# Heat Transfer

AcuSolve supports various features for the modeling of the heat transfer phenomenon.

• Convective Heat Transfer: Heat transfer due to the motion of molecules within a fluid. This includes both natural and forced convection.
• Conjugate Heat Transfer: Heat transfer between solids and fluids.
• Radiation Heat Transfer: Heat transfer due to radiation between surfaces having high temperatures.
• Solar Radiation: Heat transfer due to incident solar radiation on a surface.
• Additional features: Thermal shells, viscous heating and compression heating.

## AcuSolve Heat Transfer Methodology

AcuSolve uses an advective-diffusive equation governing the transport of enthalpy. (1)

This approach permits the conservation of energy with the Galerkin Least Squares formulation.The temperature is derived from the state equation specified for the flow.

There are two common approaches used for the solution of thermal problems in AcuSolve.
• Segregated approach: The enthalpy equation is solved separately from the flow equations. This approach can be further divided into two parts:
• Sequential solving: The enthalpy equation is solved in a sequential manner after solving the coupled flow equations using feed- forward of the velocity field into the enthalpy equation. This approach is appropriate for cases where the temperature field does not affect the flow field significantly. This can offer benefits in speed since the iterations on flow field are not carried along as part of the temperature equation. If the flow field has already converged sufficiently, additional iterations are wasted. This approach also enables “frozen flow” thermal simulations where the flow field is held constant, but different scenarios of the temperature field are investigated.
• Concurrent solving: The enthalpy equation is solved in a sequential manner after the flow equations but there is feedback of the temperature dependent variable from the enthalpy equation back to the flow equations during each iteration. This approach is appropriate for cases involving temperature dependent material properties.
• Coupled approach: The enthalpy equation is coupled into the global system and solved in conjunction with the flow equations. This approach is often times more stable for cases exhibiting a large degree of coupling between the temperature and flow fields, that is, buoyancy driven flows.

AcuSolve simulates the surface to surface radiation using a two step approach:
• View factor computation: In radiative heat transfer view factor is the proportion of radiation incident on one surface due to another surface. The view factors for each facet defining the radiation enclosure are computed using the hemicube algorithm and smoothed using least squares method as a pre-processing step. The view factors are not recomputed during the simulation.
• Heat flux addition: The radiative heat flux, based on view factors, computed using the Stefan-Boltzmann law and the total radiosity, based on Kirchoff's law, is added to the enthalpy transport equation during the solver run.

The enclosure radiation model is supported only on fluid mediums, that is, the fluid side of the fluid/solid surface.

View factor computation is an important point when dealing with moving mesh simulation. Since the view factors are not recomputed during the solver run the boundary elements forming the enclosure should not deform.

AcuSolve uses an ideal grey surface solar radiation model to calculate the solar heat flux. The fluxes are computed using a ray trace algorithm and five optical properties of the surface, specular transmissivity, diffuse transmissivity, specular reflectivity, diffuse reflectivity and absorbtivity. These properties are constrained by the relation: (2)
${\tau }_{s}\left(\theta \right)+{\tau }_{d}\left(\theta \right)+{\rho }_{s}\left(\theta \right)+{\rho }_{d}\left(\theta \right)+\alpha \left(\theta \right)=1$

## AcuSolve Thermal Shell Methodology

AcuSolve permits the definition of thermal shells as material medium to simulate heat transfer in thin solid mediums. This is particularly useful when the thickness of the component makes it inconvenient to use it as a solid medium. The shell medium supports only bricks and wedges.

AcuSolve can model shell layers of varying thickness and material properties. There are two mechanisms by which the heat transfer equation is solved in the thermal shell depending on the number of layers:
• Single Layer: The elements in a single layer shell are considered as full volume elements. The coordinates of the nodes on the opposite faces of the shell are offset by the thickness provided and the heat transport equations are used to determine the heat transfer within the shell. For this scenario the thermal shell includes all heat transfer effects that are present in a three dimensional volume element.
• Multiple Layer: The nodes on the opposite faces have the same coordinates but have different node numbers in order to support the temperature differences across the shell. A one dimensional heat equation, through the shell thickness at the element nodes, is derived by neglecting thermal inertia and conduction parallel to the surface. This means that the heat flux is the same in all the layers. If each layer material model has a constant conductivity then this simplified heat equation is solved exactly. If any of the conductivities is a function of temperature then a two-pass procedure is used to approximately solve the resulting nonlinear system. Once the temperatures are known at the corners of all the layers in each element they can be interpolated to the quadrature points in the usual manner.

## Viscous and Compression Heating

AcuSolve can simulate heat transfer due to viscosity and compressibility of the fluid in case of highly viscous flows and variable density flow models. The terms corresponding to viscous and compression heating are added to the enthalpy transport equation: (3)
where
• $\tau :\nabla \stackrel{\to }{u}$ is the viscous heating term
• $\frac{Dp}{Dt}$ is the material derivative of pressure which describes the compression heating