Chemical Engineering and Processing 46 (2007) 198–209
Enhancing sugar cane process performance through optimalproduction scheduling
H. Heluane a, M. Colombo a, M.R. Hernandez a, M. Graells b, L. Puigjaner b,∗a Departamento de Ingenierıa Quımica, Universidad Nacional de Tucuman, Av. Independencia 1800, 4000 Tucuman, Argentinab Chemical Engineering Department, Universitat Politecnica de Catalunya, ETSEIB, Av. Diagonal 647, 08028 Barcelona, Spain
Received 4 January 2006; received in revised form 15 May 2006; accepted 15 May 2006Available online 9 June 2006
bstract
Process design and operation is concerned with the optimal selection and efficient utilization of resources along time. The operational efficiency ofquipment units depends strongly on the maintenance policy employed. This work addresses critical operational issues in the sugar cane industry suchs the problem of determining the optimal cyclic cleaning policy in the evaporation section and the corresponding optimum steam consumption pro-le of both evaporation and crystallization sections. A main feature of this problem is the performance decay with time of each evaporation unit which
ust be restored by appropriate cleaning operations. in this paper, a detailed mixed integer nonlinear programming (MINLP) performance modelhich includes the effect of fouling on the overall heat-transfer coefficient is considered. The problem formulation can also handle multiple-unit par-
llel evaporation lines. Problem solution provides for each production line the optimal cleaning schedule, mass flow to be processed, and vapor bleeds.2006 Elsevier B.V. All rights reserved.
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eywords: Cyclic production and cleaning scheduling; Evaporation; Continuou
. Introduction
In the sugar cane industry, a substantial cost reduction can bechieved through efficient inventory management, reduction ofhe excess manufacturing capacity and rational use of resources.mportant cost reductions can also be achieved by coordinatinghe use of the manufacturing resources and process operations.he key role of effective tools for scheduling and planningctivities within the process industries has gained increasingecognition in recent years especially because improved cus-omer service, reduced inventory, lower manufacturing costs,nd global operations are achieved [1].
Heating processes are commonly employed in chemicallants in order to achieve product specific properties. In the spe-ific case of sugar cane production the juice is concentrated invaporator units by heating until sucrose crystals are obtained.
n this process, solid deposits are formed on the heating sur-aces (fouling) with a consequent increase of the heat transferesistance with time and a dramatic decrease in the overall per-
ormance of the evaporator. Therefore, additional costs are addedike the increase of operating costs due to a frequent cleaning ofhe equipment to restore its original performance.
Fouling affects nearly every plant relying on heat exchangersor its operation. The common practice to mitigate fouling iso implement cleaning-in-place (CIP) operations. This is espe-ially applicable to processes affected by rapid fouling, such ashat occurring in the production of milk, sugar cane juice, lemonuice, etc. [2]. During evaporator operation, as a consequence ofhe formation of solid deposits on heating surfaces, evapora-ion rates decrease with time whenever the driving force is keptonstant. Consequently, the evaporator must be shutdown to beleaned and the cycle must be restarted.
Several works have proposed methods for the optimization ofleaning schedules for a single heat exchanger [3–5]. However,n process plants, multiple interconnected heat transfer units aresed and the operating conditions of each equipment affect theverall heat exchange performance. Therefore, a rational main-enance policy must be applied to the heat exchangers network in
rder to accomplish the desired production at a minimum cost.
Scheduling of process operations has been addressed by manyuthors for different scenarios with special emphasis in batchrocess applications. But less attention has been paid to the
cheduling of continuous processes [6–8]. Jain and Grossmann9] studied the scheduling of multiple feeds on parallel unitsnd developed a mixed integer nonlinear programming modelMINLP). Georgiadis and Papageorgiou [2] considered a cyclicleaning scheduling on heat exchanger networks and proposedmixed integer linear programming model (MILP). Alle et al.
10] addressed the cyclic scheduling of cleaning and produc-ion operations in continuous plants. In all cases performanceecay with time was considered, but the additional complex-ty of multiple-unit parallel evaporation lines in the sugar canendustry has not been contemplated yet.
Otherwise, a wide range of chemical engineering prob-ems can be framed as mixed integer nonlinear programmingMINLP) like process synthesis problems (e.g., heat recoveryetworks, separation systems, reactor networks) and processperations problems (e.g., scheduling and design of batch pro-esses) [9–12].
