Applications of the Formulas 85 



Repeating the calculations for 3, 4 and so on, coverings, we 

 are lead to the general formula : 



2T.Q R<*) (t0) 

 M= 



886. Returning to the formula relative to a single covering: 



z*R'QC(t-0) 

 C' + QR' m (log R' log R) 



if we suppose C' to be very small relatively to Q R' N, the for- 

 mula reduces to 



2 TT C (tQ) 

 m (log R log R' ) 



an expression independent of Q and decreasing as R' increases; 

 thus in this case the transmission does not change with the nature 

 of the surface. If, on the contrary, the value of C' was very 

 large relatively to the following term we would have M= 2 K Q 

 R' ( t 0J, an expression independent of (T'and increasing in 

 proportion to the increase of R' . 



The first supposition is realized with a covering of wool ; the 

 second if we suppose the covering to have almost the conductiv- 

 ity of the metals. 



887. The relation of this value of Mto the quantity of heat 

 which would be transmitted under the same circumstances by a 

 bare pipe is evidently equal to 



C_ R_ 



R C' + QR' m (log R'log R) 



An inspection of this formula will show that it is not always 

 advantageous, as regards the loss of heat, to apply a covering, 

 even one of low conductivity, for the value of this expression is 

 not necessarily less than unity; and for a given value of C' it 

 varies with R and R' . There are certain values of C', belonging 

 to bodies reputed to be poor conductors, which give for ^/values 

 greater than those for a bare pipe, and consequently for these 

 bodies the increase of surface has more effect than the resistance 

 to the transmission of heat through their thickness. 



888. I<et us take for an example a horizontal cast iron 

 pipe, 2.25" radius and one foot long, filled with steam at 212 

 F. ( and covered successively with different thicknesses of hair 



