AN ANALYSIS OF A NONISOTHERMAL ONE-COMPONENT SORPTION IN A SINGLE ADSORBENT PARTICLE.pdf

8/14/2019 AN ANALYSIS OF A NONISOTHERMAL ONE-COMPONENT SORPTION IN A SINGLE ADSORBENT PARTICLE.pdf

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AN ANALYSIS OF A NONISOTHERMAL ONE-COMPONENTSORPTION IN A SINGLE ADSORBENT PARTICLEA BRUNOVSti,t V HLAVkEK,S J ILAVSK? and J VALtiNI

Department of Chemical Engmeenng aad Inorgamc Technology. Slovak Techmcal Unwerstty,8gO37 Bratrslava, &&a Street1. CzechoslovakmRecewed 5 kxm bc r l V7, a c c e p t e d 21 Mad 1978

-Heat and mass balances describingthe process of a nomsothermal smpbon a an adsorbent part%zleare formulated Apphcatlon to gas-sohd adsorption systems where s&cant thermal effects occur ISpresentedThe govermng equ&ons representa set of couple-d strongly nonhear parabohc equahcms A fimtedtfferencemethod based onexphclt-uaphcfiprocedure spropo&toapproxlmatepproxlmatesctofequatloasheeqll&onsarcsolvedfor a case of a strong adaorpt~~~ccompamcd vntb mcaut heat gencrat~~~.~ffects for mokcuhu and KaudsenchEus1011t 1s shownthat for the molecular dlffuslon the mbaparhcle temperature Merences may be of the order ofmagmtude lo-WC1 INTRoMJcTlm

The adsorption of a component of a gas mixture on aporous solid adsorbent may be represented as a three-step process (1) Ddfuslon of the adsorbate from the bnlkflow of the gas phase to the external sohd surface of theparhcle. (u) Pore dlfEuslon m the adsorbent particle, and(tu) Adsorption on the pore surfaceIn a general case all three steps may be unportant andcan contnbute to the overall rate of the process Basedupon certam assumptions analytIcal solutions have beenobtamed[l-S] Of course, hnearrty must exist to permitaualyt~cal solution and cumbersome expresslons are m-volved even for a sunpIe eqmhbnum retation However,Isothermal operation of a fixed-bed adsorbers for gasseparauon occurs only for weakly adsorbed componentsor for very low adsorbate concentrations Thus thenomsothermal operation of M-bed adsorbersreprbsents a more typical practice Under these cmum-stances the isothermal cnterra suggested for an assess-ment may be apphed only wtth some careA general solutmn to the problem would reqmre asunultaneous solunon of mass and energy balances wtththe equihbrmm relations Since rates of heat and masstransfer, pore Musmn and adsorption are each charac-tertzed by Mermg temperature and concentrationdependenctes It m&t be anticipated that no smgle modeof rate control should necessardy prevarl throughout anomsothermally operated fixed-bed adsorker Indeed itmay be expected that varymg degrees of nnxed ratecontrot anal take place between bed entrance and exrtOnly a few attempts are avadable m the hterature forthe nomsothermal case The case of admbatic sorptionhas been considered by Leawtt[6]. Pan andBasmad#a.n[7l and others[l, 91 The nomsothermai-nonadmbauc case has been dlscussed by Weber etITo whom all correspondence should beaddressed*IkparUwnt of Chcamcal Engmeenng, Institute of CbemlcaiTechnology, 16628 Rsye 6, Suchbhrova Street 1903, Cze-chos~ovakla.

