Heat Transfer study on Numerical Analysis Methodology

Jesse Thomas4/3/15

Introduction:The transfer of thermal energy along temperature gradients is the basis of heat transfer. Described

mathematically with the combined heat equation, real objects can be simulated and the heat transfer anticipated. It is the application of linear algebra and numerical methods that allows engineers to draw conclusions between simulations and complex geometries or initial conditions.

This report explores the heat equation in three dimensions using the vector matrix capabilities of MatLab. The simulated object is a long, rectangular bar with cross-sectional dimensions of 100mm x 150mm with uniform temperatures on each side. In this report, the transfer of heat and interactions inside of the cross-section are investigated using the Gauss-Seidel method of solving linear partial differentiable equations. The Gauss-Seidel method is also explored in some detail, varying tolerances, resolution and runtime for the creation of a standardized format for use with the approach.

Method:The initial conditions used, beginning with side temperatures from the top and clockwise were

75, 60, 85 and 30 degrees C. The initial guess for uniform cross-sectional temperature being a hazard of 55 degrees C to begin with. Tolerances were set to Res > 0.0001 and resolution beginning at 9. Throughout the experiment, all initial conditions were varied to create a robust code and provide a practical standard.

Parting from the initial conditions, the first variable tested was the resolution of the output. To achieve the initial condition of 9 resolution, Nx and Ny were both initialized at a value of 3. Nx and Ny were varied from the initial 3 to 31 with a cursory runtime test at 41 and then 51 to test the viability and effects of very high resolution. Runtime was measured using a digital stopwatch and result quality was measured by its effect on center node temperature and subjective aesthetics such as smoothness of charted data. A shortlist of resolutions was recorded for further experimentation for use as the ideal standard.

Using the values Nx=21, Ny=21, the residual tolerance was varied between E-4 and E-5 at increments of 5E-6 to assess the influence of tolerance on the number of iterative cycles performed by using the Gauss-Seidel method as well as the responsiveness and runtime of the computations. As a benchmark of improvement and diminishing returns, previously acquired trends of central node temperature from varied resolution were used to compare results in context. These results were applied to resolution values on the short list to resolve a practical standard for application to other heat transfer problems.

Using the identified standard, boundary conditions at the four known surfaces of constant temperature were changed to test predictable, two dimensional heat transfer problems as well as three dimensional simulations that could be easily estimated. In this way, the program was verified as robust and acceptable for application to various heat transfer problems as well as it was provided another opportunity to examine the standard parameter set for adequate tolerance and resolution.

Results and Conclusions:The results of experimentation with the grid resolution and number of nodes in the Gauss-Seidel

method of linear differential systems of equations showed that there was a great leap in accuracy in initial growth of the grid. However, the method of increasing precision by manipulating resolution alone was demonstrated to be inadequate for a number of reasons. Firstly, an increased number of nodes required a longer runtime to solve. With an acceptably tight tolerance, very high resolutions took an amount of time to compute that was untenable. For the purposes of this study, only up to Nx=Ny=51 was attempted at residual > E-3 because of its consequentially insupportable runtime. Secondly, very high resolution grids without adequately strict tolerances will show diminishing returns from smaller grids due to the

magnified errors in calculation. An example of comparably high resolution with very low tolerance resulted in a choppy image and unreliable data. As seen in figures 1.1 and 1.2, smooth data with low gap difference between points effectively show the transfer of heat through the cross-sectional area without a high load or runtime on the machine.

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Figures 1.1 and 1.2: High resolution Gauss-Seidel approximations with medium tolerance produce acceptable results. (Problem statement initial conditions. Nx=Ny=30).

Additional benefits beyond visual evidence due to higher resolution are demonstrated in figure 1.3 which shows the asymptotic approach of the approximation to a true value for center-position nodes in several resolutions. It was found that even with very high runtimes upwards of 1.5 minutes (Nx=51), the precision did not appreciably increase compared to 10 second runtimes at much lower resolutions (Nx=20) while center node temperature accuracy decreased if tolerance remained constant.

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Figure 1.3: The asymptotic approach of the Gauss-Seidel method to a true value.

Resolutions between Nx=Ny equal to 15 and 21 were recorded for future application as standards.

The problem of runtime apart from resolution-evoked increase was found to be a consequence of tighter tolerance. Because the Gauss-Seidel numerical method is asymptotic in nature, a small increase in tolerance requires many more iterations to converge. By decreasing allowable tolerance by factors of 10 and holding the resolution constant, the relationship in figure 2.1 was discovered.

