The Heating Coefficients of Rheostats. 
preferable to employ the approximate formula 2 assuming A! to he constant 
for the purpose of integration only. We then obtain 
db_ i^ftsl - ATlb 
dt~ GGcsl ^• 
Integrated 
,^|^^,_,-CG.| 8. 
This is the well-known law of the heating curve. 
The integration holds for constant A' and p. The variation of p for 
most resistance materials is negligible, but that of A' may be 20 or more per 
cent. To be able to integrate we should use an average value or proceed in 
steps. In practice it is the maximum temperature rise, which is of import- 
ance, for which A' may be more easily assumed correctly. 
When Ph=Pr no further rise takes place, and we get 
P„=Ph=A'Sc(S^-5,)=A'ScS„ 9.- 
whence 
10. 
and 
Ph _ IV^ _ _i'^s 
^A'Sc~A'P/6~ATs~AT ' 
"ca."l .... 11 
5 = ■ 
Let 
_CGrs_CGsZ_CGs?^_heat required to reach final temp. 
"""A'P^A'PZ^A'P/S ~ rate of generation of heat 
=heat required to reach actual final temperature if there were no cooling. 
Owing to the latter, the actual temperature rise only reaches ^1 — ^^,or 
63*4 per cent, of its steady value in time t^. We call the time constant. 
Hence 
t 
b^K[l- ^ " C| 13- 
It will be obvious from these equations that the accuracy of the pre- 
calculation of the temperature rise of a rheostat, or the size of its conductors 
for a given rise and specified load conditions, will entirely depend upon the 
correct assumption of A' or ^ These can, however, be obtained from tests 
alone, and even then great care has to be exercised, as A' may vary consider- 
ably when simply turning the rheostat over, whereby the ventilation is 
affected. 
In figs. 2, 3, 4, and 5 values of A' and ^ have been plotted from tests 
carried out. 
