Complete Equation-Oriented Approach for Process Analysis and Optimization of a Cryogenic Air Separation Unit

Transcription

1 hs s an open access artcle publshed under an ACS AuthorChoce cense, whch permts copyng and redstrbuton of the artcle or any adaptatons for non-commercal purposes. pubs.acs.org/iecr Complete Equaton-Orented Approach for Process Analyss and Optmzaton of a Cryogenc Ar Separaton Unt Qwen Fu, ngyu Zhu,*, and X Chen*, State Key ab of Industral Control echnology, College of Control Scence and Engneerng, Zhejang Unversty, 38 Zheda Road, Hangzhou , Chna College of Chemcal Engneerng, Zhejang Unversty of echnology, Hangzhou , Chna Downloaded va on June 14, 2018 at 22:46:04 (UC). See for optons on how to legtmately share publshed artcles. ABSRAC: A cryogenc ar separaton unt produces large volumes of hgh-purty oxygen, ntrogen, and argon through dstllaton. Such a complex process wth a heat-couplng desgn, a hgh purty requrement, and a large-scale feature s dffcult to analyze and optmze. In ths study, a complete equaton-orented (EO) model, whch ncludes a unt model and a thermodynamc model, s developed for cryogenc ar separaton nvolvng a low-pressure column, a hgh-pressure column, and an argon sde arm column. he EO approach s then appled to deal wth the followng three ssues n ar separaton: thermodynamc parameter estmaton, process analyss wth heat-couplng desgn, and process optmzaton wth varyng load demands. he proposed EO method s superor to tradtonal sequental modular based commercal software n terms of convergence performance. 1. INRODUCION A cryogenc ar separaton unt (ASU) s an mportant ndustral process that generates hgh-purty gaseous and lqud products. 1,2 It provdes an effcent and cost-effectve method of supplyng a large amount of gases to processes n metallurgy, chemcal and petrochemcal ndustres, and power engneerng. When equppng an ASU to ron and steel plants, the load must be frequently changed because of the batch operatng feature of such plants. Process analyss and optmzaton are thus mportant for operatng an ASU; but t s also challengng for such a complex process wth a heat-couplng desgn, a hgh purty requrement, and a large-scale feature. A thermo-couplng desgn s a key feature of cryogenc ar separaton. Numerous studes have focused on the synthess of thermo-couplng flowsheets and structures to save energy and mprove economc performance. Kansha et al. 3 proposed a novel cryogenc ar separaton technque based on self-heat recuperaton to reduce energy consumpton. Fu et al. 4 extended the self-heat recuperaton technology to an ntegrated gasfcaton combned cycle system. Rong et al. 5,6 proposed several specfc thermally coupled confguratons of multple component mxtures. Huang et al. 7,8 consdered nternal heat ntegraton n the desgn of reactve dstllaton columns. Incremental coupled heat and materal streams complcate optmzaton because of ther narrowed feasble regon and strct specfcatons. Search methods based on rgorous models were also nvestgated. Zhu et al. 9 appled a homotopy-based backtrackng method for smulatng and optmzng ASUs to ensure convergence. me consumpton s not a sgnfcant ssue n process desgn or synthess; however, calculaton tme s always consdered for real-tme optmzaton n operatons, such as durng a rapd and effectve search for a new operaton pont generated by a frequent fluctuaton n product demand. Other researchers proposed varous methods to smplfy the rgorous model for computaton. Srdeshpande et al. 10 represented the mass and energy balances for ar separaton by usng a smplfed algebrac model. Ban et al. 11 developed a reducedorder dynamc models for the upper column of an ar separaton plant. Kamath et al. 12 proposed a group method for complex dstllaton systems n large-scale flowsheets. Improvng computatonal effcency s mportant to analyze and optmze a complex system. he EO approach s preferred over the SM approach for handlng large-scale complex flowsheets wth nested recycle loops and mplct desgn specfcatons. In the latter, problems are solved teratvely by tearng the recycle streams n whch an optmzer s coupled wth a smulator. 13 In the former, all equatons and varables that descrbe the unt operatons are smultaneously solved. 14 he EO approach s superor to the SM approach f the Jacoban and Hessan matrces are drectly avalable. In addton, EO modelng realzes powerful decomposton methods for large-scale problems. 15,16 he dffculty n applyng the EO approach to ar separaton depends on the thermodynamcs calculaton because the equaton-of-state (EOS) method manly requres solvng a cubc equaton, 17,18 the roots of whch are computed under logcal f-else condtons, resultng n gradent dscontnuty. herefore, Kamath et al. 19 proposed a general EO approach to handle the cubc EOS by selectng the approprate root n accordance wth the desred phase. Recently, Dowlng et al. 20 have dscussed the presence of false equlbrum solutons n the supercrtcal regon followng the approach proposed by Kamath et al. 19 and then modfed the cubc EOS model. Dowlng et al. 21 optmzed cryogenc systems for coal oxycombuston power generaton. hese studes also presented a framework 22 for effcent large-scale flowsheet optmzaton, Receved: July 27, 2015 Revsed: November 6, 2015 Accepted: November 12, 2015 Publshed: November 12, Amercan Chemcal Socety 12096

