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minlphix.gms : Heat Integrated Distillation Sequences

This is a direct MINLP formulation of the model MINLPHI.
References:
Small Model of Type: MINLP
$Title Heat Integrated Distillation Sequences - MINLP (MINLPHIX,SEQ=227) $Ontext This is a direct MINLP formulation of the model MINLPHI. Morari, M, and Grossmann, I E, Eds, Chemical Engineering Optimization Models with GAMS. Computer Aids for Chemical Engineering Corporation, 1991. Floudas, C A, and Paules IV, G E, A Mixed-Integer Nonlinear Programming Formulation for the Synthesis of Heat Integrated Distillation Sequence. Computers and Chemical Engineering 12, 6 (1988), 531-546. This formulation provides the Optimal Heat Integrated Distillation Sequence with Pressure as a continuous variable for a three component separation. Components: a == Hexane b == Benzene c == Heptane total feed to superstructure == 396 kgmol/hr multicomponent feed composition: a = 0.80 b = 0.10 c = 0.10 A Superstructure of the form ... _______ _______ _|_ | _|_ | / \ ( ) / \ ( ) | |___|__ A | |___|___ B | | | | |---------| 1 | | 3 | | | | ----------| | | | | | | | | | |_______| | | | \___/ | BC \___/_______ C F | | ( ) | | -------->| |____| |----( ) (ABC) | | _______ _______ | _|_ | _|_ | | / \ ( ) / \ ( ) | | |___| AB | |___|___ A | | | |_____________| | |---------| 2 | | 4 | | | | | | | | | | |______ C | |_______ B \___/ | \___/ | | ( ) | ( ) |____| |_____| is used with binary variables representing : a_ the existence of columns in the sequence. b_ the selection of heat exchangers for heat integration. c_ the selection of hot and cold utilities. Associated Reference: "A Mixed-Integer Nonlinear Programming formulation for the synthesis of Heat-Integrated Distillation Sequences" C.A. Floudas and G.E. Paules IV, 1988. Computers and Chemical Engineering vol. 12 no. 6 pp. 531-546 $Offtext Sets i condensers-columns / c-1*c-4 / j reboilers / r-1*r-4 / hu hot utilities / lp, ex / cu cold utilities / cw / n index / a,b / m intermediates / ab,bc/ pm(i,m) products / c-1.bc, c-2.ab/ fm(i,m) feeds / c-3.bc, c-4.ab/ Alias (ip,i); Alias (jp,j); *===================================================================== * Definition of "z" sets for conditional control of model * used to map permissible matches between condensers and reboilers * and the position of columns in the superstructure *===================================================================== sets zlead(i) leading columns in superstructure /c-1, c-2 / zcrhx(i,j) condenser to reboiler allowable matches / c-1.r-3, c-2.r-4, c-3.r-1,c-4.r-2 / zlim(i,j) direction of heat integration zcr(i,j) reboiler-condenser pairs; zlim(i,j) = zcrhx(i,j) and (ord(i) lt ord(j)); zcr(i,j) = ord(i) eq ord(j); Parameter spltfrc(i,m) split fraction of distillation columns /c-1.bc 0.20 c-2.ab 0.90/; Parameter tcmin(i) minimum condenser temperatures /c-1 341.92 c-2 343.01 c-3 353.54 c-4 341.92/; Parameter trmax(j) maximum reboiler temperatures; trmax(j) = 1000; *==================================================================== * scaled cost coefficients for distillation column fits * nonlinear fixed-charge cost model * cost = fc*y + vc*flow*temp * scaling factor = 1000 *==================================================================== Parameter fc(i) fixed charge for distillation columns /c-1 151.125 c-2 180.003 c-3 4.2286 c-4 213.42/; Parameter vc(i) variable charge for distillation columns /c-1 0.003375 c-2 0.000893 c-3 0.004458 c-4 0.003176/; Parameter thu(hu) hot utility temperatures /lp 421.0 ex 373.0/; * hot utility cost coeff - gives cost in thousands of dollars per year * ucost = q(10e+6 kj/hr)*costhu(hu) Parameter costhu(hu) hot utility cost coefficients /lp 24.908 ex 9.139/; Table kf(i,n) coeff. for heat duty temperature fits a b c-1 32.4 0.0225 c-2 25.0 0.0130 c-3 3.76 0.0043 c-4 35.1 0.0156 Table af(i,n) coeff. for column temperature fits a b c-1 9.541 1.028 c-2 12.24 1.050 c-3 8.756 1.029 c-4 9.181 1.005 Scalars totflow total flow to superstructure / 396 / fchx fixed charge for heat exchangers scaled / 3.392 / vchx variable charge for heat exchangers scaled / 0.0893/ htc overall heat transfer coefficient / 0.0028/ dtmin minimum temperature approach / 10.0 / tcin inlet temperature of cold water / 305.0 / tcout outlet temperature of cold water / 325.0 / costcw cooling water cost coefficient / 4.65 / beta income tax correction factor / 0.52 / alpha one over payout time factor in years / 0.40 / u large number for logical constraints / 1500 / uint upper bound for integer logical / 20 /; Variables zoau objective function value Positive Variables f(i) flowrates to columns qr(j) reboiler duties for column with reboiler j qc(i) condenser duties for column i qcr(i,j) heat integration heat transfer qhu(hu,j) hot utility heat transfer qcu(i,cu) cold utility heat transfer tc(i) condenser temperature for column with cond. i tr(j) reboiler temperature for column with reb. j lmtd(i) lmtd for cooling water exchanges sl1(i) artificial slack variable for lmtd equalities sl2(i) artificial slack variable for lmtd equalities s1(i) artificial slack variable for reb-con equalities s2(i) artificial slack variable for reb-con equalities s3(i) artificial slack variable for duty equalities s4(i) artificial slack variable for duty equalities; Binary Variables yhx(i,j) heat integration matches condenser i reboiler j yhu(hu,j) hot utility matches hot utility hu reboiler j ycu(i,cu) cold utility matches condenser i cold util cu ycol(i) columns in superstructure; Equations nlpobj nlp subproblems objective tctrlo(i,j) prevent division by 0 in the objective lmtdlo(i) prevent division by 0 in the objective lmtdsn(i) nonlinear form of lmtd definition tempset(i) sets temperatures of inactive columns to 0 (milp) artrex1(i) relaxes artificial slack variables (nlp) artrex2(i) relaxes artificial slack variables (nlp) material(m) material balances for each intermediate product feed feed to superstructure matlog(i) material balance logical constraints duty(i) heat duty definition of condenser i rebcon(i,j) equates condenser and reboiler duties conheat(i) condenser heat balances rebheat(j) reboiler heat balances dtminlp(j) minimum temp approach for low pressure steam dtminc(i) minimum temp allowable for each condenser trtcdef(i,j) relates reboiler and condenser temps of columns dtmincr(i,j) minimum