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2 atch Dstllaton: Thermodynamc Effcency 4 José C. Zavala-Loría * and Astera Narváez-García Unversdad Autónoma del Carmen (UNACAR), DES-DAIT: Facultad de Químca, Cd. del Carmen, Cam. Méxco 1. Introducton The batch dstllaton s a separaton processes that requres great amounts of energy. Due to hgh energy costs, the study of energy consumpton s of great nterest n ths process (Zavala et al. 27). Accordng to Luyben (199), the energy consumpton ncreases when operaton occurs n a batch manner. Determnng how effcent s the heat transfer under specfc conons and to modfy them n order to fnd how effcent the heat s used, s an mportant task. The analyss of thermodynamc effcency n a batch dstllaton column has been presented by Km and Dwekar (2), Zavala-Loría (24), Zavala et al. (27), Zavala & Coronado (28) and Zavala et al. (211). The frst work only developed expressons to calculate thermodynamc effcency whle the rest of the works developed expressons to calculate thermodynamc effcency and have appled them to a new problem of optmal control: Maxmum thermodynamc effcency. 2. Descrpton of the process For ths study we used a conventonal batch dstllaton column consstng of: Reboler Tray column Condenser Reflux drum Recever Fgure 1 Shows a conventonal batch column lke the one used for ths study. Mass balances resultng from the process shown n Fgure 1, allow us to obtan the mathematcal model n Table 1. The model consders the followng elements: Theoretcal stages Total condenser Neglble pressure drop * Correspondng Author

3 92 Dstllaton Advances from Modelng to Applcatons Fg. 1. Conventonal batch dstllaton column. Constant flows: Vapor Lqud. Constant lqud holdup: Stages Condenser-Reflux drum. Adabatc column d Dt V R 1 t (1) dx V L x y x1 x V ; 1,, n (2) dx j V L y y x x H j1 j j1 j j V ; 1,, n ; j 1,, N (3) V dxd yn xd H D ; 1,, n (4) n n yj Kjxj 1 ; 1,, n ; j 1,, N (5) 1 1 Table 1. Mathematcal model of a batch dstllaton column (n=number of components and N=number of equlbrum stages).

4 atch Dstllaton: Thermodynamc Effcency 93 Consderng deal mxtures, the vapor-lqud equlbrum (VLE) can be obtaned from Raoult s law equaton: For non-deal mxtures, we can use the followng equaton: sat y P K x P ; = 1, 2,,n (6) sat y P K ; = 1, 2,,n (7) x P Where K () s the constant of the VLE, and y () s the mole fracton of the vapor phase, x () s the mole fracton of the lqud phase, represents the actvty coeffcent, P sat s the saturaton pressure and denotes the partal fugacty coeffcent, all referrng to component, and P represents the system pressure. Actvty coeffcents can be obtaned wth a soluton model (Wlson, NRTL, etc.) or the Chao-Seader correlaton. The fugacty coeffcents can be obtaned wth an equaton of state (Redlch-Kwong, Soave, Peng-Robnson, etc.) At standard or low pressures P sat can be obtaned wth Antone s equaton. sat ln P A T C ; = 1, 2,,n (8) Coeffcents A, and C appear n lterature related to ths area of study. 3. Thermodynamc effcency y defnton, the thermodynamc effcency ( t ) of a process based on avalablty or exergy s defned as: Wmn Wmn t Wtotal Wmn LW sep sep (9) where W s the work and LW s the total loss of work. Snce the mnmum work s determned by changes n exergy, they can be determned usng the Frst and Second Law of Thermodynamcs. Fgure 2 shows the control volume used to obtan the equatons that represent thermodynamc effcency n batch dstllaton. Fgure 2 shows that the process can exchange energy wth the envronment but does not perform any mechancal work. The energy balance (enthalpy) gven by the Frst Law of Thermodynamcs, consderng the reboler, the column of trays and the condenser-reflux drum, s: m di n dhtit d I sst d H I 1 D rt (1) where:

