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International Journal of Mechanics and Thermodynamics.

Volume 3, Number 2 (2012), pp. 63-68

International Research Publication House

http://www.irphouse.com

New Concept for an Effective Reynolds Number in

Mixed Convection Heat Transfer across Horizontal

Tube

1Ahmed A. Hanafy and

2Wael M. El-Maghlany

1Arab Academy for Science, Technology and Maritime Transport,Alexandria, Egypt

E-mail: [email protected] of Engineering, Suez Canal University, Ismailia, Egypt

E-mail: [email protected]

Abstract

The aim of this study was to investigate in a method for mixed convection

analysis to combined natural and forced convection in right manner. The

natural convection effect will be replaced by an equivalent forced convection

Reynolds number in the direction of natural convection, then it will be easily

to combine the forced convection Reynolds number with the equivalent

natural convection Reynolds number to generate new forced convection

Reynolds number in direction dependent on the angle between the forced and

natural convection directions. The new effective Reynolds number transfers

the problem from mixed convection to an equivalent pure forced convection

state with the new effective Reynolds number.

Keywords: Natural convection, mixed convection, Effective Reynolds

number

NomenclatureGr Grashof number

Nu Nusselt number

Ra Rayleigh number

Re Reynolds number

Reeff Effective Reynolds number

Reeq Equivalent Reynolds number

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64 Ahmed A. Hanafy and Wael M. El-Maghlany

Greek symbols

Angle between the upward vertical direction of the free convection flowand the direction of the forced flow

IntroductionThe two dimensional combined convection from an infinite horizontal isothermal

circular cylinder is very difficult to solve analytically. One concept for the evaluation

of mixed convection Nusselt number in cross flow is by the vectorial addition of the

heat transfer Nusselt numbers of the natural and forced convections victorially as

follows:

free

2

forced

2

mixed NuNuNu += (1)

B.G. Van der Hegge Zijnen [1] introduced the above equation. The agreement

between Nusselt numbers calculated from this equation and his experimental values

was unsatisfactory. Another approach to combine the separate effects of the natural

and forced convections together, is by the converting the natural convection to an

equivalent forced convection by admitting a Reynolds number in the case of natural

convection equals, and adding it vectorially to the pure forced convection Reynolds

number to give a total flow effective Reynolds number for the mixed convection as

follows:

]Re

Ra85.2Cos

Re

Ra4.31[ReRe 222eff2 ++=

(2)

The above equation was proposed by A.P. Hatton et al [2]. In which is the anglebetween the upward vertical direction of the free convection flow and the direction of

the forced flow. G.K. Sharma and S.P. Sukhatme [3] determined experimentally

Nusselt number for mixed convection in cross flow to air. Oosthuizen, P.H., and

S.Madan [4] measured the effect of the angle of attack of the forced convection on the

limits of the forced convection and pure natural convection between which the

assisting, cross and counter flow mixed convections play role for air (Pr=0.7), water

(Pr = 6.3) and glycerin (Pr = 63) ,the last was obtained by B. Gebhart and L.pera [5]

experimentally. The numerical results for cross combined flow over horizontalisothermal cylinders placed in fluid of Pr=0.7 have been reported by Badr [6]. Nakia

and Okazaki [7] obtained the average Nusselt number for cross flow mixed

convection flow on thin horizontal wires in the very low Reynolds and Grashof

numbers. A trial was made by B.F.Armaly, T.S. Chen and N.Ramachandran [8] to

combine the separate correlating equations for assisting and cross flows covering the

different Re and Gr ranges into a single correlation equation for both flows. Bassam

A. and K. Abu Hijleh [9] proposed correlation for the mixed convection in cross flow

of air; they also proposed a correlation for mixed convection Nusselt number at

different angles of attack as a function of the mixed convection Nusselt number for

cross flow.

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New Concept for an Effective Reynolds Number 65

The Effective Reynolds NumberThe flow around the cylinder due to pure natural convection is always vertically

upward, while the pure forced convection can assume any direction according to the

angle at which the flow attacks the cylinder. Therefore, for combined convection, the

flow is the resultants of the two flows as shown in Fig.1. In order to handle combined

convection, it can be suggested that the total flow is obtained by summing the pure

natural and the pure forced flows vectorially to produce a total flow having a certain

Reynolds number which was named as the effective Reynolds number.

Figure 1: The Concept of Vectorial Summation of the Natural and Forced convectionFlows.

Solution ProceedingIn this proceeding, equivalence between the Nusselt number for the free convection

with a certain value of Grashof number to Nusselt number for the forced convection

to obtain the equivalent forced convection Reynolds number in the direction of the

free convection. The forced convection Nusselt number correlation proposed by

Churchill and Bernstein [10] is very useful in rang of Reynolds number of 102

to 107

as follow

5

4

8

5

4

1

3

2

3

1

2

1

282000

Re1

Pr

401

PrRe62030

+

+

+=

.

