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CM3120 Module 2 Lecture I 1/4/2021
1
CM3120: Module 2
© Faith A. Morrison, Michigan Tech U.1
Unsteady State Heat Transfer
I. IntroductionII. Unsteady Microscopic Energy Balance—(slash and burn)III. Unsteady Macroscopic Energy BalanceIV. Dimensional Analysis (unsteady)—Biot number, Fourier
numberV. Low Biot number solutions—Lumped parameter analysisVI. Short Cut Solutions—(initial temperature 𝑇 ; finite ℎ),
Gurney and Lurie charts (as a function of position, 𝑚Bi, and Fo); Heissler charts (center point only, as a function of 𝑚 1/Bi, and Fo)
VII. Full Analytical Solutions (stretch)
CM3120: Module 2
© Faith A. Morrison, Michigan Tech U.2
Professor Faith A. Morrison
Department of Chemical EngineeringMichigan Technological University
www.chem.mtu.edu/~fmorriso/cm3120/cm3120.html
Lecture I:Introduction to Unsteady State
Heat Transfer
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CM3120 Module 2 Lecture I 1/4/2021
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© Faith A. Morrison, Michigan Tech U.3
HEAT TRANSFERSteady
Heat Transfer: Steady vs. Unsteady
Unsteadyvs.
To get started, let’s contrast the steady and unsteady cases in a familiar problem:
1D, heat conduction, in the 𝒙‐direction,
in a slab
Newton’s law of Cooling boundary conditions
𝑇
𝑇
𝑇 𝑇
1D, rectangular geometry:• Independent of time• Flux constant• linear temperature profile• Steady resistance to heat
transfer at both boundaries
© Faith A. Morrison, Michigan Tech U.4
Heat Transfer at Steady State
Aqx H
W
B x𝑥
𝑇
B
𝑇
𝑇
𝑇 𝑇
𝑇ℎ
𝑇ℎ
(Newton’s law of cooling BCs)
.
.𝑇
𝑇
Resistance to heat transfer
1/ℎ
Resistance to heat transfer 1/ℎ
What’s the temperature
profile?
Heat Transfer: Steady vs. Unsteady
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CM3120 Module 2 Lecture I 1/4/2021
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© Faith A. Morrison, Michigan Tech U.5
Heat Transfer at Steady State
Aqx H
W
B x𝑥
𝑇
B
𝑇
𝑇
𝑇 𝑇
𝑇ℎ
𝑇ℎ
(Newton’s law of cooling BCs)
.
.𝑇
𝑇
Resistance to heat transfer
1/ℎ
Resistance to heat transfer 1/ℎ
1D, rectangular geometry:• Independent of time• Flux constant• linear temperature profile• Steady resistance to heat
transfer at both boundaries
Heat Transfer: Steady vs. Unsteady
© Faith A. Morrison, Michigan Tech U.6
Heat Transfer at Steady State(Newton’s law of cooling BCs)
𝑥
𝑇
B
𝑇
𝑇 .
.𝑇
𝑇
Resistance to heat transfer
1/ℎ
Resistance to heat transfer 1/ℎ
𝑞𝐴 ℎ 𝑇 𝑇 ℎ 𝑇 𝑇𝑘
𝑑𝑇𝑑𝑥
Flux constant
Temperature distribution:
Independent of time
𝑇 𝑥
𝑇 𝑇 1ℎ𝑞𝐴
Newton’s law BC:
Heat Transfer: Steady vs. Unsteady
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CM3120 Module 2 Lecture I 1/4/2021
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© Faith A. Morrison, Michigan Tech U.7
Unsteady Heat Transfer
• We must specify the state of the system at some point in time (initial conditions)
• We must specify what then happens to cause heat to start to transfer (the scenario).
There are many circumstances that cause unsteady heat transfer.
To imagine a case where heat transfer is unsteady:
Heat Transfer: Steady vs. Unsteady
© Faith A. Morrison, Michigan Tech U.8
Can you think of any real engineering situations (unsteady heat xfer)?Can you write them in terms of:
• initial conditions and • a modeling scenario? You try.
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CM3120 Module 2 Lecture I 1/4/2021
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© Faith A. Morrison, Michigan Tech U.9
Unsteady Heat Transfer
𝑥
𝑇
B
.
.𝑇
𝑇
Example: A wide, tall slab initially uniformly at 𝑇is suddenly subjected to flowing fluid on its two broad faces. The left fluid is at 𝑇 and its heat transfer to the wall is characterized by heat transfer coefficient ℎ , while the right side is at 𝑇and characterized by ℎ . What is the temperature distribution across the slab as a function of time?
