Friction resistance of adiabatic two-phase flow in narrow rectangular ductunder rolling conditions
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Frictional resistance of adiabatic two-phase flow in narrow rectangular duct under rolling conditionsDianchuan Xing ⇑,Changqi Yan,Licheng Sun,Guangyuan Jin,Sichao TanNational Defense Key Subject Laboratory for Nuclear Safety and Simulation Technology,Harbin Engineering University,Harbin 150001,Chinaa r t i c l e i n f o Article history:Received 8May 2012Received in revised form 28September 2012Accepted 28September 2012Available online 28November 2012Keywords:Rolling motion Two-phase flowFrictional pressure drop Narrow rectangular duct Correlations evaluation New correlationa b s t r a c tFrictional resistance of air-water two-phase flow in a narrow rectangular duct subjected to rolling motion was investigated experimentally.Time-averaged and transient frictional pressure drop under rolling con-dition were compared with conventional correlation in laminar flow region (Re l <800),transition flow region (8006Re l 61400)and turbulent flow region (Re l >1400)respectively.The result shows that,despite no influence on time-averaged frictional resistance,rolling motion does induce periodical fluctu-ation of the pressure drop in laminar and transition flow regions.Transient frictional pressure drop fluc-tuates synchronously with the rolling motion both in laminar and in transition flow region,while it is nearly invariable in turbulent flow region.The fluctuation amplitude of the Relative frictional pressure gradient decreases with the increasing of the superficial velocities.Lee and Lee (2002)correlation and Chisholm (1967)correlation could satisfactorily predict time-averaged frictional pressure drop under rolling conditions,whereas poorly predict the transient frictional pressure drop when it fluctuates peri-odically.A new correlation with better accuracy for predicting the transient frictional pressure drop in rolling motion is achieved by modifying the Chisholm (1967)correlation on the basis of analyzing the present experimental results with a great number of data points.Crown Copyright Ó2012Published by Elsevier Ltd.All rights reserved.1.IntroductionWith the extensive application of nuclear power system in mar-ine transportation,effects of ocean condition (rolling,heaving,pitching,inclination etc.)on a flow and heat transfer system have attracted growing interests in recent years.From a fluid mechanics point of view,the main difference between land-based and barge-mounted equipment is that the latter is inevitable from the effect of sea wave shocks and winds (Ishida et al.,1995).The thermal hydraulic behavior of shipborne equipment is influenced by roll-ing,heaving and pitching motions,leading to the occurrence of un-steady flow as mentioned by Pendyala et al.(2008)and Tan et al.(2009a).A number of previous studies regarding single-phase flow behaviors under ocean condition have been performed in recent years.Studies of Gao et al.(1997),Ishida and Yoritsune (2002),Murata et al.(2002),Tan et al.(2009a,b)and Yan and Yu (2009)indicated that the flow rate of a natural circulation system will oscillate sinusoidally in rolling motion,whereas almost keeps constant for a forced circulation loop.Rolling parameters,flow rates and the component layout in the experimental loop have strong effects on the thermal hydraulic behavior of a naturalcirculation system.Cao et al.(2006),Xing et al.(2012)and Zhang et al.(2009)performed a series of experiments to investigate the effect of rolling parameters,flow rates and tube radius on single-phase forced circulation in pipes.Their results indicated that the frictional pressure drop of single-phase flow oscillates periodically in rolling motion.New empirical correlations for calculating the single-phase friction factor in rolling pipes were achieved from their experimental data.Studies of Pendyala et al.(2008)indicated that heaving movement can lead to the fluctuation of a forced sin-gle-phase flow.The mean friction factor in heaving motion was considered to be greater than that under immobile condition.Yan et al.