Energy Conversion and Management 50 (2009) 2266–2274

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Energy Conversion and Management

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Experimental study of turbulent forced convection in vertical eccentric annulus

R. Hosseini *, M. Ramezani, M.R. MazaheriMechanical Engineering Department, Amirkabir University of Technology, 424 Hafez Ave., P.O. Box 15875-4413, Tehran, Iran

a r t i c l e i n f o

Article history:Received 28 October 2007Received in revised form 26 September2008Accepted 16 May 2009Available online 17 June 2009

Keywords:Turbulent forced convectionEccentric annulusExperimental

0196-8904/$ - see front matter � 2009 Elsevier Ltd. Adoi:10.1016/j.enconman.2009.05.002

* Corresponding author. Tel.: +98 21 66405844; faxE-mail address: hoseinir@cic.aut.ac.ir (R. Hosseini)

a b s t r a c t

Forced convection in an open-ended vertical eccentric annulus with different eccentricities has beenexperimentally studied with several heat fluxes and inlet air velocities. Constant heat flux outer tubeand insulated inner tube boundary conditions were used. The wall temperature of the outer tube wasmeasured along the length for different eccentricities. Results indicate that with the increase of eccentric-ity, the convection heat transfer coefficient increases. Also, as the Reynolds number is increased a crest isformed in the temperature variation along the length. An empirical relation is derived for the Nusseltnumber as a function of the Reynolds number and eccentricity.

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1. Introduction

Convection heat transfer in eccentric annulus has become ofgreat interest to the researchers in the recent years. Such interestarises from the presence of eccentricity in mechanical equipments,electrical motors and generators, which in many cases is the causeof design tolerances and manufacturing limits. Further applicationsare the drilling and completion of an oil well, heat exchangerindustries, and the insulation for buried tubes in district heatingand cooling of underground electric cables.

Since an annulus contains two surfaces on which thermal con-ditions may be independently specified, there are a large numberof heat transfer problems of significant interest. Some years ago,Reynolds et al. [1] completely solved the problem of heat transferto fully developed laminar flow in concentric annulus. Theyrealized that with the usual assumptions, the energy equation islinear. Therefore a temperature field satisfying arbitrary boundaryconditions could be determined by simply superposing appropriatemultiples of temperature fields satisfying certain fundamentalboundary conditions. Four fundamental boundary conditions weredefined and the temperature fields satisfying these boundaryconditions were called fundamental solutions.

Perhaps the first problem that came to attention among otherswas the single cylinder fluid flow and heat transfer [2–5]. In all ofthese researches the flow was determined to be laminar for freeconvection heat transfer. In [6] for the analytical approach to thefree convection case, the assumption of laminar fully developed

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flow is adopted. The developing flow and heat transfer in concen-tric annulus is examined both numerically and experimentally [7–10].

Advancing the knowledge about the phenomenon, severalauthors worked on the problem of eccentric annulus. Most of theresearch on this subject has been done by numerical and/or analyt-ical methods and relatively few experimental studies have beencarried out. Therefore, to get insight into the physics of the prob-lem, further experimental studies will be valuable, as it is consid-ered here. The earliest work on eccentric annuli could beattributed to Heyda [10], who determined the velocity profile bysolving the momentum equation. Trombetta [11] has presentedan approximate solution to laminar forced convection in eccentricannulus for a wide range of eccentricities and radius ratios. Hestudied the effect of different eccentricities and radius ratios onthe Nusselt number. His research is limited to hydrodynamicallyand thermally fully developed flows. He developed fundamentalsolutions of the first, second, and fourth kinds. Cheng and Hwang[12] have given a numerical solution for fully developed laminarflow with uniform heat sources in eccentric annular ducts underthe thermal boundary conditions of uniform heat flux from eachwall. By means of a numerical solution to the problem, the princi-ple of a maximum flow rate induction by free convection also cameto attention [13,14]. Further experiments revealed the presence ofan optimum eccentricity which maximizes the overall heat trans-fer coefficient [7,14]. Using numerical methods Mokheimer andEl-Shaarawi [15] have solved for the developing velocity profiles,axial variation of pressure, full development length, and heat trans-fer parameters. They chose the thermal boundary conditions ofhaving one of the annulus tubes at a constant temperature andthe other insulated.

