Continuous recuperative mode parametric pumping

CONTINUOUS

RECUPERATIVE PUMPING

MODE

PARAMETRIC

PHILLIP C WANKAT

School of Chemrcal Engmeenng, Purdue Umverslty. West Lafayette, IN 47907, U S A (Recerued 22 February 1977, accepted 27 July 1977, receruedfor publrcatron 6 October 1977)

Al&raft-The local eqmhbnum model IS used to study the cychc steady state of open recuperative mode parametnc pumps When the thermal wave velocrty ISgreater than the concentration wave velocities (usual case for hqmds), the recuperative pump can often produce a pure bottom product, cB = 0 Mth cB = 0 both recuperattve and duect mode pumps gave the same separafions regardless of whether mIxed or unmutad rese.rvous are used The dvect mode has the advantage that It can have c, = 0 under conditions where the recupetatwe mode cannot The recuperatwe mode requues sq&cantly less energy than the direct mode, and the energy reqmrements for the recuperative mode can be decreased by usmg unmued reservous with selective reRux If c, >O unnuxed reservous wdl Increase the separation obtamed When the concentration wave veloclhes are greater than the thermal wave velocity (possible for gases) the bottom product wdl not be pure Also for these gas systems, a reverse separation IS pre&cted with cho, c c=,,~ Examples of chemical systems w&h should Illustrate the dtfferent predicted behavrors are given

INTRoDucrKBN

Parametlrc pumpmg has frequently been suggested as a method for mdustial separations and the recently commerclahzed Suotherm process[l] has some features common to parametnc pumpmg Despite tlus suggested use most of the research (recently revlewed by &ce[2]) has been on the batch, direct thermal mode of parametrrc pumping, whale mdustrml use would certamly be an open system operatmg in contmuous or seml-contmuous fashion m a cychc steady state Also, m large scale operation, the duect mode (where the entue column IS heated or cooled) mu&t be replaced by the recuperative mode (where the entermg fled 1s heated or cooled) since this mode makes heat recovery sunpler and would be easier to construct m large sues The open systems have been studed m a series of papers121 but the recuperative mode has been almost ignored smce Rolke and Wdhelm [3] obtamed dlsappomtmgIy small separations The resultmg bad “press” has severely hnuted research on the recuperative mode despite the extensive research141 on the related travelmg wave mode of cyclmg zone adsorption where separations are usually better than the direct mode process Recently, Sweed and agaudeau [S] theoretically studied batch operation of the recuperative mode and showed that large separation factors can be obtamed m thus mode of operation In tkus paper the local eqmhbrmm model161 wdl be used to study open, recuperative mode paramevlc pumpmg operatmg m the cychc steady state The characteristic solutions wffl be obtamed for a variety of different cases Then the external mass and energy balance equations wdl be developed for both completely nuxed and unnuxed reservows The separations obtamed and the energy reqmrements wdl be compared for the recuperative and drrect modes of operation A varrety of operating procedures have been developed for dvect mode open systems121 which could be extended to the recuperative mode We wdl consider 723

the one column system with top feed shown m Fe 1 although other arrangements may be better m some circumstances. Smce the column has top feed, the total fluid displaced m the downward due&on must be greater than m the upward duecuon Thus the cycle porttons must be of unequal length and/or the flmd velocltles must be drfferent m the two duections The system shown m Fu 1 IS semlcontmuous smce feed is mjected and bottom product 1s withdrawn only durmg downward flow, and top product IS withdrawn only durmg upward flow Reservous are requued to store the reflux matenal and are not shown on Fw 1 Contmuous operation can be obtamed by usmg two or more adsorbers and cychng between them When thy IS done, the reservou sizes are reduced or can be ehmmated, and the eqmpment configuration is sun&u to that of pressure-swing adsorbers[fl The analysis of each portlon of the cycle IS the same regardless of whether one or more adsorbers are used The system shown m Fe 1 is recuperative mode since the feeds are cooled or heated We wdl assume that the heat exchangers are operated so that the outlet fhud temperatures, T, and T,, are constant

FF.C..p

Fqg I Schemawall

top

of open. recuperatrve mode paramemc

feed (a) Downaow pardon of cycle Tune PerId, (b) Upftow porhon of cycle Tote per&. r.

pump t,

724

PHILLlP

LocALFQulLIBlMJMMoDEL

The local eqmhbnum model developed by Plgford et al [6] has been widely applied to cychc systems smce It usually gves reasonable predictions with a muumum of effort The cases where the predlctlons are unreasonable usuaIly atlse from the use of bnear Isotherms and not from the assumptions of local eqmhbnum and neglwble dlsperslon This model was extended to batch recuperative mode parametrrc pumpmg by Sweed and figaudeau[5] although the local eqmhbrmm model was applied only for linear isotherms, a nonequdlbnum model was used with nor&near isotherms They studied the case where the thermal wave velocity IS greater than the concentration wave velocttles Far each portion of the cycle the model IS sun&u to the apphcation of the local equlhbrmm model to the traveling dave mode of cycling zone adsorption@] Smce the local eqmhbrmm model has been presented m deli elsewhere[6,8], only an abbreviated presentanon wdl be gtven The development and nomenclature of Baker and Ptgford[8] will be followed The model disregards radial gradients m velocity, concentration, and temperature, and assumes plug flow of fluid through the column at a constant mterstrtlal vefoclty u The rates of heat and mass transfer are assumed to be high enough so that the sohd and fluid phases are locally in eqtuhbrmm In addition, the column IS assumed to be adiabatic, heat of adsorption effects are assumed to be neglwble, and axial dtsperston IS neglected With these assumptions the Mass balance for a single solute m a non-adsorbed carrier slMpllfies to

(1)

c

WANKAT

waves, etc For the system shown m Fig 1 square temperature waves wdl occur since T, and TH are constants during each portion of the cycle For square temperature waves the rtght hand stde of eqn (1) IS zero everywhere except at the thermal wave boundanes With this further slmplfficatton the mass balance ISeasily solved by the method of charactenstlcs which says that concentration IS constant along characteristics Bven by