temp approach for heat integration dtminex(j) minimum temp approach for exhaust steam hxclog(i,j) logical constraint for heat balances hxhulog(hu,j) logical constraint for heat balances hxculog(i,cu) logical constraint for heat balances qcqrlog(i) logical constraint for con-reb duties * these are the pure binary constraints of the minlp sequen(m) restricts superstructure to a single sequence lead sequence control limutil(j) limits columns to have a single hot utility hidirect(i,j) requires a single direction of heat integration heat(i) logical integer constraint; nlpobj.. zoau =e= alpha*( sum(i, fc(i)*ycol(i) + vc(i)*(tc(i)-tcmin(i))*f(i)) + sum(zcrhx(i,j), fchx*yhx(i,j) + (vchx/htc)*(qcr(i,j)/(tc(i)-tr(j)+1-ycol(i)))) + sum((i,cu), fchx*ycu(i,cu) + (vchx/htc)*(qcu(i,cu)/(lmtd(i)+1-ycol(i)))) + sum((hu,j), fchx*yhu(hu,j) + (vchx/htc)*(qhu(hu,j)/(thu(hu)-tr(j))))) + beta *( sum((i,cu), costcw*qcu(i,cu)) + sum((hu,j), costhu(hu)*qhu(hu,j))); * limit the denominator in the second line of the objective away from zero tctrlo(zcrhx(i,j)) .. tc(i)-tr(j)+1-ycol(i) =G= 1; * lmtd and ycol from being 0 and 1 at the same time to prevent divding * by 0 in the objective lmtdlo(i).. lmtd(i) =G= 2 * ycol(i); lmtdsn(i).. lmtd(i) =e= (2/3)*sqrt((tc(i)-tcin)*(tc(i)-tcout)) + (1/6)*((tc(i)-tcin)+(tc(i)-tcout)) + sl1(i) - sl2(i); artrex1(i).. s1(i) + s2(i) + sl1(i) =l= u*(1-ycol(i)); artrex2(i).. s3(i) + s4(i) + sl2(i) =l= u*(1-ycol(i)); material(m).. sum(pm(i,m), spltfrc(i,m)*f(i)) =e= sum(fm(i,m), f(i)); feed.. sum(zlead(i), f(i)) =e= totflow; duty(i).. qc(i) =e= (kf(i,"a") + kf(i,"b")*(tc(i)-tcmin(i))) + s3(i) - s4(i); rebcon(zcr(i,j)).. qr(j) =e= qc(i); conheat(i).. qc(i) =e= sum(zcrhx(i,j), qcr(i,j)) + sum(cu,qcu(i,cu)); rebheat(j).. qr(j) =e= sum(zcrhx(i,j), qcr(i,j)) + sum(hu,qhu(hu,j)); trtcdef(zcr(i,j)).. tr(j) =e= (af(i,"a") + af(i,"b")*(tc(i)-tcmin(i))) + s1(i) - s2(i); dtminlp(j).. dtmin - (thu("lp") - tr(j)) =l= 0; dtminex(j).. dtmin - (thu("ex") - tr(j)) - u*(1-yhu("ex",j)) =l= 0; tempset(i).. tc(i) + lmtd(i) + sum(zcr(i,j),tr(j)) =l= u*ycol(i); matlog(i).. f(i) =l= u*ycol(i); dtminc(i).. tcmin(i) - tc(i) =l= u*(1-ycol(i)); dtmincr(zcrhx(i,j)).. tr(j) - tc(i) - u*(1-yhx(i,j)) + dtmin =l= 0; hxclog(zcrhx(i,j)).. qcr(i,j) =l= u*yhx(i,j); hxhulog(hu,j).. qhu(hu,j) =l= u*yhu(hu,j); hxculog(i,cu).. qcu(i,cu) =l= u*ycu(i,cu); qcqrlog(i).. qc(i) + sum(j$zcr(i,j),qr(j)) =l= u*ycol(i); sequen(m).. sum(pm(i,m), ycol(i)) =e= sum(fm(i,m), ycol(i)); lead.. sum(zlead(i), ycol(i)) =e= 1; limutil(j).. sum(hu, yhu(hu,j)) =l= 1; * only one of the mutual heat integration binaries can be 1 hidirect(zlim(i,j)).. yhx(i,j) + sum((ip,jp)$(ord(ip) eq ord(j) and ord(jp) eq ord(i)),yhx(ip,jp)) =l= 1; * if a column doesn't exist then all binary variables associated * with it must also be set to zero heat(i).. sum(zcrhx(i,j), yhx(i,j) + sum((ip,jp)$((ord(ip) eq ord(j)) and (ord(jp) eq ord(i))), yhx(ip,jp))) + sum((hu,zcr(i,j)),yhu(hu,j)) + sum(cu, ycu(i,cu)) =l= uint*ycol(i); tc.lo("c-1") = tcout+1; tc.up("c-2") = tcin-1; tc.lo("c-3") = tcout+1; tc.up("c-4") = tcin-1; tr.up(j) = trmax(j); model skip / all /; option domlim=100; Solve skip using minlp minimizing zoau;