Post on 29-Mar-2020
transcript
CM3110 Heat Transfer Lecture 3 11/6/2017
1
© Faith A. Morrison, Michigan Tech U.
CM3110 Transport IPart II: Heat Transfer
1
One-Dimensional Heat Transfer - Unsteady
Professor Faith Morrison
Department of Chemical EngineeringMichigan Technological University
© Faith A. Morrison, Michigan Tech U.2
Example 1: Heat flux in a rectangular solid – Temperature BC
Example 2: Heat flux in a rectangular solid – Newton’s law of cooling
Example 3: Heat flux in a cylindrical shell – Temperature BC
Example 4: Heat flux in a cylindrical shell – Newton’s law of cooling
Example 5: Heat conduction with generation
Example 6: Wall heating of laminar flow
SUMMARY
Steady State Heat Transfer
Conclusion: When we can simplify geometry, assume steady state, assume symmetry, the solutions are easily obtained
CM3110 Heat Transfer Lecture 3 11/6/2017
2
© Faith A. Morrison, Michigan Tech U.3
Example 1: Heat flux in a rectangular solid – Temperature BC
Example 2: Heat flux in a rectangular solid – Newton’s law of cooling
Example 3: Heat flux in a cylindrical shell – Temperature BC
Example 4: Heat flux in a cylindrical shell – Newton’s law of cooling
Example 5: Heat conduction with generation
Example 6: Wall heating of laminar flow
SUMMARY
Steady State Heat Transfer
Conclusion: When we can simplify geometry, assume steady state, assume symmetry, the solutions are easily obtained
What about non‐steady‐state
problems (quite common in heat
transfer)?
Example 1: Unsteady Heat Conduction in a Semi‐infinite solid
A very long, very wide, very tall slab is initially at a temperature To. At time t = 0, the left face of the slab is exposed to an environment at temperature T1. What is the time‐dependent temperature profile in the slab? The slab is a homogeneous material of thermal conductivity, k, density, , and heat capacity, Cp.
© Faith A. Morrison, Michigan Tech U.4
1D Heat Transfer: Unsteady State
CM3110 Heat Transfer Lecture 3 11/6/2017
3
Example 1: Unsteady Heat Conduction in a Semi‐infinite solid
A very long, very wide, very tall slab is initially at a temperature To. At time t = 0, the left face of the slab is exposed to an environment at temperature T1. What is the time‐dependent temperature profile in the slab? The slab is a homogeneous material of thermal conductivity, k, density, , and heat capacity, Cp.
© Faith A. Morrison, Michigan Tech U.5
This is code for:“Newton’s law of cooling boundary conditions:”
|flux|
1D Heat Transfer: Unsteady State
Example: Unsteady Heat Conduction in a Semi‐infinite solid
x
y
z
HD
© Faith A. Morrison, Michigan Tech U.
H, D, very large
Unsteady State Heat Transfer
6
CM3110 Heat Transfer Lecture 3 11/6/2017
4
x
y
oTTt 0
oTTt 0
1
0TT
t 0
( , )
t
T T x t
x
y
© Faith A. Morrison, Michigan Tech U.
Initial Condition:
7
1D Heat Transfer: Unsteady State
General Energy Transport Equation(microscopic energy balance)
V
n̂dSS
As for the derivation of the microscopic momentum balance, the microscopic energy balance is derived on an arbitrary volume, V, enclosed by a surface, S.
STkTvtT
Cp
2ˆ
Gibbs notation:
see handout for component notation
© Faith A. Morrison, Michigan Tech U.8
1D Heat Transfer: Unsteady State
CM3110 Heat Transfer Lecture 3 11/6/2017
5
General Energy Transport Equation(microscopic energy balance)
see handout for component notation
rate of change
convection
conduction (all directions)
source
velocity must satisfy equation of motion, equation of continuity
(energy generated per unit volume per time)
STkTvt
TCp
2ˆ
© Faith A. Morrison, Michigan Tech U.9
1D Heat Transfer: Unsteady State
Note: this handout is on the web: www.chem.mtu.edu/~fmorriso/cm310/energy2013.pdf
Equation of energy for Newtonian fluids of constant density, , andthermal conductivity, k, with source term (source could be viscous dissipation, electricalenergy, chemical energy, etc., with units of energy/(volume time)).
CM310 Fall 1999 Faith Morrison
Source: R. B. Bird, W. E. Stewart, and E. N. Lightfoot, Transport Processes, Wiley, NY,1960, page 319.
