Practice to Informatics for Energy and Environment

Transcription

1 Practice to Informatics for Energy and Environment Part 3: Finite Elemente Method Example 1: 2-D Domain with Heat Conduction Tutorial by Cornell University In this tutorial, you will step through the creation of a 2-D domain using the DesignModeler by ANSYS, through setting up a Finite Element problem and solving it. It is recommended that you use the link above for the tutorial since there are videos included. 1

2 1. Problem Description: The Biot number (Bi) is a dimensionless quantity used in heat transfer calculations. It gives a simple index of the ratio of the heat transfer resistances inside of and at the surface of a body. This ratio determines whether or not the temperatures inside a body will vary significantly in space, while the body heats or cools over time, from a thermal gradient applied to its surface. In general, problems involving small Biot numbers (much smaller than 1) are thermally simple, due to uniform temperature fields inside the body. Biot numbers much larger than 1 signal more difficult problems due to non-uniformity of temperature fields within the object. In this case following dimensional values are chosen: H = 2 m thermal conductivity = 1 W/mK W= 1 m heat transfer coefficient h = 5 W/m 2 K T_wall_bottom = 1 C T_infinity = 0 C 2

3 2D Steady Conduction - Pre-Analysis & Start-Up - SimCafe about:reader?url= 1 von :27 confluence.cornell.edu John Matthew Singleton Jr The figure below illustrates the given non-dimensional boundary value problem. Click Here for Higher Resolution ANSYS solves the dimensional form of the boundary value problem as shown below.

4 2D Steady Conduction - Pre-Analysis & Start-Up - SimCafe about:reader?url= 2 von :27 Click Here for Higher Resolution Here x_d and y_d are the dimensional coordinates. We will choose the dimensions and boundary condition inputs such that the dimensional problem matches the non-dimensional one. Then, T values in Celsius that ANSYS reports can be interpreted as Θ For the geometry, we pick W = 1m, H = 2m. For the boundary conditions we pick T_0 = 1, k = 1, T_infinity = 0, h = Bi These are the inputs we'll use while setting up the problem in ANSYS. Launch ANSYS Workbench from the Start menu as shown below.

5 2D Steady Conduction - Pre-Analysis & Start-Up - SimCafe about:reader?url= 3 von :27 Management of Screen Real Estate This tutorial is specially configured so that the user can have both the tutorial and ANSYS open at the same time as shown below. It will be beneficial to have both ANSYS and your internet browser displayed on your monitor simultaneously. Your internet browser should consume approximately one third of the screen width while ANSYS should take the other two thirds as shown below. Click Here for Higher Resolution If the monitor you are using is insufficient in size, you can press the Alt and Tab keys simultaneously to toggle between ANSYS and your internet browser. The problem at hand is a steady state thermal problem, thus the steady-state thermal analysis system will be used. Click on Steady-State Thermal(ANSYS) as shown in the image below.

6 2D Steady Conduction - Pre-Analysis & Start-Up - SimCafe about:reader?url= 4 von :27 Continue to hold down the left mouse button and drag Steady-State Thermal(ANSYS) into the Project Schematic area. You will then see a green rectangle appear as shown below. Drag Steady-State Thermal(ANSYS) to the green box which will then turn red and contain text saying "Create standalone system".

7 2D Steady Conduction - Pre-Analysis & Start-Up - SimCafe about:reader?url= 5 von :27 Now, release the left mouse button to create the standalone system. Your Project Schematic window should look comparable to the image below. Lastly, rename the system to "2D Steady Conduction", as shown below.

8 2D Steady Conduction - Pre-Analysis & Start-Up - SimCafe about:reader?url= 6 von :27 In this section the material properties that appear in the boundary value problem will be specified. In our case, the only material property that appears in the boundary value problem is k, the coefficient of thermal conductivity. ANSYS requires a name for the material, so we'll call it "Cornellium". First (Double Click) Engineering Data. Then click in the cell that contains the text "Click here to add a new material" and type in Cornellium as shown below. Click Here for Higher Resolution Next, (Expand) Thermal and (Double Click) Isotropic Thermal Conductivity, as shown below. Then, set the value of the Isotropic Thermal Conductivity to 1 W/(m*C) as shown below.

