Page 77 - Thermal Hydraulics Aspects of Liquid Metal Cooled Nuclear Reactors

52 Thermal Hydraulics Aspects of Liquid Metal Cooled Nuclear Reactors

The Prandtl number does not appear explicitly in Eqs. (3.1.11), (3.1.12), but it is the
ratio of P eclet and Reynolds, and is therefore implicitly in the nondimensional form of
the equations.

3.1.2.2 Natural convection for pool-type investigation

In passive heat removal conditions, a mass flow established as a consequence of a bal-
ance between frictional pressure losses of the hydraulic loop and Archimedes forces
induced by the heating and cooling power (Welander, 1967). The pressure head gen-
erated can be defined as (Todreas and Kazimi, 1990)

ΔP B ¼ gHβρ T h T c Þ: (3.1.13)
ð
Natural convection flows are complex and often characterized by large gradients of
velocity, thus predicting a value of characteristic velocity V ch may not be trivial
for such flows (Ieda et al., 1985). Because in natural convection loop (NCL) systems,
the buoyant pressure head should balance the pressure drops (Todreas and Kazimi,
1990), the expression for the characteristic velocity V ch can be determined by
balancing the pressure term with the buoyant term:

r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð
ð
Hgβ T h T c Þ 1 ΔP ch Hgβ T h T c Þ
¼ ¼ Eu ¼) V ch ¼ , (3.1.14)
V 2 ρ V 2 Eu
ch 0 ch
where Eu is the Euler number for pressure drops already defined in the previous par-
agraph. In the NCL, flow rate, pressure drops, and buoyancy forces are not indepen-
dent quantities, thus it is not possible to know a priory V ch and ΔP ch . In the rigorous
nondimensional analysis, in order to escape this problem, the characteristic velocity
V ch (or the characteristic pressure drops ΔP ch ) is chosen in order to obtain Eu ¼ 1. For
0.5
the present case, Eu ¼ 1 implies V ch ¼ (HgβΔT) . Replacing this expression of V ch in
the normalized equations of momentum and energy (3.1.11), (3.1.12) we obtain the
following nondimensional equations for the steady state:

r ffiffiffiffiffiffiffiffiffi 2 2 2
∂U ∂U ∂P 1 H ∂ U ∂ U
U + V ¼ + + , (3.1.15)
2
∂X ∂Y ∂X Gr H L ∂X 2 ∂Y 2
r ffiffiffiffiffiffiffiffiffi 2 2 2
∂V ∂V ∂P 1 H ∂ V ∂ V
U + V ¼ + + , (3.1.16)
2
∂X ∂Y ∂Y Gr H L ∂X 2 ∂Y 2
∂θ ∂θ ∂P r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi H ∂ θ ∂ θ
2
2 2
1
U + V ¼ + + : (3.1.17)
2
∂X ∂Y ∂X Pr Gr H L ∂X 2 ∂Y 2
The appearing dimensionless groups are
2
2
ND 1 ¼ A Gr 0:5 , ND 2 ¼ A Pr Gr H Þ 0:5 , (3.1.18)
ð
R H R