Page 116 - Analysis and Design of Energy Geostructures

88 Analysis and Design of Energy Geostructures

The rate equation governing convection is Newton’s law of cooling. According to
, reads
this law, the heat flux density generated by convection, _q
conv
5 h c T s 2 T N Þ ð3:5Þ
ð
_ q
conv
where h c is the convection heat transfer coefficient (also termed boundary or film con-
ductance), T s is the surface temperature and T N is the fluid temperature. In this case it
is assumed that the convection heat flux is positive if heat is transferred from the sur-
face (T s . T N ) and negative if heat is transferred to the surface (T s , T N ). However,
5 h c T N 2 T s Þ.
ð
the heat flux may also be expressed as _q
conv
Eq. (3.5) is typically employed in the context of the analysis of internal and exter-
nal flows. In the context of the analysis of seepage flows, Newton’s law of cooling is
expressed as
q_ 5 ρ c p;f v rf ;i T s 2 T N Þ ð3:6Þ
ð
conv;i f
where c p;f and ρ are the specific heat and density of the fluid, respectively, and v rf ;i is
f
the average relative velocity vector of the fluid with respect to the solid skeleton.
Eq. (3.6) is employed in the analysis of groundwater seepage to characterise the heat
transported by water motion with reference to the specific heat and density of the
water, c pw and ρ , respectively, and the average relative velocity of water with respect
w
to the solid skeleton, v rw (cf. Fig. 3.8). A similar equation can be obtained for an air-
filled medium by replacing the material parameters of the considered gas as well as its
relative velocity with respect of the solid skeleton, v ra .
The density of fluids, similar to the density of solid materials, varies with tempera-
ture (Bergman et al., 2011). Because density variations can influence convection mass
transfer, the considered feature represents an example of the coupling between heat
and mass transfers. Such a feature may be considered in the analysis and design of
energy geostructures.

Figure 3.8 Sketch of the convection heat transfer in a geomaterial.