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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. The following cases are treated:

1. Forced Convection Flow Inside a Circular Tube
2. Forced Convection Turbulent Flow Inside Concentric Annular Ducts
3. Forced Convection Turbulent Flow Inside Non-Circular Ducts
4. Forced Convection Flow Across Single Circular Cylinders and Tube Bundles
5. Forced Convection Flow over a Flat Plate
6. Natural Convection
7. Film Condensation
8. Nucleate Pool Boiling

• List of Symbols
• References

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)

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, u_m = 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 < Re_l < 10⁷ and 0.6 < Pr < 1000

4-2 Tube bundle

Transverse pitch ratio

Longitudinal pitch ratio

Void ratio for b > 1

for b < 1

Nu_0,bundle = f_ANu_l,0 (Gnielinski)

Nu_l,0 according to section 4-1 with instead of Re_l.

Arrangement factor f_A 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 Re_L < 2·10⁵, 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)

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 T_s

7-1 Laminar film condensation

Vertical wall or tube:

(Nusselt)

T_w = mean wall temperature

Horizontal cylinder:

(Nusselt)

T_w = const.

7-2 Turbulent film condensation

For vertical wall

Re = C A^m

Re_crit = 350

turbulent film: (Grigull)

8 Nucleate Pool Boiling

T_w = temperature of heating surface

T_s = 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:

(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
	density
	surface tension

Subscripts

Dimensionless numbers

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-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).
7. Nusselt, W.: Die Oberflächenkondensation 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 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. 3^rd 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, 7^th edition, Düsseldorf 1994.

By: Dr. Bernhard Spang, Associate Content Writer (read the author's Profile)
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