At the thermal wave bounties the solute redlsmbutes between the fluld and the adsorbent The concentration change ISdetermmed from a Mass balance on segment of hetght AZ and over a ttme At =Atluth This choice of control volume ensures that all Maternal undergoes the temperature change Thts balance IS

c

u+E(l--a)-E

I

(C,+,-C,)+(l+a)(l--)P, x(qi+l-qJ=o

where I f 1 refers to condmons after the temperature change For linear isotherms of the form q = A(T)c eqn (6) sunphfies to --- 1 cr+1_ &nc, -G --- 1

1 uth 1

U -=,+a

uth

?J

kmc

=

l+1-a -r++l a

kmc

=

1+1-4~+1-4 - a a(1

(2) - .,P,s

and a very general isotherm of the form q = q(T, c) IS Implied The energy balance slmpliftes to

where V

Uth = 1+ (1 -

a)b.CAl

- 4 + PJ-c+l/prcp

(4)

U EDnC and uth are the velocItles of propagation of the concentration and thermal waves, respectively urn 1s a constant, and I(,,,, 1s temperature dependent for all isotherms and concentration dependent for nonhnear

(7)

and the concentration wave velocity becomes

where v

(6)

(8) -e&A(T)

For linear isotherms this solution predicts that concentratlon waves are propagated through the column without changmg shape or concentration until they mtersect a temperature boundary Then the concentration changes according to eqn (7) Any characteristic which undergoes a change from cold to hot and then later from hot to cold ends up at its Mttml concentration For the more realistic non-hnear Isotherms a concentration wave can change shape since its velocity depends on concentration For favorable isotherms a ‘Y~fTuse wave” of gradually decreasmg concentration results d a solution IS displaced by a more dilute solution A sharp “shock wave” will occur d a solution IS displaced by a more concentrated solution The shock wave results from the intersection of two sets of charactensttcs, and Its velocity, u.~, IS determined from a Macroscopic mass balance around the shock front

isotherms

Equation (3) IS easily solved for any shape of mput temperature wave It predicts that thermal waves will move through the cohunn at a velocity i&h and with thev shape unchanged Thus square waves remam square

kh

=

1+l-ae+ a

where subscrtpt 1 refers to condltlbns before the shock

725

Contmuous recuperative mode parametx pumprng

and I+ 1 after the shock K%,,wdl he between the value c, and the value of II,,, at of &m, at concentration concentration c,, , These results are valid for both the recuperative mode and for the direct mode where u*,, IS i&ute The solution IS easily obtamed graphlcally by plotting the characteristics on a graph of axtal distance z vs time t The thermal waves have a slope Us,,, concentration waves have a slope u,,, and the shock waves have a slope u,,, The concentration and shock wave slopes are temperature and concentration dependent These results are time dependent since concentraton and temperature change throughout the cycIe. and smce the concentration and temperature can change from cycle to cycle However, open cychc systems will reach a cychc steady state where each cycle IS the repeat of the previous cycle[9] Since large parametnc pumps would be operated m the cyclic steady state we will hmrt our results to thts cychc steady state and show only one complete cycle The startup condltlons can be obtained by combining the procedures used here with those of Sweed and Rgaudeau[5] The charactenstlcs are sticlent to study the Internal changes durtng each portron of the cycle To complete the study of the parametric pump the external equations must also be mcluded We will defer the study of the external equations until after the charactenstics have been presented STEADY

STATE CH

c

f3lxmll~S

The charactenstlc solutions for the concentration changes depend upon whether or not the thermal waves breakthrough and on the value of u,,, compared to uth To help classify the resulting separations it IS helpful to first look at the thermal charactenstics In Ag 2 three possible cases are shown for the thermal waves In Fig 2(a) the thermal wave breaks through in neither up nor downflow of one cycle However, the cold wave must eventually breakthrough since the downflow dlsplacement is greater than upflow displacement In Ftg 2(b) the cold wave breaks through dunng downflow of one cycle, but the hot wave does not breakthrough At the cyclic steady state the temperature patterns m the bed are slmthar for Figs 2(a) and (b) In Fig 2(c) both cold and hot thermal waves breakthrough The condition for the cold wave to break through IS that

TIME

TIME

Fw 2 Temperature charactensucs (a) Ulc,,,> t,, (b) & > UU~,,> t. breaks through m downflow thermal wave (c) t, > t. > L/u, Thermal wave breaks through m up and down flow

tu

+tD

(10) and the condltlon

for the hot wave to break through IS

Since Figs 2(a) and (b) result in sumhar temperature profiles m the bed at the cychc steady state, theu concentration charactenstlcs WIU be smnhzu Thus we 41 study the thermal characteristics m FU 2(b) and apply these results to case 2(a) also Fwe 3(a) shows the concentration and thermal charactenstics for hnear ISO-

CDJI ’ Fi 3 Charactenshc solunons for lukear Isotherms for systems where only cold wave breaksthrough (a) utb> afi > rc, No Separation (b) Uh> II,> Ye &?paratloa OCCUTSC, = C&. Cd,1= Y%Y
726 therms

PHlLLlP c

when

only the cold wave breaksthrough and The top product has the same concentration as the feed since tlus material undergoes no temperature change An overall mass balance shows that c, = cr. and no separation occurs Thus parametric pumpmg produces no separation If the hot wave does not breakthrough and uth > u#,> u, Use of nonlmear lsotherms does not change this conclusion The condition that uth > uh > u, would be typical for most hqmd systems For gas systems we could easily have uk > u, > &h The linear Isotherm analysis for this case IS shown m Fig 3(b) In this case some of the exltmg top product does undergo a temperature change and a separation does occur Nonlinear isotherms will also predict a separation m thus case The total separation that occurs depends on the type of reservoirs used and the reflux ratios employed This involves the external equations and IS consldered later Figure 3(b) shows an unusual separation effect which has nor been reported previously Usually, fluid that undergoes a temperature change from cold to hot has its concentration increased since the amount adsorbed decreases In Fig 3(b) the reverse occurs and fluid gomg from cold to hot has Its concentration decreased If we write eqn (7) as, u,hpuh)&