The objective of this work is to address the scheduling ofroduction and cleaning operations in a sugar plant with perfor-ance decay. A detailed mixed integer nonlinear programming
MINLP) model including the effect of fouling on the overalleat-transfer coefficient is presented. Multiple-unit parallel linesre modeled for the evaporation section. The cyclic nature ofhe cleaning operations is also taken into account. The objectiveunction to be minimized considers the costs of the evapora-ion and the crystallization sections and other facilities (i.e. heatxchangers) that require vapor (or eventually steam) to oper-te. The problem solution provides the following information:he cleaning (maintenance) frequency, the mass flow to be pro-essed by each line, vapor bleed as energy source for externaleat requirements and the starting time (scheduling) for eachleaning (maintenance) task in each line.
.1. Problem statement
The evaporation and crystallization sections of the typicalugar cane plant considered in this work are shown schematicallyn Fig. 1.
As seen in Fig. 1, the evaporation system, the crystalliza-ion stage and other operations (i.e. heat exchangers) are steamonsumers. Heat exchangers are used for pre-heating the juiceefore being fed to the first unit of the evaporation line. The
wion
Fig. 1. Use of steam and extracted
and Processing 46 (2007) 198–209 199
o called “other operations” can be operated with either vaporenerated at the evaporation and/or steam depending on plantvailability.
This paper seeks enhanced process integration in sugar plantsy considering the simultaneous roles of the evaporation andrystallization sections as material processors as well as energyuppliers.
In particular, the objective of this work is to determine theptimal production schedule that minimizes the plant cost asso-iated to cleaning and steam consumed by the evaporation andrystallization sections, and by other steam-consuming opera-ions (“other operations”).
The problem can be formally stated as follows:
given:(i) the amount of material to be processed during a certain
time period(ii) the equipment models, parameters and initial status
(iii) the individual equipment performance as a time function(iv) product (sugar) concentration(v) other steam related requirements (other operations’ heat
requirements)determine:(i) the cleaning (maintenance) frequency
(ii) the mass flow to be processed by each line(iii) starting time for each cleaning (maintenance) task(iv) flows of vapor extracted from the evaporation (“bleeds”).
.2. Cost considerations
As explained by Heluane et al. [13], the aim of evaporationnd crystallization processes at a sugar factory is to eliminateater from the juice and, thus, to obtain crystals of sucrose.he evaporation process is economically more effective than
he crystallization process due to the multiple-effect schememployed (several evaporators working in series). In multiple-ffect evaporation with I units, the water extracted from the juices approximately I times the steam used in the process. Other-
ise, at the crystallization stage the water is extracted roughly
n a proportion 1:1 with the consumed steam. Therefore, thebjective function (to be minimized) has to take into accountot only the additional cost due to the evaporator fouling, but
vapor for the system studied.
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itmdie
U
idm1
lf(
x
IE
λ
FVo
xj = xj−1 + αθjAj�θj
λjFj−1(1 + bjt)1/2 = xj−1 + βj
Fj−1(1 + bjt)1/2
(6)
00 H. Heluane et al. / Chemical Engine
lso the crystallization cost, a necessary although economicallyess efficient operation. So it is important to consider the neces-ary trade-off between evaporation and crystallization sectionsn the overall cost study.
The evaporation process leads to the formation of foulingn the inner surface of the evaporator tubes. The rate ofouling formation is dependent on the nature of the feed, ands particularly significant for the case of liquid feeds. Foulingeposits inside the tubes act as insulation thus causing highereat-transfer resistance. It is convenient to clean the equipmenteriodically in order to restore conditions of high heat-transferate. If a high concentration of the product is desired, thenhe evaporators have to be cleaned frequently which wouldncrease costs. Thus, there is also a compromise between juiceoncentration and cleaning costs.
Special consideration must be given to the vapor producedy an evaporation unit, which is mainly used for two purposes
(a) vapor source for the following evaporator unit, andb) vapor source for heating purposes other than the evaporation
units. This vapor is named as bleed in the sugar industry.
If the bleed is not enough to meet heating targets then moreteam must be generated at the boiler with the consequentncrease of operating costs. Usually, vapor produced by theast units of evaporation lines and the vapor produced at therystallization stage are not used as energy source but they areondensed in the so called “barometric condenser” to maintainppropriate vacuum conditions in the system.
