Correlations for Convective Heat Transfer

In many cases it's convenient to have simple equations for estimation of heat transfer coefficients. Below is a collection of recommended correlations for single-phase convective flow in different geometries as well as a few equations for heat transfer processes with change of phase. Note that all equations are for mean Nusselt numbers and mean heat transfer coefficients.

1 Forced Convection Flow Inside a Circular Tube

All properties at fluid bulk mean temperature (arithmetic mean of inlet and outlet temperature).
Nusselt numbers Nu0 from sections 1-1 to 1-3 have to be corrected for temperature-dependent fluid properties according to section 1-4.
1-1 Thermally developing, hydrodynamically developed laminar flow (Re < 2300)

Constant wall temperature:
Constant wall heat flux:
1-2 Simultaneously developing laminar flow (Re < 2300)

Constant wall temperature:
Constant wall heat flux:
which is valid over the range 0.7 < Pr < 7 or if Re Pr D/L < 33 also for Pr > 7.
1-3 Fully developed turbulent and transition flow (Re > 2300)

Constant wall heat flux:
(Petukhov, Gnielinski)
Constant wall temperature:
For fluids with Pr > 0.7 correlation for constant wall heat flux can be used with negligible error.
1-4 Effects of property variation with temperature

Liquids, laminar and turbulent flow:
Subscript w: at wall temperature, without subscript: at mean fluid temperature
Gases, laminar flow:
Nu = Nu0
Gases, turbulent flow:
Temperatures in Kelvin
2 Forced Convection Flow Inside Concentric Annular Ducts, Turbulent (Re > 2300)

Dh = Do - Di

All properties at fluid bulk mean temperature (arithmetic mean of inlet and outlet temperature).

Heat transfer at the inner wall, outer wall insulated:
(Petukhov and Roizen)
Heat transfer at the outer wall, inner wall insulated:
(Petukhov and Roizen)
Heat transfer at both walls, same wall temperatures:
3 Forced Convection Flow Inside Non-Circular Ducts, Turbulent (Re > 2300)

Equations for circular tube with hydraulic diameter

4 Forced Convection Flow Across Single Circular Cylinders and Tube Bundles

D = cylinder diameter, um = free-stream velocity, all properties at fluid bulk mean temperature. Correction for temperature dependent fluid properties see section 4-4.
4-1 Smooth circular cylinder

Valid over the ranges 10 < Rel < 107 and 0.6 < Pr < 1000
4-2 Tube bundle

Transverse pitch ratio
Longitudinal pitch ratio
Void ratio for b > 1
for b < 1
Nu0,bundle = fANul,0 (Gnielinski)
Nul,0 according to section 4-1 with instead of Rel.
Arrangement factor fA depends on tube bundle arrangement.
In-line arrangement:
Staggered arrangement:
4-3 Finned tube bundle

In-line tube bundle arrangement:
Staggered tube bundle arrangement:
4-4 Effects of property variation with temperature

Subscript w: at wall temperature, without subscript: at mean fluid temperature.

Temperatures in Kelvin.
5 Forced Convection Flow over a Flat Plate

All properties at mean film temperature
Laminar boundary layer, constant wall temperature:
valid for ReL < 2·105, 0.6 < Pr < 10
Turbulent boundary layer along the whole plate, constant wall temperature:
Boundary layer with laminar-turbulent transition:
6 Natural Convection

All properties at
L = characteristic length (see below)
"Length" L
Vertical wall
Horizontal cylinder

For ideal gases: (temperature in K)
(Churchill, Thelen)
valid for 10-4 < Gr Pr < 4·1014,
0.022 < Pr < 7640, and constant wall temperature
7 Film Condensation

All properties without subscript are for condensate at the mean temperature
Exception: = vapor density at saturation temperature Ts
7-1 Laminar film condensation

Vertical wall or tube:
Tw = mean wall temperature
Horizontal cylinder:
Tw = const.
7-2 Turbulent film condensation

For vertical wall

Re = C Am

Recrit = 350
turbulent film: (Grigull)
8 Nucleate Pool Boiling

Tw = temperature of heating surface
Ts = saturation temperature
Heat transfer at ambient pressure:
(Stephan and Preußer)
' saturated liquid
'' saturated vapor
Bubble departure diameter
Angle = rad for water = 0.0175 rad for low-boiling liquids= 0.611 rad for other liquids
For water in the range of 0.5 bar < p < 20 bar and 104 W/m2 < < 106 W/m2
the following equation may be applied:
List of Symbols

cpspecific heat capacity at constant pressure D, ddiameterggravitational acceleration hmean heat transfer coefficient enthalpy of evaporation Hheightkthermal conductivity Llengthheat flux Ttemperatureuflow velocitythermal diffusivity coefficient of thermal expansion dynamic viscosity kinematic viscosity density surface tension
Dimensionless numbers
GrGrashof numberNumean Nusselt numberPrPrandtl numberReReynolds number

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  2. Fritz, W.: In VDI-Wärmeatlas, Düsseldorf (1963), Hb2.
  3. Gnielinski, V.: Neue Gleichungen für den Wärme- und den Stoffübergang in turbulent durchströmten Rohren und Kanälen. Forschung im Ingenieurwesen 41, 8-16 (1975).
  4. Gnielinski, V.: Berechnung mittlerer Wärme- und Stoffübergangskoeffizienten an laminar und turbulent überströmten Einzelkörpern mit Hilfe einer einheitlichen Gleichung. Forschung im Ingenieurwesen 41, 145-153 (1975).
  5. Grigull, U.: Wärmeübergang bei der Kondensation mit turbulenter Wasserhaut. Forschung im Ingenieurwesen 13, 49-57 (1942).
  6. Hausen, H.: Neue Gleichungen für die Wärmeübertragung bei freier und erzwungener Strömung. Allg. Wärmetechnik 9, 75-79 (1959).
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  8. Petukhov, B.S.: Heat transfer and friction in turbulent pipe flow with variable physical properties. Adv. Heat Transfer 6, 503-565 (1970).
  9. Petukhov, B.S. and L.I. Roizen: High Temperature 2, 65-68 (1964).
  10. Pohlhausen, E.: Der Wärmeaustausch zwischen festen Körpern und Flüssigkeiten mit kleiner Reibung und kleiner Wärmeleitung. Z. Angew. Math. Mech. 1, 115-121 (1921).
  11. Shah, R.K.: Thermal entry length solutions for the circular tube and parallel plates. Proc. 3rd Natnl. Heat Mass Transfer Conference, Indian Inst. Technol Bombay, Vol. I, Paper HMT-11-75 (1975).
  12. Stephan, K.: Wärmeübergang und Druckabfall bei nicht ausgebildeter Laminarströmung in Rohren und ebenen Spalten. Chem.-Ing.-Tech. 31, 773-778 (1959).
  13. Stephan, K.: Chem.-Ing.-Tech. 34, 207-212 (1962).
  14. Stephan, K. and P. Preußer: Wärmeübergang und maximale Wärmestromdichte beim Behältersieden binärer und ternärer Flüssigkeitsgemische. Chem.-Ing.-Tech. 51, 37 (1979).
  15. VDI-Wärmeatlas, 7th edition, Düsseldorf 1994.
By: Dr. Bernhard Spang, Associate Content Writer

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