The issue of June, 1925 of Industrial and Engineering Chemistry published a famous paper entitled

Transcription

1 Revsta Cêncas & Tecnologa Reflectons on the use of the Mccabe and Thele method GOMES, João Fernando Perera Chemcal Engneerng Department, IST - Insttuto Superor Técnco, Torre Sul, Av. Rovsco Pas, 1, Lsboa, Portugal Chemcal Engneerng Department, ISEL - Insttuto Superor de Engenhara de Lsboa, R. Conselhero Emído Navarro, 1, Lsboa, Portugal Emal: jgomes@deq.sel.pl.pt Abstract Ths paper reflects on the contnuty of teachng the McCabe and Thele method on chemcal engneerng courses for understandng the vapour-lqud equlbrum phenomena nvolved n dstllaton. Some exercses comparng the results obtaned by ths method as well as by other accurate analytcal methods are presented and solved, as they are taught n ISEL courses. Because of ts accuracy and easness of use, and also because of the fact as ths method allows an easy vsualsaton of the whole operaton tself, the McCabe and Thele method contnues to be wdespread used and students show a remarkable adherence to t. Introducton The ssue of June, 1925 of Industral and Engneerng Chemstry publshed a famous paper enttled Graphcal Desgn of Fractonatng Columns by.l. McCabe and E.. Thele, of the Massachusetts Insttute of Technology [1], descrbed as A rapd and accurate graphcal method ( ) for computng the theoretcal number of plates requred n a gven separaton of a bnary mxture ( ). Although the precse desgn of columns for separatng bnary mxtures of volatle components dates back to a paper by Sorel n 1889 [2], assumng that equlbrum s establshed between the lqud on each plate and the vapour rsng from t, and therefore calculatng the enrchment from plate to plate by heat and materal balances. Ths model s applcable to a wde range of contnuous, steady state and equlbrum-stage dstllaton operatons, but s somewhat tedous, whch resulted n further smplfcatons made by several authors such as Lews [3] and Peters [4], among others. Apart from these analytcal methods, graphcal methods were also ntroduced by Ponchon [5], Rodebush [6], Savart [7] and, of course, McCabe and Thele. hat s strkng s the fact that, after the mathematcal model of Sorel was publshed, these other methods were proposed for solvng the related equatons graphcally or algebracally durng an 18-year perod from 1920 to 1938, pror to the avalablty of dgtal computers, whch allowed even to solve equatons for multcomponent dstllaton. Today, the only method from that era that remans n consderable wdespread use, stll beng taught n graduate chemcal engneerng courses all over the world and descrbed n the major textbooks dealng wth unt operatons [8], s, n fact, the method of McCabe and Thele. The latter method has some attrbutes that are, and probably wll be, responsble for ts contnung popularty: t shows clearly the ease or dffculty of each specfc separaton, as pnched regons can be readly seen, as well as the effect of feed locaton, n spte of beng accurate. Also t can be even appled to more complex problems such as separaton of azeotropes, ternary mxtures, etc. Apart from dstllaton tself, ths graphcal method has been appled wth success to other unt operatons such as gaseous absorpton and strppng where the basc phenomena nvolved are bascally the same. Ths method can even be appled to lqud-lqud extracton for constructng auxlary curves wthn the applcaton of the Hunter-Nash method [9], and also to the desgn of multcomponent dstllaton mxtures. Ano 1 n. 1 julho-dezembro

2 Centro de Cêncas e Tecnologa Calculatons on dstllaton In order to llustrate ths some exercses are currently solved n the classes of Separaton Processes as taught n the graduate course of Chemcal Engneerng of ISEL Insttuto Superor de Engenhara de Lsboa, where from smple examples t s ntended to show to the student the easness and accuracy of the McCabe and Thele method, partcularly when appled to more complex stuatons other than smple bnary dstllaton. The followng example s partcularly useful n order to assess the applcablty of the McCabe and Thele method, especally when compared wth other analytcal methods, such as Fenske [10] and Krkbrdge [11] methods. Let us consder the followng problem, ntroduced by Seader and Henley [8]: a 25 mol% mxture of acetone (A) n water () s to be separated by dstllaton at an average pressure of 130 kpa (975 torr), nto a dstllate contanng 95 mol% acetone and a bottoms contanng 2 mol% acetone. The nfnte-dluton actvty coeffcents are, respectvely: γ acetone = 8.12 γ water = 4.13 Usng the McCabe and Thele t s possble to calculate the number of equlbrum stages requred. Apart from that let us also calculate the same by usng the Fenske equaton and then draw conclusons. Prelmnary analyss of thermodynamc data As the pressure s close to 1 atm, the vapour-lqud equlbrum compostons can be estmated from the modfed Raoult s law for deal gases: K = γ L Ps / P Eq. (1) Vapour pressures can be obtaned from Antone equatons and lqud-phase actvty coeffcents over the entre composton range from fttng the nfnte dluton coeffcents to the two-constant van Laar equaton: Thus: ln γ 1 = A 12 / [1+(x 1 A 12 )/(x 2 A 21 ] 2 Eq. (2) ln γ 2 = A 21 / [1+(x 2 A 21 )/(x 1 A 12 ] 2 Eq. (3) K A = y A /x A = P s / P Eq. (4) and: K = y x = γ P s / P Eq. (5) Solvng equatons (2) and (3) to the actvty coeffcents: = exp ( A A / [1+ (x A A A /x A A ) ] Eq. (6) γ = exp ( A A / [1+ (x A A /x A A A ) ] Eq. (7) and, as: A A = ln γ = ln (8.12) = A A A = ln γ = ln (4.13) = Substtutng these van Laar coeffcents nto equatons (6) and (7), we obtan: = exp ( / [1+ (1.477 xa /(1- xa ) ] 2 Eq. (8) γ = exp ( / [1+ ( (1- xa )/ xa ] 2 Eq. (9) 2 - Unversdade Católca de Pernambuco