(2010,2011)gave the velocity distribution of single-phase flow in tubes under rolling condition,and showed that roll-ing motion influences only the velocity distribution near the chan-nel wall but not influence its mean frictional resistance.From afore-mentioned work,it is clear that the single-phase fluid flow in an oscillating pipe is rather different from that in a pipe at rest.However,few related studies deal with two-phase flow character-istics in rolling motion,and the summarizations are listed as follow.Cao et al.(2006)studied the frictional resistance of single-phase and two-phase flow in pipes under rolling condition,and demon-strated that the predicted friction factor by traditional correlations deviates dramatically from the experimental results.They also pro-posed a new correlation according to homogeneous flow model.0306-4549/$-see front matter Crown Copyright Ó2012Published by Elsevier Ltd.All rights reserved./10.1016/j.anucene.2012.09.024Corresponding author.Tel./fax:+8645182569655.E-mail address:spiderxdc@ (D.Xing).The time-averaged frictional resistance of bubblyflow in rolling motion was predicted by their correlation with an accuracy of ±25%.The effects of rolling motion on air-water two-phaseflow pattern transition were investigated experimentally by Luan et al.(2007)and Zhang et al.(2007).Their results showed that the rolling period,rolling amplitude and channel size affect the transitions betweenflow patterns,especially from the bubble to slugflow and churn to annularflow.Yan et al.(2008)measured the volume-averaged void fraction at a certain rolling angle with the help of quick-closing valves method.The result showed that rolling motion results in the decrease of the void fraction,but no correlation for such a condition was achieved.Tan et al.(2009c) experimentally studied the two-phaseflow instability of natural circulation under rolling condition,and their result indicated that rolling motion causes the early occurrence of two-phaseflow instability.The above reviewed researches regarding two-phaseflow behavior in rolling motion were all performed for circular tube, whereas frictional resistance in narrow rectangular duct under rolling condition has not been studied in detail so far.In addition, none of the above work gives the transient frictional pressure drop. With demands for higher heat transfer efficiency and less space requirement in practical applications,rectangular duct is one of the choices as the heat transfer tube in a compact heat exchanger. Therefore,researches on thermal hydraulic characteristics of two-phaseflow in narrow rectangular ducts have been received increasing attention over the last few decades(Lee and Lee, 2002;Ma et al.,2010;Mishima et al.,1993;Sadatomi et al., 1982;Wang et al.,2011a;Zhou and Wang,2011).Most studies on two-phaseflow resistance in narrow ducts are concerning motionless condition,few can be found in rolling motion.Wang et al.(2011b)investigated two-phaseflow patterns under rolling conditions and obtained theflow pattern map for a narrow rectan-gular duct having cross section of40mmÂ1.6mm.Recently, Hong et al.(2012a,b)and Wei et al.(2011)studied the onset of nucleate boiling and the bubble behaviors in subcooledflow boil-ing under ocean condition.According to the authors’knowledge, the frictional resistance of two-phaseflow in rectangular duct un-der rolling condition has not been studied carefully.To better understand the effect of rolling motion on two-phaseflow resis-tance,a series of experiments was performed by using a narrow rectangular duct having cross section of43mmÂ1.41mm.The ef-fects of rolling motion on time-averaged and transient frictional pressure drop were investigated in differentflow regions.2.Experimental apparatus2.1.Description of the rolling platformThe rolling movement of a ship was simulated by a simple har-monic motion.The rolling platform,a2.5mÂ3.5m rectangular plane,rotates with the central shaft(O–O)as shown in Fig.1.Roll-ing movement with required rolling period and amplitude is con-trolled by an automatic system(Wang et al.,2011b;Xing et al., 2012).Consequently,the rolling amplitude could be expressed as follow:h¼h m sinð2p ftÞð1ÞClockwise direction is defined as the positive direction of the rolling movement as