CES-实验

Experimental analysis of forced convective heat transfer in novel structured packed beds of particlesJian Yang a,Jing Wang a,Shanshan Bu a,Min Zeng a,Qiuwang Wang a,n,Akira Nakayama ba Key Laboratory of Thermo-Fluid Science and Engineering,Ministry of Education,Xi’an Jiaotong University,Xi’an,Shaanxi710049,PR Chinab Department of Mechanical Engineering,Shizuoka University,3-5-1Johoku,Hamamatsu432,Japana r t i c l e i n f oArticle history:Received31December2010Received in revised form7November2011Accepted6December2011Available online13December2011Keywords:Packed bedEllipsoidal particleConvective transportHydrodynamicsHeat transferExperimental analysisa b s t r a c tFollowed by the numerical study of Yang et al.(2010),the macroscopic hydrodynamic and heat transfercharacteristics in some novel structured packed beds are experimentally studied in this paper,wherethe packings of ellipsoidal or non-uniform spherical particles are investigated for thefirst time withexperiments and some important results are obtained.For present experiments,the interstitial heattransfer coefficient in the packed bed is determined using an inverse method of transient single-blowtechnique.The effects of packing form and particle shape are carefully investigated and the experi-mental and numerical results(Yang et al.,2010)are also compared in detail.Firstly,it is discoveredthat,the computational method reported by Yang et al.(2010)might be appropriate for heat transferpredictions in structured packings,while it might underestimate the friction factors,especially whenthe porosity is relatively low.Secondly,it is found that,the traditional Ergun’s and Wakao’s equationsmight overpredict the friction factors and Nusselt numbers for the structured packings,respectively,and some experimental modified correlations are obtained.Furthermore,it is also revealed that,boththe effects of packing form and particle shape are significant to the macroscopic hydrodynamic andheat transfer characteristics in structured packed beds.With proper selection of packing form,such assimple cubic packing(SC)and particle shape,such as ellipsoidal particle,the pressure drops in thestructured packed beds can be greatly reduced and the overall heat transfer performances will beimproved.These experimental results would be reliable and useful for the optimum design in industryapplications.&2011Elsevier Ltd.All rights reserved.1.IntroductionPacked beds,due to their high surface area-to-volume ratio,are widely used in a variety of industries,such as catalyticreactors,absorption towers,packed bed regenerators,high tem-perature gas-cooled nuclear reactors and heat accumulators,etc.In the past decades,theflow and heat transfer in the packedbeds,including random and structured packings,were exten-sively investigated by lots of researches.For example,for randompacking,Tsotsas(2010a)has well summarized the axial heatdispersion characteristics in the packed tubes.It is demonstratedthat,axial dispersion of heat in packed tubes withfluidflow is notdue to the effective thermal conductivity alone,but to a combina-tion of heat transport in the direction offlow,heat transferbetween particle surface andfluid,and heat conduction insideparticles.For low Peclet number,axial dispersion of heat is mainlydue to the effective thermal conductivity;for middle Pecletnumber,the effect offluid-to-particle heat transfer should bedominate;and for high Peclet number,the heat conduction in theparticles prevails.Tsotsas(2010b)has also well summarized theeffective thermal conductivity models for the packed beds.Itshows that,the effective thermal conductivity of packed beds isrelated to a variety of factors,including thermal conductivities ofparticles andfluid,porosity of packed bed,particle shape,particlesize distribution,mechanical properties of particles,thermody-namic properties offluid,etc.The solid-particle heat transfer inthe packed beds was also well summarized by Gnielinski(2010),where the definitions,determination methods and the modelvalidities of the solid-particle heat transfer coefficients in thepacked beds were carefully presented and discussed.Further-more,Carpinlioglu and Ozahi(2008)have experimentally studiedthe pressure drop characteristics of a variety of randomly packedbeds in turbulentflow of air.The measured pressure drops werecompared with the well-known Ergun’s equation(Ergun,1952)and the deviations were within an acceptable error margin.Asimplified correlation for the measured pressure drop wasobtained and it was observed to be correlated in terms of particleContents lists available at SciVerse ScienceDirectjournal homepage:/locate/cesChemical Engineering ScienceAbbreviations:SC,simple cubic;BCC,body center cubic;FCC,face center cubicn Corresponding author:Tel./fax:þ862982663502.E-mail address:wangqw@(Q.Wang).Chemical Engineering Science71(2012)126–137sphericity,porosity and Reynolds nfrey et al.