CONTINUOUS ELECTRIC CALORIMETRY 
111 
but it is possible to estimate the order of magnitude of the differences involved, 
which is all that is really required for our purpose. 
Assuming that the internal and external diameters of the flow-tube are 2 and 
G millims. respectively, and length 50 centims., and that the conductivity of the 
glass is ‘0020 cal. C.G.S., it is easy to calculate that the mean difference of 
temperature between the internal and external surfaces would be of the order of 
one-tenth of a degree only when the final rise of temperature is 8°, and the heat loss 
•050 watt per degree, as in the majority of our experiments. It will therefore 
evidently be unnecessary to take account of this in any of the calculations. 
To find the radial distribution of temperature in the liquid, assuming the flow to 
be linear, I will take first the case of the metallic flow-tube, which is much the 
simplest. The differential equation of the radial distribution of temperature, neglecting 
tbe minute effect of longitudinal conduction, is 
d (kr ddI dr) I dr — + vcr dd/dx .(1), 
in which k is the thermal conductivity of the liquid, and c the specific heat per unit 
volume, v the velocity of the stream, d the temperature, r the distance from the axis, 
and x the distance along the tube. The velocity v is a function of r, which can be 
easily calculated if the viscosity is assumed constant. As a matter of fact, both 
the viscosity and the conductivity vary rapidly with change of temperature. The 
viscosity at 100° is nearly six times less than at 0° C., and its variation is accurately 
known. But if we assume both conductivity and viscosity constant (as we are 
practically compelled to do, since the variation of conductivity with temperature is 
quite uncertain) we shall obtain a solution which is sufficiently simple to be useful, 
and which can be strictly applied to small changes of temperature. 
To simplify the solution still further, I shall assume the longitudinal temperature 
gradient dd/dx constant over the cross-section of the tube at any point, and equal 
to O'/l, where d' is the rise of temperature observed in a length I. This will not 
be true near the inflow end of the tube, where the radial distribution of temperature 
is rapidly changing, but it will very fairly represent the limiting state, which is 
attained when the liquid has flowed along the tube for some distance. 
If the flow is linear, and the viscosity constant, the velocity at any point of the 
cross-section is given in terms of r by the equation 
v 
= 2V (1 - (r/r 0 Y) = 2Q (I - (r/r 0 ) 2 )/nr 0 
( 2 ), 
where V is the mean velocity, Q the flow in cub. centims. per second, and r 0 the 
internal radius of the flow-tube. 
Making this substitution in (l) and integrating, we obtain 
k r dd/dr 
06/ / U 
7T/ W 
( 3 )- 
