Correlations for Convective Heat Transfer
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)
1
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
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
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
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:
5
Staggered arrangement:
4-3 Finned tube bundle
6
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.
7
5.) Forced Convection Flow over a Flat Plate
All properties at mean film temperature
Laminar boundary layer, constant wall temperature:
(Pohlhausen)
valid for ReL < 2·105, 0.6 < Pr < 10
Turbulent boundary layer along the whole plate, constant wall temperature:
(Petukhov)
Boundary layer with laminar-turbulent transition:
(Gnielinski)
8
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 < 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
9
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:
10
(Stephan and Preuكer)
' 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
cp specific heat capacity at constant pressure
D, d diameter
g gravitational acceleration
h mean heat transfer coefficient
enthalpy of evaporation
H height
k thermal conductivity
L length
heat flux
T temperature
u flow velocity
thermal diffusivity
coefficient of thermal expansion
dynamic viscosity
kinematic viscosity
11
density
surface tension
Subscripts
h hydraulic
i inside
m mean
o outside
s saturation
w wall
Dimensionless numbers
Gr Grashof number
Nu mean Nusselt number
Pr Prandtl number
Re Reynolds number
12
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-Wنrmeatlas, Düsseldorf (1963), Hb2.
3. Gnielinski, V.: Neue Gleichungen für den Wنrme- und den Stoffübergang in
turbulent durchstrِmten Rohren und Kanنlen. Forschung im Ingenieurwesen
41, 8-16 (1975).
4. Gnielinski, V.: Berechnung mittlerer Wنrme- und
Stoffübergangskoeffizienten an laminar und turbulent überstrِmten
Einzelkِrpern mit Hilfe einer einheitlichen Gleichung. Forschung im
Ingenieurwesen 41, 145-153 (1975).
5. Grigull, U.: Wنrmeübergang bei der Kondensation mit turbulenter
Wasserhaut. Forschung im Ingenieurwesen 13, 49-57 (1942).
6. Hausen, H.: Neue Gleichungen für die Wنrmeübertragung bei freier und
erzwungener Strِmung. Allg. Wنrmetechnik 9, 75-79 (1959).
7. Nusselt, W.: Die Oberflنchenkondensation 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 Wنrmeaustausch zwischen festen Kِrpern und
Flüssigkeiten mit kleiner Reibung und kleiner Wنrmeleitung. 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.: Wنrmeübergang und Druckabfall bei nicht ausgebildeter
Laminarstrِmung 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. Preuكer: Wنrmeübergang und maximale
Wنrmestromdichte beim Behنltersieden binنrer und ternنrer
Flüssigkeitsgemische. Chem.-Ing.-Tech. 51, 37 (1979).
15. VDI-Wنrmeatlas, 7th edition, Düsseldorf 1994.
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