Mass transfer characteristics of citric acid extraction by hollow fiber renewal liquid membrane

OZONE MASS TRANSFER IN WATER AND WASTEWATER TREATMENT:EXPERIMENTAL OBSERVATIONS USING A2D LASER PARTICLE DYNAMICS ANALYZERHONGDE ZHOU 1*and DANIEL W.SMITH 2M1School of Engineering,University of Guelph,Guelph,Ontario,Canada N1G 2W1and 2Department of Civil and Environmental Engineering,University of Alberta,Edmonton,Alta.,Canada T6G 2G7(First received 1October 1998;accepted in revised form 1April 1999)Abstract ÐOzone mass transfer in water and wastewater was studied by using a 2D laser particle dynamics analyzer (PDA)to simultaneously measure bubble size distribution and mixing intensity.Three types of water samples,including deionized water,tap water and pulp mill e uents were tested to demonstrate the enhancement of ozone absorption with the presence of chemical reactions.Bubble size distribution and apparent mass transfer rates were used to estimate individual mass transfer parameters and examine the e ects of operating conditions and selected gas spargers.It was shown that gas bubbles in ozonation systems varied over a broad range of sizes and their distributions usually deviated from a simple normal distribution.Among the factors examined,the gas sparger and water types were identi®ed as the most important factors a ecting mean bubble size,whereas the impacts of ozone dose and gas ¯ow rate on bubble size distribution were minor.Due to the higher gas holdup,however,the speci®c surface area was strongly dependent on the gas ¯ow rate as well.Consequently,the mass transfer coe cient increased as the gas ¯ow rate increased.A further enhancement of mass transfer was observed for pulp mill wastewaters as compared to tap and deionized water samples,because the occurrence of chemical reactions depleted the dissolved ozone within the gas±liquid ®lm.#2000Elsevier Science Ltd.All rights reservedKey words Ðozone,ozonation,water treatment,pulp mill e uent,mass transfer,bubble size distri-bution,laser particle dynamics analyzerINTRODUCTIONIn water and wastewater treatment,ozonation is usually practiced by dissolving gaseous ozone into water in order to react with target contaminants or inactivate pathogenic microorganisms.Thus,an accurate quanti®cation of ozone mass transfer into water and a better understanding of associated mechanisms become critical for the proper design and operation of process facilities.Much of the ozone mass transfer knowledge has been derived from chemical engineering principles.Based on its low solubility,ozone mass transfer has been recognized to be controlled within the liquid ®lm immediately adjacent to gas±liquid interface.Consequently,the overall mass transfer coe cient can be approximated by the local liquid mass trans-fer coe cient (Danckwerts,1970).The overall rate of ozone mass transfer will be a ected by di erent operating conditions,water quality and facility setup.Grasso and Weber (1989)measured liquid-side overall mass transfer coe cient k L a in a gas-sparged,completely mixed contactor.A mathemat-ical model was formulated to correlate it to the impeller Reynolds number,power input and gas ¯ow rates.Rakness et al .(1988)derived mass trans-fer correlations at four full-scale drinking water ozonation treatment facilities by empirically cali-brating their model to measured absorption e nglais et al .(1991)recommended several power law type correlations of k L a values to the super®cial gas velocity for the practical design and operation of ®ne bubble ozone contactors in drink-ing water treatment.The applicability of these empirical correlations was experimentally supported by Le Sauze et al .(1993);Roustan et al .(1987);Rakness et al .(1988)and Zhou et al .(1994),although the constants to de®ne these correlations may vary from one speci®c ozone±water contacting system to another.The application of these corre-lations,along with ozone decay kinetics,has been recently demonstrated by incorporating them into various hydrodynamic models in an attempt to pre-dict the performance of ozonation processes (Marin as et al .,1993;Zhou et al .,1994;Zhou et al .,1995).It has also been recognized that the occurrence of rapid chemical reactions may enhance the massWat.Res.Vol.34,No.3,pp.909±921,2000#2000Elsevier Science Ltd.All rights reservedPrinted in Great Britain0043-1354/00/$-see front matter909/locate/watresPII:S0043-1354(99)00196-7*Author to whom all correspondence should be adressed.transfer from gas phase into liquid phase (Danckwerts,1970).Such phenomena have become evident recently during the ozonation of waste-waters as the competition reactions become moresigni®cant (Beltran et al .,1995;Zhou and Smith,1997).For the purpose of simplicity,Zhou and Smith (1997)proposed two general kinetic schemes for ozone absorption based on the relative rates of ozone physical absorption to those of chemical reactions.As shown in Fig.1,slow kinetic regimes occur when the ozone chemical reactions result in only lowering the concentration of dissolved ozone in bulk liquid,thereby,increasing the driving force for ozone mass transfer from gas phase into liquid phase.As the rate of chemical reactions becomes faster,the ozone mass transfer may shift to the fast or instantaneous kinetic regime.Unlike the slow kinetic regime,dissolved ozone is completely depleted within the liquid ®lm beside the gas±liquid interface.The apparent rate of ozone mass transfer may even exceed the maximum rate of physical gas±liquid mass transfer because of steeper dis-solved ozone concentration pro®les.Consequently,the occurrence of chemical reactions will further enhance the local mass transfer ``k L ''.To quantify this enhancement,an enhancement factor ``E ''was de®ned as the actual rate of mass transfer with the occurrence of chemical reactions divided by the maximum rate of physical absorption without any chemical reaction.However,due to the intrinsic interrelationship of mass transfer and chemical enhancement,only the lumped mass transfer parameter Ek L a was measured for process simulation.To better under-stand ozone mass transfer mechanisms and provide the theoretical basis for the development of more e cient ozone contactors,it is necessary to measure individual mass transfer parameters and elucidate their e ects on ozone absorption.Thus,the objec-tive of this study was to use 2D laser particle dynamics analyzer (PDA)to simultaneously measure bubble size distribution and mixing inten-sity during the course of ozonation.An additional objective was to compare the performance of com-monly used gas di users in terms of bubble speci®c surface area and the mass transfer coe cient.The e ects of operating conditions were then examined by varying gas ¯ow rate,ozone dose and hydraulic detention time.Three types of water samples includ-ing deionized water,tap water and pulp mill e u-ents were tested to demonstrate the enhancement of mass transfer due to the occurrence of chemical reactions.METHODSThree di erent types of experiments were conducted in order to estimate individual mass transfer parameters.First,ozonation tests were conducted in a semi-batch con-tactor to measure the lump mass transfer parameter according to the procedure described previously (Zhou and Smith,1997).Then,test samples were preozonated at a series of selected consumed ozone doses.The prepared samples were aerated by oxygen gas through each selected gas di user to calculate the physical mass transfer coe -cients.Finally,bubble size distribution tests were con-ducted by using a PDA over a range of operating conditions commonly encountered in ozonation practice.From these results,the speci®c surface area ``a '',local mass transfer coe cient ``k L ''and the enhancement factor ``E ''were calculated.Figure 2shows a simpli®ed schematic of the test appar-atus described in previous studies (Zhou and Smith,1997).It consists of a plexiglas column with an inside diameter of 100mm and a height of 1750mm.The ozonized gas was generated using a PCI laboratory ozone generator (model C2P-9C-4,PCI Ozone and Control Systems,NJ)and then introduced by either one of four porous di users or a venturi-type injector commonly used in ozonation systems (see Table 1).In later case,both ozone gas was intaken by ¯owing water to generate ®ne bubbles in con-tinuous mode.In the ozonation tests,two solid PVC plates (10mm thick each)were inserted face-to-face inside thecontactorFig.1.Ozone mass transfer schemes in water and wastewater treatment.Hongde Zhou and Daniel W.Smith910(see Fig.2).They were sealed by compressing a silicone O-ring on the contactor wall at 550mm from the bottom to provide the e ective contactor volume of 4L with a head-space around 0.3L.A 8-mm hole was drilled through the plate to introduce the o -gas to a low concentration ozone gas monitor (model LC,PCI Ozone and Control Systems,NJ).The inlet ozone concentration was moni-tored by using a high concentration PCI ozone gas moni-tor monitor (model HC,PCI Ozone and Control Systems,NJ).Gas ¯ow rates,inlet ozone concentrations and test water types were varied over a broad range to determine their e ects on the ozone mass transfer rate.The recorded data were then used to calculate the overall rate of ozone mass transfer with the occurrence of chemical reactions (Ek L a )according to the procedure described previously (Zhou and Smith,1997).Main assumptions underlying such calculations include (1)near perfect mixing in liquid phase,(2)plug ¯ow in gas phase,(3)negligible decay of gaseous ozone and (4)Henry's law applies.These assump-tions can be considered valid due to the low solubility of ozone gas in water,low ratio of liquid height to diameter (H :D =5.1:1.0),and vigorous mixing induced by gas ¯ow.The local mass