2 Industral & Engneerng Chemstry Research Fgure 1. Flow dagram of the cryogenc ar separaton unt. whch combnes advanced process optmzaton formulatons wth state-of-the-art algorthms. In ths study, a complete EO model s developed for cryogenc ar separaton. Compared wth prevous work, the present study addresses three ssues of the ASU, namely, thermodynamc parameter estmaton, process analyss wth thermo-couplng desgn, and process optmzaton wth varyng load demands. An argon column s also ntroduced, whch adds complexty to process analyss and optmzaton. he proposed EO method demonstrates a superor convergng performance to the SM method. 2. EO FORMUAION FOR ASU FOWSHEE hs study focuses on the process analyss and optmzaton of typcal cryogenc ar separaton. he process flowsheet s shown n Fgure 1. he process conssts of three dstllaton columns, namely, a low-pressure column (PC), a hgh-pressure column (HPC), and an argon sde arm column (ASC). he followng four products wth mpurtes at the ppm level are generated: gaseous oxygen (GOX), lqud oxygen (OX), gaseous ntrogen (GAN), and gaseous argon (GAR). hree fractons of clean ar, that s, hgh-pressure ar (HPA), man ar (MA), and turbne ar (A), are sent nto the HPC after beng cooled and compressed to specfc states. he ar s separated nto a hgh-purty lqud ntrogen and an oxygen-rch lqud stream n ths column. hese streams are then throttled and fed nto the PC for further dstllaton and producton of hgh-purty GAN at the top and OX at the bottom. he majorty of the lqud oxygen s wthdrawn as a lqud oxygen product, and the remander s pumped and vaporzed nto GOX. he waste ntrogen (WN) s extracted as a reflux n the man heat exchanger to contrbute to a coolng capacty. he thrd column ASC s desgned to produce GAR at the top and return the oxygen-rch lqud to the PC at the bottom. As shown n Fgure 1, two heat ntegraton strateges are employed n the desgn to mnmze energy consumpton: (1) HPC and PC are desgned to share a common condenser/ reboler, that s, the condensng stream at the top of the HPC provdes heat to the lqud at the bottom of the PC; and (2) the bottom stream of the HPC, lqud ar 38-IAIR, s desgned to provde a condensng duty for the ASC after throttlng. Energy consumpton s reduced through the heat couplng desgn at the cost of complcatons encountered durng process analyss and optmzaton. o smplfy the calculaton, a coupled heat exchanger s usually smulated as two separate unts, and the heat dfference between the two unts s mnmzed toward zero as much as possble. herefore, two heat streams are used to denote the heat couplng relaton for the coupled heat exchangers. hese heat streams are denoted as DQ1 and DQ2 wth dotted lnes n Fgure 1. As wth other chemcal processes, the cryogenc ASU s characterzed by ts large scale, nonlnearty, and strong couplng. Smulatng and optmzng ASU present challenges n terms of effcency and convergence. In the current study, the EO approach s appled to formulate a rgorous ASU model. Several case studes are conducted to demonstrate the excellent convergence performance of the model. Each tray n a dstllaton column facltates separaton; the lqud stream from the upper tray and the vapor stream from the lower tray meet at the tray, exchange heat and mass, and then separate from each other wth enhanced purty. he ASU s formulated wth a rgorous tray-by-tray MESH model that conssts of the component mass balance (eq 1), equlbrum equaton (eq 2), summaton equaton (eq 3), and enthalpy balance (eq 4) of three components at the N stages. x + y + Fx = x + y j 1 j 1, j+ 1 j+ 1, j F j j, j j, (1) 12097

3 Industral & Engneerng Chemstry Research Fgure 2. Comparson of lqud and vapor compresson factor calculaton usng the SM and EO methods. y = K x j, j, j, (2) C C xj, j, = 1 = 1 = 1, y = 1 H + H = H + H + Q j 1, j 1 j+ 1, N+ 1 j, j j, j j (4) { N, Ar, O }, j {1,..., N} 2 2 where s the component ndex and j s the tray ndex. he equlbrum constants K j, and the enthalpy of streams n the lqud and vapor phases (H,j and H,j ) are obtaned usng the thermodynamc property method. In ths study, the Peng Robnson (PR) 17 cubc equatons of state (CEOS) are ntegrated nto the thermodynamc method for modelng. hs method s founded n the EOS of van der Waals and can effectvely predct thermodynamc propertes through ts relable mxng rules and alpha functons. he thermodynamc method s wdely used because of ts advantages, ncludng smple applcaton, convenence, and low parameter requrements. Detals of the PR model calculaton are presented n Appendx A. As n the PR model, the followng cubc equaton holds for the compresson factor: 3 2 z + az 1 + az 2 + a3 = 0 (5) Obtanng the correct cubc equaton roots s essental n the thermodynamc calculaton. Eq 5 yelds three roots, one of whch s always real, whereas the other two roots could be real and dstnct, real and nonunque, or magnary. Dependng on the number of real roots, each phase may ether be n the oneroot or three-root regon. hs phenomenon s dscussed n a recent study by Dowlng et al. 20 Eq 5 s formulated for both the lqud and vapor phases; thus, one phase may be n the threeroot regon, and the other may be n the one-root regon for any gven, P, and x (y). If three real roots are nvolved, the largest root represents the compresson factor of the vapor phase, whereas the smallest root represents the lqud phase. If only one real root exsts then such s the desred root