The objective is to minimize the steam cost at the evaporationnd crystallization sections as well as in other operations, i.e.eat exchangers, and the evaporator cleaning costs. Eq. (1) wille used as objective function.
.2. Model assumptions
The following assumptions have been considered to formu-ate the model presented in this work:
i. Negligible sensitive heat and boiling point increase at theevaporator units.
ii. No sugar loss during juice processing.ii. Constant fouling factor during evaporation.iv. Fixed operating conditions for the equipment units.
.3. Fouling model for evaporation units
In order to calculate the global heat-transfer coefficient, anmpirical expression that depends on juice temperature (θ) and
and Processing 46 (2007) 198–209
uice concentration (x) is used [14]. This expression is frequentlysed in sugar industry calculations within the typical range ofemperature and juice concentration and is known as the Swedishormula
= αθ
x(2)
here α is a proportionality constant.During evaporator operation, and as a consequence of foul-
ng, global heat transfer coefficient gradually decreases withime. In order to handle this situation, time dependence of U
ust be taken into account in Eq. (2). The model that betterescribes the decreasing behavior of U is given by the follow-ng expression, which was determined fitting several models toxperimental data from a local sugar plant
= αθ
x(1 + bt)1/2 (3)
The temperature (θ) of the boiling juice inside the evaporators a parameter of the problem and is usually maintained constanturing equipment operation. In Eq. (3) juice concentration (x)ust be expressed as Brix (Bx) defined as grams of solids per
00 g of water.As sugar mass remains constant at every evaporator (no sugar
oss), the outlet juice concentration for unit j can be obtainedrom the mass balance under a pseudo steady state conditionsee Fig. 2)
j = xj−1 + xj−1Vj
Fj
(4)
f sensible-heat is neglected from the evaporator energy balance,q. (5) is obtained
jVj = UjAj�θj (5)
or a given operating time t, by substituting U from Eq. (3) andj from Eq. (5) into Eq. (4) the following expression for theutlet juice concentration from unit j is derived
Fig. 2. Scheme of two evaporation units working in series.
H. Heluane et al. / Chemical Engineering and Processing 46 (2007) 198–209 201
on lin
F
β
Bbc
x
wt
2
ti
rt
S
Bfit
S
wot
2
r
S
Bol
S
wM
o
x
Swt
x
Nc(
Fig. 3. Scheme of multiple-effect evaporati
or sake of simplicity, a new variable βj is introduced, as follows
j = αθjAj�θj
λj
(7)
y operating with sugar mass balance for unit j and xj giveny Eq. (6), the following expression is derived for the outletoncentration of the juice leaving unit M at operating time t:
M = x0
M∏j=1
(1 + βj
F0x0(1 + bjt)1/2
)(8)
here F0 and x0 are the mass flow and concentration, respec-ively of the juice fed to the evaporation system.
.4. Steam for the evaporation section
As seen in Fig. 3, for each evaporation line i steam is fed onlyo the first unit (j = 1) while for the j following units the energys provided by the vapor produced at the previous one, j − 1.
Under the hypothesis mentioned above, the total steam, SE,equired for the evaporation system with N lines for an operatingime ti is given by Eq. (9)
E =N∑
i=1
sei =N∑
i=1
Vi1ti (9)
y substituting the corresponding conservation balances for therst evaporation unit of a line i in Eq. (9), the steam required by
he whole evaporation stage can be expressed as
E =N∑
Fi0
(1 − xi0
xi1
)ti (10)
i=1
here Fi0 is the flow with a concentration xi0 fed to the first unitf line i and xi1 is the average concentration of the flow leavinghe unit.
2
p
e i and the following crystallization stage.
.5. Steam for the crystallization section
Under the hypothesis mentioned above, the total steamequired for the crystallization section is given by Eq. (11)
C =N∑
i=1
sci =N∑
i=1
VCiti (11)
y substituting the mass balances for crystallization and evap-ration sections in Eq. (11), the steam required for the crystal-ization stage for N lines can be expressed as
C =N∑
i=1
Fi0xi0
(xT − xθi
xTxθi
)ti (12)
here xθi is the average concentration of the juice leaving unitfor a line i.Eq. (8) can be adapted to express the concentration of the
utlet flow of an evaporation line i with Mi units as follows:
θi = xi0
Mi∏j=1
(1 + βij
Fi0xi0(1 + bijti)1/2
)(13)
ugar concentration of the juice leaving the evaporator decaysith time due to the fouling of the heat-exchange surface. Hence,
he average concentration of the concentrated juice is given by
θi =∫ t2t1
xθi dt
t2 − t1(14)
ote that for calculating the average concentration of the con-entrated juice when the evaporation line starts operating cleanmaximum heat exchange capacity) t1 = 0.