1 --++E+ c

more Important than decreased adsorptlon when u,, > M=> +, but IS less Important than adsorption when u,~ > u,, > u, Other reverse separations have been reported m parametrtc pumpmg[2] but they were due to radial temperature gradients An addrtional phenomena occurs d i+, > u,,, > u, and IS dlscussed later Before leavmg Fig 3(b) we wdl note that although separation can occur It WLU be small and parametric pump operation wlthout breakthrough of the hot wave IS not good practice When both the cold and the hot thermal waves breakthrough, the recuperative parametric pump can a&eve large separations The charactenstlcs for these cases are shown in FQS 4-6 In Fig 4 the characterlshcs are shown for a hquld system with lrnear isotherms where

1 (12)

--uh

WANKAT

&h

we see that y > 1 If I(,,,> u, > u, which 1s the expected result However, d u, > u, > u#,.,then y -=c1 and c, C cc which 1s what occurs In Fig 3(b) For favorable nonhnear isotherms eqn (6) agrees with this result If uh > uti > uc eqn (12) predicts the physically unposslble result that 7 uc and 1s discussed later Since hquld systems usually have uth > uh > u,, they behave m the “normal” fashion and have ch > c, Gas systems can have u,, > u= z I(~ If the gas heat capacity and denszty are low For example for the adsorption of methane from air on activated carbon the data of Blum[lO] can be used to calculate the thermal wave velocity as uth = 6 3 x IO-’ u at 25°C and one atmosphere At very low concentrations the concentration wave velocities are calculated to be u, = 0 016 v at 25°C and u,, = 0 038 tl at 70°C For this system the parametric pump would have cJcc= 0 977 calculated from eqn (12), despite the fact that less material IS adsorbed at the higher temperature Thus there must be a competmg mechanism occurring which forces This ch > cc mecharusm IS that faster movement of solute molecules tends to decrease therr concentration m the tluld d adsorption IS relatively ummportant A simple analogy IS to consider automobdes on a h&way In a residential sectron with a low speed lunlt the cars move slowly and are concentrated together Once the superhahway IS reached all the cars move faster, they spread out and become less concentrated Tlus phenomena becomes

L

\ l!izD t-+-c~

1

\lC \’ \

c / // /n

t

(Cl

H

OO

p-cCe=o--I

!O

+o+tlJ

I-tCrt . . (0)

F’rg 4 Charactenstlc solutton for ltnear Isotherms w&h both thermal waves breakmg through and with utb > II,,> I&. (a) c, = 0 but have direct breakthrough of bottom reflux (b) c, = 0 but have eventual breakthrough of bottom retlux (c) cg = 0, maxlmum top product concentration (d) c, > 0, top feed eventually breaks through tnto bottoms

Contmuous

recuperative

k-cF+fl d-

t-CSSC,!L+CFj

Fig 5 Charactenstic

solution for linear Isotherms with both thermal waves breaking through and u,, > II,> uth tF 1

csll kc,--l

I-

+---Cl3 =0--j FQ 6 Solution with nonlinear isotherms for case where u& > u*,,> u, > u,, for a senes of cases where the tune for upflow 1s progressively decreased Thus IS eqmvalent to decreasing the refiux catlo at the bottom of the column In FUS 4(a)-(c) the pump IS operated so that all charactenstlcs ongmatmg at the top of the column exit at the top Thus all solute entenng m the feed must eventually leave m the top product and c, = 0 In Fe 4(a) the uptlow penod 1s too long and breakthrough of some of the CB = 0 matenal occurs m the top product ddutmg that product As t. IS decreased tlus breakthrough of matenal 1s reduced until m Ftg 4(b) no mate& passes dvectly from the bottom reservou to the top without undergoing a temperature change However, there IS stall matenal which exits at the top with a concentration of c, = 0 since It undergoes two temperature changes Further reduction m the bottom reflux ratio increases the top concentration whde cs = 0 until the situation m Fig 4(c) results Any further reduction m t,, wdl cause eventual breakthrough of solute mto the bottom product as 1s shown m Fu 4(d) The matenal which breaks through m the bottom has undergone an even number of temperature changes and 1s at Its nutial concentratron cF Because of this breakthrough, cB > 0 and complete solute removal no longer occurs Figure 4(c) thus represents a local optimum smce c, = 0 and the top product IS maxunally concentrated In Fig 4(c) the concentratron of the intermediate band whtch never leaves the column wdl depend upon the nuti condltlons and startup procedure USd. With h@ly nonlmear Isotherms the parametrx may not be able to produce separations where

pump

c, = 0 even through both hot and cold waves break through Ilus was dustrated by Camero and Sweed[l l] who studed the local eqmhbrmm theory for both batch and