for the gven phase. o ths end, the most extensvely used method s based on an analytc calculaton that nvolves f-else loops, as shown n Fgure 2. he upper two blocks llustrate the pseudocode for dervng the lqud and vapor compresson factors by usng the tradtonal SM method. Although a soluton (3) can easly be obtaned, the presence of such loops results n dscontnuous dervatves, whch may generate problems for smulaton and optmzaton n an EO framework. Consderng the dffcultes that accompany procedure-based analytc calculatons, Kamath et al. 19 and Dowlng et al. 20 proposed a method of selectng the approprate CEOS root based on the EO approach. he proposed method uses the mathematcal propertes of the cubc equaton n combnaton wth physcal nsghts regardng the nature of vapor and lqud roots. he frst- and second-dervatve constrants of the cubc equaton exhbt several regulartes that can be appled to solate and determne the desred roots. On the bass of the dervatve propertes of the vapor and lqud roots, a precse mathematcal defnton s developed; ths defnton can be adopted when solvng thermodynamc models n accordance wth the CEOS and s expressed as follows: f ( z ) = 0, f ( z ) 0, f ( z ) 0 (6) f ( z ) = 0, f ( z ) 0, f ( z ) 0 (7) where z and z are the CEOS roots n the lqud and vapor phases, respectvely. he effectveness of the dervatve-based constrants s verfed by several numercal cases based on flash vessels and dstllaton columns. In ths study, ar separaton usng the thermodynamc property method based on the PR CEOS s formulated by followng the EO framework. he basc thermodynamc calculaton model remans unchanged, as descrbed n Appendx A; however, the f-else judgment s replaced wth the dervatve constrants expressed n eqs 6 and 7. he lower two blocks n Fgure 2 llustrate the mplementaton of EO equatons for the ASU process n comparson wth the SM method. hermodynamc calculaton s performed on a ternary system composed of ntrogen, oxygen, and argon to valdate the accuracy of the EO-based PR method. he calculaton s based on a dstllaton column n ar separaton, and the results are compared wth those of Aspen Plus, n whch the equvalent thermodynamc model s embedded. he constants for physcal and thermodynamc propertes are retreved from the physcal database of Aspen Plus. Gven the temperature, pressure, and mole flow rate of streams n the vapor and lqud phases throughout the column under specfed condtons, the equlbrum constants and enthalpes can be obtaned wth 12098

4 Industral & Engneerng Chemstry Research EO formulatons for PR functons, as ntroduced prevously. he results are depcted n Fgures 3 and 4. he dstrbuton Fgure 3. Comparson of the enthalpy calculaton results. equatons and the thermodynamc model. hs study presents the advantage of EO modelng for thermodynamc parameter estmaton. Bnary nteractve parameters (BIPs) descrbe the degree of nteracton between two components n equlbrum. hese parameters are hghly sgnfcant for the accurate smulaton of a chemcal process. hree BIPs, k j (,j {N 2, AR,O 2 }, j), are ncluded n ths system. he orgnal values retreved from the Aspen Plus physcal database fal to smulate the hgh purty requrement of the argon column. A new set of BIPs provded by the ndustry s adopted n ths study. In ths secton, we demonstrate the advantages of the EO method over the tradtonal SM method n dealng wth the parameter estmaton usng process data. Aspen Plus wth the new BIP settng s used to generate the process data. A parameter estmaton task s then executed to evaluate whether or not the correspondng BIPs can be found wth the generated data. hs task can be converted to an optmzaton task. he objectve s to mnmze the concentraton dfference of the three components n n outlet streams between the generated dstllaton process data and the model-based predcton data. he three BIPs, k j, are the decson varables. he prevous MESH equatons and PR CEOS are used as constrants to represent the process model. he optmzaton problem s expressed as follows: n 3 x * pq, xpq, mn k x* pq, j p= 1 q= 1 2 (8) s. t. : MESH equatons for the flowsheet model (9) PR equatons for the thermodynamc model (10) Fgure 4. Comparson of the equlbrum constant calculaton results. results of the complete EO model overlap wth the results obtaned wth Aspen Plus, thus valdatng the accuracy of the PR thermodynamc method n an EO framework. On the bass of the EO formulatons for the ASU model, the followng three sectons are presented to demonstrate the advantage of the EO method through dfferent aspects of process analyss and optmzaton. he EO method s run wth AMP (32-bt verson) and Interor Pont OPmzer verson he tolerance s set to he SM approach wth Aspen Plus 7.2 s also used for the comparson study. he SQP algorthm s selected for the SM smulaton n Aspen Plus, wth the tolerance set to All computatons are conducted on a PC wth a 2.3 GHz Intel Core CPU, 6 GB memory, and Wndows 7 system. 3. HERMODYNAMIC PARAMEER ESIMAION In the ASU process analyss, thermodynamc propertes such as equlbrum constants and enthalpes must be determned. EO modelng facltates the smultaneous calculaton of the unt where x p,q * s the observed value of the mole fracton of component q n the outlet stream p from the dstllaton column and x p,q s the calculated value of the parameter estmaton optmzaton problem. he SM and EO methods are both tested for the parameter estmaton. Frst, Aspen Plus s used for the SM test. he test nvolves two-nested loops, wth the BIP search n the outer loop and the process smulaton n the nner loop. he orgnal BIPs from the Aspen Plus database are used as the ntal guess for the optmzaton. he teratve results of the BIPs are shown n Fgure 5 wth a 3D plot. he curve wth astersks represents Fgure 5. Comparson of BIP searchng paths of the EO method and Aspen Plus

