1 HEAT TRANSFER EQUATION SHEET Heat Conduction Rate Equations (Fourier's Law) Heat Flux : ′′ = − 2 k : Thermal Conductivity ∙ Heat Rate : = ′′ A c : Cross-Sectional Area Heat Convection Rate Equations (Newton's Law of Cooling) Heat Flux: ′′ = ℎ(− ∞ ) 2 h : Convection Heat Transfer Coefficient 2 ∙ Heat Rate: = ℎ (− ∞ ) A s : Surface Area 2 Heat Radiation emitted ideally by a blackbody surface has a surface emissive power: = 4 2 Heat Flux emitted : = 4 2 where ε is the emissivity with range of 0 ≤ ≤ 1 and = 5.67 × 10 −8 2 4 is the Stefan-Boltzmann constant Irradiation: = but we assume small body in a large enclosure with = so that = 4 Net Radiation heat flux from surface: ′′ = = () − = (4 − 4 ) Net radiation heat exchange rate: = (4 − 4 ) where for a real surface 0 ≤≤ 1 This can ALSO be expressed as: = ℎ (− ) depending on the application where ℎ is the radiation heat transfer coefficient which is: ℎ = (+ )(2 + 2 ) 2 ∙ TOTAL heat transfer from a surface: = + = ℎ (− ∞ )+ (4 − 4 ) Conservation of Energy (Energy Balance) ̇ + ̇ − ̇ = ̇ (Control Volume Balance) ; ̇ − ̇ = 0 (Control Surface Balance) where ̇ is the conversion of internal energy (chemical, nuclear, electrical) to thermal or mechanical energy, and ̇ =0 for steady-state conditions. If not steady-state (i.e., transient) then ̇ = Heat Equation (used to find the temperature distribution) Heat Equation (Cartesian): � � + � � + � � + ̇ = If is constant then the above simplifies to: 2 2 + 2 2 + 2 2 + ̇ = 1 where = is the thermal diffusivity Heat Equation (Cylindrical): 1 � � + 1 2 � � + � � + ̇ = Heat Eqn. (Spherical): 1 2 � 2 � + 1 2 sin 2 � � + 1 2 sin � sin � + ̇ = Thermal Circuits Plane Wall: ,= Cylinder: ,= ln� 2 1 � 2 Sphere: ,= ( 1 r 1 − 1 r 2 ) 4
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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
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: