Learning Goals

Learning Goals In this exercise, you will 

Develop the solution to a 2D heat conduction problem in ANSYS Mechanical

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Gain fundamental insights into finite-element analysis by connecting the ANSYS steps to concepts covered in the Big Ideas: Finite Element Analysis section Problem Specification: Recall that we had solved a 1D heat conduction problem in the Big Ideas: Finite Element Analysis section. In this exercise, you'll implement the finite-element solution to this problem in ANSYS Mechanical. One can solve 2D and 3D conduction problems in ANSYS Mechanical but not 1D problems. So you'll implement the 1D problem as a 2D one and specify boundary conditions such that there will be no temperature variation and heat flow in the y-direction. In other words, the mathematical model you'll input into ANSYS is 2D but the solution will essentially be 1D. Then, one can directly compare the ANSYS temperature and heat flux results with those shown in the Big Ideas: Finite Element Analysis section. Consider heat conduction in a two-dimensional rectangular domain shown in the figure below. The governing equation and boundary conditions are shown in the figure. To get zero heat flow in the y-direction, we impose zero heat flux boundary conditions at the top and bottom boundaries. At the right boundary, qxdenotes the heat flux in the xdirection.

You need to solve this boundary value problem in ANSYS Mechanical and find the temperature and the heat flux distribution. You need to follow a similar workflow to the 2D Conduction module - please refer to it as necessary. We'll guide you through key aspects of the solution process in ANSYS as you answer the questions in the following pages.

Pre-Analysis Refer to the following image from the Problem Specification and answer the following Pre-Analysis questions.

Expected Trend at Top Boundary (2 points possible)

Select true or false. At the top boundary, there will be no heat flow in the y-direction. True

False

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Expected Trend at Left Boundary (2 points possible)

Select true or false. The direction of the heat flow at the left boundary will be in the xdirection (either positive or negative). True

False

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Geometry Create the geometry by following the same steps as in the Geometry section of the 2D conduction module. Recall that we need to create a 2D sketch and create an area from that sketch. Don't forget to set the analysis type to 2D in the properties menu at the start.

Creating a Rectangle (2 points possible)

Select true or false. When we create a rectangle in ANSYS in the geometry step, we are not affecting the mathematical model. True

False

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Mesh Create a mesh using the following tips:

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You need to have 3 elements total as shown in the figure below. For sizings, use 3 equal divisions in the x-direction and 1 division in y-direction As mentioned before, due to the boundary conditions at the top and bottom boundaries, there will be no temperature variation in the ydirection. So you only need to have 1 division in the y-direction. You will refine the mesh in the x-direction later in the problem.

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You may get two warnings that "Some local face...". Recall from the 2D conduction module > Create Mesh that you have to turn off Advanced Size Function to get rid of this warning.

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Use elements with linear interpolation i.e. WITHOUT midside nodes. Turn off the midside nodes that ANSYS defaults to. To do this, select Mesh > Advanced > Element Midside Nodes > Dropped as shown in the figure below. You should get 4 nodes at the bottom boundary which are highlighted and numbered in the figure above. Similarly, you should get four nodes at the top boundary for a total of 8 nodes. Double-check under Mesh Statistics that you have a total of 8 nodes and 3 elements. (Later, in the verification section, you'll redo the solution with quadratic or secondorder interpolation i.e. with midside nodes.)

Now that you have 3 linear elements, similar to the first example in the Big Ideas: Finite Element Analysis section, you can continue with the analysis.

Temperature Values (2 points possible) Our mesh has 8 nodes and 3 elements. How many temperature values are we asking the ANSYS solver to determine directly? Please input your answer as an integer (e.g. 1, or 8, or 15). unanswered

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Model Setup Complete the model setup in ANSYS following a similar procedure to the 2D conduction module. Here, the governing equation has a heat generation or source term. This can be specified by selecting SteadyState Thermal > Heat > Internal Heat Generation as shown in the figure below.

At the right boundary, the problem specifies that the heat flux in the xdirection is 0.5 Wm2. When you apply heat flux at a boundary in ANSYS Mechanical, it assumes that the associated direction is along the inward normal which at the right boundary is along the negative x direction. Due to this convention in ANSYS, you need to specify the heat flux boundary condition at the right boundary as -0.5 Wm2 (note the negative sign).