5 Industral & Engneerng Chemstry Research Fgure 6. Comparson of the -x-y vapor lqud equlbrum dagrams of O 2 Ar. (a) BIPs set at the termnaton pont of Aspen Plus. (b) BIPs set at the converged pont obtaned by the EO method. the search path of BIPs calculated n Aspen Plus. he calculaton fals after fve teratons. he message shows that the solver n ths software can no longer handle the optmzaton problem. In addton, the test reveals that the falure does not depend on any algorthm settngs such as the tolerance. he EO method s subsequently tested. Aspen Plus also supports the EO mode, t cannot be appled to the BIP estmaton because the thermodynamc model s not open n Aspen Plus. he complete EO method developed n ths study s used. All EO varables are ntalzed wth the Aspen SM results obtaned at the BIPs provded by the Aspen Plus physcal database. he results of the search path denoted by the curve wth dots are also shown n Fgure 5. After 40 teratons and a computatonal tme of s, the process successfully converged to the optmal soluton, agreeng wth the values used for the data generaton. o explan why Aspen Plus faled n ths case, we further analyze the termnatng pont wth the -x-y apor qud Equlbrum (E) dagram, as llustrated n Fgure 6a. he dew pont and bubble pont curves meet at a pont, ndcatng the exstence of an azeotrope so that the mxture components can no longer be separated from one another at the current state. However, n realty, the azeotrope does not exst n ar separaton systems. hs clam can be valdated by observng the E dagram obtaned by settng the BIPs at the converged pont of the EO method, as shown n Fgure 6b. he E dagram result reveals that the parameter estmaton task executed n Aspen Plus traverses nto an nfeasble thermodynamc regon wheren the desred separaton s unachevable. Aspen Plus fals to contnue the teraton under the nested teratve calculaton wheren a thermodynamc model s embedded n the nner loop. he prevous results show that the SM method, unlke the EO method, s trapped nto an nfeasble thermodynamc regon. he response of the EO method f t traverses nto the nfeasble thermodynamc regon remans undetermned. o verfy that the EO method s not dependent on luck, further testng s conducted usng the EO method from the pont where Aspen Plus faled. We set the BIPs of the EO model to the faled termnaton pont and repeat the EO optmzaton. Fgure 7 shows the BIP search path generated usng the EO 12100

6 Industral & Engneerng Chemstry Research Fgure 7. BIP searchng path wth a swtch from Aspen Plus to the EO method. method wth the ntals nherted from the termnatng pont of Aspen Plus. hs tme, the EO method performs 29 teratons and uses s to successfully obtan the same optmal soluton. he dfferent modelng methods of Aspen Plus and the complete EO approach, as presented n Fgure 8, result n Fgure 8. Illustraton of the dfferent modelng structures. dfferent calculaton processes of the parameter estmaton. he crcle on the rght represents the EO-based optmzaton model, n whch the functons of the thermodynamc physcal method are ntegrated wth MESH functons. he AMP platform allows the access to the frst and second dervatves of all equatons, ncludng the CEOS, va automatc dfferentaton. hus, an effcent large-scale nonlnear programmng (NP) algorthm that contrbutes to an excellent calculatng performance can be appled n such an EO framework. On the contrary, the crcle on the left represents the nested teratve calculaton of Aspen Plus wth ts thermodynamc module embedded n the nner loop. he dervatves for the mplct models are not drectly avalable and are typcally determned based on a fnte dfference. Convergence s dffcult to acheve usng the Aspen Plus optmzer because of ts round-off errors and the conventonal procedure-based analytc calculaton process appled n solvng the CEOS. We repeat the test wth dfferent ntals to demonstrate that the trappng of Aspen Plus s not rare. As shown n able 1, cases 1 to 4 compare the SM and EO methods. he EO method succeeds wth the same optmal soluton n each test. By contrast, Aspen Plus fals to solve the same problem, as descrbed prevously. he correct soluton s obtaned every tme the search s swtched to the EO method at the faled ponts. hese results clearly demonstrate the advantage of the EO method over the tradtonal SM method n dealng wth BIP parameter estmaton. Another test s also conducted to systematcally explore the effect of dfferent ntal ponts on the BIP estmaton wth the EO method. he ntal guesses of the three BIPs are extended to a wde range as [ 1, 1]. he range of each BIP s evenly dvded nto 10 portons, and 11 equally spaced ponts are selected. hus, a total of = 1331 BIP ntals s obtaned. Each tme, we ntalze the BIPs as one combnaton; the other varables are set to the ntals of case 1. A total of 1331 estmatons are fnally tested wth the EO method. he 3D plot n Fgure 9 demonstrates the solvng status of the 1331 estmatons. he fve-ponted star marks the optmal BIP pont. he represents the successful ntal ponts, and the represents the ones that faled to converge. are manly located far from the optmal pont, ndcatng that the EO method performs well when the ntal pont s not far. In the prevous test, all data are generated va smulaton wthout any nose. Further testng s conducted by addng maxmal 5% whte nose to the measurement of case 1 to verfy the robustness of the EO method. he results are presented as case 5 n able 1. Comparson of cases 1 and 5 shows that the estmated parameters are slghtly changed, provng the robustness of the EO method. 