(2010) recently have developed a theoretical model for the tortuosity of fixed bed randomly packed with identical particles.They found that,the tortuosity was proportional to a packing structure factor, which could well capture the balancing effect between porosity and particle sphericity.As porosity or particle sphericity decreased,the tortuosity increased and it did not depend on the particle size.A comparison between the performance inflow and heat transfer estimation of different turbulence models in a randomly packed bed were also performed by Guardo et al. (2005).It was found that,the Spalart–Allmaras turbulence model was better than the two-equation RANS model and the computa-tional results of pressure drop and heat transfer coefficient could fit well with the traditional empirical correlations(Ergun,1952; Wakao and Kaguei,1982).Some other recent studies for random packing were also reported by Nijemeisland and Dixon(2004)and Reddy and Joshi(2010).On the other hand,the investigations for structured packing were also popular,and theflow and heat transfer characteristics were found to be quite different.Susskind and Becker(1967)have experimentally measured the pressure drops of water in an ordered packed bed of stainless steel ball bearings.It was found that,as the relative horizontal spacing of balls increased,the pressure drop in the packed bed would be greatly decreased.Nakayama et al.(1995)have numerically studied theflow in a three-dimensional spatially periodic array of cubic blocks.It was discovered that,the macroscopic hydro-dynamic correlation obtained by their model couldfit well with that of Ergun’s equation(Ergun,1952),but the inertia coefficient was much lower.Calis et al.(2001)and Romkes et al.(2003)have investigated theflow and heat transfer characteristics in a variety of composite structured packed beds of spheres.It was revealed that,theflow and heat transfer performances in the composite structured packed beds were significantly affected by the packing form.With composite structured packings,the pressure drop could be greatly lowered and the traditional correlations(Ergun, 1952;Wakao and Kaguei,1982)were unavailable for structured packings.Furthermore,the localflow distributions at the pore scale of an ordered packed bed with single phaseflow were also measured by Lee and Lee(2009)and Dumas et al.(2010).The detail velocityfields in the packed bed were obtained with PIV (Lee and Lee,2009)and tri-segmented microelectrodes techni-ques(Dumas et al.,2010),respectively.It was found that,the local flow distributions inside packed bed were closely related to the internal pore structures.Similar investigations for structured packing were also well presented by Gunjal et al.(2005)and Hassan(2008).All these studies demonstrate that not only local behavior,but also macroscopic characteristics offlow and heat transfer are significantly affected by the internal structural properties of packed beds.The hydrodynamic and heat transfer performances in random and structured packings are quite different.The tortuosity and pressure drop in randomly packed bed are usually much higher and the overall heat transfer performance may not be optimal.While in structured packings,the pressure drops are usually much lower and the overall heat transfer performances may be better.Furthermore,it was noted that,the traditional correlations(Ergun,1952;Wakao and Kaguei,1982)could appro-priately formulate theflow and heat transfer characteristics in randomly packed beds,but it was questionable for structured packings.Based on these reasons,a direct simulation offlow and heat transfer process inside small pores of some novel structured packed beds with different packing forms and particle shapes was recently performed by Yang et al.(2010),where the packings of ellipsoidal or non-uniform spherical particles were investigated particle shape(such as long ellipsoidal particle),the pressure drops in the structured packed beds could be greatly reduced and the overall heat transfer performances were improved.The traditional correlations(Ergun,1952;Wakao and Kaguei,1982) were found to overpredict the pressure drops and Nusselt numbers for all the structured packings,and some modified correlations were obtained.Thesefindings would be of great help for further understanding theflow and heat transfer character-istics in the structured packed beds and they would also be useful for the optimal design of packed bed reactors.However,in this study(Yang et al.,2010),the forced convections in the structured packed beds were only investigated with numerical method and no experimental works were performed.This is far from sufficient for practical applications.Furthermore,during the computational process,in order to avoid generating poor quality meshes at contact points between particles,the particles were assumed to be stacked with small gaps(1%of particle diameter),which may lead to underestimations of friction factors and heat transfer coefficients in the packed beds.Therefore,in the present work,followed by the study of Yang et al.