transfer coe cients (k L a )were measured according to standard procedure developed by ASCE (1992)using the same experimental setup.Initially,4L of each prepared sample was pumped into the reactor.Nitrogen gas was introduced to reduce the concentration of dissolved oxygen less than 0.5mg/L.Immediately after the termination of nitrogen gas,pure oxygen gas was introduced from a gas cylinder.During the oxygenation phase,the dissolved oxygen was monitored by three oxy-gen meters (model 50B,YSI)located at equal intervals of elevation.A data acquisition system recorded the measured dissolved oxygen on a six-channel Lakewood 1data logger (model CP-XA,Lakewood Systems,Edmonton,Canada)after the signals were ampli®ed and later downloaded to a dedicated computer.An examin-ation of recorded data showed that three oxygen probesconsistently generated almost identical readings,with a relative error less than 5%.As a result,only the average dissolved oxygen values were used for further data analy-sis.The collected data were then analyzed to calculate the k L a values by using the nonlinear least square regression method.In doing so,the dissolved oxygen was used as a response against the oxygenation time according to follow-ing equation:C L C ÃL À C ÃL ÀC L ,0 exp Àk L at ,1where:C L Ãis the saturated concentration of oxygen inwater (mg/L),C L the concentration of dissolved oxygen in the bulk liquid phase at time t (mg/L),C L,0the initial con-centration of dissolved oxygen (mg/L),k L a the liquid mass transfer coe cient (s À1)and t the oxygenation time (s).The measured k L a values for oxygen were then cor-rected for ozone according to Danckwert's (1970)surface renewal theory:k L a O 2k L a O 3&D O 2D O 3'0X 5,2where D O 2is the di usivity of oxygen in water,(=2.09Â10À5cm 2/s)and D O 3is the di usivity of ozone in water (=1.74Â10À5cm 2/s).The validation of this relationship has been con®rmed previously for di erent gases (Sherwood et al .,1975)and for ozone in water (Zhou et al .,1995).In measuring the bubble size distribution,the ozone contactor was wrapped outside by a square box made of clear acrylic organic plastics.The box was ®lled with deio-nized water so that the laser beams could project on a ¯at surface.The elevation of the box was then adjusted to determine the e ects of contactor height on the bubble size distribution.The data were collected following the achievement of steady-state conditions and then processedTable 1.Gas di users and spargers used in the studyDi usersDimension (b ,in mm)Pore size (m m)Fused crystalline aluminia di using stone 15H 25Fitted glass discs medium 2010±15coarse2040±60extra coarse 20170±220Gas injectorVenturi-typeFig.2.Experimental set-up.Ozone mass transfer in water and wastewater treatment 911for at least 1500validated gas bubbles using dedicated software (Dantec Sizeware 1,version 2).In this way,the size distribution of gas bubbles and their mean diameters based on count,surface area,volume and volume-to-sur-face area,were calculated.The basic principle of the 2D particle dynamics analyzer (Dantec model 55X)is based on the fact that the diameter of spherical particles can be linearly correlated to the phase di erences detected from di erent locations at di erent scattering angles.Figure 3presents an experimen-tal setup of 2D particle dynamics analyzer.It consists of a 300-mW Ar-ion laser generator (model 5500A-00),digital multimeter (model 8050A),and a modular LDA transmit-ting and receiving optical system (model 55X),PDA signal processor (model 58N10).The laser generator was oper-ated at 514.5nm,and the focal length of both front and receiving lens was 600mm.The laser beams were focused at about 1/3distance between the center and internal wall of the column.The receiving optics were set at a 708scat-tering angle from a true forward scatter mode.To allow measurement of reversing ¯ows,the frequency of one beam was shifted by 40MHz via a Bragg cell.Prior to measurement,the instrument was diagnosed by adjusting the optical alignment,receiving angle,beam spacing and bandwidth to obtain the high fringe counts and signal-to-noise levels as evidenced by the high percentage of vali-dated samples.Table 2summarizes the main experimental conditions for the bubble size measurements.The samples were pre-pared by ozonating 12L of raw wastewater at a gas ¯ow rate of 1500mL/min and an inlet ozone concentration of around 2.5wt%.In total,®ve di erent ozone doses were applied,with a range from 0to 150mg/L.The preozo-nated samples were then collected into 16-L plastic con-tainers and stored in the 48C cool room.The bubble size distribution measurements were made at ®ve di erent gas ¯ow rates,ranging from 500to 2000mL/min.The test waters included tap water,deionized water and three pulp mill e uents to represent both ``clean''andcontaminated waters.The pulp mill e uents were sampled from a kraft pulp mill at three di erent locations:the inlet,midpoint and outlet of its two-cell aerated lagoon treatment system.Most analyses were conducted according to Standard Methods (APHA±AWWA±WPCF,1989).Speci®cally,COD was analyzed using potassium dichro-mate as the oxidant in sulfuric acid with silver ions as the catalyst.AOX was measured using an AOX analyzer (model ECS 1000,Euroglas BV,The Netherlands)accord-ing to DIN 38409H14procedure recommended by the manufacturer.Total organic carbon (TOC)was analyzed by using a Dohrmann carbon analyzer (model DC-80).Color was measured according to the Method H5.P (Canadian Pulp and Paper Association,1974).Table 3summarizes the main characteristics of the test samples.RESULTS AND DISCUSSIONBubble size distribution and gas holdupFigure 4presents typical plots of bubble size dis-tribution in terms of count and volume,respect-ively.The results were obtained for tap water with the fused crystalline aluminia di using stone as the gas di user at a gas ¯ow rate of 500mL.However,similar plots were obtained for other test waters under other operating conditions.As shown,the PDA was able to measure the gas bubble size ran-ging from a few microns up to 4mm,with an accu-racy of better than 3%(data bot shown).In addition,both count and volume based size distri-bution plots revealed serious skews to the normal distribution reported previously (Roustan et al .,1996).However,they di er in that the count based size distribution was skewed to the side ofsmallFig.3.PDA system.Hongde Zhou and Daniel W.Smith912bubble sizes while the volume based size distri-bution to that of large bubble sizes.This re¯ects the fact that each of the larger bubbles contribute more gas volume as compared to small gas bubbles.Di erent approaches have been used to express mean bubble size.Those commonly used include:.Count mean diameter:d c 1NN i d i3.Surface area mean diameter:d s 1NN i d 2ir 4.Volume mean diameter:d v1N N i d 3i3r 5.Volume-to-surface area mean diameter:d vsN i d 3iN i d 2i,6where:N is the total number of gas bubbles per volume (counts/m 3),N i the number of gas bubbles having diameter d i per volume (counts/m 3)and d i the individual bubble diameter (m).Figure 5illustrates a comparison between various mean diameters as a function of gas ¯ow rates.In general,the mean bubble diameters vary in an ascending order from d c ,d s ,d v to d vs ,corresponding to their di erent dependence on larger bubbles as indicated by Eqs (3±6).Nevertheless,all the mean diameters varied slightly with the gas ¯ow rates.Similar observations were reported by Shah et al .(1982)for various ®ne bubble reactor systems.They suggested that the bubble diameter was only slightly dependent on the gas velocity.Roustan et al .(1996)studied the hydrodynamics and mass transfer phenomenon during ozonation.They observed the variation of volume-to-surface area to the super®cial gas velocity by a power of as low as 0.0751.As the gas ¯ow rate increased,however,the count mean diameter decreased whereas the volume-to-surface area mean diameter increased,indicating that larger bubbles were formed at higher gas ¯ow rate.Langlais et al .(1991)suggested that ozonation systems should be designed to generate the gas bubbles with a mean diameter around 1to 3mm,as observed in this study.However,Roustan et al .(1996)used a photographic technique to measure the bubble size distribution and gas holdup in a pilot ®ne bubble ozone contactor.The authors found that in the dispersed bubble regime,the bubble size distribution varied from 1.75to 5.25mm with the volume-to-surface area mean di-ameters ranging from 3.4to almost 4.0mm.These larger bubble sizes may be partially attributedtoFig.4.Typical plots of gas bubble distributions using a25-mm crystalline alumina di using stone (pore size H 25m m).(a)Count distribution (%)and (b)volume distri-bution (%).Table 2.Main experimental conditions for bubble size measurementsParameterSemi-batch a Continuous Reactor column dimension (mm)b 100Â1750b 100Â1750Water heightEk L a tests (mm)510N/A b k L a tests (mm)510N/A b Bubble size tests (mm)1530±17351530±1735E ective water volume (L)412±13.6Water ¯owrate (mL/min)N/A b1000±3000Laser beam height (mm from bottom)480,840,1250Gas ¯owrate (mL/min)500±2000Consumed ozone dose (mg/L)0±134Temperature (8C)2122a A solid PVC plate was inserted inside the column at 550mm from the bottom to reduce the headspace during Ek L a measurement tests.bN/A means not applicable.Ozone mass transfer in water and wastewater treatment913the di erence in operating conditions,experimental apparatuses and measurement techniques.Shah et al .