Significance of core diameter variation in the lateral solid transfer model for the upper dilute region of circulating fluidized beds

Significanceof core diameter variation in the lateral solid transfer model for the upper dilute region of circulating fluidizedbeds MASAYUKI HORIO andHONGWEI LEI andTechnology, Department ofChemical Engineering, Tokyo University of Agriculture 184, Koganei, Tokyo Japan Received 20August 13November 1997 1997; accepted Abstract-The formulation ofanaxial solid concentration intheupper dilute ofcirculating profile region fluidized beds isanalyzed based onthelateral solid transfer model which includes mass transfer of between thecore andtheannulus, tothecountercurrent The particles analogous absorption process. ismapped onaphase ofsolid concentration inthecore versus thatin operating region plane region theannulus Itisdemonstrated that ifthecore isassumed diameter themodel faces constant, region. aserious ofhaving either a negative solid fraction orapositive paradox decay factor/positive εpC,∞ solid fraction This was found tobesolved of decay factor/negative paradox bytheintroduction εpC,∞. avariable core diameter The calculated results ofsolid concentration theriser with assumption. along Werther correlation forcore diameter distribution showed a fairagreement with previous experimental results. NOMENCLATURE a factor (m-1) decay sectional area(m2) A diameter ofparticles (m) dp column diameter D (m) Froud number Fr (-) Uol(gD)0.5 = 9.81 9 gravitation acceleration, g m/s2 solidflux(kg/m2 Gs s) lateral solidtransfer orsoliddeposition coefficient k (m/s) K constant,=K (-) kAC/kcA equilibrium riserheight L (m) Re Re= DUop//L number, (-) Reynolds Stokes S= dgUoPp/18/LD s number, (-) turbulent inequation fluctuation u' (11)(m/s) velocity

108 Va t) v() w z

(m/s) superficial gasvelocity solidvelocity (m/s) solidvelocity, vo= G> /pp (m/s) superficial solidflowrate(kg/s) (m) height

Greek a' s JL p

dimensionless corediameter, a = Dc/ Dt (-) volume (-) voidage (Pa s) viscosity density (kg/m3)

Subscripts 1 A C d eq p t vex)

riserexit annulus core dense region, deposition equilibrium particle tube above TDH(transport disengaging height)

1.INTRODUCTION i.e.solidupflow inthecenter Core-annulus anddownflow inthewall flow, (core) is thebasicfeature ofthemacroscopic flowstructure in boundary layer(annulus), fluidized beds(CFBs). Theexistence ofcore-annulus flowatleastinthe circulating dilute ofCFBs hasbeenconfirmed forbothlaboratory-scale andlargeupper region scaleunits[1 ].Itwasfound thattheformation ofthecore-annulus structure isdue totheradial distribution ofturbulent inthedilute asinvestigated intensity region by Horio al. et[5]andPemberton andDavidson ofbubbling [6]forthefreeboard region fluidized beds.It canbeeasily thattheturbulent nearthewall postulated intensity should besmaller thanthatintheregion fromthewallboundary once away layer; orclusters arecaptured diffuse backout bytheboundary particles layer, theycannot ofthislayer.Accordingly, lateral solidtransfer fromthecoretotheannulus takes thesolidfraction From sucha radial evidence, gradient. placeagainst experimental movement ofclusters tothewallboundary hasbeenvisualized layerinCFBs bythe lasersheet [7]. technique Sofar,hydrodynamic models tocharacterize thesolid distribution proposed hold-up inCFBs asa function ofgasvelocity, solidmassflux,risergeometry andparticle