Nomenclature

A empirical constantDh hydraulic diameter (Do � Di)Di outside diameter of inner tubeDo inside diameter of outer tubeE eccentricitye distance between center line of tubesf0 correction factorGrD mean Grashof number based on hydraulic diameterGrl mean Grashof number based on lengthGr�D modified Grashof number based on hydraulic diameterg gravityhm average heat transfer coefficientk thermal conductivity of fluidL lengthNuD average Nusselt number based on hydraulic diameterNul average Nusselt number based on lengthN number of thermocouplesn empirical constantPr Prandtl number

q00 heat flux (W/m2)r* annulus radius ratio, Ri/Ro

ReD Reynolds number based on hydraulic diameterRe�D modified Reynolds number based on hydraulic diameterRi outer radius of inner tubeRo inner radius of outer tubeTamb ambient temperatureTf mean film temperatureT0 inlet fluid temperatureTs average wall temperatureTs local wall temperatureu inlet air velocity

Greek symbolsb volumetric coefficient of thermal expansionq fluid densityl dynamic fluid viscosity

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In this experimental study we will be concerned with turbulentforced convection in an open-ended vertical annulus for a range ofdifferent eccentricities, different inlet velocities, and different heatfluxes. The outer wall is heated by constant heat flux and the innerwall is maintained insulated. No experimental work on turbulentforced/free convection in eccentric annulus was found in the liter-ature to the best of our knowledge.

2. Experimental setup

A schematic view of the experimental setup is shown in Fig. 1.Basically, the setup consists of two tubes of different diameters andsame lengths (1660 mm). The inner tube has a 12 mm OD, and1 mm thickness, and is made of brass. The outer tube has a50 mm ID, and 2 mm thickness, and made of copper. The radius ra-tio of the annulus is 0.24. An electric wire heater is wound aroundthe outer tube’s external surface and carefully insulated with a25 mm layer of asbestos. Over this layer of insulation another wireheater is wound as ‘‘guard heater” as suggested in [14], and a50 mm layer of ceramic wool covers the entire length of the heater(Fig. 2).

To check for steady state condition, four pairs of thermocouplesare used, each pair resting at a specific elevation (240, 540, 1040,and 1340 mm) and different radius such that each thermocoupleof one pair rests on one side of the guard heater. When steady statecondition is reached, each pair shows the same temperature,ensuring that there is no heat flow to the environment throughthe insulation. Temperature measurement was done using 44chromel–alumel thermocouples which are installed on the outsidesurface of the outer tube and digital thermometers. Ten thermo-couples on each end of the tube are in 20 mm distances, and theother 24 are separated in 50 mm distances.

To reduce the time required to reach adiabatic boundary condi-tion on the inner tube, it is filled with asbestos and both ends areinsulated. The inner tube is suspended inside the outer tube bymeans of headings connected to the two ends of the outer tube(Fig. 2a and b). The headings are designed and calibrated so as toprovide high accuracy positioning of the inner tube at differenteccentricities using power screws.

Electric power needed for both the main and guard heaterswere provided with variable power supplies. The input voltageand current were measured and monitored. Forced flow is pro-

vided by use of an electric fan, placed at the bottom end of theannulus, connected below the bottom heading (Fig. 2b). The flowrate was controlled by controlling the input power of the fan.The flow velocity was measured by use of a rotary anemometer.The whole experimental setup was installed in a large laboratoryroom with a controlled temperature and humidity environment.A list of measured/calculated parameters and their accuracies ispresented in Table 1.

For the purpose of comparison with previous works and verifi-cation of results, experiments were carried out for single tube(without the inner tube installed), concentric annulus, and eccen-tric annulus.