What do we think will happen?• Will there be heat transfer
resistance at the boundaries?• Will there be a linear
temperature profile in the slab?• Femtoseconds after the change,
what does the profile look like?• What will the solution trend
towards as time goes on → ∞ ?
𝑇
0
Heat Transfer: Steady vs. Unsteady
© Faith A. Morrison, Michigan Tech U.10
Unsteady Heat Transfer
𝑥
𝑇
B
.
.𝑇
𝑇
What do we think will happen?
You try.𝑇
0
𝑡 10 𝑠
𝑓𝑒𝑚𝑡𝑜𝑠𝑒𝑐𝑜𝑛𝑑 10 𝑠
Example: A wide, tall slab initially uniformly at 𝑇is suddenly subjected to flowing fluid on its two broad faces. The left fluid is at 𝑇 and its heat transfer to the wall is characterized by heat transfer coefficient ℎ , while the right side is at 𝑇and characterized by ℎ . What is the temperature distribution across the slab as a function of time?
Heat Transfer: Steady vs. Unsteady
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CM3120 Module 2 Lecture I 1/4/2021
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© Faith A. Morrison, Michigan Tech U.11
Unsteady Heat Transfer
𝑇
.
.𝑇
𝑇
𝑇𝑡 10 𝑠
𝑥B
What do we think will happen?• Will there be heat transfer
resistance at the boundaries?• Will there be a linear
temperature profile in the slab?• Femtoseconds after the change,
what does the profile look like?• What will the solution trend
towards as time goes on → ∞ ?0
• Femtoseconds after the change
ℎ
ℎ
Unsteady Heat TransferExample: A wide, tall slab initially uniformly at 𝑇is suddenly subjected to flowing fluid on its two broad faces. The left fluid is at 𝑇 and its heat transfer to the wall is characterized by heat transfer coefficient ℎ , while the right side is at 𝑇and characterized by ℎ . What is the temperature distribution across the slab as a function of time?
Heat Transfer: Steady vs. Unsteady
Heat Transfer: Steady vs. Unsteady
𝑞𝐴ℎ 𝑇 𝑇
𝑞𝐴ℎ 𝑇 𝑇
© Faith A. Morrison, Michigan Tech U.12
B 𝑥
𝑇
.
.𝑇
𝑇
𝑇
𝑡 10 𝑠
𝑞𝐴
𝑞𝐴
𝑞𝐴 0
Unsteady
Heat flows in the positive 𝑥‐ direction →
𝑇 𝑇 1ℎ𝑞𝐴
Newton’s law BC:
𝑞𝐴 ∝
∝ 𝑞𝐴
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CM3120 Module 2 Lecture I 1/4/2021
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𝑞𝐴ℎ 𝑇 𝑇
𝑞𝐴ℎ 𝑇 𝑇
© Faith A. Morrison, Michigan Tech U.13
B 𝑥
𝑇
.
.𝑇
𝑇
𝑇
𝑡 10 𝑠
𝑞𝐴
𝑞𝐴
𝑞𝐴 0
Unsteady
Heat flows in the positive 𝑥‐ direction →
What do we think will happen?
What’s next?
𝑞𝐴 ∝
∝ 𝑞𝐴
Heat Transfer: Steady vs. Unsteady
𝑞𝐴
𝑞𝐴
© Faith A. Morrison, Michigan Tech U.14
B 𝑥
𝑇
.
.𝑇
𝑇
𝑇
𝑡 𝑡
𝑞𝐴ℎ 𝑇 𝑇 , 𝑞𝐴
ℎ 𝑇 , 𝑇
𝑞𝐴 𝑓 𝑥, 𝑡
and varying with 𝑡
Unsteady
𝑞𝐴 ∝
∝ 𝑞𝐴
Heat Transfer: Steady vs. Unsteady
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CM3120 Module 2 Lecture I 1/4/2021
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© Faith A. Morrison, Michigan Tech U.15
B 𝑥
𝑇
.
.𝑇
𝑇
𝑇
𝑡 𝑡
𝑞𝐴ℎ 𝑇 𝑇 , 𝑞𝐴
ℎ 𝑇 , 𝑇
𝑞𝐴 𝑓 𝑥, 𝑡
Unsteady
𝑞𝐴 ∝
∝ 𝑞𝐴
𝑞𝐴
𝑞𝐴
and varying with 𝑡
Heat Transfer: Steady vs. Unsteady
© Faith A. Morrison, Michigan Tech U.16
B 𝑥
𝑇
.
