2D Model For Steady State Temperature Distribution Finite Element - PowerPoint PPT Presentation

2D Model For Steady State Temperature Distribution Finite Element Method Vinh Nguyen, Giuliano Basile, Christine Rohr University of Massachusetts Dartmouth September 23, 2010

Introduction Advisor Dr. Nima Rahbar: Civil Engineering Project Objective To learn the fundamentals of matrices and how to analyze them. To learn how to use Matlab and finite element method to construct a 2D computer model for temperature distribution. Nguyen, Basile, Rohr 2D Model For Temperature Distribution

Temperature Distribution in Materials At steady state different materials have different temperature distributions; This is due to different atomic structures Metals – Crystalline = high thermal conductivity Ceramics – Amorphous = low thermal conductivity Polymers – Chains = low thermal conductivity This knowledge can be used to choose the correct materials for engineering designs Nguyen, Basile, Rohr 2D Model For Temperature Distribution

Thermal Distribution in Materials Table: Thermal Conductivity of Materials (W/m*K) Materials Values Wood 0.04-0.4 Rubber 0.16 Polypropylene 0.25 Cement 0.29 Glass 1.1 Soil 1.5 Steel 12.11-45.0 Lead 35.3 Aluminum 237.0 Gold 318.0 Silver 429.0 Diamond 90.0-2320.0 Nguyen, Basile, Rohr 2D Model For Temperature Distribution

Why Do We Study 2D Temperature Distribution? To generate new understanding and improve computer methods for calculating thermal distribution. 2D computer modeling is cheap fast to process gives accurate numerical results parallel method can be used for higher efficiency Nguyen, Basile, Rohr 2D Model For Temperature Distribution

Description This project is a PDE problem: δ 2 ϑ δ x 2 + δ 2 ϑ δ y 2 = 0 , Ω = { 0 < x < 5; 0 < y < 10 } (1) With boundary conditions: ϑ ( x , 0) = 0 0 < x < 5 (2) ϑ ( y , 0) = 0 0 < y < 10 (3) 100 sin( π x ϑ ( x , 10) = 10 ) 0 < x < 5 (4) δϑ δ x (5 , y ) = 0 0 < y < 10 (5) The Exact Solution is given: ϑ ( x , y ) = 100 sinh( π y 10 ) sin( π x 10 ) (6) sinh( π ) Nguyen, Basile, Rohr 2D Model For Temperature Distribution

Description-Building The Mesh The problem was first approached by creating 25 node- 32 element triangular mesh. The nodes are built from left to right and bottom up. An element is formed by connecting 3 nodal points. No heat is applied to the sides and the bottom. Heat is applied at the top of the plate: ϑ = 100 sin( π x 10 ) (7) Nguyen, Basile, Rohr 2D Model For Temperature Distribution

25 Nodes (32 Elements) — Plate vs. MatLab Solution Temperature Distribution 10 90 9 85 8 80 7 75 6 Vertical Side 70 5 65 4 60 3 55 2 50 1 45 0 0 1 2 3 4 5 Horizontal Side Nguyen, Basile, Rohr 2D Model For Temperature Distribution

81 Nodes (128 Elements) — Plate vs. MatLab Solution Temperature Distribution 10 9 90 8 7 85 6 Vertical side 5 80 4 75 3 2 70 1 0 0 1 2 3 4 5 Horizontal side Nguyen, Basile, Rohr 2D Model For Temperature Distribution

324 Nodes (512 Elements) — Plate vs. MatLab Solution Temperature Distribution 10 96 9 94 8 7 92 6 Vertical side 90 5 88 4 3 86 2 84 1 82 0 0 1 2 3 4 5 Horizontal side Nguyen, Basile, Rohr 2D Model For Temperature Distribution

900 Nodes (1682 Elements) — Plate vs. MatLab Solution Temperature Distribution 10 98 9 97 8 96 7 95 6 Vertical Side 94 5 4 93 3 92 2 91 1 90 0 0 1 2 3 4 5 Horizontal Side Nguyen, Basile, Rohr 2D Model For Temperature Distribution

Temperature Distribution (Right Side) � π y � π x � � δ ( x , y ) = 100 sinh sin 10 10 sinh( π ) 100 90 80 70 Temperature 60 32 50 Elements 40 30 20 10 0 0 1 2 3 4 5 6 7 8 9 10 Y � Axis Nguyen, Basile, Rohr 2D Model For Temperature Distribution

Temperature Distribution (Right Side) � π y � π x � � δ ( x , y ) = 100 sinh sin 10 10 sinh( π ) 100 90 80 70 Temperature 60 50 128 elements 40 30 32 elements 20 10 0 0 1 2 3 4 5 6 7 8 9 10 Y � Axis Nguyen, Basile, Rohr 2D Model For Temperature Distribution

Temperature Distribution (Right Side) � π y � π x � � δ ( x , y ) = 100 sinh sin 10 10 sinh( π ) 100 90 80 70 128 elements 60 Temperature 50 32 elements 40 30 512 elements 20 10 0 0 1 2 3 4 5 6 7 8 9 10 Y � Axis Nguyen, Basile, Rohr 2D Model For Temperature Distribution

Temperature Distribution (Right Side) � π y � π x � � δ ( x , y ) = 100 sinh sin 10 10 sinh( π ) 100 Temperature at The 90 Right Side of The 80 Plate 70 Temperature 60 50 The temperature lines 40 converge to a smooth line as 30 the number of elements 32 elements increases 128 elements 20 512 elements 10 1682 elements 0 0 1 2 3 4 5 6 7 8 9 10 Y � Axis Nguyen, Basile, Rohr 2D Model For Temperature Distribution

Error Computing The exact solution is shown: ϑ ( x , y ) = 100 sinh( π y 10 ) sin( π x 10 ) (8) sinh( π ) Error is calculated by: Solution − Nodal Error = Exact Point temperature . 100 (9) Exact Solution Nguyen, Basile, Rohr 2D Model For Temperature Distribution

Maximum Error Plot 32 elements 128 elements 7 2 6 1.5 5 Percentage Error Percentage Error 4 1 3 2 0.5 1 0 0 0 1 2 3 4 5 0 1 2 3 4 5 X � axis X � axis 512 elements 1682 elements 0.45 0.14 0.4 0.12 0.35 0.1 Percentage Error Percentage Error 0.3 0.08 0.25 0.2 0.06 0.15 0.04 0.1 0.02 0.05 0 0 0 1 2 3 4 5 0 1 2 3 4 5 X � axis X � axis Nguyen, Basile, Rohr 2D Model For Temperature Distribution

Building The Mesh For Hole Defect Model Mesh 10 20 21 22 23 24 (21) (22) (23) (24) 9 (17) (18) (19) (20) 8 15 16 17 18 19 7 (15) (16) 6 (13) (14) Y � axis 5 11 12 13 14 (11) (12) 4 (9) (10) 3 6 7 8 9 10 2 (5) (6) (7) (8) 1 (1) (2) (3) (4) 0 1 2 3 4 5 0 1 2 3 4 5 X � axis Nguyen, Basile, Rohr 2D Model For Temperature Distribution

Hole Defect Model vs. Original Model Temperature Distribution Temperature Distribution 10 10 95 95 9 9 90 90 8 8 85 85 7 7 80 6 6 80 75 Y � axis Y � axis 5 70 5 75 65 4 4 70 60 3 3 65 55 2 2 60 50 1 1 45 55 0 0 0 1 2 3 4 5 0 1 2 3 4 5 X � axis X � axis Nguyen, Basile, Rohr 2D Model For Temperature Distribution

References [Civil Engineer] Dr. Nima Rahbar Fundamental Matrix Algebra University of Massachusetts Dartmouth , Summer 2010. [Thermal Conductivity of some common Materials] Thermal Conductivity of Materials www. engineeringtoolbox. com , July 2010 Cu Atomic Structure Crystalline Atomic Structure http: // www. webelements. com , July 2010 Ceramic Atomic Structure Amorphous Atomic Structure http: // www. bccms. uni-bremen. de , July 2010 Polymer Atomic Structure Chain Atomic Structure http: // www. themolecularuniverse. com , July 2010

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