mode parametrrc pumpmg

727

contmuous direct mode thermal parametnc pumps where the isotherm was of the Langmuu form for a smgle solute or had a constant separauon factor for competmg ions The cntena for complete separatmn developed by Camero and Sweedill] for the dvect mode system showed that some ion exchange systems can not be completely separated even rf local equlllbnum IS assumed Since the direct mode system can produce a lower value of c, than the recuperative system when c, (0, these Ions wdl also be mcompletely separated by a recuperative pump Incomplete separation will also occur for single solute systems such as acetic acldwater-activated carbon where the isotherms are highly no&near The lunit cycle con&Qons for these cases can be calculated by domg the startup calculations until they converge{51 or by adapting the duect mode cntenatll] Incomplete separations, ca > 0, are also predicted by nonequlllbnum models when mass transfer rates are low [5] In Fig 5 the characteristics are shown when &, > K, > I(~,, Once agam reverse separation occurs but more separation IS achieved than was shown m Fig 3(b) The bottoms concentration, ce, can be reduced and cT mcreased by decreasing t, and td, but the thermal waves must break through or the solution reverts to Fig 3(b) In Fig 6 the nonlinear solution IS shown for the case If a 1mea.ranalysis ISused for where uklCF> &I, > u& this system it predicts mfimte concentration of solute at the thermal wave boundary during upward flow This results from “trappmg” of the solute at the thermal wave smce both hot and cold charactenstrcs pomt into the thermal wave When the effect of no&near isotherms IS mcluded, the concentration wave velocity becomes concentration dependent The increase m concentration at the thermal wave boundary causes IA to mcrease untd II.2’ %I The concentration increase can be obtained from eqn (6) when the appropriate isotherm equation IS inserted for q Smce this concentration is greater than C, a shock wave will result m front of the thermal wave The shock wave velocity can be calculated from eqn (9) where c, = C, and c,,, 1s the value obtained as c,+, from eqn (6) Figure 6 also shows a shock wave dunng downflow When the flow IS reversed, this shock wave creates a diffuse wave untd the thermal wave IS mtersected If cg was greater than zero there would also be a dtiuse wave after the hot concentration characteristic dunng downflow Smce cs = 0 m Fig 6 this characteristic has its mnumum velocity, uhle+ and no diffuse wave can occur Increasing t,, will decrease the top concentration while decreasmg t. wdl cause eventual breakthrough of solute into the bottom product These effects wdl be smular to the sequence shown in Fig 4 For most hquld systems the thermal wave velocrty will be greater than the concentration wave velocltles whale for gas systems the thermal wave velocity will be slower than the concentration wave velocltles The shock wave which appears when u,l, > uth > ~~1,~can be obtained by slowmg down the thermal wave velocity m a hqmd system by adding mert solid particles of high heat capacity to the column For gas systems increased by operatmg at high pressure

uu can be

728

PHILUP c

WhNKAT

When vartables other than temperature are used as the ships between total flow per half cycle and fluld veloclty withm cychc vanable, the wave velocity of the cychc vanable, the column and PB = PT = (u&L& uvuhblcr may naturally fall between the concentfatmn 1%(pA.d~ Equation (17) shows that we are not free to wave velocities and the solution shown m Fig 6 wdl choose arbitrary values for cycle tames, flows and bottom reflux ratio apply This was observed m the cychng zone adsorption experunents of Dare and Wankat[l2] who separated To determine the separation achieved solute balances fructose and glucose from water on a dlhydroxyborylare required A solute balance on the entire system for one cycle IS phenylsuccmamylammoethyl cellulose usmg pH as the cychc varmble The pH wave velocity was expenmentally determmed-as II= 0 528 v The fructose eqtic-E2 = RB) + c+l R,) (18) ibnum isotherms were fit to a Langmulr adsorption 6% 1sothermIl2] At the feed-concentration of 0 25 mg frucIf c, > 0 addltronal information IS requued to determme tose/ml the concentration wave velocities predlcted from the separation obtamed However, for the Important eqn (2) are 0575~ at pH 50, 0401~ at pH 735 and special case where c, = 0 eqn (18) can be solved Im0 253 v at pH 8 5 In this calculation LL and l were mediately for C~ estimated as 0 45 and 0 6 respectively, other values were obtamed from Dare and Wankat[l2] In a recuperatwe mode parametnc pump the fructose-water system would UWc, _ cm41 -R&d (19) CT= (1 - RT)PT - (1 - R,)R, have a soluuon sun&u to that shown m Fig 6 d It were operated between pH’s 5 0 and 7 35 The value of csh m When c, = 0 the solution IS independent of the type of Ftg 6 can be calculated from eqn (6) with uVmblc reservous used Examples of chemical systems where replacing uu, and a pH Jump from 5 0 (c, = 0) to 7 35 The the bottoms concentration wdl approach zero If sufficient calculated value of cl+, = cti is 5 27 mg/ml which is reflux IS used are the separation of toluene from nconsiderably larger than the feed concentration of heptane on s&ca gel[21 and the removal of fructose from 0 25 mg/ml, The veloctiy of the shock wave calculated water on DBAE cellulose[12] Mixed and completely from eqn (9) 1s &h = 0 541 v &ce & Is only slightly greater than uVmble, the highly concentrated regmn III unmured reservous with selective reflux gve the same solution but the energy usage wtil differ As long as FQJ 6 urlll be of short duration In practice, the peak c, = 0 the value of cT IS the same whether an approxiconcentration obtamed WIUbe lower due to dmpersion of mate lmear isotherm is used or an exact nonlmear ISOthe pH and concentrauon waves Theoretically, the therm 1s used When cB = 0, the top product concenrecuperative mode pH parameflc pump can produce a tratlon depends only upon the fresh feed concentration. bottom product with c, = 0 for tfus sugar solution cppr and the mternal reflux ratios Equation (19) shows that cl. wffl be increased d small values of R, and/or External mass balances large RT are used The charactenstlc solutmn will hmlt To complete the analysis of the separation achieved the mmunum value of RB for which c, = 0 Increasing the external balance equations are requued Refemng to RT mcreases the total flows and thus requires a larger the flows shown m Fu 1, we can w&e a mass balance at diameter column and more energy use the top of the column as When c, = 0, the external equations ignore the exact pattern of the charactenstics Thus when c, = 0 eqn (19) (FF) + RTP, = PB 1s vahd for both the recuperative and the duect modes, and exactly the same separation 1s predlcted for these and at the bottom two modes The du;ect mode does have the advantage that it can obtam a VdUe of cB 7 0 when u#,> rc, > uti where the recuperative mode has c, > 0 Also, in some cases such as Figs 4(a)-(c) the drrect mode can operate where RT and Rs are the top and bottom mtemal reflux at slightly lower values of RB and st~Uhave c, = 0 Then ratios Combimng eqns (13) and (14) we obtain c7 would be somewhat higher However, at the same values of cpn RB and RT the two modes predict exactly P7. = RBW.) the same separations If cB = 0 If the charactenstics m 1 - R,R, Fig 4 are drawn so that the cold concentration characterlstlc on downflow startmg at t = 0, z = L intersects (16) z = 0 at &,, and the hot concentration charactenstic on upl?ow startmg at t = t& z = 0 lntemects z = L at td + t., The bottom reflux rat10 can also be related to the up and then both recuperative and duect modes are operatmg at downflow tunes shown rn the charactetrstlc &&grams theu optimum and the maxunum value of cT IS the same for both modes These conclusions have to be moddied when drsperslve effects are Included and are dlscussed r (17) later When c, IS greater than zero, the choice of operatmg Equation (17) IS obtamed from eqn (14) and the relationmode, the type of reflux reservou employed and the