9 2D Steady Conduction - Pre-Analysis & Start-Up - SimCafe about:reader?url= 7 von :27 Click Here for Higher Resolution Lastly, click Return to Project, the Project Schematic window., in order to return to It would be in our best interest to save the project at this point. Click on the Save As.. button,, which is located on the top of the Workbench window. Save the project as "SteadyConduction". When you save in ANSYS, a file and a folder will be created. For instance if you save as "SteadyConduction", a "SteadyConduction.wbpj" file and a folder called "SteadyConduction_files" will appear. In order to reopen the ANSYS files in the future you will need both the ".wbpj" file and the folder. If you do not have BOTH, you will not be able to access your project. Go to Step 2: Geometry Go to all ANSYS Learning Modules

10 2D Steady Conduction - Geometry - SimCafe 1 von :28 confluence.cornell.edu John Matthew Singleton Jr Turn on Autoconstraints by following the instructions at this link. In some versions such as ANSYS 15.0, this feature is turned off by default. If Autoconstraints is not turned on, vertices and lines in your sketches will not be coincident with the coordinate axes. This can cause problems in your solution later on in which case you have to start from scratch which is never fun. The default Analysis Type is 3D, which must be changed, considering the problem at hand is 2D. This tells ANSYS to use the 2D version of the heat equation as the governing equation in our boundary value problem. In order to make this change first (Right Click) Geometry > Properties, as shown below.

11 2D Steady Conduction - Geometry - SimCafe 2 von :28 Click Here for Higher Resolution Then set Analysis Type to 2D as shown in the following image.

12 2D Steady Conduction - Geometry - SimCafe 3 von :28 In the geometry step, we specify the domain for our boundary value problem which is a rectangle. We'll do this in DesignModeler, the geometry engine in ANSYS Workbench. DesignModeler lets you create a geometry from scratch or import it from a CAD package. We'll do the former for our simple geometry. In order to start DesignModeler (Double Click) Geometry,. Twiddle your thumbs for a bit. After DesignModeler opens, select meter as the desired length unit. The sketching will be done in the XY plane, so (Click) XY Plane,, then click on the face plane button,. In this section a rectangle will be sketched on the XY plane with one corner at the origin. First click on the Sketching tab,, then click on the Rectangle button,. Next,move the mouse over the origin until a "P" appears and click once, then move the cursor somewhere else in the first quadrant and click again. The "P" indicates that the cursor is coincident with a point (in our case, the origin). If you don't see the "P", you need to turn Autoconstraints on.

13 ANSYS - Turning on Auto Contraints - SimCafe 1 von :10 confluence.cornell.edu Chiyu Jiang It is very important to check that the Auto Constraints feature is turned on before creating any sketches in DesignModeler. Otherwise, vertices and lines in your sketches will not be coincident with the coordinate axes. This can cause problems in your solution later on. The Auto Constraint feature is not turned on by default in ANSYS This tip demonstrates how to turn on the Auto Constraint feature in DesignModeler. Procedure Before creating a sketch, click on the "sketching" tab.

14 ANSYS - Turning on Auto Contraints - SimCafe 2 von :10 Next, click on Contraints and keep scrolling untill Auto Contraints appear. Finally, click on Auto Contraints and check the boxes next to Global and Cursor. Okay, all set! Have fun sketching and modelling!

15 2D Steady Conduction - Geometry - SimCafe 4 von :28 The dimensions of the rectangle will now be specified. First, click on the Dimensions tab,. Next, click on the left vertical line of your rectangle, move the mouse to the left, then click again. This will create a label for this edge's dimension. We'll specify the actual value of the dimension a little later. Similarly, click on the top horizontal line of your rectangle, move the mouse up and click again. Your screen should now look similar to the image below. Click Here for Higher Resolution Next, set V1 to 2 and set H2 to 1 as shown below.

16 2D Steady Conduction - Geometry - SimCafe 5 von :28 Click Here for Higher Resolution In ANSYS, you apply boundary conditions to a "body", not to a sketch. So we need to turn our sketch into a "body". In this case, the "body" is a 2D surface which ANSYS, somewhat awkwardly, calls a "surface body". To create a surface from our sketch, click on the Modeling tab,. Next, (Click)Concept > Surface From Sketches, as shown below. Then, Expand XY Plane and click on Sketch 1, as shown below. Then (click) Apply under "Details of SurfaceSk1" as shown below.

