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

Constant wall temperature:
(Stephan)
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)
where
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:
(Stephan)
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

(Gnielinski)
where
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:
(Paikert)
Staggered tube bundle arrangement:
(Paikert)
4-4 Effects of property variation with temperature

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

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





All properties at mean film temperature
Laminar boundary layer, constant wall temperature:
(Pohlhausen)
valid for ReL < 2105, 0.6 < Pr < 10
Turbulent boundary layer along the whole plate, constant wall temperature:
(Petukhov)
Boundary layer with laminar-turbulent transition:
(Gnielinski)
6 Natural Convection


All properties at
L = characteristic length (see below)
Nu0
"Length" L
Vertical wall
0.67
H
Horizontal cylinder
0.36
D
Sphere
2.00
D

For ideal gases: (temperature in K)
(Churchill, Thelen)
valid for 10-4 < Gr Pr < 41014,
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:
(Nusselt)
Tw = mean wall temperature
Horizontal cylinder:
(Nusselt)
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 Preuer)
' 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:
(Fritz)
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
Subscripts
hhydrauliciinsidemmeanooutsidessaturationwwall
Dimensionless numbers
GrGrashof numberNumean Nusselt numberPrPrandtl numberReReynolds number
References


  1. Churchill, S.W.: Free convection around immersed bodies. Chapter 2.5.7 of Heat Exchanger Design Handbook, Hemisphere (1983).
  2. Fritz, W.: In VDI-Wrmeatlas, Dsseldorf (1963), Hb2.
  3. Gnielinski, V.: Neue Gleichungen fr den Wrme- und den Stoffbergang in turbulent durchstrmten Rohren und Kanlen. Forschung im Ingenieurwesen 41, 8-16 (1975).
  4. Gnielinski, V.: Berechnung mittlerer Wrme- und Stoffbergangskoeffizienten an laminar und turbulent berstrmten Einzelkrpern mit Hilfe einer einheitlichen Gleichung. Forschung im Ingenieurwesen 41, 145-153 (1975).
  5. Grigull, U.: Wrmebergang bei der Kondensation mit turbulenter Wasserhaut. Forschung im Ingenieurwesen 13, 49-57 (1942).
  6. Hausen, H.: Neue Gleichungen fr die Wrmebertragung bei freier und erzwungener Strmung. Allg. Wrmetechnik 9, 75-79 (1959).
  7. Nusselt, W.: Die Oberflchenkondensation des Wasserdampfes. VDI Z. 60, 541-546 and 569-575 (1916).
  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 Wrmeaustausch zwischen festen Krpern und Flssigkeiten mit kleiner Reibung und kleiner Wrmeleitung. 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.: Wrmebergang und Druckabfall bei nicht ausgebildeter Laminarstrmung 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. Preuer: Wrmebergang und maximale Wrmestromdichte beim Behltersieden binrer und ternrer Flssigkeitsgemische. Chem.-Ing.-Tech. 51, 37 (1979).
  15. VDI-Wrmeatlas, 7th edition, Dsseldorf 1994.
By: Dr. Bernhard Spang, Associate Content Writer