CHT 000 - Governing equations

What you’ll learn

• The governing equations behind CHT simulations: incompressible Navier-Stokes, advection-diffusion, and heat diffusion.
• How Laminar flow, Heat fluid, and Heat solid physics couple together, and why the temperature field is continuous across fluid-solid interfaces.

Prerequisites: this is a theory reference — no simulation setup is needed. Familiarity with basic fluid dynamics and heat transfer concepts is helpful.

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,

$$\nabla \cdot \boldsymbol{v}_{\mathrm{f}} = 0 \\[5pt]$$ $$\rho_{\mathrm{f}} \left( \frac{\partial \boldsymbol{v}_{\mathrm{f}}}{\partial t} + \boldsymbol{v}_{\mathrm{f}} \cdot \nabla \boldsymbol{v}_{\mathrm{f}} \right) = - \nabla p_{\mathrm{f}} + \nabla \cdot \left( \mu_{\mathrm{f}} \left( \nabla \boldsymbol{v}_{\mathrm{f}} + \left( \nabla \boldsymbol{v}_{\mathrm{f}} \right)^T \right) \right),$$

where

• $\rho_{\mathrm{f}} ~ \rm [kg/m³]$ is fluid density,
• $\mu_{\mathrm{f}} ~ \rm [Pa \cdot s]$ is the dynamic viscosity of the fluid,
• $p_{\mathrm{f}} ~ \rm [Pa]$ is pressure, and
• $\boldsymbol{v}_{\mathrm{f}} ~ \rm [m/s]$ 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,

$$\rho_{\mathrm{f}} \, C_{\mathrm{p,f}} \left(\frac{\partial T}{\partial t} + \boldsymbol{v}_{\mathrm{f}} \cdot \nabla T \right) = \nabla \cdot \left( k_{\mathrm{f}} \nabla T \right),$$

where

• $T ~ \rm [K]$ is fluid temperature,
• $C_{\mathrm{p,f}} ~ \rm [J/(kg \cdot K)]$ is the specific heat capacity of the fluid at constant pressure (assumed constant), and
• $k_{\mathrm{f}} ~ \rm [W/(m \cdot K)]$ 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,

$$\rho_{\mathrm{s}} \, C_{\mathrm{p,s}} \frac{\partial T}{\partial t} = \nabla \cdot \left( k_{\mathrm{s}} \nabla T \right),$$

where

• $\rho_{\mathrm{s}} ~ \rm [kg/m³]$ is the density of the solid,
• $T ~ \rm [K]$ is the solid temperature,
• $C_{\mathrm{p,s}} ~ \rm [J/(kg \cdot K)]$ is the specific heat capacity of the solid at constant pressure (assumed constant), and
• $k_{\mathrm{s}} ~ \rm [W/(m \cdot K)]$ is the thermal conductivity of the solid.

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

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