Heat Transfer Booklet PDF

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1

HEAT TRANSFER EQUATION SHEET Heat Conduction Rate Equations (Fourier's Law)

 Heat Flux : 𝑞𝑥′′ = −𝑘 𝑑𝑥 𝑚2 ′′  Heat Rate : 𝑞𝑥 = 𝑞𝑥 𝐴𝑐 𝑊 Heat Convection Rate Equations (Newton's Law of Cooling) 𝑑𝑑

𝑊

 Heat Flux: 𝑞 ′′ = ℎ(𝑇𝑠 − 𝑇∞ ) 𝑚2  Heat Rate: 𝑞 = ℎ𝐴𝑠 (𝑇𝑠 − 𝑇∞ ) 𝑊 𝑊

𝑊

k : Thermal Conductivity 𝑚∙𝑘 Ac : Cross-Sectional Area

h : Convection Heat Transfer Coefficient As : Surface Area 𝑚2

𝑊

𝑚 2 ∙𝐾

Heat Radiation emitted ideally by a blackbody surface has a surface emissive power: 𝐸𝑏 = 𝜎 𝑇𝑠4  Heat Flux emitted : 𝐸 =

𝜀𝜎𝑇𝑠4

𝑊

𝑚2

𝑊

𝑚2

where ε is the emissivity with range of 0 ≤ 𝜀 ≤ 1

and 𝜎 = 5.67 × 10−8 is the Stefan-Boltzmann constant 𝑚2 𝐾 4  Irradiation: 𝐺𝑎𝑏𝑠 = 𝛼𝐺 but we assume small body in a large enclosure with 𝜀 = 𝛼 so that 𝐺 = 𝜀 𝜎 𝑇𝑠𝑠𝑠4 𝑊

4 ) ′′  Net Radiation heat flux from surface: 𝑞𝑠𝑎𝑑 = 𝐴 = 𝜀𝐸𝑏 (𝑇𝑠 ) − 𝛼𝐺 = 𝜀𝜎(𝑇𝑠4 − 𝑇𝑠𝑠𝑠 4 ) where for a real surface 0 ≤ 𝜀 ≤ 1  Net radiation heat exchange rate: 𝑞𝑠𝑎𝑑 = 𝜀𝜎𝐴𝑠 (𝑇𝑠4 − 𝑇𝑠𝑠𝑠 This can ALSO be expressed as: 𝑞𝑠𝑎𝑑 = ℎ𝑠 𝐴(𝑇𝑠 − 𝑇𝑠𝑠𝑠 ) depending on the application 𝑞

where ℎ𝑠 is the radiation heat transfer coefficient which is: ℎ𝑠 = 𝜀𝜎(𝑇𝑠 + 𝑇𝑠𝑠𝑠 )(𝑇𝑠2 + 𝑇𝑠𝑠𝑠2 ) 𝑚 2 ∙𝐾 4 4  TOTAL heat transfer from a surface: 𝑞 = 𝑞𝑐𝑐𝑐𝑐 + 𝑞𝑠𝑎𝑑 = ℎ𝐴𝑠 (𝑇𝑠 − 𝑇∞ ) + 𝜀𝜎𝐴𝑠 (𝑇𝑠 − 𝑇𝑠𝑠𝑠 ) 𝑊 Conservation of Energy (Energy Balance) 𝐸󰇗𝑖𝑐 + 𝐸󰇗𝑔 − 𝐸󰇗𝑐𝑠𝑜 = 𝐸󰇗𝑠𝑜 (Control Volume Balance) ; 𝐸󰇗𝑖𝑐 − 𝐸󰇗𝑐𝑠𝑜 = 0 (Control Surface Balance) where 𝐸󰇗𝑔 is the conversion of internal energy (chemical, nuclear, electrical) to thermal or mechanical energy, and 𝑊

𝑑𝑑 𝐸󰇗𝑠𝑜 = 0 for steady-state conditions. If not steady-state (i.e., transient) then 𝐸󰇗𝑠𝑜 = 𝜌𝜌𝑐𝑝

Heat Equation (Cartesian):

�𝑘 𝜕𝑥

𝜕

Heat Equation (used to find the temperature distribution)

𝜕𝑑

�+ 𝜕𝑥

If 𝑘 is constant then the above simplifies to: Heat Equation (Cylindrical): Heat Eqn. (Spherical):

𝜕2 𝑑 𝜕𝑥 2

�𝑘𝑘 𝜕𝑠 � + 𝜕𝑠 𝜕𝑑

1 𝜕 𝑠

𝜕𝑑 𝜕 �𝑘 � 𝜕𝜕 𝜕𝜕

1 𝜕 2 𝜕𝑑 � �𝑘𝑘 2 𝑠 𝜕𝑠 𝜕𝑠

Plane Wall: 𝑅𝑜,𝑐𝑐𝑐𝑑 =

𝐿 𝑘𝐴

+

+

𝜕2 𝑑

𝜕𝜕 2

�𝑘 𝜕𝜕

1 𝜕

𝑠2

1

+ 𝑠 2 sin 𝜃 2

𝜕

+

𝜕𝑑 𝜕 � �𝑘 𝜕𝜕 𝜕𝜕 𝜕2 𝑑

𝜕𝜕 2

𝜕𝑑 � 𝜕𝜕

�𝑘 𝜕𝜕

𝑞󰇗

𝜕𝑑 � 𝜕𝜕

1 𝜕𝑑

𝜕𝑑

𝛼 𝜕𝑜

where 𝛼 = 𝜌𝑐 is the thermal diffusivity

1

𝜕

𝑘

𝑝

�𝑘 𝜕𝜕 � + 𝑞󰇗 = 𝜌𝑐𝑝 𝜕𝑜 𝜕𝜕

𝜕

+

𝑟 ln� 2 � 𝑟1

𝜕𝑑

𝑠 2 sin 𝜃

Thermal Circuits

Cylinder: 𝑅𝑜,𝑐𝑐𝑐𝑑 =

+ 𝑞󰇗 = 𝜌𝑐𝑝 𝜕𝑜

+𝑘 =

+

𝑑𝑜

2𝜋𝑘𝐿

𝜕𝑑

�𝑘 sin 𝜃 𝜕𝜃

� + 𝑞󰇗 = 𝜌𝑐𝑝 𝜕𝑜 𝜕𝜃

𝜕𝑑

𝜕𝑑

Sphere: 𝑅𝑜,𝑐𝑐𝑐𝑑 =

1 r1

1 r2

( − ) 4𝜋𝑘

2 ℎ1 𝐴 𝑅𝑜,𝑐𝑐𝑐𝑐 = 𝑅𝑜,𝑠𝑎𝑑 = 𝑟 _____________________________________________________________________________________________________________ ℎ𝐴 General Lumped Capacitance Analysis 1

4 )]𝐴 𝑞𝑠′′𝐴𝑠,ℎ + 𝐸𝑔󰇗 − [ℎ(𝑇 − 𝑇∞ ) + 𝜀𝜎(𝑇 4 − 𝑇𝑠𝑠𝑠 𝑠(𝑐,𝑠) = 𝜌𝜌𝑐

Radiation Only Equation

𝑑=

𝜌𝜌𝑐

4 𝜀 𝐴𝑠,𝑟 𝜎

3 𝑑𝑠𝑠𝑟

𝑑𝑇 𝑑𝑑

�ln �𝑑𝑠𝑠𝑟−𝑑� − ln �𝑑𝑠𝑠𝑟−𝑑 𝑖� + 2 �tan−1 � 𝑑 � − tan−1 � 𝑑 𝑖 ��� 𝑑

𝑠𝑠𝑟

𝑑

+𝑑

𝑠𝑠𝑟

𝑑

+𝑑

𝑑

𝑠𝑠𝑟

𝑖

𝑠𝑠𝑟

Heat Flux, Energy Generation, Convection, and No Radiation Equation 𝑏

𝑑−𝑑∞ − � 𝑎� 𝑏 𝑎

𝑑𝑖 − 𝑑∞− � �

ℎ𝐴

𝑠

𝜌𝜌𝑐

Convection Only Equation

𝜃 ℎ𝐴𝑠 𝑇 − 𝑇∞ = exp �− � = � 𝑑� 𝜃𝑖 𝑇𝑖 − 𝑇∞ 𝜌𝜌𝑐

𝜏𝑜 = � ℎ𝐴 � (𝜌𝜌𝑐) = 𝑅𝑜 𝐶𝑜 1

𝑞𝑠′′𝐴𝑠,ℎ + 𝐸󰇗 𝑔

𝑠,𝑐 � and 𝑏 = = exp(−𝑎𝑑) ; where 𝑎 = � 𝜌𝜌𝑐

;

𝑄 = 𝜌𝜌𝑐 𝜃𝑖 �1 − exp �− 𝜏 �� 𝑜

𝐵𝐵 =

𝑡

ℎ𝐿𝑐

;

𝑄𝑚𝑎𝑥 = 𝜌𝜌𝑐 𝜃𝑖

𝑘

If there is an additional resistance either in series or in parallel, then replace ℎ with 𝑈 in all the above lumped capacitance equations, where 𝑈=

1

𝑅𝑡 𝐴𝑠

�𝑚2 ∙𝐾�

𝑅𝑅 =

; 𝑈 = overall heat transfer coefficient, 𝑅𝑜 = total resistance, 𝐴𝑠 = surface area.