109 classified intothree concerned characteristics canberoughly (1)those only categories: theradial withtheaxial [8-11]; (2)those profile bythe voidage profile approximating models division oftheflowintotwoormoreregions, [3,12-22]; e.g.core-annulus onthefundamental conservation flow based and(3)those equapredicting two-phase fluiddynamic tionsandwiththeaidofcomputational [23-25]. techniques toosimple ortoocomplicated for them, (1)and(3)areeither Among categories Incategory flowstructure hasfound (2),core-annulus practical design purposes. butfurther examination isrequired notonlyforthefeasibility acceptance, widespread ofthemodel Inprevious models ofthiskind, butalsoforthereliability parameters. andBrereton model as10parameters [17],asmany (atleastthree e.g.intheSenior Inthemodel needtobedetermined. ofBerruti etal.[19,21,22],the tentative ones) tobedivided intoanacceleration anda fullydeveloped riserwassupposed region thismodel, astheyalsodeclared, cannotaccount for'thesituaHowever, region. thismodel bedexists atthebaseoftheriser'[22].Moreover, tionswhere a dense theeffect oftheacceleration termonthepressure the drop;however, emphasized lessthan1%compared withthegravity term acceleration contribution onlyremains theirlowsolid theirhighsolidfluxconditions 210kg/m2 let under s),alone decrease itis difficult toexplain thesolidfraction fluxconditions. Therefore, along withthemodel ofBerrutial. etThethree-phase model ofKunii and theriserheight thanthetwo-phase model butthenecessity of [20]maybemorerealistic Levenspiel suchacomplication isdoubtful. Inaddition, these aconstant models, among previous corediameter wasassumed fortheupper dilute which is [3,12-15,18-21] region incontrast withtheexperimental reviewed [1]andBietal.[26]. findings byWerther indeveloping wewillfocusontheeffect ofcorediameter theCFB Accordingly, solidtransfer model between thecoreand andthenpropose a simple lateral model, inthedilute ofCFBs canbe theannulus sothatthesolid concentration profile region more reliably predicted. LATERAL SOLID TRANSFER MODEL 2.FORMULATION OFTHE THE FOR DILUTE REGION OFCFBS thedownflow ofsolidintheannulus andupflow Inthislateral solidtransfer model, in Fig.1. Thefollowing threelumping in thecoreareconsidered asillustrated forthepresent formulation: aremade assumptions uniform inthecoreandtheannulus. fractions andvelocities areradially (1)Solid fluxfromregion x (x = C or A)to y (y = A orC)is (2)Thesolidtransfer tothesolid fraction inregion solidfluxfromthe x,sothenetradial proportional iswritten as coretotheannulus solidtransfer coefficient fromregiontoxregion where y. kxyisthelateral are in thecoreandintheannulus, ucandVA, respectively, (3)Solidvelocities overtheheight. assumed constant

110

model. 1.Lateral solid transfer Figure fraction inthecoreinequilibrium withthatintheannulus, Defining thesolid as wecanrewrite (1)as equation sincethenetparticle transfer is fromthecoretotheannulus, K mustbelessthan unity. as Theoverall solidmassbalance ateachelevation canbewritten thesolidupflow rateinthecoreandthedownflow ratein where wcandwAdenote solidvelocity(x Thearea-averaged theannulus, vx= CorA)canbe respectively. andthedimensionless interms ofwx,thearea-averaged solidfraction defined Epx corediameter a as

as Thenequation (4)canberewritten

111 solidvelocity. massbalances forthesolid where Therefore, vois thesuperficial as anddownflow aregiven upflow

theradial solidfraction Asanadditional relationship, average 8pcanbeexpressed as: Aftertheintroducing ofequation ofequa(5)intoequation (7)andrearrangement ofthepresent aresummarized thegoverning tion(6),respectively, equations system as and therelationship between (10)expresses Equation -,c and-pA-Inthelateranalysis, theoperating lineofEpcandspAforCFBs, forconvenience, (10)istermed equation oftheoperating line. where theslope (1 - (2) ( - v A)isItermed œ2 Vc 3.PREDICTION OFMODEL PARAMETERS which Intheabove wehave K,UC, anda asmodel formulation, VA parameters, aredetermined inthefollowing manners. 3.1.Lateral solidtransfer kcA (coretoannulus) coefficient Inthedilute ofa bubbling fluidized soliddiffusion to bed,thelateral region appears beinduced caused or,inother words, bytheturbulence bythebubble eruption bythe eddies of'ghost bubbles' Sherwood number4:(forturbulent [5, 6]. flow) Assuming = 4/Dt,Pemberton thedeposition andeddywave number andDavidson [6]related fluctuations andproposed coefficient totheamplitude ofvelocity

where Stokes numberwas S defined as

112

Uo[m/s] superficial gasvelocity 2.Comparison ofexperimental results fordeposition coefficient from Bolton and Figure (reproduced Davidson [27]). Forcirculating fluidized andDavidson theamplitude of beds,Bolton [27]modified turbulent fluctuation u' by velocity theReynolds number where Rewasdefined as

