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Heat Transfer Equation Sheet
Conduction: qx = - kA
𝑑𝑑𝑑𝑑
𝑑𝑑𝑑𝑑
Convection: q = hA(Ts - T∞)
Radiation:
©2015
J. M. Hill
Heat Exchangers:
ΔTLM = {(THo – TCo) - (THi – TCi)}/{ln[(THo – TCo)/ (THi – TCi)]} Parallel flow
q = UA ΔTLM
ΔTLM = {(THo – TCi) - (THi – TCo)}/{ln[(THo – TCi)/ (THi – TCo)]} Counter flow
qr = εAσT4
NTU = UA/Cmin
ε = q/qmax qmax = Cmin (THi – TCi)
Cr = Cmin/Cmax
Diffusion equations:
πœ•πœ• 2 𝑇𝑇
πœ•πœ•π‘₯π‘₯ 2
1 πœ•πœ•
πœ•πœ• 2 𝑇𝑇
+ πœ•πœ•π‘¦π‘¦ 2 +
πœ•πœ•πœ•πœ•
πœ•πœ• 2 𝑇𝑇
πœ•πœ•π‘§π‘§ 2
1 πœ•πœ•
π‘žπ‘žΜ‡
1 πœ•πœ•πœ•πœ•
+ π‘˜π‘˜ = 𝛼𝛼 πœ•πœ•πœ•πœ•
πœ•πœ•πœ•πœ•
(π‘Ÿπ‘Ÿ πœ•πœ•πœ•πœ• ) + π‘Ÿπ‘Ÿ 2 πœ•πœ•πœ•πœ• (πœ•πœ•πœ•πœ•) +
π‘Ÿπ‘Ÿ πœ•πœ•πœ•πœ•
πœ•πœ• 2 𝑇𝑇
πœ•πœ•π‘§π‘§ 2
uniform properties, α =
π‘žπ‘žΜ‡
1 πœ•πœ•πœ•πœ•
π‘˜π‘˜
πœŒπœŒπΆπΆπ‘π‘
+ π‘˜π‘˜ = 𝛼𝛼 πœ•πœ•πœ•πœ• Cylindrical coordinates
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π‘žπ‘ž =
Ohms Law Analogy,
Plane wall: 𝑅𝑅𝑑𝑑,𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 =
Cylinder: 𝑅𝑅𝑑𝑑,𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 =
𝐿𝐿
π‘˜π‘˜π‘˜π‘˜
π›₯π›₯π›₯π›₯
, 𝑅𝑅𝑑𝑑,𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 =
π‘Ÿπ‘Ÿ
ln( 2οΏ½π‘Ÿπ‘Ÿ1 )
2πœ‹πœ‹πœ‹πœ‹πœ‹πœ‹
Ts = T∞ + π‘žπ‘žΜ‡ L/h
Generation: Plane wall,
𝑅𝑅𝑑𝑑
1
β„Žπ΄π΄
Cylinder, T(r) =
1
, 𝑅𝑅𝑑𝑑,𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 = 2πœ‹πœ‹πœ‹πœ‹πœ‹πœ‹β„Ž
𝑇𝑇− 𝑇𝑇𝑠𝑠
π‘‡π‘‡π‘œπ‘œ −𝑇𝑇𝑠𝑠
π‘žπ‘žΜ‡ π‘Ÿπ‘Ÿπ‘œπ‘œ2
4π‘˜π‘˜
(1 −
π‘Ÿπ‘Ÿ
π‘Ÿπ‘Ÿ 2
π‘Ÿπ‘Ÿπ‘œπ‘œ2
To = Ts + π‘žπ‘žΜ‡ L2/(2k)
) + Ts
T s = T∞ +
= 1 – ( )2
𝑇𝑇− π‘‡π‘‡π‘œπ‘œ
𝑇𝑇𝑠𝑠 −π‘‡π‘‡π‘œπ‘œ
= (x/L)2
π‘žπ‘žΜ‡ π‘Ÿπ‘Ÿπ‘œπ‘œ
2β„Ž
π‘Ÿπ‘Ÿπ‘œπ‘œ
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Fins of uniform cross section: θ = T - T∞
M = (hPkAc)½θb
πœƒπœƒ = πœƒπœƒπ‘π‘ 𝑒𝑒 −π‘šπ‘šπ‘šπ‘š
Infinite fin:
Adiabatic tip:
Active tip:
π‘žπ‘žπ‘“π‘“ = 𝑀𝑀
πœƒπœƒ = πœƒπœƒπ‘π‘
cosh[π‘šπ‘š(𝐿𝐿−π‘₯π‘₯)]
cosh(π‘šπ‘šπ‘šπ‘š)
πœƒπœƒ = πœƒπœƒπ‘π‘
(β„ŽοΏ½π‘šπ‘šπ‘šπ‘š ) sinh[π‘šπ‘š(𝐿𝐿−π‘₯π‘₯)]+cosh[π‘šπ‘š(𝐿𝐿−π‘₯π‘₯)]
