1
HEAT TRANSFER EQUATION SHEET Heat Conduction Rate Equations (Fourier's Law)
Heat Flux : ππ₯β²β² = βπ ππ
ππ₯ π
π2 k : Thermal Conductivity π
πβπ
Heat Rate : ππ₯ = ππ₯β²β²π΄π π Ac : Cross-Sectional Area
Heat Convection Rate Equations (Newton's Law of Cooling)
Heat Flux: πβ²β² = β(ππ β πβ) ππ2 h : Convection Heat Transfer Coefficient
ππ2βπΎ
Heat Rate: π = βπ΄π (ππ β πβ) π As : Surface Area π2
Heat Radiation emitted ideally by a blackbody surface has a surface emissive power: πΈπ = π ππ 4 π
π2
Heat Flux emitted : πΈ = ππππ 4 π
π2 where Ξ΅ is the emissivity with range of 0 β€ π β€ 1
and π = 5.67 Γ 10β8 ππ2πΎ4 is the Stefan-Boltzmann constant
Irradiation: πΊπππ = πΌπΊ but we assume small body in a large enclosure with π = πΌ so that πΊ = π π ππ π π 4
Net Radiation heat flux from surface: ππ ππβ²β² = π
π΄= ππΈπ(ππ ) β πΌπΊ = ππ(ππ
4 β ππ π π 4 )
Net radiation heat exchange rate: ππ ππ = πππ΄π (ππ 4 β ππ π π
4 ) where for a real surface 0 β€ π β€ 1
This can ALSO be expressed as: ππ ππ = βπ π΄(ππ β ππ π π ) depending on the application
where βπ is the radiation heat transfer coefficient which is: βπ = ππ(ππ + ππ π π )(ππ 2 + ππ π π
2 ) ππ2βπΎ
TOTAL heat transfer from a surface: π = πππππ + ππ ππ = βπ΄π (ππ β πβ) + πππ΄π (ππ 4 β ππ π π
4 ) π
Conservation of Energy (Energy Balance)
οΏ½ΜοΏ½ππ + οΏ½ΜοΏ½π β οΏ½ΜοΏ½ππ π = οΏ½ΜοΏ½π π (Control Volume Balance) ; οΏ½ΜοΏ½ππ β οΏ½ΜοΏ½ππ π = 0 (Control Surface Balance)
where οΏ½ΜοΏ½π is the conversion of internal energy (chemical, nuclear, electrical) to thermal or mechanical energy, and
οΏ½ΜοΏ½π π = 0 for steady-state conditions. If not steady-state (i.e., transient) then οΏ½ΜοΏ½π π = ππππππππ
Heat Equation (used to find the temperature distribution)
Heat Equation (Cartesian): π
ππ₯οΏ½π ππ
ππ₯οΏ½ + π
πποΏ½π ππ
πποΏ½ + π
πποΏ½π ππ
πποΏ½ + οΏ½ΜοΏ½ = πππ
ππππ
If π is constant then the above simplifies to: π2πππ₯2 + π2π
ππ2 + π2πππ2 + οΏ½ΜοΏ½
π= 1
πΌππππ
where πΌ = ππππ
is the thermal diffusivity
Heat Equation (Cylindrical): 1π
πππ
οΏ½ππ ππππ
οΏ½ + 1π 2
πππ
οΏ½π ππππ
οΏ½ + πππ
οΏ½π ππππ
οΏ½ + οΏ½ΜοΏ½ = πππππππ
Heat Eqn. (Spherical): 1
π 2π
ππ οΏ½ππ2 ππ
ππ οΏ½ + 1
π 2 sin π2π
πποΏ½π ππ
πποΏ½ + 1
π 2 sin π πππ
οΏ½π sin π ππππ
οΏ½ + οΏ½ΜοΏ½ = πππππππ
Thermal Circuits
Plane Wall: π π,ππππ = πΏππ΄
Cylinder: π π,ππππ =lnοΏ½π2
π1οΏ½
2πππΏ Sphere: π π,ππππ =
( 1r1
β 1r2
)
4ππ
2 π π,ππππ = 1
βπ΄ π π,π ππ = 1
βππ΄
_____________________________________________________________________________________________________________
General Lumped Capacitance Analysis
ππ β²β²π΄π ,β + πΈοΏ½ΜοΏ½ β [β(π β πβ) + ππ(π4 β ππ π π
4 )]π΄π (π,π ) = πππππππ
Radiation Only Equation
π = πππ4 π π΄π ,π π ππ π π