Thentheyusedthedeposition coefficient themasstransfer coefficient kdtoestimate in theupperdiluteregion ofCFBs.Based onthecorrelations in theliterakCA ture[28-32], theexperimental conditions ofBolton andDavidson wereconfirmed to fallinthefastfluidization theirexperimental dataforkdscattered However, regime. asshown inFig.2 (relative deviation: -35to+170%) andtheeffect ofthe widely solidfluxon kd wasnotincluded. inthepresent a different work, kCA Accordingly, is applied intoaccount oftheeffect ofsolidflux(GSfrom11.7to byalsotaking 251kg/m2 s intheliterature forboth [2-4]),i.e.kcA *kd= 0.02m/sis assumed cases ofWeinstein etal.[2]andHorio al. et[3],which liesinthetolerance of range thiscorrelation andtheestimated (+50%), kcA= kd= 0.30m/sisusedforthecase ofArena al. et[4]. 3.2.Equilibrium constant K There hasbeanlittleresearch between thesolidfraction inthe ontheequilibrium coreandthatintheannulus. theinvestigation ofWirth andSeiter[33] However, andGohla between and thattherewasa linear [34]showed relationship spc.eq/lp

113

solid fractions inthecore andtheannulus from Wirth andSeiter 3.Equilibrium (data [33] Figure and[34]). Table 1. K Calculated constant equilibrium

fromwhich asshown inFig.3forvarious conditions, operating equilibrium EpA,eq/Sp constants wereestimated andsummarized inTable 1.Ifwejusttakeaccount ofthem thatthe intheupper dilute i.e.z >4.18mforGohla's case,it canbenoted region, forUo= 2-7m/sand constant Kliesalmost intherange of0.35-0.75 equilibrium conditions Kisalmost theriserheight under thesameoperating keptconstant along K = 0.35is chosen forthelastpairofdata.Inthefollowing calculation, except K = 0.5forthatof fortheanalysis ofWeinstein etal.'s[2]experimental result, et ul.'s[4] Horioetul.'s[3]experimental resultandK = 0.10forthatofArena result under theconsideration ofthesolidfluxeffect. experimental

114 Table 2. Calculated based onexperimental data vc,VaAndVCIVA

3.3.Thechoice solidvelocity ratiovc/vA ofarea-averaged Insome oftheprevious thesolidvelocity intheannulus wasassumed conmodels, m/ss in theSenior andBrereton model orastheparticle terminal stant, e.g. -1.1 [17], andKalogerakis model which wasnotconsistent with velocity, [13], e.g.intheBerruti theexperimentally observed results. From (10)andthelaterstated equation boundary thatwejustneedtoknow thevalue ofuc/uA other (27),it canbefound equation thaneachspecific value ofucorvAsinceucisdetermined fromtheoperating solid fluxandtheexitsolidfraction. Table 2 liststheexperiment results intheliterature forvc,VA andtheratiouc/L'AItcanbefound thatv?/vA liesintheregion of -1.75 to-2.61forUo= 1.17-4.0 m/sandGs= 11.7-98 ofthe kg/m2For s. simplicity model theratioofthesolidvelocity inthecoretothatintheannulus is calculation, = -2. chosen as-2, i.e.VC/VA 3.4.Dimensionless corediameter a alongtheheight With tothecorediameter correlations have beenproposed Dc,several respect [1,26, aresummarized intheformofdimensionless corediameter a as 39,40],which follows:

115 AsforthePatience andChaouki theeffects ofgasvelocity, solidcircorrelation, culation fluxandparticle weretaken intoaccount, butit cannot the density predict corediameter variation withheight, which thata didnotvaryalong theriser implied Inthelatter twocorrelations ofBaietal.andBietal.,thesolid fraction inthe height. risershould beknown first.IntheWerther theeffects ofsolid circulation correlation, fluxandexitrestriction arenotincluded, butthiscorrelation hasbeenconfirmed also i.e.Duisburg boilers, bythedatafromlarge-scale plant(Dt= 8 m)[1]andEDF E.Huchet tobea goodcandidate. plant(Dt= 9.65m,125MW)[41],soit seems inthepresent Werther's correlation isadopted. calculation, Accordingly, 4.SOLUTION FOR ACONSTANT CORE DIAMETER AND ITSASSOCIATED DIFFICULTY between as (2)and(10),wecanrewrite Eliminating equations spA EpC,eq

