Page 145 - Analysis and Design of Energy Geostructures

Heat and mass transfers in the context of energy geostructures 117

The variation of the heat transfer coefficient as a consequence of the development
of the boundary layers has noteworthy consequences for heat transfer analyses.
Varying heat transfer coefficients are associated with different heat transfer rates by
convection. In the context of energy geostructures, relevant heat transfer rates by con-
vection can take place, for instance, between the surface of the energy geostructure,
such as energy walls and tunnels, and adjacent underground built environments, as
well as between the heat carrier fluid circulating in the pipes embedded in energy
geostructures and the surroundings.

3.15 Momentum conservation equation
3.15.1 General

The momentum conservation equation expresses the principle of balance of linear
momentum. This conservation equation is also termed Cauchy’s first law of motion or
the Cauchy momentum equation. When applied to the analysis of problems of internal
and external flows in which due account is made of the equilibrium of the moving
fluid, the momentum conservation equation is typically expressed in the form of the
Navier Stokes equations.

3.15.2 Navier Stokes equations

The Navier Stokes equations can be derived in a similar fashion to the conservation
of mass (see for further details, e.g. Lewis et al. (1996)). For a viscous incompressible
flow they read

2
dρ v rf ;i
f
ð3:53Þ
52 rp f 1 μ r v rf ;i 1 ρ b i
f f
dt
where b i is the vector of body forces. In indicial form
2

ρ @v rf ;k @v rf ;k 52 @p f 1 μ @ v rf ;k ð3:54Þ
2 1 ρ b i
1 v rf ;j
f f f
@t @x j @x k @x
k
The Navier Stokes equations can be simplified to yield the Euler equations for
describing inviscid flows. Together with the mass conservation equation, the
Navier Stokes equations allow the describing of internal and external flow pro-
blems in which due account of the equilibrium of the fluid in motion is made
under isothermal conditions. Under nonisothermal conditions, the energy conserva-
tion equation must be added to the previous equations to solve the problem
addressed.