A SHORT METHOD TO CALCULATE REACTIVE RESIDUE CURVE MAPS

Transcription

1 Dstllaton Absorpton 00 A.B. de Haan, H. Koojman and A. Górak (Edtors) All rghts reserved by authors as per DA00 copyrght notce A SHORT METHOD TO CALCULATE REACTIVE RESIDUE CURVE MAPS M. Carrera-Rodríguez, J. G. Segova-Hernández and A. Bonlla-Petrcolet Departamento de Ingenería Químca, Dvsón de Cencas Naturales y Eactas, Unversdad de Guanajuato, Campus Guanajuato, Nora Alta S/N, Guanajuato, Gto , Méco, Emal: gsegova@qujote.ugto.m Insttuto Tecnológco de Aguascalentes, Departamento de Ingenería Químca, Av. López Mateos 80, Aguascalentes, Ags. 056, Méco, Emal: petrcolet@hotmal.com Abstract Reactve resdue curve maps (RRCM) are useful for the desgn of reactve dstllaton columns as a tool to establsh feasble zones of reacton-separaton. The calculaton of RRCM usually nvolves great computatonal effort due to the nonlnearty of the model uatons and ts teratve nature for the determnaton of reactve phase ulbrum. In ths study, a smplfed method for the generaton of RRCM s presented. Ths method s based on the applcaton of reacton-nvarant composton varables and assumes that the phase ulbrum constants and the relatve volatltes are ndependent of the temperature. These assumptons allow avodng teratve calculatons for obtanng a good appromaton of RRCM. Several case studes are used to show the effectveness of the proposed method. Keywords: reactve resdue curve maps, reactve phase ulbrum, separaton boundares, reacton-nvarant composton varables. Introducton A resdual curve represents the change n composton wth respect to tme of the phases durng a smple dstllaton. The resdue curve maps (RCM) are an mportant tool n the ntal stage of the process desgn for dentfyng, n a fast form, the nfeasble suences. Ths s because RCM provde the possblty of determnng the estence of dstllaton boundares and, as a consuence, determnng dfferent potental zones of operaton. Once establshed the zone of feasble operaton and, dependng on the feed composton, t s possble to predct the dfferent components obtaned as dstllate and bottom products. It s mportant to note that several numercal dffcultes are nvolved n the modelng and desgn of reactve dstllaton (RD) systems. These dffcultes have ther orgn manly n the multcomponent nature of the problems consdered, the nonlnearty of the thermodynamc models caused by the presence of smultaneous chemcal and physcal ulbrum, and also by the type of varables nvolved n defnng the problem, whch are generally composton varables n molar unts and etents of reacton 3. In ths contet, the premse of usng the concept of transformed composton varables for obtanng RCM s that the uatons that characterze a RD system are epressed mathematcally n the same form that those reported for non-reactve dstllaton system. Usng a transformed varable approach, the soluton space s restrcted to compostons that are already at chemcal ulbrum and the problem dmenson s also reduced 3. These advantages allow studyng a varety of real and comple reactve systems, because there are several combnatons between the number of reactons (R) and the number of components (C) that can be analyzed n ternary dagrams. Therefore, the analyss of RRCM can be performed n the same form as n smple dstllaton wthout chemcal reactons. Untl now, only few methods have used transformed composton varables for the calculaton of RRCM. However, current methods may show a sgnfcant computer tme for the calculaton of RRCM 3,4. In ths study, we take advantage of the characterstcs of reacton-nvarant composton varables 4 to develop a short method for calculatng RRCM. Specfcally, we propose a smplfed approach for determnng RRCM by dscardng the effect of temperature on the reactve phase ulbrum constants, but preservng the composton effect. Although ths assumpton has been used for nonreactve mtures 5, t has not been appled for the study of reactve mtures. Our results ndcate that the use of transformed composton varables and the applcaton of smplfed phase ulbrum constants (.e., not dependent of temperature) avod the teratve calculaton of the mture bubble temperature, resultng n an effectve and faster strategy for calculatng RRCM. Fnally, the performance of our short method s compared wth those reported usng a rgorous method. 46