shown in Fig.1.Accordingly,the angular velocity and angular acceleration are deduced as follow:x¼2p f h m cosð2p ftÞð2Þb¼À4p2f2h m sinð2p ftÞð3ÞNomenclatureGeneral symbolsf rolling frequency(Hz)T rolling period(s)t time(s)h m rolling amplitude(rad)D P t total pressure drop(kPa)D P f frictional pressure drop(kPa)D P g gravitational pressure drop(kPa)D P add additional pressure drop(kPa)D q density difference between phases(kg/m3)j time-averaged superficial velocity(m/s)g gravity acceleration(m/s2)L length between pressure taps(m)h height of the duct(m)w width of the duct(m)j superficial velocity(m/s)dP f/dz two-phase frictional pressure gradient(kPa/m) D P pressure drop(kPa)d(P f)g/dz gas frictional pressure gradient(kPa/m)d(P f)g/dz liquid frictional pressure gradient(kPa/m)/2 1frictional multiplier factor,Eq.(11)x mass qualityX Martinelli parameter,Eq.(13)U0rolling velocity,Eq.(20)(m/s)l the distance between the test section and the rolling shaft(m)Re Reynolds number(Re=jd e/c)d e hydraulic diameter of the test section(m)Greek lettersh rolling angle(°)x angular velocity(rad/s)b angular acceleration(rad/s2)qfluid density(kg/m3)a void fractione ratio of duct height to width(e=h/w)c kinematic viscosity(m2/s)l dynamic viscous(Pa s)r surface tension(N/m)k single-phase friction factorSubscriptsroll under rolling conditiong gas phaseflows alone through the same pipe with itsmassflow ratel liquid phaseflows alone through the same pipe with its massflow ratel0liquid phaseflows only through the same pipe with to-tal massflow rate1,2start and end pointspred predictionexp experimentSuperscript0relative coordinate110 D.Xing et al./Annals of Nuclear Energy53(2013)109–119rolling frequency and period respectively effects of rolling parameters,five following included(h m is the rolling amplitude and T is 8s;h m10°T12s;h m10°T16s;h m15°T16s;instrumentsof the experimental loop is also shown solid and broken line denotes the water and airflow pipeline respectively.Purifiedplied and introduced into a mixing chamber and air-compressor respectively.Thethrough the test section in which theThe pressure at the inlet of the test section pressure of0.2MPa by a pressure regulator. cates vertically on the rolling platform anding shaft.The two pressure taps,locatedwall as shown in Fig.2,are spaced1.5which(p1)is0.3m from the entrance.Theis vented and therein air and water is separated. the test section is normal to the angularshown in Fig.2.Stainless steelflexible pipeline on rolling platform to that of immobile The test section is fabricated with theand its cross-section is illustrated in Fig.two parts,a substrate on which a channelFig.1.Schematic diagram of the experimental facility.plate,which are made of 10.0mm thick apparent plexiglas.The width and the length of the duct are 43mm and 2000mm,respec-tively.After finishing all experiments,the test section was brea-ched to measure the gap size with a clearance gauge.11measurement locations along flow direction and three locations in transverse direction were set uniformly as shown in Fig.3.The gap size is 1.41mm with an uncertainty of ±0.01mm.Mass flow meters (Promass 83)are used to measure the gas and liquid flow rate.They have adjustable spans in 0–4000kg/h with the uncertainty of 0.1%.Two pressure transducers are used to mea-sure the local pressure (PR35X),with the accuracies of 0.2%and measurement range of 0–250kPa (P 1)and 0–100kPa (P 2),respec-tively.The temperature measurement is performed at the outlet of the test section by standard thermometer with a measured error of ±0.1°C.All the test signals are recorded by NI data acquisition sys-tem (sampling frequency is 256Hz and the uncertainty is ±0.1%)except for the water and air temperatures.The range of the present experiment is as follows.Superficial water velocity,j l 0.16–3.73m/s Superficial air velocity,j g0.58–31.44m/s Liquid Reynolds number,Re l 543–13,250Gas Reynolds number,Re g 221–83103.Data processingThe total pressure drop in rolling motion (D P t )for adiabatic two-phase flow consists of frictional pressure drop (D P f ),gravita-tional pressure drop (D P g )and additional pressure drop (D P add ):D p t ¼D p g þD p f þD p addð4ÞIn which the gravitational pressure