5 94 Dstllaton Advances from Modelng to Applcatons d mi sst Q Q Q DI C D (11) Accordng to the Second Law of Thermodynamcs, the entropy balance s: m ds n dh S d S sst d H S 1 t t D rt (12) where: dms sst Q Q QC DSD ds T T T C rr (13) Fg. 2. Control volume for batch process. I represents enthalpy, S s entropy, Q s the amount of heat, C represents the condenser, D represents the dstllate, represents the bottoms, m s the system mass, s the reference state, t s the stage or tray and T s the temperature. The avalable work can be obtaned f we combne equatons (11) and (13). To do so, equaton (13) must be multpled by the temperature of the reference state T (t s consdered that the reference state s lqud at 25 C and 1. atm): m dmt S dmt S T T T T sst Q Q QC DTSD Td Srr C d I sst T T T T sst Q Q QC DID Q Q QC DTSD Td Srr C

6 atch Dstllaton: Thermodynamc Effcency 95 dm I T S T T T T 1 Q 1 QC D ID TSD Td Srr C (14) the avalablty s represented as A I TS, then: d ma sst T T 1 Q 1 QC DA D TdSrr (15) T TC masst d T T 1 Q 1 QC DA D LW (16) T TC Where the loss of work s defned as LW TdSrr. The terms of the rght hand sde of Equaton (16) represent the dfference between the exergy or the avalablty of flows enterng and leavng the system. The term on the left hand sde s the change of exergy n the system. Fndng the values n Equaton (16), the work loss s: sst T T d ma sst LW 1 Q 1 QC DA D (17) T TC dma can be calculated, f dscretsed, by multplyng Equaton (12) by T mnus Equaton (1): ma t sst n A H A H A t t D rt 1 dma Another way to estmate the value of s by applyng an exergy balance at the bottom of the column. Applyng an exergy balance n the reboler, we have: sst t A A A (18) dm sst d d A d (19) Introducng Equaton (1) from Table (1) n Equaton (19), we obtan: A da d DA (2) Accordng to Km and Dwekar (2) and Zavala-Loría and Coronado-Velasco (28), the total exergy can be calculated from ts physcal component ( A ) and ts chemcal component ( A chem ),.e.: phs chem AA A (21) phs

7 96 Dstllaton Advances from Modelng to Applcatons where A phs consders the physcal processes that nvolve thermal nteractons wth the surroundngs and A chem consders the mass and heat transfer wth the surroundngs. The man contrbuton to ths energy s due to mxng effects and can be estmated from the chemcal potental at low pressures (Km and Dwekar, 2). A s relatvely lower than A chem. Thus, the physcal component can be regarded as constant for all chemcal speces and the dervatve of ths term s elmnated. The chemcal component of exergy for an deal mxture can be expressed as: whereas a non-deal mxture can be calculated as: A chem RT x ln x (22) A chem RT x ln x (23) and the exergy exchange n the reboler for an deal mxture can be calculated as: for a non-deal mxture: A A A ln phs A chem RT x phs x (24) A A A ln phs A chem RT x phs, x (25) Takng the dervatve of Equatons (24) and (25) yelds: phs d x ln x d d A A chem RT d x ln, x d d A A chem RT (26) (27) The dervatve of the term on the rght hand sde of Equatons (26) and (27) can be represented n terms of the mathematcal model of the column; thus, substtutng Equaton (2) of Table 1 yelds: d x ln x dx 1 lnx V L x y x1 x V (28)

8 atch Dstllaton: Thermodynamc Effcency 97 d x ln, x dx x d, 1 ln, x, V L x y x1 x 1 ln, x V x d,, (29) Substtutng Equaton (28) on the rght hand sde of Equaton (2) for an deal mxture, and Equaton (29) on the rght hand sde of Equaton (2) for a non-deal mxture yelds: Ideal Mxture: A L d VRT x y x x 1ln x 1 V DAphs, RT x ln x (3) Non-deal Mxture: A L d VRT x y x1 x 1 ln, x V x d, RT, Dt A, phs RT x ln, x (31) The exergy of the current producton (dome) that wll be used n Equaton (6) for an deal mxture can be calculated as: for a non-deal mxture: AD A D, phs RT x D ln x D (32) AD A D, phs RT xd ln, Dx D (33) The exergy transfer assocated wth the transmsson of energy as heat n the process can be calculated by the energy balances n the reboler and condenser. Consderng that H vap s the same for each component and s not related to the temperature of the process, the Clausus-Clapeyron equaton can be used to calculate H vap, and the frst two terms on the rght hand sde of Equaton (16) can be calculated by the followng equaton;

