Finite element solution of conjugate heat transfer problems with and without the use of gap elements

Abstract

This paper describes the finite element solution of conjugate heat transfer problems with and without the use of gap elements. Direct and iterative methods to incorporate gap elements into a general finite element program are presented, along with their advantages and disadvantages of the two gap element treatments in the framework of finite elements. The numerical performance of the iterative gap element treatment is discussed in detail in comparison with analytical solutions for both 2‐ and 3‐D gap conductance problems. Numerical tests show that the number of iterations depends on the non‐dimensional number Bi = hL/k, and it increases approximately linearly with Bi for Bi≥0.6. Here, for gap heat transfer problems, h is taken to be the inverse of the contact resistance. This conclusion holds true for both 2‐ and 3‐D problems, for both linear and quadratic elements and for both transient and steady state calculations. Further numerical results for conjugate heat transfer problems encountered in heat exchanger and micro chemical reactors are computed using the gap element approach, the direct numerical simulations and analytical solutions whenever solvable. The results reveal that for the standard heat exchanger designs, an accurate prediction of temperature distribution in the moving streams must take into consideration the radial temperature distribution and the accuracy of the calculations depends on the non‐dimensional number Bi = hR/2k. From gap element calculations, it is found that classical analytical solutions are valid for a heat transfer analysis of an exchanger system, only when Bi<0.1. This important point so far has been neglected in virtually all the textbooks on heat transfer and must be included to complete the heat transfer theory for heat exchanger designs. Results also suggest that for thermal fluids systems with chemical reactions such as micro fuel cells, the gap element approach yields accurate results only when the heat transfer coefficient that accounts for the chemical reactions is used. However, when these heat transfer coefficients are not available, direct numerical simulations should be used for an accurate prediction of the thermal performance of these systems.