Download 2D Steady State Temperature Distribution

2D Steady State Temperature Distribution
Matrix Structural Analysis
Giuliano Basile
Vinh Nguyen
Christine Rohr
University of Massachusetts Dartmouth
July 21, 2010
Introduction
Advisor
Dr. Nima Rahbar: Civil Engineer
Project Description
Learning the fundamentals for creating matrices. We will be working
with 2 Dimensional frames. Constructing elements and nodes, which
will be used to study temperature distribution through out our
specimen.
Application of Research
Study the thermal distribution
Test different types of materials
Compare Numerical vs. Analytical results
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July 21, 2010
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Objectives
Use Matlab to calculate the 2D Steady State Temperature
Distribution
Consider the boundary conditions (will be discussed)
Use Triangular Elements
Compare your numerical solution with the exact analytical solution
Calculate number of nodes, elements needed for accurate results
Compute Errors
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Thermal Distribution in Materials
We consider all materials to be at Steady State
Different materials have different temperature distributions;
This is due to different atomic structures
Metals – Crystalline
Ceramics – Amorphous
Polymers – Chains
Atomic structure leads to different Thermal Conductivity
(how heat travels throughout)
This knowledge can be to choose the correct material for engineering
designs
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July 21, 2010
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Metals
Figure: Crystalline Atomic Structure
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Ceramics
Figure: Amorphous Atomic Structure
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Polymers
Figure: Chain Atomic Structure
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Thermal Distribution in Materials
Table: Thermal Conductivity of Materials (Watts/meter*Kelvin)
Basile, Nguyen, Rohr (UMD)
Materials
Values
Wood
Rubber
Polypropylene
Cement
Glass
Soil
Steel
Lead
Aluminum
Gold
Silver
Diamond
0.04-0.4
0.16
0.25
0.29
1.1
1.5
12.11-45.0
35.3
237.0
318.0
429.0
90.0-2320.0
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Why study 2D Thermal 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
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Short Description
Here we are modeling heat
flux for a 2D plate
No heat is applied to the x
and y axis (x-nodes =
y-nodes = 0)
Flux is also considered
zero on the right side of
the plate
Steady heat is being
applied at the top of the
plate:
θ = 100 sin(
πx
)
10
(1)
!
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Matrix Structural Analysis
July 21, 2010
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What’s Included
Elements
We start with 32 triangular elements
Numbering left to right; bottom to top
Each element has 3 local and global nodes
Number of Elements and Global Nodes will change
Nodes
Local nodes are used to indicate Global nodes
Nodes are used to define elements
Independent Element Number ien (3,5) = 17
3 is the Local node number
5 is the Element number
17 is the Global node number
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Matrix Structural Analysis
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25 Nodes (32 Elements) — Plate vs. MatLab Solution
Temperature Distribution
10
90
9
85
8
80
Vertical Side
7
75
6
70
5
65
4
60
3
55
2
50
1
45
0
0
1
2
3
4
Horizontal Side
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5
Matrix Structural Analysis
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81 Nodes (128 Elements) — Plate vs. MatLab Solution
Temperature Distribution
10
9
90
8
Vertical side
7
85
6
5
80
4
75
3
2
70
1
0
0
1
2
3
4
Horizontal side
Basile, Nguyen, Rohr (UMD)
5
Matrix Structural Analysis
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324 Nodes (512 Elements) — Plate vs. MatLab Solution
Temperature Distribution
10
96
9
94
8
Vertical side
7
92
6
90
5
88
4
3
86
2
84
1
0
82
0
1
2
3
4
Horizontal side
5
Basile, Nguyen, Rohr (UMD)
Matrix Structural Analysis
July 21, 2010
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900 Nodes (1682 Elements) — Plate vs. MatLab Solution
Temperature Distribution
10
98
9
97
8
96
Vertical Side
7
95
6
5
94
4
93
3
92
2
91
1
90
0
0
1
2
3
4
Horizontal Side
5
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Temperature Distribution (Right Side)
100 sinh πy
10 sin
δ(x, y ) =
sinh(π)
πx
10
100
90
80
Temperature
70
60
32
Elements
50
40
30
20
10
0
0
1
Basile, Nguyen, Rohr (UMD)
2
3
4
5
Y!Axis
6
Matrix Structural Analysis
7
8
9
10
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Temperature Distribution (Right Side)
100 sinh πy
10 sin
δ(x, y ) =
sinh(π)
πx
10
100
90
80
Temperature
70
60
50
128
elements
40
30
32
elements
20
10
0
0
1
Basile, Nguyen, Rohr (UMD)
2
3
4
5
Y!Axis
6
Matrix Structural Analysis
7
8
9
10
July 21, 2010
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Temperature Distribution (Right Side)
100 sinh πy
10 sin
δ(x, y ) =
sinh(π)
πx
10
100
90
80
Temperature
70
128 elements
60
50
32 elements
40
30
512 elements
20
10
0
0
1
Basile, Nguyen, Rohr (UMD)
2
3
4
5
Y!Axis
6
Matrix Structural Analysis
7
8
9
10
July 21, 2010
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Temperature Distribution (Right Side)
100 sinh πy
10 sin
δ(x, y ) =
sinh(π)
πx
10
100
Temperature at The
Right Side of The
Plate
90
80
Temperature
70
60
50
The temperature lines
converge to a smooth line
as
the number of elements
increases
40
30
32 elements
128 elements
512 elements
1682 elements
20
10
0
0
1
Basile, Nguyen, Rohr (UMD)
2
3
4
5
Y!Axis
6
Matrix Structural Analysis
7
8
9
10
July 21, 2010
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Maximum Error Computed
32 elements
7
128 elements
2
5
Percentage Error
Percentage Error
6
4
3
2
1.5
1
0.5
1
0
0
1
2
3
4
0
5
0
1
2
X!axis
512 elements
0.45
3
4
5
4
5
X!axis
1682 elements
0.14
0.4
0.12
Percentage Error
Percentage Error
0.35
0.3
0.25
0.2
0.15
0.1
0.08
0.06
0.04
0.1
0.02
0.05
0
0
1
2
3
4
5
0
0
X!axis
Basile, Nguyen, Rohr (UMD)
1
2
3
X!axis
Matrix Structural Analysis
July 21, 2010
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What’s Next???
Goals
Continue modeling temperature change
Add defect to material and relate it to original material
Add hole to the specimen
to be continued...
Basile, Nguyen, Rohr (UMD)
Matrix Structural Analysis
July 21, 2010
21 / 24
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
Thank You for Listening
We would like to take this time to thank some very special people
during this whole learning process.
Dr. Gottlieb
Dr. Davis
Dr. Kim
Dr. Rahbar
Dr. Hausknecht
CSUMS Staff
Daniel Higgs
Zachary Grant
Charels Poole
Sidafa Conde
CSUMS Students
Questions?
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Basile, Nguyen, Rohr (UMD)
Matrix Structural Analysis
July 21, 2010
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