Page 159 - Analysis and Design of Energy Geostructures

Heat and mass transfers in the context of energy geostructures 131

The Cauchy’s boundary condition also allows prescribing a radia-
tion boundary condition for conduction problems. In this case, the
considered condition takes the form

@T
2 λ ð H; tÞ 5 Eσ SB ðT ðtÞ 2 T ðH; tÞÞ
4
4
N
@n i
where the term on the left side of the equation represents the heat flux
2

vector [W/m ]in n i direction, T N ðtÞ [ C] and T H;tÞ [ C] are the tem-
ð
peratures of the source and surface, respectively, σ SB is the
4
Stefan Boltzmann constant [W/(m C )] and E [ ] is the surface
2
emissivity.
dd. The temperature drop across the interface between materials can be
appreciable: this phenomenon is attributed to the presence of a thermal
contact resistance. The existence of a finite contact resistance is due
principally to surface roughness effects, where the contact spots are
separated by gaps filled with a fluid. When these gaps are filled with a
gas, heat transfer is due to conduction across the actual contact area
and to conduction and/or radiation across the gaps. When the gaps are
filled with a liquid, heat transfer is due to conduction across the actual
contact area and to conduction and/or convection across the gaps.
For a unit area of a considered interface, the thermal contact resis-
tance is defined as:
2
T A 2 T B m C
00
R 5 _ q x W :
c
where T A and T B [ C] are the temperature of the surfaces separated by

2
the gap and _q [W/m ] is the heat flux in x-direction.
x
ee. As it can be observed in the cases where a contact resistance is present,
the contact area is typically small, and, particularly for rough surfaces,
the main contribution to the resistance is due to the gaps. The contact
resistance can be reduced by increasing the area of the contact spots in
the case of solids with a thermal conductivity exceeding that of the
interfacial fluids. This increase is affected by increasing the joint pres-
sure and/or by reducing the roughness of the mating surfaces.
Selecting an interfacial fluid of large thermal conductivity can also
reduce the contact resistance. With respect to this, no fluid (an evacu-
ated interface) eliminates conduction across the gap, increasing the
contact resistance.