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Energy and Water Conservation in Batch Processes
Energy and Water Conservation in Batch Processes Thokozani Majozi, Wits University, Johannesburg, South Africa 2017 Elsevier Inc. All rights reserved.
Problem Statement The problem addressed in this article can be stated as follows: Given: Scheduling data i. Production recipe for each product ii. Available units and their capacities iii. Task durations iv. Time horizon of interest or predetermined production v. Costs of raw materials vi. Selling price of final products Heat integration data (i) Hot duties for tasks requiring heating and cold duties for tasks that require cooling (ii) Operating temperatures of heat sources and heat sinks (iii) Minimum allowable temperature differences (iv) Costs of hot and cold utilities (v) Design limits on heat storage Wastewater minimization data (i) Mass load of contaminants (ii) Maximum inlet and outlet concentrations of contaminants (iii) Washing time for each unit (iv) Maximum storage available for reuse (v) Cost of freshwater (vi) Cost of effluent treatment Determine an optimal production schedule that achieves a maximum profit or minimum makespan, requiring the minimum amount of external utilities and freshwater use.
Mathematical Formulation The mathematical formulation consists of three modules, that is, production scheduling, heat integration, and wastewater minimization modules. For production scheduling, the objective is either profit maximization or minimization of makespan. During production, certain tasks require cooling or heating, for instance, exothermic reactions that require cooling or endothermic reactions that require heating. The requirements for heating or cooling afford opportunities for heat integration. In multipurpose batch plants, particularly at the completion of a processing task in a unit, the equipment unit is washed before performing subsequent tasks. This is to ensure product integrity through prevention of contamination. The washing operations present an opportunity for wastewater minimization. Sets C {c|c ¼ contaminant} J {j|j ¼ processing unit} Jc {jc|jc ¼ processing unit which may conduct tasks requiring heating} 4J Jh {jh|jh ¼ processing unit which may conduct tasks requiring cooling} 4J P {p|p ¼ time point} S {s|s ¼ any state} Sin,j {sin,j|sin,j ¼ input state to a processing unit} 4S Sin {Sin|Sin ¼ input state into any unit} Sout {sout|s out ¼ output state from any unit} Sin;j {sin;j sin;j ¼ effective state into processing unit} 4Sin Sout,j {Sout, j|Sout,j ¼ output stream from a processing unit} 4S U {u|u ¼ heat storage unit}
Encyclopedia of Sustainable Technologies, Volume 4
http://dx.doi.org/10.1016/B978-0-12-409548-9.10140-X
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Energy and Water Conservation in Batch Processes
Continuous variables B(sin,j) Batch size, either fixed or variable cin(sin,j,c,p) Inlet concentration of contaminant c, to unit j at time point p cout(sout,j,c,p) Outlet concentration of contaminant c, from unit j at time point p CL sin;jh Cooling load for hot state csin(c, p) Inlet concentration of contaminant c, to storage at time point p csout(c, p) Outlet concentration of contaminant c, from storage at time point p cw sin;jh ; p External cooling required by unit jh conducting the task corresponding to state sin;jh at time point p d(s, p) Amount of state delivered to customers at time point p dur(sin,j, p) Duration of task, either fixed or dependent on batch size H Time horizon of interest, optimization variable for makespan minimization problem HL sin;jc Heating load for cold state MB(sin,j, c, p) Mass load of contaminant c in unit j at time point p after processing sin,j that is added to the water stream mwin(sin,j, p) Mass of water into unit j for cleaning state sout at time point p mwout(sout,j, p) Mass of water produced at time point p from unit j mwf (sin,j, p) Mass of freshwater into unit j at time point p mwe(sout,j, p) Mass of effluent water from unit j at time point p mwr sout;j;j0 ; p Mass of water recycled to unit j0 from j at time point p msin(j, p) Mass of water transferred from unit j to storage at time point p msout(j, p) Mass of water transferred from storage to unit j at time point p Q(sin,j, u, p) Heat exchanged with heat storage unit u at time point p q sin;jh ; sin;jc ; p Amount of heat exchanged during direct heat