(2010),the forced convections in the similar structured packed beds are further investigated with experimental method, which would give us a more reliable apprehension of transport characteristics in the structured packed beds.According to the authors’knowledge,almost no such experimental studies were performed before(structured packings of ellipsoidal or non-uni-form spherical particles)and some important results would be obtained.For present experiments,the interstitial heat transfer coefficient in the packed bed is determined using an inverse method of transient single-blow technique(Younis and Viskanta, 1993;Hwang et al.,2002;Jiang et al.,2006).And the effects of Reynolds number,packing form and particle shape are investi-gated in detail.2.Experimental setup2.1.Experimental systemThe experimental system for investigation of macroscopic hydrodynamic and heat transfer performances in the structured packed beds is shown in Fig.1.It consists of an airflow circuit,a test section and several instruments.In the present study,air is induced to the wind tunnel by a centrifugal suction blower and the inlet temperature is read by a thermometer with precision of 70.11C.Before entering the test packed bed,the airflow is heated by passing through a removable electric heater(0–6kW)and then transverses the test packed bed,where the particles inside are heated by the hot air(T f r851C).After the packed bed temperature increases to701C,the electric heater is turned off and moved away.When the packed bed temperature is stabilized (60–651C),the cold air is sucked into the channel and the packed bed is cooled down until its temperature decreases to the ambient temperature(25–281C).During the cooling process,the experi-mental data are measured and recorded simultaneously.The volumetricflow rate through the test section is measured by a parallelflow meter system,which is situated at the downstream of the test section.Thisflow meter system is composed of three different rotameters(LZB-25: 2.78Â10À4–2.78Â10À3m3sÀ1, LZB-40:1.67Â10À3–1.67Â10À2m3sÀ1and LZB-80:1.39Â10À2–6.94Â10À2m3sÀ1)and their precisions are75%,72.5%and 74%,respectively.The static pressure difference across the test section is displayed by a micro-differential meter(Dywer-Ms-111-LCD:0–1000Pa)combined with a U-tube water columnJ.Yang et al./Chemical Engineering Science71(2012)126–137127constantan thermocouples with precision of 70.11C and the transient temperature signals are transformed and recorded by a real-time hybrid recorder (Keithley-2700)with a sample rate of 100Hz.2.2.Test sectionAs shown in Fig.2,the test channel is made of Plexiglas plates (thickness of 10mm)and the particles are orderly stacked inside.In present study,the test packed bed is composed of 10(x )Â5(y )Â5(z )packed cells,which would guarantee the fully developed flow and heat transfer inside.Continued with the study of Yang et al.(2010),five different structured packings are constructed,including SC (simple cubic packing with uniform spherical particles),SC-1(simple cubic packing with uniform ellipsoidal particles),BCC (body center cubic packing with uniform spherical particles),BCC-1(body center cubic packing with non-uniform spherical particles)and FCC (face center cubic packing with uniform spherical particles)packings.The non-uniform packing (BCC-1)is composed of eight big spherical particles (d p ¼12mm)at eight corners and one small spherical particle (d p ¼9mm)at body center,where the small particle is closely contacted with all big particles.Meanwhile,in order to reduce the the average inlet and outlet airflow temperatures are gauged by using two thermocouple racks (each rack with five beads)and the particle temperatures are monitored with the thermocouples embedded in the selected particles (each particle with one bead inside).The real photos for the test packed beds are shown in Fig.3,where the particles fixed with thermocouples in the central packed channel are also clearly presented.Furthermore,the geometric parameters and particle properties for