84 Applications of the Formulas 



per foot of length, in one hour, R and R' the internal and exter- 

 nal radii of the cylindrical covering in feet, / and /' the tempera- 

 tures of the internal and external surfaces, and the temperature 

 of the surroundings. When the regime is established the quantity 

 of heat which passes through the covering is equal to that which 

 at the same time traverses an infinitely thin annular element of 

 radius r, this quantity being equal to the product of the surface 

 2 *r of this element, the conductivity C' of a thickness of one 

 foot of the material, and the temperature difference dt of the two 

 surfaces, divided by the distance apart dr of these surfaces. We 

 will have then : 



M= ; / from which C 1 d t= 



dr 2 iir 



The minus sign shows that the variations of the tempera- 

 ture and of the radius of the covering take place in opposite 

 directions. Integrating the last equation between the limits / 

 and /' for d t, and R and R' for d r it becomes : 



C (t f ) = M -m (logR'logR) 



27T C (tf) 



and M= j^- 



where m is the modulus of the table of logarithms and equals 

 2.3025 and N represents m (log R' log R). 



But we also know that M=2 - R' Q ( t' t) ) : eliminating /' 

 from these equations we have : 



_ 27T R' Q (t0) 



>Q*'N ' (a) 



C' 



If there are two contiguous coverings, designating by x the 

 temperature of the common surface, we have: 



M .. ~ , , M ,. 



C (tx)= N; C,' ( xt' ) = N' 



27T 2 7T 



and M=z * R" Q ( t' ) equations which give, after eliminat- 

 ing*.- 



M= 2 -^ *" " * } 



