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

Heat Flux : 𝑞𝑥′′ = −𝑘 𝑑𝑑

𝑑𝑥 𝑊

𝑚2 k : Thermal Conductivity 𝑊

𝑚∙𝑘

Heat Rate : 𝑞𝑥 = 𝑞𝑥′′𝐴𝑐 𝑊 Ac : Cross-Sectional Area

Heat Convection Rate Equations (Newton's Law of Cooling)

Heat Flux: 𝑞′′ = ℎ(𝑇𝑠 − 𝑇∞) 𝑊𝑚2 h : Convection Heat Transfer Coefficient

𝑊𝑚2∙𝐾

Heat Rate: 𝑞 = ℎ𝐴𝑠(𝑇𝑠 − 𝑇∞) 𝑊 As : Surface Area 𝑚2

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

𝑚2

Heat Flux emitted : 𝐸 = 𝜀𝜎𝑇𝑠4 𝑊

𝑚2 where ε is the emissivity with range of 0 ≤ 𝜀 ≤ 1

and 𝜎 = 5.67 × 10−8 𝑊𝑚2𝐾4 is the Stefan-Boltzmann constant

Irradiation: 𝐺𝑎𝑏𝑠 = 𝛼𝐺 but we assume small body in a large enclosure with 𝜀 = 𝛼 so that 𝐺 = 𝜀 𝜎 𝑇𝑠𝑠𝑠4

Net Radiation heat flux from surface: 𝑞𝑠𝑎𝑑′′ = 𝑞

𝐴= 𝜀𝐸𝑏(𝑇𝑠) − 𝛼𝐺 = 𝜀𝜎(𝑇𝑠

4 − 𝑇𝑠𝑠𝑠4 )

Net radiation heat exchange rate: 𝑞𝑠𝑎𝑑 = 𝜀𝜎𝐴𝑠(𝑇𝑠4 − 𝑇𝑠𝑠𝑠

4 ) where for a real surface 0 ≤ 𝜀 ≤ 1

This can ALSO be expressed as: 𝑞𝑠𝑎𝑑 = ℎ𝑠𝐴(𝑇𝑠 − 𝑇𝑠𝑠𝑠) depending on the application

where ℎ𝑠 is the radiation heat transfer coefficient which is: ℎ𝑠 = 𝜀𝜎(𝑇𝑠 + 𝑇𝑠𝑠𝑠)(𝑇𝑠2 + 𝑇𝑠𝑠𝑠

2 ) 𝑊𝑚2∙𝐾

TOTAL heat transfer from a surface: 𝑞 = 𝑞𝑐𝑐𝑐𝑐 + 𝑞𝑠𝑎𝑑 = ℎ𝐴𝑠(𝑇𝑠 − 𝑇∞) + 𝜀𝜎𝐴𝑠(𝑇𝑠4 − 𝑇𝑠𝑠𝑠

4 ) 𝑊

Conservation of Energy (Energy Balance)

�̇�𝑖𝑐 + �̇�𝑔 − �̇�𝑐𝑠𝑜 = �̇�𝑠𝑜 (Control Volume Balance) ; �̇�𝑖𝑐 − �̇�𝑐𝑠𝑜 = 0 (Control Surface Balance)

where �̇�𝑔 is the conversion of internal energy (chemical, nuclear, electrical) to thermal or mechanical energy, and

�̇�𝑠𝑜 = 0 for steady-state conditions. If not steady-state (i.e., transient) then �̇�𝑠𝑜 = 𝜌𝜌𝑐𝑝𝑑𝑑𝑑𝑜

Heat Equation (used to find the temperature distribution)

Heat Equation (Cartesian): 𝜕

𝜕𝑥�𝑘 𝜕𝑑

𝜕𝑥� + 𝜕

𝜕𝜕�𝑘 𝜕𝑑

𝜕𝜕� + 𝜕

𝜕𝜕�𝑘 𝜕𝑑

𝜕𝜕� + �̇� = 𝜌𝑐𝑝

𝜕𝑑𝜕𝑜

If 𝑘 is constant then the above simplifies to: 𝜕2𝑑𝜕𝑥2 + 𝜕2𝑑

𝜕𝜕2 + 𝜕2𝑑𝜕𝜕2 + �̇�

𝑘= 1

𝛼𝜕𝑑𝜕𝑜

where 𝛼 = 𝑘𝜌𝑐𝑝

is the thermal diffusivity

Heat Equation (Cylindrical): 1𝑠

𝜕𝜕𝑠

�𝑘𝑘 𝜕𝑑𝜕𝑠

� + 1𝑠2

𝜕𝜕𝜕

�𝑘 𝜕𝑑𝜕𝜕

� + 𝜕𝜕𝜕

�𝑘 𝜕𝑑𝜕𝜕

� + �̇� = 𝜌𝑐𝑝𝜕𝑑𝜕𝑜

Heat Eqn. (Spherical): 1

𝑠2𝜕

𝜕𝑠�𝑘𝑘2 𝜕𝑑

𝜕𝑠� + 1

𝑠2 sin 𝜃2𝜕

𝜕𝜕�𝑘 𝜕𝑑

𝜕𝜕� + 1

𝑠2 sin 𝜃 𝜕𝜕𝜃

�𝑘 sin 𝜃 𝜕𝑑𝜕𝜃

� + �̇� = 𝜌𝑐𝑝𝜕𝑑𝜕𝑜

Thermal Circuits

Plane Wall: 𝑅𝑜,𝑐𝑐𝑐𝑑 = 𝐿𝑘𝐴

Cylinder: 𝑅𝑜,𝑐𝑐𝑐𝑑 =ln�𝑟2

𝑟1�

2𝜋𝑘𝐿 Sphere: 𝑅𝑜,𝑐𝑐𝑐𝑑 =

( 1r1

− 1r2

)

4𝜋𝑘

2 𝑅𝑜,𝑐𝑐𝑐𝑐 = 1

ℎ𝐴 𝑅𝑜,𝑠𝑎𝑑 = 1

ℎ𝑟𝐴

_____________________________________________________________________________________________________________

General Lumped Capacitance Analysis

𝑞𝑠′′𝐴𝑠,ℎ + 𝐸�̇� − [ℎ(𝑇 − 𝑇∞) + 𝜀𝜎(𝑇4 − 𝑇𝑠𝑠𝑠

4 )]𝐴𝑠(𝑐,𝑠) = 𝜌𝜌𝑐𝑑𝑇𝑑𝑑

Radiation Only Equation

𝑑 = 𝜌𝜌𝑐4 𝜀 𝐴𝑠,𝑟 𝜎 𝑑𝑠𝑠𝑟

3 �ln �𝑑𝑠𝑠𝑟+𝑑𝑑𝑠𝑠𝑟−𝑑

� − ln �𝑑𝑠𝑠𝑟+𝑑𝑖𝑑𝑠𝑠𝑟−𝑑𝑖

� + 2 �tan−1 � 𝑑𝑑𝑠𝑠𝑟

� − tan−1 � 𝑑𝑖𝑑𝑠𝑠𝑟

���

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

𝑑−𝑑∞− �𝑏𝑎�

𝑑𝑖− 𝑑∞− �𝑏𝑎�

= exp(−𝑎𝑑) ; where 𝑎 = �ℎ𝐴𝑠,𝑐

𝜌𝜌𝑐� and 𝑏 = 𝑞𝑠

′′𝐴𝑠,ℎ+ �̇�𝑔

𝜌𝜌𝑐

Convection Only Equation

𝜃𝜃𝑖

=𝑇 − 𝑇∞

𝑇𝑖 − 𝑇∞= exp �− �

ℎ𝐴𝑠

𝜌𝜌𝑐� 𝑑�

𝜏𝑜 = � 1ℎ𝐴𝑠

� (𝜌𝜌𝑐) = 𝑅𝑜𝐶𝑜 ; 𝑄 = 𝜌𝜌𝑐 𝜃𝑖 �1 − exp �− 𝑜𝜏𝑡

�� ; 𝑄𝑚𝑎𝑥 = 𝜌𝜌𝑐 𝜃𝑖

𝐵𝐵 = ℎ𝐿𝑐𝑘

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 = 1𝑅𝑡𝐴𝑠

� 𝑊𝑚2∙𝐾

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

Convection Heat Transfer

𝑅𝑅 = 𝜌𝜌𝐿𝑐𝜇

= 𝜌𝐿𝑐𝜈

[Reynolds Number] ; 𝑁𝑁���� = ℎ�𝐿𝑐𝑘𝑓

[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, �̇� is the mass flow

rate, ℎ� is the average convection coefficient, and 𝑘𝑓 is the fluid thermal conductivity.

3 Internal Flow

𝑅𝑅 = 4 �̇�𝜋𝜋𝜇

[For Internal Flow in a Pipe of Diameter D]

For Constant Heat Flux [𝑞𝑠ʺ = 𝑐𝑐𝑐𝑐𝑑𝑎𝑐𝑑]: 𝑞𝑐𝑐𝑐𝑐 = 𝑞𝑠

ʺ(𝑃 ∙ 𝐿) ; where P = Perimeter, L = Length

𝑇𝑚(𝑥) = 𝑇𝑚,𝑖 +𝑞𝑠

ʺ · 𝑃�̇� ∙ 𝑐𝑝

𝑥

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

= 𝑅𝑥𝑒 �− 𝑃∙𝑥�̇�∙𝑐𝑝

ℎ��

If there are multiple resistances between the outermost temperature, 𝑇∞, and the mean fluid temperature, 𝑇𝑚, use