3. Governing equations

To obtain the convection heat transfer coefficient, mean tem-perature of the fluid flowing inside the annulus and the annuluswall temperature are required. To be able to compare results withprevious works done by researchers, all calculations were carriedout based on the inlet air temperature as suggested by Al-Arabiet al. [8]. The average convection heat transfer coefficient can becalculated by Eq. (1).

hm ¼q00

Ts � T0ð1Þ

where T0 is the inlet air temperature and Ts is the average wall tem-perature of the outer tube. Ts can be derived from Eq. (2).

Ts ¼1L

Z L

0Tsdy �

PNi¼1Tsi

Nð2Þ

Here, Tsi is the local temperature measured along the surface ofthe outer tube, and N is the number of thermocouples used formeasuring the wall temperature.

The average Nusselt number is defined as follows:

NuD ¼hmDh

kð3Þ

where Dh is the hydraulic diameter and k is the thermal conductiv-ity of the fluid.

The eccentricity is defined as:

Fig. 1. Experimental setup (schematic diagram).

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E ¼ eRo � Ri

ð4Þ

where e is the distance between center lines of the tubes.In places where free convection flow is considered, the modified

Grashof number is used which is defined as Eq. (5).

Gr�D ¼ GrDDh

Lð5Þ

The Grashof number itself, is defined as:

GrD ¼gbðTs � T0ÞD3

hq2

l2 ð6Þ

For the case of forced convection flow, the Reynolds number isused in place of the Grashof number.

ReD ¼quDh

lð7Þ

The modified Reynolds number is defined as:

Re�D ¼ ReDDh

Lð8Þ

All physical properties of air are read at film temperature, de-scribed as:

Tf ¼Ts þ T0

2ð9Þ

4. Experimental procedure

The experiment setup was placed in a laboratory withcontrolled temperature and humidity. To record each data set,the heaters (main heater and guard heater) were turned onto aspecific power by means of a power supply. Input voltages andcurrents were measured and controlled to provide a steadyinput power. Also the fan was turned on when forced flow wasdesired. After about 2 h the guard heater thermometers werechecked to give the same temperature readings. In case of anydifference, the guard heater power should have been readjustedto a new value, and another 2 h had to pass to reach steady state.This process was done until there was almost no difference be-tween the temperatures of the guard heater thermometers. Inthis case the assumption of adiabatic outer wall would beacceptable, and temperatures, inlet velocity, ambient tempera-ture, and input power were recorded. After another 2 h datarecording was done again to ensure steady state condition isreached.

Fig. 2. Experimental setup (isometric view): top heading (a) and bottom heading (b).

Table 1A list of measured/calculated parameters and their accuracies.

Parameter Device Accuracy Unit

Temperature K-type thermocouples 0.1 �CVelocity Rotary anemometer 0.1 m/sEccentricity Power screw 0.1 mmVoltage Volt meter 1 VoltCurrent Current meter 0.02 APower Multi meter 1 WNusselt number – 13% –

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5. Results and discussion

5.1. Temperature curve

At each specified heat flux, when the steady state condition waspreserved, all temperature readings were recorded. The wall

Fig. 3. Effect of heat flux on annulus wall temperature in

dimensionless temperature versus the dimensionless length ofthe outer tube is illustrated in Fig. 3 for four different heat fluxesfor turbulent forced convection. It is quite clear that the generalshape of temperature distribution is the same for different heatfluxes. The temperature increases with the length up to about5 � Dh from the bottom of the annulus, it remains constant up toabout 10 � Dh, and starts to increase until it reaches a crest pointat about 40 � Dh. The phenomenon can be explained by Fig. 4.

At the entrance (point a), the thickness of the hydrodynamicboundary layer is zero, and starts to increase until it fills the annu-lus. Therefore, the convection heat transfer coefficient decreasesand the wall temperature increases until the flow is fully devel-oped. Since in this region the boundary layer is forming, this regionis called the entrance region. At point b, the temperature is ex-pected to vary along the bc line, but it is not happening, and thetemperature measurement history coincides on the line bc0. Itmay be interpreted that at point b, the transition region begins

turbulent forced convection (ReD � 9000, E = 0.54).

Fig. 4. General shape of temperature curves.

