HEAT FROM SMALL CYLINDERS IN A STREAM OF FLUID. 
393 
Section 12. On Possible Sources of Error in the Calculation of Temperatures. 
(i.) Calculation of the Radial Temperature Gradient. 
The question arises as to whether under conditions of rapid rate of cooling there 
may not exist a temperature gradient in the wire of sufficient magnitude to cause the 
temperature as calculated from the apparent resistance of the wire to differ appreciably 
from the surface-temperature.( 34 ) This point can be settled by the following 
analysis :— 
Let p be the specific resistance of the wire at the temperature under consideration, 
e the voltage drop per unit length, constant at all points of the cross-section. The 
heat generated per unit volume is e 2 // 5 and the equation for the radial conduction of 
heat is 
1 8 /V 00\ e 2 , 00 
rC Kr Sr) = -p e U‘ 
(40) 
where K is the heat-conductivity and c the specific heat per unit volume. If H is the 
heat-loss per unit length at the boundary r — a , we have 
H=-2x»K(^.(41) 
Also, if R is the apparent resistance of the wire at temperature 0 we have 
l/R = I (2? rr/p). dr, giving from (40) and (41) when d6/dt — 0, the result H = —e 2 /R 
Jo 
which holds independently of radial temperature gradients or variations of specific 
resistance and heat-conductivity with temperature. The radial temperature gradient 
may be calculated to a first approximation by assuming that the gradient is so small 
that Iv and p may be considered constant throughout the wire. We then have 
l/R = 77 -ct 2 Ip and therefore H = wa 2 . e 2 /p. The integral of (40) consistent with (41) is 
K (90/0?’) = — Hr/27r« 2 , which integrates further to 
0 — 0« — H/4ttIv . (l—r 2 /a 2 ), .(42) 
0 a being the surface temperature at r = a. The maximum temperature difference is 
that between the centre of the wire and the surface and is given by 0 — d a = H/4ttK. 
For platinum the heat-conductivity is K = 0'1664 calories per cm. 2 per sec., or K = 0'698 
watts cm. 2 sec. at 0° C. ; in the case of the largest wire H does not exceed 5 watts 
per cm. at 1000° C., so that we have 0 — 9 a < 5/(47 r x0'698) or 0 — 6 n < 0‘6' C. We 
note that this small temperature gradient justifies the simplifying assumption regarding 
the constancy of K and p over the cross-section of the wire, and conclude, finally, that 
the error in the calculation of the surface temperature introduced by a radial tempe¬ 
rature gradient is within the limits of accuracy of the other measurements. 
( 34 ) A similar problem is dealt with by Kelvin, ‘Roy. Soc Proc.,’ June 10,^1875; also ‘Collected 
Works,’ vol. 3, p. 245. 
3 E 
VOL. CCXIV.-A. 
