Page 141 - Analysis and Design of Energy Geostructures

Heat and mass transfers in the context of energy geostructures 113

flow gradients for higher temperatures. This is another feature characterising the cou-
pling between the thermal and hydraulic behaviours of materials.

3.12 Mass conservation equation
3.12.1 General

The mass conservation equation expresses the principle of conservation of mass. Such
an expression, in particular, establishes a relation between the kinematic characteristics
of a fluid’s motion and the density of the fluid. This conservation equation is also
termed the continuity equation.

3.12.2 Mass conservation equation

The mass conservation equation can be derived for a representative volume in which
mass flows in and out, subjected to arbitrary hydraulic conditions on its surfaces with
internal volumetric mass generation _q per unit time (cf. Fig. 3.19). The balance for
v
the elementary volume, as performed for the energy conservation equation, reads

Rate of mass entering through 1 Rate of mass 5 Rate of mass
the bounding surfaces of a volume generation in a volume storage in a volume

Accordingly, the mass conservation equation reads
@ρ

2 r ρ v rf ;i 1 _q 5 f ð3:41Þ
f v
@t
In many practical cases, no volumetric mass generation is considered. Often the
fluid is also assumed incompressible. The hypothesis of incompressibility indicates that
the density of the fluid remains constant in space and over time. Based on the above,
Eq. (3.41) can be rewritten as

r v rf ;i 5 0 ð3:42Þ

dx
dz
y
dy
ρ υ ∂( ρ υ )
f rf,x dydz ρ υ rf,x + f rf,x
f dx dydz
∂x
x
z
Figure 3.19 Balance of variables over the representative volume.