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104 Analysis and Design of Energy Geostructures

where T N ðtÞ and T H;tÞ are the temperature of the source and surface, respectively.
ð
This boundary condition renders the analytical solution of the problem extremely dif-
ficult so that two simplifications are often used (Boley and Weiner, 1997) to reduce
Eq. (3.27) to Eq. (3.26), provided that T N ðtÞ is very high compared to TH; tÞ during
ð
the period of interest or neither temperature varies over too wide a range.

3.8.6 Interface boundary condition
If two solid bodies are in perfect thermal contact (cf. Fig. 3.17A), their temperature at
the surface contact must be the same. Moreover the heat flux leaving one body
through the contact surface must be equal to that entering the other body. In this
case, for a point H on the contact surface

T 1 H;tð Þ 5 T 2 ðH;tÞ ð3:28Þ

@T 1 @T 2
λ 1 @n i ð H;tÞ 5 λ 2 @n i ð H;tÞ ð3:29Þ

where 1 and 2 are the labels for the two bodies and n i is the common normal to the
contact surface at H.
If two solid bodies are not in perfect thermal contact (cf. Fig. 3.17B), the concept
00 00
of contact resistance, R (or contact conductance h c 5 1=R ), is often used. The equal-
c c
ity of heat fluxes must still be enforced but a difference between the two surface tem-
peratures, proportional to the heat flux, will now exist (Boley and Weiner, 1997).
The appropriate boundary conditions are in this case

@T 1 1
ð
λ 1 ð H;tÞ 5 T 2 H;tð Þ 2 T 1 H;tÞ ð3:30Þ
00
@n 1 R
c
@T 1 @T 2
λ 1 ð H;tÞ 5 λ 2 ð H;tÞ ð3:31Þ
@n 1 @n 1
where n 1 is the normal to the contact surface at H referred to body 1.

(A) (B) Body 1
Body 1
Mathematical
Body 2 Body 2 common
boundary
Figure 3.17 Schematic representation of two solid bodies (A) in perfect contact and (B) not in
perfect contact.