Chapter 5 UMERICAL METHODS IN HEAT CONDUCTION

Transcription

1 Heat Transfer Chapter 5 UMERICAL METHODS IN HEAT CONDUCTION Universitry of Technology Materials Engineering Department MaE216: Heat Transfer and Fluid

2 bjectives Understand the limitations of analytical solutions of conduction problems, and the need for computationintensive numerical methods Express derivates as differences, and obtain finite difference formulations Solve steady one- or two-dimensional conduction problems numerically using the finite difference method Solve transient one- or two-dimensional conduction problems using the finite difference method

3 Y NUMERICAL METHODS? In Chapter 2, we solved various heat conduction problems in various geometries in a systematic but highly mathematical manner by (1) deriving the governing differential equation by performing an energy balance on a differential volume element, (2) expressing the boundary conditions in the proper mathematical form, and (3) solving the differential equation and applying the boundary conditions to determine the integration

4 itations Analytical solution methods are limited to highly simplified problems in simple geometries. The geometry must be such that its entire surface can be described mathematically in a coordinate system by setting the variables equal to constants. That is, it must fit into a coordinate system perfectly with nothing sticking out or in. Even in simple geometries, heat transfer problems cannot be solved analytically if the thermal conditions are not sufficiently simple. Analytical solutions are limited to problems that are simple or can be simplified with

5 tter Modeling attempting to get an analytical solution hysical problem, there is always the ncy to oversimplify the problem to make athematical model sufficiently simple to nt an analytical solution. fore, it is common practice to ignore any that cause mathematical complications s nonlinearities in the differential ion or the boundary conditions earities such as temperature dence of thermal conductivity and the ion boundary conditions). hematical model intended for a numerical n is likely to represent the actual m better. umerical solution of engineering ms has now become the norm rather

6 lexibility ineering problems often require extensive parametric studies nderstand the influence of some variables on the solution in r to choose the right set of variables and to answer some t-if questions. is an iterative process that is extremely tedious and timeuming if done by hand. puters and numerical methods are ideally suited for such ulations, and a wide range of related problems can be solved inor modifications in the code or input variables. y it is almost unthinkable to perform any significant ization studies in engineering without the power and flexibility mputers and numerical methods.

7 mplications problems can be solved analytically, e solution procedure is so complex and sulting solution expressions so icated that it is not worth all that effort. he exception of steady one-dimensional sient lumped system problems, all heat ction problems result in partial ntial equations. g such equations usually requires matical sophistication beyond that ed at the undergraduate level, such as onality, eigenvalues, Fourier and e transforms, Bessel and Legendre ns, and infinite series. h cases, the evaluation of the solution, often involves double or triple ations of infinite series at a specified

8 man Nature Analytical solutions are necessary because insight to the physical phenomena and engineering wisdom is gained primarily through analysis. The feel that engineers develop during the analysis of simple but fundamental problems serves as an invaluable tool when interpreting a huge pile of results obtained from a computer when solving a complex problem. A simple analysis by hand for a limiting case can be used to check if the results are in the proper range. In this chapter, you will learn how to formulate and solve heat transfer problems numerically using one or more approaches.

9 ITE DIFFERENCE FORMULATION DIFFERENTIAL EQUATIONS umerical methods for solving differential ions are based on replacing the ntial equations by algebraic equations. case of the popular finite difference d, this is done by replacing the tives by differences. we demonstrate this with both first- and d-order derivatives. AMPLE

10 finite difference form of the first derivative Taylor series expansion of the function f about the point x, The smaller the x, the smaller the error, and thus the more accurate the approximation.

11 er steady one-dimensional heat conduction in a plane wall of thickness L at generation. Finite difference representation of the second derivative at a general internal node m. no heat generation

12 Finite difference formulation for steady twodimensional heat conduction in a region with heat generation and constant thermal

13 -DIMENSIONAL STEADY HEAT DUCTION ection we develop the finite difference tion of heat conduction in a plane wall e energy balance approach and how to solve the resulting equations. ergy balance method is based on ing the medium into a sufficient of volume elements and then g an energy balance on each element.

14 equation is applicable to each of the interior nodes, and its application M - 1 equations for the determination peratures at M + 1 nodes. wo additional equations needed to for the M + 1 unknown nodal eratures are obtained by applying the y balance on the two elements at the

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16 ndary Conditions ary conditions most commonly encountered in practice are the ied temperature, specified heat flux, convection, and radiation ary conditions, and here we develop the finite difference formulations m for the case of steady one-dimensional heat conduction in a plane f thickness L as an example. ode number at the left surface at x = 0 is 0, and at the right surface at t is M. Note that the width of the volume element for either boundary is x/2. ified temperature boundary condition

17 other boundary conditions such as the specified heat flux, convection, tion, or combined convection and radiation conditions are specified at a dary, the finite difference equation for the node at that boundary is obtained iting an energy balance on the volume element at that boundary. inite difference form of various ary conditions at the left boundary:

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19 Schematic for the finite difference formulation of the interface boundary condition for two

20 ting Insulated Boundary Nodes as Interior Nodes: Mirror Image Concept The mirror image approach can also be used for problems that possess thermal symmetry by replacing the plane of symmetry by a mirror. Alternately, we can replace the plane of symmetry by insulation and consider only half of the medium in the solution. The solution in the other half of the medium is simply the mirror image of the solution obtained.

