Description
This model designs a heat exchanger network which operates at minimal annual cost and satisfies heating and cooling require- ments. The superstructure consists of two stages with eight possible exchangers.
Small Model of Type : MINLP
Category : GAMS Model library
Main file : synheat.gms
$title Simultaneous Optimization for HEN Synthesis (SYNHEAT,SEQ=117)
$onText
This model designs a heat exchanger network which operates at
minimal annual cost and satisfies heating and cooling require-
ments. The superstructure consists of two stages with eight
possible exchangers.
Yee, T F, and Grossmann, I E, Simultaneous Optimization of Models for
Heat Integration - Heat Exchanger Network Synthesis. Computers and
Chemical Engineering 14, 10 (1990), 1151-1184.
Keywords: mixed integer nonlinear programming, chemical engineering, heat exchanger
network
$offText
Set
i 'hot streams' / 1*2 /
j 'cold streams' / 1*2 /;
Scalar nok 'number of stages in superstructure' / 2 /;
Set
k 'temperature locations nok + 1' / 1*3 /
st(k) 'stages';
Singleton Set
firstK(k) 'first temperature location'
lastK(k) 'last temperature location';
st(k) = yes$(ord(k) < card(k));
firstK(k) = yes$(ord(k) = 1);
lastK(k) = yes$(ord(k) = card(k));
Parameter
fh(i) 'heat capacity flowrate of hot stream'
fc(j) 'heat capacity flowrate of cold stream'
thin(i) 'supply temp. of hot stream'
thout(i) 'target temp. of hot stream'
tcin(j) 'supply temp. of cold stream'
tcout(j) 'target temp. of cold stream'
ech(i) 'heat content hot i'
ecc(j) 'heat content cold j'
hh(i) 'stream-individual film coefficient hot i'
hc(j) 'stream-individual film coefficient cold j'
hucost 'cost of heating utility'
cucost 'cost of cooling utility'
unitc 'fixed charge for exchanger'
acoeff 'area cost coefficient for exchangers'
hucoeff 'area cost coefficient for heaters'
cucoeff 'area cost coefficient for coolers'
aexp 'cost exponent for exchangers'
hhu 'stream-individual film coefficient hot utility'
hcu 'stream-individual film coefficient cold utility'
thuin 'inlet temperature hot utility'
thuout 'outlet temperature hot utility'
tcuin 'inlet temperature cold utility'
tcuout 'outlet temperature cold utility'
gamma(i,j) 'upper bound of driving force'
a(i,j,k) 'area for exchanger for match ij in interval k (chen approx.)'
al(i,j,k) 'area calculated with log mean'
acu(i) 'area coolers'
ahu(j) 'area heaters'
tmapp 'minimum approach temperature'
costheat 'cost of heating'
costcool 'cost of cooling'
invcost 'investment cost';
Binary Variable
z(i,j,k)
zcu(i)
zhu(j);
Positive Variable
th(i,k) 'temperature of hot stream i as it enters stage k'
tc(j,k) 'temperature of cold stream j as it leaves stage k'
q(i,j,k) 'energy exchanged between i and j in stage k'
qc(i) 'energy exchanged between i and the cold utility'
qh(j) 'energy exchanged between j and the hot utility'
dt(i,j,k) 'approach between i and j at location k'
dtcu(i) 'approach between i and the cold utility'
dthu(j) 'approach between j and the hot utility';
Variable cost 'hen and utility cost';
Equation
eh(i,k) 'energy exchanged by hot stream i in stage k'
eqc(i) 'energy exchanged by hot stream i with the cold utility'
teh(i) 'total energy exchanged by hot stream i'
ec(j,k) 'energy exchanged by cold stream j in stage k'
eqh(j) 'energy exchanged by cold stream j with the hot utility'
tec(j) 'total energy exchanged by cold stream j'
month(i,k) 'monotonicity of th'
montc(j,k) 'monotonicity of tc'
monthl(i) 'monotonicity of th k = last'
montcf(j) 'monotonicity of tc for k = 1'
tinh(i) 'supply temperature of hot streams'
tinc(j) 'supply temperature of cold streams'
logq(i,j,k) 'logical constraints on q'
logqh(j) 'logical constraints on qh(j)'
logqc(i) 'logical constraints on qc(i)'
logdth(i,j,k) 'logical constraints on dt at the hot end'
logdtc(i,j,k) 'logical constraints on dt at the cold end'
logdtcu(i) 'logical constraints on dtcu'
logdthu(j) 'logical constraints on dthu'
obj 'objective function';
teh(i).. (thin(i) - thout(i))*fh(i) =e= sum((j,st), q(i,j,st)) + qc(i);
tec(j).. (tcout(j) - tcin(j))*fc(j) =e= sum((i,st), q(i,j,st)) + qh(j);
eh(i,k)$st(k).. fh(i)*(th(i,k) - th(i,k+1)) =e= sum(j, q(i,j,k));
ec(j,k)$st(k).. fc(j)*(tc(j,k) - tc(j,k+1)) =e= sum(i,q(i,j,k));
eqc(i).. fh(i)*(th(i,lastK) - thout(i)) =e= qc(i);
eqh(j).. fc(j)*(tcout(j) - tc(j,firstK)) =e= qh(j);
tinh(i).. thin(i) =e= th(i,firstK);
tinc(j).. tcin(j) =e= tc(j,lastK);
month(i,k)$st(k).. th(i,k) =g= th(i,k+1);
montc(j,k)$st(k).. tc(j,k) =g= tc(j,k+1);
monthl(i).. th(i,lastK) =g= thout(i);
montcf(j).. tcout(j) =g= tc(j,firstK);
logq(i,j,k)$st(k).. q(i,j,k) - min(ech(i), ecc(j))*z(i,j,k) =l= 0;
logqc(i).. qc(i) - ech(i)*zcu(i) =l= 0;
logqh(j).. qh(j) - ecc(j)*zhu(j) =l= 0;
logdth(i,j,k)$st(k)..