integration qws(p) Amount of water stored in storage at time point p st sin;jc ; p External heating required by unit jc conducting the task corresponding to state sin;jc at time point p t0(u, p) Initial temperature in heat storage unit u at time point p tf (u, p) Final temperature in heat storage unit u at time point p t0(sin,j, u, p) Time at which heat storage unit commences activity tf (sin,j, u, p) Time at which heat storage unit ends activity tu(sin,j, p) Time at which a state is used in unit j tout(sout,j, p) Time at which a state is produced from unit j at time point p twin(sin,j, p) Time at which water is used at time point p in unit j twout(sout,j, p) Time at which water is produced at time point p from unit j twr sout;j;j0 ; p Time at which water is recycled from unit j to unit j0 at time point p tsin( j, p) Time at which water is transferred from unit j to storage at time point p tsout( j, p) Time at which water is transferred from storage to unit j at time point p W(u) Capacity of heat storage unit u G(sin,j, u, p) Glover transformation variable J(sin,j, u, p) Reformulation-Linearization variable Binary variables x sin;jc ; sin;jh ; p Binary variable associated with heat integration between unit jc conducting the task corresponding to state sin;jc and unit jh conducting the task corresponding to state sin;jh , at time point p y(s * in,j, p) Binary variable associated with usage of state s in unit j at time point p ywr sout;j;j0 ; p Binary variable showing usage of recycle from unit j to unit j0 at time point p yw(sin,j, p) Binary variable showing usage of water in unit j at time point p ysin(j, p) Binary variable showing transfer of water from unit j to storage at time point p ysout(j, p) Binary variable showing transfer of water from storage to unit j at time point p z(sin,j, u, p) Binary variable associated with heat integration between unit j conducting the task corresponding to state sin,j with heat storage unit u at time point p Parameters A(c) Contaminant loading (g contaminant/kg batch) a Constant coefficient of processing time b Variable coefficient of processing time cp Specific heat capacity of heat storage fluid CPstate(sin,j) Specific heat capacity of state CU in(sin,j, c) Maximum inlet concentration of contaminant c in unit j
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CU out(sout,j, c) Maximum outlet concentration of contaminant c from unit j CE Cost of effluent water treatment (c.u./kg water) CF Cost of freshwater (c.u./kg water) Cost_cw Cost of cooling water Cost_st Cost of steam CP(s) Selling price of product s, s ¼ product CSoout(c) Initial concentration of contaminant in storage E(sin,j) Fixed amount of heat required by or removed from unit j conducting the task corresponding to state sin,j H Time horizon of interest, for profit maximization problem M(sin,j, c) Mass load of contaminant c in unit j after processing sin,j that is added to the water stream MM Any large number MwU(sin,j) Maximum inlet water mass of unit j Qwos Initial amount of water in storage QwU s Maximum capacity of storage T(sin,j) Operating temperature for processing unit j conducting the task corresponding to state sin,j, for constant temperature processes Tin(sin,j) Inlet temperature for state sin,j Tout(sin,j) Outlet temperature for state sin,j T L Lower bound for heat storage temperature T U Upper bound for heat storage temperature DTmin Minimum allowable thermal driving force s(sin,j) Duration of the task corresponding to state sin,j conducted in unit j sw(sout,j) Duration of washing for unit j W L Lower bound for heat storage capacity W U Upper bound for heat storage capacity The necessary heat integration and wastewater minimization constraints are embedded in the scheduling framework. The heat integration constraints are presented first, followed by the necessary wastewater minimization constraints. The scheduling framework as used in the models by Adekola and Majozi (2011) and Stamp and Majozi (2011), was used in the proposed model. The model uses the state sequence network recipe representation and an uneven discretization of the time horizon. This has proven to result in fewer binary variables compared to models based on other representations. The mathematical model is based on the superstructure in Fig. 1. Fig. 1A is the superstructure for heat integration while Fig. 1B is the superstructure for water optimization.
