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HEAT TRANSFER EQUATION SHEET
Heat Conduction Rate Equations (Fourier's Law)
Heat Flux: 𝑞𝑞
𝑥𝑥
′′
𝑑𝑑𝑑𝑑
𝑑𝑑𝑥𝑥
𝑊𝑊
𝑚𝑚
2
k : Thermal Conductivity
𝑊𝑊
𝑚𝑚∙𝑘𝑘
Heat Rate: 𝑞𝑞
𝑥𝑥
𝑥𝑥
′′
𝑐𝑐
𝑊𝑊 A
c
: Cross-Sectional Area
Heat Convection Rate Equations (Newton's Law of Cooling)
Heat Flux: 𝑞𝑞
′′
𝑠𝑠
∞
𝑊𝑊
𝑚𝑚
2
h : Convection Heat Transfer Coefficient
𝑊𝑊
𝑚𝑚
2
∙𝐾𝐾
Heat Rate: 𝑞𝑞 = ℎ𝐴𝐴
𝑠𝑠
𝑠𝑠
∞
) 𝑊𝑊 A
s
: Surface Area 𝑚𝑚
2
Heat Radiation emitted ideally by a blackbody surface has a surface emissive power: 𝐸𝐸
𝑏𝑏
𝑠𝑠
4
𝑊𝑊
𝑚𝑚
2
Heat Flux emitted: 𝐸𝐸 = 𝜀𝜀𝜎𝜎𝑇𝑇
𝑠𝑠
4
𝑊𝑊
𝑚𝑚
2
where ε is the emissivity with range of 0 ≤ 𝜀𝜀 ≤ 1
and 𝜎𝜎 = 5.67 × 10
−
𝑊𝑊
𝑚𝑚
2
𝐾𝐾
4
is the Stefan-Boltzmann constant
Irradiation: 𝐺𝐺
𝑎𝑎𝑏𝑏𝑠𝑠
= 𝛼𝛼𝐺𝐺 but we assume small body in a large enclosure with 𝜀𝜀 = 𝛼𝛼 so that 𝐺𝐺 = 𝜀𝜀 𝜎𝜎 𝑇𝑇
𝑠𝑠𝑠𝑠𝑠𝑠
4
Net Radiation heat flux from surface: 𝑞𝑞
𝑠𝑠𝑎𝑎𝑑𝑑
′′
𝑞𝑞
𝐴𝐴
𝑏𝑏
𝑠𝑠
𝑠𝑠
4
𝑠𝑠𝑠𝑠𝑠𝑠
4
Net radiation heat exchange rate: 𝑞𝑞
𝑠𝑠𝑎𝑎𝑑𝑑
𝑠𝑠
𝑠𝑠
4
𝑠𝑠𝑠𝑠𝑠𝑠
4
) where for a real surface 0 ≤ 𝜀𝜀 ≤ 1
This can ALSO be expressed as: 𝑞𝑞 𝑠𝑠𝑎𝑎𝑑𝑑
𝑠𝑠
𝑠𝑠
𝑠𝑠𝑠𝑠𝑠𝑠
) depending on the application
where ℎ
𝑠𝑠
is the radiation heat transfer coefficient which is: ℎ
𝑠𝑠
𝑠𝑠
𝑠𝑠𝑠𝑠𝑠𝑠
𝑠𝑠
2
𝑠𝑠𝑠𝑠𝑠𝑠
2
𝑊𝑊
𝑚𝑚
2
∙𝐾𝐾
TOTAL heat transfer from a surface: 𝑞𝑞 = 𝑞𝑞
𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐
𝑠𝑠𝑎𝑎𝑑𝑑
𝑠𝑠
𝑠𝑠
∞
𝑠𝑠
𝑠𝑠
4
𝑠𝑠𝑠𝑠𝑠𝑠
4
Conservation of Energy (Energy Balance)
𝑖𝑖𝑐𝑐
𝑔𝑔
𝑐𝑐𝑠𝑠𝑜𝑜
𝑠𝑠𝑜𝑜
(Control Volume Balance) ; 𝐸𝐸̇
𝑖𝑖𝑐𝑐
𝑐𝑐𝑠𝑠𝑜𝑜
= 0 (Control Surface Balance)
where 𝐸𝐸̇
𝑔𝑔
is the conversion of internal energy (chemical, nuclear, electrical) to thermal or mechanical energy, and
𝑠𝑠𝑜𝑜
= 0 for steady-state conditions. If not steady-state ( i.e. , transient) then 𝐸𝐸̇
𝑠𝑠𝑜𝑜
𝑝𝑝
𝑑𝑑𝑑𝑑
𝑑𝑑𝑜𝑜
Heat Equation (used to find the temperature distribution)
Heat Equation (Cartesian):
𝜕𝜕
𝜕𝜕𝑥𝑥
𝜕𝜕𝑑𝑑
𝜕𝜕𝑥𝑥
𝜕𝜕
𝜕𝜕𝜕𝜕
𝜕𝜕𝑑𝑑
𝜕𝜕𝜕𝜕
𝜕𝜕
𝜕𝜕𝜕𝜕
𝜕𝜕𝑑𝑑
𝜕𝜕𝜕𝜕
𝑝𝑝
𝜕𝜕𝑑𝑑
𝜕𝜕𝑜𝑜
If 𝑘𝑘 is constant then the above simplifies to:
𝜕𝜕
2
𝑑𝑑
𝜕𝜕𝑥𝑥
2
𝜕𝜕
2
𝑑𝑑
𝜕𝜕𝜕𝜕
2
𝜕𝜕
2
𝑑𝑑
𝜕𝜕𝜕𝜕
2
𝑞𝑞̇
𝑘𝑘
1
𝛼𝛼
𝜕𝜕𝑑𝑑
𝜕𝜕𝑜𝑜
where 𝛼𝛼 =
𝑘𝑘
𝜌𝜌𝑐𝑐
𝑝𝑝
is the thermal diffusivity
Heat Equation (Cylindrical):
1
𝑠𝑠
𝜕𝜕
𝜕𝜕𝑠𝑠
𝜕𝜕𝑑𝑑
𝜕𝜕𝑠𝑠
1
𝑠𝑠
2
𝜕𝜕
𝜕𝜕𝜕𝜕
𝜕𝜕𝑑𝑑
𝜕𝜕𝜕𝜕
𝜕𝜕
𝜕𝜕𝜕𝜕
𝜕𝜕𝑑𝑑
𝜕𝜕𝜕𝜕
𝑝𝑝
𝜕𝜕𝑑𝑑
𝜕𝜕𝑜𝑜
Heat Eqn. (Spherical):
1
𝑠𝑠
2
𝜕𝜕
𝜕𝜕𝑠𝑠
2
𝜕𝜕𝑑𝑑
𝜕𝜕𝑠𝑠
1
𝑠𝑠
2
sin 𝜃𝜃
2
𝜕𝜕
𝜕𝜕𝜕𝜕
𝜕𝜕𝑑𝑑
𝜕𝜕𝜕𝜕
1
𝑠𝑠
2
sin 𝜃𝜃
𝜕𝜕
𝜕𝜕𝜃𝜃
�𝑘𝑘 sin 𝜃𝜃
𝜕𝜕𝑑𝑑
𝜕𝜕𝜃𝜃
𝑝𝑝
𝜕𝜕𝑑𝑑
𝜕𝜕𝑜𝑜
Thermal Circuits
Plane Wall: 𝑅𝑅
𝑜𝑜,𝑐𝑐𝑐𝑐𝑐𝑐𝑑𝑑
𝐿𝐿
𝑘𝑘𝐴𝐴
Cylinder: 𝑅𝑅
𝑜𝑜,𝑐𝑐𝑐𝑐𝑐𝑐𝑑𝑑
ln�
𝑟𝑟
𝑟𝑟
�
2𝜋𝜋𝑘𝑘𝐿𝐿
Sphere: 𝑅𝑅
𝑜𝑜,𝑐𝑐𝑐𝑐𝑐𝑐𝑑𝑑
(
1
r
−
1
r
)
4𝜋𝜋𝑘𝑘
𝑜𝑜,𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐
1
ℎ𝐴𝐴
𝑜𝑜,𝑠𝑠𝑎𝑎𝑑𝑑
1
ℎ
𝑟𝑟
𝐴𝐴