c$$‘(l-

-

Contmuous

recuperattve

exact nature of the charactensttcs ah make a drfference Examples where ca wrll necessarrly be greater than zero for a recuperattve pump are gas separatrons such as the removal of SO, from arr on stltca gel[13] and the separatron of ions on ion exchange resms where mass transfer IS slow [3,51 Constder iirst the par~ular case shown m Ftg 5 where the recuperattve mode IS used to separate a system with II,, > u, > uth In this case c, > 0 regardless of the values of RB and R, For completely muted reservous the top and bottom concentrauons are an average of the exrtmg concentrattons A mass balance around the bottom of the column shows ce =

ab adB

+bcc +Cd ada acF

-_c

(201

where cs IS the average bottoms composruon, c,, = ‘yc, wtth 7-z 1 for thus system, ob, ad etc are dtstances shown on Ftg 5, and cF IS the average feed to the column cF IS determmed from a mass balance around the top mtxmg pomt as,

mode parametnc

cFF

4

L --ef urn G=t.’

fg

(22)

+-p

L u,

fs z--

1 -&u

1

I-R, CT=--CFF c, ( RB > Rs

Equauons (18), (2O)-(22) are four equawhere cr = C& trons wrth the three unknowns c,, cP and cT Thus only three of these equattons are independent We WINsolve eqns (18), (21) and (22) The ratto d&zig represents the fractron of trme that the top product leaves wtth a concentratton of CF Thts and the other fracttons can be calculated from the pomts of mtersectton of the charactenshcs

de L -=-9 dg u,t,

[

(for R, = 0)

(26)

“-+j-

A mass balance arround the top of the column grves,

de

729

rattos If the results for other charactertstrc dmgrams such as Ftgs 3(b) and 4(d) are desued, eqns (13)-(22) are unchanged Equatton (23) wdl be d&rent and some of the ratros may be zero Thts wtlt change eqn (24) but eqn (25) IS unchanged The analysts IS stratghtforward for these other cases The results for completely muted reservorrs with open drrect mode parametrrc pumps are considered elsewhere121 The value of cT obtamed m Ftg 5 can be increased by decreasing t. (eqmvalent to decreasmg RB) untd t, = L/ue This ISa special case where fg/dg = 0, but eqn (24) is stall valid Reducing td (which increases R,) Wffl decrease the value of cB obtamed Increasing the top reflux ratro RT WIU Increase both c, and cT Thus d a muumum value of c, IS desued the parametnc pump shown m Ftg 5 should be operated with RT = 0, t, = L/urh, and a htgh value of Re When no top regux IS used, RT = 0, eqns (24) and (25) become much simpler

=, = CFFVF) + RT~T+ FF + R,P,

CT =&cF+gl

pumpmg

-- L *I6 ufh tU

(for R, = 0)

(27)

For the charactenshcs shown m Figs 3(b) and 4(d) mcreasmg R, also increases c, and c= so operation w&out top regux may be advantageous dependmg upon the purpose of the separation For cases where c, = 0 top reflux does not affect c, but does Increase c, so top reflux 1s destred

L

1

(23) 0

Combmmg eqns (IS), (21) and (23) wtth the defiruuon of 7, we obtam after algebrtuc mampuIauon

Once c, 1s known cT can be found from the overall balance eqn (18) as

cT

=

(I- ~BRT-~F__(1 -R&s R&-R,)

R&l - Ry)

(25)

and c, can be obtamed from eqn (21) An ahemauve IS to carefully construct the characterrstrc dragram and _ _ oalcttlate the mass balance from actually measured

Fra 7 Linear

where

Isotherm charactensbcs for unmixed reservows u,, P U, ZF-II,,, and C, w 0 (a) RB = 617. R, = Z/3, Y = 213 _ -.- -.-.^. _

(b) Increased r,. KB5W3,

&=2#3.

Jf=2/3

PmluP

730

c

An alternatIve method of mcreasmg the separation when cB > 0 IS to use unnuxed reservoirs and withdraw the most ddute matenal as the bottom product and the most concentrated mate& as the top product The bottom reflux wdi then be more concentrated and the top reflux less concentrated Thus scheme of unnuxed reservows was used by Thompson and Bowen[l41 for batch dvect mode paramemc pumps In an open system there are a variety of ways thrs techmque could be employed and no general solution applicable to alI techmques can be developed We wdl dlustrate three spectic rnstances for the recuperauve mode parametnc pump where u,, > %>%ll In Fig 7(a) a recuperative parametnc pump with unnuxed reservoPs IS shown The reflux ratlo, Re, cdculated from eqn (17) IS6/7, y calculated from eqn (12) IS 2/3 The top reflux ratlo was chosen so aI1of the mated of cancentratlon cF is refluxed Thus RT = L./u& = 213 The bottom product IS withdrawn at a concentration of

c,=ycjq+?p The more concentrated mated IS recycled so that It wdl undergo a temperature change and leave as part of the top product The top product has a concentration