Heat Integration Constraints The heat integration constraints are based on the superstructure in Fig. 1A. In Fig. 1A, each processing unit may operate using either direct or indirect heat integration. Direct heat integration refers to the use of heat generated from a processing unit to supply a processing unit requiring heat without use of storage. Indirect heat integration refers to the use of heat previously stored in a heat storage vessel to supply a processing unit requiring heat. The heat is stored through a heat transfer fluid medium, for example, water. Processing units may also operate in standalone mode, using only external utilities. This may be required for control reasons or when thermal driving forces or time do not allow for heat integration. If either direct or indirect heat integration is not sufficient to satisfy the required duty, external utilities may make up for any deficit. Constraints (1)–(26) constitute the heat integration model, useful for multipurpose batch processes. The formulation is based on the model by Stamp and Majozi (2011) and includes direct and indirect heat integration. Constraints (1) and (2) are active simultaneously and ensure that one hot unit will be integrated with one cold unit when direct heat integration takes place, in order to simplify operation of the process. Also, if two units are to be heat integrated at a given time point, they must both be active at that time point. However, if a unit is active, it may operate in either integrated or standalone mode. X x sin;jc ; sin;jh ; p y sin;jh ; p ; cp ˛ P; sin;jh ˛ sin;j (1) Sin;jc
X x sin;jc ; sin;jh ; p y sin;jc ; p ;
cp ˛ P; sin;jc ˛ sin;j
(2)
Sin;jh
Constraint (3) ensures that only one hot or cold unit is heat integrated with one heat storage unit at any point in time. This is to simplify and improve operational efficiency in the plant. X X x sin;jc ; u; p þ x sin;jh ; u; p 1; cp ˛ P; u ˛ U (3) Sin;jc
Sin;jh
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Fig. 1
Energy and Water Conservation in Batch Processes
Superstructure for mathematical formulation: (A) when units perform heating/cooling tasks and (B) when units perform washing tasks.
Constraints (4) and (5) ensure that a unit cannot simultaneously undergo direct and indirect heat integration. This condition simplifies the operation of the process. X x sin;jc ; sin;jh ; p þ z sin;jc ; u; p 1; cp ˛ P; sin;jc ˛ Sin;j ; u ˛ U (4) Sin;jh
X x sin;jc ; sin;jh ; p þ z sin;jh ; u; p 1;
cp ˛ P; sin;jh ˛ Sin;j ; u ˛ U
(5)
Sin;jc
Constraints (6) and (7) quantify the amount of heat received from or transferred to the heat storage unit, respectively. There will be no heat received or transferred if the binary variable signifying use of the heat storage vessel, z(sin,j, u, p), is zero. These constraints are active over the entire time horizon, where p is the current time point and p 1 is the previous time point. Q sin;jc ; u; p 1 ¼ W ðuÞcp T0 ðu; p 1Þ Tf ðu; pÞ z sin;jc ; u; p 1 ; cp ˛ P; p > p0; sin;jc ˛ Sin;j ; u ˛ U (6) Q sin;jh ; u; p 1 ¼ W ðuÞcp Tf ðu; pÞ T0 ðu; p 1Þ z sin;jh ; u; p 1 ;
cp ˛ P; p > p0; sin;jh ˛ Sin;j ; u ˛ U
(7)
Constraint (8) quantifies the heat transferred to the heat storage vessel at the beginning of the time horizon. The initial temperature of the heat storage fluid is T0(u, p1). Q sin;jh ; u; p0 ¼ W ðuÞcp Tf ðu; p1Þ T0 ðu; p1Þ z sin;jh ; u; p0 ; csin;jh ˛ Sin;j ; u ˛ U (8) Constraint (9) ensures that the final temperature of the heat storage fluid at any time point becomes the initial temperature of the heat storage fluid at the next time point. This condition will hold regardless of whether or not there was heat integration at the previous time point. T0 ðu; pÞ ¼ Tf ðu; p 1Þ;
cp ˛ P; u ˛ U
(9)
Constraints (10) and (11) ensure that temperature of heat storage does not change if there is no heat integration with the heat storage unit, unless there unit. MM is any large number, thereby resulting in an overall “Big M” is heat loss from the heat storage formulation. If either z sin;jc ; u; p 1 or z sin;jh ; u; p 1 is equal to one, Constraints (10) and (11) will be redundant. However, if these two binary variables are both zero, the initial temperature at the previous time point will be equal to the final temperature at the current time.
