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TRANSCRIPT
11/18/2019
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11/18/2019 PHY 711 Fall 2019 -- Lecture 33 1
PHY 711 Classical Mechanics and Mathematical Methods
10-10:50 AM MWF Olin 103
Plan for Lecture 33:
Chapter 11 in F & W:
Heat conduction
1. Basic equations
2. Boundary value problems
11/18/2019 PHY 711 Fall 2019 -- Lecture 33 2
11/18/2019 PHY 711 Fall 2019 -- Lecture 33 3
Conduction of heat
jh
T(r,t)
30 0 0
Enthalpy of a system at constant pressure
non uniform temperature ,
mass density and heat capacity
,
,
p
p
V
p
T t
c
H c T t T d r H T p
r
r
1
2
3
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11/18/2019 PHY 711 Fall 2019 -- Lecture 33 4
Note that in this treatment we are considering a system at constant pressure p
Notation: Heat added to system --
External work done on system --
Internal energy --
dQ TdS
dW pdV
dE dQ dW T
Entropy --
Enthalpy -- ( )
dS pdV
dS
dH d E pV TdS Vdp
3
Heat capacity at constant pressure:
p
p pp
p
p
dQ H SC T
dT T
c
T
C d r
More generally, note that cp can depend on T; we are assuming that dependence to be trivial.
11/18/2019 PHY 711 Fall 2019 -- Lecture 33 5
Conduction of heat -- continued
30 0
3 3
0, (
Time rate of change of enthapy:
,
, )p
V
p h
V A V
H c T t T d r H T
T tdHc d r d qd r
dt t
p
r
rj A
heat flux heat source
,p h
T tc q
t
r
j
11/18/2019 PHY 711 Fall 2019 -- Lecture 33 6
Conduction of heat -- continued
2
,
Empirically: ,
, ,
p h
h th
p
th
p
T tc q
tk T t
T t qT t
t c
k
c
r
j
j r
rr
https://www.engineersedge.com/heat_transfer/thermal_diffusivity_table_13953.htm
Typical values (m2/s)Air 2x10-5
Water 1x10-7
Copper 1x10-4
thermal diffusivity
4
5
6
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11/18/2019 PHY 711 Fall 2019 -- Lecture 33 7
Boundary value problems for heat conduction
a
T0
b
c
2
2
0
,,
,Without source term: , 0
Example with boundary values: 0, , , , , ,
p
T t qT t
t c
T tT t
tT y z t T a y z t T
rr
rr
x
11/18/2019 PHY 711 Fall 2019 -- Lecture 33 8
Boundary value problems for heat conduction
a
T0
b
c
2
0
,, 0
0, , , , , ,
,0, , , , ,0
, ,0, , , ,0
T tT t
tT y z t T a y z t T
T x z t T x b z t
y y
T x y t T x y c t
z z
rr
0
Let
,,, : varablesof Separation
222
22
22
2
22
2
2
0
Zdz
ZdY
dy
YdX
dx
Xd
ezZyYxXTtzyxT t
Assuming thermally insulated boundaries
x
11/18/2019 PHY 711 Fall 2019 -- Lecture 33 9
Boundary value problems for heat conduction
a
T0
b
c
0
2 2 2
, , ,
0 0 sin
00 cos
00 cos
0
t
nmp
T x y z t T X x Y y Z z e
m xX X a X x
a
dY dY b n yY y
dy dy b
dZ dZ c p zZ z
dz dz c
n m p
a b c
x
7
8
9
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11/18/2019 PHY 711 Fall 2019 -- Lecture 33 10
Boundary value problems for heat conduction
a
T0
b
c
0
2 2 2
Full solution:
, , , sin cos cos mnp tnmp
nmp
nmp
m x n y p zT x y z t T C e
a b c
n m p
a b c
x
11/18/2019 PHY 711 Fall 2019 -- Lecture 33 11
0
Full solution:
, , , sin cos cos mnpt
nmpnmp
m x n y p zT x y z t T C e
a b c
1, 0, 0m n p 1, 1, 0m n p /b /b/a /a
11/18/2019 PHY 711 Fall 2019 -- Lecture 33 12
Oscillatory thermal behavior
z=0 z
0( 0, ) i tT z t T e
2
2
2
2
Assume: ( , ) ( ) i t
T T
T z t f z e
d fi f
d
t
z
z
32 /2
L
2
et ( )
1
z
i
f z Ae
ie
i
10
11
12
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11/18/2019 PHY 711 Fall 2019 -- Lecture 33 13
Oscillatory thermal behavior -- continued
z=0 z
0( 0, ) i tT z t T e
(
/0
1 ) /
Physical solution:
( , )
2where
( , ) c s o
i z i
z
tT z t Ae e
zT z t T e t
11/18/2019 PHY 711 Fall 2019 -- Lecture 33 14
0/( , ) co sz z
T z t T e t
z
11/18/2019 PHY 711 Fall 2019 -- Lecture 33 15
Initial value problem in an infinite domain; Fourier transform
tTqt
tT
fT
fredf
tTredtT
fT
tTt
tT
i
i
,~,
~
~0,
~
~
,,~
:Let
0,
0,,
2
3
3
2
rq
rq
rr
rr
rq
rq
tqeTtT2
0,~
13
14
15
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11/18/2019 PHY 711 Fall 2019 -- Lecture 33 16
Initial value problem in an infinite domain; Fourier transform
tTqedtTtTredtT ii ,~
2
1, ,,
~ 33
3 qrrq rqrq
tqeTtT2
0,~
tqi eTqedtT2
0,~
2
1, 3
3
qr rq
rqq rq fredfT i3~0,
~
tqi eqedtG
TtGrdtT
2'33
3
2
1,' with
0,','',
rrqrr
rrrr
11/18/2019 PHY 711 Fall 2019 -- Lecture 33 17
Initial value problem in an infinite domain; Fourier transform
t
tqi
et
tG
eqedtG
TtGrdtT
4/'
2/3
'33
3
2
2
4
1,'
2
1,' with
0,','',
rr
rrq
rr
rr
rrrr
11/18/2019 PHY 711 Fall 2019 -- Lecture 33 18
Heat equation in half-space
0for ),0(
0for 0)0,(
0for 0),(
:aluesboundary v and initial with ),(,
0,,
0
2
tTtT
zzT
ztzT
tzTtT
tTt
tT
r
rr
x
u duex
t
zTT
22erfc where
2erfc :Solution 0
16
17
18
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11/18/2019 PHY 711 Fall 2019 -- Lecture 33 19
Heat equation in half-space -- continued
0,,
2
2
z
tzT
t
tzT
x
u duex
t
zTT
22erfc where
2erfc :Solution 0
22 22erfc that Note x
x
u eduedx
d
dx
xd
3
4/2
2
3
4/
4
2
2erfc
4
2
2erfc
2
2
t
ze
t
z
z
t
ze
t
z
t
tz
tz
11/18/2019 PHY 711 Fall 2019 -- Lecture 33 20
t=0.01
t=3
t=50
0 erfc2
zT T
t
11/18/2019 PHY 711 Fall 2019 -- Lecture 33 21
Temperature profile
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20
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