.6. Steam requirements for other operations
Many heating operations are met by making use of the vaporroduced by the evaporators. As different vapors have differ-
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w
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02 H. Heluane et al. / Chemical Engine
nt temperature levels, those vapors are selectively used (i.e.apors from first units are used exclusively as heating supplyor requirement E1). All vapors except those from the last unitay be used for heating operations. Last unit vapors are sent tobarometric condenser to assure vacuum conditions in the unitsf the line (see Fig. 3). When vapor is not enough as heatingupply, steam is used.
Let us assume that Ej is the energy demand during the operat-ng time of line i, ti, by operations classified as “other operations”hat can be supplied with vapor from thermal level j. This is totalapor produced by the units in the jth position in each line. Thus
j =N∑
i=1
λijVBijti + λssrj (15)
here VBij represents the flow of vapor (bleed) extracted fromhe evaporator j on line i, λs is the heat of vaporization of steamnd srj is the amount of steam used when vapor from units j isot enough to supply energy demand. Due to temperature levels,hen vapor is used, only that from unit j can be used to supplyj requirements.
Hence, steam requirements can be obtained from Eq. (15)
rj = Ej
λs−
N∑i=1
λij
λsVBijti, j = 1, 2, . . . , M − 1 (16)
hen
N
i=1
λijVBijti ≥ Ej (17)
rj must be set equal to zero because bleed is enough to supplyther heating requirements.
Therefore, for a certain operation time ti, the total steamonsumption (sr) for “other operations” in a system with N evap-ration lines will be expressed as
r
⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩
0 ifN∑
i=1
λijVBijti ≥
M−1∑j=1
srj =M−1∑j=1
(Ej
λs−
N∑i=1
λij
λsVBijti
)otherwise
.7. Cleaning costs
SR
⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩
0
M−1∑j=1
SRj =M−1∑j=1
(EH
j
λs− η
N∑i=1
λij
λsV
Evaporators are cleaned by line, thus, all evaporators belong-ng to a certain line are stopped at the same time and cleaningperations are performed. For a certain time period the clean-ng costs (Cc) for N evaporation lines can be calculated as
Ψ
j
and Processing 46 (2007) 198–209
j = 1, 2, . . . , M − 1
(18)
ollows:
c = cc
N∑i=1
ni (19)
here cc is the cost of cleaning one evaporation line; ni theumber the of cleanings of a line i during a certain period.
.8. Cycle
The cyclic nature of the scheduling may be taken into accounty the mathematical model. The model allows determining theperation schedule for one cycle of TC hours [9]. This cycle cane repeated until the desired production level is achieved. If Hs the time horizon, then the number of evaporation cycles cane calculated by the following equation:
= H
TC(20)
he steam consumed during a time horizon H will be
team = η
(N∑
i=1
Vi1 +N∑
i=1
VCi
)ti + SR (21)
here
if η
N∑i=1
λijVBijti ≥ EHj , j = 1, 2, . . . , M − 1
i
)otherwise
(22)
Hj is the energy demand (obtained from vapor from units j
nd/or steam) for the time horizon H.Therefore the objective function can be expressed as follows:
in FO = csu
(η
(N∑
i=1
Vi1 +N∑
i=1
VCi
)ti
+M−1∑j=1
Ψj(TC, VBij, ti)
⎞⎠+ ccη
N∑i=1
Ni (23)
ith
j(TC, VBij, ti) = max
(0,
EHj
λs− η
N∑i=1
λij
λsVBijti
),
= 1, 2, . . . , M − 1 (24)
ering
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L
j
0
0
w
η
u
wt
Ψ
Ψ
2
cl
N
∑
Ibte
2
ni
x
2
(e
F
Mf
x
V
F
I(ot
I
T
V
Iect
I
2
asTstloag
H. Heluane et al. / Chemical Engine
he equivalent mathematical representation of Eq. (24) is theollowing:
λsVBijti, where j = 1, 2, . . ., M − 1, respectively. UP2 is
pper bound on Ψj(TC, VBij, ti) −(
EHj
λs− η∑N
i=1λij
λsVBijti
),
here j = 1, 2, . . ., M − 1. UP3 is upper bound on Ψ j(TC, VBij,i) j = 1, 2, . . ., M − 1. zj is a binary variable. If zj = 1 then,
j(TC, VBij, ti) =(
Ej
λs− η∑N
i=1λij
λsVBijti
), and if zj = 0 then
j(TC, VBij, ti) = 0.