drop could be calculated from:D p g ¼q gLcos hð5Þwhere g is the gravity acceleration;q is the fluid density,which can be expressed as:q ¼q g a þq l ð1Àa Þð6Þa denotes the area-averaged void fraction for which Jones and Zuber(1979)proposed drift flux correlation:j g =a ¼C 0j þð0:23þ0:13h =w ÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiD q gw =q lq ð7Þwhere the distribution parameter C 0could be calculated by the cor-relation of Ishii (1977):C 0¼1:35À0:35ffiffiffiffiffiffiffiffiffiffiffiffiq g =q lq ð8ÞTable 1Time-averaged additional and gravitational pressure drop.Rolling amplitude (°)Rolling period (s)Time-averagedadditional pressure drop (kPa)Time-averagedgravitational pressure drop (kPa)1080.017814.58710120.007914.58810160.004414.58815160.009914.44930160.039713.708For the bubbly flow,Eq.(7)is not valid for rectangular duct.the drift velocity is calculated by Ishii (1977):=a ¼C 0j þffiffiffi2p r g D q q 2l0:25ð1Àa Þ1:75ðIn Eqs.(7)–(9),j is the superficial velocity;h and w denote the height and width of the test section,respectively as shown in Figs.and 3;q g ,q l and D q represent the densities of gas and liquid well as their difference;r is the surface tension.Gao et al.(1997)analyzed the forces acting on coolant under rolling condition,proved the existence of additional pressure drop and gave its theoretical expression:P add ¼q ðz 022Àz 021Þ2Á4p 2h 2mT 2cos 22p T t þq y 01ðz 02Àz 01ÞÁ4p 2h m T2sin 2p T t ð10z 01and y 01denote the coordinates in the relative coordinate sys-tem (o 0x 0y 0z 0)fixed on the rolling platform as shown in Fig.2.The first and the second term on the right-hand side of Eq.(10)repre-sent the additional pressure drop caused by centrifugal inertial force and tangential inertial force respectively (Xing et al.,2012).Lockhart and Martinelli (1949)had put first forward correlation for calculating the two-phase frictional pressure gradient by introducing a Martinelli parameter X to the two-phase multiplier /2l :¼dP fdzd ðP f Þl dzð11Chisholm (1967)gave the following correlation to calculate /¼1þC X þ1X2ð12The parameter C depends on the flow regimes of the liquid and gas phases,ranging from 5to 20,and the Martinelli parameter X defined by¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffid ðP f Þl dz d ðP f Þgdzs ð13where d (P f )g /dz and d (P f )l /dz represent the gas and liquid frictionalpressure gradient as each phase flows alone through the same pipe with its mass flow rate,respectively.In present experiments,the gas and water flow rate fluctuate slightly,so traditional correlations are applied to calculate the single-phase flow friction factor Fig.5.Experimental results of two-phase flow in the immobile pipe.Fig.6.Time-averaged frictional resistance under rolling condition:(a)laminar flow region;(b)transition flow region;(c)turbulent flow region.where e is the ratio of duct height to duct width.For Re P2500,cor-relation of Sadatomi et al.(1982)was used:k¼0:3164Á½ð0:0154C V=64À0:012Þ1=3þ0:85 ÁReÀ0:25ð15Þwhere C V=k Re is given by Eq.(14).The experimental uncertainties of/2land X under non-rolling condition are±14.5%and±13.56% respectively based on Eqs.(11)and(13).4.Results and discussion4.1.Pressure drop component in rolling motionThe transient frictional and additional pressure drop of two-phaseflow oscillates with the same period of rolling motion, whereas the gravitational pressure dropfluctuates with the half rolling period,as shown in Fig.4.It is indicated that thefluctuation amplitude of additional pressure drop is1–2orders of magnitude smaller than that of gravitational pressure drop.As the mass qual-ity increases,gravitational and additional pressure drop decreases as the result of decreasing two-phase density as shown in Eqs.(5) and(10).Meanwhile,much larger frictional loss is resulted as shown in Fig.4.Therefore the proportion of gravitational and addi-tional pressure drop to the total pressure drop decreases sharply as the mass quality increases.Table1summarizes time-averaged gravitational and additional pressure drop in rolling motion by integrating Eqs.