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CHT 000 - Conjugate heat transfer equations for incompressible flow

Conjugate heat transfer (CHT) simulations require three physics,

  • Laminar flow
  • Heat fluid
  • Heat solid

as well as their strongly coupled interactions.

Incompressible Navier-Stokes equations

The Laminar flow physics solves the incompressible Navier-Stokes equations,

∇⋅vf=0\nabla \cdot \boldsymbol{v}_{\mathrm{f}} = 0 \\[5pt] ρf(∂vf∂t+vf⋅∇vf)=−∇pf+∇⋅(μf(∇vf+(∇vf)T)),\rho_{\mathrm{f}} \left( \frac{\partial \boldsymbol{v}_{\mathrm{f}}}{\partial t} + \boldsymbol{v}_{\mathrm{f}} \cdot \nabla \boldsymbol{v}_{\mathrm{f}} \right) = - \nabla p_{\mathrm{f}} + \nabla \cdot \left( \mu_{\mathrm{f}} \left( \nabla \boldsymbol{v}_{\mathrm{f}} + \left( \nabla \boldsymbol{v}_{\mathrm{f}} \right)^T \right) \right),

where

  • ρf\rho_{\mathrm{f}} [kg/m³] is fluid density,
  • μf\mu_{\mathrm{f}} [Pa⋅\cdots] is the dynamic viscosity of the fluid,
  • pfp_{\mathrm{f}} [Pa] is pressure, and
  • vf\boldsymbol{v}_{\mathrm{f}} [m/s] is the flow velocity of the fluid.

Advection-Diffusion equation

The Heat fluid physics solves the advection-diffusion equation for the temperature field in the fluid domain,

ρf Cp,f(∂T∂t+vf⋅∇T)=∇⋅(kf∇T),\rho_{\mathrm{f}} \, C_{\mathrm{p,f}} \left(\frac{\partial T}{\partial t} + \boldsymbol{v}_{\mathrm{f}} \cdot \nabla T \right) = \nabla \cdot \left( k_{\mathrm{f}} \nabla T \right),

where

  • TT [K] is fluid temperature,
  • Cp,fC_{\mathrm{p,f}} [J/(kg⋅\cdotK)] is the specific heat capacity of the fluid at constant pressure (assumed constant), and
  • kfk_{\mathrm{f}} [W/(m⋅\cdotK)] is the thermal conductivity of the fluid.

Diffusion equation

The Heat solid physics solves the diffusion equation for the temperature field in the solid domain,

ρs Cp,s∂T∂t=∇⋅(ks∇T),\rho_{\mathrm{s}} \, C_{\mathrm{p,s}} \frac{\partial T}{\partial t} = \nabla \cdot \left( k_{\mathrm{s}} \nabla T \right),

where

  • ρs\rho_{\mathrm{s}} [kg/m³] is the density of the solid,
  • TT [K] is the solid temperature,
  • Cp,sC_{\mathrm{p,s}} [J/(kg⋅\cdotK)] is the specific heat capacity of the solid at constant pressure (assumed constant), and
  • ksk_{\mathrm{s}} [W/(m⋅\cdotK)] is the thermal conductivity of the solid.

The temperature field TT is continuous across the fluid and the solid domains. Thus, the interaction is strongly coupled and no material specific suffixes are required.