_____________________________________________________________________________________________________________
General Lumped Capacitance Analysis
𝑠𝑠
′′
𝑠𝑠,ℎ
𝑔𝑔̇
[
∞
4
𝑠𝑠𝑠𝑠𝑠𝑠
4
)]
𝑠𝑠
( 𝑐𝑐,𝑠𝑠
)
Radiation Only Equation
𝜌𝜌𝜌𝜌𝑐𝑐
4 𝜀𝜀 𝐴𝐴
𝑠𝑠,𝑟𝑟
𝜎𝜎 𝑑𝑑
𝑠𝑠𝑠𝑠𝑟𝑟
3
�ln �
𝑑𝑑
𝑠𝑠𝑠𝑠𝑟𝑟
+𝑑𝑑
𝑑𝑑
𝑠𝑠𝑠𝑠𝑟𝑟
−𝑑𝑑
� − ln �
𝑑𝑑
𝑠𝑠𝑠𝑠𝑟𝑟
+𝑑𝑑
𝑖𝑖
𝑑𝑑
𝑠𝑠𝑠𝑠𝑟𝑟
−𝑑𝑑
𝑖𝑖
� + 2 �tan
−
𝑑𝑑
𝑑𝑑
𝑠𝑠𝑠𝑠𝑟𝑟
� − tan
−
𝑑𝑑
𝑖𝑖
𝑑𝑑
𝑠𝑠𝑠𝑠𝑟𝑟
Heat Flux, Energy Generation, Convection, and No Radiation Equation
𝑑𝑑−𝑑𝑑
∞
− �
𝑏𝑏
𝑎𝑎
�
𝑑𝑑
𝑖𝑖
− 𝑑𝑑
∞
− �
𝑏𝑏
𝑎𝑎
�
= exp(−𝑎𝑎𝑑𝑑) ; where 𝑎𝑎 = �
ℎ𝐴𝐴
𝑠𝑠,𝑐𝑐
𝜌𝜌𝜌𝜌𝑐𝑐
� and 𝑏𝑏 =
𝑞𝑞
𝑠𝑠
′′
𝐴𝐴
𝑠𝑠,ℎ
𝑔𝑔
𝜌𝜌𝜌𝜌𝑐𝑐
Convection Only Equation
𝑖𝑖
∞
𝑖𝑖
∞
= exp �− �
𝑠𝑠
𝑜𝑜
1
ℎ𝐴𝐴
𝑠𝑠
𝑜𝑜
𝑜𝑜
𝑖𝑖
� 1 − exp �−
𝑜𝑜
𝜏𝜏
𝑡𝑡
𝑚𝑚𝑎𝑎𝑥𝑥
𝑖𝑖
ℎ𝐿𝐿
𝑐𝑐
𝑘𝑘
If there is an additional resistance either in series or in parallel, then replace ℎ with 𝑈𝑈 in all the above lumped capacitance
equations, where
1
𝑅𝑅
𝑡𝑡
𝐴𝐴
𝑠𝑠
𝑊𝑊
𝑚𝑚
2
∙𝐾𝐾
� ; 𝑈𝑈 = overall heat transfer coefficient, 𝑅𝑅
𝑜𝑜
= total resistance, 𝐴𝐴
𝑠𝑠
= surface area.
Convection Heat Transfer
𝜌𝜌𝜌𝜌𝐿𝐿
𝑐𝑐
𝜇𝜇
𝜌𝜌𝐿𝐿
𝑐𝑐
𝜈𝜈
[Reynolds Number] ; 𝑁𝑁𝑁𝑁
ℎ
�𝐿𝐿
𝑐𝑐
𝑘𝑘
𝑓𝑓
[Average Nusselt Number]
where 𝜌𝜌 is the density, 𝜌𝜌 is the velocity, 𝐿𝐿 𝑐𝑐
is the characteristic length, 𝜇𝜇 is the dynamic viscosity, 𝜈𝜈 is the kinematic viscosity, 𝑚𝑚̇ is the mass flow
rate, ℎ
�
is the average convection coefficient, and 𝑘𝑘
𝑓𝑓
is the fluid thermal conductivity.
