where Pe 
> B, sin@, cos 6, 
enn a ae ee [127] 
1-5 B, sin20, 
1 
n= 
z 0° 2 co 2 
Ato) = (: - > B, sin? ) a S B, sin@, cos *) {128] 
n=1 n=1 
tan @, = —® [129] 
In preparing to make the calculations for the time constants it was observed that 
'$*(b) was very small if k, 2100 «, and if u2b2<0.01. Also k,/o,, is small for the first 
few values of n. Therefore, most of the calculations were made for Ky = Kp/ On =0% 
(x,/x,) 5, = 0 and $*(5) = 0. For this approximation the problem is independent of yu? b? 
and 
Mau AA LIC, (CB Gin) 
2 [130] 
Z(C,r)= 2 Cy, Cy 8.057) 
The functions C'),(z,y) and Cy, (z, y) are tabulated in Tables 1, 2, and 3 for three coating 
thicknesses: a/b =1.5, 2, and 3. The data for obtaining the time constants are presented 
in Tables 9 through 11. Curves showing how the time constant depends upon the coating 
thickness and P/k, are presented in Figure 8. ' 
If the thermal conductivity of the wire is not large compared with that of the coating, 
k,/o, and (x,/k,) 5, cannot be neglected. In order to find out how the thermal conductivity 
of the coating affects the wire response, calculations of the time constant were made for 
a/b = 2, u? 62 = 0.01, and for kK, =100 ky and 10 Kae The data for these calculations are 
given in Tables 12 and 13. The results for x, = 100 x, were almost the same as the results 
for xk, = ©. The curves of Figure 9 show how the time constant depends upon P/x, and 
K,/ Ko. 
The response of the bare constant-temperature hot wire to a step-like change in con- 
vective cooling is itself a step-function with an initial overshoot and appears to be quite 
unlike the response of the coated wire. Inthis case a time constant has no meaning and the 
frequency response is flat. If Q/x, is replaced by its series expansion in Equation [99], 
the bare-wire response is 
31 
