Page 187 - Thermal Hydraulics Aspects of Liquid Metal Cooled Nuclear Reactors

System thermal hydraulics for liquid metals 159

4.1.1 Turbulent Prandtl number

In turbulent flow, the shear stress at wall surface is obtained as the sum of molecular
and turbulent contributes:

τ w ¼ μ ∂w + ρε m ∂w ¼ ρν + ε m Þ ∂w
ð
∂y ∂y ∂y
y¼0 y¼0 y¼0
where ε m is the eddy diffusivity for momentum transfer (turbulent kinematic viscosity).
Similarly, the wall heat flux is the sum of the molecular and turbulent components:

∂T ∂T ∂T ε m ∂T
00
q ¼ k ¼ ρc p α +
w + ρc p ε q ¼ ρc p α + ε q
∂y ∂y ∂y Pr t ∂y
y¼0 y¼0 y¼0 y¼0
where the ε q is the eddy diffusivity for heat transfer and where the turbulent Prandtl
number, Pr t , is defined as the ratio between heat and momentum eddy diffusivity.
The common assumption that the turbulent Prandtl number, Pr t , is constant and is in
the range of 0.85–1, frequently used in CFD codes, is not valid for liquid metals. In
fact, for liquid metals, the turbulent Prandtl number seems to be more strongly
influenced by the Reynolds and molecular Prandtl number and only slightly by the
distance from the wall (Chen et al., 2013). Many models have been developed for
the turbulent Prandtl number for liquid metals, starting from the one proposed by
Aoki (1963) and the more recent correlation of Cheng and Tak (2005) for liquid metals
and for circular tube geometry:

1 0:45 0:2 1
¼ 0:014 Re Pr 1 exp ð AokiÞ
Pr t 0:014Re 0:45 Pr 0:2
8
4:12 Pe 1000
>
>
>
0:01Pe
>
>
>
> 1000 < Pe 2000 ð Cheng and TakÞ
<
0:8 4 1:25
Pr t ¼ 0:018Pe 1:6+ 9 10 Pe
>
>
0:01Pe
>
>
> 2000 < Pe 6000
>
>
1:25
0:018Pe 3:4
: 0:8
The influence of the Peclet number on the turbulent Prandtl number can be visually
appreciated on Fig. 4.3, where Pr t was reported as a function of the Peclet number for
both the correlations previously mentioned.
4.1.2 Convective heat transfer correlations
The convective heat transfer correlations for a liquid metal flowing inside a circular
pipe can be derived analytically using one of the correlations that gives the turbulent
Prandtl number (see, e.g., Taler, 2016), or it can be obtained experimentally, as in the
case of well-known Lyon (1951) and Seban and Shimazaki (1950) correlations: