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Homework 7 - BEE 4530

Homework #7 with required graphs and figures for BEE 4530.
Course

Computer-Aided Engineering: Applications To Biomedical Processes (MAE 4530)

3 Documents
Students shared 3 documents in this course
Academic year: 2015/2016
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Cornell University

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Homework #7 10.16 The computations for this problem were implemented in an excel sheet. The largest time step that yields the correct, physically accurate trend is 500 seconds, given by the following analysis: α ∆t ≤0 ( ∆ x )2 ∆t ≤ 0 ( )2 0 .05 2 ∆ x = −7 =500 α 5 10 ( ) Increasing the time step in general resulted in less refined results, as predicted. As shown in the plot below, at a time step of 900 seconds, the final solution (at t=1800 s) fails to converge to the right trend. 40 35 Temperature (C) 30 25 20 Δt=200 s Δt=600 s Δt=900 s 15 10 5 0 0 0 0 0 0 0 0 0 0 0 0 Location in Slab (cm) 10.16 (1) Yes there is a region where the solution is unphysical, (2) occurring at earlier times within the duration of five minutes. See the figure below. (3) The unphysical values occur at earlier times because at earlier times there is an observed discrete difference in the left and right node temperature as the left-most node is the only one at a surface temperature of 5 ˚C, and thus the software overshoots the second nodes temperature. For the same reason, the region nearest to the boundary condition exhibits the most error. Increasing the node density near the left boundary can reduce this error. The new mesh is shown below: 5cm On the following page the solution obtained by the new mesh (top) is plotted next to the solution of the evenly distributed mesh (bottom). As can be observed, a larger error is observed in the solution of the evenly distributed mesh. element order: cubic Finally, reducing the relative and absolute tolerances an order of magnitude (from 0 to 0 and from 0 to 0 respectively) had a negligible reduction effect on the error. Relative Tolerance Change Absolute Tolerance Change In summary, changing the mesh element density in regions with large temperature changes between nodes reduced error slightly, and using first element order had the largest effect on error reduction.

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Homework 7 - BEE 4530

Course: Computer-Aided Engineering: Applications To Biomedical Processes (MAE 4530)

3 Documents
Students shared 3 documents in this course

University: Cornell University

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Homework #7
10.16.6
The computations for this problem were implemented in an excel sheet. The largest
time step that yields the correct, physically accurate trend is 500 seconds, given by the
following analysis:
α ∆t
(
∆ x
)
2≤0.5
∆ t ≤ 0.5
α
(
∆ x
)
2=0.5
10−7
(
.05
5
)
2
=500
Increasing the time step in general resulted in less refined results, as predicted. As
shown in the plot below, at a time step of 900 seconds, the final solution (at t=1800 s)
fails to converge to the right trend.
0 0.01 0.01 0.02 0.02 0.03 0.03 0.04 0.04 0.05 0.05
0
5
10
15
20
25
30
35
40
Δt=200 s
Δt=600 s
Δt=900 s
Location in Slab (cm)
Temperature (C)
10.16.10
(1) Yes there is a region where the solution is unphysical, (2) occurring at earlier times
within the duration of five minutes. See the figure below.

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