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HEAT TRANSFER EQUATION SHEET
Heat Conduction Rate Equations (Fourier's Law)
Heat Flux : 𝑞𝑥
′′=−𝑘𝑑𝑑
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∞) 𝑊
𝑚2
h
: Convection Heat Transfer Coefficient 𝑊
𝑚2∙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 𝑊
𝑚2𝐾4 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 𝑇𝑠𝑠𝑠
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Net Radiation heat flux from surface: 𝑞𝑠𝑎𝑑
′′ =𝑞
𝐴= 𝜀𝐸𝑏(𝑇𝑠)− 𝛼𝐺= 𝜀𝜎(𝑇𝑠4− 𝑇𝑠𝑠𝑠
4)
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) 𝑊
𝑚2∙𝐾
TOTAL heat transfer from a surface: 𝑞=𝑞𝑐𝑐𝑐𝑐+𝑞𝑠𝑎𝑑 =ℎ𝐴𝑠(𝑇𝑠−𝑇∞)+𝜀𝜎𝐴𝑠(𝑇𝑠4− 𝑇𝑠𝑠𝑠
4) 𝑊
Conservation of Energy (Energy Balance)
𝐸𝑖𝑐+𝐸𝑔− 𝐸𝑐𝑠𝑜 = 𝐸𝑠𝑜 (Control Volume Balance) ; 𝐸𝑖𝑐 − 𝐸𝑐𝑠𝑜 = 0 (Control Surface Balance)
where 𝐸𝑔 is the conversion of internal energy (chemical, nuclear, electrical) to thermal or mechanical energy, and
𝐸𝑠𝑜 = 0 for steady-state conditions. If not steady-state (i.e., transient) then 𝐸𝑠𝑜 =𝜌𝜌𝑐𝑝𝑑𝑑
𝑑𝑜
Heat Equation (used to find the temperature distribution)
Heat Equation (Cartesian): 𝜕
𝜕𝑥�𝑘𝜕𝑑
𝜕𝑥�+ 𝜕
𝜕𝜕�𝑘𝜕𝑑
𝜕𝜕� + 𝜕
𝜕𝜕�𝑘𝜕𝑑
𝜕𝜕�+ 𝑞=𝜌𝑐𝑝𝜕𝑑
𝜕𝑜
If 𝑘 is constant then the above simplifies to: 𝜕2𝑑
𝜕𝑥2+ 𝜕2𝑑
𝜕𝜕2+ 𝜕2𝑑
𝜕𝜕2+𝑞
𝑘=1
𝛼𝜕𝑑
𝜕𝑜 where 𝛼=𝑘
𝜌𝑐𝑝 is the thermal diffusivity
Heat Equation (Cylindrical): 1
𝑠𝜕
𝜕𝑠�𝑘𝑘𝜕𝑑
𝜕𝑠�+1
𝑠2𝜕
𝜕𝜕�𝑘𝜕𝑑
𝜕𝜕� + 𝜕
𝜕𝜕�𝑘𝜕𝑑
𝜕𝜕�+ 𝑞=𝜌𝑐𝑝𝜕𝑑
𝜕𝑜
Heat Eqn. (Spherical): 1
𝑠2𝜕
𝜕𝑠�𝑘𝑘2𝜕𝑑
𝜕𝑠�+1
𝑠2sin𝜃2𝜕
𝜕𝜕�𝑘𝜕𝑑
𝜕𝜕� + 1
𝑠2sin𝜃 𝜕
𝜕𝜃�𝑘sin 𝜃𝜕𝑑
𝜕𝜃�+ 𝑞=𝜌𝑐𝑝𝜕𝑑
𝜕𝑜
Thermal Circuits
Plane Wall: 𝑅𝑜,𝑐𝑐𝑐𝑑 =𝐿
𝑘𝐴 Cylinder: 𝑅𝑜,𝑐𝑐𝑐𝑑 =ln�𝑟2
𝑟1�
2𝜋𝑘𝐿 Sphere: 𝑅𝑜,𝑐𝑐𝑐𝑑 =
(1
r1−1
r2)
4𝜋𝑘