𝑇∞ − 𝑇𝑚(𝑥)𝑇∞ − 𝑇𝑚,𝑖

= 𝑅𝑥𝑒 �−𝑃 ∙ 𝑥

�̇� ∙ 𝑐𝑝𝑈� = 𝑅𝑥𝑒 �−

1�̇� ∙ 𝑐𝑝 ∙ 𝑅𝑜

�

Total heat transfer rate over the entire tube length:

𝑞𝑜 = �̇� ∙ 𝑐𝑝 ∙ �𝑇𝑚,𝑐 − 𝑇𝑚,𝑖� = ℎ� ∙ 𝐴𝑠 ∙ ∆𝑇𝑙𝑚 𝑐𝑘 𝑈 ∙ 𝐴𝑠 ∙ ∆𝑇𝑙𝑚 ; 𝑇𝑠 = 𝑐𝑐𝑐𝑐𝑑𝑎𝑐𝑑

Log mean temperature difference: ∆𝑇𝑙𝑚 = ∆𝑑𝑜−∆𝑑𝑖

ln�∆𝑇𝑜∆𝑇𝑖

� ; ∆𝑇𝑐 = 𝑇𝑠 − 𝑇𝑚,𝑐 ; ∆𝑇𝑖 = 𝑇𝑠 − 𝑇𝑚,𝑖

Free Convection Heat Transfer

𝐺𝑘𝐿 = 𝑔𝑔(𝑑𝑠−𝑑∞)𝐿𝑐3

𝜈2 [Grashof Number]

𝑅𝑎𝐿 = 𝑔𝑔(𝑑𝑠−𝑑∞)𝐿𝑐3

𝜈𝛼 [Rayleigh Number]

Vertical Plates: 𝑁𝑁����𝐿 = �0.825 + 0.387 𝑅𝑎𝐿1/6

�1+�0.492𝑃𝑟 �

9/16�

8/27�

2

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

- For better accuracy for Laminar Flow: 𝑁𝑁����𝐿 = 0.68 + 0.670 𝑅𝑎𝐿1/4

�1+�0.492𝑃𝑟 �

9/16�

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 = 𝐴𝑠𝑃

where As = Surface Area and P = Perimeter

- Upper surface of Hot Plate or Lower Surface of Cold Plate:

𝑁𝑁����𝐿 = 0.54 𝑅𝑎𝐿1/4 (104 ≤ 𝑅𝑎𝐿 ≤ 107) ; 𝑁𝑁����𝐿 = 0.15 𝑅𝑎𝐿

1/3 (107 ≤ 𝑅𝑎𝐿 ≤ 1011) - Lower Surface of Hot Plate or Upper Surface of Cold Plate:

𝑁𝑁����𝐿 = 0.27 𝑅𝑎𝐿1/4 (105 ≤ 𝑅𝑎𝐿 ≤ 1010)

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

met: 𝜋𝐿

≥ 35

𝐺𝑠𝐿1/4

Long Horizontal Cylinders: 𝑁𝑁����𝜋 = �0.60 + 0.387 𝑅𝑎𝐷1/6

�1+�0.559𝑃𝑟 �

9/16�

8/27�

2

; 𝑅𝑎𝜋 ≲ 1012 [Properties evaluated at Tf]

Spheres: 𝑁𝑁����𝜋 = 2 + 0.589 𝑅𝑎𝐷1/4

�1+�0.469𝑃𝑟 �

9/16�

4/9 ; 𝑅𝑎𝜋 ≲ 1011 ; 𝑃𝑘 ≥ 0.7 [Properties evaluated at Tf]

Heat Exchangers

Heat Gain/Loss Equations: 𝑞 = �̇� 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�

ln��𝑇ℎ,𝑖−𝑇𝑐,𝑖�

�𝑇ℎ,𝑜−𝑇𝑐,𝑜��

[Parallel-Flow Heat Exchanger]

Log-Mean Temperature Difference: ∆𝑇𝑙𝑚,𝐶𝑃 = �𝑑ℎ,𝑖−𝑑𝑐,𝑜�−�𝑑ℎ,𝑜−𝑑𝑐,𝑖�

ln��𝑇ℎ,𝑖−𝑇𝑐,𝑜�

�𝑇ℎ,𝑜−𝑇𝑐,𝑖��

[Counter-Flow Heat Exchanger]

For Cross-Flow and Shell-and-Tube Heat Exchangers: ∆𝑇𝑙𝑚 = 𝐹 ∆𝑇𝑙𝑚,𝐶𝑃 ; where 𝐹 is a correction factor

Number of Transfer Units (NTU): 𝑁𝑇𝑈 = 𝑈𝐴𝐶𝑚𝑖𝑚

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

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

[Heat Capacity Ratio]

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

5

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

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7

8

9

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