
for distances far downstream from the beginning of the heated zone. Here, we give the
complete solution to the partial differential equation as well as the asymptotic solution
for short distances. That is, the system shown in Fig. 10.8-2 is discussed from three view-
points in this book:

a Complete solution of the partial differential equation by the method of separa-
0
tion of variables (Example 12.2-1).
b. Asymptotic solution for short distances down the tube by the method of combi-
nation of variables (Example 12.2-2).
c. Asymptotic solution for large distances down the tube (§10.8).

1 M. Jakob, Heat Transfer, Vol. I, Wiley, New York (1949), pp. 451^64.
2 H. Grober, S. Erk, and U. Grigull, Die Grundgesetze der Warmeiibertragung, Springer, Berlin (1961),
Part II.
3 R. K. Shah and A. L. London, Laminar Flow Forced Convection in Ducts, Academic Press, New York
(1978).
4 L. C. Burmeister, Convective Heat Transfer, Wiley-Interscience, New York (1983).
5
L. D. Landau and E. M. Lifshitz, Fluid Mechanics, Pergamon, Oxford (1987), Chapter 5.
6
L. G. Leal, Laminar Flow and Convective Transport Processes, Butterworth-Heinemann (1992),
Chapters 8 and 9.
W. M. Deen, Analysis of Transport Phenomena, Oxford University Press (1998), Chapters 9
7
and 10.
L. Graetz, Ann. Phys. (N.F.), 18, 79-94 (1883), 25, 337-357 (1885); W. Nusselt, Zeits. Ver. deutch. Ing.,
8
54,1154-1158 (1910). For the "extended Graetz problem," which includes axial conduction, see E.
Papoutsakis, D. Ramkrishna, and H. C. Lim, Appl. Sci. Res., 36,13-34 (1980).
9
E. N. Lightfoot, C. Massot, and F. Irani, Chem. Eng. Progress Symp. Series, Vol. 61, No. 58 (1965),
pp. 28-60.
10
R. B. Bird, R. C. Armstrong, and O. Hassager, Dynamics of Polymeric Liquids, Wiley-Interscience
(1987), 2nd edition, Vol. 1, §4.4.
11
R. J. Nunge and W. N. Gill, AIChE Journal, 12, 279-289 (1966).
H. C. Brinkman, Appl. Sci. Research, A2,120-124 (1951); R. B. Bird, SPE Journal, 11, 35-40 (1955);
12
H. L. Toor, Ind. Eng. Chem., 48, 922-926 (1956).