Page 135 - Analysis and Design of Energy Geostructures

Heat and mass transfers in the context of energy geostructures 107

this critical number the flow is stable and any potential perturbation triggered tends
to vanish. In contrast, for higher values than the critical number the flow is
unstable and any perturbation triggered (even if minimal) can degenerate in a turbu-
lent mechanism.
For flows over plane surfaces, such as the surface of an energy wall, the critical
5
6
value of the Reynolds number lies in the range 10 # Re c # 3 3 10 (Bergman
et al., 2011). For flows within pipes the critical value of the Reynolds number is
approximately Re c 2000 (Bergman et al., 2011). For seepage flows within soils,
the critical value of the Reynolds number lies in the range 2000 # Re c # 3000,
although Khalifa et al. (2002) report this value to lie in the range 1 # Re c # 10. The
reason for the different aforementioned values is a consequence of the characteristic
length that is considered to describe the analysed problem and may vary in different
situations.
A noteworthy simplification involved with laminar flows is that the contribution
of velocity head can be neglected with respect to the contribution of the elevation
and pressure heads, that is h v {h z 1 h p , and Eq. (3.32) reduces to

H 5 h z 1 h p 5 z 1 p w ð3:34Þ
γ 5 h
w
where h is the piezometric head. The previous simplification cannot be employed
when dealing with problems of turbulent flow because the contribution of the veloc-
ity head significantly characterises the flow process.
The flow of the heat carrier fluid circulating in the pipes of energy geostructures
can be laminar or turbulent. The flow of air in underground built environments adja-
cent to energy geostructures such as energy walls and energy tunnels may also occur
in laminar or turbulent conditions. In contrast, the seepage flow of groundwater in
soils typically occurs under laminar conditions. Turbulent conditions may arise in
highly permeable soils or through fractured rocks.
An example of the negligible magnitude of the velocity head h v ,in contrastto
the piezometric head h 5 h z 1 h p , for seepage flows in laminar conditions can be
reported following the considerations of Vulliet et al. (2016). For a coarse-grained
soil fully saturated with water characterised by pore size diameters at maximum
equal to 5 mm as is usually encountered in practice, the assumption of a critical
Reynolds number of Re c 5 2000 leads to the critical water velocity v w;c 5 0:56 m/s
at which a transition between laminar and turbulent flow conditions occurs (refer-

ence is made to the usual values of water properties at a temperature of T 5 20 C).
2
The velocity head associated with this critical velocity reads h v 5 v =2g 16 mm.
w;c
Therefore as the typical values of piezometric head h in the analysis of groundwater
flow are of the order of metres, the velocity head can be considered negligible
under laminar conditions.