Fig. 5. Effect of eccentricity on annulus wall

Fig. 6. Variation of the Nusselt number (NuD�f0) v

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and this increases the convection heat transfer coefficient, result-ing in drop of the wall temperature. Another interesting part of thiscurve is concerned with the portion beyond the point bc0. In thispart, the wall temperature first increases rapidly and then de-creases. This incident validates the fact that a rotating flow existsin the eccentric annulus as denoted in [7]. The temperature dropafter point d can be attributed to the presence of downflow andrecirculation as stated by Sparrow et al. [16].

5.2. Effect of eccentricity

Fig. 5 illustrates the effect of eccentricity on the annulus surfacetemperature, and the variation of the Nusselt number (NuD.f0) ver-sus Re�D � Pr is illustrated in Fig. 6 in logarithmic scale. It is clearlyobserved that the Nusselt number and hence the convection heattransfer coefficient is increased as the eccentricity increases. Thisvariation can be represented by the following equation:

NuD ¼Af 0

Re�DPr� �n ð10Þ

temperature (ReD � 4500, q00 = 185 W).

ersus ReD�Pr for different eccentricity ratios.

Fig. 7. Curves of empirical constants A and n, versus the eccentricity ratio.

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where A and n are empirical constants, and f0 is a correction factorsuggested by Al-Arabi et al. [2] to consider for the change of phys-ical properties with temperature along the tube radius.

f 0 ¼ 1þ 0:05Ts � To

Toð11Þ

By fitting curves on the empirical constants, A and n, as in Fig. 7,the following relations are derived:

A ¼ 0:034Eþ 0:06 ð12Þn ¼ �0:084Eþ 1:262 ð13Þ

Substituting Eqs. (12) and (13) in Eq. (10), an empirical relationis derived for calculating the Nusselt number as a function of theReynolds number and the eccentricity ratio:

NuD ¼0:034Eþ 0:06

f 0Re�ð�0:084Eþ1:262Þ

D ð14Þ

Eq. (14) is derived for air (Pr � 0.7), and annulus radius ratio ofr* = 0.24, and holds for Reynolds numbers in the range of1036 ReD 6 104 and may be used for all eccentricities of the annu-

lus. Eq. (14) can be used for all eccentricities. This equation showsthat in the eccentric vertical annulus, where turbulent forced con-vection is considered, the Nusselt number increases by increasingthe Reynolds number.

Fig. 8. At each Reynolds number, the Nusselt num

5.3. Effect of heat flux

Experiments were done for four different heat fluxes. The maineffect of heat flux in vertical motion of fluid is to change the fluiddensity and hence the Grashof number which affects buoyancyforces to make the flow. The ratio GrD=Re2

D must be less than 0.1to ensure buoyancy effects are negligible on the flow [17]. For allcases of the forced flow that is presented in this article this ratiois less than 0.04.

Fig. 8 illustrates the Nusselt number versus Re�DPr in logarithmicscale. It is obvious from this figure that at each Reynolds number,the Nusselt numbers are concentrated around a specific point. Thisis because the Nusselt numbers do not have a significant variationby varying the heat flux. Furthermore, this slight variation of theNusselt number can be attributed to the end effects of the annulus.So it is concluded that the heat flux has a negligible effect on theconvection heat transfer coefficient, although increasing the heatflux increases the annulus wall temperature (Fig. 3).

5.4. Effect of Reynolds number

To consider for the effects of Reynolds number on the convec-tion heat transfer coefficient, results were recorded for free convec-tion and laminar flow as well as for turbulent forced flow. Toevaluate the Reynolds number, the velocity was measured at theannulus outlet, and all fluid properties were read at filmtemperature.

