Transshipment LP Model for Minimizing the Utility Cost in a Heat Exchanger Network

Transcription

1 Trasshipet LP Model for Miiizig the Utility ost i a eat Exchager Networ Niolaos V. ahiidis ad Igacio E. Grossa Departet of heical Egieerig aregie Mello Uiversity 1. Proble tateet Give is a process that ivolves the followig set of hot ad cold streas: trea F cp (/) T i () T out () The followig utilities are available for satisfyig heatig ad coolig requireets: Utility Teperature() ost($/-yr) Maxiu available() Fuel P stea LP stea oolig water The goal is to predict the iiu utility cost for a heat exchager etwor that has a iiu teperature approach of 10. trea 1 is ot to be allowed to exchage ay heat with strea Forulatio e are ased to fid a way of exchagig heat aog the process streas ad the utilities so that the target teperatures are et for the process streas ad the total utility cost is iiized. The data of the proble are displayed i Table 1, where heat cotets of the hot ad cold processig streas are show at each of the teperature itervals which are based o the ilet, ad highest ad lowest teperatures give. Iterval T hot T cold eat otets Table 1: eat cotets of process streas 1

2 The flows of the heat cotets ca be represeted i the heat cascade diagra of Fig. 1. ere the heat cotets of the hot streas are itroduced i the correspodig itervals, while the heat cotets of the cold streas are extracted also fro their correspodig itervals. The variables 1 to 5 represet heat residuals, while Fuel, P, LP, represet the heatig ad coolig loads respectively. The usefuless of the heat cascade diagra i Fig. 1 is that it ca be odeled as a trasshipet proble which we ca forulate as a liear prograig proble. I ters of the trasshipet odel hot streas are treated as source odes, ad cold streas as destiatio odes. eat ca the be regarded as a coodity that ust be trasferred fro the sources to the destiatios through soe iterediate warehouses which correspod to the teperature itervals which guaratee feasible heat exchage. he ot all of the heat ca be allocated to the destiatios (cold streas) at a give teperature iterval, the excess is cascaded dow to lower teperature itervals through the heat residuals. The trasshipet odel for predictig the iiu utility cost give a arbitrary uber of hot ad cold utilities, ad hot ad cold streas, ca be forulated as follows. First, we cosider that we have teperature itervals that are based o the ilet teperatures of the process streas ad of the iterediate utilities whose ilet teperatures fall withi the rage of the teperatures of the process streas. e assue that the itervals are ubered fro the top to the botto. e ca the defie the followig idex sets: = {i hot strea i supplies heat to iterval } (1) = {j cold strea j deads heat fro iterval } (2) = { hot utility supplies heat to iterval } (3) = { cold utility extracts heat fro iterval } (4) = set of hot utilities (5) = set of cold utilities (6) he we cosider a give teperature iterval, we will have the followig ow paraeters ad variables (see Fig. 2): ow paraeters:, heat cotets of hot streas i ad cold strea j i iterval i j c, c uit costs of hot utility ad cold utility Variables: s, heat loads of hot utility ad cold utility heat residual exitig iterval 2

3 Fig. 1: Trasshipet odel for heat flows i i j j Fig. 2: eat flows i iterval. 3

4 The iiu utility cost for a give set of hot ad cold processig streas ca the be forulated as the followig LP (Papoulias ad Grossa, 1983): i s. t , 0 0 1,2,3... c c (7) (9) (10) 0 (11) i i j j 1,2,3... (8) The above odel is ow i the literature as the codesed trasshipet odel. Its objective fuctio represets the total utility cost, while the equatios are heat balaces aroud each teperature iterval. If =0 at the optial solutio, the the teperature iterval correspods to a pich poit. I the above forulatio it is easy to ipose upper liits o the heat loads that are available fro soe of the utilities (eg. Maxiu heat fro high pressure stea). O the other had, it is ot possible to use it as it is i order to eforce costraits that exclude atches betwee ay give pair of hot ad cold streas. This could be due to the fact that the streas are too far apart, or because of other operatioal cosideratios such as cotrol, safety or start-up. For this reaso, Papoulias ad Grossa (1983) have forulated a expaded trasshipet odel which we preset ext. The codesed LP trasshipet odel i (7) to (11) iplicitly assues that ay give pair of hot ad cold streas ca exchage heat sice there is o iforatio as to which pairs of streas actually exchage heat. I order to develop a LP forulatio where we do have that iforatio, we ca cosider withi each teperature iterval, ali for the heat exchage betwee a give pair of hot ad cold streas, where the cold strea is preset at that iterval, ad the hot strea is either also preset, or else it is preset i a higher teperature iterval. For this reaso, withi a iterval we will defie the variable ij to deote the heat exchage betwee hot strea i ad a cold strea j. Liewise, we ca defie siilar variables for the exchage betwee process streas ad utilities. hereas i the codesed LP trasshipet odel we assiged a sigle overall heat residual exitig at each teperature iterval, i the expaded trasshipet odel we will assig idividual heat residuals i, l for each hot strea i ad each hot utility that are preset at or above the teperature iterval. Fig. 3 illustrates that above ideas for a iterval where we cosider a hot strea i ad a cold strea j. 4

