Transport Phenomena.pdf

Problems 403
themselves at a reasonably steady temperature. Let the temperature in the ground be T(y, t),
where у is the depth below the surface of the earth and t is the time, measured from the time
of maximum temperature T o
. Further, let the temperature far beneath the surface be T^, and
let the surface temperature be given by
7X0, t) - Т ж
= 0 for t < 0
T(0, t) - T x
= (T o
- T x
) cos 0 (12C.5-1)
Here (o = 2ir/t peT
, in which t per
is the time for one full cycle of the oscillating temperature—
namely, 24 hours. Then it can be shown that the temperature at any depth is given by
(12С.5-2)
This equation is the heat conduction analog of Eq. 4D.1-1, which describes the response of the
velocity profiles near an oscillating plate. The first term describes the "periodic steady state"
and the second the "transient" behavior. Assume the following properties for the soil: 8 p =
1515 kg/m 3 , к = 0.027 W/m • K, and C p
= 800 J/kg • K.
(a) Assume that the heating of the earth's surface is exactly sinusoidal, and find the amplitude
of the temperature variation beneath the surface at a distance y. To do this, use only the
periodic steady state term in Eq. 12C.5-2. Show that at a depth of 10 cm, this amplitude has
the value of 0.0172.
(b) Discuss the importance of the transient term in Eq. 12C.5-2. Estimate the size of this contribution.
(c) Next consider an arbitrary formal expression for the daily surface temperature, given as a
Fourier series of the form
' ' °° = 2 (я„ cos ritot + b n
sin ntot) (12C.5-3)
i 0
- 1 K n
=0
How many terms in this series are used to solve part (a)
12C.6. Heat transfer in a falling non-Newtonian film. Repeat Problem 12B.4 for a polymeric fluid
that is reasonably well described by the power law model of Eq. 8.3-3.
12D.1. Unsteady-state heating of a slab (Laplace transform method).
(a) Re-solve the problem in Example 12.1-2 by using the Laplace transform, and obtain the
result in Eq. 12.1-31.
(b) Note that the series in Eq. 12.1-31 does not converge rapidly at short times. By inverting
the Laplace transform in a way different from that in (a), obtain a different series that is
rapidly convergent for small times. 9
(c) Show how the first term in the series in (b) is related to the "short contact time" solution
of Example 12.1-1.
12D.2. The Graetz-Nusselt problem (Table 12D.2).
(a) A fluid (Newtonian or generalized Newtonian) is in laminar flow in a circular tube of radius
R. In the inlet region z < 0, the fluid temperature is uniform at T v
In the region z > 0, the
wall temperature is maintained at T o
. Assume that all physical properties are constant and
8
W. M. Rohsenow, J. P. Hartnett, and Y. I. Cho, eds., Handbook of Heat Transfer, 3rd edition,
McGraw-Hill (1998), p. 2.68.
9
H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, 2nd edition, Oxford University Press
(1959), pp. 308-310.