HEAT FROM SMALL CYLINDERS IN A STREAM OF FLUID. 
381 
When the variable x is large, equation (24) enables us to write approximately 
y = l/(27r) + \A/(27r).(29) 
Thus when the term n/3 0 is sufficiently large and the term l/(8w/3 0 ) is small in 
comparison with unity, equation (21) takes the form 
H = k6 0 + k 6 0 \/m(3 0 = k0 o + \/itks<t/3 0 "V^6 0 .(30) 
Expressions (27) and (30) hold for cylinders of any shape for which the constant /3 0 
can be calculated from the hydrodynamical problem. It is easily proved in the case 
of an elliptic cylinder of semi-axes (a, b) that 
/3 0 = 2{a + b) .(31) 
This result holds good independently of the direction of the axes of the cross-section 
with that of the stream and may be utilized in the special cases of calculating the 
heat-loss from a circular cylinder or a strip of breadth 2 a. The case of a circular 
cylinder of radius a represents the conditions of the experiments described in Part II. 
of the present paper; writing /3 0 = 4 a we obtain finally the approximate formulae ( 12 ) 
for the heat-loss 
Small velocities . . . H = 27r/c0 o /[log (&/«)],.(32) 
Large velocities . . . H = k6 0 + 2v w/cso-a V^@ 0 .(33) 
The limits within which these approximate formulae represent the values of the heat- 
loss given by the exact expression (21) is examined in the description of Diagram I. 
( 12 ) It is interesting to compare (32) with Langmuir’s formula for free convection, 
H = 2»r (<f> 2 - <fo)/[log b/a], 
where 4> denotes a function of the thermal conductivity and the temperature given by the relation 
<p = * dd. This result has been shown by Langmuir to represent with fair accuracy the results of his 
Jo 
experiments on the free convection from small platinum wires (‘Phys. Rev.,’ 34, p. 401, 1912). 
Interpreted in the light of equation (32) the term b of Langmuir’s equation represents a term depending 
on the “ effective ” velocity Y of the free convection current set up by the heated wire. 
Equation (33) may be compared with that derived by Kennelly (‘Trans. A.I.E.E.,’ 26, p. 969, 1907, 
and ‘Trans. A.I.E.E.,’ 28, p. 368, 1909) as a result of his experiments on the forced convection of heat 
from small copper wires. In the notation of the present paper Kennelly’s formula may be written 
in the form 
H = (C + B d) 8 0 JY + v Q , 
where C and B are constants given by C = 300 x 10~ 7 , B = 5 - 8 x 10~ 8 , the heat-loss H being measured 
in watts per unit length, the temperature difference 6 0 being expressed in degrees C. and the velocity Y in 
cm./sec. v 0 is a constant whose value is v 0 = 25 cm./sec., and d is the diameter of the wire in cm. It 
must be noticed that in deriving the above formula no correction was made in the experiments for 
the “ swirl ” of the rotating arm in the determination of the velocity. 