9 98 Dstllaton Advances from Modelng to Applcatons Ideal Mxture: T T vap Q 1 QD VH T T TD TC T (34) and consderng constant relatve volatlty: 1 T T xd Q 1 QD VRT ln T 1 TD x 1 11 (35) Non-deal Mxture: T T 1, D1, K1, 1 Q 1 QD VRT ln T T D 1, 1, DK 1, D (36) where, K s the lqud-vapor equlbrum constant and Φ s defned as: k sat Vk P Pk k sat k exp RT ; k = 1, 2,,n (37) Therefore, the term for exergy loss or work loss, LW n Equaton (17) for an deal mxture, can be calculated wth Equaton (35) and for a non-deal mxture wth Equaton (36): Ideal Mxture: Non-deal Mxture: 1 x D 1 11 LW VRTln 1 DRT x ln x x D ln x D x L VRT x y x1 x 1lnx V D A A (38), phs D, phs 1, D1, K1, LW VRTln DtRT x ln, x xd ln, Dx D 1, 1, DK 1, D D L VRT x y x1 x 1 ln, x V x d, RT, t, phs D, phs A A (39) Consderng that A, A,, then: phs D phs

10 atch Dstllaton: Thermodynamc Effcency 99 Ideal Mxture: 1 x D ln x DRT x ln x x D ln x D L VRT x y x1 x 1lnx V LW VRT (4) Non-deal Mxture: 1, D1, K1, LW VRTln DRT x ln, x xd ln D, x D 1, 1, DK 1, D L x d, VRT x y x1 x 1 ln, x RT V, (41) the mnmum work can be calculated as: Ideal Mxture: Wmn DRT xd ln x D x ln x L VRT x y x1 x 1ln x V (42) Non-deal Mxture: Wmn DRT xd ln D, x D x ln, x L VRT x y x1 x 1 ln, x V x d, RT, (43) and the thermodynamc effcency [Equaton (9)] results n: Ideal Mxture: t x ln D x D x ln x Rx1 y x y 1 ln x 1 x D 1 11 R 1ln 1 x 1 11 (44)

11 1 Dstllaton Advances from Modelng to Applcatons Non-deal Mxture: t x d, xd lnd, xd x ln, x Rx1 y x y 1 ln, x V, K1, 1, 1, D R 1ln K1, D 1, D 1, (45) Equaton (45) can be reduced f one consders that the changes of actvty coeffcents n the reboler are so small that they can be neglected. Then: t x xd lnd, xd R x1 y y 1 ln, x K1, 1, 1, D R 1ln K1, D 1, D 1, If the gas phase s deal or close to dealty, then Equaton (46) can be reduced even more: t x xd lnd, xd R x1 y y 1 ln, x K1, 1, D R 1ln K1, D 1, The soluton of Equaton (45) or ts smplfcatons [Equaton (46) and (47)] requre the calculaton of actvty coeffcents of the mxtures, whch can be obtaned wth an approprate soluton model (Wlson, NRTL, UNIQUAC, UNIFAC, etc.). Accordng to Equatons (44), (45), (46) and (47), the thermodynamc effcency not only depends on the reflux rato but on the concentraton of components n both phases and the knd of mxture. A convenent way to express thermodynamc effcency s the average thermodynamc equaton used by Zavala Loría (24), Zavala et al. (27), Zavala and Coronado (28). and Zavala et al. (211) to solve the maxmum thermodynamc effcency problem: (46) (47) t t average (48) t 4. Results and dscusson Usng the Frst and Second Law of Thermodynamcs, ths work has developed an expresson for calculatng the thermodynamc effcency of a batch dstllaton process [Equatons (44) and (45)]. To solve the mathematcal model, the feedng was ntroduced at the top of the column at bolng temperature, neglectng the accumulaton of vapor n each stage. The process was performed wth total condenser, atmospherc pressure and adabatc column. To solve the model, two mxtures were consdered. The frst one s an deal mxture of Hexane/enzene/Chlorobenzene (HC); the second one s a non-deal mxture of Ethanol- Water (EW).

