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:
𝜕𝜕 2 𝑇𝑇
𝜕𝜕𝑥𝑥 2
1 𝜕𝜕
𝜕𝜕 2 𝑇𝑇
+ 𝜕𝜕𝑦𝑦 2 +
𝜕𝜕𝜕𝜕
𝜕𝜕 2 𝑇𝑇
𝜕𝜕𝑧𝑧 2
1 𝜕𝜕
𝑞𝑞̇
1 𝜕𝜕𝜕𝜕
+ 𝑘𝑘 = 𝛼𝛼 𝜕𝜕𝜕𝜕
𝜕𝜕𝜕𝜕
(𝑟𝑟 𝜕𝜕𝜕𝜕 ) + 𝑟𝑟 2 𝜕𝜕𝜕𝜕 (𝜕𝜕𝜕𝜕) +
𝑟𝑟 𝜕𝜕𝜕𝜕
𝜕𝜕 2 𝑇𝑇
𝜕𝜕𝑧𝑧 2
uniform properties, α =
𝑞𝑞̇
1 𝜕𝜕𝜕𝜕
𝑘𝑘
𝜌𝜌𝐶𝐶𝑝𝑝
+ 𝑘𝑘 = 𝛼𝛼 𝜕𝜕𝜕𝜕 Cylindrical coordinates
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
𝑞𝑞 =
Ohms Law Analogy,
Plane wall: 𝑅𝑅𝑡𝑡,𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 =
Cylinder: 𝑅𝑅𝑡𝑡,𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 =
𝐿𝐿
𝑘𝑘𝑘𝑘
𝛥𝛥𝛥𝛥
, 𝑅𝑅𝑡𝑡,𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 =
𝑟𝑟
ln( 2�𝑟𝑟1 )
2𝜋𝜋𝜋𝜋𝜋𝜋
Ts = T∞ + 𝑞𝑞̇ L/h
Generation: Plane wall,
𝑅𝑅𝑡𝑡
1
ℎ𝐴𝐴
Cylinder, T(r) =
1
, 𝑅𝑅𝑡𝑡,𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 = 2𝜋𝜋𝜋𝜋𝜋𝜋ℎ
𝑇𝑇− 𝑇𝑇𝑠𝑠
𝑇𝑇𝑜𝑜 −𝑇𝑇𝑠𝑠
𝑞𝑞̇ 𝑟𝑟𝑜𝑜2
4𝑘𝑘
(1 −
𝑟𝑟
𝑟𝑟 2
𝑟𝑟𝑜𝑜2
To = Ts + 𝑞𝑞̇ L2/(2k)
) + Ts
T s = T∞ +
= 1 – ( )2
𝑇𝑇− 𝑇𝑇𝑜𝑜
𝑇𝑇𝑠𝑠 −𝑇𝑇𝑜𝑜
= (x/L)2
𝑞𝑞̇ 𝑟𝑟𝑜𝑜
2ℎ
𝑟𝑟𝑜𝑜
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Fins of uniform cross section: θ = T - T∞
M = (hPkAc)½θb
𝜃𝜃 = 𝜃𝜃𝑏𝑏 𝑒𝑒 −𝑚𝑚𝑚𝑚
Infinite fin:
Adiabatic tip:
Active tip:
𝑞𝑞𝑓𝑓 = 𝑀𝑀
𝜃𝜃 = 𝜃𝜃𝑏𝑏
cosh[𝑚𝑚(𝐿𝐿−𝑥𝑥)]
cosh(𝑚𝑚𝑚𝑚)
𝜃𝜃 = 𝜃𝜃𝑏𝑏
(ℎ�𝑚𝑚𝑚𝑚 ) sinh[𝑚𝑚(𝐿𝐿−𝑥𝑥)]+cosh[𝑚𝑚(𝐿𝐿−𝑥𝑥)]
(ℎ�𝑚𝑚𝑚𝑚) sinh(𝑚𝑚𝑚𝑚)+cosh(𝑚𝑚𝑚𝑚)
𝜃𝜃 =
Prescribed tip:
All fins:
m2 = hP/kAc
𝑞𝑞𝑓𝑓 = 𝑀𝑀 𝑡𝑡𝑡𝑡𝑡𝑡ℎ(𝑚𝑚𝑚𝑚)
𝜃𝜃
( 𝐿𝐿�𝜃𝜃 ) sinh(𝑚𝑚𝑚𝑚)+sinh[𝑚𝑚(𝐿𝐿−𝑥𝑥)]
𝑏𝑏
𝜃𝜃𝑏𝑏
sinh(𝑚𝑚𝑚𝑚)
Effectiveness,
𝑞𝑞𝑓𝑓
𝜀𝜀𝑓𝑓 = ℎ𝐴𝐴
𝑏𝑏 𝜃𝜃𝑏𝑏
Efficiency, 𝜂𝜂𝑓𝑓 =
𝑞𝑞𝑓𝑓 =
𝑞𝑞𝑓𝑓 = 𝑀𝑀
𝑞𝑞𝑓𝑓
ℎ𝐴𝐴𝑓𝑓 𝜃𝜃𝑏𝑏
𝜃𝜃
cosh(𝑚𝑚𝑚𝑚)−( 𝐿𝐿�𝜃𝜃 )
𝑏𝑏
𝑀𝑀
sinh(𝑚𝑚𝑚𝑚)
sinh(𝑚𝑚𝑚𝑚)+(ℎ�𝑚𝑚𝑚𝑚)cosh(𝑚𝑚𝑚𝑚)
cosh(𝑚𝑚𝑚𝑚)+(ℎ�𝑚𝑚𝑚𝑚)sinh(𝑚𝑚𝑚𝑚)
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Lumped Capacitance: Bi = hLc/k < 0.1
Fo = αt/(Lc)2
𝑇𝑇−𝑇𝑇∞
𝑇𝑇𝑖𝑖 −𝑇𝑇∞
= 𝑒𝑒 −𝑡𝑡/𝜏𝜏𝑡𝑡
𝜏𝜏t =
𝜌𝜌𝜌𝜌𝐶𝐶𝑝𝑝
ℎ𝐴𝐴𝑠𝑠
Q = 𝜌𝜌𝜌𝜌𝐶𝐶𝑝𝑝 𝜃𝜃𝑖𝑖 (1 - 𝑒𝑒 −𝑡𝑡/𝜏𝜏𝑡𝑡 )
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Semi-infinite solid:
imposed temperature:
𝑇𝑇−𝑇𝑇𝑠𝑠
= erf(
𝑥𝑥
)
𝑇𝑇𝑖𝑖 −𝑇𝑇𝑠𝑠
2√𝛼𝛼𝛼𝛼
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Finite difference methods:
𝑝𝑝+1
Explicit, 1-D unsteady: 𝑇𝑇𝑚𝑚
𝑝𝑝
𝑝𝑝
𝑝𝑝
Fo = αΔt/(Δx)2 ≤ ½ for stability
= Fo(𝑇𝑇𝑚𝑚+1 +𝑇𝑇𝑚𝑚−1) + (1 – 2Fo) 𝑇𝑇𝑚𝑚
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
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
�����
𝐶𝐶
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
�����
𝐶𝐶𝑓𝑓,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.
�����
𝑁𝑁𝑁𝑁𝐿𝐿 = 0.664 ReL1/2 Pr1/3
Nux = 0.0296 Rex4/5 Pr1/3
���� = (0.037Re L4/5 – 871)Pr1/3
𝑁𝑁𝑁𝑁