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Resistance Supplement 11/7/2019
1
CM3110 Transport IPart II: Heat Transfer
1
Heat Transfer Resistances (Supplement)
Professor Faith Morrison
Department of Chemical EngineeringMichigan Technological University
© Faith A. Morrison, Michigan Tech U.
These slides are incorporated into the slides from lectures 14-16, but are assembled here to tell the heat-transfer resistance
story all together.
© Faith A. Morrison, Michigan Tech U.
2
Thermal conductivity 𝑘 and heat transfer coefficient ℎ may be thought of as sources of resistance to heat transfer.
These resistances stack up in a logical way, allowing us to quickly and accurately determine the effect of adding insulating layers, encountering pipe fouling, and other applications.
1D Heat Transfer – Resistance Supplement
𝑇
𝑘
r
𝑘
𝑇2𝑅
2𝑅
2𝑅
𝐵
x
0
𝑇
𝑇
𝐵/2 𝐵
𝑘
𝑘
𝑇
Resistance Supplement 11/7/2019
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3
Using the solution: Composite Door:
For an outside door, a metal is used (𝑘 for strength, and a cork 𝑘is used for insulation. Both are the same thickness 𝐵/2. What is the temperature profile in the door at steady state? What is the flux? The inside temperature of the metal is 𝑇 and the outside temperature of the cork is 𝑇 .
© Faith A. Morrison, Michigan Tech U.
1D Heat Transfer
Let’s try.
𝐵
x
0
𝑇
𝑇
𝐵/2 𝐵
𝑘 𝑘
𝑘 ≫ 𝑘
© Faith A. Morrison, Michigan Tech U.
4
See handwritten notes.
https://pages.mtu.edu/~fmorriso/cm310/selected_lecture_slides.html
Note: in the hand notes the temperatures from left to right are 𝑇 ,𝑇 ,𝑇 .
Resistance Supplement 11/7/2019
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5
© Faith A. Morrison, Michigan Tech U.
𝑇 𝑥𝑇 𝑇𝐵/2
𝑥 𝑇
SOLUTION:
1D Heat Transfer
Example 1b: Composite Door (two equal width layers) 𝐵
x
0
𝑇
𝑇
𝐵/2 𝐵
𝑘
𝑘
𝑘 material: 0 𝑥 𝐵/2
𝑇 𝑥𝑇 𝑇𝐵/2
𝑥 2𝑇 𝑇
𝑘 material: 𝐵/2 𝑥 𝐵
𝑞𝐴
𝑇 𝑇𝐵2 𝑘 𝑘
𝑘 𝑘
𝑇𝑘 𝑇 𝑘 𝑇𝑘 𝑘
𝑇
6
© Faith A. Morrison, Michigan Tech U.
1D Heat Transfer
𝐵
x
0
𝑇
𝑇
𝐵/2 𝐵
𝑘
𝑘
𝑇
𝑞𝐴
𝑇 𝑇ℛ ℛ
driving forceresistance
Let: ℛ ≡
SOLUTION:
Example 1b: Composite Door (two equal width layers)
Each of the layers contributes a resistance, added in series (like in electricity).
𝑞𝐴
𝑇 𝑇𝐵/2𝑘
𝐵/2𝑘
Resistance Supplement 11/7/2019
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7
What is the steady state temperature profile in a rectangular slab if the fluid on one side is held at Tb1 and the fluid on the other side is held at Tb2?
Assumptions:•wide, tall slab•steady state•h1 and h2 are the heat transfer coefficients of the left and right walls
Tb1
Tb1>Tb2
H
W
B
Tb2
x
Newton’s law of cooling boundary
conditions
Bulk fluid temperature on left
Bulk fluid temperature on right
© Faith A. Morrison, Michigan Tech U.
Example 2: Heat flux in a rectangular solid – Newton’s law of cooling BC
1D Heat Transfer
© Faith A. Morrison, Michigan Tech U.
8
See handwritten notes (in class, also on web).
https://pages.mtu.edu/~fmorriso/cm310/algebra_details_N_law_cooling.pdf
https://pages.mtu.edu/~fmorriso/cm310/selected_lecture_slides.html
Resistance Supplement 11/7/2019
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9
Solution: (temp profile, flux)
21
1
21
1
11
1
hkB
h
hkx
TT
TT
bb
b
21
2111
hk
B
h
TT
A
q bbx
Rectangular slab with Newton’s law of cooling BCs
Temperature profile:
(linear)
Flux:
(constant)
© Faith A. Morrison, Michigan Tech U.
Example 2: Heat flux in a rectangular solid – Newton’s law of cooling BC
1D Heat Transfer
10
Temperature profile:
(linear)
© Faith A. Morrison, Michigan Tech U.
𝑇 𝑇 𝑥
1D Heat Transfer
Solution: (temp profile, flux)
Example 2: Heat flux in a rectangular solid – Newton’s law of cooling BC
Resistance due to heat transfer at boundaryResistance due to finite thermal conductivity
21
1
21
1
11
1
hkB
h
hkx
TT
TT
bb
b
Resistance Supplement 11/7/2019
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11
R1
Example 3: Heat flux in a cylindrical shell – Temp BC
Assumptions:•long pipe•steady state• k = thermal conductivity of wall
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)
Cooler wall at T2
Hot wall at T1
R2
r
L
(very long)
Material of thermal conductivity k
© Faith A. Morrison, Michigan Tech U.
1D Heat Transfer – Radial
© Faith A. Morrison, Michigan Tech U.
12
See handwritten notes in class.
https://pages.mtu.edu/~fmorriso/cm310/selected_lecture_slides.html
Resistance Supplement 11/7/2019
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13
Solution for Cylindrical Shell:
1
2
2
12
2
ln
ln
RRrR
TT
TT
Note that 𝑇 𝑟 does not
depend on the thermal conductivity, 𝑘 (steady state)
Pipe with temperature BCs
© Faith A. Morrison, Michigan Tech U.
