THERMAL METHODS 969 



From Equation 2 



For convenience, set 



k^ = U.; k^ = Uy; k^=U, (4) 



dx dy ^ ds ^ ^ 



In terms of the new variable, the rate of heat flow into the parallelepiped 

 across the face OABC is : 



Ua Ay A0 



The rate of heat flow out of the parallelopiped across the face DEFG is : 



Therefore, the net rate of flow out of the element in the x direction is : 



UigAyAs— I Ua,-^ -^-^ Ax j Ay As = ^-^ Ax Ay As 



Similarly, the net outward flow in the y and z directions are : 



r-*^ Ax Av As and — - — Ax Ay As 



dy -^ ds -^ 



respectively. 



The total rate of flow out of the element is : 





Replacing U^, Uy, Uz, by their equivalents from Equation 4, the total rate 

 of flow out of the element is : 





But the outward flow is equal in magnitude and opposite in sign to the 

 absorption of heat in the volume element ; therefore, 



k yf^ + ^ + ^) i^i^y^z = ccj'^ AxAyAz 



or 



Af3!I + ^ + 3!l\ Av2T=3i: (5) 



c a\ dx^ dy^ ds^ ) c a "dt 



Equation 5 states that the time rate of the change of temperature at any point 



k 

 in an isotropic homogeneous medium is equal to times the Laplacian of 



C a 



the temperature. Thus, the temperature distribution expressed by Equa- 

 tion 5 involves the specific heat, the thermal conductivity, and the density 



