968 



EXPLORATION GEOPHYSICS 



tion 2 by replacing -r— by the temperature gradient appropriate to three 

 dimensions. This yields : 



9i 

 dt 



< 



dT _j^dT , dT 

 dx dy dz 



) 



(3) 



This equation implies that heat flow may be represented by a vector 

 whose X, y, and z components are, respectively : 







d2 



It is likewise apparent from the equation that the flow field, being a gradi- 

 ent, is everywhere perpendicular to the isothermal surfaces, T = constant. 



Rate of Temperature Change 



Equation 3 can be simplified by transforming it into a second order 

 differential equation in which T is the only independent variable. To 

 carry out this transformation, it is necessary to introduce the specific heat, 



c, of the material in which the heat flow 

 is assumed to occur. The specific heat 

 is defined as the quantity of heat ex- 

 pressed in calories required to raise a 

 unit mass of the material by 1°C. 



Consider a volume element having 

 the shape of a rectangular parallelo- 

 piped and sides of lengths A,r, Ay, and 

 A^, located in an infinite, homogeneous, 

 isotropic medium of density a, specific 

 heat c and heat conductivity k. (Fig- 

 ure 594). When the temperature of 

 the volume element increases by an 

 amount AT, the quantity of heat ab- 

 sorbed by the volume element changes 

 by an amount equal to the product of 

 the specific heat, the mass, and the 

 change in temperature. Hence, the time rate of change of absorption is 



dT 

 Ca Ax Av A^r -;^— 

 -^ dt 



If it is assumed that the volume element does not contain any heat 

 sources or sinks, the rate at which heat flows into the volume element 

 must equal the rate at which heat flows out of the volume element. The 

 rate at which heat flows into the volume element across the face OABC is : 



Z 



Fig 



594. 



Sketch illustrating three-dimen- 

 sional heat flow. 



UJ- 



Ay A^r 