4. PROCESS ANAYSIS WIH HERMA-COUPING DESIGN As shown n Fgure 1, the complex flowsheet s desgned for ar separaton. he process can be smulated gven the total ar feed nformaton and the manpulated varables. he manpulated varables nclude the flow rate of the oxygen n PC (F 35 OX ), the flow rate of the lqud ntrogen reflux n HPC (F 40 IN ), the flow rate of the AR sde draw n PC (F 47 ARC ), the flow rate of the WN sde draw n PC (F 52 WN ), the flow rate fracton of the turbne ar n the total ar feed [f A =F A /(F A +F HPA +F MA )], the flow rate fracton of the hgh-pressure ar [f HPA =F HPA / (F A + F HPA +F MA )] n the total ar feed, the vaporzaton fracton of the heat exchanger HEA PC (ψ HEA PC ), the fracton of lqud oxygen n the total oxygen product [f OX = F OX /(F OX+GOX )], and the reflux rato of ASC (R ASC ). All manpulated varables are lsted n able 2. Once the manpulated varables are fxed, all of the other process varables are theoretcally determned through process smulaton. he process analyss determnes the sutable settngs of the manpulated varables for the ASU process. he hgh-purty specfcatons of the products n ths process are also lsted n able 2. he manpulated varables should be adjusted to meet the specfcatons. Meanwhle, the heat-couplng feature of ths process should also be ncluded n the smulaton. hs system ncludes two man heat ntegraton desgns: (1) HPC and PC are desgned to share a common condenser/reboler, that s, the condensng stream at the top of the HPC provdes heat to the lqud at the bottom of the PC; and (2) the bottom stream of the HPC, lqud ar 38-IAIR, s desgned to provde condensng duty for the ASC after the throttle. he coupled heat exchangers should deally be smulated for the heatntegrated system. However, convergence s dffcult to attan 12101

7 Industral & Engneerng Chemstry Research able 1. BIP Estmaton Results k O2AR k O2N 2 k N2AR status case 1 ntals results EO model converged AspenPlus calculatons wth errors EO from faled pont converged case 2 ntals results EO model converged AspenPlus calculatons wth errors EO from faled pont converged case 3 ntals results EO model converged AspenPlus calculatons wth errors EO from faled pont converged case 4 ntals results EO model converged AspenPlus calculatons wth errors EO from faled pont converged case 5 Intals EO results converged Fgure 9. Solvng status of BIP estmaton problem wth the EO method from dfferent ntal guesses. wth such a desgn n tradtonal commercal software. As dealt wth n Aspen Plus, ths process hardly converges wth a complete thermal-couplng constrant whle meetng hghpurty product specfcatons. In practce, a coupled heat exchanger s replaced wth two separate heat exchangers, and ther heats are connected as two heat streams to observe the heat dfference. As shown n Fgure 1, the heat streams are denoted as dotted lnes to dstngush them from the physcal streams. he heat dutes of the four unts n the two pars of thermally coupled desgn are also lsted n able 2: Q condenserhpc for the HPC condenser, Q rebolerpc for the PC reboler, Q condenserasc for the ASC condenser, and Q HEA PC for the preheater of the PC, HEA PC. DQ1 and DQ2 represent the heat resdues between the two pars of thermally coupled unts, deally wth values equal to zero. he thermal-couplng desgn of the heat-ntegrated dstllaton columns n cryogenc ar separaton can maxmze heat/ coolng ntegraton and reduce overall energy consumpton. However, the desgn also complcates the convergence of process smulaton. he hgh purty specfcatons n ths process narrow the feasble regon and ncrease the dffculty of calculaton. Establshng an operatng condton that can meet all ndustral specfcatons s challengng. If the heat-couplng specfcatons are enforced, Aspen Plus fals to fnd a feasble soluton. he manpulated varables can be manually adjusted to reduce the heat dfference whle gradually meetng the hghpurty product specfcatons; however, ths manual approach depends on expert experence and s tme-consumng. he best result obtaned by the authors wth Aspen Plus s lsted n the thrd column of able 2. he two pars of thermally coupled unts are not fully coupled n terms of heat resduals. In addton, the purtes of some ar products have not met the specfcatons under the operatng condtons generated by 12102