After intoequation for-pcwitha constant core (9)andintegrating introducing EpC.eq weobtain diameter,

where dense and zdisthelower region height

116

inthephase ofFpA versus forconstant a. 4.Paradox ofoperating lines Figure plane Epc between isobtained (8)and(10)andintroducing byeliminating equations EpA spoo (22)intoequation (8)forz-+ 00as equation

(20)hasthesameformastheempirical expression proposed by Although equation inequations ZenzandWeil[42],thereisa serious (21)and(22)(or(23)). difficulty factor wemayhaveeither Thatisifuo,ucand(-vA)areallpositive, negative decay areallpositive a ornegative solidfraction Botha and onlywhen which is thesolidflowintheannulus isupward orthenetsolidfluxis downward, asfollows: analyzed Inequations (21)and(22),thesame partis

Ifa and and

areallpositive,

117 should havethesame ornegative. Because thesolid flowisupward signforpositive . inthecoreofa CFB, i.e.vc> 0,wehavethepositive

Therefore, should bepositive. Thiswillleadtotwosituations: should (1)when voispositive, VA bepositive flowVA intheannulus isupward), which there isno (i.e.thesolid implies soliddownflow intheannulus and(2)when solidflowintheannulus isdownward benegative, which thecirculating solidfluxinthe (i.e.VA< 0),uoshould implies riserisdownward solid fluxintheriser). 4 (i.e.thereisnoupward circulating Figure demonstrates thisparadox whenvo,vcand(-vA)areallpositive. Foroperating <0 while lineI, a > 0;foroperating lineII,eP?,?> 0 while a <0. 5.SOLUTION TOVARIABLE CORE DIAMETER INTHE LATERAL SOLID TRANSFER MODEL Fromequations linefora variable a casesatisfies the (9)and(10),theoperating tothederivation inAppendix (refer I): following equation

where theslope oftheoperating linechanges withtheheight incontrast to Accordingly, thefollowing constant forthecaseofconstant a slope

Since besolved it is numerically with (24)cannot equation analytically, integrated Werther's correlation (i.e.equation (16)).Inthiscase,theoperating line,i.e.EpA versus a curve instead ofa straight one. Epcforms

118 Theboundary atz = L,which conditions arederived fromequations (2)and(16) andlimitanalysis onequations in (6),(8),(10)and(24)(referto thederivation II),areasfollows: Appendix

values ofsolidfractions andEpC.d (8)and(10),theinitial Bysolving equations epa.d atthedense areobtained from Zdforintegration region height

Forthetimebeing, atZdisdetermined fromexperimental data.Thecomputation 8pd isshown inFig.5. procedure

5.Calculation ofthelateral model forvariable a. solid transfer Figure procedure

119 6.DEMONSTRATION OFVARIABLE OFTHE SIGNIFICANCE CORE DIAMETER LATERAL INTHE SOLID TRANSFER MODEL Tovalidate themodel derived a comparison wasmade forthecalculated and above, axialsolidfraction Theoperating conditions andexperimental experimental profile. results intheliterature inTable 3.Thenumerical forthe [2-4]arelisted integration solidfraction wasperformed withtheRunge-Kutta method fromthedense region conditions andinitial values aredetermined from Zdtotheriserexit;boundary height (26)-(31). equations Thecalculated results aregiven inTable 4. Figure 6ashows thephase and plane ofthecalculated withtheobserved axialprofile of8pforHorio etal., comparison etal.andFig.6cforArena etal.Forthevariable corediameter, Fig.6bforWeinstein theoperating linebends itproceeds when totheequilibrium lineintheupper region asshown intheinsets inFig.6.Theaxialsolid fraction show fairagreement profiles withtheexperimental results. Thisindicates thatthedecay inthedilute phenomenon ofCFBs canbeinterpreted lateral solidtransfer region bythepresent two-region model. Inthecaseofa variable corediameter, thenetsolidfluxis always tothe equal ifthecorediameter isassumed itturnsoutthat Gs.However, constant, experimental thenetsolidfluxbecomes withtherealistic value ofdspc/dspA asanalyzed negative above. tothenumerical results ofHorio al. et[3]inTable of 4,theslope According fromequation (24)as: spAversus Epcatz =Zdiscalculated

theabove 'realistic' oftheoperating lineatz = zdandsubstituting it Adopting slope intoequation corediameter weobtain (25)fortheconstant model,