the test packed beds are presented in Tables 1and 2,respectively.In Table 1,a ,b and c are the length,width and height of packed cell,respectively;L p ,W p and H p are the inner length,width and height of packed bed,respectively;f is the porosity;d p and d h are the equivalent particle diameter and pore scale hydraulic diameter,respectively,and their definitions are as follows:d h ¼4f 1Àf V pA p cell ;d p ¼23V p 4p1=3ð1Þwhere V p is the particle volume;A p is the area of particle surface;the subscript ‘‘cell’’means the value is obtained within packed cell.3.Measurement of interstitial heat transfer coefficient (h sf )Fig.2.Test packed bed:1.Thermocouples rack,2.Wire fence,3.Particle with thermocouple,4.Packed cell,5.Central packed channel and 6.Plexiglasplate.Fig.1.Experimental system:1.Thermometer,2.Removable electric heater,3.Thermocouples,4.Stabilization channels,5.Test packed bed,6.Differential meter,(Dwyer:MS-111-LCD),7.U-tube manometer,8.Data acquisition instrument (Keithley-2700),9.By-pass,10.Rotameter (LZB-25),11.Rotameter (LZB-40),12.Rotameter (LZB-80),13.Blower and puter.J.Yang et al./Chemical Engineering Science 71(2012)126–137128hard to directly measure the interstitial heat transfer coefficient in the packed bed.Therefore,an inverse method of transient single-blow technique is finally used,which is considered to be suitable and convenient for determining the interstitial heat transfer coeffi-cients in porous media (Younis and Viskanta,1993;Hwang et al.,2002;Jiang et al.,2006).Meanwhile,in order to reduce the wall effect as possible,the heat transfer characteristics is only investi-gated for the central packed channel (see Fig.2),which is evaluated from the outlet airflow temperature and double rechecked by the particle temperature distributions inside.The mathematical model is based on the following assumptions:(1)The airflow velocity and temperature of central packedchannel are uniform.The airflow and particle temperatures in the central packed channel only change along the flow direction and with time.Therefore,it is a transient,one-dimensional problem.(2)The airflow is incompressible and its influence on axialresistance of particles is also neglected.Therefore,both the airflow and particle’s thermal physical properties are consid-ered to be constants.(3)There is no local thermal equilibrium between airflow andparticles.With these assumptions,the transient,one-dimensional,non-equilibrium energy equations for fluid and solid phases in the central packed channel are established as follows:Fluid phase :f ðr c p Þf @T f þðr c p Þf u D @T f ¼f k f @2T f þh sf e ðT s ÀT f ÞSolid phase :ð1Àf Þðr c p Þs @T s @t ¼ð1Àf Þk s @2T s @x 2þh sf e ðT f ÀT s Þ8<:ð2Þwhere u D is the Darcy velocity in main flow direction;f is the porosity;T f and T s are the temperatures of fluid and solid phases,respectively;h sf is the area heat transfer coefficient of particle to fluid;e is the surface area-to-volume ratio within packed cell,e ¼ðA p =V p Þcell .The closed boundary conditions for airflow and particle phases based on experiments are used and the corresponding initial and boundary conditions are as follows:t ¼0,0r x r L p T f ¼T fi ,T s ¼T si t 40,x ¼0T f ¼T r ,@T s @x ¼0t 40,x ¼L p@T f @x ¼@T s @x ¼08>><>>:ð3ÞFig.3.Real photos for different test packed beds with thermocouples inside:(a)SC (uniform sphere);(b)SC-1(uniform ellipsoid);(c)BCC (uniform sphere);(d)BCC-1(non-uniform sphere)and (e)FCC (uniform sphere).Table 1Geometric parameters for the test packed beds.Packing modela/mm b /mm c /mm L p /mm W p /mm H p /mm fd p /mm d h /mm SC (uniform sphere)12.0012.0012.00132.0072.0072.000.47712.007.30SC-1(uniform ellipsoid)39.1011.7211.72430.1070.3270.320.47717.518.78BCC(uniform sphere)13.8613.8613.86150.6083.1671.160.32112.00 3.78BCC-1(non-uniform sphere)12.1212.1212.12133.2072.7260.720.27810.71 2.80FCC (uniform sphere)16.9716.9716.97181.7096.8584.850.26012.002.81Table 2Particle properties for the test packed beds.Particle shapeMaterialr /kg mÀ3c p /J Kg À1K À1k /W m À1K À1Sphericalparticle Bearing steel (GCr15)7810553.040.1Ellipsoidal particleGlass2500750.90.68J.Yang et al./Chemical Engineering Science 71(2012)126–137129ambient reference temperature of airflow,which are all obtained from experimental measurements.Introducing the following dimensionless variables:y f ¼T f ÀT r s ,i r ;y s ¼T s ÀT rs ,i r ;X ¼x p ;t ¼t p D ;Re ¼ðu D =f Þd h n f Pr ¼ðr c p Þf m f r f k f;St ¼h sfeLp ðr c p Þf u D;C 1¼d hL p;C 2¼d h k s ðr c p Þf L p k f ðr c p Þs;C 3¼ðr c p Þfðr c pÞs8<:ð4ÞThe governing energy equations (Eq.