Gibbs notation (vector notation)
pp C
ST
C
kTv
tT
ˆˆ2
Cartesian (xyz) coordinates:
ppzyx C
S
z
T
y
T
x
T
C
kzT
vyT
vxT
vtT
ˆˆ 2
2
2
2
2
2
Cylindrical (rz) coordinates:
ppzr C
S
z
TT
rrT
rrrC
kzT
vT
rv
rT
vtT
ˆ11
ˆ 2
2
2
2
2
Spherical (r) coordinates:
pr
r
T
rrT
rrrC
kTr
vTrv
rT
vtT
sin
1sin
sin
11ˆsin 222
22
© Faith A. Morrison, Michigan Tech U.10
CM3110 Heat Transfer Lecture 3 11/6/2017
6
Example 1: Unsteady Heat Conduction in a Semi‐infinite solid
A very long, very wide, very tall slab is initially at a temperature To. At time t = 0, the left face of the slab is exposed to an environment at temperature T1. What is the time‐dependent temperature profile in the slab? The slab is a homogeneous material of thermal conductivity, k, density, , and heat capacity, Cp.
© Faith A. Morrison, Michigan Tech U.
Newton’s law of cooling BC’s:
11
x
y
oTTt 0
oTTt 0
1
0TT
t 0
( , )
t
T T x t
x
y
1D Heat Transfer: Unsteady State
ppzyx C
S
z
T
y
T
x
T
C
k
z
Tv
y
Tv
x
Tv
t
Tˆˆ 2
2
2
2
2
2
Microscopic Energy Equation in Cartesian Coordinates
pC
kˆ
thermal diffusivity
what are the boundary conditions? initial conditions?
© Faith A. Morrison, Michigan Tech U.12
1D Heat Transfer: Unsteady State
CM3110 Heat Transfer Lecture 3 11/6/2017
7
© Faith A. Morrison, Michigan Tech U.13
Example 7: Unsteady Heat Conduction in a Semi‐infinite solid
You try.
x
y
oTTt 0
oTTt 0
1
0TT
t 0
( , )
t
T T x t
x
y
Initial Condition:
1D Heat Transfer: Unsteady State
2
2
2
2
ˆ x
T
x
T
C
k
t
T
p
Initial condition:
Boundary conditions:
xTTt o ,0
0,0 1 tTThAqx x
Unsteady State Heat Conduction in a Semi‐Infinite Slab
tTTx o ,
© Faith A. Morrison, Michigan Tech U.© Faith A. Morrison, Michigan Tech U.14
x
y
oTTt 0
oTTt 0
1
0TT
t 0
( , )
t
T T x t
x
y1D Heat Transfer: Unsteady State
CM3110 Heat Transfer Lecture 3 11/6/2017
8
2
2
2
2
ˆ x
T
x
T
C
k
t
T
p
Initial condition:
Boundary conditions:
xTTt o ,0
0,0 1 tTThAqx x
tTTx o ,
© Faith A. Morrison, Michigan Tech U.© Faith A. Morrison, Michigan Tech U.15
x
y
oTTt 0
oTTt 0
1
0TT
t 0
( , )
t
T T x t
x
y
“for all ”
Unsteady State Heat Conduction in a Semi‐Infinite Slab
1D Heat Transfer: Unsteady State
© Faith A. Morrison, Michigan Tech U.© Faith A. Morrison, Michigan Tech U.
2
2
2
2
ˆ x
T
x
T
C
k
t
T
p
Initial condition:
Boundary conditions:
xTTt o ,0
0,0 1 tTThAqx x
tTTx o ,
k
th t
x
2
16
x
y
oTTt 0
oTTt 0
1
0TT
t 0
( , )
t
T T x t
x
y
Unsteady State Heat Conduction in a Semi‐Infinite Slab
1D Heat Transfer: Unsteady State
CM3110 Heat Transfer Lecture 3 11/6/2017
9
Geankoplis 4th ed., eqn 5.3‐7, page 363
k
th t
x
2
complementary error function of
error function of
© Faith A. Morrison, Michigan Tech U.17
erfc erfc
erfc ≡ 1 erf
erf ≡2
′
x
y
oTTt 0
oTTt 0
1
0TT
t 0
( , )
t
T T x t
x
y
Solution:
Unsteady State Heat Conduction in a Semi‐Infinite Slab
1D Heat Transfer: Unsteady State
Geankoplis 4th ed., eqn 5.3‐7, page 363
k
th t
x
2
complementary error function of
error function of
© Faith A. Morrison, Michigan Tech U.18
erfc erfc
erfc ≡ 1 erf
erf ≡2
′
x
y
oTTt 0
oTTt 0
1
0TT
t 0
( , )
t
T T x t
x
y
To make this solution easier to use, we can
plot it.