Table 3. conditions and results Operating experimental

120 Table 4. core diameter results forthevariable Calculated

solid transfer model and with ofvariable a inthelateral 6.Significance comparison experimenta Figure etal.[2]; etal.[4]. results: etal.[3]; (b)forWeinstein (c)forArena (a)forHorio as Thenthenetsolidfluxiscalculated

model. corediameter Thisisanexample oftheproblems fortheconstant

121

6.(Continued). Figure 7.EFFECTS OFMODEL PARAMETERS AND ONTHE OPERATING kCA uc /uA PHASE PLANE AND AXIAL SOLID FRACTION PROFILES InFig.7,theeffect ofthelateral solidtransfer coefficient kcAontheoperating phase is shown.If kCA is toobig,thecontribution of planeandsolidfraction profiles variable corediameter ifitistoosmall, issuppressed andevennegative appear; 8p,oo theeffect ofthevariable corediameter is sosignificant thatthe alongtheheight willbereached inthemiddle oftheriserheight. Accurate of equilibrium prediction

122

ofkeA onoperating lines and axial solid 7.Effect voidage profiles. Figure

lines andaxial solid 8.Effect ofUC/VA onoperating voidage profiles. Figure thesolidfraction inthelateral solidtransfer model topredict kcAis thusnecessary inthedilute region. is Theeffectof v?/vAontheoperating phaseplaneandsolidfraction profile shown inFig.8. It canbeknown fromequations (30)and(31)thatthevariation inthecoreandtheannulus atthedense leadstodifferent solidfractions ofVCIVA forceofthelateral solid to theinitial Zdwhich driving region height correspond If uc/(-uA) thedriving forceatZdis toobig,sothatthe transfer. is toosmall,

123 solidtransfer attheriserexit;ontheotherhand,if cannot bereached equilibrium istoolarge, theinitial forceatZdistoosmall sothesolidtransfer uc/(-uA) driving reaches theequilibrium much below theriserexit.Sinceucinthismodel isjust determined conditions thevariation ofVC/(-VA) (seeequation (27)), bytheoperating thevariation of-Up.Sucha variation reflects influence of -vAwillstrongly particle andclustering intheannulus, andthenaffects thelateral solidtransfer agglomeration fromthecoretotheannulus. Therightdirection ofCFB research should betofocus ontheclustering behavior of asalready stressed suspension byIshiietal.[14]andsucha meso-scale suspension structure hasbeenrecently ina moreprecise waybythelasersheet investigated models arestillpreferred tocomplete thepresent [7,42];some technique simplified model ofpredicting theaxialdistribution conditions. capable onlyfromoperating 8.CONCLUSIONS Alateral solidtransfer model based onthemasstransfer between thecoreandthe annulus inthedilute ofCFBs is solved forbothconstant andvariable core region diameter cases.Thevariable corediameter model withanempirical corediameter correlation canpredict thesolid volume fractions well.Theconstant corediamfairly etermodel, failstosatisfy alltherequirements ofthepositive however, equilibrium solidfraction, thepositive factor andthepositive solid massflux.Itwasmade decay clear thatinthecore-annulus solid transfer model thecorediameter variation isakey factor toobtain realistic which indicates thatthevariable corediameter predictions, hasa significant roleindeveloping theCFBmodel. Further isneeded investigation forthedetermination ofkCA andVCIVA. REFERENCES 1.J.Werther. Fluid mechanics oflarge-scale CFB units. In:Circulating Bed Fluidized lV, Technology A.A.Avidan New York AIChE, (Ed.). (1993), pp.1-4. 2.H.Weinstein, R.A.Graff, M.Meller and M.J.Shao. The influence oftheimposed pressure drop across afast fluidized bed. In:Fluidization and R.Toei lV,D.Kunii (Eds). Foundation, Engineering New York (1984), pp.299-306. 3.M.Horio, K.Morishita, O.Tachibana and N.Murata. Solid distribution and movement incirculating beds. In:Circulating fluidized Bed . and J.F.Large Fluidized II,PBasu Technology (Eds). Pergamon Oxford Press, (1988), pp.147-154. 4.U.Arena, A.Marzocchella, L.Massimilla and A.Malandrino. ofcirculating fluidized Hydrodynamics beds with risers ofdifferent andsize. Powder Technol. 70, 237-247 (1992). shape 5.M.Horio, A.Taki, Y.S.Hsieh and I.Muchi. Elutriation and the freeboard particle transport through ofagas-solid fluidized bed. In:Fluidization and J.M.Masten Plenum III,J.R.Grace Press, (Eds). 509-518. New York (1980), pp. - II.Disengagement 6.S.T.Pemberton and J.F.Davidson. Elutriation from fluidized beds ofparticles from Chem. Sci. 41,253-262 (1986). Eng. gasinthefreeboard. 7.M.Horio and H.Kuroki. Three-dimensional flow ofdilutely solids inbubbling visualization dispersed and fluidized beds. Chem. Sci. 2413-2421 49, (1994). Eng. circulating . and 8.YLi M.Kwauk. The offast fluidization. In:Fluidization and J.M.MasIII,J.R.Grace dynamics ten(Eds). Plenum New York Press, (1980), pp.537-544.

