Chapter 5NUMERICAL METHODS IN HEAT

CONDUCTION

Heat Transfer

Universitry of Technology Materials Engineering DepartmentMaE216: Heat Transfer and Fluid

ObjectivesUnderstand the limitations of analytical solutions of conduction problems, and the need for computation-intensive 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

HY NUMERICAL METHODS?In Chapter 2, we solved various heat conduction problems in variousgeometries 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

t t

mitationsAnalytical 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 thevariables equal to constants. That is, it must fit into a coordinate systemperfectly 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

bl i ti

etter Modelingn 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.

efore, it is common practice to ignore any s that cause mathematical complicationsas nonlinearities in the differential tion or the boundary conditionsinearities such as temperature ndence of thermal conductivity and the tion boundary conditions).

thematical model intended for a numericalon is likely to represent the actual em better.

numerical solution of engineering ems has now become the norm rather

lexibility

gineering problems often require extensive parametric studiesnderstand the influence of some variables on the solution in er to choose the right set of variables and to answer some at-if” questions.

s is an iterative process that is extremely tedious and time-suming if done by hand.

mputers and numerical methods are ideally suited for such culations, and a wide range of related problems can be solved minor modifications in the code or input variables.

ay it is almost unthinkable to perform any significant mization studies in engineering without the power and flexibility omputers and numerical methods.

omplicationse problems can be solved analytically, e solution procedure is so complex and

esulting solution expressions so licated that it is not worth all that effort.

the exception of steady one-dimensional nsient lumped system problems, all heat uction problems result in partialential equations.

ng such equations usually requires ematical sophistication beyond that red at the undergraduate level, such as gonality, eigenvalues, Fourier and ce transforms, Bessel and Legendre ons, and infinite series.

ch cases, the evaluation of the solution, h often involves double or triple mations of infinite series at a specified

uman Nature Analytical solutions are necessary because insight to the physical phenomena and engineering wisdomis 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.

NITE DIFFERENCE FORMULATIONDIFFERENTIAL EQUATIONS

numerical methods for solving differential tions are based on replacing the ential equations by algebraic equations. case of the popular finite differenceod, this is done by replacing the atives by differences. w we demonstrate this with both first- and nd-order derivatives.

XAMPLE

finite difference form of the first derivative

Taylor series expansion of the function f about the point x,

The smaller the x, the smallerthe error, and thus the more accurate the approximation.

der steady one-dimensional heat conduction in a plane wall of thickness L eat generation.

Finite difference representationof the second derivative at a general internal node m.

no heat generation

Finite difference formulation for steady two-dimensional heat conduction in a region withheat generation and constant thermal conductivity in rectangular coordinates

E-DIMENSIONAL STEADY HEAT NDUCTIONsection we develop the finite difference ation of heat conduction in a plane wall he energy balance approach and s how to solve the resulting equations. ergy balance method is based on iding the medium into a sufficient r of volume elements and then g an energy balance on each element.

equation is applicable to each of the interior nodes, and its application

s M - 1 equations for the determination mperatures at M + 1 nodes.

two additional equations needed to e for the M + 1 unknown nodal peratures are obtained by applying the gy balance on the two elements at the

undary Conditionsdary conditions most commonly encountered in practice are theified temperature, specified heat flux, convection, and radiationdary conditions, and here we develop the finite difference formulations em for the case of steady one-dimensional heat conduction in a plane of thickness L as an example. node number at the left surface at x = 0 is 0, and at the right surface at it is M. Note that the width of the volume element for either boundary is x/2.

cified temperature boundary condition

n other boundary conditions such as the specified heat flux, convection,ation, or combined convection and radiation conditions are specified at andary, the finite difference equation for the node at that boundary is obtainedriting an energy balance on the volume element at that boundary.

finite difference form of various dary conditions at the left boundary:

Schematic for the finite

differenceformulation of the interface

boundarycondition for two mediums A and

ating Insulated Boundary Nodes as Interior Nodes:Mirror Image Concept

The mirror image approach can also be used for problems that possess thermalsymmetry 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.

XAMPLE

ode 1

ode 2

ct solution:

finite difference formulation of dy heat conduction problemslly results in a system of N braic equations in N unknown al temperatures that need to be ed simultaneously.e are numerous systematic oaches available in the literature, they are broadly classified as ct and iterative methods. direct methods are based on a number of well-defined steps that t in the solution in a systematic ner. iterative methods are based on an l guess for the solution that is ed by iteration until a specified ergence criterion is satisfied.

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

O-DIMENSIONAL STEADY HEAT NDUCTION

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.

uare mesh:

undary Nodes

region is partitioned between the es by forming volume elementsnd the nodes, and an energy nce is written for each boundary e.

energy balance on a volumeent is

assume, for convenience in ulation, all heat transfer to be into the me element from all surfaces except pecified heat flux, whose direction is ady specified.

Node 2

Node 1

EXAMPLE

Node 3

4

Node 5

Node 6

Nodes 7, 8

e 9

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 irregulargeometry 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.

ANSIENT HEAT CONDUCTIONite difference solution of transient ms requires discretization in time inn to discretization in space.

done by selecting a suitable time step solving for the unknown nodal atures repeatedly for each t until the n at the desired time is obtained.

sient problems, the superscript i is used index or counter of time steps, with i = 0ponding to the specified initial condition.

t method: If temperatures at the previousep i is used.t method: If temperatures at the new time

+ 1 is used.

sient Heat Conduction in a Plane Wall

mesh Fourier number

emperature of an interior node at ew time step is simply the average temperatures of its neighboring

s at the previous time step.

at generation and = 0.5

bility Criterion for Explicit Method: Limitation on txplicit method is easy to use, but it suffers n undesirable feature that severely restricts ty: the explicit method is not unconditionally , and the largest permissible value of the time t is limited by the stability criterion.

ime step t is not sufficiently small, the ons obtained by the explicit method may te wildly and diverge from the actual solution.

oid such divergent oscillations in nodal ratures, the value of t must be maintained a certain upper limit established by the

ity criterion.

Example

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.

Node 1

Node 2

cit finite difference formulation

EXAMPLE

Node 1

Node 2

cit finite difference formulation

-Dimensional Transient Heat Conduction

Stability criterion

Explicit formulation

1

AMPLE

Node 2

Node 3

Node 4

Node 5

Node 6

Nodes 7, 8

Node 9

teractive SS-T-CONDUCT Software

e SS-T-CONDUCT (Steady State and Transient Heat nduction) software was developed by Ghajar and his -workers and is available from the online learning nter (www.mhhe.com/cengel) to the instructors and dents.

e software is user-friendly and can be used to solve any of the one- and two-dimensional heat conduction oblems with uniform energy generation in rectangular ometries discussed in this chapter.

r transient problems the explicit or the implicit solution ethod could be used.

SummaryWhy numerical methods?Finite difference formulation of differential equationsOne-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