Page 414 - Bird R.B. Transport phenomena

396 Chapter 12 Temperature Distributions with More Than One Independent Variable

points at distances у г and y 2 from the periodically heated surface. Show that the thermal dif-
fusivity may then be estimated from the formula

2

(b) Calculate the thermal diffusivity a when the sinusoidal surface heat flux has a frequency
0.0030 cycles/s, if y - У\ = 6.19 cm and the amplitude ratio A- /A 2 is 6.05.
[
2
Answer: a = 0.111
12A.6. Forced convection from a sphere in creeping flow. A sphere of diameter D, whose surface is
maintained at a temperature T , is located in a fluid stream approaching with a velocity v x and
o
temperature Т . The flow around the sphere is in the "creeping flow" regime—that is, with the
ж
Reynolds number less than about 0.1. The heat loss from the sphere is described by Eq. 12.4-34.
(a) Verify that the equation is dimensionally correct.
(b) Estimate the rate of heat transfer, Q, for the flow around a sphere of diameter 1 mm. The
fluid is an oil at T x = 50°C moving at a velocity 1.0 cm/sec with respect to the sphere, the sur-
face of which is at a temperature of 100°C. The oil has the following properties: p = 0.9 g/cm ,
3
C = 0.45 cal/g • КД = 3.0 X 10" cal/s • cm • K, and /x = 150 cp.
4
p
126.1. Measurement of thermal diffusivity in an unsteady-state experiment. A solid slab, 1.90 cm
thick, is brought to thermal equilibrium in a constant-temperature bath at 20.0°C At a given
instant (t = 0) the slab is clamped tightly between two thermostatted copper plates, the sur-
faces of which are carefully maintained at 40.0°C. The midplane temperature of the slab is
sensed as a function of time by means of a thermocouple. The experimental data are:
f(sec) 0 120 240 360 480 600
T(C) 20.0 24.4 30.5 34.2 36.5 37.8
Determine the thermal diffusivity and thermal conductivity of the slab, given that p =
3
1.50 g/cm and C p = 0.365 cal/g • С
2
3
4
Answer: a = 1.50 X 10~ cm /s;к = 7.9 X 10~ cal/s-cm-Cor 0.19 Btu/hr • ft • F
12B.2. Two-dimensional forced convection with a line heat source. A fluid at temperature T x
flows in the x direction across a long, infinitesimally thin wire, which is heated electrically at
a rate Q/L (energy per unit time per unit length). The wire thus acts as a line heat source. It is
assumed that the wire does not disturb the flow appreciably. The fluid properties (density,
thermal conductivity, and heat capacity) are assumed constant and the flow is assumed uni-
form. Furthermore, radiant heat transfer from the wire is neglected.
(a) Simplify the energy equation to the appropriate form, by neglecting the heat conduction
in the x direction with respect to the heat transport by convection. Verify that the following
conditions on the temperature are reasonable:

^
T->T X as у oo for all x (12B.2-1)
P s: C (T T - = T X x x a t x < 0 for all x > 0 (12Б.2-2)
all у
(12B.2-3)
for
= Q/L
TJ\ v dy
p
(b) Postulate a solution of the form (for x > 0)
T(x, y)~T = f(x)g(rj) where 77 = y/8(x) (12B.2-4)
K
Show by means of Eq. 12B.2-3 that f(x) = C /8(x). Then insert Eq. 12B.2-4 into the energy
}
equation and obtain

(с) Set the quantity in brackets in Eq. 12B.2-5 equal to 2 (why?), and then solve to get 8(x).