Analytical Methods for Heat Transfer and Fluid Flow Problems

Fluid flow and heat transfer problems are present in all of our daily life. For example, if we walk along a river and look at the water flowing with high speed over the river-bed, we actually observe a fluid mechanics problem. If we put some sugar into our coffee and stir it, we are faced with a complicated heat and mass transfer problem. In particular, convective heat transfer problems are present everywhere in our world. Most of the problems encountered in fluid mechanics or heat transfer are described by partial differential equations. One good example of such equations are the Navier-Stokes equations and the energy equation for an incompressible flow with constant fluid properties. If we consider a three-dimensional, steady-state problem, these equations read where Φ _Dis denotes the dissipation function in the energy equation. This function is given by

Several important heat and fluid flow processes in technical applications and in nature can approximately be described by linear partial differential equations. As stated in the previous chapter, linear partial differential equations are normally much simpler to solve than nonlinear partial differential equations. In addition, a large body of literature exists on how to solve linear PDEs.

The method of separation of variables has been briefly explained in the last chapter by solving two simple examples. In general, this solution method leads to eigenvalue problems which have to be solved either analytically or numerically. For most technical problems, where analytical solutions based on the method of separation of variables are still obtainable, these eigenvalue problems might become quite difficult.

The numerical method for obtaining the eigenvalues and eigenfunctions of Eq. (3.25) is explained in Chap. 3 and Appendix C. This numerical procedure gets cumbersome and difficult if the eigenvalues become large, because of the increasing number of zero points of the eigenfunction. This is elucidated in Fig. 4.1, which shows several eigenfunctions for a turbulent pipe flow with constant wall temperature.

In Chapters 3–4, we have been concerned with heat transfer in hydrodynamically fully developed flow in a pipe or a planar channel. There we made the assumption that the axial heat conduction within the fluid can be ignored. This assumption implies that the Peclet number (Pe _D = Re _D Pr) = for the problem under consideration has to be large enough. If one neglects the axial heat conduction term in the energy equation, the partial differential equation (3.5) changes its type from elliptic to parabolic. This means that the original energy equation (3.5) is elliptic, whereas the simplified equation (e.g. Eq. (3.14)) is parabolic. This has the consequence that a boundary condition for the parabolic problem has to be prescribed at the entrance into the heated section at x = 0 (see Fig. 3.1). This shows clearly, that for parabolic problems no information can be transferred upstream of the heated zone into the unheated zone (x < 0). As stated above, this is a good approximation, if the Peclet number is large (Pe _D > 100) and thus axial heat conduction effects in the fluid can be neglected. However, for several technical applications the Peclet number is small, so that the axial heat conduction within the fluid has to be taken into account. A typical situation for this case is a compact heat exchanger using liquid metals as working fluid. For liquid metals, the Prandtl number is very low (0.001 < Pr < 0.1) and therefore the Peclet number may attain very small values. In addition to the above mentioned example, there exist other types of applications, where one has to include the effect of axial heat conduction within the fluid, even for larger Peclet numbers.

In the previous chapters, we discussed the solution of linear partial differential equations. Special focus was given to the solution of internal heat transfer problems in duct flows. However, in most technical applications, problems are often described by nonlinear partial differential equations. For a lot of these applications, the equations have to be solved by numerical methods. In contrast to the large amount of literature dealing with the solution of linear partial differential equations, much less literature exists on the solution of nonlinear partial differential equations. One of the major difficulties arising in the solution of nonlinear problems is that we are no longer able to use the powerful superposition method for constructing solutions as for linear problems. Sometimes, the equations under consideration may be linearised by using perturbation methods. An example on how to use this sort of method is shown in Chap. 4 for the solution of eigenvalue problems. In the present chapter, we do not discuss this solution approach. The interested reader is referred to the books of van Dyke (1964), Kevorkian and Cole (1981), Simmonds and Mann (1986), Aziz and Na (1984) and Schneider (1978) for many interesting applications of the perturbation method to fluid dynamics and heat transfer problems. In the following sections, we intend to provide a short overview on some selected solution methods for nonlinear partial differential equations occurring in heat transfer and fluid flow problems. The solution approaches discussed here include, for example, the method of separation of variables, the Kirchhoff transformation, and special solutions of the energy equation.