Mkm Sharifpur Lecture 1
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Transcript of Mkm Sharifpur Lecture 1
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Lecture 1
COMPUTATIONAL
MECHANICS MKM410
Dr M. Sharifpur
Department ofMechanical and Aeronautical Engineering
University ofPretoria
February 24th, 2011
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Computers and Humans
"Computers are incrediblyfast,accurate, and stupid;
humans are incredibly slow,
inaccurate and brilliant;together they are powerful
beyond imagination."
1921 Nobel Prize in Physics for photoelectric effect. In 1999 Einstein
was named Time magazine's "Person of the Century", and a poll of
prominent physicists named him the greatest physicist of all time
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Wrong Info
Wrong Initial & BC
Wrong Answer
The important thing is:
The problem The answer
Did you solve your own problem?
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First order linear differential equation
Solution;
Example;
Simple
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Sometimes we have the experimental data, and we wantto find the generalized solution.
Sometimes we have the Mathematical model but we do
not have the exact analytical solution.
t
Tce
z
Tk
zy
Tk
yx
Tk
x
gen
Example; General Heat Conduction Equation
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t
Tce
z
Tk
zy
Tk
yx
Tk
xgen
In your Text book this equation (General Heat Conduction Equation)
is represented by equation 2.22(page 24) as
zxandyxxxeq gen 321 ,,
Therefore:
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t
Tcez
Tkzy
Tkyx
Tkx
gen
and Initial condition
Initial and Boundary conditions
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8
Boundary and Initial Conditions
t
Tc
k
e
y
Tb
x
Ta
x
T gen
2
21) BC=1, I=1
3) BC=2, I=0
4) BC=2, I=15) BC=3, I=0
6) BC=3, I=1
7) BC=4, I=08) BC=4, I=1
0
gene
z
Td
y
Tc
x
Tk
x
02
2
gene
z
Td
y
Tc
x
T
x
k
x
Tk
(Mathematically we call them Conditions)
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9
Boundary and Initial Conditions
t
T
k
e
y
T
x
T gen
12
2
2
2 B=4, I=1
Mechanically
h, T
Insulated
a
b
h, T
1
23
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10
Boundary and Initial Conditions
2 2 2
2 2 2
1geneT T T T
x y z k t
t
T
k
eT
z
T
y
T
x
T gen
12
2
2
2
2
21
x
T
t
T
k
e
z
T
y
T
y
T gen
02
2
k
e
y
T
x
T gen BC=3, I=0
BC=4, I=1
BC=6, I=1
BC=5, I=1
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Unannounced Test 1Allowance Time 2 minutes
01
2
2
t
T
k
eT
z
Td
y
Tc
y
Tb
x
Ta
gen
For solving following partialdifferential equation analytically,
How many Initial and Boundary
Conditions do we need?
Specify how many at all , and how
many in each direction? and why?
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01
2
2
t
T
k
eTz
Tdy
Tcy
Tbx
Ta
gen
Initial Condition: 1
Initial andBoundary Conditions:
Boundary Conditions:
x- direction: 1
y- direction: 2
z- direction: 1
BC = 4 at all
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and Initial condition
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For Heat transfer and Fluid flow
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General Heat Conduction Equation:
t
Tce
z
Tk
zy
Tk
yx
Tk
x
gen
In rectangularcoordinates
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16
Special Cases
2 2 2
2 2 2
1geneT T T T
x y z k t
2 2 2
2 2 2 0
geneT T T
x y z k
2 2 2
2 2 2
1T T T T
x y z t
2 2 2
2 2 20
T T T
x y z
Two-dimensional
Three-dimensional
1) Steady-state with heat generation
2) Transient, no heat generation:
3) Steady-state, no heat generation:
Constant thermal conductivity:
t
Tce
z
Tk
zy
Tk
yx
Tk
xgen
General:
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17
Special Cases
For Plain Wall and constant thermal conductivity:
tTce
zTk
zyTk
yxTk
xgen
General:
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In Cylindrical
Coordinates
2
1 1gen
T T T T T rk k k e c
r r r r z z t
General Heat Conduction Equation:
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Usually for homogenous material, with
symmetry boundary condition
2
1 1gen
T T T T T rk k k e c
r r r r z z t
Special Cases in CylindricalCoordinates
2
1 1gen
T T T T T rk k k e c
r r r r z z t
+for long cylinder (L>>D)
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In Spherical
Coordinates
2
2 2 2 2
1 1 1sin
sin singen
T T T Tkr k k e c
r r r r r t
General Heat Conduction Equation:
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2
2 2 2 2
1 1 1sin
sin singen
T T T Tkr k k e c
r r r r r t
Special Case in SphericalCoordinates
Usually for homogenous material, withsymmetry boundary condition
2
2 2 2 2
1 1 1sin
sin singen
T T T Tkr k k e c
r r r r r t
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22
Boundary Conditions
Specified Temperature Boundary Condition
Specified Heat Flux Boundary Condition Convection Boundary Condition
Radiation Boundary Condition
Interface Boundary Conditions
Generalized Boundary Conditions
t
Tce
z
Tk
zy
Tk
yx
Tk
xgen
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23
Specified Temperature Boundary Condition
For one-dimensional heat transfer
through a plane wall of thickness
L, for example, the specified
temperature boundary conditionscan be expressed as
T(0, t) = T1T(L, t) = T2
t
Tce
z
Tk
zy
Tk
yx
Tk
xgen
t
Tc
x
Tk
2
2
BC
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24
Specified Heat Flux Boundary Condition
dTq k
dx
Heat flux in
the positive
x-direction
The sign of the specified heat flux is determined byinspection:positiveif the heat flux is in the positive
directionof the coordinate axis, and negativeif it is in
the oppositedirection.
The heat flux in the positivex-direction anywhere in the medium,
including the boundaries, can be
expressed by Fouriers lawof heat
conduction as
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Two Special Cases
Insulatedboundary Thermalsymmetry
(0, ) (0, )0 or 0
T t T t k
x x
0),0( tQx
,20
LT t
x