3 οΏ½ln οΏ½ππ π π+πππ π πβπ
οΏ½ β ln οΏ½ππ π π+ππππ π πβππ
οΏ½ + 2 οΏ½tanβ1 οΏ½ πππ π π
οΏ½ β tanβ1 οΏ½ ππππ π π
οΏ½οΏ½οΏ½
Heat Flux, Energy Generation, Convection, and No Radiation Equation
πβπββ οΏ½πποΏ½
ππβ πββ οΏ½πποΏ½
= exp(βππ) ; where π = οΏ½βπ΄π ,π
ππποΏ½ and π = ππ
β²β²π΄π ,β+ οΏ½ΜοΏ½π
πππ
Convection Only Equation
πππ
=π β πβ
ππ β πβ= exp οΏ½β οΏ½
βπ΄π
ππποΏ½ ποΏ½
ππ = οΏ½ 1βπ΄π
οΏ½ (πππ) = π ππΆπ ; π = πππ ππ οΏ½1 β exp οΏ½β πππ‘
οΏ½οΏ½ ; ππππ₯ = πππ ππ
π΅π΅ = βπΏππ
If there is an additional resistance either in series or in parallel, then replace β with π in all the above lumped capacitance
equations, where
π = 1π π‘π΄π
οΏ½ ππ2βπΎ
οΏ½ ; π = overall heat transfer coefficient, π π = total resistance, π΄π = surface area.
Convection Heat Transfer
π π = πππΏππ
= ππΏππ
[Reynolds Number] ; πποΏ½οΏ½οΏ½οΏ½ = βοΏ½πΏπππ
[Average Nusselt Number]
where π is the density, π is the velocity, πΏπ is the characteristic length, π is the dynamic viscosity, π is the kinematic viscosity, οΏ½ΜοΏ½ is the mass flow
rate, βοΏ½ is the average convection coefficient, and ππ is the fluid thermal conductivity.
3 Internal Flow
π π = 4 οΏ½ΜοΏ½πππ
[For Internal Flow in a Pipe of Diameter D]
For Constant Heat Flux [ππ ΚΊ = ππππππππ]: πππππ = ππ
ΚΊ(π β πΏ) ; where P = Perimeter, L = Length
ππ(π₯) = ππ,π +ππ
ΚΊ Β· ποΏ½ΜοΏ½ β ππ
π₯
For Constant Surface Temperature [ππ = ππππππππ]:
If there is only convection between the surface temperature, ππ , and the mean fluid temperature, ππ, use
ππ βππ(π₯)ππ βππ,π
= π π₯π οΏ½β πβπ₯οΏ½ΜοΏ½βππ
βοΏ½οΏ½
If there are multiple resistances between the outermost temperature, πβ, and the mean fluid temperature, ππ, use
πβ β ππ(π₯)πβ β ππ,π
= π π₯π οΏ½βπ β π₯
οΏ½ΜοΏ½ β ππποΏ½ = π π₯π οΏ½β
1οΏ½ΜοΏ½ β ππ β π π
οΏ½
Total heat transfer rate over the entire tube length:
ππ = οΏ½ΜοΏ½ β ππ β οΏ½ππ,π β ππ,ποΏ½ = βοΏ½ β π΄π β βπππ ππ π β π΄π β βπππ ; ππ = ππππππππ
Log mean temperature difference: βπππ = βππββππ
lnοΏ½βππβππ
οΏ½ ; βππ = ππ β ππ,π ; βππ = ππ β ππ,π
Free Convection Heat Transfer
πΊππΏ = ππ(ππ βπβ)πΏπ3
π2 [Grashof Number]
π ππΏ = ππ(ππ βπβ)πΏπ3
ππΌ [Rayleigh Number]
Vertical Plates: πποΏ½οΏ½οΏ½οΏ½πΏ = οΏ½0.825 + 0.387 π ππΏ1/6
οΏ½1+οΏ½0.492ππ οΏ½
9/16οΏ½
8/27οΏ½
2
; [Entire range of RaL; properties evaluated at Tf]
- For better accuracy for Laminar Flow: πποΏ½οΏ½οΏ½οΏ½πΏ = 0.68 + 0.670 π ππΏ1/4
οΏ½1+οΏ½0.492ππ οΏ½
9/16οΏ½
4/9 ; π ππΏ β² 109 [Properties evaluated at Tf]
Inclined Plates: for the top and bottom surfaces of cooled and heated inclined plates, respectively, the equations of the vertical
plate can be used by replacing (g) with (π cos π) in RaL for 0 β€ π β€ 60Β°.