17 2D Steady Conduction - Geometry - SimCafe 6 von :28 Click Here for Higher Resolution Now ANSYS knows which sketch to use to create the surface. Lastly, (click) Generate,, in order to create the surface body. At this point the rectangle should have become filled in as shown in the following image. At this point you can close DesignModeler. Then, save the project in the Project Schematic window. Go to Step 3: Mesh Go to all ANSYS Learning Modules

18 2D Steady Conduction - Mesh - SimCafe von :29 confluence.cornell.edu John Matthew Singleton Jr In this step, we'll discretize or chop up the domain to create a mesh containing many "elements". During the numerical solution, ANSYS will obtain a discrete version of our boundary value problem on this mesh. We will generate a regular face mesh with 200 elements. The horizontal edges will have 10 divisions and the vertical edges will have 20 divisions. ANSYS Mechanical is the application where we'll generate a mesh, apply the material properties and boundary conditions, and obtain a numerical solution to our boundary value problem. In order to start ANSYS Mechanical (Double Click) Model,. When Mechanical opens, make sure that the units are properly defined to be in metric. Do this by clicking on Units, in the top toolbar, and selecting Metric (m, kg, N, s, V, A). We will create a nice, regular mesh which is done using the "mapped face meshing" option. A mapped face mesh is a mesh that can be mapped to a rectangular domain, hence the name. Our domain is already rectangular, thus mapped faced meshing will

19 2D Steady Conduction - Mesh - SimCafe von :29 yield a rectangular grid as the mesh. First (Click) Mesh, in the model tree. Next, (Click) Mesh Control > Mapped Face Meshing, as shown below. Resolution Click Here for Higher Then, click on the rectangle and it should be highlighted in green. If it does not highlight in green, click on the face selection filter button, at the top of the GUI,then click on the rectangle. Once the rectangle has been selected, (Click) Apply in the "Details of Mapped Face Meshing" table as shown below. Resolution Click Here for Higher ANSYS now knows that a mapped face mesh has to be generated for the rectangle. Next, (Click) Update,, in order to generate the mesh. You should obtain the following mesh.

20 2D Steady Conduction - Mesh - SimCafe von :29 Resolution Click Here for Higher Let's specify the number of divisions to be used along each edge rather than have ANSYS pick them automatically. ANSYS calls this step "edge sizing". We will use 20 divisions along the vertical edges and 10 divisions along the horizontal edges. We will work with the horizontal edge first. In order to implement the edge sizing, first (Click) Mesh,. Next, (Click) Mesh Control > Sizing, as shown below.

21 2D Steady Conduction - Mesh - SimCafe von :29 Resolution Click Here for Higher Next, click on the edge selection filter,, located at the top of the GUI. Then click on the bottom edge of the rectangle, ctrl-click on the top of the rectangle (to select both of them), and (Click) Apply in the "Details of Sizing" table as shown below. Resolution Click Here for Higher Next, set Type to Number of Divisions and set Number of Divisions to 10, as shown below. Click Here for Higher Resolution You will notice that under Number of Divisions there is a row labeled Behavior which is defaulted to Soft. Change this

22 2D Steady Conduction - Mesh - SimCafe von :29 option to Hard. Soft Behavior means that the mesher will take your sizing just as a guideline, not an absolute law. For our simple geometry, we know exactly what the mesh should look like, and don't want the mesher taking any creative interpretations of our sizing options. This completes the sizing for the horizontal edge. Now, create a new edge sizing, apply it to the right and left vertical edges (ctrl-click to keep both of them selected.) And set the Number of Divisions to 20. After, you have properly implemented the two edge sizing commands, (Click) Update,, in order to generate the new mesh. You should then obtain the following mesh. Important note: If ANSYS modifies the number of divisions you give it, change the setting for Behavior in the above menu from Soft to Hard.