..Nu (3)

While the free convection Nusselt number correlation proposed by Churchill and

Chu [11] is very useful in rang of Raleigh number of 10-5

to 1012

as follow

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6

1

9

16

16

9

2

1

Pr

559.01

Pr387.060

+

+=Gr

.Nu (4)

By equating Nu in equation (3) and Nu in equation (4)

5

4

8

5

4

1

3

2

3

1

2

1

2

6

1

9

16

16

9 282000

Re1

Pr

401

PrRe62030

Pr

559.01

Pr387.060

+

+

+=

+

+ eqeq

.

..

Gr.

(5)

The above equation gives a relation between natural convection Gr and forced

convection Reeq . The relation between natural convection Gr and forced convection

Reeq is plotted in the Fig.2, this plot is correlated to obtain simple relation between Grand Reeq as follow

Greq=Re (6)

And hence the effective Reynolds number in mixed convection will be

cosReReReReRe 22eqeqeff

++= (7)

The above new effective Reynolds number will be a simple method to relate the

mixed convection heat transfer Nusselt number to, not to both forced convection Re

and free convection Gr for any angle between them (mixed cross (=90) , mixedassistance (=0), mixed opposing (=180), and any angle).

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New Concept for an Effective Reynolds Number 67

1E+004 1E+005 1E+006 1E+007 1E+008 1E+009 1E+010

Gr

100

1000

10000

100000

Re

eq

Reeq=Gr0.5

Figure 2: The Relation between Free Convection and Equivalent Forced Convection

(Pr=0.7)

ConclusionNew effective Reynolds number has been obtained as a simple method to relate the

mixed convection heat transfer Nusselt number to, not to both forced convection Re

and free convection Gr for any angle between them (mixed cross (=90) , mixed

assistance (=0), mixed opposing (=180), and any angle).

References

[1] B.G. van der Hegge Zijnen , "Modified Correlation Formula for the HeatTransfer by Natural and by Forced Convection From Horizontal Cylinders . " ,

1956 , Applied Scientific Research , Series A , Vol. 6 , pp. 129-140 .

[2] A.P. Hatton , D.D. James and H.W. Swire , " Combined Forced and NaturalConvection With Low Speed Air Flow Over Horizontal Cylinder . " , 1970 ,

ASME Journal of Fluid Mech., Vol. 42 , part 1 , pp. 17-31 .

[3] Sharma, G.K., and Sukhatma , " Combined Free and Forced Convection HeatTransfer from a Heated Tube to a Transverse Air Stream. ", 1969 , ASME

Journal of Heat Transfer , Vol. 91 , pp. 457-459.

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[4] P.H. Oosthuizen & S. Madan , " Combined Convection Heat Transfer FromHorizontal Cylinders in Air . " , 1970 , ASME Journal of Heat Transfer , Feb. ,

pp. 194-196 .

[5] B. Gebhart & L. Pera , " Mixed Convection From Long Horizontal Cylinders ." , 1970 , J. of Fluid Mechanics , Vol. 45 , Part 1 , pp. 49-64

[6] H.M. Badr , " A Theoretical Study of Laminar Mixed Convection from aHorizontal Cylinder in Cross Stream . ", 1983, Int. J. Heat Mass Transfer, Vol.

26, pp. 639-653.

[7] B.F. Armaly , T.S. Chen & N. Ramachandran , " Correlations for MixedConvection Flows Across Horizontal Cylinders and Spheres . " , 1988 , ASME

Journal of Heat Transfer , Vol. 110 , pp. 511-514 .

[8] Seiichi Nakai , Takuro Okazaki , " Heat Transfer From a Horizontal CircularWire at Small Reynolds and Grashof Numbers - II. " , 1975 , Int. J. Heat MassTransfer , Vol. 18 , pp. 397-413.

[9] .Bassam A. and K.Abou-Hijleh , " Laminar Mixed Convection Correlations foran Isothermal Cylinder in Cross Flow at Different Angles of Attack . " , 1999 ,

J. Heat Transfer Vol. 42 , pp. 1383-1388 .

[10] Churchill, S. W., and M. Bernstein. A Correlating Equation for ForcedConvection from Gases and Liquids to a Circular Cylinder in Cross flow, J.

Heat Transfer, vol. 99,pp. 300306, 1977.

[11] Churchill, S. W., and H. H. S. Chu. Correlating Equations for Laminar andTurbulent Free Convection from a Horizontal Cylinder, Int. J. Heat Mass

Transfer, vol. 18, p. 1049, 1975.

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