.𝑇
𝑇
𝑇
𝑡 𝑡
𝑞𝐴ℎ 𝑇 𝑇 , 𝑞𝐴
ℎ 𝑇 , 𝑇
𝑞𝐴 𝑓 𝑥, 𝑡
Unsteady
𝑞𝐴 ∝
∝ 𝑞𝐴
…
𝑞𝐴
𝑞𝐴
and varying with 𝑡
Heat Transfer: Steady vs. Unsteady
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CM3120 Module 2 Lecture I 1/4/2021
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© Faith A. Morrison, Michigan Tech U.17
B 𝑥
𝑇
.
.𝑇
𝑇
𝑇
𝑡 ∞
𝑞𝐴ℎ 𝑇 𝑇 , 𝑞𝐴
ℎ 𝑇 , 𝑇
constant
𝑞𝐴
𝑞𝐴
𝑞𝐴𝑇
𝑇
Steady
𝑞𝐴 ∝
∝ 𝑞𝐴and constant
Heat Transfer: Steady vs. Unsteady
© F
aith
A. M
orris
on, M
ichi
gan
Tech
U.
18B 𝑥
𝑇
.
.𝑇
𝑇
𝑇decreases with time
also decreases with time
SteadyUnsteady
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CM3120 Module 2 Lecture I 1/4/2021
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© F
aith
A. M
orris
on, M
ichi
gan
Tech
U.
19B 𝑥
𝑇
.
.𝑇
𝑇
𝑇decreases with time
also decreases with time
SteadyUnsteady
Q: Does it always happen like this?
© F
aith
A. M
orris
on, M
ichi
gan
Tech
U.
20B 𝑥
𝑇
.
.𝑇
𝑇
𝑇decreases with time
also decreases with time
SteadyUnsteady
Q: Does it always happen like this?
A: Mmmm. No. It depends. . . .
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CM3120 Module 2 Lecture I 1/4/2021
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© Faith A. Morrison, Michigan Tech U.21
What are the various cases that are seen?
𝑥
𝑇
B
.
.𝑇
𝑇
𝑇
0
Let’s think.
• ℎ are large
Heat Transfer: Steady vs. Unsteady
© Faith A. Morrison, Michigan Tech U.22
• ℎ are largeHeat Transfer: Steady vs. Unsteady
𝑥
𝑇
B
.
.𝑇
𝑇
𝑇
0
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CM3120 Module 2 Lecture I 1/4/2021
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© Faith A. Morrison, Michigan Tech U.23
What are the various cases that are seen?• 𝑘 is large
𝑥
𝑇
B
.
.𝑇
𝑇
𝑇
0
Let’s think.
Heat Transfer: Steady vs. Unsteady
© Faith A. Morrison, Michigan Tech U.24
• 𝑘 is largeHeat Transfer: Steady vs. Unsteady
𝑥
𝑇
B
.
.𝑇
𝑇
𝑇
0
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CM3120 Module 2 Lecture I 1/4/2021
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© Faith A. Morrison, Michigan Tech U.25
What are the various cases that are seen?• Neither slab conduction
nor fluid convection dominates
𝑥
𝑇
B
.
.𝑇
𝑇
𝑇
0
Let’s think.
Heat Transfer: Steady vs. Unsteady
© Faith A. Morrison, Michigan Tech U.26
• Neither slab conduction nor fluid convection dominates
Heat Transfer: Steady vs. Unsteady
𝑥
𝑇
B
.
.𝑇
𝑇
𝑇
0
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CM3120 Module 2 Lecture I 1/4/2021
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B 𝑥
𝑇
.
.𝑇
𝑇
𝑇decreases with time
also decreases with time
© Faith A. Morrison, Michigan Tech U.27
What are the various cases that are seen?• If ℎ is large, the wall temp is
just the bulk temperature (fast convection)
• If 𝑘 is large, the temp profile is always straight (quasi‐steady state in the slab) and the convection works to keep up (heat transfer is limited by ℎ ; fast conduction in slab)
• If neither mechanism dominates, it’s complicated!
• If the boundary conditions vary with 𝑡, 𝑥, it’s complicated!
Heat Transfer: Steady vs. Unsteady
© Faith A. Morrison, Michigan Tech U.28
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