23 2D Steady Conduction - Mesh - SimCafe von :29 This completes the meshing process for this problem. Save the project. Do not close Mechanical yet. Go to Step 4: Physics Setup Go to all ANSYS Learning Modules

24 2D Steady Conduction - Physics Setup - SimCafe 1 von :29 confluence.cornell.edu John Matthew Singleton Jr At this point, the material, "Cornellium", will be assigned to the geometry. Then, ANSYS will know the value of k, the coefficient of thermal conductivity, to use in the boundary value problem. To assign the material, expand Geometry,, in the tree outline. Next, click on Surface Body,. Then set Assignment to Cornellium in the "Details of Surface Body" table, as shown below. Click Here for Higher Resolution

25 2D Steady Conduction - Physics Setup - SimCafe 2 von :29 The top and left edges of the rectangular domain are perfectly insulated. In order to incorporate these boundary conditions, first (Right Click) Steady-State Thermal > Insert > Perfectly Insulated, as shown below. Click Here for Higher Resolution Next, hold down Control key and click on the top and left edges of the rectangle. Holding down the Control key lets you select multiple items. Then, (Click) Apply in the "Details of Heat Flow" table. The bottom edge of the rectangular domain has a constant non-dimensional temperature of =1. Recall that we are specifying the dimensional problem in ANSYS such that the dimensional and non-dimensional values of temperature are the same. To implement this boundary condition, (Right Click) Steady-State Thermal > Insert > Temperature, as shown below.

26 2D Steady Conduction - Physics Setup - SimCafe 3 von :29 Click Here for Higher Resolution Next, click on the bottom side of the rectangle and (Click) Apply in the "Details of Temperature" table. Then, set Magnitude to 1 degree Celsius as shown below. Click Here for Higher Resolution The right side of the rectangle has a convection boundary condition. To implement this boundary condition, (Right Click) Steady-State Thermal > Insert > Convection, as shown below.

27 2D Steady Conduction - Physics Setup - SimCafe 4 von :29 Click Here for Higher Resolution Next, click on the right side of the rectangle and (Click) Apply in the "Details of Convection" table. Then, set the Film Coefficient (i.e. convection coefficient, h) to 5 W/(m^2 C) and set the Ambient temperature to 0 degrees Celsius, as shown below. As we saw in Pre-Analysis, these dimensional settings correspond to a Biot number of 5. Click Here for Higher Resolution This concludes the setup of the boundary value problem. We can move on to obtaining a numerical solution. Save the project now. Do not close Mechanical.

28 2D Steady Conduction - Physics Setup - SimCafe 5 von :29 Go to Step 5: Numerical Solution Go to all ANSYS Learning Modules

29 2D Steady Conduction - Numerical Solution - SimCafe 1 von :30 confluence.cornell.edu John Matthew Singleton Jr In the solve step, ANSYS will solve our boundary value problem numerically and obtain the temperature at each node. It is important to keep in mind that all other results are derived from the nodal temperatures through interpolation. Before we hit Solve, we can tell ANSYS what results we would like to look at. ANSYS will extract these results from the nodal temperatures and report them. We could alternatively add in these results after we obtain the numerical solution. Below, we do the former and add in some result requests before hitting Solve. To obtain the temperature distribution in the domain, (Right Click) Solution > Insert > Thermal > Temperature, as shown below.

30 2D Steady Conduction - Numerical Solution - SimCafe 2 von :30 Click Here for Higher Resolution To obtain the distribution of the total heat flux magnitude in the domain, (Right Click) Solution > Insert > Thermal > Total Heat Flux, as shown below. Click Here for Higher Resolution (Click) Solve,. ANSYS will:

31 2D Steady Conduction - Numerical Solution - SimCafe 3 von :30 calculate the element stiffness matrices assemble them into the global stiffness matrix invert the global stiffness matrix to obtain the temperature at the nodes populate the results requested from the nodal temperatures Save the project now. Do not close Mechanical. Go to Step 6: Numerical Results Go to all ANSYS Learning Modules

32 2D Steady Conduction - Numerical Results - SimCafe von :30 confluence.cornell.edu John Matthew Singleton Jr To view the temperature distribution over the surface, select Solution > Temperature from the tree on the left. Click Here for Higher Resolution In order to view the contours as isolines, select the viewing button, and change from Contour Bands into Isolines.