.9. Integrality constraints for the number of subcycles
Each evaporation line may be cleaned many times during oneycle time (TC). This fact determines subcycles (Ni) for eachine
i =K∑
k=1
kyik, ∀i (28)
K
k=1
yik = 1, ∀i (29)
f the number of subcycles for evaporation line i is k then theinary variable yik is one. Note that for any evaporation linehe number of subcycles will be at least one, therefore all thevaporators will operate during the cycle time.
.10. Last evaporation unit outlet flow concentration
Given the operation time for each subcycle (ti/Ni) Eq. (14)eeds to be solved for each particular case (the set Mi) for obtain-ng outlet juice average concentration
θi = x0
�t
∫ t2
t1
M∏j=1
⎛⎜⎝1 + βj
F0x0
(1 + bj
tiNi
)1/2
⎞⎟⎠ (30) v
and Processing 46 (2007) 198–209 203
.11. Mass balance
For a system of N lines, the total mass flow of juiceF) fed to the evaporation system must be processed in thevaporators
Tc =N∑
i=1
Fi0ti (31)
ass and energy balances for each unit can be expressed by theollowing equations:
ij = xij−1 +
(2βij
(1 + b ti
Ni
)1/2 − 1
)bFij−1
tiNi
, ∀i, ∀j (32)
ij = Fij−1
(1 − xij−1
xij
), ∀i, ∀j (33)
ij = Fij−1 − Vij, ∀i, ∀j (34)
f a unit does not exist for an evaporator system βij will be zerosee Eq. (7)). On the other hand, when βij is not zero an amountf vapor is generated in unit (i, j) and is available to be used athe next unit of the line. Therefore
f βij �= 0 then VPij = Vij+1, ∀i, j = 1, 2, . . . , M − 1
(35)
he “bleed” can be calculated as follows:
Bij = Vij − VPij, ∀i, j (36)
f for a given unit (i, j) no “bleed” is required, VBij will bequal to zero for that unit. An additional constant is used Bij.Thisonstant takes the value 1 when the unit has a “bleed”, otherwisehe constant’s value is 0
f Bij = 0 then VBij = 0, ∀i, j (37)
.12. Storage tank
The implementation of the results of this model will requirestorage tank because the inlet flows (Fi0) to the evaporation
ystem remain constant during TC (operation + cleaning times).herefore, when line i is shut down to be cleaned, the corre-ponding Fi0 is diverted to a storage tank until the operation ofhe evaporation line i is re-established. As operating times areonger than cleaning times, it is possible to implement a sequencef cleaning in such a way that no overlapping of cleaning oper-tions occurs. Hence, the minimum desirable tank volume isiven by the following equation:
ol = maxi
⎛⎝τi
⎛⎝F −
∑l �=i
Fl0
⎞⎠⎞⎠ (38)
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t
2
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a
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2
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x
Fa
t
ts3at7waS
The case was implemented in GAMS [15] using DICOPT++as a solver. The results are given in Tables 2 and 3.
The optimal value obtained for the objective function is$4,076,400 for the time horizon and the cycle time is 154 h.
The MINLP model has Eq. (23) as objective function and thebove constraints.
The formulation is flexible enough to model multiple unit Mnits) and parallel N lines) evaporator systems. It can also modelituations where extraction of vapor (bleed) from the evaporationnits is needed to supply other operations.