al [lo, 111 and Rutheven et al [12] Usually the masstransfer rate IS represented by a hnear dnvmg forceequauon and the external flutd film resistance and m-traparticle Muslon are considered as a lumpedparameter [ 121 Moreover, the temperature dflerencebetween gas and pellet 1s considered neable Thetwo-phase model suggested by Weber et uf [lo, 1 ] andvon Rosenberg[9] is described by not&near parttalddferentml equatrons of great complexity, the extent towbch fixed-bed adsorber performance IS affected bypa&c&r processes may be reahzed only by recourse toa computer solution smce nonhneanty precludes anyanaly&al approachIn the two-phase model the pticle Itself IS consideredas a contmuum uubedded m a concentratton andtemperature field To our knowledge no detadednumerical calculatmn have been made m order toanalyze the behavtour of a single nomsothennal parucleIn an old paper by W~cke 19] the resistance agamstheat transfer 1s lumted to the external fhmi iilm and anaveraged mtrapartsck temperature IS consldered Wlckepomted out that for certam systems the externaltermperature gradients cannot be neglected Theattempts pubhshed m the hterature to mtegrate ana@&-ally the govemmg set of dtierentml equations yieldcumbersome expressions which are of httle value fordestgn purposes[13-161The present paper 1s concerned with a detadednumerical analysis of a smgle nomsothermal paRlclewhen sorption processes occur

In order to slmpbfy the theoretical analyst thefoUowmg approxunaaons w&h are reasonable for manypractical systems are introduced1 IntraparMe mass transfer may be descmbed bymeans of the Flcks law. the dtffusrvlty IS assumedconcenmon and temperature mdependent2 One component system ts considered3 Intraparhck heat transfer IS represented by the1385


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1386 A BRU NOVSKAt a lFouner law, the effective thermal conductlvlty of thepartlcle IS temperature mdependent

4 Constant average heat capacities charactenze thesystem, heat of adsorptton IS assumed Independent ofconcentration and temperature5 Gas-to-sohd mass and heat transfer IS absent

6 There IS an equtilbnum between the gas phase andsohd phase concentrations everywhere m the parttcle7 Only the gas phase dtiuslon IS consideredWith these assumptions the mass and heat balancesare, respectively

(1)

PC cmnaT=A 3+x%p ar [ 1 +(-A&g (2)The nutml and boundary conditions are

t =o, 0=xZZR a = al, T= T , (3)t>O, x=R a = @(co, Ti) = a,*, T = T,

x=0 aa_o aLOax- ax-For a general eqmhbnum asotherm c = ~(a, T) eqn (I)may be rewritten

(5)The govermng equattons, eqns (l)-(5), are represented bya set of strongly no&near coupled parabohc equationsThese equations are rendered dimensionless by thefollowmg subsfitutlons

a - aiq=- &ATao* _ at 0 = 5, = (-A&,,)(~* - a,)

azQ ae+$$QQ+&) +2 3a8-=ar

+Ve+ f

ae2

(6)

(7)

(8)

The boundary conditions are7=0, OI~11, q=o, 8=0 (9)7>0, f=O 3=0, $0

5-l q=l. e-0 (10)

We may note that the problem contams two parameters Sand L,e The additional parameters are m the eqmllbnumrelation

For the Langmulr IsothermKPa=a.l+ (11)

using the above presented transformations we obtamQ= [ f J+K2]&jexp( )-m (12)

Herea_* aiKI =- ada* - &*Ia, - a1 lc* = a,(&* - Ui)p = Lx q&* - =*)T&P

3. APPUCATlON TO VARKHJS OPEUATING SE?TEM3We shall discuss the type of gas-sohd adsorptionsystems where slgmficant thermal effects occur m the

particle The above formulated problem IS characterizedby SIX parameters a, /3, 8, Le, K~ and ~2 The parameterit, the dnnenslonless adsorption energy, represents thetemperature dependence of the Langmurr constant KWe may estimate an uppermost hmit of a to be encoun-tered, with (-AH,) = 170 kJ /mol and T, = 293 K (a =35) The heat evolution parameter p IS determmed by thefollowmg propefies (-AH,&, k*, Ti and pC, Estunat-ing the maxunum value a,+ = 3mol/l and pC, =17 J /cm3K, then /3 ==O6 For the Lewis number themagnitudes of D and A are necessary The upper boundto the magmtude of posstble dtiuslvtues IS the free gasphase dlffuslvlty corrected for the fraction of votd spacem the sohd and for turtuoslty Thus a reasonable uppermagmtude IS of the order D = 0 2 x x f = 5x lo- cm%at atmosphenc pressure Sohds with pore structure ofsmall enough dnnenslons to be operatmg m Knudsend8uslon range have typlcal effective dtffuslvltles D - l-5 X lo+ cm*/s Typical conductlvlty of the porousmater& IS A -4-g x 10-3J /cmKs Wrth tlus m mmd ourcalculation covers the range Le=OO5-009 for the freegas phase dlffuslvlty and Le = 0 5-4 5 for the Knudsendtiuslon For an adsorptmn process on a fresh adsorbenta, = 0 and the value of the parameter K( IS m the rangeK,P(O 15-090) Supposmg ai =0 the parameter 8,


