Page 420 - Bird R.B. Transport phenomena

402 Chapter 12 Temperature Distributions with More Than One Independent Variable

What is the final expression for Z(f)? (Note: In this problem it has been assumed that a phase
change occurs instantaneously and that no supercooling of the liquid phase occurs. It turns
out that in the freezing of many liquids, this assumption is untenable. That is, to describe the
solidification process correctly, one has to take into account the kinetics of the crystallization
6
process. )
7
12C4. Viscous heating in oscillatory flow. Viscous heating can be a disturbing factor in viscosity
measurements. Here we see how viscous heating can affect the measurement of viscosity in
an oscillating-plate system.
A Newtonian fluid is located in the region between two parallel plates separated by a
distance b. Both plates are maintained at a temperature T . The lower plate (at z = 0) is made
o
to oscillate sinusoidally in the z direction with a velocity amplitude v 0 and a circular fre-
quency o). Estimate the temperature rise resulting from viscous heating. Consider only the
high-frequency limit.
(a) Show that the velocity distribution is given by

/sinh я(1 - О cos a{\ - f) sinh a cos a \
\+ sin a(\ — f) cosh a(\ — f) sin a cosh a)
I - sin a(\ — £) ( - £) £) sinh a cos a\
£) cosh a(\
(
h
{\
h ( l
v (x, t) \+smhfl(l - О cos a{\ - f) sin a cosh a) )
О
f)
z
v 2 2 2 2
o sinh a cos a + cosh a sin a
2
where a = \/ро)Ь /2/х and f = x/fr.
(b) Next calculate the dissipation unction^ for the velocity profile in Eq. 12C.4-1. Then ob-
f
tain a time-averaged dissipation function Ф , by averaging over one cycle. Use the formulas
у
2
2
cos cot = sin (at = \ and sin wt cos cot = О (12С.4-2)
which may be verified. Then simplify the result for high frequencies (i.e., for large values of a)
to obtain
Ф, (large a>) = Ajje'^ (12C.4-3)
(c) Next take a time average of the heat conduction equation to obtain
0 = к <Ц- + /xO y (12C.4-4)
dx l
in which Г is the temperature averaged over one cycle. Solve this to get

e
2
2
T
T ~ ° = \^Г Ш ~ e" *) ~ 0- ~ ~ °W (12C.4-5)
This shows how the temperature in the slit depends on position. From this function, the max-
imum temperature rise can be calculated. For reasonably high frequencies, T — T l
o
12C.5. Solar heat penetration. Many desert animals protect themselves from excessive diurnal tem-
perature fluctuations by burrowing sufficiently far underground that they can maintain

6 H. laneschitz-Kriegl, Plastics and Rubber Processing and Applications, 4,145-158 (1984);
H. Janeschitz-Kriegl, in One-Hundred Years of Chemical Engineering (N. A. Peppas, ed.), Kluwer Academic
Publishers, Dordrecht (Netherlands) (1989), pp. 111-124; H. laneschitz-Kriegl, E. Ratajski, and G. Eder,
Ind. Eng. Chem. Res., 34, 3481-3487 (1995); G. Astarita and J. M. Kenny, Chem. Eng. Comm., 53, 69-84
(1987).
7 R. B. Bird, Chem. Eng. Prog. Symposium Series, Vol. 61, No. 58 (1965), pp. 13-14; see also F. Ding,
A. J. Giacomin, R. B. Bird, and C-B Kweon, J. Non-Newtonian Fluid Mech., 86,359-374 (1999).