12 atch Dstllaton: Thermodynamc Effcency 11 Case of study 1: Hexane/enzene/Chlorobenzene mxture The conons and data used to solve the mathematcal model of thermodynamc effcency are gven n Table 2. Relatve volatltes were calculated wth the Soave-Redlch-Kwong equaton of state. Varable/Parameter Amount Unts Feed (F) Vapor flow (V) Condenser lqud holdup (H D ) Tray lqud holdup (H T ) atch tme Z Hexane Z enzene Z Chlorobenzene α Hexane α enzene Number of trays Reflux rato (Constant) kmol kmol/h kmol kmol h Table 2. Data used to solve the mathematcal model of thermodynamc effcency The frst step for the resoluton of the mathematcal model was to reach the steady state to extract the product durng one hour. Fgure 3, shows the concentratons at the top of the column (ncludng the steady state) Hexane enzene Clorobenzene Tme, h Fg. 3. Concentratons at the top of the column.

13 12 Dstllaton Advances from Modelng to Applcatons The average thermodynamc effcency s 17.58% for ths producton tme whle the average concentraton of the most volatle component at the top of the column s 88.55% mol. Table 3, shows the varatons observed when the reflux rato s changed. Reflux rato average (%) 1 x (%) D, average Table 3. Thermodynamc effcency behavor based on varatons n the reflux rato. Fgure 4 shows the thermodynamc effcency behavor n the product obtanng Tme, h Fg. 4. Thermodynamc effcency behavor n the product obtanng. Such varatons make evdent the nfluence of the reflux rato over the thermodynamc effcency; n other words, the thermodynamc effcency of the process s smaller when the reflux rato s hgher. We also could observe that the reflux rato affects the product concentraton. If the reflux rato ncreases the product concentraton ncreases as well.

14 atch Dstllaton: Thermodynamc Effcency 13 Case of study 2: Ethanol/Water mxture The conons used for solvng the mathematcal model of thermodynamc effcency for a non deal mxture (Ethanol/Water) are gven n Table 4. Wlson s equaton was used to obtan the actvty coeffcents; the vapor phase s consdered to have an deal behavor. The steady state was reached when Ethanol presented a purty level of 89.61% mol obtaned usng Wlson s equaton hours was the perod of tme needed to reach the steady state, we obtaned an average thermodynamc effcency of 36.96% and the product presented an average concentraton of 86.74% mol of the most volatle component. Varable/Parameter Amount Unts Feed (F) Vapor flow (V) Condenser lqud holdup (H D ) Tray lqud holdup (H T ) atch tme Z Ethanol Z Water α Ethanol α Water Number of trays Reflux rato (Constant) kmol kmol/h kmol kmol h Table 4. Data used to solve the mathematcal model of thermodynamc effcency Ethanol Water Tme, h Fg. 5. Concentraton profles at the top of the column.

15 14 Dstllaton Advances from Modelng to Applcatons Fgure 5 shows the behavor of the concentratons at the top of the column, whle the behavor of punctual thermodynamc effcency s shown n Fgure 6. Table 5 shows the varatons observed when dfferent reflux ratos are used Tme, h Fg. 6. Thermodynamc effcency profle (obtanng product). Reflux rato average (%) 1 x (%) D, average Table 5. Thermodynamc effcency behavor wth regards to the reflux rato varaton. As n the last case, we observe that when the reflux rato ncreases, the effcency decreases; however, the concentraton of the most volatle component presents also an ncrement. We can formulate the followng heurstc rule: If the varables of the process are mantaned, the reflux rato wll have an nverse effect on the thermodynamc effcency of the process. 5. Concludng remarks Usng the Frst and Second Law of Thermodynamcs (exergy concept), ths work has developed an expresson for calculatng the thermodynamc effcency of a batch dstllaton

