Modeling and Approximation in Heat Transfer

Convection is heat conduction into a medium that moves. The motion is generally nonuniform, and we are often most concerned with heat flow through a wall perpendicular to the direction of the flow.

The modeling of convection heat transfer always involves two elements. The first is to determine the local rate of heat transfer from wall to fluid. The second is to determine the streamwise energy balance on the fluid. Usually, these two elements are coupled because streamwise temperature variations affect the local temperature differences that drive heat transfer. Only occasionally will the rate of heat transfer be set independently, such as by electrical heating of the wall.

The determination of convection heat transfer rates always begins with the identification of the flow field, and, in particular, it requires one to distinguish between laminar and turbulent flow. Laminar flows often admit relatively simple modeling, in which an estimate of the local boundary layer thickness may be used to determine local convection thermal resistance by analogy to heat conduction. In turbulent flow, thermal resistance has a less obvious relationship to boundary layer thickness, and empirical data must usually be applied. The local laminar thermal resistance is relatively sensitive to the upstream thermal history and wall boundary conditions, whereas the turbulent thermal resistance is not.

For internal flows, bulk flow energy conservation provides a powerful tool for assessing streamwise variations in temperature and other properties. For external flows, the streamwise growth of the boundary layer is of primary importance, requiring consideration of the changing velocity field as well as the temperature history. Such problems can sometimes be approached using the integral energy conservation equation.

In addition to situations that have a mean flow past a wall, some situations of technical importance involve stirred tanks or free surfaces above turbulent pools in which heat transfer to the bulk is driven by turbulent mixing with negligible mean flow. Surface renewal models can be applied to estimate heat transfer coefficients in these cases if certain information about the mixing process is available.

Buoyancy-driven flows add a further level of complexity, in that the fluid velocity depends entirely on the temperature differences that are present during convection. Here the velocity and temperature fields must usually be analyzed simultaneously.