Page 161 - Analysis and Design of Energy Geostructures

Heat and mass transfers in the context of energy geostructures 133

kk. In analogy with the heat transfer, where the potential variable is the
temperature, total head, H [m], characterises mass transfer.
ll. The expression of the total head, for a fluid such as water, reads
p v 2
H 5 h z 1 h p 1 h v 5 z 1 w 1 w
γ w 2g

where z is the elevation of a considered fluid particle above a reference
plane [m], p is the fluid pressure [Pa], γ is the unit weight of the
w
w
3
fluid [N/m ], v w is the velocity of the fluid at a point on a streamline
2
[m/s] and g is the gravitational acceleration [m/s ].
mm. When the contribution of the velocity head can be neglected with
respect to the contribution of the elevation and velocity heads (i.e.
laminar flow) the total head can be reduced to the piezometric head as
p
H 5 h z 1 h p 5 z 1 w 5 h
γ w

where z [m] is the elevation of a considered fluid particle above a refer-
3
ence plane, p [Pa] is the fluid pressure, γ [N/m ] is the unit weight
w
w
of the fluid and h [m] is the piezometric head.
nn. The Reynolds number can be mathematically expressed as
ρ v N x
Re x 5 f
μ f

3
where ρ [kg/m ] is the density of the fluid, v N [m/s] is the character-
f
istic velocity of the fluid, x [m] is the characteristic length of the prob-
lem and μ [Pa s] the dynamic viscosity of the fluid.
f
oo. The Reynolds number represents the ratio of the inertia to viscous
forces. If the Reynolds number is small, inertia forces are insignificant
relative to viscous forces. The disturbances are then dissipated, and the
flow remains laminar. For a large Reynolds number, however, the
inertia forces can be sufficient to amplify the triggering mechanisms,
and a transition to turbulence occurs.
pp. Knowing that the volumetric flow rate can be expressed as _ V 5 Av z
3
[m /s], the velocity in z-direction can be calculated as
V _ 3:5
v z 5 5 5 7m=s
A 0:5
2
2
From the formula of a circle cross-sectional area A 5 πd =4[m ],
p
the diameter of the pipe can be calculated as