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Number of Gauss-Seidel Iterations as a Function of Tolerance, Nx=Ny=21

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Figure 2.1: The relationship between tightened tolerance and the number of iterations of the Gauss-Seidel approximation required for convergence.

With very loose tolerance at Residual > 10, only two iterations were required while beyond Residual > 1, many hundreds of iterations were completed before the solution converged. This decrease in tolerance had an effect on runtime but its linear nature caused the increased calculations to effect runtime much less than similar increases in the exponentially behavioral resolution property.

Similar to the increased precision due to an increase in resolution, decreased tolerances showed great initial increases in precision as seen in figure 2.2 but without the loss of accuracy from the high resolution-loose tolerance example.

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Center Node Temperature as a function of Tolerance, Nx=Ny=21

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Figure 2.2: increases in solution precision due to tightened tolerance.

Diminishing returns are apparent much earlier in the variance of tolerance which is the consequence of the limits of Nx=Ny=21 resolution.

More simply explained with respect to numerical analysis: the number of iterations required to obtain a convergent solution was directly related to the tolerance in the Gauss-Seidel convergence solution. When allowed a greater tolerance for error, the function discontinued iterations at a solution in fewer cycles than runs with a more strict tolerance. In figure 2.3 the number of iterations and the core node temperature (for additional context) have been compared to their respective tolerance values – Residual > 0.0001 and 0.00001. The difference between the two is that with a tighter tolerance, the program continued to solve the Gauss-Seidel approximation until the solution was much further on in the asymptotic approach to a true solution, resulting in more iterations.

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Iterations and Center Node Temperature as a function of Gauss-Seidel tolerance

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Figure 2.3: Iterations required for convergence with respect to tightened tolerances with center node temperatures shown for context. See figure 2.2 for complete center node temperature analysis. (Matlab)

Executing the program using tight tolerances and high resolution was shown to heavily burden the computer processor resulting, in extreme cases, in computational runtimes up to five or ten minutes with resolutions only in the few thousands. For this reason, a standard Nx, Ny resolution and tolerances were identified for comparative and accurate results fit for adequately rigorous analyses. Nx, Ny for this standard were both held at 21 with a relatively tight tolerance of Residual > E-5. Computational runtime at this setting on a standard Iowa Engineering machine are within acceptable ranges of 25s to one minute depending on several factors. With this standard, it was possible to vary several important boundary conditions for comparison and presentation without spending too much downtime in computation and still providing enough smoothness in data and fidelity to the resolution that inherent variance did not interfere with side-by-side comparison (e.g. center node temperature at high resolutions and high tolerances).

As an example of the standard chosen above, the boundary conditions and initial temperatures were reconfigured to Tbottom = 96, Ttop = 15, Tleft = 13, Tright = 55 degrees C. As seen in figures 3.1-3.4, the smoothness of data with high predictability and resolution provide excellent demonstrations of heat transfer in three dimensions with a runtime of only 20s in this instance. The experiment and code was thus shown to be adequately rigorous.

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Figures 3.1 and 3.2: Example boundary conditions at standard operating resolution and tolerances (Matlab).

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Figures 3.3 and 3.4: Application of the standard developed parameters to a predictable heat transfer problem (Tl=Tr=Tb=0, Tt=90 C). (Matlab)

Appendix:Begin Matlab code -

clear allclose all %Specify grid sizeNx = 21; % you select (and vary) NxNy = 21; % you select (and vary) Ny %Specify boundary conditions (insert values as per problem assignment)Tbottom = 0;Ttop = 0;

