Managing Financial Risk in the Planning of Heat Exchanger Cleaning

Transcription

1 Managing Financial Risk in he Planning of Hea Exchanger Cleaning Javier H. Lavaja and Miguel J. Bagajewicz * Deparmen of Chemical Engineering and Maerials Science Universy of Oklahoma, E. Boyd S., T-335, Norman OK 739. USA Absrac In his paper we exend a recenly presened mixed ineger linear model for he planning of hea exchanger cleaning in chemical plans o an uncerain model. We compare he sochasic soluions o he deerminisic and heurisic soluions and discuss financial risk managemen opions. Keywords: Hea Exchanger Nework Cleaning, Operaions Planning, Risk Managemen.. Inroducion Fouling migaion and managemen is an imporan problem in indusry (he oal cos of fouling in highly indusrialized naions has been projeced a.25% of he GNP; he oal annual cos of fouling in he U.S. is esimaed a $8 billion). For his reason, deermining which exchanger o clean and when, during operaions, is of paramoun imporance. On one hand, cleaning resuls in less energy coss over he ime horizon afer is cleaned, bu also implies ha he exchanger needs o be pu off-line during cleaning and herefore in his period of ime he energy cos acually increases. Thus, while cleaning is advanageous, doing oo ofen may no be economically advisable afer all. Several models have been developed o deermine he opimal schedule of cleaning in hea exchanger neworks. Of all hese, he model by Smaïli e al. (22) is he one ha inroduces he leas simplifying assumpions, as does no resor for example o any hypohesis of cyclic cleaning and models he exchanger hrough deailed and rigorous equaions. We have recenly developed an MILP formulaion ha is even more rigorous (Lavaja and Bagajewicz, 23) which could in principle render global opimaly, conrasing wh previous models (hey are non-linear). The paper also makes a leraure review, which we om here. For cases where compuaion ime is exensive (cases where several subopimal soluions exiss), all models (linear or no) have difficulies in idenifying he global opimum. For hese cases, in he aforemenioned paper we presen an efficien decomposion procedure ha renders beer soluions han oher formulaions, ouperforms heurisic approaches and proves ha soluions based on moving horizons (a simplificaion arising from he need o speed up compuaion) are * Auhor o whom correspondence should be addressed : bagajewicz@ou.edu

2 ofen no opimal. In his paper, we make he model sochasic and we discuss opions o manage financial risk. 2. MILP Deerminisic Model Consider he hea exchanger nework (HEN) of a crude disillaion un (Figure, reproduced from Smaïli e al, 22) where hea is recovered from a disillaion column producs and pump-around sreams. We consider ha ime is discreized in inerval periods (ypically monhs); and each one his is subdivided ino a cleaning sub-period and an operaion one. Thus, he objecive is o deermine which exchanger is o be cleaned in which period given oher resricions and resource availabily so ha he ne presen value is maximized. The soluion should also ake ino accoun he possibily of changing any nework flow rae and/or fluid for any operaion period. Throughpu losses due o pressure drops are beyond he scope of his projec and hus hey are no considered. The clean and acual hea ransfer coefficien in period relaed o he fouling facor (r ) by: c ( U i and U respecively) are r U c Ui = () Smaïli e al (22) use a linear and exponenially asympoic fouling model. We concenrae here on he exponenially asympoic fouling model, given by: r ( e ) K = r (2) waer Desaler F l a s h Furnace Figure. HEN from Smaïli e al (22). We define a binary variable ha idenifies when and which each exchanger is cleaned as follows: if he h hea exchanger is cleaned in period Y = (3) oherwise

3 The clean and acual hea ransfer coefficien for each sub-period can be wren in erms he binary variable and he fouling facor as follows (Lavaja and Bagajewicz, 23): = ecp c U aik k = j= k + ( ) c c Yij + b Y + c ( Yip ) i, = eop o U aik k = j= k + ( ) o o Yij + b Y + c ( Yip ) i, (4) (5) where a, and are consans ha are a funcion of he differen parameers. ik b c c These equaions are subsued in he equaions corresponding o he hea exchanger hea balance o render an expression for he ho oule emperaure (Th Th 2 =, R e ( R ) + ( Y ) b Y + c ( Y ) e R di aik k = j= k + ij + R R Tc Th e Tc ip di aik ij k = j= k + + di aik ij k = j= k ). ( Y ) b Y + c ( Y ) ( Y ) b Y + c ( Y ) ip ip i (6) where = Ai d ( R ) Fc Cc. The expression can be easily linearized hrough sandard ricks (Lavaja and Bagajewicz, 23). The model minimizes he expeced ne presen value (hroughou ime horizon) of he operaing coss arising from he rade-off beween furnace exra fuel coss due o fouling, and hea exchanger cleaning coss (which include man power, chemicals and mainenance). NPC = cl ( Ef Ef ) d CEf + d η f Y Ccl i where NPC is he ne presen cos, consumpion, Ef cl Ef (7) is he acual furnace s energy is he furnace s energy consumpion for clean condion, CEf is he furnace s fuel cos, C cl is he cleaning cos, η f is he furnace efficiency, and d is he discoun facor.