(5)and(10) respectively,assuming that the density is1000kg/m3.It is shown from Table1that for the case of larger rolling amplitude and smal-ler rolling period,rolling motion gives rise to the increasing of time-averaged additional pressure drop.In addition,time-aver-aged gravitational pressure drop only increases as the rolling amplitude decreases,which is nearly independent of the rolling period.The time-averaged additional pressure drop is2–3orders of magnitude smaller than the gravitational pressure drop,much smaller than frictional pressure drop,so the effect of time-aver-aged rolling inertial force onflow resistance could be neglected.4.2.Time-averaged frictional resistance in rolling motionThe experimental/2lvarying with X in vertical non-rolling con-dition is plotted in Fig.5,in which threeflow regions are exhibited apparently against the liquid Reynolds number(It is defined as the Reynolds number assuming the liquid phaseflows alone in the same pipe with its massflow rate,Re l=j l d e/c l),namely,laminar flow region(Re l<800),transitionflow region(8006Re l61400) and turbulentflow region(Re l>1400).Wang et al.(2011a)and Zhou and Wang(2011)obtained the similar characteristics of fric-tional resistance in mini/micro rectangular ducts.Therefore,the method is used to classify the experimental data in present study. Chisholm parameter C strongly depends on the liquid and gas phaseflow condition and the present experimental data in steady condition varies in the range of(C=5–20)as shown in Fig.5.The time-averaged two-phase frictional multiplier changing with X in rolling motion is also divided into three regions as illus-trated in Fig.6.In eachflow region,the rolling period and ampli-tude nearly have no influence on time-averaged frictional resistance.Section4.1have concluded the neglectable time-aver-aged rolling inertial force,additionally,the time-averagedflow rate and frictional pressure drop under rolling condition are very close to that under non-rolling condition.Therefore,we can conclude that the effect of rolling motion in present condition nearly has no influence on time-averaged frictional resistance of adiabatic two-phaseflow.Similar results have been achieved concerning sin-gle-phaseflow resistance in rolling motion as shown in recent lit-eratures of Xing et al.(2012)and Yan et al.(2010,2011).Lee and Lee(2002)presented a correlation to calculate air–water two-phaseflow frictional pressure drop in narrow rectangu-lar duct.The gap size ranges from0.4to4mm while the width beingfixed to20mm.The corresponding aspect ratio ranges from 0.02to0.2,in which the present aspect ratio is included.The Chis-holm parameter C in their correlation could be expressed as:C¼Al2lqlr d eq lljrrRe sl0ð16ÞThe constant A and exponent(q,r and s)are determined by experimental data with the corresponding value referred to Lee and Lee(2002),Mishima et al.(1993)conducted experiments to study theflow regime,void fraction,bubble velocity and the pres-sure loss in narrow rectangular duct.Their results indicated that the Chisholm parameter C depends on the hydraulic diameter, decreasing from21to0as the hydraulic diameter decreases from 10to0.1mm.The also proposed a correlation both for round tubes and rectangular ducts using an appropriate hydraulic diameter, which could be expressed as:C¼21½1ÀexpðÀ0:27d eÞ ð17ÞKandlika(2002)differentiated the conventional channel,mini-channel and microchannel associated with the hydraulic diameter. The channels employing hydraulic diameter ranging from200l m to3mm are referred as minichannels.When the gap width of nar-row channel is less than3mm,it has been shown that effect of sur-face tension onflow and heat transfer characteristics is very significant.Therefore the critical duct height differentiated narrow channel and conventional channel is considered to be3mm in some literatures(Huang et al.,2009;Xu et al.,2008).In present study,the test section is thought to be narrow or mini rectangular duct from the above discussion.Table2shows the comparison of the time-averaged frictional pressure drop with predictions by Lee and Lee(2002)correlation, Chisholm(1967)correlation and Mishima et al.