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Tleft = 0;Tright = 90; % initialize coefficient matrix and constant vector with zerosA = zeros(Nx*Ny);C = zeros(Nx*Ny,1); % initial ’guess’ for temperature distribution (uniform distribution OK -- you specify value)T(1:Nx*Ny,1) = 55; % Build coefficient matrix and constant vector% inner nodesA = zeros (Nx,Ny);for n = 2:(Ny-1) for m = 2:(Nx-1); i = (n-1)*Nx + m;%DEFINE COEFFICIENT MATRIX ELEMENTS HERE A(i,i+Nx) = 1; A(i,i-Nx) = 1; A(i,i+1) = 1; A(i,i-1) = 1; A(i,i) = -4; endend % Edge nodes% bottomfor m = 2:(Nx-1) %n = 1 i = m;%DEFINE COEFFICIENT MATRIX AND CONSTANT VECTOR ELEMENTS HERE A(i,i+Nx) = 1; A(i,i+1) = 1; A(i,i-1) = 1; A(i,i) = -4; C(i) = -Tbottom;end %top:for m = 2:(Nx-1) % n = Ny i = (Ny-1)*Nx + m;%DEFINE COEFFICIENT MATRIX AND CONSTANT VECTOR ELEMENTS HERE A(i,i-Nx) = 1; A(i,i+1) = 1; A(i,i-1) = 1; A(i,i) = -4; C(i) = -Ttop; end %left:for n=2:(Ny-1) %m = 1 i = (n-1)*Nx + 1;%DEFINE COEFFICIENT MATRIX AND CONSTANT VECTOR ELEMENTS HERE

A(i,i-Nx) = 1; A(i,i+Nx) = 1; A(i,i+1) = 1; A(i,i) = -4; C(i) = -Tleft; end %right:for n=2:(Ny-1) %m = Nx i = (n-1)*Nx + Nx;%DEFINE COEFFICIENT MATRIX AND CONSTANT VECTOR ELEMENTS HERE A(i,i-Nx) = 1; A(i,i+Nx) = 1; A(i,i-1) = 1; A(i,i) = -4; C(i) = -Tright; end % Corners%bottom left (i=1):i=1;%DEFINE COEFFICIENT MATRIX AND CONSTANT VECTOR ELEMENTS HEREA(1,Nx+1) = 1 ;A(1,2) = 1;A(1,1) = -4;C(i) = -(Tbottom + Tleft); %bottom right:i = Nx;%DEFINE COEFFICIENT MATRIX AND CONSTANT VECTOR ELEMENTS HEREA(i,i+Nx) = 1;A(i,i-1) = 1;A(i,i) = -4; C(Nx) = -(Tbottom + Tright); %top left:i = (Ny-1)*Nx + 1;%DEFINE COEFFICIENT MATRIX AND CONSTANT VECTOR ELEMENTS HEREA(i,i-Nx) = 1;A(i,i+1) = 1;A(i,i) = -4;C(i) = -(Ttop + Tleft); %top right:i = Nx*Ny;%DEFINE COEFFICIENT MATRIX AND CONSTANT VECTOR ELEMENTS HEREA(i,i-Nx) = 1; A(i,i-1) = 1;A(i,i) = -4;C(i) = -(Ttop + Tright); %Solve using Gauss-Seidel

residual = 100; % set residual to a big number so loop executes at least onceiterations = 0; % set the iteration counter to zerowhile (residual > .000001) % The residual criterion is 0.0001 in this example % You can test different values % increment the iteration counter iterations = iterations+1; %Transfer the previously computed temperatures to an array Told Told = T; %Update estimate of the temperature distribution%INSERT GAUSS-SEIDEL ITERATION HERE % Transfer the previously computed temperatures to an array Told for n=1 : Ny for m=1 : Nx i = (n-1)*Nx + m; Told(i) = T(i); end end % iterate through all of the equations for n=1 : Ny for m=1 : Nx i = (n-1)*Nx + m; %sum the terms based on updated temperatures sum1 = 0; for j=1 : i-1 sum1 = sum1 + A(i,j)*T(j); end %sum the terms based on temperatures not yet updated sum2 = 0; for j=i+1 : Nx*Ny sum2 = sum2 + A(i,j)*Told(j); end % update the temperature for the current node T(i) = (1/A(i,i)) * (C(i) - sum1 - sum2); end end %compute residual % deltaT is a vector of differences deltaT = abs(T - Told); residual = max(deltaT);end iterations % report the number of iterations that were executed %Now transform T into 2-D network so it can be plotted.delta_x = .100/(Nx+1) % Lx is the width of the bar (insert value)delta_y = .150/(Ny+1) % Ly is the height of the bar (insert value)for n=1:Ny for m=1:Nx i = (n-1)*Nx + m; T2d(m,n) = T(i); x(m) = m*delta_x; y(n) = n*delta_y; endend T2d

% plotting the transpose T2d correctly aligns with x and y vectorssurf(x,y,T2d)xlabel('x')ylabel('y')zlabel('Temperature')figurecontour(x,y,T2d)xlabel('x')ylabel('y')