.15. Case study I
The following example is based on a sugar plant located inucuman, Argentina. Five parallel evaporation lines are consid-red and each line is a quintuple effect system. Concentrationf the juice leaving evaporation line i will be expressed by Eq.45) which was obtained by integrating Eq. (13) with time for aine with five evaporation units and assuming identical foulingoefficient (b) for all evaporators
θi = xi0 + 2
b
((1 + b ti
Ni
)1/2 − 1
)tiNi
∑Mij=1βij
Fi0
+ln(
1 + b tiNi
)tiNi
∑Mij=1∑
r>j(βijβir)
bF2i0xi0
+ 2
b
((1 + b ti
Ni
)1/2 − 1
)tiNi
(1 + b ti
Ni
)1/2
∑Mij=1∑
r>j
∑s>r(βijβirβis)
F3i0x
2i0
345
N
and Processing 46 (2007) 198–209
+∑Mi
j=1∑
r>j
∑s>r
∑t>s(βijβirβisβit)
F4i0x
3i0
(1 + b ti
Ni
)
+ 2
3b
∏Mij=1βj
F5i0x
4i0
(1 + b ti
Ni
)3/2 − 1(1 + b ti
Ni
)3/2 (45)
or this particular case also the operating time is imposed to bet least six times the cleaning time of each line
i ≥ 6τiNi, ∀i (46)
It is desired to determine a configuration and cycle scheduleo process 800 t/h of 16 Bx juice. The final concentration of theugar (xT) must be 99 Bx. As vapors extracted from units 1, 2,, and 4 have different enthalpy conditions, they will be usedt different stages of the process. The energy demand of eachype of vapor (EH
j ) is 42,200 MW h for a time horizon (h) of20 h. If any vapor is not enough to meet the requirements, steamill be used. Cost of cleaning one evaporation line (cc) was
ssumed $4500 and steam cost per mass unit (csu) 8.386 $/t.ome parameters of the problem are shown in Table 1.
H. Heluane et al. / Chemical Engineering and Processing 46 (2007) 198–209 205
Cvo
t
1
2
oa
2
wtcoiasoum
pt
t6iui
tooo
mf
2
aoir
2
mwas solved using Minos5.
The resulting MINLP for the case study had 60 binary vari-ables, 227 continuous variables and 263 equations. The solutionwas obtained in 0.70 CPU seconds on a Pentium I.
Fig. 4. Gantt chart of the optimal cleaning time distribution.
leaning cost is $105,194. As seen from Table 2 the amount ofapor generated in the evaporation stage is not enough to supplyther energy requirements (SR1, SR2, SR3 and SR4 > 0).
The cleaning policy showed in Fig. 4 will be adopted in ordero avoid the following inconveniences:
. Under-utilization of cleaning resources, in particular man-power.
. The superposition of the cleaning times at the end of theschedule yields a great mass accumulation that would requirestorage until the beginning of a new cycle.
This sequence allows using the same manpower for cleaningperation and, on the other hand, avoids the storage of juice forlong time that would cause a decrease in sugar yield.
.16. Storage requirements
The Gantt chart shown in Fig. 4 presents 10 time intervalshere the variations of total processed mass flow are due to
he cleaning policy and different feed conditions. If the pro-essed flows are analyzed at each interval, two situations arebserved. When one line is being cleaned, the mass flow arriv-ng to the evaporation section exceeds the flow being processed,nd when the five lines are operating simultaneously the oppo-ite occurs. Therefore, it is necessary to contemplate the storagef juice so that the evaporation section could be operated contin-ously (which is not explicitly taken into account in the MINLPodel).As shown in Fig. 5, the accumulation of juice is 13,742 m3
er each cycle when all units are stopped and cleaned at the sameime.
When the cleaning task for a unit starts immediately afterhe previous one is finished, the volume of juice accumulated is
840 m3 (see Fig. 5). In the situation shown in Fig. 4, the juices accumulated while one line is being cleaned and immediatelysed in the next time interval where the five lines are work-ng together. In that case, the storage requirement is reduced F
Fig. 5. Storage of juice per cycle for different situations.
o 1716 m3 as shown in Fig. 5. Any storage tank of a volumef 1716 m3 or higher will allow the operation of the proposedptimum scheduling but higher storage capacities gives moreperational flexibility.
Fig. 6 shows the variation of the value of the optimum (mini-al) costs defined by Eq. (1) with storage tank volume available
or that purpose at the plant.