(1993)correlation. The Mean Absolute Error(MAE)is defined as:MAE¼1nX j D P predÀD P exp jÂ100D P expð18Þwhere n is the number of the data points.Table2Evaluations of some conventional correlations with the time-averaged frictional pressure drop.Correlations Conditions Vertical h m10°T8s h m10°T12s h m10°T16s h m15°T16s h m30°T16s Data number171163153148151122Lee and Lee(2002)MAE(%)16.9717.1115.9018.1117.6616.41 Proportion(%)90.0687.7397.3988.5186.0986.89 Chisholm(1967)MAE(%)18.7412.4913.0712.8614.5613.31 Proportion(%)80.1295.7192.8191.8990.0794.26 Mishima et al.(1993)MAE(%)24.1419.2916.5817.3619.4118.37 Proportion(%)70.1980.9883.0181.0876.1681.15114 D.Xing et al./Annals of Nuclear Energy53(2013)109–119Fig.7.Transient frictional pressure gradient in rolling motion:(a)laminarflow region;(b)transitionflow region;(c)turbulentflow region.D P pred and D P exp are the predicted frictional pressure drop and the experimental results,respectively.The experimental results under verticalflow condition agree favorably with the Lee and Lee(2002)correlation and Chisholm(1967)correlation,which im-ply the accuracy and reliability of our experimental data.It can be 4.3.Transient frictional pressure gradient in rolling motionThe transient frictional pressure gradientfluctuates with the same period of rolling motion both in laminar and in transition flow region,whereas,it is nearly invariable in turbulentflow re-parison of the predicted C with the experimental results:(a)laminarflow,Re g62000;(b)laminarflow,Re g>2000;(c)transitionflow,Re g62000;(d)transition flow,Re g>2000.D.Xing et al./Annals of Nuclear Energy53(2013)109–119117influence onfluctuation amplitude of frictional pressure gradient than that of superficial liquid velocity.Fig.7also indicated that the phase lag between the rolling motion and transient frictional pressure gradient could be neglected.Fig.8presents the comparison of experimental transient fric-tional pressure gradient with Lee and Lee(2002)correlation and Chisholm(1967)correlation in rolling motion.It can be seen from Fig.8that the both correlations poorly predict the transient fric-tional pressure gradient under rolling condition in laminar and transitionflow regions because in these regions frictional pressure gradientfluctuates strongly with the same period of rolling mo-tion.The parameter C of Chisholm(1967)correlation is determined by the gas and liquid phaseflow conditions,varying discretely in5, 10,12and20.If the gas and liquid phaseflow with the Reynolds numberfluctuating around1000,the predicted transient frictional pressure gradient changes abruptly and deviates dramatically from the experiments,as shown in Fig.8b.In turbulentflow region,the frictional pressure gradient oscillates randomly,so the Lee and Lee (2002)correlation and Chisholm(1967)correlation work better as shown in Fig.8c.Lee and Lee(2002)correlation and Chisholm (1967)correlation are developed based on experimental data un-der non-rolling condition.It works for the time-averaged condition as shown in Table2,but not for transient pressure gradient which oscillates periodically.4.4.New correlation for transient frictional resistance in rolling motionUp to now,no correlation is suitable for calculating transient frictional pressure drop of two-phaseflow in rolling motion,so new correlation is required for engineering application.Murata et al.(2002),Xing et al.(2012)and Zhang et al.(2009)took into ac-count of the influence of rolling motion by adopted the rolling Rey-nolds number,which denoted a ratio between the inertial force caused by rolling motion and viscous force,and could be expressed as:Re roll¼U h l=cð19Þwhere c represents the kinematic viscosity.The rolling velocity U h is the scale of the rolling motion,and is described as:U h¼4h m l=Tð20Þwhere l is the distance between the test section and the rolling shaft (l¼y01).Based on Chisholm(1967)correlation,we modified the parameter C,which is affected by many factors and could be expressed:C¼fðb l2= j2;x2ld e= j2;Re roll;Re g;Re l;X;xÞð21Þwhere the dimensionless parameters of b l2= j2and x2ld e= j2denote the ratio of tangential and the centrifugal inertial force caused by rolling motion toflow inertial force respectively.Thefirst three groups represent the effect of rolling motion on frictional resis-tance.As discussed in study of Xing et al.