8 Industral & Engneerng Chemstry Research able 2. Comparson of the Smulaton Results smulaton n Aspen Plus smulaton wth EO approach Aspen Plus verfcaton manpulated varables F 40 IN (kmol/h) F 47 ARC (kmol/h) F 52 WN (kmol/h) F 35 - OX (kmol/h) f A f HPA R ASC ψ Heat PC purty specfcaton y GOX, O2 ( 0.999) a y GOX, N2 ( 8 ppm) y GAN, O2 ( 8 ppm) y GAN, N2 ( ) a y ARC, N2 ( 800 ppm) coupled heat (MJ/h) Q condenserhpc Q rebolerpc Q condenserasc Q HEA PC DQ a DQ a a he purtes of some ar products. Fgure 10. Comparson of the column temperature profles of the EO method and Aspen Plus. Aspen Plus. he volated constrants are marked wth superscrpt a as shown n able 2. o avod the tedous work of tunng the manpulated varables n the process analyss of the ASU, the followng optmzaton problems can be formulated: mn : eu + ep + du + dp (11) s. t. ep eu = Q + Q condenserhpc rebolerpc (12) dp du = Q + Q condenserasc HEA PC (13) Δ 1 = 1.5 K tophpc bottompc (14) Δ 2 = 3 K topasc HEA PC (15) y GOX,O2 (16) y y 8ppm GOX,N2 (17) 8ppm GAN,O2 (18) y GAN,N2 (19) y 800 ppm ARC,N2 (20) eu, ep, du, dp 0 (21) 12103

9 Industral & Engneerng Chemstry Research Flowsheet rgorous MESH models are provded (eqs 1 4). he equatons of thermodynamcs PR models are n Appendx A, where eu, ep, du, and dp are non-negatve numbers that express the thermal-coupled unt desgn. he combnaton of the objectve functon (eq 11) and the frst two constrants (eqs 12 and 13) s well-desgned to generate the thermo-couplng feature of the process. he objectve functon can mnmze the values of the four nonnegatve varables to zero as much as possble; meanwhle, eqs 12 and 13 mnmze the heat resdues of the coupled heat pars. Heat s fully coupled once the value of the objectve functon, whch s expressed as eu + ep + du + dp, equals zero. At ths pont, the values of the heat resdues, whch are obtaned usng eqs 12 and 13, are also zero. Eqs 14 and 15 are attached as thermal-couplng constrants to ensure that heat s transferred from the hot stream to the cold stream. Eqs 16 to 20 are constrants that ensure hgh purty of the ar products. he process models, ncludng the flowsheet unt model based on the MESH equatons and the thermodynamc model based on PR CEOS, are also ncluded n the constrants. Wth the above optmzaton formulaton, the manpulated varables need not be adjusted manually. he optmal soluton provdes the best thermal-couplng desgn whle meetng the hgh purty specfcatons. However, ths formulaton s dffcult to mplement wthout the EO modelng formulaton. As n Aspen Plus, both SM and EO modes have dffcultes n dealng wth the constrants eqs Meanwhle, the complete EO approach offers advantages n descrbng and solvng ths optmzaton problem. On the bass of the EO modelng dea, the bult process model nvolves equatons and varables. o smulate a process, the eght manpulated varables shown n able 2 must be fxed. When the four auxlary varables (eu, ep, du, and dp) and the two equaltes (eqs 12 and 13) are ntroduced nto the EO problem, the EO model consders equatons and varables. Solver IPOP s used to calculate ths large-scale optmzaton problem. Wth the Aspen Plus results n able 2 as the ntal guess, the optmal soluton can be obtaned after 149 teratons and a computatonal tme of s. he results are also presented n able 2. he fourth column represents the process condtons generated by EO optmzaton. he two pars of unts are fully coupled, and the hgh purty requrements are met. he manpulated varables are altered to a certan extent n comparson wth the prevous condton. A comparson of the temperature profles of each column between the two condtons s llustrated n Fgure 10, whch shows the devatons between the two condtons. o confrm the accuracy of the optmal results obtaned usng the proposed EO method, another process smulaton s conducted n Aspen Plus by settng the manpulated varables at the optmal results and ntalzng the other varables wth the EO soluton. In ths process, Aspen Plus converges successfully wth the results, as lsted n the last column of able 2. he results n the last two columns are almost smlar, ndcatng that the EO-based rgorous model descrbes the ar separaton process as precsely as Aspen Plus. In the current state, the purty and coupled heat are clearly mproved compared wth the prevous Aspen results. he heat resdues between the two pars of thermally coupled unts are very close to zero, ndcatng the extremely tght thermal couplng feature. In addton, all purty requrements are strctly satsfed. he case study on determnng the desred operatng condton demonstrates the excellent convergence performance of the EO approach, partcularly for such a large-scale NP wth strongly coupled varables and strct constrants. he EO-based model can precsely analyze the ASU process through CEOSbased thermodynamc calculaton. In addton, the study hghlghts the convenence of swtchng from smulaton to optmzaton wth the EO model. 5. PROCESS OPIMIZAION FOR ARYING OAD DEMANDS Operatng condton fluctuaton commonly occurs n ndustral producton. oad change s a frequently encountered scenaro able 3. Computatonal Results of Optmzaton wth oad Change gas oxygen load (Nm 3 /h) , tme costs (s) teratons n whch the ndustry ams to match changng customer demands by adjustng manpulated varables. hs case s often observed n ASUs equpped to ron and steel plants because such processes are followed by batch operatons. An ASU should change ts operatng pont n lne wth varyng loads to avod oxygen release and thus save energy. he hgh-purty requrement should always be met durng load change. hus, the process should be optmzed for varyng load demands. Nonetheless, the operatonal optmzaton entals dffculty because of the large-scale, heat-couplng, and hgh-purty features of the process. hs study addresses ASU optmzaton under oxygen load change by usng the complete EO model. he results demonstrate