dt(i,j,k) =l= th(i,k) - tc(j,k) + gamma(i,j)*(1 - z(i,j,k));
logdtc(i,j,k)$st(k)..
dt(i,j,k+1) =l= th(i,k+1) - tc(j,k+1) + gamma(i,j)*(1 - z(i,j,k));
logdthu(j).. dthu(j) =l= (thuout - tc(j,firstK));
logdtcu(i).. dtcu(i) =l= th(i,lastK) - tcuout;
obj..cost =e= unitc*(sum((i,j,st),z(i,j,st))
+ sum(i,zcu(i)) + sum(j,zhu(j)))
+ acoeff*sum((i,j,k),(q(i,j,k)*((1/hh(i)) + (1/hc(j)))
/ (((dt(i,j,k)*dt(i,j,k+1)*(dt(i,j,k) + dt(i,j,k+1))/2
+ 1e-6)**0.33333) + 1e-6) + 1e-6)**aexp)
+ hucoeff*(sum(j,(qh(j)*((1/hc(j)) + 1/hhu))
/ (((thuin - tcout(j))*dthu(j)*((thuin - tcout(j) + dthu(j))/2)
+ 1e-6)**0.33333) + 1e-6)**aexp)
+ cucoeff*sum(i,(qc(i)*((1/hh(i)) + (1/hcu))
/ (((thout(i)-tcuin)*dtcu(i)*(thout(i) - tcuin + dtcu(i))/2
+ 1e-6)**0.33333) + 1e-6)**aexp)
+ sum(j,qh(j)*hucost) + sum(i,qc(i)*cucost);
* Process streams
* hot
thin('1') = 650; thout('1') = 370; fh('1') = 10; hh('1') = 1;
thin('2') = 590; thout('2') = 370; fh('2') = 20; hh('2') = 1;
* cold
tcin('1') = 410; tcout('1') = 650; fc('1') = 15; hc('1') = 1;
tcin('2') = 350; tcout('2') = 500; fc('2') = 13; hc('2') = 1;
* costs and coefficients
hucost = 80; hucoeff = 150; thuin = 680; thuout = 680; hhu = 5;
cucost = 15; cucoeff = 150; tcuin = 300; tcuout = 320; hcu = 1;
unitc = 5500; acoeff = 150;
aexp = 1; tmapp = 10;
* bounds
dt.lo(i,j,k) = tmapp;
dthu.lo(j) = tmapp;
dtcu.lo(i) = tmapp;
th.up(i,k) = thin(i);
th.lo(i,k) = thout(i);
tc.up(j,k) = tcout(j);
tc.lo(j,k) = tcin(j);
* initialization
th.l(i,k) = thin(i);
tc.l(j,k) = tcin(j);
dthu.l(j) = thuout - tcin(j);
dtcu.l(i) = thin(i) - tcuout;
ech(i) = fh(i)*(thin(i) - thout(i));
ecc(j) = fc(j)*(tcout(j) - tcin(j));
gamma(i,j) = max(0,tcin(j) - thin(i), tcin(j) - thout(i),tcout(j) - thin(i), tcout(j) - thout(i));
dt.l(i,j,k) = thin(i) - tcin(j);
q.l(i,j,k)$st(k) = min(ech(i),ecc(j));
Model super/ all /;
option optCr = 0, limRow = 0, limCol = 0, solPrint = off, sysOut = off;
solve super using minlp minimizing cost;
* Areas by chen approximation
a(i,j,k)$st(k) = q.l(i,j,k)*((1/hh(i)) + (1/hc(j)))
/ (2/3*sqrt(dt.l(i,j,k)*dt.l(i,j,k+1))
+ 1/6*(1e-8 + dt.l(i,j,k) + dt.l(i,j,k+1)));
* Areas by log mean temperature
al(i,j,k)$st(k) = (q.l(i,j,k)*((1/hh(i)) + (1/hc(j))))
/ (dt.l(i,j,k)*dt.l(i,j,k+1)
* (dt.l(i,j,k) + dt.l(i,j,k+1))/2)**0.33333;
display a, al;
* Areas of heaters and coolers
ahu(j) = (qh.l(j)*((1/hc(j)) + (1/hhu))/(((thuin-tcout(j))*dthu.l(j)
* ((thuin-tcout(j)+dthu.l(j)))/2) + 1e-6)**0.33333);
acu(i) = (qc.l(i)*((1/hh(i)) + (1/hcu))/(((thout(i) - tcuin)*dtcu.l(i)
* (thout(i) - tcuin + dtcu.l(i))/2 + 1e-6)**0.33333));
display acu, ahu;
* Utility costs
costheat = sum(j,qh.l(j)*hucost);
costcool = sum(i,qc.l(i)*cucost);
display costheat, costcool;
* Investment cost
invcost = cost.l - costheat - costcool;
display invcost;