21 AMPLE de 1 de 2

22 t solution:

23 inite difference formulation of y heat conduction problems lly results in a system of N raic equations in N unknown l temperatures that need to be d simultaneously. are numerous systematic aches available in the literature, hey are broadly classified as t and iterative methods. irect methods are based on a number of well-defined steps that t in the solution in a systematic er. terative methods are based on an guess for the solution that is d by iteration until a specified rgence criterion is satisfied.

24 of the simplest iterative methods is the Gauss-Seidel iteration.

25 -DIMENSIONAL STEADY HEAT DUCTION Sometimes we need to consider heat transfer in other directions as well when the variation of temperature in other directions is significant. We consider the numerical formulation and solution of two-dimensional steady heat conduction in rectangular coordinates using the finite difference method.

26 are mesh:

27 ndary Nodes egion is partitioned between the s by forming volume elements d the nodes, and an energy ce is written for each boundary. ergy balance on a volume ent is ssume, for convenience in lation, all heat transfer to be into the e element from all surfaces except ecified heat flux, whose direction is dy specified.

28 EXAMPLE Node 1 Node 2

29 Node 3 4 Node 5 Node 6

30 9 Nodes 7, 8

31 gular Boundaries Many geometries encountered in practice such as turbine blades or engine blocks do not have simple shapes, and it is difficult to fill such geometries having irregular boundaries with simple volume elements. A practical way of dealing with such geometries is to replace the irregular geometry by a series of simple volume elements. This simple approach is often satisfactory for practical purposes, especially when the nodes are closely spaced near the boundary. More sophisticated approaches are available for handling irregular boundaries, and they are commonly incorporated into the commercial software packages.

32 NSIENT HEAT CONDUCTION te difference solution of transient s requires discretization in time in to discretization in space. done by selecting a suitable time step olving for the unknown nodal tures repeatedly for each t until the at the desired time is obtained. ient problems, the superscript i is used ndex or counter of time steps, with i = 0 onding to the specified initial condition.

33 t method: If temperatures at the previous p i is used. t method: If temperatures at the new time 1 is used.

34 sient Heat Conduction in a Plane Wall mesh Fourier number

35 t generation and =0.5 mperature of an interior node at time step is simply the average emperatures of its neighboring at the previous time step.

36 ility Criterion for Explicit Method: Limitation on t plicit method is easy to use, but it suffers n undesirable feature that severely restricts y: the explicit method is not unconditionally and the largest permissible value of the time is limited by the stability criterion. me step t is not sufficiently small, the ns obtained by the explicit method may e wildly and diverge from the actual solution. id such divergent oscillations in nodal atures, the value of t must be maintained a certain upper limit established by the ty criterion. Example

37 The implicit method is unconditionally stable, and thus we can use any time step we please with that method (of course, the smaller the time step, the better the accuracy of the solution). The disadvantage of the implicit method is that it results in a set of equations that must be solved simultaneously for each time step. Both methods are used in practice.

38 EXAMPLE it finite difference formulation Node 1 Node 2

39 it finite difference formulation Node 1 Node 2

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41 Dimensional Transient Heat Conduction

42 Stability criterion Explicit formulation

43 MPLE 1

44 Node 2 Node 3 Node 4

45 Node 5 Node 6 Nodes 7, 8

46 Node 9

47 eractive SS-T-CONDUCT Software SS-T-CONDUCT (Steady State and Transient Heat duction) software was developed by Ghajar and his workers and is available from the online learning ter ( to the instructors and dents. software is user-friendly and can be used to solve ny of the one- and two-dimensional heat conduction blems with uniform energy generation in rectangular metries discussed in this chapter. transient problems the explicit or the implicit solution thod could be used.

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49 Summary Why numerical methods? Finite difference formulation of differential equations One-dimensional steady heat conduction Boundary conditions Treating Insulated Boundary Nodes as Interior Nodes: The Mirror Image Concept Two-dimensional steady heat conduction Boundary Nodes Irregular Boundaries Transient heat conduction Transient Heat Conduction in a Plane Wall Stability Criterion for Explicit Method: Limitation on t Two-Dimensional Transient Heat Conduction