𝑊

𝜌𝜌𝐿𝑐 𝜇

=

𝜌𝐿𝑐 𝜈

Convection Heat Transfer

[Reynolds Number]

;

���� = ℎ 𝐿𝑐 𝑁𝑁 𝑘 �

𝑓

[Average Nusselt Number]

where 𝜌 is the density, 𝜌 is the velocity, 𝐿𝑐 is the characteristic length, 𝜇 is the dynamic viscosity, 𝜈 is the kinematic viscosity, 𝑚󰇗 is the mass flow rate, ℎ� is the average convection coefficient, and 𝑘𝑓 is the fluid thermal conductivity.

3

Internal Flow

𝑅𝑅 =

4 𝑚󰇗 𝜋𝜋𝜇

[For Internal Flow in a Pipe of Diameter D]

For Constant Heat Flux [𝑞𝑠ʺ = 𝑐𝑐𝑐𝑐𝑑𝑎𝑐𝑑]:

𝑞𝑐𝑐𝑐𝑐 = 𝑞𝑠ʺ (𝑃 ∙ 𝐿) ; where P = Perimeter, L = Length

𝑞𝑠ʺ · 𝑃 𝑇𝑚 (𝑥) = 𝑇𝑚,𝑖 + 𝑚󰇗 ∙ 𝑐 𝑥 𝑝 For Constant Surface Temperature [𝑇𝑠 = 𝑐𝑐𝑐𝑐𝑑𝑎𝑐𝑑]: If there is only convection between the surface temperature, 𝑇𝑠 , and the mean fluid temperature, 𝑇𝑚, use 𝑑𝑠 −𝑑𝑚 (𝑥) 𝑑𝑠 −𝑑𝑚,𝑖

𝑃∙𝑥 = 𝑅𝑥𝑒 �− 𝑚󰇗 ∙𝑐 ℎ� � 𝑝

If there are multiple resistances between the outermost temperature, 𝑇∞, and the mean fluid temperature, 𝑇𝑚, use 𝑇∞ − 𝑇𝑚 (𝑥) 1 𝑃∙𝑥 𝑈� = 𝑅𝑥𝑒 �− � = 𝑅𝑥𝑒 �− 𝑚󰇗 ∙ 𝑐𝑝 𝑚󰇗 ∙ 𝑐𝑝 ∙ 𝑅𝑜 𝑇∞ − 𝑇𝑚,𝑖

Total heat transfer rate over the entire tube length: 𝑞𝑜 = 𝑚󰇗 ∙ 𝑐𝑝 ∙ �𝑇𝑚,𝑐 − 𝑇𝑚,𝑖 � = ℎ� ∙ 𝐴𝑠 ∙ ∆𝑇𝑙𝑚 𝑐𝑘 𝑈 ∙ 𝐴𝑠 ∙ ∆𝑇𝑙𝑚

Log mean temperature difference:

∆𝑇𝑙𝑚 = 𝐺𝑘𝐿 =

𝑅𝑎𝐿 =

Vertical Plates:

����𝐿 = �0.825 + 𝑁𝑁

; 𝑇𝑠 = 𝑐𝑐𝑐𝑐𝑑𝑎𝑐𝑑

; ∆𝑇𝑐 = 𝑇𝑠 − 𝑇𝑚,𝑐 ; ∆𝑇𝑖 = 𝑇𝑠 − 𝑇𝑚,𝑖

∆𝑑𝑜 −∆𝑑𝑖

∆𝑇 ln� ∆𝑇𝑜 � 𝑖

Free Convection Heat Transfer 𝑔𝑔(𝑑𝑠 −𝑑∞)𝐿𝑐3 𝜈2

[Grashof Number]

𝜈𝛼

[Rayleigh Number]

𝑔𝑔(𝑑𝑠 −𝑑∞)𝐿𝑐3

0.387 𝑅𝑎𝐿

1/6

8/27 0.492 9/16

�1+� 𝑃𝑟 �

�

2

� ; [Entire range of RaL; properties evaluated at Tf]

���� 𝐿 = 0.68 + - For better accuracy for Laminar Flow: 𝑁𝑁

1/4

0.670 𝑅𝑎𝐿

0.492 9/16 � �1+� 𝑃𝑟 �

4/9

; 𝑅𝑎𝐿 ≲ 109 [Properties evaluated at Tf]

Inclined Plates: for the top and bottom surfaces of cooled and heated inclined plates, respectively, the equations of the vertical plate can be used by replacing (g) with (𝑔 cos 𝜃) in RaL for 0 ≤ 𝜃 ≤ 60°.

