Introduction to Applied Mathematics
Introduction to Applied Mathematics MA 325
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This 6 page Class Notes was uploaded by Braeden Lind on Thursday October 15, 2015. The Class Notes belongs to MA 325 at North Carolina State University taught by Robert White in Fall. Since its upload, it has received 6 views. For similar materials see /class/223679/ma-325-north-carolina-state-university in Mathematics (M) at North Carolina State University.
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Date Created: 10/15/15
Lecture 6 Diffusion in a Cooling Fin Introduction In this lecture we will consider heat diffusion in a thin plate The model of the temperature will have the form newu A oldu b for the time dependent case and u A u b for the steady state case In general the matrix A can be extremely large but it will also have a special structure with many more zeros than nonzero components Previously we considered the model of heat diffusion in a thin wire so that there was diffusion in only one direction Now we will extend the model to heat diffusion in two directions One could attribute this to heat diffusion in the radial direction for the thin wire problem or to heat diffusion for a thin cooling fin that might be used to cool a computer chip or an amplifier or a motor Model The model is derived via the Fourier heat law It can be formulated as either a continuous model or as a discrete model A model for heat diffusion in a thin 2D plate where there is diffusion in both the X and y directions but any diffusion in the z direction is minimal and ignored The objective is to determine the temperature in the interior of the fin given the initial temperature and the temperature on the boundary Problems similar to this come from the design of cooling fins or from the manufacturing of large metal objects which must be cooled so as to not damage the interior of the object In order to generate a 2D time dependent model for heat transfer diffusion the Fourier heat law must be applied in both the X and y directions The continuous and discrete 2D models are very similar to the 1D versions for the wire In the continuous 2D model the temperature u will dependent on three variables uXyt Figure Diffusion in Two Directions In order to model the temperature we will first assume the temperature is given along the 2D boundary and that the thickness T is small Consequently there will be diffusion in just the X and y directions Consider a small mass within the above plate whose volume is AX Ay T This volume will have heat sources or sinks via the two AX T surfaces two Ay T surfaces and two AX Ay surfaces as well as any internal heat equal to f heat vol time The top and bottom surfaces will be cooled by a Newton like law of cooling to the surrounding region whose temperature is usur The Fourier heat law applied to each of the two directions will give the heat flowing through the four vertical surfaces change in heat of AX Ay T pcuXytAt 7 uXyt m fXy dX dy T At 2 AX Ay At Cusur 7 uXyt AX T At Kuyxy Ayt KuyXyt Ay T At Kuxx AXyt KuxXyt This approximation gets more accurate as AX Ay and At go to zero So divide by AX Ay T At and let AX Ay and At go to zero This gives a differential equation with partial derivatives and 11 is an example of a partial differential equation 2D Diffusion Model for Cooling Fin pcutXyt fXyt 2CTusur 7 uXyt t KUXXYJX KuYXyaty 11 for Xy in 0LX0W fXy is the internal heat source p is the density c is the speci c heat K is the thermal conductivity T is the small thickness of the plate and C is the ability to transfer heat to the surrounding region uXy0 given initial condition and 12 uXyt given on the boundary of the n 13 Explicit Finite Difference 2D Model of Heat Transfer De ne 0c Kpc Ath2 and let uiik1 be the approximation of uihjh klAt uijk1 Atpc f 2CT usur 0cui1vik ui1ik uuiyi1k uiyi1k l Atpc 2CT 40c uiyik 21 ij lnl k 0 manl uiio given for ij l n1 and 22 uijk given for k lman and ij 0 or n 23 Stability Condition for 2 l Atpc 2CT 40c gt 0 and at gt 0 Method In order to execute the discrete model there must be three nested loops The outer loop must be the time loop uses k and the two inner loops are for the space grid uses i and j The order the i X direction and the j y direction is not important so long as they are both within the k time loop Implementation The following Matlab code is for heat diffusion on a thin plate which has initial temperature equal to 70 and with temperature at boundary X 0 equal to 370 for the first 120 time steps and then set equal to 70 after 120 time steps The other temperatures on the boundary are always equal to 70 In the code we have used C 0 so that no heat is lost via the large top and bottom surfaces of the fin the heat ows from the hot mass through the fin to the other three smaller edges of the fin The code in heat2dm generates a 3D array with the temperatures for space and time The code mov2dheatm generates a sequence of 3D plots of the temperature versus space One can see the heat moving from the hot side into the interior and then out the cooler boundaries You may find it interesting to vary the parameters and also change the 3D plot to a contour plot by replacing mesh by contour Matlab Codes for 2D Heat Flow heat2dm and m0v2dheatm Heat 2D Diffusion in File heat2dm clear L 10 W L Tend 80 maxk 300 dt Tendmaxk n 20 ulnllnllmaxkl 70 dx Ln dy Wn n dx b dthh cond 002 spneat 10 rho 1 a condspneatrno alpha ab 39 lznl ilh ilh for klmaxkl timek k ldt for jlnl uljk 300klt120 70 end end for klmaxk for j 22n for i 22n uijkl Odtspheatrho l 4alphauijk alpha ui ljkuiljk uij lkuijlk end end meshxyumaxk39 Generates Sequence of 3D plots in File m0V2dheatm heatZd lim 0 l O l O 400 for kl5200 meshxyuk 39 title 39heat versus space at different times39 axislim k waitforbuttonpress end AssessmentThe heat conduction in a thin plate has a number of approximations Different mesh sizes in either the time or space van39able will give different numerical results However ifthe stability conditions holds and the mesh size decreases then the numerical computations will differ by smaller amounts Also the stability condition on the step sizes can still be a serious constraint An alternative is to use an implicit time discretization but this generates a sequence oflinear prob1eins one for each time step Homework 1 Experiment with the parameters in the 2D Matlab code Observe the stability condition 2 Observe the steady state solutions for both f 0 cunent code value and f not equal to 0 3 Modify the 2D Matlab code so that c is not zero Experiment with di erent values ofC to see how fast the fin cools