HW_Diffusion_BME2000

pdf

School

Cornell University *

*We aren’t endorsed by this school

Course

MISC

Subject

Mechanical Engineering

Date

Jan 9, 2024

Type

pdf

Pages

Uploaded by ColonelScience102929

1 Name: netID: Collaborators: BME 2000 Homework #9 Diffusion Due: Wednesday, November 8, 11:59 pm on GradeScope Please show your work. If you use code or spreadsheets, please upload a copy. Simulation of oxygen diffusion in a metastatic liver lesion using Ansys Problem Specification Consider the steady-state diffusion problem discussed in class where we solve for the concentration of oxygen in a spherical metastatic liver lesion. Consider the case where ࠵? =1/2 where ࠵? is defined as ࠵? = ࠵? ! 6 |࠵? " | |࠵? ",$ | 1 |࠵? & | There are two parts to this HW. In part I, you’ll perform a 3D Ansys simulation using the rectangular coordinate system (RCS). In part II, you’ll perform a 2D axisymmetric Ansys simulation using the cylindrical coordinate system (CCS). The RCS solution was discussed in class by Dr. Bhaskaran and is also covered in the following tutorial: https://courses.ansys.com/index.php/courses/3d-steady-diffusion/ The videos and documents available in this tutorial are helpful for completing this homework. Note the videos don’t have audio. For generating the Ansys simulations, use the file called Diffusion_Geometry_Mesh.wbpz provided on Canvas in Modules>Unit 6. When you load this file into Ansys Workbench, you will see that we have provided you with the following: • Geometry for the RCS solution: This means you can skip the Geometry part of the Geometry & Mesh video in the tutorial. • Geometry and mesh for the (Cylindrical Coordinate System) CCS solution: This means you can start with the Setup step in Ansys Workbench. Note there is not a tutorial specifically for the CCS case. Answer the questions that follow in the sections below. Part I: 3D solution in rectangular coordinate system (RCS) 1. Governing equations in RCS a. State the governing equation for the RCS in scalar form. Include only non-zero terms. (2 pts) b. What is the unknown variable in the governing equation that we need to determine through the simulation? (2 pts)

2 2. Boundary conditions in RCS: There is only one boundary in this case i.e. the surface of the sphere. What is the boundary condition at the surface of the sphere for the ࠵? = ’ ! case? (2 pts) 3. Numerical solution strategy: As we have seen, the idea of discretization involves determining the primary unknowns at selected points (the cell centers in the Fluent solver) rather than everywhere in the domain. Briefly explain the strategy used to determine the primary unknowns at the cell centers. Hint: This has been covered in class and is in Dr. Bhaskaran’s slides. (2 pts) 4. Expected trends: The same problem when cast in the spherical coordinate system (SCS) is a 1D problem that can be solved by hand. This “hand solution” or analytical solution was presented in class. From this solution, where do you expect the concentration of oxygen to be the lowest? What is the concentration at this location in the hand solution? (2 pts) You will later be checking your 3D Ansys results against these results from the 1D hand solution. 5. Mesh for RCS : Generate the mesh for the sphere using the guidance in the tutorial. Present a snapshot of your mesh. How many cells (i.e. elements) does your mesh contain? (2 pts) 6. Mathematical model setup in Fluent for RCS : In Fluent, you need to turn on the energy equation and turn off the flow equations (and set the viscous model to laminar). This is covered in the online tutorial. Why are we turning off the flow equations and turning on the energy equation? Explain briefly. (2 pts)

3 Complete the setup of the governing equations and boundary conditions in Fluent by following the videos in the tutorial. 7. Numerical solution in Fluent for RCS: Obtain the numerical solution to the governing equation and boundary condition(s) on the mesh that you generate. a. During the numerical solution, how many algebraic equations does the Fluent solver need to generate and solve to find the primary unknown(s) at the cell centers? Explain briefly. (2 pts) b. Present a snapshot of iterative convergence that is produced by Fluent after you hit Solve. What conclusion can you draw from this plot? (2 pts) 8. Post-processing in CFD Post for RCS: Present the following results. a. Filled contours of the concentration (i.e. temperature) variation in the xy plane. Please include a screenshot. (2 pts) Where is the concentration lowest and highest in your Ansys simulation results? Do the location and value of the lowest concentration match your expectations from question 4 above? (2 pts) In the 1D hand solution, we have assumed that the solution is only a function of ࠵? , the distance from the sphere center. Explain briefly if the Ansys contours are consistent with this assumption. (2 pts) b. The variation of oxygen concentration along a line from ࠵? = 0 to ࠵? = ࠵? , ࠵? being the sphere radius. You can generate this plot in CFD Post by creating a line and then adding a “Chart” plot. Plot ࠵? (or equivalently ࠵? ) on the horizontal axis and concentration on the vertical axis. (2 pts) Add the corresponding curve for the 1D analytical solution in the SCS provided as a CSV file in the tutorial under the Verification step. Distinguish between the curves by providing a legend. (2 pts)

Your preview ends here

Eager to read complete document? Join bartleby learn and gain access to the full version

Access to all documents
Unlimited textbook solutions
24/7 expert homework help

4 Briefly comment on the comparison between the Ansys result and the 1D analytical solution. What would you have to do in Ansys to make the comparison better? You don’t actually have to do it in the interest of time. (2 pts) Export these curves as a csv file using Export option in CFD post. You will need this later to these results to those obtained in the CCS. (2 pts) Part II: 2D axisymmetric solution in cylindrical coordinate system (CCS). 9. Governing equations in CCS a. State the governing equation for the CCS in scalar form. Include only non-zero terms. (2 pts) b. What is the unknown appearing in the governing equation that we need to determine through the simulation? (2 pts) 10. Boundary conditions in CCS: What are the boundary conditions in terms of the unknown variable? (2 pts) 11. Mathematical model setup in Fluent for CCS : Complete the setup of the governing equations and boundary conditions in Fluent. Note there are no videos for this but you should be able to figure it out based on what you have done above in RCS and also previously for axisymmetric flow in a cylindrical duct. What do you have to do in Fluent to switch to CCS? (2 pts) 12. Numerical solution in Fluent for CCS: Obtain the numerical solution to the governing equation and boundary condition(s) on the mesh that you generate. a. During the numerical solution, how many algebraic equations does the Fluent solver need to generate and solve to find the primary unknown(s) at the cell centers? Explain briefly. (2 pts) 13. 14. Post-processing in CFD Post for CCS: Present the following results. a. Filled contours of the concentration (i.e. temperature) variation in the 2D domain. (2 pts)

5 Explain briefly if the concentration variation you get in CCS matches what you got above using the RCS. (2 pts) b. The variation of oxygen concentration along a line from ࠵? = 0 to ࠵? = ࠵? , ࠵? being the sphere radius. Recall from the cylindrical duct case that you have to interpret ࠵? as the axial coordinate while using CCS. (2 pts) Add the corresponding curves for the 1D analytical solution in the SCS and the 3D Ansys solution in RCS to this plot. You can do this by reading in the csv file you exported above in question 8b. Distinguish between the curves by adding a legend. (2 pts) Briefly comment on the comparison between the two Ansys results and the 1D analytical solution. What is the advantage of using the CCS compared to the RCS? Hint: Think about the number of algebraic equations solved in each case. (2 pts)