the satsfactory convergence performance and fast computaton capablty of the EO approach. As oxygen load changes, the key varables of the process should be adjusted n a tmely manner to ensure that the output of the process matches the demand. he same manpulated varables lsted n able 2 are chosen for optmzaton. Sales revenues and producton cost are two man aspects consdered n the ndustry. In ths process, the energy consumpton of three nlet streams s the man source of cost, and lquefed oxygen s the only product sold; the ar products are for nternal use n the steel plant n accordance wth the load demand. hus, the objectve functon of ths optmzaton problem can be expressed as follows: objfun = F prce F cost {OutStr} j {FeedStr} (22) where {OutStr} represents lquefed oxygen; {FeedStr} represents a set of nlet streams, namely, A, MA, and HPA; F or F j denotes the flow rates of feed or product streams (kmol/h); prce refers to the unt prce of lquefed oxygen ($/kmol); and cost j s the unt energy consumpton of the three nlet streams ($/kmol). he default oxygen load of the ASU s desgned as N m 3 /h. hs load can vary from to N m 3 /h as demanded. he other two gaseous products, ntrogen and argon, are constraned above a certan output level. F F GOX GAN = oxygen load (Nm /h) (Nm /h) 3 3 j j (23) (24) 12104

10 Industral & Engneerng Chemstry Research able 4. Key arable Results of Optmzaton wth oad Change gas oxygen load (Nm 3 /h) key varables purty specfcaton y GOX, O2 ( 0.999) y GOX, N2 ( 8 ppm) 9.97e 11 1e e e e e 10 y GAN, O2 ( 8 ppm) 7.99e e e e e e 6 y GAN, N2 ( ) y ARC, N2 ( 800 ppm) 8e 4 8e 4 8e 4 8e 4 8e 4 8e 4 coupled heat DQ DQ product flow rates (Nm 3 /h) F GOX F OX F GAN F GAR flow rates of key streams (kmol/h) FEED F 40 IN F 47 ARC F 52 WN F 35 - OX F 46 ARR feed splt ratos f A f HPA reflux rato vapor rato R ASC ψ Heat PC F GAR 500 (Nm /h) 3 (25) he thermally coupled desgn s realzed by mnmzng the objectve functon expressed wth auxlary varables (eqs 12 13) n the process analyss secton. However, these varables are restraned by equalty constrants n ths optmzaton problem as follows: Q + Q = 0 condenserhpc rebolerpc (26) Q + Q = 0 condenserasc HEA PC (27) In addton to thermal-couplng and hgh purty constrants (eqs 14 21), other constrants of equpment and safe consderaton on the flow rates of nlet streams are also ncluded to ensure normal operaton. FHPA* * FHPA FHPA* FGOX FGOX FGOX * (28) 0.6F * F 1.5F * FEED FEED FEED (29) 0.6( FHPA*+ FA* ) FHPA + FA 1.5( FHPA*+ FA* ) (30) 0.6F * F 1.5F * HPA HPA HPA (31) 0.6F * F 1.5F * A A A (32) Eq 28 s the safety consderaton for the rato between the HPA and GOX. Eqs descrbe the upper and lower capacty lmts of the each stage of the multstage compressor. he rgorous flowsheet model (eqs 1 4) and the thermodynamcs model (Appendx A wth eqs 6 and 7) are also ncluded as constrants. he optmzaton problem nvolves equatons and varables. he dfference between ths model and the model n the prevous secton s that the flow rate of the gas oxygen s specfed each tme n accordance wth the load change demand; the total ar flow rate s also freed as a manpulated varable. In addton to the equatons n the process model, eqs 26 and 27 are ncorporated to defne the heat-couplng specfcaton. As ndcated n able 3, sx cases wth dfferent GOX load demands are examned. Solver IPOP s appled to solve ths large-scale NP. he ntals of ths model are also from the Aspen Plus SM results, same as that used n the prevous secton on process analyss. he convergence results are lsted n able 3. he optmzatons are successful n all cases. Moreover, the computatonal tmes and teratons of optmzaton dffer under vared oxygen loads. Overall, the computatonal costs of handlng such a large-scale, nonlnear, and strong-couplng optmzaton problem are less than 20 s, demonstratng the satsfactory convergence performance and hgh effcency of the EO method. As the oxygen load ncreases from to N m 3 /h, the manpulated varables are adjusted n a specfc range to meet the demand of output and operatonal specfcatons. he optmzed results of the key varables for the sx case studes are presented n able 4. All cases converge to optmal solutons whle satsfyng the purty specfcatons and the heat-couplng desgn. On the bass of these key varables, several suggestons are prepared to gude operatons: (1) ASC reflux rato ncreases along wth the vapor fracton of HEA PC to compensate for the addtonal energy demand caused by the ncrease n feed ar whle mantanng the tght couplng of heat; (2) the rato of lqud oxygen n the total oxygen outlets decreases as oxygen product load ncreases; and (3) the other flows of sde-draw streams ncrease n correspondence wth the rsng feed ar n the process. 6. CONCUSIONS In ths study, ASU process analyss and optmzaton are conducted usng the EO approach to acheve satsfactory convergence performance wthn an EO framework. he EO formulatons, whch nclude dervatve-based constrants for PR CEOS thermodynamcs, are mplemented wth several dstllaton columns. Parameters are then estmated to 12105