(2))can be re-written in the dimensionless form as follows:Fluid phase :@y f @t þ1f @y f @X ¼C 1f @2y f @X þSt f ðy s Ày f ÞSolid phase :@y s @t ¼C 2f PrRe @2y s @X 2þC 3St ð1Àf Þðy f Ày s Þ8<:ð5ÞThe corresponding dimensionless initial and boundary condi-tions are as follows:t ¼0,0r X r 1y f ¼y s ¼1t 40,X ¼0y f ¼0,@y s @X ¼0t 40,X ¼1@y f @X ¼@y s @X ¼08>><>>:ð6ÞThe flowchart for determination of heat transfer coefficient (h sf )is shown in Fig.4.Firstly,a guessed value of h sf is used together with the prescribed values of geometric properties,physical properties,volumetric flow rate,measured inlet and initial temperatures.Then the solid and fluid temperatures (y s and y f )are numerically solved from Eq.(5).If the absolute deviation between the predicted and measured outlet airflow temperatures is less than 0.01,the predicted h sf is regarded to be equal to the measured h sf .Otherwise,the value of h sf is modified and the energy equations (Eq.(5))are resolved until the convergence condition is finally satisfied.In present study,the dimensionless energy equations (Eq.(5))are solved based on the finite controlvolume method and the transient finite difference form of the energy equations is solved implicitly with line-by-line iterative method.For convergence criteria,the relative variation of tem-perature between two successive iterations are demanded to be smaller than the previously specified accuracy levels of 10À6.Before proceeding further,the effects of grid number and time-step are checked first.The simple cubic packing with uniform spheres (SC)is selected for the test (see Figs.2and 3(a))and the predicted outlet airflow temperatures (y f,out )with different sets of grids (40,70,100and 150)and time-steps (2,1,0.5and 0.2)are presented in Table 3.It shows that,the computational results with grid number of 100(D X ¼0:01)and time-step of 0.5are satisfactory.Therefore,the sets of D X ¼0:01and D t ¼0:5are finally adopted for the following studies.Furthermore,the com-parisons of measured and predicted temperatures of fluid and solid phases for representative packings are also presented in Fig.5.It shows that,both the predicted temperatures of fluid and solid phases for different packings can fit well with those from experimental measurements,which indicates that,the inverse method of transient single-blow technique presented here is suitable and capable of predicting heat transfer coefficient (h sf )in the packed beds.4.Data reductionIn the present study,the pressure drop (D p =D x ),friction factor (f ),Nusselt number of particle to fluid (Nu sf )and overall heat transfer efficiency (g )in the test packed bed are defined as follows:D p D x ¼D p expL p;f ¼D p exp =L pð1=2Þr f ð9u D 9=f Þ2=d h;Nu sf ¼h sf d Pk f;g ¼h sfD p exp =L pð7Þwhere D p exp is the total static pressure difference across the test packed bed;L p is the total length of the test packed bed;u D is the Darcy velocity in main flow direction;h sf is the area heat transfer coefficient of particle to fluid;d p and d h are the equivalent particle diameter and pore scale hydraulic diameter,respectively,which are defined as in Eq.(1).According to the previous studies of Ergun (1952),Wakao and Kaguei (1982),Calis et al.(2001),Romkes et al.(2003)and Yang et al.(2010),the friction factor (f )and Nusselt number (Nu sf )in the packed beds can also be formulated with following correla-tions f ¼c 1þc 2ð8ÞNu sf ¼a 1þa 2Pr1=3Rend Pd hfn ð9Þwhere c 1,c 2are the friction factor constants,with c 1¼133and c 2¼2.33in Ergun’s equation (Ergun,1952);a 1,a 2and n are the heat transfer model constants,with a 1¼2.0,a 2¼1.1and n ¼0.6in Wakao’s equation (Wakao and Kaguei,1982).Fig.4.Flowchart for determination of heat transfer coefficient (h sf ).Table 3Predicted y f,outfor SC packing (sphere)with different grids and time-steps (Pr ¼0.7,Re ¼1203).Grid (D t ¼0.5)Time-step (D X ¼0.01)Experiment4070100150210.50.2y f ,t ¼21140.2720.3110.3230.3250.2810.3170.3230.3240.329J.Yang et al./Chemical Engineering Science 71(2012)126–137130Furthermore,according to the book of Nield and Bejan (2006)and Eq.(8),the permeability (K )and inertial coefficient (c F )for fluid flow in the packed bed porous media can be defined as follows:K ¼d 2h f c 1=2;c F ¼c 2=2ffiffiffiffiffiffiffiffiffiffic 1=2p f3=2ð10Þwhere c 1,c 2are the friction factor constants as presented in Eq.(8).5.Results and discussion5.1.Experimental results for different structured packing forms Firstly,the experimental results for different structured pack-ing forms are presented.Three different kinds of packing forms are studied in this section,including SC,BCC and FCC packings (see Figs.2and 3(a),(c)and (e)).The variations of measured pressure drops (D p =D x )and fric-tion factors (f )for different packing forms are presented in Fig.6.In Fig.6(a),it shows that,as Re increases,the pressure drops of with the numerical analysis of Yang et al.