(1982)noted that the gas spargers,the speci®c gas±liquid system and its properties with respect to coalescence would all a ect bubble size.In addition,the original bubble size distribution may di er markedly between di erent measurement tech-niques.This is particularly true for the photo-graphic technique because it can only detect the bubbles near the contactor wall.In contrast,the PDA technique can measure bubble size distri-bution at any speci®c location rapidly and with high accuracy.Although the relative bubble size distribution can be measured accurately by the PDA technique,its ability to measure gas bubble concentration was much less satisfactory,with a typical error around 30%.To get around this,gas bubble concen-trations were calculated based on the gas holdup from the liquid level increase after the gas was introduced.The liquid levels were read directly from an attached tape measure with a typical error less than 7%.Ne g p 6H dn i d 3in i I e 1a 37N i N n i n i,8where E g is the gas holdup (m 3/m 3),n i the countedgas bubbles having diameter d i .From the resultant bubble concentrations,the speci®c surface area for ozone mass transfer was thus calculated asaN i p d 2i X9Figures 6±8show the e ects of various operating conditions on the count mean bubble diameter and speci®c surface area as a function of gas ¯ow rates.Although the mean bubble sizes only slightly decreased as the gas ¯ow rates increased,the speci®c surface areas increased almost proportion-ally with gas ¯ow rates,suggesting that the gas ¯ow rate was the most important factor that would a ect the ozone mass transfer.The phenomenon can be attributed to the substantial increase in gas holdup as discussed later.Figure 6shows the e ects of gas spargers on the mean bubble size and speci®c surface area as observed using di erent gas di users.The fused crystalline aluminia di using stone resulted in the lower count mean diameter and higher speci®c sur-face area as compared to porous glass di using discs.Among three di using discs,the mean bubble diameter and speci®c surface area decreased as the di user pore size decreased,illustrating the import-ance in selecting a proper gas di user for e cient ozone dissolution.However,it should be noted that any small pore di user will have higher risk of being clogged by the particles in water andwaste-parison of mean diameters for ozonation systems.Table 3.Selected characteristics of test pulp mill wastewatersParameters In¯uent Midpoint E uent COD (mg/L)954639609BOD 5(mg/L)2102816AOX (mg/L)13.68.88.1Color (TCU)100011001050pH7.77.857.62Hongde Zhou and Daniel W.Smith914Fig.6.E ects of gas di users on mean bubble diameter and speci®c surface area.(a)Porous di users in semi-batch ozonation systems.(b)Comparison in count mean diameters between porous di usersand venturiinjector.Fig.7.E ect of water on mean bubble diameter and speci®c surface area.Ozone mass transfer in water and wastewater treatment 915water.Thus,a tradeo may have to be made to maximize the mass transfer without causing the op-erational problems.From this study,it appears that ozonation systems,equipped with a porous di user of 170to 220m m,could still generate the desired gas bubbles less than 3mm in diameter.To eliminate the clogging problem with porous di users and to e ectively dissolve high concen-trations ozone into water,venturi-type injectors are gaining wider application in water treatment.As a result,further tests were conducted to compare the performance between porous di users and a ven-turi-type injector.These tests were conducted in a continuous mode with the supply of both gas and liquid phases from the bottom of the contactor.As shown in Fig.6(b),the bubble size distributions fol-low similar pattern for both types of gas spargers.However,the gas bubbles generated from the injec-tor were much smaller even as compared to the fused alumina di using stone.This di erence was noticeable particularly in the case of the ozonation of pulp mill e uents at lower gas ¯owrates,although it is less conclusive for the ozonation of drinking water.As expected,high mixing intensity in the injector resulted in the instant formation of smaller gas bubbles.As the coalescence of gas bubbles will occur,the e ects of gas spargers on bubble formation will become less signi®cant.However,it should be pointed out that due to the limited side-stream pump capacity,the injector was operated at a liquid-to-gas ratio much lower than those recommended by the manufacturer.It is sus-pected that the advantages of using this type of injector may become more signi®cant,although more data must be collected to con®rm this expec-tation.Figure 7shows the e ects of di erent test waters on the mean bubble size and speci®c surface area.Three test waters (lagoon midpoint,tap water and deionized water)all had similar mean bubble size and varied from 1.1to 1.4mm as the gas ¯ow rate increased.However,the speci®c surface areas were found much higher for the lagoon midpoint waste-water as compared to the tap and deionized water.Again,this re¯ects higher gas holdup caused by the presence of surfactants and other organic com-pounds in pulp mill wastewaters.These organic compounds change the liquid viscosity and the bubble surface tension dependent on their chemical structure,thereby,hindering their rising and form-ing small bubbles.It is well known that surfactants orient themselves with their hydrophobic group toward the gas phase,and the generated dipole layer suppresses the coalescence of gas bubbles.However,it should be noted that not all the surfac-tants will suppress bubble coalescence,and their ability to reduce mass transfer rate has been observed for wastewater aeration as well.Much less profound e ects on gas bubble sizes and speci®c surface area were observed from the degree of ozonation.As shown in Fig.8,preozona-tion of pulp mill e uents generally resulted in a slight decrease in the speci®c surface area,but no more than 20%.As the degree of oxidation increases,however,these organic compounds become oxidized or destroyed partially.Consequently,their e ects on bubble size and gas holdup become very small and,in turn,on speci®c surface area.Figures 9±11show the e ect of operating con-ditions on the gas holdup in ozone bubble contac-tors.The gas holdup was calculated by measuring the increase in liquid height after introducing the ozone gas.In general,the observed gas holdups were less than 2.5%,corresponding to low gas-to-liquid ratios often encountered in ozonation sys-tems.Unlike the mean bubble diameter,the gas holdup strongly depended on the gas ¯ow rate,in addition to gas spargers and source water type.Similar observations were made by Roustan et al.Fig.8.E ects of ozonation on mean bubble diameter and speci®c surface area.Hongde Zhou and Daniel W.Smith916(1996),who proposed that the gas holdup almost linearly increased as the gas¯ow rate increases. Less e ect on the gas holdup were observed from the degree of ozonation.This is reasonable as the gas holdup was mainly determined by the chemical composition of test samples.Ozonation usually has little impact on the composition of inorganic con-stituents.In addition,it is di cult to completely mineralize organic compounds although partial oxi-dation will result in modi®cation of the organic molecular structure.Mass transfer in water and wastewaterFigures12±14present the e ects of operating conditions on physical ozone mass transfer coe -cient k L a values.The experiments were conducted by injecting oxygen gas into preozonated samples in order to avoid the interference from ozone chemical reactions.The correction was made by accounting for the di usivity di erence between ozone and oxygen based on the surface renewal model.As can be seen,all the measured k L a values fall within a range of0.5±1.5minÀ1,which are very comparable to those reported in the literature(Farooq and Ahmed,1989;Le Sauze et al.,1993;Roustan et al., 1996).In addition,the gas¯ow rate was identi®ed as the most important factor a ecting the ozone mass transfer coe cient.As the gas¯ow rate increases,the k L a values increase.Based on the similar trends,it is believed that this increase can be attributed to the increase in speci®c surface area. At a certain gas¯ow rate,physical mass transfer coe cients were also a ected by gas spargers and source water type,while little impact was observed as the consumed ozone dose increased.As a matter of fact,this conclusion can be inferred from the bubble size distribution measurements.As discussed above,both the mean bubble diameter and speci®c surface area were almost independent of ozone dose.Consequently,the e ects of ozone dose on physical mass transfer would be minimal.In con-trast,the®ner pore di users will form smaller gas bubbles,thereby increasing the speci®c surface areas as observed.This increase was particularly important when surfactants and/or other organic compounds are present in pulp millwastewaters.Fig.10.E ect of water type on gasholdup.Fig.9.E ect of gas di user type on gas holdup.Ozone mass transfer in water and wastewater treatment917。

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探索与争鸣191科技资讯 SCIENCE & TECHNOLOGY INFORMATIONDOI：10.16661/ki.1672-3791.2009-5042-1975马克思主义社会科学方法论及其当代价值①吕微(吉林师范大学 吉林长春 130000)摘 要：马克思主义社会科学方法论具有科学性和先进性，在社会科学活动领域，马克思主义也有创造性发展，深层次地理解马克思主义社会科学方法论，了解其特点和意义、当代价值，有利于在当代中国发展过程中，用马克思主义社会科学方法论作为思想武器。

不仅如此，深化理解马克思主义社会科学方法论，还可以更好地直接作用于指导具体的社会科学研究，变成人类改造世界的行动指南。

该文重点探讨了马克思主义社会科学方法论地及其当代价值，以期提供参考借鉴。