3 Revsta Cêncas & Tecnologa Now, we have to consder the correlaton between temperature and pressure of the gaseous phase for both substances, whch can be obtaned from Perry s Chemcal Engneers Handbook [12], as follows: log P s = / (T ) Eq. (10) A log P s = / (T ) Eq. (11) where: P s s the vapour pressure of component (torr) and T s the temperature ( C). Determnaton of the mnmum stages by the Fenske equaton In order to apply the Fenske equaton, t s necessary, frst, to determne the relatve volatltes at the top and at the bottom of the column. Then the average value can be used to compute the Fenske equaton. Bubble pont for the dstllate composton The bubble pont condton s: Σ x D K = Σ x D (γ P s / P) = 1 Eq. (12) In the dstllate, xa = 0.95 and x = From equatons (6) and (7): = exp ( / [1+ (0.95x2.094 / 0.05x1.418) ] 2 = γ = exp ( / [1+ (0.05x1.418 / 0.95x2.094) ] 2 = 3.75 Then, equaton (12) becomes: 0.95 (1.003 P s /975) (3.75 A Ps /975) = 1, where the vapour pressure s expressed n torr. Now, usng equatons (10) and (11), by a tral and error calculaton, we obtan a dstllate temperature of 64 C. Then, from equatons (4) and (5), α A, = K A /K = 1.48 Bubble pont for the bottoms composton In the bottoms x A = 0.02 and x = Agan, from equatons (6) and (7): = exp ( / [1+ (0.02x2.094 / 0.98x1.418) ] 2 = 7.19 γ = exp ( / [1+ (0.98x1.418 / 0.02x2.094) ] 2 = Equaton (12) now becomes: 0.02 (7.19 P s /975) (1.001 A Ps /975) = 1, where the vapour pressure s expressed n torr. Now, usng equatons (10) and (11), agan by a tral and error calculaton, we obtan a dstllate temperature of 95 C. Then, from equatons (4) and (5), α A, = K A /K = The average volatlty s the geometrc mean average volatlty: (1.48x27.7) 1/2 = 6.40 Calculaton of the mnmum number of stages The mnmum number of stages can now be computed from the Fenske equaton: N mn = log [(x, N+1 /x, 1 ) / (x j,1 /x j, N+1 ) ] / log α j Eq (13) N mn = log [(x Da /x Ba ) / (x Bw /x Dw ) ] / log α A, = log [(0.95/0.02)/(0.98/0.05)]/log 6.40 = 3.7 Ano 1 n. 1 julho-dezembro

4 Centro de Cêncas e Tecnologa Determnaton of the mnmum stages by the McCabe and Thele method In order to apply the McCabe and Thele method, t s necessary to obtan the y-x equlbrum curve at the operatng pressure of the column (975 torr) n terms of acetone molar fractons. Ths can be obtaned by performng the bubble-pont calculatons, as above, for a set of ponts n the range of lqud compostons between the dstllate and bottoms compostons: for each pont, the K-value and the acetone vapour mole fracton are obtaned from equaton (1). The obtaned results are presented on table 1. Table 1 Data for y-x equlbrum curve The McCabe and Thele plot referrng these data s shown n fgure 1. The equlbrum stages for total reflux are stepped off between the equlbrum curve and the 45 lne, whch represents the operatng lne, from the acetone x D = 0.95, down to the acetone x B = From ths plot t can easly be seen that there are 5 mnmum stages, whch s 35% hgher than the 3.7 mnmum stages that were obtaned by the Fenske equaton. However, t should be noted that the Fenske equaton s not relable for hghly non-deal systems, such as ths one. 4 - Unversdade Católca de Pernambuco

