Conduction Shape Factor and Dimensionless Conduction Heat Rate
Details
In many instances, two-dimensional or three-dimensional conduction problems may be rapidly solved by utilizing existing solutions to the heat diffusion equation. These solutions are reported in terms of a shape factor S or a steady-state dimensionless conduction heat rate qss*. That is, the heat transfer rate may be expressed as:
where ΔT1−2 is the temperature difference between boundaries, as shown in Fig1. It also follows that a two dimensional conduction resistance may be expressed as:
Shape factors have been obtained analytically for numerous two- and three-dimensional systems, and the results for some common configurations are summarized in the references Conduction Shape Factors and Dimensionless Conduction Heat Rates.
The shape factors and dimensionless conduction heat rates reported in the references Conduction Shape Factors and Dimensionless Conduction Heat Rates are associated with objects that are held at uniform temperatures. For uniform heat flux conditions, the object's temperature is no longer uniform but varies spatially with the coolest temperatures located near the periphery of the heated object. Hence, the temperature difference that is used to define S or qss* is replaced by a temperature difference involving the spatially averaged surface temperature of the object (1 − T2), or the difference between the maximum surface temperature of the heated object and the far field temperature of the surrounding medium, (T1,max − T2). For the uniformly heated geometry of Case 10 in the reference Conduction Shape Factors (a disk of diamter D in contact with a semi-infinite medium of thermal conductivity k and temperature T2), the values of S are 3π2D/16 and πD/2 for temperature differences based on the average and maximum disk temperatures, respectively.