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ME428: Numerical Methods
创建日期：2024-04-10 16:46:00
所属分类：数学mathematics
标签： ME428
Numerical Methods

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Final Project
ME428: Numerical Methods

The goal of this project is to numerically and analytically solve the interesting problem of heat transfer
through human tissue in cold surroundings. We will consider the cases of analytical solution from heat
transfer (from the book), a numerical solution where we assume the human to be composed of purely
muscle, and the interesting case where heat transfer is occurring through layers of muscle and skin.
1 Problem Statement
Consider the general form of the bioheat equation with boundary conditions [1],
d2T
dx2
+
q˙m + ωρbcb (Ta − T )
k
= 0 (1a)
T |x=0 = Ta, (1b)
− h [T |x=L − T∞] = −k
dT
dx
∣∣∣∣
x=L
(1c)
where T is the temperature, x is the distance, Ta is the arterial blood temperature, k is the thermal
conductivity, q˙m is the metabolic heat generation rate within the muscle, ω is the perfusion rate, and ρb
and cb are the blood density and specific heat, respectively. Equation 1a is the governing equation for
heat transfer, Eq. (1b) is the temperature inside the boundary, and Eq. 1c is the convective heat transfer
boundary condition due to exposure to the environment. A schematic of the problem appears in Fig. 1 and
all parameters for the simulation appear in Tab. 1.
We wish to determine the heat transfer, temperature distribution, and rate of heat loss to the environment
of a human under cold conditions with their skin exposed. We will only consider a layer of muscle covered
by a layer of skin/fat, exposed to the elements. First we will consider the condition where the human is
exposed to frigid temperatures of 0C in air, then we will consider the case where the human is submerged
in freezing water of 0C.
Tab. 1: Heat Transfer Parameters
Parameter Symbol Value
Thickness of muscle layer Lm 30 mm
Thickness of skin layer Lsf 3 mm
Arterial blood temperature Ta 37 C
Thermal conductivity (muscle) km 0.5 W/m ·K
Thermal conductivity (skin/fat) ksf 0.3 W/m ·K
Metabolic heat generation q˙m 700 W/m
3
Perfusion rate ω 0.0005 s−1
Blood density ρb 1000 kg/m
3
Blood specific heat cb 3600 J/kg ·K
Ambient temperature T∞ 0 C
Convection coefficient (air) hair 2 W/m
2K
Convection coefficient (water) hwater 200 W/m
2K
Copyright 2020, University of Illinois at Chicago, all rights reserved.
No distribution without express written permission.
1
Fig. 1: Schematic of heat transfer through human muscle tissue and skin.
2 Statement on Academic Dishonesty
Academic dishonesty, including but not limited to cheating, will be reported to the Office of the Dean of
Students. 
Cheating: Either intentionally using or attempting to use unauthorized materials, information, people, or
study aids in any academic exercise; providing to, or receiving from another person, any kind of unauthorized
assistance on any examination or assignment.
Examples of academic dishonesty include, but are not limited to: 1) using a third-party website to request
help in the solution of this problem, 2) being given code or using another individuals code to complete the
project, 3) using figures or data from another individual and presenting it as your own, 4) using code or
reports from other professors/students from other sections or previous occurrences of this class, even if not
for the exact same problem, and many more. If you are not sure if what you are doing is considered academic
dishonesty, please consult with the TA or professor.
Copyright 2020, University of Illinois at Chicago, all rights reserved.
No distribution without express written permission.
2
3 Assignments
The following sections are the assignments you are expected to complete and place in the report in a logical
manner.
3.1 Analytical Solution of the Idealized Case
As we know, the solution of this ODE is not straightforward with the boundary conditions considered.
However, it is possible to determine the skin surface temperature using one-dimensional conduction models,
such as those available in Incroprera [1]. Find the analytical solution using example 3.12 from Incroprera’s
textbook as a reference for the analytical solution.
Hints:
• The solution to a very similar problem is available in Incroprera “Heat transfer” Example 3.12 on pp.
163–164.
3.2 Numerical Solution for Convection in Air
Solve Eq. (1) using a second-order central difference scheme with the parameters shown in Tbl. 1 for air.
Use a varying number of solution points from n = 5, 11, 51, 101. Plot the results of the varying step sizes
on a single graph. Determine the difference between the analytical solution, discuss, then decide the correct
step size to get an accurate solution (convergence study). Discuss the effect of step size on the skin surface
temperature (Tn). How cold does the human’s skin get if exposed to freezing cold?
Hints:
• This is a boundary value problem. Use the most logical method for the solution of BVPs.
• The left-hand side boundary condition (Eq. 1b for body temperature) is a Dirichlet type boundary
condition and does not need to be discretized. Simply set T1 = Ta.
• You will need to track x as you build your coefficient matrix. Since the total length is L = 33mm, at
distances 0 ≤ x ≤ 30, you should use km for muscle, and at distances 30 ≤ x ≤ 33 you should use ksf
for skin in the matrix.
• The right-hand side boundary condition (Eq. 1c for convection) requires discretization, and you should
use second-order or higher to ensure an accurate solution.
3.3 Numerical Solution for Convection in Water
Solve Eq. (1) using a second-order central difference scheme with the parameters shown in Tbl. 1 for water.
Use a varying number of solution points from n = 11, 51, 101, 201. Plot the results of the varying step sizes
on a single graph. Determine the difference between the analytical solution, discuss, then decide the correct
step size to get an accurate solution (convergence study). Discuss the effect of step size on the skin surface
temperature (Tn). How cold does the human’s skin get if submerged in frigid water?
Hints:
• This is a boundary value problem. Use the most logical method for the solution of BVPs.
• The left-hand side boundary condition (Eq. 1b for body temperature) is a Dirichlet type boundary
condition and does not need to be discretized. Simply set T1 = Ta.
Copyright 2020, University of Illinois at Chicago, all rights reserved.
No distribution without express written permission.
3
• You will need to track x as you build your coefficient matrix. Since the total length is L = 33mm, at
distances 0 ≤ x ≤ 30, you should use km for muscle, and at distances 30 ≤ x ≤ 33 you should use ksf
for skin in the matrix.
• The right-hand side boundary condition (Eq. 1c for convection) requires discretization, and you should
use second-order or higher to ensure an accurate solution.
3.4 Determining the rate of heat transfer from the body
Use a second-order scheme to determine the rate of heat loss from the body,
Q = −kA dT
dx
∣∣∣∣
x=0
, (2)
for the cases in air and water. Assume A = 1cm2, and state the heat loss per square centimeter of exposed
skin. Place this data in a table and discuss.
3.5 Comparison of Convection in Water and Air
Discuss the differing results between a human exposed to cold air compared to submerged in frigid water.
Produce a plot of the temperature profiles (x vs. T ) for the cases with the best results (grid independent
and converged) from Sections 3.2 and 3.3.
4 EXTRA CREDIT: Another layer of insulation
Consider the case where the human has a layer of insulation on them (for example a jacket or wetsuit).
Use the value kins = 0.026 for this layer. Determine the necessary thickness of this final layer to achieve a
comfortable skin temperature above 33C. What is the heat transfer from the body (Q) in this case?
Hints:
• The skin surface temperature will be inside the domain at x = 33mm, not at Tn. You must account
for this when determining the skin surface temperature.
• The insulation will add length to the domain, make sure to account for this in your code.
References
[1] F. P. Incroprera, D. P. Dewitt, T. L. Bergman, and A. S. Lavine. Introduction to heat transfer. chapter
One-dimensional, steady-state conduction, pages 96–168. John Wiley & Sons, Hoboken, 5 edition, 2007.
Copyright 2020, University of Illinois at Chicago, all rights reserved.
No distribution without express written permission.
4
5 Report Format
This is not exact, and you can add/remove/move sections as necessary. But in general, your report should
be formatted as such:
1. Title
Must include the project title, your name, and affiliation.
2. Abstract
An overview of the work presented with description of results and the conclusion of the work.
3. Table of Contents
Should be on its own page and contain page numbers and titles for all sections/subsections of the
report. This can be generated automatically by Microsoft Word or LaTeX.
4. Table of Figures
Should be on its own page. Should contain captions and page numbers of all figures in report. This
can be generated automatically by Microsoft Word or LaTeX.
5. Table of Tables
Should be on its own page. Should contain captions and page numbers of all tables in report. This
can be generated automatically by Microsoft Word or LaTeX.
6. Introduction
Should give the problem statement as well as review what the problem is, why we are doing it, some
background information (read up about this problem and its applications for example), and an overview
of the methods we are using. This is where Eq. ?? should appear in the report along with a figure
explaining the configuration of our problem and its boundary conditions. Show a free body diagram
to explain.
7. Analysis
This is where you should show your analysis of the
(a) Analytical Solution
This is where you should show the form of the analytical solution of the equation.
8. Numerical Method
This section will show all of our numerical methods.
(a) Discretization of Governing Equations
This section should contain the discretized form of the governing equation. It should also show
how the equations are arranged and discuss the solution of these equations (what method did you
use, how does it work, how is it applied).
9. Results
Place all of your results in this section. They should be clearly organized with nice figures and tables
that are adequately discussed. You should address all points in the assignments.
10. Conclusions and Future Work
Discuss what happened in this report. Summarize your results. Reiterate interesting points that were
brought up earlier in the report. Go back to your problem statement in the introduction and explain
how this report fulfilled that problem.