WANKAT

reservows for the system shown m Fag 7(a) but with no top reflux, RT = 0 Now fresh feed IS used for the entue downflow potion and the top product contams a large amount of matertal of concentration c, The bottom product 1s unchanged for the unmixed reservou system and c, can easily be found from eqn (18) 2 c, =-c,, 3

For completely nuxed reservolcs c,(mlxed) = 0 75c,,

1 2 7 I&=-c&y+-cFp=-CFE; 3 3 6 This calculation detemunes the fractron of the top product wluch undergoes one temperature change and the fraction that undergoes two temperature changes whch cancel each other Alternatively cF can be determmed from eqn (18) Equaaons (15) and (16) can be used to calculate PT and Pm Then the top and bottom product rates are

With these product flowrates and the cornposItions calculated. the overall solute balance eqn (18) IS satisfied The values of cB and cT for completely mured reservoirs can be calculated from eqns (24) and (25) for the same values of y, RB, RT etc These concentrations are c,(nuxed) = 0 794~~

c,(nuxed) = 1 103~~

Thus the use of umnlxed reservors does mcrease the separation A second example 1s to compare mrxed and unrmxed

c,(mured) = 1 042c,,

The unmured reservoir system agam aves a larger separation Note that use of no top reflux reduces c, for mvred reservous, but does not change c, for unmixed reset-vows Use of top reflux m an unnuxed reservor system will have no effect on c, as long as only mate& of concentration c, is refluxed at a posItIon where It eventually becomes bottom product The easiest way to Increase the unmixed reservoir separation shown m Fig 7(a) IS to mcrease &, keepmg all other varmbles except Rs constant When thts 1s done It IS possible to withdraw all of the materml of concentration ycm as bottom product and make al1 of the top product have a concentration c&y This IS shown m Fig 7(b) where RB = 2/3 Then for the unmixed reservous 2 ca = -cm 3

or

cT = 1056c,

3 CT = -cF, 2

For a shghtiy longer cycle FF has been Increased by a factor of S/3, bottom product flowrate has been tnpled and the top product flowrate IS unchanged but the top product IS much more concentrated Thus Increase m separation IS due to the abtity of the unmured reservou system to take the deszred streams as products and reflux the undesued streams The totally mixed reservoir system obtams a poorer separation for the system shown m F?g 7(b) c,(nuxed) = 0 893c,

c,(mlxed) = I 16c,,

The top product concentration in Fe 7(b) can be Increased further by mcreasmg RT This Increase m top reflux and decrease In I;I; will cause some of the materml of concentration cJy to be refluxed It can pass through the downflow unchanged and then be concentrated agam on upflow Thus part or all of the top product wdl be of concentration cpdy’ If a value of R-,- = 13/H ISused for the system shown m Fig 7(b) then ail of the top product wdl be at a concentration c&y2 Further increases m RT WIU cause material to be concentrated three or more tunes with proper placement of the reflux streams The results obtamed here for systems w1t.h c, Z-0 show that the separation achieved can be unproved by usmg unrmxed reservors Thus conclusion IS also vahd for duect mode systems The separations predwted here have been modest Tlus 1s true smce we have been

Contrnuous recuperatrve mode parametrlc pumpmg dlscussmg systems where u,, > u, > k,, Large S&e thermal recuperatwe mode operation with low pressure gas systems ts probably uneconomic beCauSe of the low separations which are obtamed Unmixed reservoirs wdi increase the separation obtamed with hqutd systems where uth > u,, > u, d the solute follows a favorable non-linear isotherm and has cB > 0 For hqtud sys&ems with u,,, > u,, > u, and where the Isotherm IS not h@y non-ltnear the parametrtc pump would generally be operated with c, close to zero and then unmixed reservows do not affect separation EXZXENALENERGY

BALANCES

recuperative mode has the advantage over the tiect mode that energy recovery ts easier and even without energy recovery less energy IS requued for operation of the parametnc pump We wtil first compare unmated reservoirs to mixed reservous for the recuperative mode, and then compare these results to the direct mode For cases where c, = 0 the type of reservou employed has no effect on the separatmn, but tt does affect the energy use For example, consider Fu 4(c) Wtth an unmated reservou we would preferentmlly reflux all of the hot material at the bottom along wtth some of the cold matenal Smce only the cold reflux needs to be heated, an energy balance shows that the heat reqmred per cycle for heatmg the bottom reflux ts The

where L/u* IS the period of ttme that hot materml exits on downflow, C, IS assumed to be constant, and IuI ts assumed to be the same on up and downflow The term [f, -(UK,)] IS the tune period for whch cold material must be reiluxed to the column dunng upflow Smce this amount of cold fltud must be heated, eqn (28) mves the heat required at the bottom of the column At the top of the column the fresh feed may have to be cooled and any of the hot material refluxed must be cooled Smce the cold matenal IS refluxed first, no hot material wdl be refluxed d Rdu c Uu, The energy removed per cycle at the top of the column IS

731

At the top of the column

Compartson of eqns (28) and (30) shows that Qauv< and compmson of eqns (29) and (31) shows that ;::l4Qd Tb us 1ess energy IS requtred wtth the unmated reservous and reflux of the appropriate streams When c, > 0, the mated withdrawn as top and bottom product IS controlled by the requirement of maxtmlzmg separation In this case the unmixed reservou system may reqmre more energy than mlxed reservoms, but conslderably more separation IS obtamed Thus for Figs 7(a) and 7(b)

= C,U, - T,+ Q%CIM

- UutX,(Z

- T,)(k%4

(29)

for R+, > L/u, If Rd. 5 L./uu, the last term in eqn (29) is taken as zero If the feed enters at T, =d R& I L./utL no heat exchanger ts reqtnred at the top of the system If totally mixed reservous are used then materml at some average exrt temperature IS retied The exrt temperatures at bottom and top can be calculated by nuxmg all exit streams When tlus temperature is substituted mto the energy balance at the bottom of the column, the heat reqmred to heat the bottom reflux 1s