Solution:
1D Heat Transfer: Unsteady State Heat Conduction in a Semi‐Infinite Slab
CM3110 Heat Transfer Lecture 3 11/6/2017
10
© Faith A. Morrison, Michigan Tech U.19
This:
Versus this:
At various values of this:
k
th
t
x
2
erfc erfc
x
y
oTTt 0
oTTt 0
1
0TT
t 0
( , )
t
T T x t
x
y
To make this solution easier to use, we can
plot it.
1D Heat Transfer: Unsteady State Heat Conduction in a Semi‐Infinite Slab
t
x
2
o
o
TT
TT
1
Geankoplis 4th ed., Figure 5.3‐3, page 364
© Faith A. Morrison, Michigan Tech U.20
1D Heat Transfer: Unsteady State Heat Conduction in a Semi‐Infinite Slab
CM3110 Heat Transfer Lecture 3 11/6/2017
11
0 2 4 6 8 100.2
0
0.2
0.4
0.6
0.8
1
Y( ),x 0.1
Y( ),x 0.2
Y( ),x 0.5
Y( ),x 1
Y( ),x 5
Y( ),x 10
Y( ),x 50
Y( ),x 100
x x
o
o
TT
TT
1
increasing time, t
t = 0.1, 0.2, 0.5, 1, 5, 10, 50, 100
h = k = = 1
Unsteady State Heat Conduction in a Semi‐Infinite Slab
© Faith A. Morrison, Michigan Tech U.21
With modern tools, we can plot the solution directly (evaluated on Mathcad)
1D Heat Transfer: Unsteady State Heat Conduction in a Semi‐Infinite Slab
How could we use this solution?
Example: Will my pipes freeze?
The temperature has been 35oF for a while now, sufficient to chill the ground to this temperature for many tens of feet below the surface. Suddenly the temperature drops to ‐20oF. How long will it take for freezing temperatures (32oF) to reach my pipes, which are 8 ft under ground? Use the following physical properties:
Ffth
BTUk
h
ft
Ffth
BTUh
osoil
soil
o
5.0
018.0
0.2
2
2
© Faith A. Morrison, Michigan Tech U.22
1D Heat Transfer: Unsteady State Heat Conduction in a Semi‐Infinite Slab
CM3110 Heat Transfer Lecture 3 11/6/2017
12
© Faith A. Morrison, Michigan Tech U.23
k
th
t
x
2
erfc erfc x
y
oTTt 0
oTTt 0
1
0TT
t 0
( , )
t
T T x t
x
y
?
???
Both and depend on time
1D Heat Transfer: Unsteady State Heat Conduction in a Semi‐Infinite Slab
t
x
2
o
o
TT
TT
1
Geankoplis 4th ed., Figure 5.3‐3, page 364
© Faith A. Morrison, Michigan Tech U.24
(Interativesolution)
1D Heat Transfer: Unsteady State Heat Conduction in a Semi‐Infinite Slab
CM3110 Heat Transfer Lecture 3 11/6/2017
13
© Faith A. Morrison, Michigan Tech U.25
erfc erfc x
y
oTTt 0
oTTt 0
1
0TT
t 0
( , )
t
T T x t
x
y
Answer:480 20
1D Heat Transfer: Unsteady State Heat Conduction in a Semi‐Infinite Slab
2H
Example 8: Unsteady Heat Conduction in a Finite‐sized solid
x
y
zL
D
•The slab is tall and wide, but of thickness 2H•Initially at To•at time t = 0 the temperature of the sides is changed to T1
x
y
© Faith A. Morrison, Michigan Tech U.
2HT1 T1
t >0
26
1D Heat Transfer: Unsteady State
CM3110 Heat Transfer Lecture 3 11/6/2017
14
Use same microscopic energy balance eqn as before.
see handout for component notation
rate of change
convection
conduction (all directions)
source
(energy generated per unit volume per time)
STkTvt
TCp
2ˆ
Unsteady State Heat Transfer
© Faith A. Morrison, Michigan Tech U.27
1D Heat Transfer: Unsteady Heat Conduction in a Finite Solid
ppzyx C
S
z
T
y
T
x
T
C
k
z
Tv
y
Tv
x
Tv
t
Tˆˆ 2
2
2
2
2
2
Microscopic Energy Equation in Cartesian Coordinates
pC
kˆ
thermal diffusivity
what are the boundary conditions? initial conditions?
© Faith A. Morrison, Michigan Tech U.28
1D Heat Transfer: Unsteady Heat Conduction in a Finite Solid
CM3110 Heat Transfer Lecture 3 11/6/2017
15
© Faith A. Morrison, Michigan Tech U.29
You try.