2 M. Carrera-Rodríguez et.al.. Descrpton of Reactve Resdue Curve Calculaton For homogeneous mtures wth multple chemcal reactons, the calculaton of a reactve resdual curve (RRC) s based on a modfcaton of the Raylegh epresson 4. For a system of C components subject to R ndependent chemcal reactons, the followng set of uatons s used : d dτ Y () where represents the transformed composton n the lqud phase of component, Y s the transformed composton n the vapor phase of component, and τ s the dmensonless tme, respectvely. Equaton () s obtaned from a mass balance appled to a dstllaton unt and by ntroducng a dmensonless tme varable. RRCM s obtaned from the forward and backward ntegraton of Equaton () wth respect to tme startng from an ntal composton. For calculatng the RRC, the ntegraton step depends on the case study, however, a small ntegraton step s usually recommended. It s convenent to note that each RRC rures a sgnfcant number of ponts to be constructed, each one nvolvng the calculaton of the vapor-phase composton n ulbrum wth the lqud-phase composton. Therefore, bubble pont calculatons are needed to obtan the vapor-phase composton n the tradtonal approach. Because such calculatons are teratve, the generaton of RRCM rures a sgnfcant numercal effort and computer tme due to the nonlnearty of the model uatons and the use of numercal methods for calculatng the reactve phase ulbrum 3,4. In ths study, we propose the use of transformed composton varables to reduce the problem dmenson and the applcaton of smplfed phase ulbrum constants, whch are ndependent of temperature, to decrease the numercal effort for obtanng a RRCM. In a reactve system, the Gbbs free energy functon behaves as n a non-reactve system f transformed composton varables () are used nstead of the conventonal composton varables 4. Usng these transformed varables, the soluton space s restrcted to compostons that are already at chemcal ulbrum and, as a consuence, the problem dmenson s also reduced 3. So, the reactve phase dagrams look smlar to the non-reactve ones and the non-reactve flash algorthms can be easly modfed to account for the ulbrum reactons. Ung and Doherty 4 showed that the chemcal potental follows all the thermodynamc relatonshps of a non-reactve system as long as all the thermodynamc propertes are functons of the transformed composton varables. Based on ths fact, the ulbrum constant Kˆ for phase ulbrum calculatons can be defned as 6 α γ K ˆ {, Y },,C R () β γ where the transformed mole fractons and Y are gven by v N ref N ref,,c R (3) y v N y ref Y N yref,,c R (4) α where s the lqud mole fracton of component, y s the vapor mole fracton of component, γ s β the lqud actvty coeffcent of component, and γ s the vapor actvty coeffcent of component. For transformed varables, ref s the column vector of R reference component mole fractons, v s the row vector of stochometrc number of component for each reacton, v TOT s a row vector where each element corresponds to reacton R and t s the sum of the stochometrc number for all components that partcpate n reacton R, and N s a square matr formed from the stochometrc number of the reference components n the R reactons 4. 46

