Lecture 2 Heat Transfer CM3110
11/30/2015
CM3110
Transport I
Part II: Heat Transfer
One-Dimensional Heat
Transfer (continued)
Professor Faith Morrison
Department of Chemical Engineering
Michigan Technological University
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© Faith A. Morrison, Michigan Tech U.
General Energy Transport Equation
(microscopic energy balance)
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. S
dS
n̂
V
Gibbs notation:
⋅
see handout for component notation
© Faith A. Morrison, Michigan Tech U.
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Lecture 2 Heat Transfer CM3110
11/30/2015
General Energy Transport Equation
(microscopic energy balance)
convection
source
(energy generated per unit volume per time)
⋅
rate of change
velocity must satisfy equation of motion, equation of continuity
conduction (all directions)
see handout for component notation
© Faith A. Morrison, Michigan Tech U.
Note: this
handout is
also on the
web
© Faith A. Morrison, Michigan Tech U.
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Lecture 2 Heat Transfer CM3110
11/30/2015
Example 3: Heat flux in a cylindrical shell – Temp BC
What is the steady state temperature profile in a cylindrical shell (pipe) if the inner wall is at T1 and the outer wall is at T2? (T1>T2)
Assumptions:
•long pipe
•steady state
• k = thermal conductivity of wall
Cooler wall at T2
R1

r
R2
L
(very long)
Hot wall at T1
Material of thermal conductivity k
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© Faith A. Morrison, Michigan Tech U.
Example 3: Heat flux in a cylindrical shell – Temp BC
Let’s
try.
© Faith A. Morrison, Michigan Tech U.
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Lecture 2 Heat Transfer CM3110
11/30/2015
Example 3: Heat flux in a cylindrical shell – Temp BC
Solution:
qr c1

A r
c
T   1 ln r  c2
k
Not constant
Boundary conditions?
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© Faith A. Morrison, Michigan Tech U.
Example 3: Heat flux in a cylindrical shell – Temp BC
Solution for Cylindrical Shell:
NOT
constant
qr T1  T2  k 

 
A ln R2  r 
R1
R2
T2  T
r

T2  T1 ln R2
R1
ln
NOT
linear
The heat flux on, k; also DOES depend decreases as 1/r
Note that T(r) does not depend on the thermal conductivity, k (steady state)
Pipe with temperature BCs
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© Faith A. Morrison, Michigan Tech U.
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Lecture 2 Heat Transfer CM3110
11/30/2015
Example 3: Heat flux in a cylindrical shell – Temp BC
DimensionlessTemperature Profile in a pipe;
R1=1, R2=2
1
0.9
0.8
T2  T
T2  T1
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
Pipe with temperature BCs
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
r
R2
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© Faith A. Morrison, Michigan Tech U.
Example 4: Heat flux in a cylindrical shell – Newton’s law of cooling
Assumptions:
•long pipe
•steady state
•k = thermal conductivity of wall
•h1, h2 = heat transfer coefficients
What is the steady state temperature profile in a cylindrical shell (pipe) if the fluid on the inside is at Tb1 and the fluid on the outside is at Tb2? (Tb1>Tb2)
Cooler fluid at Tb2
R1

r
R2
Hot fluid at Tb1
© Faith A. Morrison, Michigan Tech U.
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Lecture 2 Heat Transfer CM3110
11/30/2015
Example 4: Heat flux in a cylindrical shell
You try.
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© Faith A. Morrison, Michigan Tech U.
Example 4: Heat flux in a cylindrical shell
Solution:
qr c1

A r
c
T   1 ln r  c2
k
Not constant
Boundary conditions?
© Faith A. Morrison, Michigan Tech U.
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Lecture 2 Heat Transfer CM3110
11/30/2015
Example 4: Heat flux in a cylindrical shell
c1
 h1 Tb1  Tw1 
R1
c1
 h2 Tw 2  Tb 2 
R2
c
Tw1   1 ln R1  c2
k
c
Tw 2   1 ln R2  c2
k
4 equations
4 unknowns;
c1 , Tw1 , c2 , Tw 2
SOLVE
© Faith A. Morrison, Michigan Tech U.
Newton’s law of
cooling boundary
conditions
Example 4: Heat flux in a cylindrical shell
Solution: Radial Heat Flux in an Annulus
ln
ln
1
1
1
ln
1
© Faith A. Morrison, Michigan Tech U.
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Lecture 2 Heat Transfer CM3110
11/30/2015
Example 5: Heat Conduction with Generation
What is the steady state temperature profile in a wire if heat is generated uniformly throughout the wire at a rate of Se W/m3 and the outer radius is held at Tw?
long wire
R
r
Tw
Se = energy production per unit volume
© Faith A. Morrison, Michigan Tech U.
Example 5: Heat conduction with generation
You try.
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© Faith A. Morrison, Michigan Tech U.
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Lecture 2 Heat Transfer CM3110
11/30/2015
Compare solutions
1.2
Temperature ratio
T  T1 1
r
 ln
T1  T2 ln R1
R1
R2
Conduction in pipe
Wire with generation
1
0.8
T  Tw
r
 1  
S e R 2 / 4k
R
0.6
2
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
r/R or r/R1
© Faith A. Morrison, Michigan Tech U.
Example 6: Wall heating of laminar flow. What is the steady state temperature profile in a flowing fluid in a tube if the walls are heated (constant flux, q1 /A) and if the fluid is a Newtonian fluid in laminar flow?
cross‐section A:
r
z
r
z
A
assume:
constant viscosity
heater
L
vz(r)
fluid
R
© Faith A. Morrison, Michigan Tech U.
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Lecture 2 Heat Transfer CM3110
11/30/2015
cross‐section A:
Example 5: Wall heating of laminar flow
r
z
r
z
A
L
heater
vz(r)
fluid
R
You try.
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© Faith A. Morrison, Michigan Tech U.
cross‐section A:
Example 5: Wall heating of laminar flow
r
z
r
z
A
heater
L
vz(r)
fluid
R
We need to solve this partial differential equation:
1
0
with the appropriate boundary conditions. To see the solution see:
•
•
R. Siegel, E. M. Sparrow, T. M. Hallman, Appl. Science Research A7, 386-392 (1958)
R. B. Bird, W. Stewart, and E. Lightfoot, Transport Phenomena, Wiley, 1960, p295.
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© Faith A. Morrison, Michigan Tech U.
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Lecture 2 Heat Transfer CM3110
11/30/2015
SUMMARY
Steady State Heat Transfer
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
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© Faith A. Morrison, Michigan Tech U.
SUMMARY
Steady State Heat Transfer
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
Conclusion: When we can simplify
geometry, assume steady state, assume
symmetry, the solutions are easily obtained
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© Faith A. Morrison, Michigan Tech U.
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Lecture 2 Heat Transfer CM3110
11/30/2015
SUMMARY
Steady State Heat Transfer
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
Unsteady State Heat Transfer
???
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© Faith A. Morrison, Michigan Tech U.
Next
© Faith A. Morrison, Michigan Tech U.
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