Example 8: Unsteady Heat Conduction in a Finite‐sized solid
1D Heat Transfer: Unsteady Heat Conduction in a Finite Solid
2
2
2
2
ˆ x
T
x
T
C
k
t
T
p
Initial condition:
Boundary conditions:
xTTt o ,0
0,2
,0
1
1
tTTHxTTx
Unsteady State Heat Conduction in a Finite Slab
© Faith A. Morrison, Michigan Tech U.30
1D Heat Transfer: Unsteady Heat Conduction in a Finite Solid
CM3110 Heat Transfer Lecture 3 11/6/2017
16
Q: How can two completely different situations give the same governing equation?A: The boundary conditions make all the difference
o
o
TT
TT
1
x
oTT
TT
1
1
x
For more solutions to this equation see Carslaw and Jeager, Conduction of Heat in Solids, 2nd Edition, Oxford, 1959
x
0
0
© Faith A. Morrison, Michigan Tech U.31
1D Heat Transfer: Unsteady Heat Conduction in a Finite Solid
Unsteady State Heat Conduction in a Finite Slab
© Faith A. Morrison, Michigan Tech U.
2
2
2
2
ˆ x
T
x
T
C
k
t
T
p
Initial condition:
Boundary conditions:
xTTt o ,0
0,2
,0
1
1
tTTHxTTx
32
1D Heat Transfer: Unsteady Heat Conduction in a Finite Solid
CM3110 Heat Transfer Lecture 3 11/6/2017
17
2
2
x
Y
t
Y
Initial condition:
Boundary conditions:
1,0 YxTTt o
00,20,0
1
1
tYTTHxYTTx
Unsteady State Heat Conduction in a Finite Slab: solution by separation of variables
)()( txXY Guess:
Let
oTT
TTY
1
1
© Faith A. Morrison, Michigan Tech U.33
1D Heat Transfer: Unsteady Heat Conduction in a Finite Solid
2
2
x
Y
t
Y
Unsteady State Heat Conduction in a Finite Slab: soln by separation of variables
)()( txXY
© Faith A. Morrison, Michigan Tech U.
)()(
)()(
)()(
)()()()(
2
2
2
2
tdx
xXd
x
Y
tdx
xdXtxX
xx
Ydt
tdxXtxX
tt
Y
34
1D Heat Transfer: Unsteady Heat Conduction in a Finite Solid
CM3110 Heat Transfer Lecture 3 11/6/2017
18
2
2
x
Y
t
Y
© Faith A. Morrison, Michigan Tech U.
2
2
2
2
)(
)(
1)(
)(
1
)()()(
)(
dx
xXd
xXdt
td
t
tdx
xXd
dt
tdxX
Substituting:
function of time only
function of position only
constant
35
The function of two variables is separable into two functions of one variable.
1D Heat Transfer: Unsteady Heat Conduction in a Finite Solid
© Faith A. Morrison, Michigan Tech U.
2
2 )(
)(
1
)(
)(
1
dx
xXd
xX
dt
td
t
Separates into two ordinary differential equations:
Solve.
Apply BCs.
Apply ICs.
36
The function of two variables is separable into two functions of one variable.
1D Heat Transfer: Unsteady Heat Conduction in a Finite Solid
CM3110 Heat Transfer Lecture 3 11/6/2017
19
H
xe
H
xe
H
xe
TT
TT
Ht
Ht
Ht
o
2
5sin
5
1
2
3sin
3
1
2sin
4
2
22
2
22
2
2
45
43
4
1
1
Geankoplis 4th ed., eqn 5.3‐6, p363
Temperature Profile for Unsteady State Heat Conduction in a Finite Slab
© Faith A. Morrison, Michigan Tech U.37
1D Heat Transfer: Unsteady Heat Conduction in a Finite Solid
© Faith A. Morrison, Michigan Tech U.38
We use solutions like this in CM3215 to back calculate from non‐steady data:
CM3110 Heat Transfer Lecture 3 11/6/2017
20
Strategy: • Look up solution in literature• solve using numerical methods
Tricky step:
solving for T field; this can be mathematically difficult
•partial differential equation in up to three variables•boundaries may be complex•multiple materials, multiple phases present•may not be separable from mass and momentum balances
© Faith A. Morrison, Michigan Tech U.
Microscopic Energy Balance – is the correct physics for many problems!
(e.g. Comsol)
39
**** Or ****• Develop correlations on complex systems by
using Dimensional Analysis
1D Heat Transfer: Unsteady State
© Faith A. Morrison, Michigan Tech U.
40
Fluid Mechanics: What did we do?
• Turbulent tube flow • Noncircular conduits• Drag on obstacles
1. Find a simple problem that allows us to identify the physics
2. Nondimensionalize
3. Explore that problem
4. Take data and correlate
5. Solve real problems
Solve. Real. Problems.
Powerful.
Works on heat transfer too.