4 3. New Decomposion Procedure Our decomposion procedure is he following (Lavaja and Bagajewicz, 23): ) Solve he firs exchanger schedule assuming all he res are no cleaned. 2) Solve he nex exchanger schedule assuming he res have he same cleaning schedule as he curren soluion. 3) Check for convergence once all exchangers have been solved. If convergence is achieved ha proceed o he nex sep. If no sar a new eraion. 4) Pick he larges number of periods for which a moving window soluion procedure would solve in a reasonable amoun of ime. Sar wh he firs monh and solve he problem whin ha horizon. Leave he scheduled cleaning ouside he horizon as hey were esablished in he las run. 5) Keep running he moving horizon unil he end of he ime horizon is reached. 6) Check for convergence. If no convergence is achieved, run he moving horizon again. 4. Sochasic Model To build a sochasic model we have only considered uncerainies in he energy prices. Oher parameers ha are uncerain bu have been kep deerminisic are he fouling coefficiens of he model, cleaning coss, plan urnaround horizons, and he processing of differen crudes a differen imes. To build he model we considered he sandard wo-sage sochasic programming model. The scenarios are consruced sampling energy prices, which are assumed o follow a cyclical average rend of seasonal variaions, based on U.S. Deparmen of Energy daa. Ten base curves were consruced around he average rend, wh increasing deviaion from for similar monhs of he year as he ime horizon increases (Figure 2). Then sampling around hese base curves was performed assuming normal disribuions. The model was solved for inial condions were fouling has already aken place in some exchangers. To obain he sochasic soluion, we used he echnique inroduced by Aseeri and Bagajewicz (23). This echnique consiss of solving each scenario independenly. Then each soluion is solved again wh he firs sage variables fixed for all he scenarios o obain all he risk curves for he res of he scenarios. We generaed 2 soluions based on he scenarios generaed using en rends. Figure 3 shows all he soluions obained. I also depics he soluion obained by he deerminisic model using he mean values of he rend. Finally, we simulaed he heurisic approach of cleaning every hea exchanger ha reaches 5%, 2%, and 25% fouling. On one hand, he resuls show ha he flucuaions around every rend do no affec he resuls significanly, ha is, all he curves obained by sampling around he same rend are very close. For example, for rend he difference in he expeced NPC beween he bes and he wors soluion is only.6%. On he oher hand, here are wo clear groups of soluions, depending on which rend is associaed o he soluion. Figure 3 shows how he heurisic soluions are far away from hose prediced by modelling. Noiceably, even in he case when he same number of cleanings is used,

5 modelling suggess beer soluions. For example, compare he deerminisic soluion obained using he overall mean rend wh he heurisic for 8%Uc: boh use 28 cleanings, bu heir coss differ significanly (see Tables and 2). Trend Trend 3 Trend 5 Trend 7 Trend 9 Mean 9 Trend 2 Trend 4 Trend 6 Trend 8 Trend CEf (US$/MMBu) (monh) Figure 2. Energy Prices rends. 5. Risk Managemen Figure 3 shows ha one group of curves and he deerminisic curve are close. This is because he rends and he overall mean are relaively close. However, for he rends ha depar significanly from he mean (especially hose ha do no cross he mean oo many imes), he resuls group in oher se of curves. Thus, if risk is of concern, hen one should choose he soluion corresponding o a curve exhibing lower cos a low cumulaive probabilies, insead of he opimal, which in his case is close o he deerminisic soluion (rends and 6 o ). Trend Trend 2.9 Trend 3 Cumulaive Probabily.8 Trend 4 Trend 5.7 Trend 6.6 Trend 7 Trend 8.5 Trend 9.4 Trend Mean (28 cleanings, ENPC=923,29) Heurisic 75% (5 cleanings, ENPC=,98,7) Heurisic 8% (28cleanings, ENPC=,2,9) Heurisic 85% (5 cleanings, ENPC=,76,77) Figure 3. Risk Curves NPC ( US$)

6 Table. Cleaning schedule for heurisic 8% U c (ENPC = US$,2,9) HEx # cleans / HEx Toal # of cleanings 28 Table 2. Cleaning schedule for deerminisic case (ENPC = US$ 923,29) HEx # cleans / HEx Toal # of cleanings Conclusions Sochasic programming was used o deermine he risk associaed o he problem of scheduling he cleaning of hea exchanger neworks when here is uncerainy in he energy price. The resuls show ha flucuaions around he rend do no aler significanly he resuls. In addion, in cerain cases, risk managemen is possible. I was also illusraed ha heurisic approaches resul in much higher coss even wh he same oal number of cleanings. Acknowledgemens We are graeful o he Universy of Oklahoma Supercompuing Cener for Educaion and Research for allowing us use heir service. References Lavaja J.H. and Bagajewicz M.J., 23, A new MILP Model for he Planning of Hea Exchanger Nework Cleaning. Submed o Indusrial and Engineering Chemisry Research. Special issue honoring Arhur Weserberg. Available a hp:// (unpublished papers). Aseeri A. and Bagajewicz M.J., 23. New Measures and Procedures o Manage Financial Risk wh Applicaions o he Planning of Gas Commercializaion in Asia. Submed o Compuers and Chemical Engineering. Available a hp:// (unpublished papers). Smaïli F., Vassiliadis V.S. and Wilson D.I., 22. Opimizaion of Cleaning Schedules in Hea Exchanger Neworks Subjec o Fouling. Chem. Engng. Comm., 89,