88 Applications of the Formulas 



other diameters, the same phenomena recur, but at other thick- 

 nesses. 



891. *. 



892. In all that precedes in regard to cylindrical coverings, 

 I have supposed the value of Q to be constant, that is to say that 

 Newton's law of cooling held true for any excess of temperature ; 

 but this is not the case as we have seen in 805, and the value of 

 Q increases quite rapidly with the excess of temperature ; thus the 

 different values of M calculated by the formula of 885 , can only 

 be considered approximations, the more exact the smaller they 

 are, since the excess of temperature decreases with M. But it is 

 easy in any particular case to obtain a value of the quantity of 

 heat which is very close to the truth. Suppose that we have first 

 calculated the value of M by the formula (885); dividing this 

 result by 2 K R' Q we will have the temperature /' of the surface, 

 and by means of the curves of figures i, 2 and 3, we may deduce 

 a new value Q l of Q, then new values M^ and / 1} and we may 

 repeat the process, until two consecutive values of M are the 

 same. 



I will take for an example a horizontal cast iron pipe 4.5" 

 outside diameter covered with one -half inch of hair felt, filled 

 with steam at 212 and surrounded by air at 59. We have 

 already found that for this case ^/equals 94.9. Since the surface 

 per foot run is 1.18 sq. feet and Q was equal to 1.28 we have 



//'_0i = 94^ =62.8 



' 1.18X1.28 



We know that ^=.75 and A"' = .53 then from figures i, 2 

 and 3, 



Qi= -75Xi. 22X1 + .53X1. 25 = i. 58 



Then ^ = 102 and (t' O) 1 =54.7 then 2 = i -55 and 

 M 2 = 101.5 



Thus the value of Afis 101.5 instead of 94.9, not a large 

 difference, and it would have been much smaller if the covering 

 had been thicker. 



893. In general when the pipes are of small diameter, the 

 second term of the denominator of the expression for M is very 

 large in relation to the first term, at least when the covering is a 



*The table given in 891 is not reproduced as it has but little value for modern condi- 

 tions. 



