Page 417 - Bird R.B. Transport phenomena

Problems 399

Fluid approaches
with velocity v^
and temperature T c

Surface of cylinder at
uniform temperature T o Fig. 12B.6. Heat transfer from a long
cylinder of radius R.

temperature T x and velocity v . The cylindrical surface is maintained at temperature T . For
Q
K
3
this system the velocity distribution has been determined by Lamb in the limit of Re << 1.
His result for the region close to the cylinder is
v R sin в
x
~ 2S (12B.6-1)
in which ф is the first polar-coordinate stream function in Table 4.2-1. The dimensionless
quantity S is given by S = \ - у + ln(8/Re), where у = 0.5772 • • • is "Euler's constant/' and
Re = DVoop/fi.
(a) For this system, determine the interfacial velocity gradient (S defined in Example 12.4-3.
(b) Determine the rate of heat loss Q from a length L of the cylinder using the method of Ex-
ample 12.4-3. Note that

.
'sin Odd = B(|,I) = 2.3963 .. (12B.6-2)
where B(m, n) = T(m)T(n)/T(m + n) is the "beta function/'
(c) Determine 8 /R at в - 0, \тт, and тг.
T
. 2v x sin в
Answers: (a) /3 = Kb
(b) Q = C(7TDL)(T 0 - Tjl ^ Д 1 (Evaluate the constant C)

( 9S 1/3
(0¥ = /(«;/= 00,1.1982,(1)
'2\l/3
R VRePr
12B.7. Timetable for roasting turkey
(a) A homogeneous solid body of arbitrary shape is initially at temperature T o throughout. At
t = 0 it is immersed in a fluid medium of temperature T v Let L be a characteristic length in
the solid. Show that dimensional analysis predicts that
© = ®(f, 7/, £,, 7, and geometrical ratios) (12B.7-1)
where в = (T - T Q ) / ^ - T ), f = x/L, rj = y/L, £ = z/L, and т = at 2 . Relate this result to
o
the graphs given in §12.1.

3
H. Lamb, Phil. Mag., (6) 21,112-110 (1911). For a survey of more detailed analyses, see
L. Rosenhead (ed.), Laminar Boundary Layers, Oxford University Press, London (1963), Chapter 4.