(32)

smce all of the cold materml ts refluxed along with some hot materml Note that eqns (28) and (32) are different and QBUM predlcted by eqn (32) IS larger The muted reservou value is given by eqn (30) For both Ftgs 7(a) and 7(b) QauM > Q-Ix, but the energy per amount of solute removed IS less for the unnuxed reservou system At the top of the column lQTl could be greater than IQ-,,I, but they are equal tn FQS 7(a) and 7(b) On the basis of energy usage per amount of solute concentrated the unwed reservoir system Is superior For open duect mode parametnc pumps the choice of a mixed or unnuxed reservou wti not matter smce all reflux matenal at the bottom must be heated and all reflux matenal at the top must be cooled In the duect mode the heatmg and cooling of the fluld occurs wlthm the column, and the entue column IS heated or cooled before flow ts reversed Then for a duect mode system stmdar to the recuperative system shown m Fe 1, the heatmg requuement for an entie cycle ts, QIislt = [U - ax1 -

lhC,

+p,c,Kl-

a)r + all

x (A&MT,, - Tc) + C,(T, - T,)P, and the coolmg requuement

+(&L

-$‘)(ihW

QcDol= RI- aIt1 xGL9(

(33)

ts,

lhC, + e-G[(1-

ak + all

Tc - Tin)+ CAT, - Tm=WI;3

+ CA Tc - Td&&

04

Smce Pr = (t, 1uIpAp) these equations can be compared lath the results for the recuperattve mode The energy requuements for the duect mode are conslderably greater although some of tis heat could be recovered The energy savmgs whrch are posstble wfl be Illustrated for the separatron of toluene from n-heptane on s&a gel Wtielm et al [15] obtamed separation factors as tugh as Id M a batch, direct thermal mode parametnc pump for this system Plgford et al (61 modeled these

732

PWILLIP

c

results with the local eqmhbrmm theory and lmear ~sotherms It Is reasonable to expect that a contmuous system could produce a bottom product where c, - 0 The specfic problem constdered IS outhned m Table 1 The temperature ranges, superfictal veloctty and feed concentration are the same as those used by Wtlheim et al [15] The system properttes shown m TabIe 1 were obtamed from fieke[16] for a shghtly different s&a gel Table 1 Example calculation Set condlllons Tk = 7OT, T, = 4T, superhal vebclty = 2 546 cm/mm, ce = 0, cT = 2cpF. FF = 1000 kg/cycle, TFF = 25°C. L = 5 m, to = 240 mm. feed 1s 20 ~01%toluene (CJ=F = 173 4 g/l) System properbes a=040,r=0383,p,=207g/cm3,C,=O122cal/goC C, = 0 48 I cal/g’C, pf = 0 7204 g/cm” Calculated values v = 6 365 cm/mm, ulb = 2 830 cm/mm, u, = 1808 cm/mm u,, = 2 323 cm/mm, (, = 201 1 mm, RB = 0 838 & =0807.PT=25845kg,P,=30845kg,A,=0701mZ. coi dla = 0 94 m =9984x 106cal =-1011x10’cal ::: Bnx=2 167xlO’cal a, = -1816X 1o’cal He,,= 154x loBcal Q,,=-1483xtoscal

value of I(~ was cakulated from eqn (4) Smce isotherm data for toluene at 4” and 7o”C, was unavailable, an approxunate fit to hnear isotherms was made An estunate of U, was made from fieke’s data[ 161 at 2°C and a concentration of 20 ~01% toluene The value of u,, was estunated from K, and the equthbrmm constant change parameter, b, used by Plgford et nl[6] We decided to operate the recuperative mode paramemc pump so that ICScharactenstlc solutton looked hke Fig 4(c) smce thts mmumtes RB but keeps c, = 0 Once this was decided the charactenstlcs startmg at t = 0, I = L can be followed and t, can be calculated Then RB was calculated from eqn (17) and RT was found by requnmg cy = 2CFF m eqn (19) PT and PB were obtamed from eqns (15) and (16), and A, was calculated from PT = ty/ulpfA& The reqmred heat duties for the recuperative mode pump with unmated reservous were obtamed from eqns (28) and (29), for the recuperatrve pump with mixed reservoirs from eqns (30) and (31), and for the duect mode pump from eqns (33) and (34) The recuperative pump with unmixed reservous requrres the smalIest amount of heatmg and coohg, the recuperative pump wtth mlxed reservous about twice tLs, and the direct mode pump about 15 times the unmixed recuperative mode pump, In ali three cases some of this energy could be recovered d addItional heat exchangers were employed

The

DISCUSSION The

local

equthbnum

model predtcts

that both the

recuperative and dvect mode systems can often remove alI solute from one of the products, and that when this occurs both systems have the same separation The