16 atch Dstllaton: Thermodynamc Effcency 15 process. The resultng equaton was used to fnd the batch dstllaton thermodynamc effcency for an deal mxture and a non-deal mxture. The equaton obtaned s a generalzaton of the equaton developed by Zavala-Loría et al. (27), Zavala and Coronado (28) and Zavala et al. (211). The results obtaned by solvng the Equatons, allowed us to observe the relatonshp between reflux and thermodynamc effcency of the process. Furthermore, varables such as the product purty are affected by the reflux rato, n other words, the purty of the product requres a greater amount of reflux to obtan a hgher concentraton. 6. Nomenclature H j H D molar hold-up on tray j molar hold-up on the condenser I enthalpy (J/mol) S entropy (J/mol K) V vapor flow rate (mol/h) L lqud flow rate (mol/h) D dstllate flow rate (mol/h) amount of moles n the reboler, mol n number of components N number of trays n the column R t reflux rato t tme (h) W work (J) LW work loss (J) z mole fracton n the feed () j x mole fracton n the lqud of component at plate j () j y mole fracton n the vapor of component at plate j () x mole fracton n the lqud of component n the reboler () D x mole fracton n the vapor of component n the dstllate () j K equlbrum constant T A t H vap Subndex temperature avalablty punctual thermodynamc effcency vaporzaton heat reference c condenser f end D dstllate boler or reboler t tme

17 16 Dstllaton Advances from Modelng to Applcatons 7. References Luyben, Wllam L. (199); Process modelng, Smulaton and Control for Chemcal Engneers; McGraw-Hll. Second eon, 725 pp. Km, K.J., Dwekar, U.M. (2); Comparng atch Column Confguratons: Parametrc Study Involvng Multple Objetves; Ache Journal, 46(12), pp Zavala-Loría, J.C. (24); Optmzacón del Proceso de Destlacón Dscontnua; Tess Doctoral, Departamento de Ingenería Químca, Insttuto Tecnológco de Celaya, Celaya, Guanajuato, Méxco. Zavala, José C., Córdova, Atl., Cerón, Rosa M. and Palí, Ramón J. (27); Thermodynamc effcences of a conventonal batch column; Int. J. Exergy, 4(4), pp Zavala, J. C. and Coronado, Crstna (28); Optmal Control Problem n atch Dstllaton Usng Thermodynamc Effcency; Ind. Eng. Chem. Res., 47, pp Zavala-Loría, J. C., Ruz-Marín, A. and Coronado-Velásco, Crstna (211); Maxmum Thermodynamc Effcency Problem n atch Dstllaton; razlan Journal of Chemcal Engneerng, 28(2), pp

18 Dstllaton - Advances from Modelng to Applcatons Eed by Dr. Sna Zereshk ISN Hard cover, 282 pages Publsher InTech Publshed onlne 23, March, 212 Publshed n prnt eon March, 212 Dstllaton modelng and several applcatons mostly n food processng feld are dscussed under three sectons n the present book. The provded modelng chapters amed both the thermodynamc mathematcal fundamentals and the smulaton of dstllaton process. The practcal experences and case studes nvolve manly the food and beverage ndustry and odor and aroma extracton. Ths book could certanly gve the nterested researchers n dstllaton feld a useful nsght. How to reference In order to correctly reference ths scholarly work, feel free to copy and paste the followng: José C. Zavala-Loría and Astera Narváez-García (212). atch Dstllaton: Thermodynamc Effcency, Dstllaton - Advances from Modelng to Applcatons, Dr. Sna Zereshk (Ed.), ISN: , InTech, Avalable from: InTech Europe Unversty Campus STeP R Slavka Krautzeka 83/A 51 Rjeka, Croata Phone: +385 (51) Fax: +385 (51) InTech Chna Unt 45, Offce lock, Hotel Equatoral Shangha No.65, Yan An Road (West), Shangha, 24, Chna Phone: Fax: