# COMSOL Multiphysics

Modeling with PDE, MA 461 Assignment 10: Heat Flow Problem Due date: 04/05/2021 In this assignment we use COMSOL Multiphysics to investigate 2-D heat flow in a heated metal plate. Email the instructor the mph file and upload your Report (see the end of the assignment) to Canvas. The Problem The uniform metal plate with thermal diffusivity 1m2 /s is represented by a rectangle with opposite corners at (−0.5, −0.8) and (0.5, 0.8). It is assumed that the plate has a hollow crack or cavity represented by the rectangle with opposite corners at (−0.05, −0.4) and (0.05, 0.4). The left side of the plate is heated to 100◦ C, while on the right side the heat is flowing out at a constant rate of 10W/m2 ; all other boundaries (the top, bottom, and interior sides of the cavity) are assumed insulated. The entire plate is initially at 0◦ C. The PDE for the temperature u(x, y, t) at time t seconds at a point in the plate with coordinates (x, y) takes the form ∂u = ∆u. ∂t The region is bounded on the outside by the boundary of the large rectangle, and on the inside by the boundary of the inner rectangle. The boundary conditions are given by the Dirichlet condition u = 100 on the left side, and the Neumann condition ∂u = −10 ∂n on the right side, and all other boundaries are assumed to take the Neumann condition ∂u = 0. ∂n COMSOL Multiphysics Specify 2D domain and choose Mathematics→Classical PDEs→Heat Equation. (Follow the procedure of A Wave Animation in the lecture notes.) Specifying the Domain Domains are constructed by the addition and subtraction of primitive domains such as rectangles, polygons, and ellipses. In the Model Builder window right click Geometry 1 and choose Rectangle to draw the cavity first. Specify Width as 0.1 and Height as 0.8. Then set the base corner with x-coordinate −0.05 and y-coordinate −0.4. Click Build Selected. Repeat the procedure to build the larger rectangle of 1 × 1.6. Next right click Geometry 1 and choose Booleans and Partitions → Difference. Activate Objects to add and choose the larger rectangle by clicking on it in the Graphics window. You should see r2 showing up as chosen. Similarly, choose the smaller one (r1) as the object to subtract. Click Build Selected. You should be able to see the intended domain in the Graphics window. Specifying the PDE and Initial Condition Click Heat Equation 1 under Heat Equation in the Model Builder window. In the Heat Equation window click Equation. Set da = 1, c = 1, and f = 0. In the Initial Values window the initial condition has been set as 0 by default. Specifying the Boundary Conditions The boundary conditions have been set as zero flux (homogenous Neumann boundary condition) by default. You need to reset the right side as −10. For this you add a node to the model builder tree by right-clicking Heat Equation and choose Flux/Source. In the Flux/Source window choose boundary 8 (by clicking the right side of the larger rectangle) and set g as −10. (Keep q as 0.) Similarly, set Dirichlet boundary condition to the left side. Also check Zero Flux 1 to make sure other boundary conditions are set correctly. Specifying the Mesh The COMSOL Multiphysics routines use the finite element method to approximate the PDE solution. This means that the region is divided into a number of triangular subregions, and the solution is then assumed to be continuous and piecewise planar on each of the subregions. This method is particularly useful when the Domain of the PDE has curved boundaries, as is common in many practical problems. COMSOL Multiphysics has automatic routines for performing the initial triangularization of the domain, and any subsequent refinements that might be needed for greater accuracy. To initialize the mesh click Mesh 1 in the Model Builder window. In the Mesh window the Element size is set as Normal by default. Change it to Fine. Then click the green button Build All to build the mesh. Solving and Plotting To set the time span of solution click Step 1: Time Dependent in the Model Builder window. In the Time Dependent window set Times as range (0,0.15,3). This is to start the computation at t = 0, finish at t = 3, with time step 0.15 to save results from the computation. You can solve the equation by pressing the “=” button in the Home tab. After solving the problem view an animation. The Report Write a short report describing the movement of heat in the plate. 1. Besides the animation you may see an arrow plot that describes the temperature gradient: In 2D Plot Group 1 tab choose Arrow Surface. To see a clear plot right-click Arrow Surface 1 in the model builder tree and select Plot In New Window. Similarly, you can view each of the surface plots under 2D Plot Group 1 in the model builder tree in a seperate window. 2. Can you see a slowdown of the increasing in temperature from the animation and draw some conclusion? From the arrow plot can you make some conclusion on the direction of the heat flow (i.e. heat flux) in terms of the temperature gradient? 3. Finally, determine the temperature at the middle of the back of the cavity (i.e. the point (−0.05, 0.0)) at time t = 3: In the Model Builder window right-click Data Sets and choose Cut Point 2D. In the Cut Point 2D window enter −0.05 for x and 0.0 for y. You may click Plot in that window to verify the point that you chose. Rightclick Derived Values in the Model Builder tree and choose Point Evaluation. In Point Evaluation window select Cut Point 2D 1 for Data set and Last for Time selection. Click Evaluate in that window. What do you find under Table 1 tab beneath the Graphics window? Hand in a surface plot, a surface plot with height and an arrow plot for the solution at t = 3 with your report. Also email me the mph file at ynzeng@uab.edu. Modeling with PDE, MA 461 Assignment 10: Heat Flow Problem Due date: 04/05/2021 In this assignment we use COMSOL Multiphysics to investigate 2-D heat flow in a heated metal plate. Email the instructor the mph file and upload your Report (see the end of the assignment) to Canvas.

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