THERMAL METHODS 



967 



in drill holes, to study the vertical distribution of temperature along the 

 drill hole. (See Chapter XI.) 



Mathematical Theory of Heat Flow 



As in the case of flow of electrical current through a solid medium, 

 it is convenient to derive the differential equation for the uni-dimensional 

 flow of heat and then generalize the equa- 

 tion for three-dimensional flow. Physically, 

 uni-dimensional flow is illustrated by the 

 flow through a sheet or slab of mate- 

 rial having two dimensions considerably 

 greater than the third dimension. 



The theoretical discussion will be con- 

 fined to a homogeneous and thermally iso- 

 tropic medium. 



Referring to Figure 593, assume that 

 the planes x = and x = I are at tempera- 

 tures To and Ti respectively and that 

 To > Ti. According to Fourier's law the 



quantity of heat which flows in the x direction across unit area in a time t is 

 proportional to the product of the time and the temperature gradient. 



z 



T. 



To^Tj 



Fig. 593. — Sketch illustrating uni- 

 dimensional heat flow in x direction. 



where 



q — kt 



To-Tt 



(1) 



To-T 



q = quantity of heat 



t = time 



/ = thickness of slab 



- = temperature gradient 



k = a. constant, called the thermal conductivity of the material. 



k may be a function of position ; in general, however it is determinate 

 at each point, when the temperature is known for a homogeneous medium 

 and for moderate ranges of temperature. 



In differential form, the equation for uni-dimensional heat flow in the 

 X direction is : 



Equation 2 states that the time rate at which heat is transported across 

 unit area of a plane perpendicular to the x direction is equal to the product 

 of the thermal conductivity and the thermal gradient in the x direction. 



The equation for three-dimensional flow may be obtained from Equa- 



