137
homogeneous, isotropic solid with uniform initial temperature placed in
a moving fluid of constant and uniform temperature can be expressed in
terms of the dimensionless temperature ratio, the Biot (Bi) and Fourier
(Fo) numbers as follows (Equation 4-4)
T(z,t) Tb
0 = T(z,O) Tb n c(z) exp(-a2Fo) (4-4)
where an is the nth eigenvalue of the following transcendental
equations for the case of a sphere, an infinite cylinder and infinite
slab.
sphere: Bi = 1 ancot(an) (4-5)
infinite cylinder: Bi = an Ji(an)/Jo(an) (4-6)
infinite slab: Bi = an tan(an) (4-7)
The transcendental equation for the infinite cylinder (Equation 4-6)
contains zero and first order Bessel functions of the first kind (JO
and J1, respectively).
The solutions to the transient heat equation are often presented
graphically in the form of semi-logarithmic plots of the temperature
ratio as a function of the Fourier number. A family of curves is
presented for a particular position within the body for various values
of the reciprocal of the Biot number (Ozisik, 1980). The Biot number
is the ratio of the product of the surface heat transfer coefficient
(hh) and some characteristic length of the sample (L) to the thermal
conductivity of the sample material (X).