Horizontal Plates: use the following correlations with πΏ = π΄π π
where As = Surface Area and P = Perimeter
- Upper surface of Hot Plate or Lower Surface of Cold Plate:
πποΏ½οΏ½οΏ½οΏ½πΏ = 0.54 π ππΏ1/4 (104 β€ π ππΏ β€ 107) ; πποΏ½οΏ½οΏ½οΏ½πΏ = 0.15 π ππΏ
1/3 (107 β€ π ππΏ β€ 1011) - Lower Surface of Hot Plate or Upper Surface of Cold Plate:
πποΏ½οΏ½οΏ½οΏ½πΏ = 0.27 π ππΏ1/4 (105 β€ π ππΏ β€ 1010)
4 Vertical Cylinders: the equations for the Vertical Plate can be applied to vertical cylinders of height L if the following criterion is
met: ππΏ
β₯ 35
πΊπ πΏ1/4
Long Horizontal Cylinders: πποΏ½οΏ½οΏ½οΏ½π = οΏ½0.60 + 0.387 π ππ·1/6
οΏ½1+οΏ½0.559ππ οΏ½
9/16οΏ½
8/27οΏ½
2
; π ππ β² 1012 [Properties evaluated at Tf]
Spheres: πποΏ½οΏ½οΏ½οΏ½π = 2 + 0.589 π ππ·1/4
οΏ½1+οΏ½0.469ππ οΏ½
9/16οΏ½
4/9 ; π ππ β² 1011 ; ππ β₯ 0.7 [Properties evaluated at Tf]
Heat Exchangers
Heat Gain/Loss Equations: π = οΏ½ΜοΏ½ ππ(ππ β ππ) = ππ΄π βπππ ; where π is the overall heat transfer coefficient
Log-Mean Temperature Difference: βπππ,ππ = οΏ½πβ,πβππ,ποΏ½βοΏ½πβ,πβππ,ποΏ½
lnοΏ½οΏ½πβ,πβππ,ποΏ½
οΏ½πβ,πβππ,ποΏ½οΏ½
[Parallel-Flow Heat Exchanger]
Log-Mean Temperature Difference: βπππ,πΆπ = οΏ½πβ,πβππ,ποΏ½βοΏ½πβ,πβππ,ποΏ½
lnοΏ½οΏ½πβ,πβππ,ποΏ½
οΏ½πβ,πβππ,ποΏ½οΏ½
[Counter-Flow Heat Exchanger]
For Cross-Flow and Shell-and-Tube Heat Exchangers: βπππ = πΉ βπππ,πΆπ ; where πΉ is a correction factor
Number of Transfer Units (NTU): πππ = ππ΄πΆπππ
; where πΆπππ is the minimum heat capacity rate in [W/K]
Heat Capacity Rates: πΆπ = οΏ½ΜοΏ½π ππ,π [Cold Fluid] ; πΆβ = οΏ½ΜοΏ½β ππ,β [Hot Fluid] ; πΆπ = πΆππππΆπππ
[Heat Capacity Ratio]
Note: The condensation or evaporation side of the heat exchanger is associated with πΆπππ₯ = β
5
If Pr β€ 10 β n = 0.37 If Pr β₯ 10 β n = 0.36
6
7
8
9
10