Fundamentals of Heat Transfer

Fundamentals of Momentum, Heat, and Mass Transfer

Chapter 14

Fundamentals of Heat Transfer

14.1 Conduction

14.2 Thermal Conductivity 14.3 Convection

14.4 Radiation

14.5 Combined Mechanisms of Heat Transfer

- 1 -

본 자료의 모든 그림, 표, 예제 등은 다음의 문헌을 참고하였습니다.

참고문헌 : J. R. Welty, G. L. Rorrer, D. G. Foster,“Fundamentals of Momentum, Heat and Mass Transfer”, 6th ed., Wiley, 2013.

Similarity among Momentum, Heat and Mass Transfers

- Momentum Transfer

Transferred Amount related to

Driving Force



yx

  

dy dv

_x

(Newton’s Law of Viscosity) - Heat Transfer

A

q

_x

  k

dx dT

(Fourier’s Law of Conductivity) - Mass Transfer

x

J

A_,

  D

_AB

dx dc

_A

(Fick’s Law of Diffusivity)







Molecular Interaction

Fluid



from Kinetic Theory of Gas

(Transfer Rate per Area)

Flux 

Proportionality



Gradient in Key Variable

- 3 -

Mechanism of Heat Transfer

A

q   k T

 

 





 





 





2 2

or

m W m

J s

for Conductive Heat Transfer

h T



for Convective Heat Transfer

T

4



 

for Radiative Heat Transfer

(1) Conduction

(2) Convection

(3) Radiation

T4

q A

 

  Stefan-Boltzmann Constant

T q A h    Heat Transfer Coefficient

T q A k   Thermal Conductivity

(Newton’s Law of Cooling)

 Heat Flux

10 8

676

5  ^

 .

 Three modes of

energy transfer

14.1 Conduction, 14.2 Thermal Conductivity

 Energy transfer by conduction : (1) Molecular interaction, (2) Free electrons

 Fourier’s first law of heat conduction

 Thermal conductivity for gases

Kinetic Theory of Gases

m T

k d

^B

3

2 2

3

1 

 

Rigid spherical molecules No intermolecular interaction Completely elastic collision

  ^J/K

10 1.38

Constant,

Boltzmann 

^-23

B





diameter molecule



d T  Absolute Temperatur e

 

^K _m

m W 



 





2 



 



 

mK

※ W

T qA k  

- 5 -

14.2 Thermal Conductivity

Chapman-Enskog Equation : accounted for attraction and repulsion forces

Lennard-Jones Constants (Appendix K)

m k T



k

 0 . 0829

₂



Diameter Collision

 

Conduction for

Integral Collision





_k

  ^A

^

 

 



 mK

W

 Thermal conductivity for gases

J. R. Welty, G. L. Rorrer, D. G. Foster,“Fundamentals of Momentum, Heat and Mass Transfer”, 6th ed., Wiley, 2013.

14.2 Thermal Conductivity

 Liquids : no rigorous theory => use experimental data in Appendix I

 Solids : no rigorous theory => use experimental data (ex: Appendix H)



 





 2.45 10^8 ₂ K W T

k L k

e

or

J. R. Welty, G. L. Rorrer, D. G. Foster,“Fundamentals of Momentum, Heat and Mass Transfer”, 6th ed., Wiley, 2013.

14.2 Thermal Conductivity

 Example 1. Heat conduction in radial direction

- cylinder material : steel

q

r

L

Q-1 : heat flow(transfer) rate per pipe length = ?

Q-2 : heat flux based on inside and outside diameter =?

- inside radius : r_i=0.94 [cm]

- wall thickness : r_o-r_i=0.391[cm] => r_o=1.331[cm]

- inside and outside surface temperature : T_i=367[K], T_o=344[K]

- 7 - J. R. Welty, G. L. Rorrer, D. G. Foster,“Fundamentals of Momentum, Heat and Mass Transfer”, 6th ed., Wiley, 2013.

14.2 Thermal Conductivity

14.2 Thermal Conductivity

 Example 2. Heat conduction in radial direction

q

r

L

Q-1 : steady-state heat-transfer rate = ? - thermal conductivity :

k ^{ k}

₀

 ^{1   T} 

- 9 - J. R. Welty, G. L. Rorrer, D. G. Foster,“Fundamentals of Momentum, Heat and Mass Transfer”, 6th ed., Wiley, 2013.

14.3 Convection



Convection

 Involves energy exchange between a surface and an adjacent fluid.