关键词：马克思主义 社会科学方法论 基本内涵 当代价值 实践运用中图分类号：G71 文献标识码：A文章编号：1672-3791(2021)02(b)-0191-03Marxist Social Science Methodologyand Its Contemporary ValueLV Wei(Jilin Normal University, Changchun, Jilin Province, 130000 China)Abstract : Marxist social science methodology is scientif ic and advanced. In the f ield of social science activities, Marxism also has creative development. A deep understanding of Marxist social science methodology,understanding of its characteristics, significance, and contemporary value is beneficial to the development process in contemporary China. Marxist social science methodology is used as an ideological weapon.What is more, deepening the understanding of Marxist social science methodology can better direct specific social science research and become a guide for mankind to transform the world. This article focuses on the Marxist social science methodology and its contemporary value in order to provide reference.Key Words : Marxism; Social science methodology; basic connotation; Contemporary value; Practical application①作者简介：吕微（1994—），女，硕士，研究方向为马克思主义哲学。

Particuology11 (2013) 421–427Contents lists available at SciVerse ScienceDirectParticuologyj o u r n a l h o m e p a g e:w w w.e l s e v i e r.c o m/l o c a t e/p a r t icIn situ preparation of hydrophobic CaCO3nanoparticles in a gas–liquid microdispersion processLe Du,Yujun Wang,Guangsheng Luo∗The State Key Laboratory of Chemical Engineering,Department of Chemical Engineering,Tsinghua University,Beijing100084,Chinaa r t i c l e i n f oArticle history:Received30April2012Received in revised form13June2012 Accepted19July2012Keywords:CaCO3nanoparticlesIn situ surface modification MicroreactorMass transfer a b s t r a c tThis study presents a novel process of in situ surface modification of CaCO3nanoparticles using a multiple-orifice dispersion microreactor.CO2/Ca(OH)2precipitation reaction was employed to prepare CaCO3 nanoparticles with sodium stearate surfactant.Synthesized CaCO3products were characterized by ther-mogravimetric analysis(TGA),infra-red(IR),X-ray diffraction(XRD),transmission electron microscopy (TEM)and Brunauer–Emmet–Teller analysis(BET).The effect of various operation parameters on nanopar-ticles and the dosage of sodium stearate were determined.The results showed that the preparation process could be precisely controlled with efficient mass transfer process.The particles were highly hydropho-bic with a contact angle of117◦and monodisperse with an average size of30nm.The adsorptions of sodium stearate and calcium ion on solid particles during the in situ surface modification process were investigated.© 2013 Chinese Society of Particuology and Institute of Process Engineering, Chinese Academy ofSciences. Published by Elsevier B.V. All rights reserved.1.IntroductionNanoparticles have been widely studied in recent decades for their special characteristics such as the surface effect and quan-tum effect(Katz&Willner,2004;Raschke et al.,2003).As an example,calcium carbonate(CaCO3)nanoparticle is an important multifunctional additive used primarily for paints,rubber,pigment, paper,plastics,etc.(Sahebian,Zebarjad,Khaki,&Sajjadi,2009; Wang,Piao,Zhai,Hickman,&Li,2010;Wang,Tang,Wu,Dai,& Qiu,2007).Several techniques have been developed to prepare hydrophilic CaCO3nanoparticles,including emulsion liquid mem-brane method(Sun&Deng,2004)and gas–liquid carbonation(Cao, Wang,&Zhang,2003).Considering the economy and practicality, the method of gas–liquid carbonation seems to be one of the best industrial processes.Carbon dioxide(CO2)gas and calcium hydrox-ide(Ca(OH)2)suspension are employed as the reactants.But the hydrophilic surfaces of common CaCO3particles are incompati-ble with the hydrophobic polymers,which cause agglomeration in the polymer matrix(Chen et al.,2006;Konopacka-Lyskawa& Lackowski,2011).Rubber,plastics and other materials with such kind offiller particles tend to fracture and age fast.Thus the sur-face modification of CaCO3nanoparticles is required to reduce the ∗Corresponding author.E-mail address:gsluo@(G.Luo).surface energy and improve the dispersion stability in the matrix. The surfaces of CaCO3particles are often modified by a variety of surfactants,such as lauric acid,stearic acid,silane coupling agent, and polyethylene glycol(Ma et al.,2008;Novokshonova et al.,2003; Wang,Lu,&Wang,1997).Among these surfactants,fatty acids and their salts are commonly used.The modification process ends up with hydrophobic alkyl chains chemisorbed to the particle sur-face,which can significantly improve the wettability and binding force between thefiller and polymer matrix(Kong et al.,2008). However,the traditional techniques for industrial production,such asfluidization,batch bubbling precipitation,batch stirred reac-tion and emulsification,cause several problems(Li,2009).With these techniques people cannot easily control the particle qual-ity,especially in the large-scale production process.Normally the surface modification is carried out in the wet process after the particles are synthesized.The process usually requires excess addi-tion of surfactants for sufficient modification(Ding,Lu,Deng,& Du,2007;Shui,2003),which causes high-energy consumption. Thus strategies of in situ modification,which are effective to solve the problems as mentioned above,have been extensively stud-ied(Wang et al.,2010;Wang,Sheng,et al.,2007;Wang,Zhao, et al.,2007;Ye&Zhang,2004).A variety of surfactants have been added with the reactants at one step,which simplifies the proce-dures and utilizes the surfactants effectively.However,for Ca(OH)2 slurry system,most of the surfactants tend to coat on the solid particles and prevent Ca2+ions from diffusing into liquid phase.1674-2001/$–see front matter© 2013 Chinese Society of Particuology and Institute of Process Engineering, Chinese Academy of Sciences. Published by Elsevier B.V. All rights reserved. /10.1016/j.partic.2012.07.009422L.Du et al./Particuology11 (2013) 421–427NomenclatureNotationA interfacial area in the reactor,m2d32average diameter of CaCO3nanoparticles,nmd32G average diameter of microbubbles,␮mF cflow rate of continuous phase,mL/minF dflow rate of disperse phase,mL/minN CO2mass transferflux density of CO2,mol/(m2s)M CO2mass transferflux of CO2,mol/sS BET specific surface area of CaCO3particles,m2/gT reaction temperature,◦Cconcentration of Ca(OH)2solution/suspension,g/L Song,Kim,and Kim(2003)studied the typical two-step reactions as follows:[Ca2+]surface+(stearate)−→[Ca(stearate)]+surface,(1)[Ca(stearate)]+surface+(stearate)−→[Ca(stearate)2]surface,(2) which caused most adsorption processes on solid particles.Fur-thermore,the addition of surfactants enhances mass transfer resistance in the liquid membrane.Therefore,a process with both in situ modification method and precise controllable operation is in demand.In the last two decades,microfluidics has exhibited excellent performance in mixing,transfer rate and controllable operating conditions(Duraiswamy&Khan,2009;Guenther,Gross,Wagner, Jahn,&Koehler,2008;Koehler,Held,Huebner,&Wagner,2007; Lee et al.,2009).Reactants with various properties in the form of water/oil/gas have been effectively utilized and the processes can be precisely controlled.In our previous study,several types of microfluidic devices have been developed and successfully used to synthesize nanoparticles in homogeneous and heterogeneous reac-tion systems(Chen,Luo,Li,Xu,&Wang,2005;Chen,Luo,Yang,Sun, &Wang,2004;Li,Xu,Wang,&Luo,2008;Luo,Du,Wang,Lu,&Xu, 2011).Surface treatment and functionalization of various particle materials have also been developed(Guo,Luo,&Wang,2003;Yang, Wang,Luo,&Dai,2008;Shen,Wang,Xu,Lu,&Luo,2012).Especially the large-scale preparation of hydrophilic CaCO3nanoparticles has been industrialized by using the membrane microdispersion reac-tors,which make the annual output up to50,000tons(Wang,Wang, Chen,Luo,&Wang,2007).In this study,an in situ preparation strategy to prepare hydrophobic CaCO3nanoparticles with a gas–liquid microdisper-sion process has been developed.A multiple-orifice dispersion microreactor was designed and applied to generate gas–liquid microdispersed systems.Calcium hydroxide suspension and car-bon dioxide/nitrogen mixed gas(29.8%volume fraction CO2)were selected as the reactants while sodium stearate(RCOONa)was selected as the modifying agent.The operation parameters were varied and their influences on the propertiesof CaCO3nanopar-ticles were investigated.The dosage of sodium stearate was optimized.CaCO3nanoparticles with high-class quality were suc-cessfully prepared.2.ExperimentalFig.1shows the experimental set-up.The major component was a six-orifice dispersion microreactor,in which the diameter of the micro-orifice was0.2mm with orifice spacing of1.5mm.The geometric size of the main channel was20mm×2mm×0.5mm (length×width×height).Fig.1.The experimental set-up for preparing CaCO3nanoparticles.The chemicals include sodium stearate(RCOONa),calcium hydroxide and mixed gas(29.8%volume fraction CO2,0.3MPa). RCOONa wasfirst mixed with Ca(OH)2suspension and stirred for 3h at80◦C before being pumped into the microreactor.Then the Ca(OH)2suspension as the continuous phase and the mixed gas as the dispersed phase were mixed in the microreactor.Pressure difference between the two sides of the dispersion media was employed as the driving force to disperse the gas phase into the continuous phase in the form of microbubbles.CaCO3precipitates were synthesized when the two phases contacted each other in the mixing chamber.To realize complete consumption of Ca(OH)2 in the continuous phase,the suspension was circulated at a cer-tain feed rate.At the beginning,the pH of the system was12.2. The reaction process was stopped when the pH was7.After an aging treatment for1h,CaCO3precipitates were separated from the suspension using a centrifugal separator(LD5-2A,Beijing Med-ical Centrifugal Separator Factory).The CaCO3precipitates were washed3times with distilled water,twice with ethanol and dried in a drying cabinet at100◦C for24h.Finally,the product of CaCO3 was obtained.In addition,experiments to observe and record the status of gas–liquid microdispersion were carried out.A microscope at mag-nifications from20×to200×with a high-speed