Page 473 - Bird R.B. Transport phenomena

Problems 453
The terms involving derivatives with respect to 77 and ф Q = -2fcf(n • VT) dS
have been omitted because they are not needed. Show that - TJJXn • V0) dS
this equation may be solved with the boundary conditions
that 0(f ) = 0 and ©(00) = 1 to obtain = 2k(T 0 - TJ ( ( R 2 cos y] sin y]di)dijj
o
Jo \TTR COS 17
ГТГ — arctan(sinh £)
0 = 1 - (14D.1-7) = 8ЩТ - TJ (14D.1-9)
0
\TT - arctan(sinh ) and that the Nusselt number is given by Nu = 16/77 =
f 0
(c) Next, specialize this result for the two-sided disk (that 5.09. Since Nu = 2 for the analogous sphere problem, we
f
is, the limiting case that 0 = 0), and show that the normal see that the Nusselt number for any oblate ellipsoid must
temperature gradient at the surface is lie somewhere between 2 and 5.09.
(d) By dimensional analysis show that, without doing any
1
= (14D.1-8) detailed derivation (such as the above), one can predict
(n-V0)| surf
77
R COS 7]
that the heat loss from the ellipsoid must be proportional
where a has been expressed as R, the disk radius. Show fur- to the linear dimension a rather than to the surface area. Is
ther that the total heat loss through both sides of the disk is this result limited to ellipsoids? Discuss.