5 i, 1 i ij j i, Fig. 3: eat exchage betwee hot ad cold strea for expaded trasshipet odel. Note that usig this represetatio, we allow to a cold strea j to exchage heat i iterval ot oly with a hot strea which is available i that iterval, but also with a hot strea i which is available at a higher iterval siply because hot strea i is trasferrig heat to iterval through its residual i. Let us ow defie soe otatio: = {i hot strea i supplies heat to iterval or at a higher iterval} (12) = { hot utility supplies heat to iterval or at a higher iterval} (13) The idex sets, are defied as i (2) ad (4), respectively. e also have the followig ew variables (see Fig. 4): ij exchage of heat of hot strea i ad cold strea j at iterval (14) j exchage of heat of hot utility ad cold strea j at iterval (15) i exchage of heat of hot strea i ad cold utility at iterval (16) i heat residual of hot strea i exitig iterval (17) heat residual of hot utility exitig iterval (18) i j Fig. 4: eat flows i iterval for expaded trasshipet odel. 5

6 The variables,, ad the paraeters,,, are idetical to the oes of the i codesed trasshipet odel. I cotrast to the copact LP trasshipet odel (7) to (11) where we siply did ad overall heat balace aroud each teperature level, i this case we have to cosider balaces at the followig poits withi each teperature iterval (poits A, B, ad D of Fig. 4): a)for the hot process ad utility streas at the iterval odes that relate the heat cotet, residuals ad heat exchages, b) for the cold process ad utility streas at the destiatio odes tha relate the heat cotet ad heat exchages. I this way the expaded LP trasshipet odel is the followig: i s. t. i i i i i0 ij i, c i, 1, 1, i ij, 0 j j j j c ij 1,2,3..., j i j j, (19) i, 1,2,3... 1,2,3... i 0 (25) (24) (23) j 1,2,3... j i (22) (21) (20) The size of this LP is obviously larger tha the oe i (7) to (11). The iportace of the expaded forulatio is however the fact that we ca easily specify costraits o the atches: a) I order to forbid a atch betwee hot i ad cold j we eed to set ij =0 for all itervals. Alteratively, we siply delete these variables fro the forulatio. b) I order to ipose a atch betwee hot strea i ad cold strea j, we specify that the total heat exchage, which is the su of ij over all itervals, lies withi soe specified lower ad upper bouds: L ij 1 ij U ij (26) e ca also siply specify a fixed value for the su i (26). The expaded trasshipet odel (19) to (26) ca be exteded to a MILP trasshipet odel which, for a give utility cosuptio, ca rigorously predict the uber of fewest uits i the etwor, as well as the strea atches that are ivolved i each uit, ad the aout of heat that they ust exchage. This is siply doe by itroducig 0-1 variables for each potetial atch as discussed i Papoulias ad Grossa (1983). Fially, the desig of heat exchager etwors with several factors (utility cost, uber of uits, heat exchager area) beig cosidered siultaeously has also bee addressed i the literature. A MINLP odel addressig this proble is preseted i aother study (Yee ad rossa, 1991) 6

7 3. esult The iput file for GAM cosist of three parts. The data are etered i the first sectio of the progra. Notice that the teperature itervals eed ot be etered sice they are calculated i the secod part of the progra alog with the heat cotets of the streas. ere we ae use of the LOOP stateet of GAM i order to repeatedly execute a set of coads. Fially, the third part of the progra defies ad solves the odel. The optial solutio show i the GAM output file idicates that by disallowig the atch betwee strea 1 ad strea 1 the iiu utility cost of 570,000 $ per year ca be achieved with the followig loads for the utilities: fuel high pressure strea cold utility Also, fro the solutio we ca see that there is o pich poit for this proble sice there are ozero residuals for all the teperature itervals. 4. uggested exercises (1) It is iterestig that by iposig costraits o atches the iiu utility cost ay or ay ot chage. osider the followig (u)restricted cases for the give proble: (a) No restrictios for heat atches. (b) trea 1 is ot allowed to exchage heat with strea 2. (c) trea 1 is ot allowed to exchage heat with either strea 1 or strea 2. (2) how that the expaded LP trasshipet odel i (19)-(25) reduces to the LP odel (7)-(11) if o costraits o atches are iposed. 5. efereces Papoulias,.A., ad I.E. Grossa, A tructural Optiizatio Approach i Process ythesis Part II: eat ecovery Networs, oputers ad heical Egieerig, 7(6), (1983). Yee, T.F., ad I.E. Grossa, iultaeous Optiizatio Model for eat Exchager Networ ythesis, this Volue (1991). 7