CCD video camera was used to record the microbubble and measure the diameter.The morphology of nanoparticles was recorded by transmission electron microscopy(TEM,JEOL-2010,120kV).The crystal form of the nanoparticles was characterized by X-ray diffraction analy-sis(XRD,Rigaku Corporation D/max-RB).Specific surface area of the particles was determined by BET(Quantachrome autosorb-1).The weight loss was measured by thermogravimetric analysis (TGA,STA409PC).The photos of the contact angle were taken using a high-speed CCD camera attached to an Olympus U-TV0-5xC-3microscope.Additionally,the contact angles were measured using DataPhysics SCA20contact angle measurement software Ver.3.12.11.Fourier transform infrared spectroscopic measurements (FTIR,BRUKER Corporation TENSOR27)were taken to record FTIR spectra.3.Results and discussion3.1.Influence of two phaseflow rates on the processIn order to test the possibility of the synthesis and select suit-able operation conditions,the preparation of hydrophilic CaCO3 nanoparticles with no addition of surfactant was conducted.L.Du et al./Particuology11 (2013) 421–427423Fig.2.TEM images of the hydrophilic CaCO3nanoparticles.Images(a)–(d)correspond to sample(Ca-1)–(Ca-4)in Table1.Table1lists concentrations of the reactants and the preparation conditions,such as the continuous phaseflow rate,F c,and the dispersed phaseflow rate,F d.TEM images of CaCO3particles are shown in Fig.2,indicating that the average particle size is about36nm with the surface area of53m2/g.The size is relatively uniform and its distribution is pared to the products of high-class standard(d≤80nm, S BET≥18m2/g),the nanoparticles prepared by the multiple-orifice microdispersion reactor are completely qualified.In addition,the particles became relatively smaller and showed better dispersi-bility with the increase of the continuous phaseflow rate.Thus higherflow rate of the continuous phase was applied in the follow-ing experiments.3.2.Influence of surfactant on the processA certain amount of RCOONa was added into Ca(OH)2suspen-sion.The experiment was carried out at20◦C,with the continuous phaseflow rate of240mL/min and the dispersed phaseflow rate of 160mL/min.The concentration of Ca(OH)2suspension was50g/L as above.The adding ratio of RCOONa was based on the weight of CaCO3product,which was0,1,1.5,2,2.5and3wt%,respectively.Fig.3exhibits the relation between RCOONa adding ratio and contact angles of the hydrophobic particles.The powders were compacted intoflakes and held a drop of deionized water on the surface.The contact angle increases with RCOONa adding ratio,from8.8◦to117.7◦.When the RCOONa adding ratio exceeds 2.5wt%,the contact angle changes unnoticeably.The process of hydrophobicity changing is considered to take place in this way: when the modification is carried out,thefirst layer(monolayer) of stearate ions is chemically adsorbed onto the particle surface, leaving a hydrophobic alkyl chain on the outermost surface(Tran, Tran,Vu,&Thai,2010).Table1Preparation conditions of hydrophilic CaCO3particles.Sample Ca-1Ca-2Ca-3Ca-4F c(mL/min)60120180240F d(mL/min)4080120160 Temperature(◦C)20Concentration of Ca(OH)2(g/L)50Fig.3.Contact angles of hydrophobicity of CaCO3nanoparticles with different RCOONa contents.Crystal structures of the particles are shown in Fig.4,given by the X-ray diffraction analysis.Main characteristic planes of(012), (104),(006),(110),(113),(202),(024),(018),(116),(211) and(1010),corresponding to2Âvalue of23.2◦,29.5◦,31.8◦,36.0◦, 39.3◦,43.0◦,47.0◦,47.5◦,48.4◦,57.3◦and58.5◦,respectively,appear in all patterns and indicate a calcite structure.The sharpcrystallineFig.4.XRD patterns of CaCO3powders.424L.Du et al./Particuology 11 (2013) 421–427Fig.5.FTIR spectra of CaCO 3nanoparticles.Fig.6.The thermogravimetric curves of CaCO 3nanoparticles.peaks demonstrate highly crystallized calcite phase even with the surfactant added.To verify the existence of the surfactant layer,FTIR spectra of CaCO 3nanoparticles are recorded,as shown in Fig.5.Three strong peaks at 1454.9,873.3and 706cm −1,corresponding to 2, 3and 4vibrations respectively,are considered to arise from CO 32−.The broad and weak peaks at 3500cm −1are assigned to the stretch-ing vibration of OH.The peak at 1557cm −1is attributed to the antisymmetric and symmetric stretching of COO .FTIR spectra of RCOONa show strong peaks at 2920.2and 2843.5cm −1corre-sponding to C H and C C vibrations respectively,which are not observed in the unmodified sample Ca-4.In addition,it is observed that the intensity of C H and C C increases with RCOONa adding ration,indicating the degree of the modification.TGA curves in Fig.6show the differences between the modi-fied and unmodified particles.The unmodified hydrophilic particles remain stable until the temperature raised beyond 700◦C and the decomposition occurs.For the modified hydrophobic particles,a significant weight loss occurs when temperature ranges from 420◦C to 500◦C.Furthermore,the value of weight loss isexactlyFig.8.Change of pH values with reaction time with/without RCOONa added in (MR:microreactor;BR:batch reactor).the same as the RCOO −dosage.However,when the RCOONa adding ratio exceeds 2.5wt%,the weight loss changes unnoticeably,which also implies the optimal adding ratio is around 2.5wt%.Fig.7exhibits some TEM images of hydrophobic CaCO 3parti-cles.With the surfactant added,more particles of large size are generated and agglomeration occurs simultaneously.Fig.8shows the change of pH value when RCOONa is added.Although the mass transfer could be enhanced by introducing the microreac-tor compared with traditional batch reactor,the whole process still required nearly 35min to be completely finished.At the same time,the reaction time was 10min shorter for the surfactant-free pro-cess.The results were far from the initial anticipation which might be caused by inefficient mixing and mass transfer process.3.3.Mechanism analysis and enhancement of mass transferTemperature is one of the most important factors for the gas–liquid system.Generally,the reaction temperature influences the solubility of reactants,the diffusion coefficient and supersatu-ration.With the increase of temperature,the dissolution of solid Ca(OH)2and the reaction rate were accelerated for the molecular thermal motion.However,the dissolution of CO 2gas was blocked,and the reduction of bubble diameter,namely the mixing scale,was hardly to achieve.Thus there is an optimal value for the efficient diffusion and reaction.The experiment results exhibited that temperature control was of great importance and the particle size was decreased signifi-cantly with the temperature decreasing,as shown in Fig.9.The specific surface area of CaCO 3particles increases apparently with the decrease of temperature,as a result of the particle size reduc-tion.It is necessary to analyze the mechanism of the processes of surface modification and carbonation.Fig.10(a)shows the adsorp-tion of RCOO −and Ca 2+ions on solid Ca(OH)2in aqueous medium.First,RCOO −and Ca 2+ions are adsorbed on the solid surfacebyFig.7.TEM images of the modified hydrophobic CaCO 3nanoparticles.L.Du et al./Particuology11 (2013) 421–427425Fig.9.Effect of temperature on particle size and specific surface area. electrostatic force(Song et al.,2003).Ca2+ions are saturated on the Stern layer,and the interaction between RCOO−and Ca2+per-sists until sufficient number of RCOO−is adsorbed.The adsorption ends up with the monolayer saturation of stearate.After the mono-layer saturation of stearate,surfactants are associated in the form of two-dimensional aggregate between hydrophobic groups of sur-factants.Finally,the adsorption of surfactant reaches a plateau corresponding to complete surface coverage and the excess pos-itive ions such as Ca2+and Na+ions are attached strongly.In this case,Ca2+ions generated from the dissolution of solid Ca(OH)2are hard to diffuse into the aqueous solution owing to the blocking by the coated stearate layers.Thus the mass transfer is inefficient and the mixing of reactants is uneven,which causes the generation of large particles.To avoid the coating effect on the solid Ca(OH)2,it is feasible to decrease the concentration of Ca(OH)2.For example,the saturated solution can be utilized instead of suspension.The concentration of RCOONa is8×10−4mol/L while the concentration of Ca(OH)2satu-rated solution is0.002mol/L,quantitatively about120times larger than RCOONa.In this case,most of Ca2+and OH−ions exist in the solution.RCOO−combined with Ca2+exists in the form of micelles, as shown in Fig.10(b).Thus mass transfer processes of Ca2+,HCO3−and CO32−are significantly improved,which are available to syn-thesize smaller particles with narrow size distribution.The related experiments also exhibited similar results.Fig.11 shows the microscope images of the microbubbles at different Ca(OH)2concentrations and TEM images of nanoparticles in the corresponding conditions.The diameter of the smallest bubble could be reduced to20–30␮m.In these conditions,the particle size was decreased to25–35nm and the monodispersity of particles was significantly improved.However,with the Ca(OH)2concen-tration decreasing,the particles became larger again,which might be caused by the decrease of the supersaturation and inefficient nucleation process.Based on the mass transferfilm theory,the mass transferflux density can be obtained.N CO2=M CO2,(3)where N CO2is the mass transferflux density of CO2(mol/(m2s)), M CO2represents the mass transferflux of CO2(mol/s),and A is the interfacial area in the reactor(m2).A could be calculated by mea-suring average diameters(d32G)of the bubbles from microscope images.Fig.12shows the relationship between particle size and mass transferflux with the concentration of Ca(OH)2.Obviously,the mass transfer rate of CO2was enhanced by decreasing Ca(OH)2 concentration,which might be caused by reduction of the coating effect mentioned before.However,with the Ca(OH)2concentration decreasing,the particle size increased and N CO2decreased sharply. The reason might be the decrease of the supersaturation and inef-ficiently nucleating with relatively low concentration.Therefore, saturated Ca(OH)2solution with2.5wt%RCOONa added was con-sidered to be the optimal condition for in situ surface modifying CaCO3nanoparticles in the microreactor.parison with other preparation methodsThe preparation of hydrophobic CaCO3nanoparticles with a microreactor was compared with some typical methods,includ-ing in situ surface modification.The results are shown in Table2. It is obvious that the microdispersion method is excellent in both synthesis and product qualities.The process can be carried out con-tinuously and is controllable with relatively simple operation.The Fig.10.Schematic representation for adding RCOONa into the system.