5 Revsta Cêncas & Tecnologa Fgure 1 Plot for obtanng equlbrum stages by the McCabe and Thele method Apart from that the McCabe and Thele method presents to the student a clear pcture of the whole operaton and also allows for some senstvty analyss as he can analyse quckly the effect of changes on the operatonal condtons such as feed condton, and the number of stages necessary n order to obtan slghtly dfferent x D or x B startng from the same pont x F. In order to assess the full extenson of the methods, we, usually proceed wth ths exercse now to determne the rato of rectfyng to strppng equlbrum stages, as gven by the Fenske equaton, the Krkbrdge equaton and the McCabe and Thele method. Then, part 2 of ths exercse, conssts n determnng the rato of rectfyng to strppng stages regardng the same mxture and operatonal condtons, usng the Fenske equaton, the Krkbrdge equaton and the McCabe and Thele method. Fenske equaton The table 2 sums up the composton of the dfferent streams. Ano 1 n. 1 julho-dezembro

6 Centro de Cêncas e Tecnologa Table 2 Composton of feed, dstllate and bottoms For computng the results usng the Fenske equaton, the relatve volatlty at feed condtons s also needed: assumng a bubble-pont lqud feed, from vapour-lqud data table 1, we obtan: α, w = [(0.789/0.250)] / [(0.211/0.750)] = 11.2 From the Fenske equaton for determnng feed-stage locaton: ( ) mn / (N S ) mn Eq. (14) or: NR / NS = log [(x LK,D /z LK,F ) (z HK,F /x HK,D ) log [(α B, α F ) 1/2 ] / / log [(z LK,F /x LK,B ) (x HK,B /Z HK,F ) log [(α D, α F ) 1/2 ] Eq. (15) Thus: = log [(0.95/0.25)(0.75/0.05)] log [(27.7x11.2) 1/2 )] / / log [(0.25/0.02)(0.98/0.75)] log [(1.48x11.2) 1/2 ] = 2.96 Krkbrdge equaton As the Fenske equaton s not relable except for farly symmetrcal feeds and separatons, a reasonably good approxmaton of optmal feed-stage locaton can be made by employng the emprcal Krkbrdge equaton [11]: = [(z HK,F /z LK,F ) (x LK,B /x HK,D ) 2 (B/D)] Eq. (16) By materal balances on acetone and water, we get: B/D = 0.753/0.247 = 3.05 and now: = [(0.75/0.25) (0.02/0.05) 2 (3.05)] = 1.08 McCabe and Thele method From the McCabe and Thele graph shown on fgure 1, the total number of mnmum stages s 5. The rato of the number of rectfyng stages to strppng stages can also be derved from the graph as: = 4.2/0.8 = 5.25 In ths partcular example, the McCabe and Thele method s the most accurate. The Fenske equaton s somewhat better than the Krkbrdge equaton, because the latter does not take nto account the drastc change n relatve volatlty occurrng from the dstllate to the bottom streams, as ponted out by Olver [13]. Concluson Ths example clearly shows the students that when the assumpton of constant molar overflow s vald n each of the two sectons of a dstllaton column, the McCabe and Thele graphcal method s convenent for determnng stage and reflux requrements. Ths method facltates the vsualzaton of 6 - Unversdade Católca de Pernambuco

7 Revsta Cêncas & Tecnologa many aspects of dstllaton and also provdes a procedure for locatng the optmal feed-stage locaton. Further examples can be drawn, showng that the McCabe and Thele method can be extended to handle Murphree stage effcency [14], multple feeds, sde streams, open steam, as well as the use of rebolers and ntercondensers. From the student s opnons on ths ISEL course, t was concluded that the effort requred to learn ths method s well justfed, manly because t provdes an unque tool for solvng chemcal engneerng problems about unt operatons dealng wth phase equlbrum. It s beleved that ths fact wll provde for the contnuty of use of the McCabe and Thele method n chemcal engneerng course. References [1] McCabe,.L.; Thele, E.. Ind. Eng. Chem., v.17, p , [2] Sorel. Comptes Rendus. v. 58, p. 1128, [3] Lews. Ind. Eng. Chem., v. 14, p.1, [4] Peters. Chem. Met. Eng., v. 28, p. 207, [5] Ponchon, M. Tech. Moderne, v. 13, p. 20, [6] Rodebush. v. 14, p. 1036, [7] Savart, R. Arts et Meters. Specal n , May, [8] Seader, J.D.; Henley, E.J. Separaton Process Prncples. 2nd Edton, John ley & Sons, New York, [9] Hunter, T.G.; Nash, A.., J. Soc. Chem. Ind., v. 53, p. 95T-102T, [10] Fenske, M.R.. Ind. Eng. Chem., v. 24, p , [11] Krkbrdge, C.G. Petroleum Refner. v.23(9), p ,1944. [12] Perry, R.H.;Green, D.. McGraw Hll,Perry s Chemcal Engneers Handbook, New York, [13] Olver, E.D. Dffusonal Separaton Processes: Theory, Desgn and Evaluaton, John ley & Sons, New York, [14] Murphree, E.V. Ind. Eng. Chem., v. 17, p. 747, Ano 1 n. 1 julho-dezembro