Page 231 - Thermal Hydraulics Aspects of Liquid Metal Cooled Nuclear Reactors

202 Thermal Hydraulics Aspects of Liquid Metal Cooled Nuclear Reactors

0

ij
u
β ¼ (5.40)
u ij
Although the velocity fluctuation is undirected and the net exchange of fluid volume
across the gap is zero, the net exchange of mass, momentum, and enthalpy between
subchannels through the gap does not vanish and is expressed in subchannel analysis
programs as below:

0
m ¼ β u ij ρ ρ s ij (5.41)
ij m i j

0
M ¼ β u ij u i ρ u j ρ s ij (5.42)
M
j
i
ij
0
H ¼ β u ij h i ρ h j ρ s ij (5.43)
H
j
i
ij
The coefficients β m , β M , and β H are the turbulent mixing coefficients for mass,
momentum, and energy, respectively. Because they cannot be directly determined
by experiments, conventional SCTH programs often substitute a single coefficient
β for all three values. This conventional coefficient β is either a mixing coefficient
derived from measurements of the velocity fluctuations or the thermal diffusion coef-
ficient derived from the enthalpy distribution in subchannels. The net exchange of
mass, momentum, and energy are defined by the following:
∂ρ

0 0
m ¼ u l t,m s ij △z (5.44)
ij ij
∂l

ð
∂ ρ uÞ
0 0
M ¼ u l t,M s ij △z, (5.45)
ij
ij
∂l

∂ ρ hÞ
ð
0 0
H ¼ u l t,H s ij △z,respectively: (5.46)
ij ij
∂l
The parameters l t,m ,l t,M , and l t,H are the equivalent turbulent mixing length of mass,
momentum, and energy, which are unknown. Also the gradient of density, momen-
tum, and enthalpy across the gap is unknown. To make Eqs. (5.44)–(5.46) applicable
to SCTH codes, the production of the gradient with the mixing length is replaced by
∂ρ
l t,m ¼ f m ρ ρ j (5.47)

i
∂l
∂ ρ uÞ
ð
l t,M ¼ f M ρ u i ρ u j (5.48)
i
j
∂l
∂ ρ hÞ
ð
l t,H ¼ f H ρ h i ρ h j (5.49)
j
i
∂l