(2012),the effect of cen-trifugal inertial force is much slighter than the tangential one,so the effect of x2ld e= j2is neglected in the new correlation.By analysis of the experimental data,the parameter C could be expressed as:C¼k0þk1Xþk2ðb l2= j2Þð22Þwhere j is the time-averaged superficial velocity,and the coefficient k i(i=0,1,2)in Eq.(22)could be written as the function of time-averaged Reynolds number for gas and liquid,rolling Reynolds number and mass quality:k i¼mRe arollðRe g=Re fÞb x dð23ÞConstant m and exponents a,b and d are determined by gas and liquid phaseflow conditions,and given in Table3by multiple regression against a large number of experimental data in laminar and transitionflow region.The comparisons of the proposed correlation and the experi-mental results are plotted in Fig.9,showing a good agreement. The proposed correlation predicts the periodical change of the transient frictional pressure drop much better than Lee and Lee (2002)correlation and Chisholm(1967)correlation,because that it takes into account of the rolling effect on frictional resistance by introducing dimensionless parameters(such as the rolling Rey-nolds number and b l2= j2).The modified correlation is validated over theflow and rolling parameter ranges:X varying from0.24to2.98;rolling period rang-ing from8s to16s and rolling amplitude ranging from10°to30°. The correlation is developed from a single duct.More data sets are required to extend the application of the proposed correlation. 5.ConclusionIn the present paper,frictional resistance of adiabatic two-phaseflow in a narrow rectangular duct under rolling condition is investigated parisons are made between conventional correlations and the experimental data.Major con-clusions of this study are summarized as follows:Frictional resistances of two-phaseflow under rolling condition are divided into three regions according to liquid Reynolds num-ber,similar with that in vertical non-rolling condition.The time-averaged additional pressure drop is so small compared with fric-tional pressure drop that the effect of rolling motion on time-aver-aged frictional resistance is neglectable.Different from characteristics of time-averaged frictional resis-tance,transient frictional pressure dropfluctuates synchronously with rolling motion both in laminar and transitionflow region. Thefluctuation amplitude of the Relative Frictional Pressure Gradi-ent decreases as the gas and liquid velocities increase.For the case of turbulentflow,transient frictional pressure drop does not oscil-late periodically.Although Lee and Lee(2002)correlation and Chisholm(1967) correlation are suitable for time-averaged frictional pressure drop under rolling condition,they cannot predict the periodical change of the transient frictional pressure drop.Finally,a new correlation applied to calculate the transient frictional pressure drop in narrow rectangular duct under rolling condition is achieved by modifying the factor of C in Chisholm(1967)correlation.The proposed corre-lation takes into account the influence of rolling motion andflow conditions,expressed as dimensionless groups,and shows good agreement with the experimental data.AcknowledgementThe authors are profoundly grateful to thefinancial supports of the National Natural Science Foundation of China(Grant Nos.: 51076034and11175050).ReferencesCao,X.X.,Yan,C.Q.,Gao,P.Z.,et al.,2006.Pressure drop correlations of single-phase and two-phaseflow in rolling rubes.In:Proceedings of the14th International Conference on Nuclear Engineering(ICONE14-89237),Miami,Florida,USA. Chisholm,D.,1967.A theoretical basis for the Lockhart-Martinelli correlation for two-phaseflow.Int.J.Heat Mass Transfer10,1767–1778.Gao,P.Z.,Pang,F.G.,Wang,Z.X.,1997.Mathematical model of primary coolant in nuclear power plant influenced by ocean condition.J.Harbin Eng.Univ.18,24–27(in Chinese).Hong,G.,Yan,X.,Yang,Y.H.,et al.,2012a.Experimental study on onset of nucleate boiling in narrow rectangular channel under static and heaving conditions.Ann.Nucl.Energy39,26–34.Hong,G.,Yan,X.,Yang,Y.H.,et al.,2012b.Experimental research of bubble characteristics in narrow rectangular channel under heaving motion.Int.J.Therm.Sci.51,42–50.118 D.Xing et al./Annals of Nuclear Energy53(2013)109–119。