3 A short method to calculate reactve resdue curve maps Then, the reactve phase ulbrum condton n terms of transformed varables, usng Equatons ()- (4), s gven by 6 Y Kˆ θ + δ,,c R (5) and v N TOT ref θ N yref (6) v N ( Kˆ y ) TOT ref ref δ,,c R (7) N yref For the transformaton procedure, the references mole fractons are calculated usng Equaton (3) or (4) and from the ulbrum constants for each reacton K by solvng a system of R nonlnear uatons gven by K n r v a r,,r (8) r where a s the actvty of component and v s the stochometrc number of component n reacton R, respectvely. When we know the reference mole fractons, the remanng mole fractons are calculated usng Equaton (3) or (4) for lqud and vapor phase, respectvely. When the effect of temperature over Kˆ s neglected, a sutable estmaton of the bubble temperature (T B ) s suffcent to obtan a good descrpton of the reactve vapor-phase ulbrum composton. In ths study, a weghted sum of the pure-component bolng temperatures ( T ) s used for estmaton T B b T B C R Tb (9) Usng Equaton (9), the transformed composton n the vapor phase s calculated applyng the Equatons () and (5)-(7). Note that a sgnfcant reducton n computer tme should be observed because the bubble pont temperature s not calculated usng an teratve procedure. 3. Results and Dscusson In ths secton, we show the effectveness of the proposed short method for the calculaton of RRCM and the results are compared to those obtaned wth the tradtonal method based on teratve bubblepont calculatons. RRCM were calculated for three reactve mtures: a multreactve deal system at.03 bar and two real systems that nvolves the synthess of methyl tert-butyl ether (MTBE) n the presence of nert at 8.04 bar, and the synthess of Tert-amyl methyl ether (TAME) wthout nert at 4.05 bar. Detals of reactve systems and thermodynamc models are provded n Table. For the sake of brevty, n ths paper we report the RCCM for all reactve systems and detaled results are analyzed for one case of study. 3. System A3 A4 A5 A4 A4 A6 wth nert A, A We consdered a hypothetcal system consstng of s components where four of them are nvolved n three ndependent chemcal reactons and the remanng two are nert. Ths system was analyzed at atmospherc pressure. The presence of lqud-vapor ulbrum was assumed, where both phases were consdered deal. The saturaton pressure of the pure compounds was calculated wth the Antone uaton, see parameters reported n Table. The components A 3, A 4 and A 5 were used as reference substances to calculate the transformed mole fractons, whch are defned as 463

4 M. Carrera-Rodríguez et.al. (0) () () These transformed mole fractons present values n the nterval (0, ). Fgure dsplays the RRCM for the multreactve system usng both our short method and the tradtonal approach. An ecellent agreement of RRCM s obtaned wth both methods and both reactve resdue curves can be consdered for process desgn. In ths reactve system, the presence of dstllaton boundares or azeotropes are not observed. In Fgure, the RRCM for the synthess of MTBE s shown. In ths case, the presence of a reactve ternary azeotrope near the pure n-butane node and a nonreactve bnary azeotrope (Methanol- Butane) are observed. Ths mture shows a dstllaton boundary that dvdes the composton dagram n two regons. In addton, pure Methanol and Isobutene provdes a stable node, and the n-butane s a saddle node. Agan, an ecellent agreement s agan observed between the RRCM calculated by both the rgorous method and our short strategy. The RRCM for the reactve mture related to the synthess of TAME s shown n Fgure 3. Two unstable bnary azeotropes (MB-Methanol and MB-Methanol) were found and there are two dstllaton boundares that dvde the composton dagram n three regons. The dstllaton boundares have a pronounced curvature. In addton, pure Methanol provdes a stable node, and the MB and MB are saddle nodes. The comple nature of ths reactve mture s reflected n the shape of ts RRCM. Although our short method does not match the results of the rgorous method as well as t does for others reactve systems, t may consder as a proper reference pont for prelmnary applcatons, because a good descrpton of the composton space s observed. Table. Reactve mtures selected to calculate RRCM usng the short method Thermodynamcs System Chemcal ulbrum constant Model Hypothetc system of three Ideal lqud and 3 K.5, K and K 5 reactons wth nert A, A Ideal gas A A A A A A6 Isobutene + Methanol MTBE wth n-butane as an nert -methyl--butene (MB) + -methyl--butene (MB) + Methanol TAME Wlson model and G o R T T T Ideal gas 7 rs / ln ( G / T ) K e T []K Wlson model and 4 (473.5 / T ) K Ideal gas 8 e T []K Table. Parameters of the Antone uaton for modelng a hypothetc multreactve system Component A B C A A A A A A log 0 P sat B A T + C P [] mmhg T [] C, 5 P sat []Pa, T []K sat 464

