, it follows from the fact that the total amount of heat
absorbed is equal to the heat received from radiation plus the heat
removed from the gases by cooling from the temperature T_{1} to the
temperature T_{2}, that

_ _
| / T \ / t \ |
E = 1600 | |----|^{4} - |----|^{4}| S^{1} + WC(T_{1} - T_{2})
|_\1000/ \1000/ _|

This formula can be used for calculating the furnace temperature when E,
t and T_{2} are known but it must be remembered that an assumption
which is probably, in part at least, incorrect is implied in using it or
in using any similar formula. Expressed in this way, however, it seems
more rational than the one proposed a few years ago by Dr. Nicholson[88]
where, in place of the surface exposed to radiation, he uses the grate
surface and assumes the furnace gas temperature as equal to the fire
temperature.

If the heat transfer rate is taken as independent of the gas temperature
and the heat absorbed by an element of the surface in a given time is
equated to the heat given out from the gas passing over this surface in
the same time, a single integration gives

Rs
(T - t) = (T_{1} - t) e^{- --}
WC

where s is the area of surface passed over by the gases from the furnace
to any point where the gas temperature T is measured, and the rate of
heat transfer is R. As written, this formula could be used for
calculating the temperature of the gas at any point in the boiler
setting. Gas temperatures, however, calculated in this way are not to be
depended upon as it is known that the transfer rate is not independent
of the temperature. Again, if the transfer rate is assumed as varying
directly with the weight of the gases passing, which is Reynolds'
suggestion, it is seen that the weight of the gases entirely disappears
from the formula and as a consequence if the formula was correct, as
long as the temperature of the gas entering the surface from the furnace
was the same, the temperatures throughout the setting would be the same.
This is known also to be incorrect. If, however, in place of T is
written T_{2} and in place of s is written S, the entire surface of the
boiler, and the formula is re-arranged, it becomes:

_ _
WC | T_{1} - t |
R = --- Log[89]| --------- |
S |_ T_{2} - t _|

This formula can be considered as giving a way