33 2D Steady Conduction - Numerical Results - SimCafe von :30 Click Here for Higher Resolution You can save this plot to a file by selecting "Image to File" in the top toolbar as shown in the snapshot below. We will view the heat flux as vectors. This will tell us the direction of heat flow within the geometry as well as at the boundaries In order to plot the heat flux vectors, select Solution > Total Heat Flux from the tree on the left. Then (Click) Vectors, near the top of the GUI. The sliders in the top bar can be used to change the size and number of vectors displayed.

34 2D Steady Conduction - Numerical Results - SimCafe von :30 At this point, the heat flux vectors should appear similar to the image below. Take a minute to think about whether the heat flow directions at the boundaries match what you expect.

35 2D Steady Conduction - Numerical Results - SimCafe von :30 Click Here for Higher Resolution We ll plot the heat flux crossing the bottom surface as a function of x. The steps involved are: 1. Create a path i.e. line corresponding to the bottom surface (boundary). The path will start at (0,0) and end at (1,0). 2. Plot the y-component of the heat flux along this path. 3. Export heat flux values to Excel for further processing These steps are demonstrated in the videos below. If the videos don't appear below, try reloading the webpage. 1. Create Path Corresponding to Bottom Surface

36 2D Steady Conduction - Numerical Results - SimCafe von :30 Summary of the above video: 1. Select Model > Construction Geometry > Path. 2. Specify number of samples and end location of path as (1,0). Start location is (0,0) which is the default. 3. Rename path (optional). 2. Plot Directional Heat Flux along Path Summary of the above video: 1. Solution > Thermal > Directional Heat Flux 2. Scope to Path 3. Specify Orientation i.e. direction as Y Axis. 4. Rename object in tree (optional) and then select Evaluate All Results 3. Export Data to Excel Summary of the above video: 1. Go into Tabular data 1. Right click > Export 2. Another option: Right click > Select All 1. Copy > Open Excel 2. Paste and rename columns

37 2D Steady Conduction - Numerical Results - SimCafe von :30 To integrate the heat flux along a boundary where temperature or convection is specified, drag the boundary condition in the tree to the Solution branch. This will add a "reaction" in the Solution Branch. Right click on this and select Evaluate All Results. Summary of the above video: 1. To calculate overall heat flux at bottom surface 1. Drag Temperature from Steady-State Thermal onto Solution tree 2. Rename Reaction Probe to "Overall heat flux crossing bottom surface" 3. Right click > Evaluate all results 2. To calculate overall heat flux at right boundary 1. Do the same steps but with Convection in step 1a The overall heat fluxes are reported as reactions at boundaries in ANSYS. This terminology comes from analogy with structural mechanics and is explained in the video below. The following video shows a couple of ways to probe the temperature in the solution domain. If the video doesn't appear below, try reloading the webpage. Summary of the above video:

38 2D Steady Conduction - Numerical Results - SimCafe von :30 1. First way to probe any value in the domain 1. Highlight Temperature 2. Click on Probe icon in the top toolbar 3. Click on a particular location 4. To delete, highlight label icon under File 5. Press delete 2. Second way to probe a particular vertex in the domain 1. Select Solution in the tree 2. Select in the Solution row toolbar Probe > Temperature 3. Click Select Vertices icon in top toolbar 4. Select the vertex of choice 5. Press Apply in Location Method 6. Press Solve To get the value of the temperature at a particular location within the model: Summary of the above video: 1. Create a co-ordinate system centered at the particular location. 1. Go to Tree > Highlight Coordinate Systems

39 2D Steady Conduction - Numerical Results - SimCafe von :30 2. Click Create Coordinate System Icon 3. Origin X and Origin Y should be 0.5 m and 1 m 2. Insert a probe that uses this co-ordinate system 1. Highlight Solution in the tree 2. Click Probe icon in toolbar > Temperature 3. For Location method > My Coordinate System 4. Click Solve Save the project now. Go to Step 7: Verification & Validation Go to all ANSYS Learning Modules

40 2D Steady Conduction - Verification & Validation - SimCafe about:reader?url= 1 von :31 confluence.cornell.edu John Matthew Singleton Jr Verification and validation can be thought of as a formal process for checking results. Each of these terms has a specific meaning which we won't get into here. The only verification check discussed below is investigation of the effect of mesh size on the results. Refine Mesh Let's check what happens to the heat flux variation along the bottom surface when we refine the mesh i.e. use more elements in the mesh. Summary of the above video: 1. Duplicate Model branch in project page 2. Bring up Mechanical window by double-clicking on Model and double edge sizings. Since we double the number of divisions in x and y directions in the refined mesh, the number of elements is increased by a factor of 4 (2x2) as compared to the original mesh. To resolve the steep temperature gradient at right, lower corner, we bias the edge sizings using the following process:

