Heat Transfer Resistance Supplement CM3110

11/20/2020

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CM3110 Transport IPart II: Heat Transfer

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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.

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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

Heat Transfer Resistance Supplement CM3110

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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.

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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 .

Heat Transfer Resistance Supplement CM3110

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© 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

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© 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

Heat Transfer Resistance Supplement CM3110

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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.

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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

Heat Transfer Resistance Supplement CM3110

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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

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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

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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.

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See handwritten notes in class.

https://pages.mtu.edu/~fmorriso/cm310/selected_lecture_slides.html

Heat Transfer Resistance Supplement CM3110

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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

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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

Heat Transfer Resistance Supplement CM3110

11/20/2020

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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

Heat Transfer Resistance Supplement CM3110

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© Faith A. Morrison, Michigan Tech U.

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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

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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.

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See handwritten notes.

https://pages.mtu.edu/~fmorriso/cm310/selected_lecture_slides.html

Heat Transfer Resistance Supplement CM3110

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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

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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

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© Faith A. Morrison, Michigan Tech U.

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