(a)Adsorption process of RCOO−and Ca2+ions on solid Ca(OH)2in aqueous medium and(b)micelles exist in Ca(OH)2saturated solution.426L.Du et al./Particuology 11 (2013) 421–427Fig.11.Microscope images of the microbubbles and TEM images of modified CaCO 3nanoparticles at different Ca(OH)2concentrations.(a) (Ca(OH)2)=1.5g/L;(b) (Ca(OH)2)=1g/L;and (c) (Ca(OH)2)=0.5g/L.Table 2Comparison with different methods to synthesize hydrophilic CaCO 3particles.DeviceMicroreactorStirred tankStirred tankStirred tankOperation Continuous/batch BatchBatch BatchModification In situFollow-upIn situFollow-upReactantsCO 2,Ca(OH)2,sodium stearate CO 2,Ca(OH)2,sodium stearate CO 2,Ca(OH)2,oleic acid CO 2,Ca(OH)2,sodium stearate,ethanol Particle size (nm)3040–6040–60200–300DispersionMonodisperse Monodisperse Agglomerated Monodisperse Contact angle (◦)117.7127108.77125ReferenceThis studyTran et al.(2010)Wang et al.(2010)Ma et al.(2008)average particle sizes are smaller than those obtained with other methods.Agglomeration,the disadvantage of in situ modification,can also be avoided for the precise control in the microreactor.The contact angles are large enough to ensure the dispersibility of CaCO 3nanoparticles in polymermaterials.Fig.12.Mass transfer flux densities of CO 2and particle sizes at different Ca(OH)2concentrations.4.ConclusionsIn situ surface modified CaCO 3nanoparticles were prepared using a multiple-orifice dispersion microreactor.The monodis-persed particles were highly hydrophobic with the average size of 30nm.An efficient mixing and fast transfer rate were achieved in the microreactor with both high supersaturation and uniform reaction environment.The dosage of the sodium stearate was opti-mized.The effects of operation conditions on mixing performance and particle size were investigated,confirming that the particle size could be controlled by varying the flow rate,reactant concentration and temperature.The possible mechanism has been presented to explain the effect of surfactant in suspension system.AcknowledgementsWe gratefully acknowledge the supports of the National Nat-ural Science Foundation of China (21036002and 20876084)and National Basic Research Program of China (2007CB714302)to this work.L.Du et al./Particuology11 (2013) 421–427427ReferencesCao,W.L.,Wang,Z.,&Zhang,J.C.(2003).Analysis of particle size control in the preparation of nano-size CaCO3particles.Chinese Journal of Chemical Engineer-ing,11(5),589–593.Chen,G.G.,Luo,G.S.,Li,S.W.,Xu,J.H.,&Wang,J.D.(2005).Experimental approaches for understanding mixing performance of a minireactor.AIChE Journal,51(11), 2923–2929.Chen,G.G.,Luo,G.S.,Yang,X.R.,Sun,Y.W.,&Wang,J.D.(2004).Preparation of ultra-fine TiO2particles using micro-mixing precipitation technology.Journal of Inorganic Materials,19(5),1163–1167.Chen,X.H.,Li,C.Z.,Xu,S.F.,Zhang,L.,Shao,W.,&Du,H.L.(2006).Interfacial adhesion and mechanical properties of PMMA-coated CaCO3nanoparticle reinforced PVC composites.China Particuology,4(1),25–30.Ding,H.,Lu,S.-C.,Deng,Y.-X.,&Du,G.-X.(2007).Mechano-activated surface modi-fication of calcium carbonate in wet stirred mill and its properties.Transactions of Nonferrous Metals Society of China,17(5),1100–1104.Duraiswamy,S.,&Khan,S. 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Estimation of heat-transfer characteristics on a vertical annular circular fin of finned-tube heat exchangers in forced convectionHan-Taw Chen *,Wei-Lun HsuDepartment of Mechanical Engineering,National Cheng Kung University,Tainan 701,TaiwanReceived 19January 2007;received in revised form 17June 2007Available online 6September 2007AbstractThe finite difference method in conjunction with the least-squares scheme and experimental measured temperatures is applied to pre-dict the average heat transfer coefficient h and fin efficiency g f on a vertical annular circular fin of finned-tube heat exchangers for various fin spacings in forced convection.The distribution of the heat transfer coefficient on the fin is assumed to be non-uniform,thus the whole annular circular fin is divided into several sub-fin regions in order to predict the h and g f values.These two predicted values can be obtained using the present inverse scheme in conjunction with the experimental measured temperatures.The results show that the effect of the fin spacing S on the h value can be negligible when the S value exceeds about 0.018m.The h value increases with increasing the air speed V air for 1m/s 6V air 65m/s (15506Re d 67760)and increasing the fin spacing for 0.005m 6S 60.018m.The present estimated results can be applied to obtain the new correlations of the Nusselt number and fin efficiency based on d o ,S and Re d .The average heat transfer coefficients obtained from this new correlation of the Nusselt number are in good agreement with those given by Hu and Jacobi [X.Hu,A.M.Jacobi,Local heat transfer behavior and its impact on a single-row,annularly finned tube heat exchanger,ASME J.Heat Transfer 115(1993)66–74].Ó2007Elsevier Ltd.All rights reserved.1.IntroductionAnnular-finned tube heat exchangers are commonly used in industry.In designing such heat exchangers,it is necessary to note the interactions between the local heat transfer and flow distribution within the fins.The previous works about the effect of the fin spacing of annular-finned tube heat exchangers were limited to the experiments [1].Thus the present study applies the hybrid inverse scheme in conjunction with experimental temperature data to esti-mate the heat-transfer characteristics of annular-finned tube heat exchangers in forced convection.The fin in heat exchangers is always applied to increase the heat flow per unit of basic surface.The analysis of a continuous plate fin pierced by a regularly spaced array of circular tubes in staggered and in-line arrays has many engineering appli-cations [2].In order to simplify the problem considered,the calculation of the standard fin efficiency usually assumed that the heat transfer coefficient was constant over the plate fin.It can be found from Refs.[3–6]that there exhibited very complex three-dimensional flow characteristics within a plate finned-tube heat exchanger.In particular,complex flow patterns caused by the horseshoe vortices at the corner junction are of interest to the study of local heat transfer enhancement mechanism.The flow accelerates around a heated horizontal annular-finned circular tube in a cross-flow and forms a low-velocity wake region behind the tube.The boundary layer over a heated horizontal tube starts to develop at the front of the tube and increases in thickness along the circumference of the tube.Thus the heat transfer coefficient in forced convection is highest on the upstream fin region and is lowest on the wake fin region,as shown in Ref.[7].This causes local variations of the heat transfer coefficient on the fin.On the other hand,the heat transfer coefficient on the fin is non-uniform.As shown in Ref.[8],0017-9310/$-see front matter Ó2007Elsevier Ltd.All rights reserved.doi:10.1016/j.ijheatmasstransfer.2007.06.035*Corresponding author.Fax:+88662352973.E-mail address:htchen@.tw (H.-T.Chen)./locate/ijhmtAvailable online at International Journal of Heat and Mass Transfer 51(2008)1920–1932the measurement of the local heat transfer coefficient on the platefin under steady-state heat transfer conditions was very difficult to perform,since the localfin temperature and local heatflux were required.Moreover,reliability is an important concept in engineering design,and the use of reliable components enables the designers to utilize more sophisticated techniques to improve the performance[9]. Thus the estimation of a more accurate heat transfer coef-ficient on thefin is an important task for the device of the high-performance heat exchangers.Heat transfer coefficients encountered in forced convec-tion are typically much higher than those encountered in natural convection because of the higherfluid velocities associated with forced convection.As a result,most of researchers tend to ignore natural convection in heat trans-fer analyses that involve forced convection.However,this error may be considerable at low velocities associated with forced convection[10].It is known that the physical quantities of the test mate-rial can be predicted using the measured temperatures inside this test material.Such problems are called the inverse heat conduction problems that have become an interesting subject recently.To date,various inverse meth-ods in conjunction with the measured temperatures inside the test material have been developed for the analysis of the inverse heat conduction problems[11,12].However, to the authors’knowledge,a few researchers performed the prediction of the local heat transfer coefficients on a vertical annular circularfin offinned-tube heat exchangers with regard for the effect of thefin spacing[3–7,13,15–17].Sung et al.[3]applied the naphthalene sublimation tech-nique to obtain the local mass transfer coefficient on a cir-cular cylinder with transverse two annularfins in a cross flow.Hu and Jacobi[6]also applied the naphthalene subli-mation technique to obtain the local convective heat and mass transfer coefficients on a circular cylinder with trans-verse annularfins in a crossflow over the Reynolds number ranging from3300to12,000.Watel et al.[14]investigated the influence of theflow velocity andfin spacing on the forced convective heat transfer from a single annular-finned tube using the Particle Image Velocimetry and infrared thermography.They obtained a valuable correlation of the mean Nusselt number on thefin as a function of the dimensionlessfin spacing,outer diameter of a circular tubeNomenclatureA f area of the annular circularfin,m2A j area of the j th sub-fin region,m2[A]global conduction matrixD H hydraulic diameter of heat exchanger,md o outer diameter of the circular tube,m[F]force matrixg acceleration of gravity,m/s2h local heat transfer coefficient,W/m2Kh unknown average heat transfer coefficient,W/m2Kh iso unknown average heat transfer coefficient underthe isothermal condition,W/m2Kh ffin height,R oÀR ihjunknown average heat transfer coefficient on thej th sub-fin region,W/m2Kk f thermal conductivity of thefin,W/m Kk air thermal conductivity of the air,W/m K‘r distance between two neighboring nodes in the r-direction,‘r=(R oÀR i)/(N rÀ1)‘h distance between two neighboring nodes in the h-direction,‘h=2p/(N hÀ1)N number of sub-fin regionsNu isod Nusselt number defined in Eq.(22)N r number of nodes in the r-directionN h number of nodes in the h-directionQ total heat transfer rate dissipated from the annu-lar circularfin,Wq j heat transfer rate dissipated from the j th sub-fin region,W r spatial coordinateR i inner radius of the annular circularfin or outer radius of the circular tube,mR o outer radius of the annular circularfin,mRe Reynolds number defined in Eq.(23)Re d Reynolds number defined in Eq.(24)Sfin spacing,mTfin temperature,KT ffilm temperature,(T o+T1)/2T j measured temperature on the j th sub-fin region, KT o outer surface temperature of the circular tube,K T1ambient temperature,K[T]global temperature matrixV air frontal air speed,m/sV max maximum air speed,m/sGreek symbolsa thermal diffusivity of the air,m2/sb volumetric thermal expansion coefficient,1/K dfin thickness,mg ffin efficiencym kinematic viscosity of the air,m2/sh spatial coordinateSuperscriptscal calculated valuemea measured dataH.-T.Chen,W.-L.Hsu/International Journal of Heat and Mass Transfer51(2008)1920–19321921and Reynolds number for25506Re d642,000.They also found that the reduction infin spacing leaded to a decrease of heat transfer for afixed Reynolds number.However,the difference of the average heat transfer coefficients on the downstream and upstreamfin regions andfin efficiency was not shown in Ref.[14].As stated by Hu and Jacobi [6],it was difficult to obtainfin efficiency measurements, because the temperature measurements within thefin or on its surface are inherently difficult to make without dis-rupting the heat transfer behavior.Mon and Gross[15] applied the three-dimensional numerical study to investi-gate the effect of thefin spacing on four-row annular-finned tube bundles in staggered and in-line arrangements.The heat transfer andfluidflow characteristics were predicted using the computationalfluid dynamics commercial code of FLUENT.It can be found from Ref.[15]that the lami-narflow between thefins and one-dimensional heat conduc-tion equation were assumed.Sparrow and Samie[16] measured Nusselt numbers and pressure loss coefficients for one-and two-row arrays of annular-finned tubes using the experimental method.Matos et al.[17]applied a theo-retical,numerical and experimental study to demonstrate thatfinned and non-finned circular and elliptic tubes heat exchangers in forced convection that can be optimized for maximum heat transfer under afixed volume constraint. Jang[18]used the numerical and experimental studies to estimate the heat transfer andfluidflow characteristics of a four-row annular-finned tube heat exchanger in a stag-gered arrangement in order to develop the high efficiency air-cooled steam condenser for the power plant.Sometimes, it is maybe difficult to measure the temperature distribu-tions on thefin of platefinned-tube heat exchangers using the infrared thermography and thermocouples for some practical heat transfer problems.Due to this reason,Chen and Hsu[13]applied thefinite difference method in conjunc-tion with the least-squares scheme and experimental mea-sured temperatures to predict the natural-convection heat transfer coefficient andfin efficiency on a vertical annular circularfin offinned-tube heat exchangers in a small open box with regard for the effect of thefin spacing.It can be found that the predicted results of the average heat transfer coefficient given in Ref.[13]agreed with those obtained from the correlations recommended by current textbooks [1,19]under the assumption of the ideal isothermalfin.This implies that the present inverse scheme has good reliability. In order to validate the reliability and accuracy of the pres-ent inverse scheme in forced convection further,the present study performs the estimations of the average heat transfer coefficient andfin efficiency on a vertical annular circularfin offinned-tube heat exchangers in forced convection.The present estimated results of the average heat transfer coeffi-cient under the ideal isothermal condition will compare with the estimated results given by Hu and Jacobi[6]and Watel et al.[14].The inverse analysis of the present study is that the wholefin area is divided into several analysis sub-fin regions and then thefin temperatures at selected measure-ment locations are measured using T-type thermocouples. Later,thefinite difference method in conjunction with the measured temperatures and least-squares method is applied to predict the average heat transfer coefficients on these sub-fin regions.Furthermore,the average heat transfer coefficient h andfin efficiency can be obtained for various air speeds andfin spacings under the given conditions of the ambient and tube temperatures.The computational procedures for the estimates of the heat transfer coefficient on each sub-fin region are performed repeatedly until the sum of the squares of the deviations between the calculated and measured temperatures becomes minimum.2.Mathematical formulationThe schematic diagram of the present problem is shown in Fig.1.Fig.2shows the physical model of the two-dimensional thin annular circularfin with measurement locations and sub-fin regions in forced convection.R o and R i denote the outer and inner radii of the annular cir-cularfin,respectively.S and d,respectively,denote thefin spacing andfin thickness.The center of the circular tube is located at r=0.T o and T1,respectively,denote the outer surface temperature of the circular tube and the ambient temperature.Due to the thinfin behavior,the temperature gradient in the z-direction(thefin thickness)is small and thefin temperature varies only in the r-and h-directions. The‘‘insulated tip”assumption can be an adequate approximation provided that the actual heat transfer rate dissipated through the tip is much smaller than the total heat transfer rate drawn from the base wall[20].It can be found from Refs.[7,13]that the heat transfer coefficient on thefin offinned-tube heat exchangers was non-uniform. Thus the heat transfer coefficient h(r,h)in the present study is also assumed to be non-uniform.The average heattrans-1922H.-T.Chen,W.-L.Hsu/International Journal of Heat and Mass Transfer51(2008)1920–1932fer coefficient on each sub-fin region can be estimated pro-vided that thefin temperatures at various measurement locations can be measured.Under the assumptions of the steady state and constant thermal properties,the two-dimensional heat conduction equation for the continuous thefin of a one-tubefinned-tube heat exchanger can be expressed aso2T o r2þ1ro To rþ1r2o2To h2¼2hðr;hÞk f dðTÀT1Þin R i<r<R o;0<h62pð1ÞIts corresponding boundary conditions areo Tðr;0Þo h ¼o Tðr;2pÞo hð2ÞTðr;0Þ¼Tðr;2pÞð3ÞTðr;hÞ¼T o at r¼R ið4Þando Tðr;hÞo r¼0at r¼R oð5Þwhere T is thefin temperature.r and h are cylindrical coor-dinates.h(r,h)is the unknown heat transfer coefficient on thefin.k f is the thermal conductivity of thefin.3.Numerical analysisIt might be difficult to measure the temperature distribu-tions on the annular circularfin using the infrared ther-mography and thermocouples for some practical heat transfer problems.Relatively,the distribution of the unknown heat transfer coefficient on afin h(r,h)is not easy to be obtained.Under this circumstance,the annular circu-larfin considered can be divided into N sub-fin regions in the present inverse scheme and then the unknown heat transfer coefficient on each sub-fin region can be assumed to be constant.Thus the application of thefinite difference method to Eq.(1)can produce the following difference equation on the k th sub-fin region asT iþ1;jÀ2T i;jþT iÀ1;j‘2rþ1R iþðiÀ1Þ‘rT iþ1;jÀT iÀ1;j2‘rþ1½R iþðiÀ1Þ‘r 2T i;jþ1À2T i;jþT i;jÀ1‘2h¼2 h kk f dðT i;jÀT1Þfor i¼2;...;N r;j¼ðN hÀ1ÞðkÀ1Þ=Nþ2;ðN hÀ1ÞðkÀ1Þ=Nþ3;...;ðN hÀ1Þk=Nð6Þwhere‘r and‘h,respectively,are the distances between two neighboring nodes in the r-and h-directions and are de-fined as‘r=(R oÀR i)/(N rÀ1)and‘h=2p/(N hÀ1).N r and N h are the nodal numbers in the r-and h-directions, respectively. h k is denoted as the average heat transfer coef-ficient on the k th sub-fin region.The application of the central difference approximation to the boundary condition(2)and then the substitution of the resulting equation into Eq.