Page 399 - Bird R.B. Transport phenomena

§12.2 Steady Heat Conduction in Laminar, Incompressible Flow 381

Fig. 12.1-4o Variation of the fluid tem-
perature with time after a sphere of ra-
dius R at temperature T is placed in a
]
well-stirred fluid initially at a tempera-
ture T . The dimensionless parameter В
o
is defined in the text following Eq.
12.1-50. [H. S. Carslaw and J. С Jaeger,
Conduction of Heat in Solids, 2nd edi-
tion, Oxford University Press (1959),
p. 241.]

0.04 0.08 0.12 0.16 0.20
a s t/R 2

It can be shown that D(p) has a single root at p = 0, and roots at \fp k = ib (with к = 1, 2,
k
3,..., oo), where the b are the nonzero roots of tan b = 3b /(3 + Bb\). The Heaviside partial
k
k
k
fractions expansion theorem may now be used with
4
N(0) 1/3 N(p k ) IB (12.1-59, 60)
2
D'(0) 1 + В D'ty*) 9(1 + B) + B b\
to get
exp (-b r)
6B2 k (12.1-61)
2 2
=i 9(1 + B) + B b k
Equation 12.1-61 is shown graphically in Fig. 12.1-4. In this result the only place where the
2
thermal diffusivity of the solid a s appears is in the dimensionless time т = a t/R , so that the
s
temperature rise of the fluid can be used to determine experimentally the thermal diffusivity
of the solid. Note that the Laplace transform technique allows us to get the temperature his-
tory of the fluid without obtaining the temperature profiles in the solid.
§12.2 STEADY HEAT CONDUCTION IN LAMINAR,
INCOMPRESSIBLE FLOW
In the preceding discussion of heat conduction in solids, we needed to use only the en-
ergy equation. For problems involving flowing fluids, however, all three equations of
change are needed. Here we restrict the discussion to steady flow of incompressible,
Newtonian fluids with constant fluid properties, for which the relevant equations of
change are:
Continuity (V • v) = О (12.2-1)
Motion p[v • Vv] = (12.2-2)
Energy pC (v-VT) = (12.2-3)
p

4
A. Erdelyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Tables of Integral Transforms, Vol. 1,
McGraw-Hill, New York (1954), p. 232, Eq. 20; see also C. R. Wylie and L. C. Barrett, Advanced Engineering
Mathematics, McGraw-Hill, New York, 6th Edition (1995), §10.9.