Energy and Water Conservation in Batch Processes 1 X X T0 ðu; p 1Þ Tf ðu; pÞ þ MM@ z sin;jc ; u; p 1 þ z sin;jh ; u; p 1 A;
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0
sin;jc
cp ˛ P; p > p0; u ˛ U
(10)
cp ˛ P; p > p0; u ˛ U
(11)
sin;jh
0
1 X X T0 ðu; p 1Þ Tf ðu; pÞ þ MM@ z sin;jc ; u; p 1 þ z sin;jh ; u; p 1 A; sin;jc
sin;jh
Constraint (12) ensures that minimum thermal driving forces are obeyed when there is direct heat integration between a hot and a cold unit. This constraint holds when both hot and cold units operate at constant temperature, which is commonly encountered in practice. An example is when there is heat integration between an exothermic and an endothermic reaction. T sin;jh T sin;jc DT min MM 1 x sin;jc ; sin;jh ; p 1 ; cp ˛ P; p > p0; sin;jc ; sin;jh ˛ Sin;j (12) Constraints (13) and (14) ensure that minimum thermal driving forces are obeyed when there is heat integration with the heat storage unit. Constraint (13) applies for heat integration between heat storage and a heat sink, while Constraint (14) applies for heat integration between heat storage and a heat source. In Constraints (13) and (14), the units operate at fixed temperatures. Tf ðu; pÞ T sin;jc DT min MM 1 z sin;jc ; u; p 1 ; cp ˛ P; p > p0; sin;jh ˛ Sin;j ; u ˛ U (13) T sin;jh Tf ðu; pÞ DT min MM 1 z sin;jh ; u; p 1 ;
cp ˛ P; p > p0; sin;jh ˛ Sin;j ; u ˛ U
(14)
Constraint (15) states that the cooling of a fixed heat source will be satisfied by either direct or indirect heat integration as well as external utility if required. X min E sin;jc ; E sin;jh x sin;jc ; sin;jh ; p ; cp ˛ P; sin;jh ˛ Sin;j ; u ˛ U (15) E sin;jh y sin;jh ; p ¼ Q sin;jh ; u; p þ cw sin;jh ; p þ sin;jc
sin;jc ;sin;jh
Constraint (16) ensures that the heating of a fixed heat sink will be satisfied by either direct or indirect heat integration as well as external utility if required. X E sin;jc y sin;jc ; p ¼ Q sin;jc ; u; p þ st sin;jc ; p þ min E sin;jc ; E sin;jh x sin;jc ; sin;jh ; p ; cp ˛ P; sin;jc ˛ sin;j ; u ˛ U (16) sin;jh
sin;jc ;sin;jh
Constraints (17) and (18) give the heating load for a cold state and cooling load for a hot state, respectively, for situations where the batch size of material processed in a unit is variable. HL sin;jc ; p ¼ B sin;jc ; p CPstate sin;jc Tout sin;jc Tin sin;jc ; cp ˛ P; sin;jc ˛ Sin;j (17) CL sin;jh ; p ¼ B sin;jh ; p CPstate sin;jh Tin sin;jh Tout sin;jh ;
cp ˛ P; sin;jh ˛ Sin;j
(18)
Constraints (19) and (20) give the heating load for a cold state and cooling load for a hot state, respectively. The heating load and cooling load are made up using heat from other units as well as external utility if required. X HL sin;jc ; p ¼ st sin;jc ; p þ q sin;jc ; sin;jh ; p ; cp ˛ P; sin;jc ˛ Sin;j (19) sin;jh
X q sin;jh ; sin;jc ; p ; CL sin;jh ; p ¼ cw sin;jh ; p þ
cp ˛ P; sin;jh ˛ Sin;j
(20)
sin;jh
Constraints (21) and (22) ensure that the times at which units are active are synchronized when direct heat integration takes place. Starting times for the tasks in the integrated units are the same. This constraint may be relaxed for operations requiring preheating or precooling and is dependent on the process. tu sin;jh ; p tu sin;jc ; p MM 1 x sin;jc ; sin;jh ; p cp ˛ P; sin;jc ; sin;jh ˛ Sin;j (21) tu sin;jh ; p tu sin;jc ; p þ MM 1 x sin;jc ; sin;jh ; p
cp ˛ P; sin;jc ; sin;jh ˛ Sin;j
(22)