 (1) Forced convection, (2) free or natural convection

  ^K

m W 



 





2

T q A

h    



 



 

K m

W

2



Newton rate equation

Convective heat-transfer

coefficient Unit of h

Heat flux for convection

*q : Rate of convective heat transfer

14.4 Radiation



Radiation

 No medium is required. (maximum in a perfect vacuum)



Rate of energy emission from blackbody

(Stefan-Boltzmann law of thermal radiation)

Heat flux for radiation

*q : Rate of radiant-energy emission

Stefan-Boltzmann constant Unit of

σ

T

4

q A

 

 

10 8

676

5  ^

 .

^{ }   ^{ } _ ^{ } _ ^

 





4 4 2

2

K m

W K

m W

- 11 -

Mechanism of Heat Transfer

A

q   k T

 

 





 





 





2 2

or

m W m

J s

for Conductive Heat Transfer

h T



for Convective Heat Transfer

σT

4 for Radiative Heat Transfer

(1) Conduction

(2) Convection

(3) Radiation

q A T q A h    Heat Transfer Coefficient

T q A k   Thermal Conductivity

 Heat Flux

 Three modes of energy transfer

 

 



 mK

W



 





K m

W

2





W

14.5 Combined Mechanisms of Heat Transfer

dx k dT A

q

_x

  

0

^{ } 

₁²

T T

L x

dx kdT

A q

 ^T

₁

^T

₂



k A L

q

_x

   ^T

₁

^T

₂



L

q

_x

 kA 

L T q

_x

 kA 

T hA q

_x

 

(1) conductive heat transfer

(2) convective heat transfer

 ^{ T  T}

₁

^{ T}

₂



 







 



















c h

T T

T

or

T T

T

2

1

kA q dx

dT  

_x

* At steady state, L ΔT

q

_x

 kA

※ Table 14.1  h value _{- 13 -} J. R. Welty, G. L. Rorrer, D. G. Foster,“Fundamentals of Momentum, Heat and Mass Transfer”, 6th ed., Wiley, 2013.

14.5 Combined Mechanisms of Heat Transfer

 ^{T T}

₁



A h

q

_x



_{h h}





_{1 2}



1

1

T T

L A

q

_x

 k 



_{2 3}



2

2

T T

L A

q

_x

 k 



_{3 4}



3

3

T T

L A

q

_x

 k 



_c



c

x

h A T T

q 

₄



 

A q h

T T

h x h

1

1





 

A k q L T

T

_x

1 1 2

1

 

 

A k q L T

T

_x

2 2 3

2

 

 

A k q L T

T

_x

4 4 3

3

 

 ^T

_h

^{ T}

_c



 





 T  T

q

_x ^{h c}

 ^



thermal

R q

_x

T

 

A q h

T T

c x c

1

4

 

 Rectangular Geometry

(Cartesian Coordinates)

fluid fluid

q

x

(I) (II) (III) (IV) (V)

(I) (II)

(III) (IV)

(V)

 

 



    

 k A h A

L A

k L A

k L A

q h

c h

x

1 1

3 3 2

2 1

1

R I  V

 

R

i

V

J. R. Welty, G. L. Rorrer, D. G. Foster,“Fundamentals of Momentum, Heat and Mass Transfer”, 6th ed., Wiley, 2013.

14.5 Combined Mechanisms of Heat Transfer

 

 

 



    

 

A h A k

L A

k L A

k L A

h

T q T

c h

c h

x

1 1

3 3 2

2 1

1

^q

^x

^  ^{ R T}

^thermal



Rectangular geometry (cartesian coordinates)



Cylindrical geometry (cylindrical coordinates)

thermal

 R

 



 



    

 

o o r r

r r r r

i i

o i r

A h L k ln L

k ln L

k ln A

h

T q T

i 1

2 2

2 1

3 2

1

2 3 1

2 1







- 15 -

J. R. Welty, G. L. Rorrer, D. G. Foster,“Fundamentals of Momentum, Heat and Mass Transfer”, 6th ed., Wiley, 2013.

14.5 Combined Mechanisms of Heat Transfer

 Overall heat-transfer coefficient

T q A h   

Heat-transfer coefficient Heat flux for convection

※ U : Overall heat-transfer coefficient

 Shape factor

Rectangular

※ S : Shape factor



 





K m

W

2

 Example 3.

q

r

L

(1) Heat loss per meter ?

14.5 Combined Mechanisms of Heat Transfer

0.0209 m 0.0267 m



^



thermal

R

q_x T R_thermal 

- 17 -

J. R. Welty, G. L. Rorrer, D. G. Foster,“Fundamentals of Momentum, Heat and Mass Transfer”, 6th ed., Wiley, 2013.