.17. Case study II
The same problem was considered when no tank is avail-ble (zero storage). Then, the Gantt chart shown in Fig. 7 isbtained. The optimal value obtained for the objective functions $4,185,186, a cycle time of 95 h, and flows fed to the evapo-ation lines of 200 t/h.
.18. Computational statistics
The GAMS modeling system was used to implement theathematical model as mentioned above. The NLP subproblem
ig. 6. Variation of total operation cost with available storage tank volume.
206 H. Heluane et al. / Chemical Engineering
ts((mw(awc
2
dotasamtwi
2
uDs2fco
2
tw
tujtt
t
S
Eq. (7). It should be also noted that even when the influence ofinlet juice concentration and sugar concentration are significant,their values could be obtained with relative accuracy from plantinformation. More significant is the relatively small dependence
Fig. 7. Gantt chart with zero storage.
The sensitivity of the solution to the initial point provided tohe solver is shown in Fig. 8. A sample of 96 points was cho-en at random. The minimum value obtained was $4,076,400solution of the problem) and the maximum was $4,120,600range: $44,200). Only 22.9% of the initial points led to theinimum cost value but for the rest of the initial points, costsere below 1% of difference from the minimum cost value
$4,076,400) although the schedules were different. This givesremarkable flexibility because the system can be operatedith different cleaning schedules and costs remain practically
onstant.
.19. Case study III
Case study III has the same parameters than case study I, theifference between them is the evaporator scheme. The evap-ration system was considered as follows: five parallel lines,hree of them with quintuple units, one with quadruple units andnother one with 3 units. Values of βij are the same as in casetudy I except β15 = 0, β54 = 0, and β55 = 0. Therefore, line 1 isquadruple unit line while line 5 is a triple unit line. The opti-al value for the objective function is $4,295,100. The lack of
hree units in the new evaporation system causes an increase ofater to be extracted at the crystallization section and hence an
ncrease of costs.
Fig. 8. Sensitivity of the solution to the initial point.
and Processing 46 (2007) 198–209
.20. Heuristic case
For comparison purposes, a heuristics based case study issed to schedule the five lines. A 7-day cycle is considered.uring the first 87 h of the cycle (time devoted to cleaning
equentially every line) each operating line is fed at a rate of00 t/h of juice. At the following period (87–168 h), all lines areairly clean and operate with the same mass flow of 160 t/h. Theost calculated in this case was $4.618E+6 which imply savingsf $542,000 for the time horizon of 720 h.
.21. Sensitivity analysis
The influence of the different parameters and variables onhe objective function, once the optimum has been achieved,as determined.The relative influence of the main parameters of the evapora-
or model such as cleaning costs per unit, steam costs per massnit, body temperature, driving force, area, fouling factor, inletuice concentration, and final sugar concentration on the objec-ive function has been studied and the corresponding results forhe case study are shown in Fig. 9.
The parameter used for such purposes is Sp defined accordingo
p = ∂Z
∂p
p
Z(47)
The influence of A, θ and �θ is the same, in agreement with
Fig. 9. Sensitivity to parameters of the model.
H. Heluane et al. / Chemical Engineering and Processing 46 (2007) 198–209 207
Fig. 10. Sensitivity to an increase of 10% in variable values.
n b, which is usually the most uncertain parameter. Otherwise,he model had shown a strong influence of steam cost per massnit, which is another uncertain parameter.
In order to reflect the loss of performance when an optimalperating condition cannot be implemented, the sensitivity dueo the decision variables, Sv, has been calculated according to
v = �Z
�v
v
Z(48)
here �v has been set to ±10%.According to Fig. 10 (�v = +10%), the objective function
hows to be especially sensitive to the cycle time Tc) and theows for every evaporator (Fi0). A similar result is obtained for�v = −10%).
.22. Technical objective function
If cleaning costs are neglected, minimum costs are obtainedaximizing the outlet juice concentration at the evaporation
tage [13]. Therefore, a new MINLP problem was solved. In thisase the constraints were maintained but the following equationas considered as objective function:
ax xT =∑N
i=1Fi0ti∑Ni=1
Fi0tixei
(49)
here xT is the average outlet juice concentration of all evapo-ation lines.