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A nontsothermal one-component sorption m a smgle adsorbent partlcle 1387representtng the ratto of the eqtibraun amount of theadsorbate 111gas phase m pores to that on the surface, 1sS = ecda,,* For adsorption of vapours at atmosphencpressure and low temperature c0 = 10P3 mol/l and themuumum value of uo* 1s aO* = 1 mol/l , supposmg c =0 5 we have for 6 an estuuate S = 0 005 In h@ pressureseparation systems (at 20 at and 200C) the maxunumvalue of c, IS c, = 0 3mol/l , the mtnunum value of uO*may be estimated a,,.+= 0 5 mol/l and thus S = 0 15

must be replaced by

4. NUMERICAL SOLUTION OF GOVERNIN G EQUATIONEquations (7) and (8) SubJect to boundary con&tlons

(9), (10) represent a set of strongly nonlinear parabolicpart& dtierentlal equations In thus set the tune denva-tlves are not separated, however, both equations contamthe derlvatlves aq/& and aO/ a7s well So far there ISlrttle mformatton reported m the hterature on numerlcalsolution of such dtfferentlal equations The proposedfimte-dtierence scheme, which proved to be very reh-able, IS based on a combmatlon of explicit and imphcltprocedure

After denoting the space derlvatlves m eqns (7) and (8)rr and rz, respectively, the time denvatlves may bereplaced by forward dtierences

j= rI (13)e ,+1 -e,i

k_qrd+I-9rl= r2

Here we have denoted(14)

AI = 1 + scP(qMB)AZ = &(q)@(e)

+$(q) = (I- K+l+ K21 - Klq1tie)= l+Beexp ( )?$i

In the set of finite ddference equations, eqns (13) and(14), the nonhnear coefficients AI and AZ as well as thenonhnear right--hand sides rl and r2 are evaluated at thepomt (I,]) The set of explicit finite-difference equationsmay be easily solved for 8,,+1 and qid+l

rl - A2r2qrj+l=qr,+ A,+A2krl +A l r 2eir+l=t5j+ A,+A k2

In the center of the part&e the operationa2 nap- s ar

(16)(17)

a2(R+lQzand accord&y eqns (7) and (8) must be mod&d Thedevelopment of exphcclt fhute-dtierence formulas ISanalogous to (16) and (17)

The explicit irute-dBerence formulas (16) and (17) areused to calculate a first guess of the new time profileJ + 1 Of course, because of stab&y these values mustbe Improved by a unphclt finite-dtierence procedure

In order to construct a feasible lmphcclt finlte-difference schema followmg lmearuatlon can beadopted

~(q)+(eVq = (cp(q)*(e))Wq)m+ = A0W+f

p q)tyqeFe = rp q)ty e))yv*e)-1 = A6p2e)m+I

(18)Here the superscnpt m represents the mth lteratlon Oncombmatlon of eqns (7) and (8) and using approxunattonsgven by eqn (18) followmg equations result

~=B,$+B~$+B++B$ (19)ae- =- _a7 (20)

Here B1 , Bz , Bs , B . a r e ce r t am coefliclents defined asfollows

B, = (n l Z)A-a+ As + AsAs+Az

A4B3=Ar AzB = -As4 A l +Az

A, = E Le lS


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1388 A BltUtKwti et alTo solve this set of linear parabohc equations the equation are ordered according to mesh pomts AfterCrank-Nicholson fimte-dtierence schema was used groupmg the finite4iTerence equatmns III a sequenceAfter approxunation of eqn (19), by fimtedlfferences (191, (2% (19), (2% etc and varutbles qt-lJ +I, O1--l~+lrfollowing equation results qw t, et., l, etc a s1x dmgonal band mamx resultsOther groupmgs yield a matrtx with a more compItcatedstructure A special algorithm for band matrices wasused to solve the set of hnear equations[l7]The expht sehema (m = 1) serves for evaluation of

the vanables in (18) denoted by the subscnpt m Theprofiles calculated by the lmphclt procedure (m = 2) areused to improve the values of the nonhnear coefficientsm (18) In ;he next iteration these coCffiClCntS iwe evah-

(21) qIJ+t/z=- (qIl+ d3 d/28, j+112 = 03 j + a(:) d/2-t

We may notEe that s1x variables occur m tis equationqr-1 1, 61-1j t, 411 1r elJ l. qi lJ l d ef lJ l The unphclt schema at the new tune profile j + 1 ISFurthermore, we can note that the varrables in this iterated as long as the dtierences of e;+, - 8;:: and

1o-

9

(a)

0.5

0

1 ~sorp~n~ns~~calpartlcle(Le=005,a=10,~=03,S-005.~,=07,~~=0) T 1,00005,2,00020.3.0~~~,~.00101,5,0~191,6.00uI1.7,00951,8,0 72.9.0260, O,0 93 (a)Concentratmnro6ks or aries7 (b)Temperaturerofilesor arious


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A nomsothcrmal one-component sorphon m a smgk adsorbent part~ck 1389&+I--4;;:: are smaller than the predetermmedtolerances and , respectively In our calculations wehave used h = 0 025 and k = 1 X lOA In the problemssolved we drd not observe mstabdlty of the fimte-dlff erence procedure

5 NunmmcALIEsuLTsAND-ONIn F&s l-3 temperature and concentration profilesw&m a parttcle are drawn for a case of a strong ad-sorptron (6 = 0 05, KI 0 7) which 1s accompamed by ahigh heat of adsorptmn (j3 = 0 3) In these figures thedependence of the temperature and concentrationproties on the value of the J _ew~snumber L.e 1s shownFor a case of the free gas dtiuswtty (Le = 0 05) we maynote very steep temperature gradtents m the partrcle witha h@~ transient hot spot temperature For Instance, for

1C

9

05

7 = 0 002 the hot spot temperature IS 8,, = 0 25, Le for7i = 300 K 1sAT = 22 5 From this result can be mfer-red that m a short time pronounced temperatureMerences occur The maxunum temperature is reachedm the parkle center, the temperature differenceamounts to approx AT = 50C In FU 4 the dependenceof the average temperature and concentration m theparticle IS drawn We can observe m this figure thatthough the partlcie IS almost saturated It 1s stffl over-heated with respect to the ambient gasFwe 2 presents the development of temperature andconcentration profiles m the particle for the Knudsendtiusion regtme (Le = 0 5) Of course, the hot spottemperature 1s lower, here the maxunum temperature IS8,,= 01. le for TI=300K IS AT=9C Fve 3reveals that for Le = 5 the temperature increase m theparkle IS very low

(a)

Fii 2 Adsorpnon m spherical pamcle (Le = 0 5,Q = 10, j3 = 0 3.6 = 0 05, x1 =07. Kz=o 7 1, 00005. 2, 00020,3.00651,4,0 0101.5.00191,6,0 0381.77.00971(a)Concentretwmprosles for vartous r (b) Temperature protiksfor varmus 7


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A BRUNOVSKAa al

LO-

0 0,25 0.5(a)

(b)FU 3 Adsorption n sphertcal part& (Le = 5, o = 10, ,9 = 0 3, 6 = 0 05, K, = 0 7, K* = 0) 7 1,o 0005, 2,0 0020,3.0 MU1,4,0 0101,X 0 0191,6.0 0381,7,0 100 (a) Concentration protiles for vartous 7 (b) Temperature profiles for

various 7

-015

0 0.2 0.3 t 04Fii 4 Average concentratton and temperature m the partxle vs I (Le = 0 OS, a = 10, J= 0 3, 6 = 0 05, K, = 0 7,