Constraints (23) and (24) ensure that if indirect heat integration takes place, the time a unit is active will be equal to the time a heat storage unit starts either to transfer or receive heat. tu sin;j ; p tu sin;j ; p MM y sin;j ; p z sin;j ; u; p cp ˛ P; u ˛ U; sin;j ˛ Sin;j (23) tu sin;j ; p t0 sin;j ; u; p þ MM y sin;j ; p z sin;j ; u; p
cp ˛ P; u ˛ U; sin;j ˛ Sin;j
(24)
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Constraints (25) and (26) state that the time when heat transfer to or from a heat storage unit is finished will coincide with the time the task transferring or receiving heat has finished processing. tu sin;j ; p 1 þ s sin;j y sin;j ; p 1 tf sin;j ; u; p MM y sin;j ; p 1 z sin;j ; u; p 1 cp ˛ P; p > p0; u ˛ U; sin;j ˛ Sin;j (25) cp ˛ P; p > p0; u ˛ U; sin;j ˛ Sin;j tu sin;j ; p 1 þ s sin;j y sin;j ; p 1 tf sin;j ; u; p þ MM y sin;j ; p 1 z sin;j ; u; p 1 (26)
Wastewater Minimization Constraints The wastewater minimization constraints are based on the superstructure in Fig. 1B. Unit j represents a water using operation in which the water used consists of freshwater, stored water, or recycled/reused water. Water from unit j can be recycled into the same unit, reused by other units, or sent to storage. Direct water reuse refers to the use of an outlet wastewater stream from a processing unit in another processing unit while indirect water reuse refers to the use of previously stored wastewater in a processing unit. Constraints (27)–(77) constitute the wastewater minimization model useful for multipurpose batch processes, which involve multiple contaminants. The formulation is based on the model by Adekola and Majozi (2011) and includes both direct water reuse and indirect water reuse due to the presence of a central storage vessel. Mass balances around each processing unit and the central storage vessel are formulated as follows.
Mass Balance Around a Unit Constraint (27) is the water balance over the inlet to a unit. Water entering the unit is a combination of reuse/recycle streams from other units, j0 , freshwater and water from storage. Constraint (28) states that the water leaving a unit could be recycled/reused, sent to storage, or discarded as effluent. Constraint (29) states that the amount of water exiting a unit must equal the amount of water entering the unit at the previous time point. This constraint captures the fact that water is neither produced nor lost in the unit during the washing operation. X mwin sin;j ; p ¼ mwr sout;j0;j ; p þ mwf sin;j ; p þ msout ð j; pÞ cj; j0 ˛ J; sin;j ˛ Sin;j ; sout;j0 ;j ˛ Sout;j0;j ; p ˛ P (27) sout;j0;j
X mwout sout;j ; p ¼ mwr sout;j;j0 ; p þ mwe sout;j ; p þ msin ð j; pÞ
cj; j0 ˛ J; sout;j;j0 ˛ Sout;j;j0 ; p ˛ P
(28)
sout;j;j0
mwin sin;j ; p 1 ¼ mwout sout;j ; p
cj ˛ J; sin;j ˛ Sin;j sout;j ; ˛ Sout;j ; p ˛ P; p > p0
(29)
Constraint (30) represents the inlet contaminant mass balance. The contaminant mass load in the inlet stream is the sum of the contaminant mass load in recycle/reuse water and that in water from storage. Constraint (31) represents the outlet contaminant mass as the mass of contaminant that entered the unit at the previous time point and the mass load of contaminant picked up in the unit during its operation. In Constraint (31), the mass load of contaminant in a unit is a given parameter. X mwin sin;j ; p cin sin;j ; c; p ¼ mwr sout;j0 ;j ; p cout sout;j ; c; p þ msout ð j; pÞcsout ðc; pÞ cj; j0 ˛ J; sin;j ˛ Sin;j ; sout;j0 ;j ˛ Sout;j0 ;j ; p ˛ P; c ˛ C sout;j0 ;j