11 Industral & Engneerng Chemstry Research determne the accurate bnary nteracton factors that best ft ths process. he essental dfference n the convergences of EO and nested teratve calculaton approaches s revealed wth respect to handlng complex chemcal processes based on CEOS thermodynamcs. Subsequently, ar separaton s analyzed on the bass of the thermal-couplng concept wth an EO-based rgorous model. he effectveness of ths model s verfed. he process wth load change s optmzed to confrm the excellent convergence performance and calculaton speed of the EO approach. he study results demonstrate that the EO approach s advantageous n handlng ASUs wth large-scale, heat-couplng, and hgh-purty features. APPENDIX A hermodynamc Model wth PR Cubc Equaton of States Equaton of state R a P = v b vv ( + b) + bv ( b) Mxng rules b = xb a = xx( aa) (1 k ) Rc b = P j Rc a = α P 0.5 j j j c 2 2 a = ( aa) (1 k ) c 0.5 j j j a = xxa, a = yya j j j b = xb, b = yb Alpha functons 0.5 R α ( ) = 1 + m(1 ) j j j m = ω ω 2 Cubc equatons of state Z (1 B ) Z + ( A 3B 2 B ) Z ( AB B 2 B 3 ) = Z (1 B ) Z + ( A 3B 2 B ) Z ( AB B 2 B 3 ) = 0 A ap ap bp bp =, A =, B =, B = R R R R Fugacty constants functons b A ln( ϕ ) = ( Z 1) ln( Z B ) b 2 2B S b + Z B ln a b Z 0.414B b ln( ϕ ) = ( Z 1) ln( Z B ) b A S b B a b Z 2 ln 2 2 Z j S = a x, S = a y Equlbrum constant j ϕ K = ϕ Enthalpy functons o ( H + H) Z 2.414B = Z 1 ln R Z 0.414B xxa (1 + M + M) j j j R2 2B o ( H + H) Z 2.414B = Z 1 ln R Z 0.414B yya (1 + M + M ) j j j R2 2B N N o o = 1 = 1 o o m H = xh, H = yh, M = 2 o 0 H =Δ fh + CP d ref B 0.414B α C,3/ C,5/ CP = C,1 + C,2 + C,4 snh( C,3/ ) cosh( C,5/ ) AUHOR INFORMAION Correspondng Authors *E-mal: zhuly@zjut.edu.cn. *E-mal: xchen@pc.zju.edu.cn. Notes he authors declare no competng fnancal nterest. ACKNOWEDGMENS We gratefully acknowledge the fnancal support from Natonal Natural Scence Foundaton of Chna (Grant ) and 973 Program of Chna (Grant 2012CB720503). We would also lke to thank Professor.. Begler at Carnege Mellon Unversty for the dscusson and constructve suggestons on ths work. REFERENCES (1) Agrawal, R. Producton of ultrahgh-purty oxygen: A dstllaton method for the coproducton of the heavy key component stream free of heaver mpurtes. Ind. Eng. Chem. Res. 1995, 34 (11), R

12 Industral & Engneerng Chemstry Research (2) Fu, C.; Gundersen,. Usng exergy analyss to reduce power consumpton n ar separaton unts for oxy-combuston processes. Energy 2012, 44 (1), 60. (3) Kansha, Y.; Kshmoto, A.; Nakagawa,.; sutsum, A. A novel cryogenc ar separaton process based on self-heat recuperaton. Sep. Purf. echnol. 2011, 77 (3), 389. (4) Fu, Q.; Kansha, Y.; u, Y.; Song, C.; Ishzuka, M.; sutsum, A. An advanced cryogenc ar separaton process for ntegrated gasfcaton combned cycle (IGCC) systems. Chem. Eng. rans. 2014, 39, (5) Rong, B. G.; Kraslawsk, A.; Nystro m,. Desgn and synthess of multcomponent thermally coupled dstllaton flowsheets. Comput. Chem. Eng. 2001, 25 (4), 807. (6) Rong, B. G.; Kraslawsk, A.; urunen, I. Synthess of heatntegrated thermally coupled dstllaton systems for multcomponent separatons. Ind. Eng. Chem. Res. 2003, 42 (19), (7) Huang, K.; Iwakabe, K.; Nakawa, M.; sutsum, A. owards further nternal heat ntegraton n desgn of reactve dstllaton columns Part I: he desgn prncple. Chem. Eng. Sc. 2005, 60 (17), (8) Huang, K.; Nakawa, M.; sutsum, A. owards further nternal heat ntegraton n desgn of reactve dstllaton columns Part II. he process dynamcs and operaton. Chem. Eng. Sc. 2006, 61 (16), (9) Zhu,.; Chen, Z.; Chen, X.; Shao, Z.; Qan, J. Smulaton and optmzaton of cryogenc ar separaton unts usng a homotopy-based backtrackng method. Sep. Purf. echnol. 2009, 67 (3), 262. (10) Srdeshpande, A. R.; Ierapetrtou, M. G.; Andrecovch, M. J.; Naumovtz, J. P. Process synthess optmzaton and flexblty evaluaton of ar separaton cycles. AIChE J. 2005, 51 (4), (11) Ban, S.; Khownj, S.; Henson, M. A.; Belanger, P.; Megan,. Compartmental modelng of hgh purty ar separaton columns. Comput. Chem. Eng. 2005, 29 (10), (12) Kamath, R. S.; Grossmann, I. E.; Begler,.. Aggregate models based on mproved group methods for smulaton and optmzaton of dstllaton systems. Comput. Chem. Eng. 2010, 34 (8), (13) Begler,..; Grossmann, I. E.; Westerberg, A. W. Systematc Methods of Chemcal Process Desgn; Prentce Hall: NJ, (14) Westerberg, A. W.; Berna,. J. Decomposton of very largescale Newton-Raphson based flowsheetng problems. Comput. Chem. Eng. 1978, 2 (1), 61. (15) Zhu, Y.; egg, S.; ard, C. D. Optmal desgn of cryogenc ar separaton columns under uncertanty. Comput. Chem. Eng. 2010, 34 (9), (16) Zhu, Y.; egg, S.; ard, C. D. Optmal operaton of cryogenc ar separaton systems wth demand uncertanty and contractual oblgatons. Chem. Eng. Sc. 2011, 66 (5), 953. (17) Peng, D. Y.; Robnson, D. B. A new two-constant equaton of state. Ind. Eng. Chem. Fundam. 1976, 15 (1), 59. (18) Soave, G. Equlbrum constants from a modfed Redlch- Kwong equaton of state. Chem. Eng. Sc. 1972, 27 (6), (19) Kamath, R. S.; Begler,..; Grossmann, I. E. An equatonorented approach for handlng thermodynamcs based on cubc equaton of state n process optmzaton. Comput. Chem. Eng. 2010, 34 (12), (20) Dowlng, A. W.; Balwan, C.; Gao, Q.; Begler,.. Optmzaton of sub-ambent separaton Systems wth Embedded Cubc Equaton of State hermodynamc Models and Complementarty Constrants. Comput. Chem. Eng. 2015, 81, 323. (21) Dowlng, A. W.; Balwan, C.; Gao, Q.; Begler,.. Equatonorented optmzaton of cryogenc systems for coal oxycombuston power generaton. Energy Proceda 2014, 63, 421. (22) Dowlng, A. W.; Begler,.. A framework for effcent large scale equaton-orented flowsheet optmzaton. Comput. Chem. Eng. 2015, 72,