41 2D Steady Conduction - Verification & Validation - SimCafe about:reader?url= 2 von :31 Summary of the above video: 1. Check direction of edges by clicking on arrow in top toolbar. 2. For first edge sizing, select the bias type and apply bias factor of 10. Rescope to just the bottom boundary. Note that the bias factor is the ratio of the largest division to the smallest division. 3. Repeat for second edge sizing, rescoping to left boundary. 4. Duplicate these edge sizings, rescope to upper/right boundaries and reverse bias type. 5. Set Behavior = Hard for all four edge sizings so that ANSYS does not mess with the edge sizing inputs provided. (You don't need to repeat this if you already did this in the meshing step.) To update prior results on the new mesh, click Solve. Highlight the column in the tabular data containing the updated heat flux values along the bottom surface and copy into the Excel file in a new column. We repeated the above process for a third mesh obtained by doubling the edge sizings on the second mesh while keeping the bias factors the same. The following figure shows that the heat flux variation along the bottom surface is almost the same on the two finer meshes (20x40 and 40x80). This demonstrates that the results on the two finest meshes are almost mesh independent and so are likely to be very close to the true solution.

42 2D Steady Conduction - Verification & Validation - SimCafe about:reader?url= 3 von :31 The above figure was generated in MATLAB from the Excel file using this MATLAB script. The Excel file called boundary_flux_tutorial.xlsx had four columns as shown below. Go to Exercises Go to all ANSYS Learning Modules

43 Example 2: Steady-State Heat Transfer: Temperature Distribution in a House Corner 1. Problem Description The temperature distribution in the corner of a house shall be calculated according to the following scheme. Following assumptions shall be used: Ambient temperature: tout = -12 C, = 15 W/m 2 K Inner temperature: tin = 20 C, = 8 W/m 2 K Heat conductivity of the wall: k = 0.15 W/mK ca. 2 m T = -12 C T = 20 C 0.4 ca. 2 m 2. Work steps Make a new project Define a new material Under Geometry choose 2D as analysis type Build the geometry with the DesignModeller: o Set the Auto Constraint 3

44 o Draw 2 rectangles with the origin as common point o Remove the line between the two rectangles by Delete and Modify/Trim o Use Dimensions to set the dimensions o Make a surface from the sketch Read the Modell and Set the correct material Read the model and create a mesh with about 10 elements through the wall Set the boundary conditions and solve for temperature and heat flux Investigate the temperature difference between corner and a position about 1 m from the corner 4

45 Example 3: Steady-State Heat Transfer: Heat Fin Model 1. Problem Description An aluminum tubular heat fin is heated along its inner radius. The heat conducts through the aluminum to the heat fin's external surface, where heat convects to the surrounding atmosphere. You will build and analyze a steady-state model of the fin. Due to symmetry of both geometry and loading, we only need to model and analyze a 1/16 section of the heat fin, which greatly reduces the problem size. (See Figure 1.) Figure 1: Heat Fin Dimensions We will construct the model of the 1/16-symmetry section by first drawing a two-dimensional (2-D) cross-section. Dimension details are in meters, as shown in Figure 2. Figure 2: Dimension Details of the 2-D Fin Cross-Section Additional model information is presented in Table 1. 5

46 Table 1: Model Information Inner Radius Temperature fixed at 170 C External Surface Thermal Conductivity of Aluminum Loses heat by convection to surrounding atmosphere 200 W/m C Ambient Temperature 25 C Convection Coefficient 130 W/m 2 C 2. Grid generation and Model Setup Create the model Make a new project Define a new material Under Geometry choose 2D as analysis type Build the geometry with the DesignModeller: o Set the Auto Constraint o Draw the geometry using two circles and lines o Use dimensions to set the length and angles o Use Modify/Intersect and Modify/Trim to get the geometry o Make a surface from the sketch Read the model Investigate the temperature distribution and the heat fluxes in all directions 6