(2010).In FCC packing,the porosity is the lowest and the airflow velocity inside would be the highest,which will lead to the highest pressure drop inside.Furthermore,in the present study,the Reynolds number is relatively high (Re 4100).The inertia effect is significant and Ergun’s equation (random packing)is found to overpredict the pressure drops for structured packings.This is also consistent with the study of Yang et al.(2010),which confirms that the hydrodynamic characteristics of structured and random packings are quite different.The tortuosities in structured packings are much lower and the pressure drops would be greatly reduced.In Fig.6(b),it shows that,as Re increases,the friction factors decrease first and then keep constant (Re 43000).The friction factor of SC packing is found to be higher than those of BCC and FCC packings.In SC packing,the porosity is rge airflow channeling and vortices would be formed,which will lead to higher local tortuosity and friction factor inside.The model constants c 1,c 2in friction factor correlation (in Eq.(8))for present study are obtained based on the experimental points with non-linear fitting method and the average fitting deviation is less than 10%.The values of c 1,c 2for different packing forms are listed in Table 4,where the results of Yang et al.(2010),Martin et al.0.00.20.40.60.8θfτ0.00.10.20.30.40.50.60.70.00.20.40.60.81.0τ0.00.10.20.30.40.50.60.7XθsθfXθsparisons of measured and predicted temperatures of fluid and solid phases for representative packings:(a)y f (SC);(b)y s (SC);(c)y f (BCC)and (d)y s (BCC).J.Yang et al./Chemical Engineering Science 71(2012)126–137131equation (random packing),while the values of c 2are much lower.This means,the viscosity effect (represented by c 1)in the packed bed is not so sensitive to the effect of packing form,while the inertial effect (represented by c 2)is quite different.The inertial effect in structured packing is much lower.Furthermore,due to the higher tortuosity inside,the value of c 2for SC packing is found to be higher than those for BCC and FCC packings.In addition,it is observed that,for SC packing,the experimental results can agree well with those reported by Martin et al.(1951)and Yang et al.(2010).While for BCC and FCC packings,the 2010),in order to avoid generating poor quality meshes at contact points between particles,the particles in the packed beds were assumed to be stacked with small gaps (1%of particle diameter).For SC packing,the porosity is relatively rge straight airflow channeling would be formed inside and the effect caused by the small gaps to the fluid flow would be small.Therefore,the numerical simulation (Yang et al.,2010)would be suitable for SC packing.While for BCC and FCC packings,the porosity is much lower.The airflow channeling inside is smaller and flexuous.Therefore,the effect caused by the small gaps would be larger and the numerical method (Yang et al.,2010)might underestimate the friction factors in BCC and FCC packings.The variations of measured Nusselt numbers (Nu sf )and overall heat transfer efficiencies (g )of particle to fluid for different packing forms are presented in Fig.7.In Fig.7(a),it shows that,as Re increases,Nusselt numbers of different packings increase.The value of Nu sf for FCC packing is the highest and it is the lowest for SC packing.In FCC packing,its internal structure is the most compact and the surface area-to-volume ratio is the highest,which would lead to the highest heat transfer capacity inside.Furthermore,Wakao’s equation (random packing)is found to overpredict the Nusselt numbers for structured packings,which is also consistent with the numerical studies (Yang et al.,2010).The model constants a 1,a 2and n in heat transfer correlation10210310101010-1Δ p Δ x / P a m-1Re10210310101fReFig.6.Variations of measured pressure drops and friction factors for different packings with spherical particles:(a)D p D x À1/Pa m À1and (b)f .Table 4Values of c 1and c 2(sphere).Packing modelfd h /mm c 1c 2SC (sphere,experiment)0.4777.30145.300.99SC (sphere,(Yang et al.,2010))0.4927.75143.880.88SC (sphere,(Martin et al.,1951))0.4777.30139.210.80BCC (sphere,experiment)0.321 3.78142.250.81BCC (sphere,(Yang et al.,2010))0.340 4.12129.810.37FCC (sphere,experiment)0.260 2.81155.000.82FCC (sphere,(Yang et al.,2010))0.282 3.14164.120.30Random (Ergun,1952)//133.002.331031010N u s fRe1010101010Reγ / W m -3 K -1 P a -1Fig.7.Variations of measured Nusselt numbers and overall heat transfer effi-J.Yang et al./Chemical Engineering Science 71(2012)126–137132。