138 
DE. J. P. JOULE ON THE SUEEACE-CONDENSATION OF STEAM. 
difference of temperatures inside and outside in a length from x to x-\-dx, v the differ- 
ence itself at any point P, k the conducting power of the metal, A the area of the tube 
per unit length, a its thickness. By integrating, we find 
- V kh.x 
Avhere V denotes the difference of temperatures at the entrance end. A will be the area 
2a 
corresponding to a mean diameter calculated by the formula g — , when the outer 
log 
D-2a 
diameter D, and the inner D — 2a differ so much that it will not do to use one or the 
other indifferently. For all practical purposes, with such tubes as are actually used, it 
will do to take as the mean diameter the arithmetic mean D — a. 
“ The truth, however, is that, except with a very great velocity of the water, there will 
be a heated film close to the metal much higher in temperature than the average tem- 
perature of the water in the same section, and the abstraction of heat mil be much 
slower than according to the preceding formula. It is not improbable, however, that 
some law of variation will still hold from point to point in the direction of flow ; and if 
so, the same formula would apply, only that for k something much smaller than the true 
conductivity of the metal must be substituted. Thus, supposing il; to be a function of w, 
smaller the smaller is w and increasing to a limit (the true conductivity of the metal), 
your experiments might give values of k for different rates of the flow of the water by 
the expression 
aw . 
It would be necessary to ascertain by experiment how nearly the geometrical law of 
decrease of the difference of temperatures along the tube holds, as there is no sufficient 
theory for convection to give any decided indication. 
“ As, the results would probably depend but little on the thickness and quality of the 
k 
metal, it would be better perhaps to take - as the thing to be determined : calling it 
C, we have 
y — CA.r 
-TT 
or v=.Yi . 
Ax ° V 
-CA 
s being the base of the nap. log, s is the fraction expressing the reduction of the 
( CA\ 
1 — ^ jlOO is the per-centage of difference 
lost per unit of length. If this be called 0, we have 
v=Y{l-d)% or log p^g=^log^, 
where log denotes any kind of log. These are, in fact, the compound interest formuhe, 
and are perhaps the most convenient for numerical reductions.” 