.23. Case study IV
The same parameters as in case study I were used. The opti-al value obtained for the objective function is 35.6 Bx. The cal-
ulated costs using optimum values obtained from the problemolved with the technical objective function has no significantifference with the cost obtained with Eq. (23) as objective func-ion. Table 4 shows the results obtained for the different cases
tudied. The optimization of a technical objective function withconomical background is useful in cases where uncertain costarameters are involved. For instance, in this work, accurate val-es of cleaning and steam costs may be difficult to determine,
ence, to find an alternative objective function is a valuable toolor process optimization.
. Conclusions
Efficient process integration in sugar cane plants andnhanced operation performance may be achieved by consider-ng the combined operation of the evaporation and crystallizationections, along with the appropriate management of their asso-iated steam bleeds for satisfying energy demands from otherlant operations. A common cost objective allows formulating aroblem for determining the optimal operating conditions underifferent scenarios.
Aimed at a practical application of the results this work seekso evaluate different operating conditions of multiple evapo-ation systems working in parallel in order to choose thoseonditions leading to minimum operating costs.
A MINLP model was developed to determine an optimalchedule for the evaporator system. The formulation is flexiblenough to model multiple units (M units) and parallel (N lines)vaporator systems, as well as network arrangements arisingrom the combination of these basic cases. The formulation maylso consider “bleed” at any unit. Results show that significantavings of steam could be achieved just operating the evapora-ion section in a different way and with no additional investmenteeded.
Although the solution of the MINLP model is sensitive tohe initial point, most of the times, costs were only about 1%igher than minimum cost (optimal solution). This situation isemarkable because it gives operational flexibility because the
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08 H. Heluane et al. / Chemical Engine
vaporation system may be operated with different schedulesithout sensible cost increase.Identical costs were determined with an alternative technical
bjective function which is convenient because the objectiveunction based on costs is very sensitive to a parameter (steamost per unit mass) that has a fairly uncertain value.
cknowledgements
This work was partially supported by Consejo de Investi-aciones de la Universidad Nacional de Tucuman (Argentina).upport received by the European Commission is also thankfullycknowledged (Project no. MRTN-CT-2004-512233).
ppendix A. Nomenclature
heat-exchange area (m2)Fouling coefficient for the evaporator
ij bleed constantc cost of cleaning one evaporation unit ($/unit)su cost of steam per mass unit ($/t)
total cost ($)cleaning costs of the cleaning operation ($)steam crystallization cost of steam of the crystallization section
($)steam evaporation cost of steam of the evaporation section ($)steam other uses cost of steam used as supply for other operations
of the process ($)energy required for other operations (MW h)total mass flow of fed juice (t/h)
i0 mass flow of juice fed to line i (t/h)Pi juice flow leaving evaporation line i (t/h)
time horizon (h)maximum expected number of cleaning tasks duringTc
i number of subcycles in line iQ mass of vapor and/or steam required for other opera-
tions (t/h)ci steam condensed at the crystallizer in the line i (t/h)ei steam condensed in the first evaporator of each line i
(t/h)l slack variablert steam required for other operations of the process for t
(t/h)p sensitivity to parametersv sensitivity to variablesC total steam condensed at the crystallization section (t/h)E total steam condensed at the evaporation section (t/h)RH steam required for other operations of the process for
H (t/h)i total operation time of line i (h)toti processing and cleaning time of line i in Tc (h)
C cycle time (h)
global heat-transfer coefficient (kW/m2 ◦C)P upper boundol storage tank volume (m3)
[
and Processing 46 (2007) 198–209
total water removed as vapor from an evaporator (t/h)B vapor removed as “bleed” (t/h)C water removed as vapor from crystallization section
(t/h)P vapor removed from an evaporator and derived to the
following one (t/h)ij outlet juice concentration at evaporation unit (i, j) (Bx)i0 inlet juice concentration at evaporation unit j = 1 (Bx)T sugar concentration of the product obtained at the crys-
tallization section (Bx)0 concentration of the juice fed to an evaporator (Bx)θi average concentration of the concentrated juice at evap-
oration line i (Bx)average sugar concentration obtained at evaporator(Bx)
ik binary variable (yi,k = 1 if unit i operates k subcycles inTc)
sli , zj binary variable
ndicesevaporation lineevaporation unit
reek lettersproportionality constant (kW Bx/(m2 ◦C2))number of evaporation cycles in the time horizonjuice temperature in the evaporator (◦C)
θ driving force (◦C)heat of vaporization of water (kWh/t)
i time devoted to clean line i (h)
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