Vertical Cylinders: the equations for the Vertical Plate can be applied to vertical cylinders of height L if the following criterion is
met:
𝜋𝜋
𝐿𝐿
35
𝐺𝐺𝑠𝑠
𝐿𝐿
1 / 4
Long Horizontal Cylinders: 𝑁𝑁𝑁𝑁
����
𝜋𝜋
= �0.60 +
- 387 𝑅𝑅𝑎𝑎
𝐷𝐷
1 / 6
�1+�
- 559
𝑃𝑃𝑟𝑟
�
9 / 16
�
8 / 27
�
2
; 𝑅𝑅𝑎𝑎
𝜋𝜋
≲ 10
12
[Properties evaluated at T
f
]
Spheres: 𝑁𝑁𝑁𝑁
����
𝜋𝜋
= 2 +
- 589 𝑅𝑅𝑎𝑎
𝐷𝐷
1 / 4
�1+�
- 469
𝑃𝑃𝑟𝑟
�
9 / 16
�
4 / 9
; 𝑅𝑅𝑎𝑎
𝜋𝜋
≲ 10
11
; 𝑃𝑃𝑘𝑘 ≥ 0.7 [Properties evaluated at T
f
]
Heat Exchangers
Heat Gain/Loss Equations: 𝑞𝑞 = 𝑚𝑚̇ 𝑐𝑐
𝑝𝑝
𝑐𝑐
𝑖𝑖
𝑠𝑠
𝑙𝑙𝑚𝑚
; where 𝑈𝑈 is the overall heat transfer
coefficient and A
s
is the total heat exchanger surface area
Log-Mean Temperature Difference: ∆𝑇𝑇
𝑙𝑙𝑚𝑚,𝑃𝑃𝑃𝑃
�𝑑𝑑
ℎ,𝑖𝑖
−𝑑𝑑
𝑐𝑐,𝑖𝑖
�−�𝑑𝑑
ℎ,𝑜𝑜
−𝑑𝑑
𝑐𝑐,𝑜𝑜
�
ln�
�𝑇𝑇
ℎ,𝑖𝑖
−𝑇𝑇
𝑐𝑐,𝑖𝑖
�
�𝑇𝑇
ℎ,𝑜𝑜
−𝑇𝑇𝑐𝑐
,𝑜𝑜�
�
[Parallel-Flow Heat Exchanger]
Log-Mean Temperature Difference: ∆𝑇𝑇
𝑙𝑙𝑚𝑚,𝐶𝐶𝑃𝑃
�𝑑𝑑
ℎ,𝑖𝑖
−𝑑𝑑
𝑐𝑐,𝑜𝑜
�−�𝑑𝑑
ℎ,𝑜𝑜
−𝑑𝑑
𝑐𝑐,𝑖𝑖
�
ln�
�𝑇𝑇
ℎ,𝑖𝑖
−𝑇𝑇𝑐𝑐
,𝑜𝑜�
�𝑇𝑇
ℎ,𝑜𝑜
−𝑇𝑇
𝑐𝑐,𝑖𝑖
�
�
[Counter-Flow Heat Exchanger]
For Cross-Flow and Shell-and-Tube Heat Exchangers: ∆𝑇𝑇
𝑙𝑙𝑚𝑚
𝑙𝑙𝑚𝑚,𝐶𝐶𝑃𝑃
; where 𝐹𝐹 is a correction factor
obtained from the figures by calculating P & R values
Effectiveness – NTU Method ( ε – NTU):
Number of Transfer Units (NTU): 𝑁𝑁𝑇𝑇𝑈𝑈 =
𝑈𝑈𝐴𝐴
𝐶𝐶
𝑚𝑚𝑖𝑖𝑚𝑚
; where 𝐶𝐶
𝑚𝑚𝑖𝑖𝑐𝑐
is the minimum heat capacity rate in [W/K]
Heat Capacity Rates: 𝐶𝐶
𝑐𝑐
𝑐𝑐
𝑝𝑝,𝑐𝑐
[Cold Fluid] ; 𝐶𝐶
ℎ
ℎ
𝑝𝑝,ℎ
[Hot Fluid]
𝑠𝑠
𝐶𝐶
𝑚𝑚𝑖𝑖𝑚𝑚
𝐶𝐶
𝑚𝑚𝑎𝑎𝑚𝑚
[Heat Capacity Ratio]
Note: The condensation or evaporation side of the heat exchanger is associated with 𝐶𝐶 𝑚𝑚𝑎𝑎𝑥𝑥
𝑐𝑐
𝑝𝑝,𝑐𝑐
𝑐𝑐,𝑐𝑐
𝑐𝑐,𝑖𝑖
ℎ
𝑝𝑝,ℎ
ℎ,𝑖𝑖
ℎ,𝑐𝑐
𝑠𝑠
𝑙𝑙𝑚𝑚
𝑚𝑚𝑎𝑎𝑥𝑥
𝑚𝑚𝑖𝑖𝑐𝑐
ℎ,𝑖𝑖
𝑐𝑐,𝑖𝑖
� where 𝜀𝜀 =
𝑞𝑞
𝑞𝑞
𝑚𝑚𝑎𝑎𝑚𝑚
Use: 𝜀𝜀 = 𝑓𝑓
𝑠𝑠
relations or 𝑁𝑁𝑇𝑇𝑈𝑈 = 𝑓𝑓
𝑠𝑠
relations as appropriate
- If Pr ≤ 10 → n = 0.
- If Pr ≥ 10 → n = 0.