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125 M.Seiter. Solids concentration and solids inthewall ofcirculating 33.K.E.Wirth and velocity region In:Proc. 11th Int.Conf. onFluidized Bed E.J.Anthony fluidized beds. Combustion, ASME, (Ed.). Canada Montreal, (1991), pp.311-315. zurBerechnung desstromungsverhaltens von zirkulierenden WirbelVier-Zonen-Modell 34.M.Gohla. Otto von Guericke Doctoral thesis, (1994). schichtfeuerungen. University, Magdeburg, Germany in a 30.5-cm-diameter circuJ.Findlay and T.D.Knowlton. Gas/solids flow 35.D.R.Bader, patterns Bed andJ.F.Large fluidized bed. In:Circulating Fluidization II,P.Basu (Eds). Technology lating Oxford Press, (1988), Pergamon pp.123-137. - first . ndersson M.R.Golriz, W.Zhang, B.AA andF.Johnson. 36.B.Leckner, Boundary layers In:Proc. inthe12MW CFB research atCharmers 11th Int.Conf. measurements plant University. Bed E.J.Anthony Canada onFluidized Combustion, ASME, Montreal, (Ed.). (1991), pp.771-776. and Solid mass fluxes incirculating fluidized beds. Powder S.Dou, K.Tuzla J.C.Chen. 37.B.Herb, Technol. 70,197-205 (1992). and for Interaction ofpressure and diameter onCFB 38.T.D.Knowlton. pressure drop hold-up. Paper andControl Systems', (1995). 'Modeling of Fluidized-bed Hamburg Workshop: intheriser ofacirculating fluidized-bed. 39.G.S.Patience and J.Chaouki. Gas phase hydrodynamic Chem. Sci. 48,3195-3205 (1993). Eng. N.Nagakawa KKato. . Distinction 40.D.R.Bai, E.Shibuya, Y.Masuda, K.Nishio, and between upward beds. Powder Technol. and downward flows incirculating fluidized 84,75-81 (1995). inorder 41.L.Lafanechere and L.Jestin. of a fluidized bed furnace behavior to scale Study circulating onFluidized Bed K.J.Heinschel itupto600 MWe. In:Proc. 13th Int.Conf Combustion, (Ed.). Orlando ASME, (1995), pp.971-977. ofparticle 42.F.A.Zenz and N.A.Weil. Atheoretical-empirical tothemechanism entrainment approach from fluidized beds. AIChE J.4,472-479 (1958). M.Ito,H.Kamiya andM.Horio. Three-dimensional ofparticle clusters in 43.M.Tsukada, imaging dilute flow. Can. J.Chem. 75, 466-470 (1997). Eng. gas-solid suspension APPENDIX I:DERIVATION OFEQUATION (24) (9),weobtain Rearranging equation

From (5)and(7),wecanwrite equations

wehave Accordingly,

126 intoequation (1.3) (1.1) yields Introducing equation

weobtain (10)andrearranging Substituting spi= '0 intoequation

intoequation (1.5) (1.4): (24)isobtained byintroducing equation Accordingly, equation

OFEQUATION II:DERIVATION APPENDIX (29) thelimitanal(10)andperforming (27)and(28)intoequation Substituting equations ysisforz L,wecanwrite

weobtain Ontheother hand,dividing (1.1) by(1/œ2)(dœ2/dz), equation

= 0. Accordingly, is When z - L, wehave(KspA8,C)z=L (11.2) equation as rewritten

127 andrearranging, wecanwrite and(11.3) (IL 1) Eliminating equations 6p,between

intoequation (29) (25)forz-j L,weobtain (11.4) equation Introducing equation asfollows: