Two-Dimensional Conduction: Finite-Difference Equations ...

1 Two-Dimensional Conduction: Finite-Difference EquationsandSolutionsChapter 4 Sections and methods analytical solutions that allow for the determination of the exact temperature distribution are only available for limited ideal cases. graphical solutions have been used to gain an insight into complex heat transfer problems, where analytical solutions are not available, but they have limited accuracy and are primarily used for Two-Dimensional problems. advances in numerical computing now allow for complex heat transfer problems to be solved rapidly on computers, , "numerical techniques . current numerical techniques include: Finite-Difference analysis; finite element analysis (FEA); and finite -volume analysis. in general, these techniques are routinely used to solve problems in heat transfer, fluid dynamics, stress analysis, electrostatics and magnetics, etc.

2 We will show the use of Finite-Difference analysis to solve conduction heat transfer Analysis numerical techniques result in an approximate solution, however the error can be made very small. properties ( , temperature) are determined at discretepoints in the region of interest-these are referred to as nodal points or the Finite-Difference technique for 2-D conduction heat transfer: in this case each node represents the temperature of a point on the surface being considered. the temperature at the node represents the average temperature of that region of the surface. algebraic expressions are used to define the relationship between adjacent nodes on the surface usually the boundary conditions are specified. by increasing the number of nodes on the surface being considered it is possible to increase the spatial resolution of the solution and to potentially increase the accuracy of the numerical solution, however this increases the number of calculation is required to obtain a solution to the ApproximationThe Nodal Network and Finite-Difference Approximation The nodal networkidentifies discrete points at which the temperature is to be determined and uses an m,nnotation to designate their is represented by the temperature determined at a nodal point,as for example, Tm,n?

3 How is the accuracy of the solution affected by construction of the nodalnetwork?What are the trade-offs between selection of a fineor a coarse mesh? Finite-Difference MethodThe Finite-Difference MethodProcedure: Represent the physical system by a nodal network , discretization of problem. Use the energy balance methodto obtain a finite -differenceequationfor each node of unknown temperature. Solve the resulting set of algebraic Equations for the unknown nodal temperatures. Use the temperature field and Fourier s Law to determine the heat transfer in the mediumFinite difference formulation of the differential equation numerical methods are used for solving differential Equations , , the DE is replaced by algebraic Equations in the finite difference method, derivatives are replaced by differences , , this is based on the premise that a reasonably accurate result can be obtained by replacing differential quantities by sufficiently small differences , letting ()()()df xf xxf xdxx+ 0()()limxdf xf xdxx = xx f ()fxx+ ()fx()fxxx+ xTangent lineFinite-Difference ApproximationFinite-Difference Formulation of Differential EquationFor example.

4 Consider the 1-D steady-state heat conduction equation with internal heat generation) , For a pointm,nwe approximate the first derivatives at points m- x and m+ x as220 Tqkx += x Finite-Difference Formulation of Differential Equationexample: 1-D steady-state heat conduction equation with internal heat generationFor a pointmwe approximate the 2nd derivative asNow the Finite-Difference approximation of the heat conduction equation isThis is repeated for all the modes in the region considered112211221122dTdTmmmmdxdxmmmmmm TTTTT xxxxxTTTx+ ++ + 11220mmmmTTTqkx+ ++= Finite-Difference Formulation of Differential EquationIf this was a 2-D problem we could also construct a similar relationship in theboth the x and Y-direction at a point (m,n) ,Now the Finite-Difference approximation of the 2-D heat conduction equation isOnce again this is repeated for all the modes in the region considered.

5 We could also derive a similar equation for the 3-D case()2,1,,122,2mnmnmnmnTTTTyy + + ()21,,1,22,2mnmn mnmnTTTTxx+ + ()()1,,1,,1,,122220m nmnm nmnmnmnTTTTTT qkxy+ + + +++= Finite-Difference Formulation of Differential EquationIf x= y, then the Finite-Difference approximation of the 2-D heat conduction equation iswhich can be reduced toand the relationship reduces toif there is no internal heat generation,Which is just the average of the surrounding node s temperatures!()21,,1,,1,,122 0m nmnm nmnmnmnqxTTTTTTk + + + + + + = ()21,1,,1,1,40m nm nmnmnmnqxTTTT Tk + + +++ + = ()2,1,1,,1,114mnm nm nmnmnqxTTTTTk + + =++++ ,1,1,,1,114mnm nm nmnmnTTTTT + + =+++ Consider this simple caseConsider this simple case100 100 10050 20050 20050 200300 300 300100 100 10050 20050 20050 200300 300 300100 100 10050 20050 20050 200300 300 300[]1321100504 TTT=+++[]21411002004 TTT=+++[]3141300504 TTT=+++[]

6 43213002004 TTT=+++What if the boundary conditions are different Energy Balance MethodDerivation of the Finite-Difference Equations - The Energy Balance Method - As a convenience that eliminates the need to predetermine the direction of heatflow, assume all heat flows are into the nodal region of interest, and express allheat rates , the energy balance becomes:0ingEE+=ii( ) Consider application to an interior nodal point(one that exchanges heat byconduction with four, equidistant nodal points):()4() ( , )1 0imniqqxy =+ = iAwhere, for example,()()()1,,1,,mn mnmn mnTTqkyx = AIs it possible for all heat flows to be into the m,nnodal region?What feature of the analysis insures a correct form of the energy balance equation despite the assumption of conditions that are not realizable?

7 ( ) A summary of Finite-Difference Equations for common nodal regions is providedin Table Balance Method (cont.)Consider an external corner with convection heat transfer.( )() ( )() ()()1,,,1,,0mnmnmnmnmnqqq ++=()()1,,,1,,,22xyhh022mnmnmnmnmnmnTTTT yxkkxyTTTT + + + = AAAAor, with , xy = 1,,1,hh2210mnmnmnxxTTTTkk ++ += ( )e_04_02e040203e040202p_04_45fig_04_06p_ 04_44e_04_04_02 Energy Balance Method (cont.) Note potential utility of using thermal resistance concepts to express rateequations. , conduction between adjoining dissimilar materials with an interfacial contact resistance.()(),1,,1,mnmnmnmntotTTqR =()(),/2/2tctotABRyyRkxx kx =++ AA A( )p_04_38 Solution MethodsSolutions methods Matrix Inversion: Expression of system of N Finite-Difference Equations forN unknown nodal temperatures as:[][][]ATC=( )CoefficientMatrix (NxN)Solution Vector(T1,T2.)

8 TN)Right-hand Side Vector of Constants(C1, )Solution [][][]1 TAC =Inverse of Coefficient Matrix( ) Gauss-Seidel Iteration: Each Finite-Difference equation is written in explicitform, such that its unknown nodal temperature appears alone on the left-hand side:()()1(1)11iNijijkkkiijjjjiiiiiiiaaC TTTaaa ==+= ( )where i =1, 2,.., N andk is the level of proceeds until satisfactory convergence is achieved for all nodes:()()1kkiiTT What measures may be taken to insure that the results of a finite -differencesolution provide an accurate prediction of the temperature field?Problem: Finite-Difference EquationsProblem : Finite-Difference Equations for (a) nodal point on a diagonalsurface and (b) tip of a cutting tool.(a) Diagonal surface(b) Cutting :ASSUMPTIONS: (1) Steady-state, 2-D conduction, (2) Constant properties Problem: Finite-Difference Equations (cont.

9 ANALYSIS: (a) The control volume about node m,n is triangular with sides x and y and diagonal (surface) of length 2 x. The heat rates associated with the control volume are due to conduction, q1 and q2, and to convection, qc. An energy balance for a unit depth normal to the page yields inE0 = 12c qq q 0++=()()()()m,n-1m,nm+1,nm,nm,nTTT Tkx1ky1h2 x1T + + = With x = y, it follows that m,n-1m+1,nm,nhxhxTT2 T22 ++ + = (b) The control volume about node m,n is triangular with sides x/2 and y/2 and a lower diagonal surface of length ()2x/2. The heat rates associated with the control volume are due to the uniform heat flux, qa, conduction, qb, and convection qc. An energy balance for a unit depth yields inabcE =0 qqq0++= ()m+1,nm,nom, x2 + + = or, with x = y, m+1,nom, + + + = Problem: Cold PlateProblem : Analysis of cold plate used to thermally control IBM multi-chip,thermal conduction : Heat dissipated in the chips is transferred by conduction through spring-loaded aluminum pistons to an aluminum coldplate.

10 Nominal operating conditions may beassumed to provide a uniformlydistributed heat flux of at the base of the cold W/moq = Heat is transferred from the coldplate by water flowing throughchannels in the cold : (a) Cold plate temperature distribution for the prescribed conditions. (b) Options for operating at larger power levels whileremaining within a maximum cold platetemperature of 40 : Cold Plate (cont.)Schematic:ASSUMPTIONS: (1) Steady-state conditions, (2) Two-Dimensional conduction, (3) Constant properties. Problem: Cold Plate (cont.)ANALYSIS: Finite-Difference Equations must be obtained for each of the 28 nodes. Applying the energy balance method to regions 1 and 5, which are similar, it follows that Node 1: ()()()()2610yxTxyTyxxyT + + = Node 5: ()()()()41050yxTxyTyxxyT + + = Nodal regions 2, 3 and 4 are similar, and the energy balance method yields a Finite-Difference equation of the form Nodes 2,3,4: ()()()()()1,1,,1,220mn mnmnmnyxTTxyTyxxyT + ++ + = Energy balances applied to the remaining combinations of similar nodes yield the following Finite-Difference Equations .