The wall temperature distribution along the outer tube for dif-ferent Reynolds numbers is shown in Fig. 9. For zero inlet velocity,where flow is due to free convection, there is no crest observed inthe temperature curve. In the case of free convection the flow islaminar (ReD 6 2300). Temperature curves for free convection inconcentric annulus are quite similar to those reported by Al-Arabiet al. [8]. It is seen in Fig. 10 that in the case of laminar forced con-vection, an optimum eccentricity exists around E = 0.5, which theconvection heat transfer coefficient becomes maximum at thiseccentricity. The same optimum eccentricity is reported in Ref.[14] by Hosseini et al. for free convection in a vertical eccentricannulus. By increasing the Reynolds number, the optimum eccen-tricity is vanished, and also a crest is formed in the temperaturecurves. This crest has not been reported in the literature. The opti-mum eccentricity is also observed in forced convection with lowinlet velocity where flow is laminar. Since the optimum eccentric-

bers are concentrated around a specific point.

Fig. 10. Effect of eccentricity on laminar forced convection (ReD � 1250, q00 = 185 W).

Fig. 9. Effect of Reynolds number on annulus wall temperature (q00 = 220 W, E = 0).

Fig. 11. Result comparison for single tube.

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Fig. 12. Comparison of results of single tube, concentric annulus and eccentric annulus.

R. Hosseini et al. / Energy Conversion and Management 50 (2009) 2266–2274 2273

ity is vanished by increasing the Reynolds number, it can be attrib-uted to the change of the flow regime from laminar in free convec-tion to turbulent in forced convection. The temperature curves fordifferent Reynolds numbers are presented in Fig. 9. When Reynoldsnumber is increased, heat transfer (Nusselt number) is also in-creased and hence, surface temperature is decreased.

As mentioned in Section 5.2, in the eccentric vertical annulus, inthe case of turbulent forced convection, the Nusselt number in-creases by increasing the Reynolds number (Fig. 6). Also the linesof different eccentricities converge as the Reynolds number is in-creased in the plot of Nusselt number versus Re�DPr in logarithmicscale. This means that as the Reynolds number increases, the effectof eccentricity is becoming negligible in the convection heat trans-fer coefficient.

5.5. Comparison with previous work

As stated before, all data was recorded for free convection, lam-inar flow, concentric annulus, and single tube as well as for turbu-lent forced convection in eccentric annulus, for comparisonpurposes and verification of experiments.

The result of free convection heat transfer for a single tubeheated from outside is quite similar to those reported by Al-Arabiet al. [8] and Hosseini et al. [14] (Fig. 11). For this case Al-Arabi andco-workers suggested n = 0.25 and A = 0.11 for 1.44 � 107 < Grl �Pr < 8.85 � 108 and 10 < L/D < 31.4. The present results indicate agood agreement with the results derived by Al-Arabi et al. [8]and Hosseini et al. [14].

In Fig. 12 results for single tube, concentric annulus, and eccen-tric annulus are shown for comparison. From this figure it is obvi-ous that the convection heat transfer coefficient is larger for anannulus in comparison with a single tube, and it is even largerfor an eccentric annulus than a concentric annulus.

6. Summary and conclusion

An experimental study was carried out to investigate free andforced convection in an open-ended vertical annulus. Results arepresented for a fluid of Pr = 0.7 in an annulus radius ratio of 0.24.Different eccentricities, inlet velocities, and heat fluxes have beeninvestigated under the thermal boundary conditions of havingthe annulus outer wall isothermally heated while the inner wallis insulated. Also results were recorded for single tube and concen-tric annulus for comparison purposes.

The analyses of the experimental results indicate that:

(1) The rate of heat transfer is greater for eccentric annulus incomparison with the concentric annulus and single tube.

(2) In an eccentric vertical annulus, when turbulent forced con-vection is considered, by increasing the eccentricity ratio,the rate of heat transfer increases. When the flow is laminaran optimum eccentricity ratio of about 0.5 is observed, bywhich the convection heat transfer coefficient is maximumat this eccentricity.

(3) Increasing heat flux has a negligible effect on the convectionheat transfer coefficient in turbulent forced convection,although increases the wall temperature.

(4) As the Reynolds number is increased, a crest is formed in thetemperature–length plots.

(5) The empirical relations to calculate the heat transfer rate aredeveloped for turbulent forced convection in an eccentricvertical annulus.

(6) As the Reynolds number increases, the effect of eccentricitybecomes negligible in the convection heat transfercoefficient.

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