Page 193 - Thermal Hydraulics Aspects of Liquid Metal Cooled Nuclear Reactors

164 Thermal Hydraulics Aspects of Liquid Metal Cooled Nuclear Reactors

From a theoretical point of view, in the averaging process of the differential
balance equations, the local instantaneous information is lost, and so, there is the
need.

l to reintroduce information on the effect of turbulent fluctuations,
l to reintroduce information on local gradients and related fluxes.
For these reasons, to “close” the problem, the phasic balance equations must be also
supplemented by.
l “jump conditions” expressing the continuity of mass, momentum, and energy across the
liquid-gas interface;
l state relationships between thermodynamic variables (equation of state, EOS);
l constitutive laws to evaluate specific terms (friction factor, convective heat transfer coeffi-
cient, etc.).
In STH codes, one-dimensional equations are used. They are obtained averaging the
3-D balance equations on the space over a conveniently short piece of a duct with
impermeable walls and variable cross section (see Fig. 4.5).
As an example, the balance equations used in RELAP5 1D STH code (RELAP5/
Mod.3.3, 2003) are typically written as reported in the following:
Continuity equations
∂ 1 ∂
ð α k ρ Þ + ð α k ρ w k AÞ ¼ Γ k ð k ¼ f, gÞ
k
k
∂t A∂z
The overall continuity consideration at the interface between liquid (f ) and vapor (g)
yields to the requirement that the liquid mass generation term must be the negative of
the vapor generation (Γ f ¼ Γ g ).
Momentum equations

1 ∂w 2 ∂p 1 f k,w
∂w k
α k ρ k + α k ρ k k ¼ α k α k ρ g cos ϑ α k ρ k w k w k j
j
k
∂t 2 ∂z ∂z 2 D
+ Γ k w k,i w k Þ + F k,i + F k,vm ð k ¼ f, gÞ
ð
In the previous equation, the subscript i refers to the phase interface, and w refers to the
wall. Several STH codes include the “virtual mass effect” (the term F k,vm ) that occurs

Fig. 4.5 Elementary control volume for a one-dimensional flow in a channel.