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ACCOUNTING BUSS1040 CHEM 181 MAT136H1 GY 6183 COMP3170 AGRI90089 PHYSICS 1002 ECON10003 IEOR E4004 ACS61012 MATH 40550 MKTG2113 TELE4653 ENSC1004 ECE 232E MK726 MTRX2700 SHEET 3 CSCA08 COMP30027 Math 2XX3 statistic MMAN3200 BISM7213 ECMM109 ECON 2022 FNCE 20005 GR5067 CSCB09 ME 3017 CS 5100 PHYS 2903 COMS 4771 COMP222 COMP 1046 CSC148H1 ISYS2038 FINS2615 OT5206 COMP2048 A6 MECH5770M MA826/20 CMPT 726 STA457 INFS6071 INFS588 integral ECON 216 IRDR 0017 MA323 PA 15213 PSYC2001 ENV222 Statistics ACCT5930 ACCT12 LAND2121 ME 5405 COMM1180 TELE3113 A7 ECE 568 ACCT2019 INFS1200 ENGG1340 CS 1400 ASST3 FINM7409 SWEN30006 INFO2222 FIN20150 SOSS1000 CSSE7231 FINS5513 COMS3200 CECN 702 ECON4060H MN3515 HW2 PSYA02H3 ECON 2112 COMP90051 CS 1652 MA30059 AF1605 MKTG7501 ECO-6003B COM4521 CTM ECON3350 COMP 3804 ELEC9762 BISM 7255 ST22 MATH20014 MNE3122 COMP343 MATH5165 ELE7029 ENGR30002 SEEM3500 ECON7350 RISK5002 FINM7405 COMP90059 MOS 3320B STAT 431 BISM7221 MATH3014 MTH783P EBA 35302 ACS133 BUS2206 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