Horizontal Plates: use the following correlations with 𝐿 =

𝐴𝑠

𝑃

where As = Surface Area and P = Perimeter

- Upper surface of Hot Plate or Lower Surface of Cold Plate: ����𝐿 = 0.54 𝑅𝑎𝐿1/4 (104 ≤ 𝑅𝑎𝐿 ≤ 107 ) ; 𝑁𝑁 ����𝐿 = 0.15 𝑅𝑎𝐿1/3 (107 ≤ 𝑅𝑎𝐿 ≤ 1011 ) 𝑁𝑁 - Lower Surface of Hot Plate or Upper Surface of Cold Plate: 1/4 ���� (105 ≤ 𝑅𝑎𝐿 ≤ 1010 ) 𝑁𝑁𝐿 = 0.27 𝑅𝑎𝐿

4

Vertical Cylinders: the equations for the Vertical Plate can be applied to vertical cylinders of height L if the following criterion is met:

𝜋 𝐿

≥

1/4 35 𝐺𝑠 𝐿

���� 𝜋 = �0.60 + Long Horizontal Cylinders: 𝑁𝑁

Spheres:

���� 𝜋 = 2 + 𝑁𝑁

1/4 0.589 𝑅𝑎𝐷

4/9 0.469 9/16

�1+� 𝑃𝑟 �

�

1/6 0.387 𝑅𝑎𝐷

�1+�

0.559

� 𝑃𝑟

; 𝑅𝑎𝜋 ≲

9/16 8/27

� 11 10

�

; 𝑅𝑎𝜋 ≲ 1012 [Properties evaluated at Tf]

2

; 𝑃𝑘 ≥ 0.7

[Properties evaluated at Tf]

Heat Exchangers

Heat Gain/Loss Equations:

𝑞 = 𝑚󰇗 𝑐𝑝 (𝑇𝑐 − 𝑇𝑖 ) = 𝑈𝐴𝑠 ∆𝑇𝑙𝑚 ; where 𝑈 is the overall heat transfer coefficient

Log-Mean Temperature Difference: ∆𝑇𝑙𝑚,𝑃𝑃 =

�𝑑ℎ,𝑖−𝑑𝑐,𝑖�−�𝑑ℎ,𝑜−𝑑𝑐,𝑜 �

Log-Mean Temperature Difference: ∆𝑇𝑙𝑚,𝐶𝑃 =

�𝑑ℎ,𝑖−𝑑𝑐,𝑜�−�𝑑ℎ,𝑜−𝑑𝑐,𝑖�

�𝑇ℎ,𝑖 −𝑇𝑐,𝑖� � �𝑇ℎ,𝑜 −𝑇𝑐,𝑜�

ln�

�𝑇ℎ,𝑖 −𝑇𝑐,𝑜 �

ln�

For Cross-Flow and Shell-and-Tube Heat Exchangers: Number of Transfer Units (NTU):

𝑁𝑇𝑈 =

𝑈𝐴

𝐶𝑚𝑖𝑚

�𝑇ℎ,𝑜 −𝑇𝑐,𝑖 �

�

[Parallel-Flow Heat Exchanger]

[Counter-Flow Heat Exchanger]

∆𝑇𝑙𝑚 = 𝐹 ∆𝑇𝑙𝑚,𝐶𝑃 ; where 𝐹 is a correction factor

; where 𝐶𝑚𝑖𝑐 is the minimum heat capacity rate in [W/K]

Heat Capacity Rates: 𝐶𝑐 = 𝑚󰇗𝑐 𝑐𝑝,𝑐 [Cold Fluid] ; 𝐶ℎ = 𝑚󰇗ℎ 𝑐𝑝,ℎ [Hot Fluid] ; 𝐶𝑠 =

𝐶𝑚𝑖𝑚

Note: The condensation or evaporation side of the heat exchanger is associated with 𝐶𝑚𝑎𝑥 = ∞

𝐶𝑚𝑎𝑚

[Heat Capacity Ratio]

5

If Pr ≤ 10 → n = 0.37 If Pr ≥ 10 → n = 0.36

6

7

8

9

10...