(β„ŽοΏ½π‘šπ‘šπ‘šπ‘š) sinh(π‘šπ‘šπ‘šπ‘š)+cosh(π‘šπ‘šπ‘šπ‘š)
πœƒπœƒ =
Prescribed tip:
All fins:
m2 = hP/kAc
π‘žπ‘žπ‘“π‘“ = 𝑀𝑀 π‘‘π‘‘π‘‘π‘‘π‘‘π‘‘β„Ž(π‘šπ‘šπ‘šπ‘š)
πœƒπœƒ
( πΏπΏοΏ½πœƒπœƒ ) sinh(π‘šπ‘šπ‘šπ‘š)+sinh[π‘šπ‘š(𝐿𝐿−π‘₯π‘₯)]
𝑏𝑏
πœƒπœƒπ‘π‘
sinh(π‘šπ‘šπ‘šπ‘š)
Effectiveness,
π‘žπ‘žπ‘“π‘“
πœ€πœ€π‘“π‘“ = β„Žπ΄π΄
𝑏𝑏 πœƒπœƒπ‘π‘
Efficiency, πœ‚πœ‚π‘“π‘“ =
π‘žπ‘žπ‘“π‘“ =
π‘žπ‘žπ‘“π‘“ = 𝑀𝑀
π‘žπ‘žπ‘“π‘“
β„Žπ΄π΄π‘“π‘“ πœƒπœƒπ‘π‘
πœƒπœƒ
cosh(π‘šπ‘šπ‘šπ‘š)−( πΏπΏοΏ½πœƒπœƒ )
𝑏𝑏
𝑀𝑀
sinh(π‘šπ‘šπ‘šπ‘š)
sinh(π‘šπ‘šπ‘šπ‘š)+(β„ŽοΏ½π‘šπ‘šπ‘šπ‘š)cosh(π‘šπ‘šπ‘šπ‘š)
cosh(π‘šπ‘šπ‘šπ‘š)+(β„ŽοΏ½π‘šπ‘šπ‘šπ‘š)sinh(π‘šπ‘šπ‘šπ‘š)
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Lumped Capacitance: Bi = hLc/k < 0.1
Fo = αt/(Lc)2
𝑇𝑇−𝑇𝑇∞
𝑇𝑇𝑖𝑖 −𝑇𝑇∞
= 𝑒𝑒 −𝑑𝑑/πœπœπ‘‘π‘‘
𝜏𝜏t =
πœŒπœŒπœŒπœŒπΆπΆπ‘π‘
β„Žπ΄π΄π‘ π‘ 
Q = πœŒπœŒπœŒπœŒπΆπΆπ‘π‘ πœƒπœƒπ‘–π‘– (1 - 𝑒𝑒 −𝑑𝑑/πœπœπ‘‘π‘‘ )
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Semi-infinite solid:
imposed temperature:
𝑇𝑇−𝑇𝑇𝑠𝑠
= erf(
π‘₯π‘₯
)
𝑇𝑇𝑖𝑖 −𝑇𝑇𝑠𝑠
2√𝛼𝛼𝛼𝛼
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Finite difference methods:
𝑝𝑝+1
Explicit, 1-D unsteady: π‘‡π‘‡π‘šπ‘š
𝑝𝑝
𝑝𝑝
𝑝𝑝
Fo = αΔt/(Δx)2 ≤ ½ for stability
= Fo(π‘‡π‘‡π‘šπ‘š+1 +π‘‡π‘‡π‘šπ‘š−1) + (1 – 2Fo) π‘‡π‘‡π‘šπ‘š
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Convection, Steady flow over a flat plate: Pr = Cpμ/k = μ/(ρα)
Laminar: δ = 5x Rex-1/2
Cf,x = 0.664 Rex-1/2
Mixed laminar and turbulent for Rex,c < Rex < 108:
Mixed laminar and turbulent with Rex,c = 5x105:
Rex = ρu∞x/μ
-1/2
οΏ½οΏ½οΏ½οΏ½οΏ½
𝐢𝐢
𝑓𝑓,𝐿𝐿 = 1.328 ReL
δ = 0.37x Rex-1/5
Nux = hx/kf
Nux = 0.332 Rex1/2 Pr1/3
Cf,x = 0.0592 Rex-1/5
οΏ½οΏ½οΏ½οΏ½οΏ½
𝐢𝐢𝑓𝑓,𝐿𝐿 = 0.074 ReL-1/5 -1742/ ReL
Note: If the laminar region is small, the constants 1742 and 871 in the above equations become zero.
οΏ½οΏ½οΏ½οΏ½οΏ½
𝑁𝑁𝑁𝑁𝐿𝐿 = 0.664 ReL1/2 Pr1/3
Nux = 0.0296 Rex4/5 Pr1/3
οΏ½οΏ½οΏ½οΏ½ = (0.037Re L4/5 – 871)Pr1/3
𝑁𝑁𝑁𝑁
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