WANKAT direct mode does have the advantage that It can often have c, = 0 w~tb smaller bottom reflux ratios, and It can have c, = 0 when u,, > U, > uth In a real system where local equthbrtum 1s not acbKved these conclustons must be modGed Since the thermal wave will be dispersed, the entire temperature dtierence IS not avtiable to force separation and less separation would be expected for the recuperative mode151 In large sues there wdl necessardy be radud temperature grtients m duect mode paramemc pumps These radml gradients reduce separatlon[t] and may force the designer to use specml and more expenstve destgns such as packmg the tubes of a shell-and-tube heat exchanger Thus each mode of operation has Its advantages and dtsadvantages The recuperative mode does use substantially less energy and this can often be further reduced by using unmixed reservoms with selective reflux Actual reserVOWSwlthout murmg or stream sphtters should be fatrly close to an Ideal unmixed reserve= If cB > 0 the separatton can be enhanced for both recuperative and direct modes tf unmixed reservous with selecttve reflux are employed When c, = 0, the type of reservor employed has no effect on the separation When one exammes Fm 4(a), (b) or (c) and sees that there are potions of the top product w&h are of higher concentration this may appear to be mcorrect However, d an unmured reservou is used the reflux material will have a lower concentratton than for a mured reservou Since c1 = yC, thrs effect cancels the advantage of removmg the most concentrated mate& It may also appear paradoxical that the recuperative and duect modes will have the same separation If c, = 0, since the charactenstic solution for the direct mode wdl not have the large tune permd where top fled exits at CF Examination of eqn (12) shows that yRecu_r,vc > y,,_, when uth > U, > u, and the concentrated regton 1s more concentrated in the recuperative m&de Calculatton of c7 by a mass balance on the top product and recuperative mode systems with the same values for L, u,,, u,. t, and td shows that the values of C, are the same for the two modes When the recuperative mode IS employed, thermodynamlc varmbles other than temperature such as pH can easdy be used The soluttons presented here are apphcable d uth IS replaced with L(_._~ and d the velocity of the thermodynamtc vmable 1s constant throughout the cycle The different regtons m the charactenstrc soluttons would refer to dtierent levels of the new thermodynamic vanable and u,, and u, would be the values of the concenmon wave velocltles at these levels The velocity of the thermodynannu varmble, u,,_,,,._ could be determrned expenmentally or from the appropnate mass balance If the separation of two adsorbed speaes 1s stu&ed the results obtamed here must be mod&d If the adsorption of the two components IS competttive and IS characterlzed by a constant separation factor the adSOrptIOn Isotherm can be wntten m the same form as a Langmuv Isotherm [ 111 The nonhnear analysis outhned here for a smgle solute can then be applied to the separation of two adsorbed spectes which compete for the adsorption sites

Continuous recuperative mode parametrtc pumping

The separation of the competmg solutes can be Increased d the feed IS m the middle of the cohunn[lll A two column recuperatwe parametnc pump utllrzlng the arrangement shown m Ftg 1 IS qute sumlar to pressure swmg adsorption[7] The bottom reflux IS analogous to purge and the 1mtu4 top rellux IS analogous to usmg blowdown for partial repressuruahon Top reflux m the parametnc pump does have the advantage that It keeps the mass transfer zone wlthm the bed and thus more of the bed wdl be active Additional top reflux may be economic m parametrrc pumpmg, but the correspondmg reflux of low pressure waste has not been used m pressure swmg adsorption because of the compresston costs For gases the recuperative mode parametnc pump tends to mve low separations because us,, IS too low, and a large scale duect mode system would tend to have low separations or low capacity because of slow radml heat transfer Thus pressure swmg or combined temperature and pressure swmg systems are preferable for gases When no top reflux 1s used, the parametnc pump becomes a more familiar adsorption cycle with countercurrent regeneration Acknowfedgemenrs-The use of facllmes at the Utuverslty of Caldornm-Berkeley while on sabbatical IS gratefully achowledgcd Thus research was partmlly supported by NSF Grant No Eng 74-02002AOl

733

Subscripts E bottom C cold cone concentration-related value D downflow drl dtluted FF fresh feed f fluld F total feed to top of column mcludmg top reflux and fresh feed h hot I + 1 condltlons after a concentration charactensuc crosses a temperature boundary 1 condttlons before a concentration characterIs& crosses a temperature boundary, d I = hot, 1+ 1 = cold, and vice versa I increased concentration J con&tlons before a shock wave conditions after a shock wave J+1 mured reservous MIX sohd S sh shock T

th IJt

top

thermal upflow unnuxed reservous

NOTATION

sohd-flrud eqmhbnum dsmbutmn parameter for linear Isotherms, q = A(T)C cross sectional area of column, cm2 heat capacity, cal/g“C flurd concentration, g/l fresh feed per cycle, g length of bed, cm total product flow per half cycle m bed, g sohd-phase concentration, g/g of dry sohd heat requued per cycle, cal Internal reflux ratio, amount refluxedlamount total flow per cycle temperature, “C tune, mm wave velocity, cm/mm cold and hot concentration wave velocltles, cm/mm mterstitlal fled velocity, cm/mln ax1a.ldistance, cm Greek symbols a mterpticle void fraction c mtra-partmle votd fraction pI structural density of sohd, g/cm’ pt fluid density, g/cm’ y concentration ratlo defined in eqn (12)

REmRBNcla

I11 Bolto 3 A, Chem Tech 1975 5 303 I(rce R G , Sepamt Punfic Mehds 1976 5 139 t:; Rolke R W and W&elm R H , Ind Engng Chem Fnndls 1%9 8 235 [41 Wankat P C , Dare J C and Nelson W C . Separar Pun@ Methods 1975 4 215 J M , m Adsorptwn and Ion 151 Sweed N H and budeau &change (E&cd by I Zvvlebel and N H Sweed), AICh E Symp Ser 1975 71 (152) 1 161 Pleford R L , Baker B and Blum. D E . Ind Engng Chem Fund/s I%9 8 144 171 Skarstrom C W , Recent Dcuelopments m Separatwn Sclcnce (Ed&xl by N LI). Vol II. pp 95-106 Cl&eland, CRC Press 1972 [a Baker B and Plgford R L , Rd Engng Chem Fundls 1971 10 283 t91 Lavle R and Reilly M J , Chem Engng Scr 1972 27 1835 (101 Blum D E, Ph D Thesis, pp 76-77 Umversity of Cabforma-Berkeley. 1971 (111 Camero A A and Sweed N H, AZCh EJ 197622 369 iI21 Dart I C and Wankat P C , Chem Engng Scr 197631921 [131 ywaF315,R, Schrodt J T and Kermode R I, &part Scr (141 Thompson D W and Bowen B D, Ind Engng Chem Fundls 1972 11415 [ISI W&elm R H , hce A W , Rolke R W and Sweed N H , Ind Engng Chem Fundls 1968 7 337 1161fieke R D , Ph D Theses, pp 31-34 Untverslty of Cabforma-Berkeley, 1972