(6)can yield the difference equations asT iþ1;1À2T i;1þT iÀ1;1‘2rþ1R iþðiÀ1Þ‘rT iþ1;1ÀT iÀ1;12‘rþ1½R iþðiÀ1Þ‘rT i;2À2T i;1þT i;NhÀ1‘h¼2 h mk f dðT i;1ÀT1Þfor i¼2;3;...;N rð7ÞH.-T.Chen,W.-L.Hsu/International Journal of Heat and Mass Transfer51(2008)1920–19321923andT iþ1;Nh À2T i;NhþT iÀ1;Nh‘rþ1R iþðiÀ1Þ‘rT iþ1;NhÀT iÀ1;Nh2‘rþ1½R iþðiÀ1Þ‘r 2T i;2À2T i;NhþT i;NhÀ1‘2h¼2 h mk f dðT i;NhÀT1Þfor i¼2;3;...;N rð8Þwhere h m is defined as h m¼ð h1þ h NÞ=2.The discretized form of Eq.(4)is given asT1;j¼T o for j¼1;2;...;N hð9ÞThe application of the central difference approximation to the boundary conditions(3)and(5)can yield their differ-ence equations asT i;1¼T i;Nhfor i¼2;3;...;N rð10ÞandT NrÀ1;j ¼T Nrþ1;jfor j¼1;2;...;N hð11ÞThe difference equations for the nodes at the interface of two neighboring sub-fin regions,as shown in Fig.3,can be expressed asT iþ1;jÀ2T i;jþT iÀ1;j‘r þ1R iþðiÀ1Þ‘rT iþ1;jÀT iÀ1;j2‘rþ1½R iþðiÀ1Þ‘r 2T i;jþ1À2T i;jþT i;jÀ1‘2h¼ hkþ h kþ1k f dðT i;jÀT1Þfor i¼2;...;N r;j¼ðN hÀ1Þk=Nþ1;k¼1;2;...;NÀ1ð12ÞRearrangement of Eqs.(6)–(12)can yield the following ma-trix equation as½A ½T ¼½F ð13Þwhere[A]is a global conduction matrix.[T]is a matrix rep-resenting the nodal temperatures.[F]is a force matrix.The nodal temperatures can be obtained from Eq.(13)using the Gauss elimination algorithm.Once the average heat transfer coefficient on each sub-fin region is obtained,the heat transfer rate dissipated from the j th sub-fin region q i and average heat transfer coeffi-cient on the wholefin h can be determined using the follow-ing expressions.The heat transfer rate dissipated from the j th sub-fin region q i is defined asqi¼2 h jZA jðTÀT1Þd A for j¼1;2;...;Nð14ÞThe average heat transfer coefficient on the wholefin h can be expressed ash¼X Nj¼1hjA j=A fð15Þwhere A f is the area of the annular circularfin.A j is the area of the j th sub-fin region.Thefin efficiency g f is defined as the ratio of the actual heat transfer rate from the wholefin to the dissipated heat transfer rate from thefin maintained at the tube tempera-ture T o.Thus thefin efficiency g f can be expressed asg f¼P Nj¼1qj2A fðT oÀT1Þ hð16ÞThe total heat transfer rate dissipated from the wholefin to the ambient Q can be written asQ¼X Nj¼1qjð17ÞIn order to estimate the unknown heat transfer coefficient on the j th sub-fin region h j,the additional information of the steady-state measured temperatures is required.The more a number of the sub-fin regions are,the more accu-rate the estimation of the unknown average heat transfer coefficient on the wholefin can be.Relatively,a more com-putational time can be required.In the present study, T-type thermocouples are used to record the temperature information at selected measurement locations.The mea-sured temperature taken from the j th thermocouple isdenoted as T meaj,j=1,...,N,as shown in Tables1–3.The least-squares minimization technique is applied to minimize the sum of the squares of the deviations between the calculated and measured temperatures at selected mea-surement locations.In the present study,the unknown average heat transfer coefficients on each sub-fin region h i can be expressed ashi¼C i for i¼1;2;...;Nð18ÞFig.3.Nodes at the interface of two-neighboring sub-fin areas.1924H.-T.Chen,W.-L.Hsu/International Journal of Heat and Mass Transfer51(2008)1920–1932Table1Temperature measurements and the present estimates for V air=1m/s and various S valuesS=0.005m S=0.01m S=0.015m S?1T0=330.63K T0=328.82K T0=330.34K T0=335.03KT1=298.03K T1=298.20K T1=299.74K T1=300.23KT mea j ðKÞT mea1¼305:67T mea1¼304:71T mea1¼305:51T mea1¼306:67T mea2¼305:83T mea2¼305:36T mea2¼305:67T mea2¼307:06T mea3¼314:30T mea3¼309:22T mea3¼313:25T mea3¼312:75T mea4¼312:93T mea4¼309:28T mea4¼311:31T mea4¼312:75T mea5¼305:91T mea5¼305:39T mea5¼307:67T mea5¼309:22T mea6¼303:70T mea6¼303:17T mea6¼304:75T mea6¼306:16h j (W/m2K) h1¼24:72 h1¼29:65 h1¼34:35 h1¼36:27h2¼35:86 h2¼31:97 h2¼45:10 h2¼41:00h3¼3:20 h3¼12:51 h3¼4:06 h3¼11:20h4¼7:49 h4¼12:51 h4¼13:98 h4¼13:62h5¼31:02 h5¼29:40 h5¼24:56 h5¼24:74h6¼45:46 h6¼48:96 h6¼48:50 h6¼45:60q j(W)q1=0.72q1=0.77q1=0.85q1=1.01q2=1.11q2=0.90q2=1.19q2=1.22q3=0.14q3=0.43q3=0.16q3=0.44q4=0.32q4=0.43q4=0.51q4=0.54q5=0.95q5=0.82q5=0.72q5=0.82q6=1.11q6=1.09q6=1.08q6=1.17hðW=ðm2KÞÞ24.6327.5028.4228.74h iso W=ðm2KÞPresent Eq.(20)18.7320.3520.6920.97Present Eq.(25)18.4919.3719.5521.02Ref.[14]18.4921.0421.9825.73Q(W) 4.35 4.44 4.51 5.20g f38%37%36%35%Table2Temperature measurements and the present estimates for V air=3m/s and various S valuesS=0.005m S=0.01m S=0.015m S?1T0=331.70K T0=331.00K T0=330.80K T0=331.60KT1=298.15K T1=299.00K T1=299.90K T1=300.15KT mea j ðKÞT mea1¼300:31T mea1¼302:46T mea1¼302:49T mea1¼302:95T mea2¼303:73T mea2¼302:75T mea2¼302:95T mea2¼303:08T mea3¼310:39T mea3¼306:36T mea3¼307:21T mea3¼306:79T mea4¼308:82T mea4¼306:92T mea4¼306:95T mea4¼307:05T mea5¼302:16T mea5¼302:79T mea5¼303:41T mea5¼303:80T mea6¼301:31T mea6¼301:21T mea6¼302:46T mea6¼302:79h j ðW=m2KÞ h1¼116:86 h1¼62:67 h1¼84:40 h1¼78:63h2¼45:01 h2¼67:67 h2¼80:43 h2¼84:43h3¼9:38 h3¼25:12 h3¼21:83 h3¼27:02h4¼14:13 h4¼20:89 h4¼25:12 h4¼26:16h5¼69:25 h5¼64:38 h5¼67:88 h5¼65:85h6¼73:28 h6¼107:52 h6¼87:12 h6¼86:73q j(W)q1=2.24q1=1.29q1=1.56q1=1.50q2=1.17q2=1.48q2=1.61q2=1.69q3=0.35q3=0.71q3=0.59q3=0.71q4=0.49q4=0.60q4=0.68q4=0.70q5=1.62q5=1.40q5=1.41q5=1.40q6=1.39q6=1.82q6=1.48q6=1.51hðW=ðm2KÞÞ54.6558.0461.1361.47h isoðW=ðm2KÞÞPresent Eq.(20)30.3732.0233.2933.51Present Eq.(25)30.9631.6731.7833.62Ref.[14]35.4339.3840.9047.10Q(W)7.267.307.337.51g f28%28%27%27%H.-T.Chen,W.-L.Hsu/International Journal of Heat and Mass Transfer51(2008)1920–19321925The error in the estimates E (C 1,C 2,...,C N )will be mini-mized in order to obtain the required estimates.E (C 1,C 2,...,C N )is defined asE ðC 1;C 2;...;C N Þ¼XN j ¼1½T cal j ÀT mea j2ð19Þwhere T cal j denotes the calculated temperature taken from the j th thermocouple location and is obtained from Eq.(13).The estimated values of C j ,j =1,2,...,N ,are deter-mined until the value of E (C 1,C 2,...,C N )is minimum.The detailed computational procedures for estimating the C j values,j =1,2,...,N ,can be found in Refs.[7,13].In order to avoid repetition,they are not shown in this paper.The computational procedures of the present study are repeated until the values of T mea j ÀT calj T mea j,j =1,2,...,N ,are lessthan 10À6.4.Experimental apparatusAn experimental configuration of the small wind tunnel used for the present problem is the same as Fig.1of Ref.[7].The experimental procedures and manners have also been shown in Ref.[7].In order to avoid repetition,they are not shown in this study.Fig.4shows the experimental configuration of the test annular fin vertically mounted ona circular tube in forced convection.The test annular circu-lar fin with 27mm in inner diameter,99mm in outer diam-eter and 1mm in thickness are made of AISI 304stainless material.It can be found from Ref.[21]that the thermal conductivity of AISI 304stainless material is 14.9W/m K.An anemometer installed at 300mm in front of the airflow entering the test specimen is used to measure the frontal air velocity.The limit of its error is ±0.4%for the velocity ranging from 0.4m/s to 30.0m/s.The ambient and test fin temperatures are measured using T-type thermocouples.The limit of error of the T-type thermocouple is ±0.4%for 0°C 6T 6350°C.Six thermo-couples,TC7,TC8,TC9,TC10,TC11and TC12,placed in the gap between the fin and the circular tube are fixed at p /6,p /2,5p /6,7p /6,3p /2and 11p /6in angle,as shown in Fig.2.In order to reduce the heat loss between the fin and the circular tube,their gap is filled with the cyanoacry-late (Satlon,D-3).The average of these six measured tem-peratures is taken as the fin base temperature and is also assumed to be the outer surface temperature of the circular tube T 0in the present study.Three thermocouples pene-trated the central line of the top surface and two lateral sur-faces are positioned at 100mm away from the test fin in order to measure the ambient temperature T 1.The aver-age of these three measured temperatures is taken as the ambient temperature T 1.In order to estimate the average heat transfer coefficient on the fin,the annular circular finTable 3Temperature measurements and the present estimates for V air =5m/s and various S valuesS =0.005m S =0.01m S =0.015m S ?1T 0=330.13K T 0=331.20K T 0=329.30K T 0=331.50K T 1=298.90KT 1=299.25K T 1=299.53K T 1=299.88K T mea jðK ÞT mea 1¼300:98T mea 1¼301:80T mea 1¼301:18T mea 1¼301:44T mea 2¼301:37T mea 2¼301:97T mea 2¼301:64T mea 2¼302:39T mea 3¼306:43T mea 3¼305:11T mea 3¼304:43T mea 3¼304:13T mea 4¼308:75T mea 4¼305:41T mea 4¼305:25T mea 4¼304:92T mea 5¼300:98T mea 5¼301:67T mea 5¼301:48T mea 5¼302:13T mea 6¼300:86T mea 6¼300:82T mea 6¼301:18T mea 6¼301:61h j ðW =ðm 2K ÞÞh 1¼102:50h 1¼83:83 h 1¼120:11 h 1¼134:30 h 2¼98:83 h 2¼90:25 h 2¼103:57 h 2¼89:38 h 3¼24:03 h 3¼33:43 h 3¼39:21 h 3¼51:64 h 4¼7:68 h 4¼29:36 h 4¼27:64 h 4¼37:36 h 5¼121:35 h 5¼98:33 h 5¼114:14 h 5¼103:20 h 6¼107:11 h 6¼137:38 h 6¼119:16 h 6¼120:46q j (W)q 1=1.79q 1=1.56q 1=1.91q 1=2.22q 2=1.87q 2=1.77q 2=1.79q 2=1.67q 3=0.68q 3=0.84q 3=0.88q 3=1.14q 4=0.24q 4=0.75q 4=0.65q 4=0.87q 5=2.24q 5=1.86q 5=1.95q 5=1.89q 6=1.67q 6=2.10q 6=1.72q 6=1.84 h ðW =ðm 2K ÞÞ76.9278.7687.3089.39 h iso ðW =ðm 2K ÞÞPresent Eq.(20)38.1539.0141.9642.74Present Eq.(25)39.2739.7939.8341.82Ref.[14]47.2052.6754.5862.44Q (W)8.498.888.909.63g f25%25%24%24%1926H.-T.Chen,W.-L.Hsu /International Journal of Heat and Mass Transfer 51(2008)1920–1932。