# Convective Heat Flux in CFD Problems

### Key Takeaways

• There are three possible heat transfer mechanisms: conduction, convection, and radiation.

• The effectiveness of forced convection depends on the convective heat transfer coefficient, which will depend on the shape and material properties of the system being examined.

• Heat transfer into a fluid causes an equilibrium temperature distribution to develop under steady flow, which must be determined using the Navier-Stokes equations. Electronics, industrial systems, and many other products may need some cooling measures designed into the system to ensure reliability. Systems engineers who remember their physics classes will know there are three mechanisms by which heat naturally moves from hot to cold media: radiation, conduction, and convection. Among these three mechanisms, convection can be driven with forced airflow or with active liquid cooling, both of which require some level of analysis to ensure sufficient heat flux away from a hot system.

When designing a system to have high convective heat flux under forced cooling, the system needs to be treated as a CFD problem, not as a heat equation problem. Only CFD problems can consider the airflow required to drive heat flux away from a system and into its surroundings or cooling media (air, liquid, etc.). In systems with complex geometry, CFD solver applications can help designers determine the level of forced airflow required to reach a convective heat flux goal in their designs. In this article, we’ll give an overview of convective heat flux problems and the solution methodology in complex systems, including systems with airflow.

## Comparing Heat Transfer Mechanisms

Heat transfer will always occur via the three fundamental heat transfer mechanisms, although systems can be designed such that one mechanism dominates over others. Typically, conduction or convection will be the dominant mechanism of heat transfer unless an object is in vacuum and/or at very high temperatures. A summary of the three fundamental heat transfer mechanisms is shown in the table below.

 Heat Transfer Mechanism Description Convection Heat transfer is mediated by a flowing fluid, which carries heat from a hot region and dissipates it into a cold region. Conduction Occurs naturally due to molecular motion when two objects are brought into direct contact. Radiation Also occurs naturally due to molecular motion, but involves the emission of radiation and does not require direct contact. Radiative heat transfer is affected by the emission of electromagnetic radiation, which is used to generate thermal images

Conduction is governed by thermal diffusion and comprehensively described using the heat equation. This equation is not specific to solids, but it is generally used when the medium is static, i.e., the medium is either solid or it is stationary (no flow). This is not the case in fluids, except possibly at very high viscosities; fluids move and provide heat transport, so fluid motion must be considered alongside conduction. The heat equation is simple enough that it can be described using finite element analysis (FEA) in the steady state or with a transient analysis technique in a finite difference time domain (FDTD) simulation.

Radiation can also be brought into a conduction problem by adding a temperature-dependent loss term in the heat equation. In these problems, where the geometry is fixed and the boundary conditions are static, it’s a simple matter to include the Stefan–Boltzmann law in the heat equation, although the problem is more difficult and still requires an FEA solver to arrive at a solution.

## Fluid Flow and Convective Heat Transfer in CFD Problems

Determining convective heat flux and the steady state temperature of a system under steady flow is more difficult. In particular, the numerical techniques involved require simultaneously solving thermal and fluid flow problems. This is due to the fact that:

1. All three mechanisms can operate simultaneously, with convection and conduction typically being of similar magnitude. In isolation, these are governed by different equations.
2. The fluid is moving along some boundary, which may cause the temperature of the boundary to change as it receives or gives off heat. Other boundaries may have fixed temperatures, which will add or remove some heat from the fluid.
3. Real systems that rely on convective heat flux for cooling can be very complex. Advanced numerical techniques that are not found in static thermal or mechanical problems are needed to solve these problems.

Convective heat transfer problems can be broken into two portions:

1. Solving the convective heat flux through a boundary.
2. Determining the temperature field in the system as the fluid carries heat away from the boundary.

### Convective Heat Flux Calculation

The convective heat flux across a boundary is determined by the convective heat transfer coefficient. This coefficient relates the thermal power drawn across a boundary and into cooling/heating fluid, the wetted contact area, and the temperature difference across the boundary: Convective heat flux equation

The convective heat flux is equal to P/A. Convective heat transfer coefficients are cataloged online and in many textbooks for various geometries. Note that if the free-stream temperature T is less than the wall temperature Twall, then P will be negative and heat will be pulled out of the fluid. This calculation assumes that radiation is negligible, otherwise, a term proportional to (T - Twall)4 would appear in the above equation.

The temperature of the free-stream flow (far from the boundary) is often assumed constant, although this might not be the case in a totally bounded flow (e.g., inside a pipe). This is also where a static or dynamic boundary condition becomes important. If the boundary condition and far-field temperature difference are fixed, then the convective heat flux across the boundary will be constant, assuming the wetted area does not change.

### Resulting Temperature Field

Determining the heat flow into the fluid and how an equilibrium temperature field develops requires solving more complex equations. The temperature field in the problem relies on several factors and must be calculated using the Navier-Stokes equations (or Euler’s equations for inviscid flow). This portion of a convective heat transfer problem is best solved using a solver application. A solver application can generate flow and temperature maps like this, where the streamlines are clearly visible and overlaid on the temperature distribution

Multiphysics systems analysis tools are needed to fully capture convective heat flux, conduction, and radiation from complex systems. The complete set of CFD simulation tools in Omnis 3D Solver are ideal for building and solving CFD problems and thermal problems in complex systems with industry-standard numerical techniques.