5 A short method to calculate reactve resdue curve maps Short method Rgorous method Fgure. The RRCM for a hypothetcal multreactve system of three reactons wth two nert n-butane Short method Rgorous method Isobutene 0.0 Methanol Fgure. RRCM for the synthess of MTBE n presence of n-butane as nert MB MB Methanol Fgure 3. RRCM for the synthess of TAME Short method Rgorous method 465

6 M. Carrera-Rodríguez et.al. It s convenent to remark that the man advantage of the proposed short method s the reducton of computer tme. Table 3 shows the number of ntegraton steps and the computer tme rured for the calculaton of 0 reactve resdue curves. The calculaton tme for the short method s 0% 6% of the tme rured for the rgorous method. Table 3. Computer tmes and ntegraton steps rured for RRCM calculatons n selected reactve systems Rato of computer tme: Integraton steps short RRMC/rgorous RRMC System Rgorous Short RRMC RRMC Ideal MTBE TAME Our results suggest that, for deal multreactve systems that not present dstllaton boundares or azeotropes, the calculaton of RRCM usng an appromate T B value produced ecellent results. However, the MTBE and TAME systems are more comple because they show azeotropes and dstllaton boundares. The bggest dfferences between the results of the short method and the rgorous method are present around these ponts. For MTBE, the azeotrope has a temperature value wthn the nterval of the bolng temperatures for the pure components. The proposed method provdes good results for RCCM n ths case. In the case of TAME, whch shows the major devatons, there s an azeotrope wth temperature outsde the range gven by the bubble ponts of the pure components. Even under these condtons, the short method provdes reactve resdue curves that satsfactorly match those obtaned by rgorous calculatons. These results suggest that the proposed method s an alternatve for obtanng a sutable estmaton of RCCM effcently. 4. Conclusons Reactve resdue curve maps (RRCM) are mportant tools to analyze the feasblty of a proposed splt for the desgn of reactve dstllaton columns n an easy, fast, and qualtatve format. In ths study, the use of smplfed phase ulbrum constants based on reacton-nvarant composton varables has been proposed to calculate these reactve resdue curves. Our results ndcate that a good appromaton of the RRCM s obtaned wth ths smplfed method, even for reactve azeotropc mtures wth more than one separatr. The bggest dfferences between the short method and the rgorous method are present around of the azeotropes and dstllaton boundares. However, ths dscrepancy does not represent a sgnfcant problem because these RRCM are used n the ntal stages of process desgn. However, f a better representaton n the neghborhood of the dstllaton boundares s rured, a hybrd method can be used: the boundary and the closer curves can be calculated wth the rgorous method and the rest of the composton space wth the short method. Usng ths approach, a sgnfcant reducton n computaton tme for the calculaton of RRCM s assured. Acknowledgements We acknowledge the fnancal support provded by Unversdad de Guanajuato, Insttuto Tecnológco de Aguascalentes, CONACyT and CONCyTEG (Meco). References. K. S. Wasylkewcz and S. Ung, Flud Phase Equlb, 75(000) S. Skogestad, V.N. Kva and E.K. Hlmen, Chem. Eng. Sc., 58(003) W.D. Seder and S. Wdagdo, Flud Phase Equlb, 3(996) S. Ung and M.F. Doherty, Chem. Eng. Sc., 50(995) C. Gutérrez-Antono, M. Vaca and A. Jménez, Ind. Eng. Chem. Res., 45(006) A. Bonlla-Petrcolet et.al, Chem. Bochem. Eng., 0(006) R.W. Maer, J.F. Brennecke and M.A. Stadtherr, Comp.Chem. Eng., 4(000) F. Chen, R.S. Huss, M.F. Doherty and M.F. Malone, Comp. Chem. Eng., 6(00)