Material Assignment (2 points possible)

Remember to assign the correct material to the part before solving! Consider the following components of the mathematical model: a. Governing equations b. Essential boundary conditions c. Natural boundary conditions

Which of these is/are affected when you assign the material to the part? (a) only

(a) and (b)

(a) and (c)

(a), (b), and (c)

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Numerical Solution Make sure your mathematical model is completely defined at this point. Then, click 'Solve' to obtain the nodal temperatures.

Number of Algebraic Equations (2 points possible)

How many algebraic equations would ANSYS need to solve simultaneously to determine the nodal temperatures? Please input answer as an integer (e.g. 1, or 8, or 15). unanswered

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Numerical Results Plot the temperature contours and heat flux vectors. Check that they make sense, agree with expected trends from Pre-Analysis and are in accordance with the boundary conditions.

Heat Flow Direction at Boundaries (2 points possible)

Select true or false. In the plot of heat flux vectors from ANSYS, the heat flow at both the left and top boundaries is in the y-direction. True

False

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Add Probes Answer the following questions by placing temperature and heat flux probes at appropriate locations -- you can review how to do this in the 'Probe Temperature' video in the 2D conduction module here. Recall you can do this by adding coordinate systems centered at the location you would like to probe, and then selecting this coordinate system as the selected location once you create the probe. For each of the following questions, refer to the image below for node numbers;

Temperature at Node 1 (2 points possible)

What is the value of temperature at node 1 (in ∘C) in the ANSYS solution? Please input your answer in decimal notation (e.g. 18.65). unanswered

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Temperature at Node 2 (2 points possible)

What is the value of temperature at node 2 (in ∘C)? Please input your answer in decimal notation (e.g. 18.65). unanswered

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Heat Flux at Node 4 (2 points possible)

What is the value of the heat flux in the x-direction at node 4 (in Wm2)? Please input your answer in decimal notation (e.g. 18.65). unanswered

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Verification and Validation

Let's now verify the results.

Energy Balance (2 points possible)

To check if the ANSYS solution satisfies total energy balance, add a Reaction Probe for the left boundary. Make sure to set the thickness to 1m. The thickness is necessary to convert from heat flux to heat flow as discussed in the 2D Conduction module. What is the reaction at the left boundary (in W)? Please input your answer in decimal notation (e.g. 18.65). unanswered

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Add a Path For the next problem, add a path along the bottom boundary as shown in the image below. To add a path, you can review the video included with the 2D Conduction problem from before, located here. Change the number of sampling points to 29.

Next, plot the variation of the heat flux in the x-direction along this path by selecting Solution > Thermal >Directional Heat Flux.

We are interested in getting the heat flux for each element obtained by differentiating the temperature interpolation. To do this, you must select the 'Unaveraged' display option under your Directional Heat Flux solution, as shown below. You can then compare this result with that shown in the Big Ideas: Finite Element Analysis module.

Variation of x-Direction Heat Flux along Path (2 points possible)

Select the scoping method as 'Path' for your directional heat flux solution, and choose the path you just created. Now observing the x-direction heat flux at various points along the path across the domain, answer the following question. Using the values from the path and the resulting ANSYS graph/table, what is the value of x-direction flux at the location x = 1m (in Wm2)? Please input your answer in decimal notation (e.g. 18.65). unanswered

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Mesh Refinement Change the number of divisions in the x-direction to 6 and regenerate the mesh. You should have six elements total as shown in the figure below. The nodes are highlighted in red in the figure.

Mesh Refinement (2 points possible)

Re-solve. Now what is the value of x-direction heat flux at the right boundary (node 7) (in Wm2)? Please input your answer in decimal notation (e.g. 18.65). unanswered

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Second-Order Interpolation

Next, investigate the effect of increasing the order of interpolation to two by adding midside nodes. Ask ANSYS to put in midside nodes by selecting the dropdown menu shown below and changing from 'Dropped' to 'Kept'.

Edit the mesh size so that there is only 1 division in x-direction. The result is one element for the whole domain.

Second-Order Interpolation (2 points possible)

Now how many algebraic equations must ANSYS solve simultaneously to determine the nodal temperatures? Please input your answer as an integer (e.g. 1, or 8, or 15). unanswered

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x-Direction Heat Flux at Right Boundary (2 points possible)

Re-solve. Again, what is the value of the x-direction heat flux at the right boundary (node 3 in above figure) (in Wm2)? Please input your answer in decimal notation (e.g. 18.65). unanswered

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