Page 158 - Analysis and Design of Energy Geostructures

130 Analysis and Design of Energy Geostructures

bb. To determine the temperature distribution in a medium, it is necessary
to solve the appropriate form of the energy conservation equation.
However, such a solution depends on the physical conditions existing
at the boundaries of the medium and, if the situation is time depen-
dent, on conditions existing in the medium at some initial time. With
regard to the boundary conditions, there are several common possibili-
ties that are simply expressed in mathematical form. Because the heat
equation is second order in the spatial coordinates, two boundary con-
ditions must be expressed for each coordinate needed to describe the
system. Because the equation is first order in time, however, only one
condition, termed the initial condition, must be specified. If in steady-
state conditions, no initial condition is needed.
cc. Usually, three types of boundary conditions are prescribed in heat
transfer.
The so-called Dirichlet’s boundary condition or boundary condition
of the first kind allows fixing the temperature of any surface as

ð
T H; tÞ 5 f ðH; tÞ

where H is a point on the considered surface, f ðH;tÞ [ C] is a pre-
scribed function and t [s] is the time.
The so-called Neumann’s boundary condition or boundary condi-
tion of the second kind allows fixing a heat input. Based on Fourier’s
law, this boundary condition takes the form

@T
2 λ ðH; tÞ 5 _qðH; tÞ
@n i
where λ [W/(m C)] is the thermal conductivity, T [ C] is the temper-

2
ature, n i is the normal to the surface at the point H and _q [W/m ]is
the heat flux.
The so-called Cauchy’s boundary condition, mixed Neumann’s
boundary condition or boundary condition of the third kind allows
prescribing a convection boundary condition for conduction problems.
In this case the considered condition takes the form

@T
2 λ ð H; tÞ 5 h c ½T N 2 T H; tފ
ð
@n i
where h c [W/(m 2 C)] is the convection heat transfer coefficient and

T N ðtÞ [ C] and T H;tð Þ [ C] are the temperature of the source and sur-
face, respectively.