FE Update - Heat transfer 101

"Why else would it be so infrequently addressed in the engineering trade press?" asks David Dearth, consulting engineer and president of Applied Analysis & Technology, Huntington Beach, Calif. "It seems most engineers are comfortable working with mechanical units and terminology," he says. "For example, units for static or dynamic analysis include force, moment, and torque and have a tangible meaning." But setting up heat-transfer problems involves less intuitive terms, such as Btu, specific heat, and film coefficients. "And the voltage and current analogy often used to explain heat flow doesn't help much either" he adds.

To reacquaint readers with its terms and calculations, here is a heat-transfer problem that will be solved two ways. One way, by hand, will provide a review of heat-transfer phenomena. The second way encourages readers to repeat the problem in their FEA programs and compare results with the hand calculation.

It's not surprising that finite-element methods are often applied to investigate the cooling performance of fans and heat sinks. FEA also allows investigating the effects of temperature changes on stress and deflections in a structure. For simplicity, this article considers only steady-state conditions and leaves transients for another time. "Before any finite-element work, I recommend doing a few sample or warm-up problems with known solutions," says Dearth.

The sample problem is a cooling fin idealized as a uniform rod. The cylindrical fin has a heat source at the base with temperature, T_b, of 250°F. The surrounding still air is 72°F. The fin is fabricated from generic aluminum with thermal conductivity, K, of 130 Btu/(hr-ft-°F). Material properties are assumed constant. Find the temperature along the fin's length and at the tip. "The problem contains the features of any real-life problem and can be found in most heat-transfer texts," he adds.

A first step finds an average heat-transfer film coefficient, h_ave, for free convection to the surrounding ambient air. Let h_ave = 1.30 Btu/(hr-ft²-°F). Detailed calculations for estimating have and other needed values can be found in a .pdf file available from the Machine Design Web site or by requesting a copy from Dearth at [email protected]. The file and textbooks recommended contain calculations for other values such as Prandtl, Grashof, and Nusselt numbers.

Differential equations for a conduction-convection system for the model are in the Holman text (Heat Transfer 5th Edition) in the chapter titled Natural-convection systems. It's a good idea to solve the equations in a math program or spreadsheet. Calculating to four decimal places minimizes round-off and gives a fin tip temperature of T_{fin tip} = 229.9090°F. FEA results, a color contour plot in this case, from MSC.Nastran for Windows, indicate an average tip temperature of 229.9119°F, a difference of only 0.00126%.

"You can solve most heat-transfer problems for simple geometry using classical equations," says Dearth. "However, in real life, geometries are not simple and temperature distributions are not always uniform," he says. One problem with real life is that heat-transfer characteristics of the selected material are often a function of temperature, or may not be the same in orthogonal directions, or both. When material properties are independent of direction along which they are measured, the material is called isotropic. When material properties differ in two or more orthogonal directions, the material is called orthotropic.