The heat flux DOES depend
on, 𝑘; also decreases as 1/𝑟
Example 3: Heat flux in a cylindrical shell – Temp BC
NOTlinear
NOTconstant
1D Heat Transfer – Radial
𝑞𝐴
𝑇 𝑇1𝑘 ln𝑅𝑅
1𝑟
14
Solution for Cylindrical Shell:
© Faith A. Morrison, Michigan Tech U.
Example 3: Heat flux in a cylindrical shell – Temp BC
NOTconstant
1D Heat Transfer – Radial
𝑞𝐴
𝑇 𝑇1𝑘 ln𝑅𝑅
1𝑟
Resistance due to finite thermal conductivity, radial
Let: ℛ ≡ ln
𝑞𝐴
𝑇 𝑇ℛ
1𝑟
driving forceresistance
Resistance Supplement 11/7/2019
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Using the solution: Insulated Pipe (Composite, radial conduction)
For a metal pipe carrying a hot liquid (𝑘 an insulation layer is added with thermal conductivity 𝑘 . What is the temperature profile in the composite pipe at steady state? What is the flux? The inside temperature of the metal pipe is 𝑇and the outside temperature of the insulation is 𝑇 .
© Faith A. Morrison, Michigan Tech U.
1D Radial Heat Transfer
𝑘 ≫ 𝑘
𝑇
𝑘
r
𝑘
𝑇2𝑅
2𝑅
2𝑅
𝑇
𝑘
r
𝑘
𝑇2𝑅
2𝑅
2𝑅
16© Faith A. Morrison, Michigan Tech U.
𝑻 𝒓NOTlinear
FLUX NOT
constant
1D Heat Transfer – Radial
SOLUTION:
Example 3b: Insulated Pipe (Composite, radial conduction)
𝑞𝐴
𝑘𝑑𝑇𝑑𝑟
constant1𝑟
𝑇 𝑟 𝑎 ln 𝑟 b
𝑘 material: 𝑅 𝑟 𝑅
𝑘 material: 𝑅 𝑟 𝑅
𝑇 𝑟 𝑎 ln 𝑟 b
Resistance Supplement 11/7/2019
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© Faith A. Morrison, Michigan Tech U.
17
See Lecture 16 Slides
https://pages.mtu.edu/~fmorriso/cm310/selected_lecture_slides.html
18© Faith A. Morrison, Michigan Tech U.
1D Heat Transfer – Radial
SOLUTION:
Example 3b: Insulated Pipe (Composite, radial conduction)
𝑞𝐴
𝑇 𝑇1𝑘 ln𝑅𝑅
1𝑘 ln
𝑅𝑅
1𝑟
𝑞𝐴
𝑇 𝑇ℛ ℛ
1𝑟
driving forceresistance
Each of the layers contributes a resistance, added in series (like in electricity).
Let: ℛ ≡ ln
Note that we can continue to add layers in terms of resistance
𝑇
𝑘
r
𝑘
𝑇2𝑅
2𝑅
2𝑅
Resistance Supplement 11/7/2019
10
R1
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
Hot fluid at Tb1
R2
r
© Faith A. Morrison, Michigan Tech U.
1D Heat Transfer – Radial
© Faith A. Morrison, Michigan Tech U.
20
See handwritten notes.
https://pages.mtu.edu/~fmorriso/cm310/selected_lecture_slides.html
Resistance Supplement 11/7/2019
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© Faith A. Morrison, Michigan Tech U.
Solution: Radial Heat Flux in an Annulus
Example 4: Heat flux in a cylindrical shell Newton’s law of cooling boundary conditions
𝑇 𝑇𝑇 𝑇 ln 𝑅
𝑟𝑘
ℎ 𝑅
𝑘ℎ 𝑅 ln 𝑅
𝑅𝑘
ℎ 𝑅
1D Heat Transfer – Radial
𝑞𝐴
𝑇 𝑇1
ℎ 𝑅1𝑘 ln 𝑅
𝑅1
ℎ 𝑅
1𝑟
Resistance ℛ due to heat transfer coefficients, radialResistance ℛ due to finite thermal conductivity, radial
𝑇 𝑟
𝑞 𝑟
© Faith A. Morrison, Michigan Tech U.
Solution: Radial Heat Flux in an Annulus
1D Heat Transfer – Radial
𝑞𝐴
𝑇 𝑇1
ℎ 𝑅1𝑘 ln 𝑅
𝑅1
ℎ 𝑅
1𝑟
Resistance ℛ due to heat transfer coefficients, radialResistance ℛ due to finite thermal conductivity, radial
Note that we can continue to add layers in terms of resistance
Resistance Supplement 11/7/2019
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23© Faith A. Morrison, Michigan Tech U.
1D Heat Transfer – Composite Structures
𝑞𝐴
𝑇 𝑇ℛ ℛ
1𝑟
driving forceresistance
Let: ℛ ≡ ln𝑇
𝑘
r
𝑘
𝑇2𝑅
2𝑅
2𝑅
𝐵
x
0
𝑇
𝑇
𝐵/2 𝐵
𝑘
𝑘
𝑇
𝑞𝐴
𝑇 𝑇ℛ ℛ
driving forceresistance
Let: ℛ ≡
Note: Geankoplis uses a different resistance. For rectangular heat flux:
𝑅 ℛ/𝐿𝑊
Note: Geankoplis uses a different resistance. For radial heat flux:
𝑅 ℛ/2𝜋𝐿
24
© Faith A. Morrison, Michigan Tech U.
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