Page 124 - Analysis and Design of Energy Geostructures

96 Analysis and Design of Energy Geostructures

Figure 3.12 (A) Cylindrical and (B) spherical coordinate systems.

Eq. (3.14) presents a unique unknown: the temperature, T. Conduction heat
transfer problems can thus be fully addressed by solving Eq. (3.14).
Because the solution of Eq. (3.14) is in general difficult to obtain, it is often
assumed that the thermal conductivity of the medium is constant throughout it so that
the equation reduces to

2 @T
ð3:15Þ
λr T 1 _q 5 ρc p
v
@t
2
2
where r is the Laplace operator. The quantity r T is as follows in various coordi-
nate systems.
• Cartesian coordinates x, y, z:
2 2 2
2
r T 5 @ T 1 @ T 1 @ T ð3:16Þ
2 2 2
@x @y @z
• Cylindrical coordinates r, θ, z (cf. Fig. 3.12A):
2 2 2
2
r T 5 @ T 1 1 @T 1 1 @ T 1 @ T ð3:17Þ
2 r @θ 2 2
2
@r r @r @z
• Spherical coordinates r, θ, φ (cf. Fig. 3.12B):
2 1 @ 1 2
2
r T 5 @ T 1 2 @T 1 sinθ @T 1 @ T ð3:18Þ
2
2
2 r sinθ @θ @θ r sin θ @φ 2
2
@r r @r
3.7.3 Fourier heat conduction equation for no volumetric thermal
energy generation
If no heat is generated within the medium, Eq. (3.15) reduces to

2
α d r T 5 @T ð3:19Þ
@t