Page 75 - Thermal Hydraulics Aspects of Liquid Metal Cooled Nuclear Reactors

50 Thermal Hydraulics Aspects of Liquid Metal Cooled Nuclear Reactors

3.1.2 Scaling and theory

The general equations of continuity, momentum transport, and energy transport for an
incompressible fluid can be expressed in the vector form

∂ρ
+ r ρ u ¼ 0; (3.1.1)
∂t
∂ u 1 g
+ u r u ¼ v rp + r νr uÞ + ; (3.1.2)
ð
∂t ρ ρ

∂T
ð
ρc p + u rT ¼r krTÞ + Q: (3.1.3)
∂t
If the pool geometry is approximated as axial symmetric, the dominating components
of velocity are v (vertical) and u (radial). Under this assumption (Spaccapaniccia,
2016), and making the hypothesis of equal heat provided and removed, steady flow,
and applying the Boussinesq approximation for which density variations are due to
temperature only, Eqs. (3.1.1)–(3.1.3) become

∂u ∂v
+ ¼ 0, (3.1.4)
∂x ∂y
2
2
∂u ∂u 1 ∂p x ∂ u ∂ u
u + v ¼ + ν + ν , (3.1.5)
∂x ∂y ρ ∂x ∂x 2 ∂y 2
0
2
2
∂v ∂v 1 ∂p y ∂ v ∂ v
u + v ¼ + ν + ν gβ T h T c Þ, (3.1.6)
ð
∂x ∂y ρ ∂y ∂x 2 ∂y 2
0
2
2
∂T ∂T ∂ T ∂ u
u + v ¼ α + , (3.1.7)
∂x ∂y ∂x 2 ∂y 2
where α is the thermal diffusivity of the fluid. The next step is to introduce character-
istic quantities in order to normalize the variables appearing in Eqs. (3.1.4)–(3.1.7):
x y u v
X ¼ ; Y ¼ ; U ¼ ; V ¼ ;
L H U ch V ch
(3.1.8)
p x p y T T c
P x ¼ 2 ; P y ¼ 2 ; θ ¼ ,
ρ U ρ V
0 ch 0 ch T h T c
where L, H are, respectively, the horizontal and vertical distance between core and
heat exchangers, U ch and V ch are characteristic velocities. The same velocities are
used to normalize the pressure. Finally, T ch and T h are the coldest and the hottest tem-
peratures achievable in the loop. Replacing the normalized quantities of Eq. (3.1.8)in
Eqs. (3.1.4)–(3.1.7), we obtain the nondimensional transport equation