K2=01


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A nomsothermal one-component sorption In a smgle adsorbent particle 13916. coNcLumoN

For a steep adsorption Isotherm and lugh heat ofadsorption effects the overheatmg of the sorbent parMemay be unportant and a model of a IsothermaI process ISnot JUSUlied This situation IS typlcal for adsorption ofgases on zeohtes The model and the numerical proce-dures as well make It possible to calculate the tempera-ture profiles m the sorbent particle The model proposedmay be easdy modffied to handle also desorptton pro-cesses In the next paper expenmental observation oftemperature gradients occumng m a particle dunng anadsorption process wffl be reported[ 181 The work on amore detaded model mcludmg. e g finite values of thegas-to-solid heat transfer IS m progress[ZO]

aaia,a.C

i;D

(-AH&K

R9?TT,ATtX

NurATloNsorbate concentramn IIIparticlemrtud sorbate concentration m partlcleeqmhbnum sorbate concentration m parUclemonolayer capacity in the Langmmr equationgas phase concentrationgas phase concentration m the bulk flowspectic heat of sorbentMuston coefficientheat of adsorptioneqmhbnum parameter m the Langmuu equa-t l O I l

L ew i s numbersymmetry parameterpar&d pressure of adsorbatedlmenslonless gas phase concentraUondunensionless sorbate concentration inpartrckcharactenstic lengthgas constanttemperatureImtial temperaturetemperature nsetunespace coordinate

Greek svmbotsdlmensloniess adsorption energy, see eqn (12)dunensronless adiabatic temperature nsechmensionless parameter. see eqn (6)porosity of particleduuensionless temperature nsednnenslonless parameters, see eqn (12)effective thermal conduchvlty of sorbentpar%ledunenslonless space coordinateapparent density of sorbent particledunensionless tune

HI121131;:;[61PI181191

;::;WI(1311141WIWIVl,:i;1201

-cmEdeskutyF J andAmundson N R J Phys Chem I 952 56148Rosen J B , J Chem Phys 1952 29 387Rosen J B , Ind Engng Chem Fundfs 1954 46 15%Masamune S and Smrth M , A Z Ch E J 1956 11 34Masamune S and Srmth J M , Znd Engng Chem Fundls1964 3 179Leawtt F W. Chem Eng~ Progr 1962 58 54Pan C Y and BasmadJum D, * hem Engng Scr lW7 22285, 1970 25 1653, 1971 2( 45Rhee H K and Amundson N RI Chem Engeg J 1970 1241. 1970 1279, 1972 3 22. 1972 3 121Sch&f D E and von Ros&erg D U , Numerical SoUronof Mcroscopx-M acroscopic Systems Fued Bed adsorp-tron of a Gaseous Componmr , presented at A I Ch EMc&ng 1972Meyer0 andWeberT W,AZChEJ 196713457Lee R G and Weber T W , Canad .I Chem Engng 1969 4760Rutheven D M. Garg D R and Crawford R M , ChemEngng SC1 1975 J o go3Todes 0 M and Lezm Yu S , DAN of USSR (m Russran)1956 & 307Zolotarev P P and Raduskevlc L V , J Phys Chem (mRussron) 1970 44 889, 1970 44 30%Zolotarev P P and Kalmcev A J I Phys Chem (mRuss&m) 197145 2849,1972 46 1130Zolotarev P P J Phys Chem m Russran ) 1972 46 1104Nimec J Personal Commumca~on (Praaue 1977)Ilavskg J , et aI, Chem Engng Sci , &IpreparationWlcke E , Kollotizeitschnft 1939 M 167Brunovske A et al, Chek Engng . & I, III preparation

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AN ANALYSIS OF A NONISOTHERMAL ONE-COMPONENT SORPTION IN A SINGLE ADSORBENT PARTICLE.pdf