DEPARTMENT OF MECHANICAL ENGINEERINGPage 1 of 2EMEC-3559: COMPUTATIONAL METHODS AND MODELING IN MECHNANICAL ENGINEERING || WINTER 2023Instructor: Dr. Muhammad KhalidPROJECT 1 SUBMISSION...

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DEPARTMENT OF MECHANICAL ENGINEERING Page 1 of 2 EMEC-3559: COMPUTATIONAL METHODS AND MODELING IN MECHNANICAL ENGINEERING || WINTER 2023 Instructor: Dr. Muhammad Khalid PROJECT 1 SUBMISSION DEADLINE: March 12, 2023 (11:59 PM) IMPORTANT NOTE: ▪ Late submissions will not be accepted and graded. Problem Statement: Consider a two-dimensional rectangular plate (made of stainless steel) of dimensions 0.1 m × 0.15 m (height × width), as shown in the figure. Relying on the initial and boundary condition described here, the following time- dependent heat conduction equation governs variation of temperature, ?(?, ?, ?), inside the plate: 1 ? ?? ?? = ?2? ??2 + ?2? ??2 Objectives & Outcomes: 1. Using the Taylor’s series expansions, convert the differential terms in the governing equation to finite difference terms with the following techniques: a. First-order accurate explicit forward-difference scheme in time b. Second-order accurate central-difference scheme in space 2. Develop a computer program in MATLAB to perform simulations of the heat conduction process inside the plate through numerically solving the governing equation. 3. Considering a mesh size of 51 × 51, perform the timestep convergence study using the following three stability criteria: a. ?Δ? ???(Δ?2,Δ?2) = 0.45 DEPARTMENT OF MECHANICAL ENGINEERING Page 2 of 2 b. ?Δ? ???(Δ?2,Δ?2) = 0.40 c. ?Δ? ???(Δ?2,Δ?2) = 0.35 You should compare the results of the three simulations by showing the contour plots (in the form of a side-by-side three figures) and extracting the steady-state (final) temperature profiles on the horizontal middle section of the plate (in a single figure with three plots). Provide the relevant timestep sizes in your report along with the results. Also, you should present your analysis for your results in the form of a paragraph by explaining which timestep size is most appropriate for an actual numerical simulation of the heat conduction process inside the plate. 4. Considering the chosen timestep size from part 3, conduct a grid-convergence study using the following three mesh sizes: a. 51 × 51 b. 101 × 101 c. 151 × 151 You should compare the results of these three simulations by showing the contour plots (in the form of a side-by-side three figures) and extracting the steady-state (final) temperature profiles on the horizontal middle section of the plate (in a single figure with three plots). Moreover, you should present your analysis for your results in the form of a paragraph by indicating which grid-size is most appropriate for an actual numerical simulation of the heat conduction process inside the plate. 5. You should submit a 2-3 pages report (in your own words) with the following subtitles for your work: a. Description of the project b. Numerical Methodology c. Timestep Convergence Study d. Grid Independence Study e. Conclusions The report should be prepared using Times New Roman font with a size of 12. You can write the headings in bold letters. The paragraphs should be setup with a single line spacing. Requirements for the Submission: Following items are needed in a single zip/rar folder (LastName_FirstName_StudentID) on the relevant D2L page. 1. Report (in the form of a pdf document) 2. MATLAB program (*.m file(s)) GOOD LUCK & HAPPY LEARNING