Page 305 - Thermal Hydraulics Aspects of Liquid Metal Cooled Nuclear Reactors

Turbulent heat transport 275

This chapter reports the status and future perspectives of different approaches to
model the turbulent heat transfer for liquid-metal flows. Section 6.2.1.1 will provide
an understanding of the peculiarities related to the modeling of the heat transfer in
liquid-metal flows. In Section 6.2.1.2, an extensive overview of the available turbulent
heat flux models will be provided along with their pros and cons with respect to
their application to the liquid-metal flows. This is followed by a summary in
Section 6.2.1.3.

6.2.1.1 Understanding the peculiarities of heat transfer
modeling in turbulent liquid metal flows

Before focusing on the turbulent heat flux modeling, it must be kept in mind that
already there are a lot of uncertainties in the turbulence modeling of the velocity fields
(for details, see Gr€ otzbach (2007)). In forced convection flows, for example, channel
and rod bundle flows, these uncertainties can be reduced by selecting the best suited
momentum transfer model. In this regard, a couple of best practice guidelines are
available; see Casey and Wintergerste (2000), Menter et al. (2002), OECD (2007),
and Roelofs (2017). Whereas, in buoyancy dominated flows, when the temperature
is no longer a passive scalar, the temperature field needs to be modeled with utmost
accuracy as it forms a source term for the velocity field.
The Reynolds-averaged momentum and energy equations governing turbulent
flows are as follows:

DU i 1∂P ∂ ∂U i
¼ F i + ν u i u j (6.2.1.1)
Dt ρ∂x i ∂x j ∂x j
DT q ∂ ν ∂T
¼ + θu j (6.2.1.2)
Dt ρc p ∂x j σ T ∂x j

where D ¼ ∂ ∂ is the material derivative, F i is the body force, and q is the inter-
Dt ∂t + U k ∂x k
nal energy source. In these equations, a closure is required for u i u j (turbulent momen-
tum flux) and θu j (turbulent heat flux). The most frequent approaches adopted to close
these equations are the following:

Eddy viscosity model for turbulent momentum flux (u i u j ):
2 ∂U i ∂U j
u i u j ¼ kδ ij ν t + (6.2.1.3)
3 ∂x j ∂x i

Eddy diffusivity model for turbulent heat flux (θu j ):
ν t ∂T
θu i ¼ (6.2.1.4)
Pr t ∂x i