(30) mwout sout;j ; p cout sout;j ; c; p ¼ M sout;j ; c yw sout;j ; p 1 þ mwin sin;j ; p 1 cin sin;j ; c; p 1 cj ˛ J; sout;j ˛ Sout;j ; p ˛ P; p > p1 ; c ˛ C (31) In the case where mass load of contaminant in a unit is defined as a function of batch size of material processed in the unit, Constraint (32) represents the outlet contaminant mass. Constraint (33) defines the variable contaminant mass load as the product of the contaminant loading and the batch size of material processed. mwout sout;j ; p cout sout;j ; c; p ¼ MB sin;j ; c yw sin;j ; p 1 þ mwin sin;j ; p 1 cin sin;j ; c; p 1 cj ˛ J; sin;j ˛ Sin;j ; sout;j ˛ Sout;j ; p ˛ P; p > p1 ; c ˛ C (32) MB sin;j ; c; p ¼ AðcÞB sin;j ; p
cj ˛ J; sin;j ˛ Sin;j ; p ˛ P; c ˛ C
(33)
Constraints (34) and (35) ensure that the inlet and outlet contaminant concentrations do not exceed the allowed maximum. Similarly, the maximum allowable water in a unit must not be exceeded. This is governed by Constraint (36). Constraint (37)
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restricts the mass of water entering the unit from recycle/reuse, to the maximum allowable for the unit. Likewise, Constraint (38) restricts the mass of water entering the unit from storage, to the maximum allowable for the unit. cin sin;j ; c; p CU cj ˛ J; sin;j ˛ Sin;j ; p ˛ P; c ˛ C (34) in sin;j ; c yw sin;j ; p cout sout;j ; c; p CU out sout;j ; c yw sin;j ; p 1
cj ˛ J; sin;j ˛ Sin;j ; sout;j ˛ Sout;j ; p ˛ P; p > p1 ; c ˛ C
mwin sin;j ; p MW U sin;j yw sin;j ; p
cj ˛ J; sin;j ˛ Sin;j ; p ˛ P
(35) (36)
mwr sout;j0 ;j ; p MwU sin;j ywr sout;j0 ;j ; p cj; j0 ˛ J; sin;j ˛ Sin;j ; sout;j0 ;j ˛ Sout;j0 ;j ; p ˛ P
(37)
msout sout;j ; p MwU sin;j ysout sout;j ; p cj ˛ J; sin;j ˛ Sin;j ; sout;j ˛ Sout;j ; p ˛ P
(38)
The maximum quantity of water into a unit is represented by Constraint (39). It is important to note that for multicontaminant wastewater the outlet concentration of the individual components cannot all be set to the maximum, since the contaminants are not limiting simultaneously. The limiting contaminant(s) will always be at the maximum outlet concentration and the nonlimiting contaminants will be below their respective maximum outlet concentrations. ( ) M sin;j ; c (39) MwU sin;j ¼ max U cj ˛ J; sin;j ˛ Sin;j ; sout;j ˛ Sout;j c ˛ C c˛C Cout sout;j ; c CU in sin;j ; c
Mass Balance Around Central Storage Constraint (40) is the water mass balance around the storage tank. The amount of water stored in the storage tank consists of water stored at the previous time point and the difference between water entering the storage tank from a processing unit and water leaving the storage tank to a processing unit. Constraint (41) defines the initial amount of water in the tank. X X msin ð j; pÞ msout ð j; pÞ cj ˛ J; p ˛ P; p > p0 (40) qws ðpÞ ¼ qws ðp 1Þ þ j
j
qws ðp0Þ ¼ Qwos
X
msout ð j; p0Þ
cj ˛ J
(41)
j
The definition of the inlet contaminant concentration to the storage tank is given in Constraint (42). The concentration of water exiting the storage tank is assumed to be equal to the concentration of water in the tank as given in Constraint (43). This condition is true in the case of perfect mixing. The initial concentration in the storage tank is given in Constraint (44). Constraint (45) ensures that the maximum capacity of the tank is not exceeded. P j msin ð j; pÞcout sin;j ; c; p P cj ˛ J; sin;j ˛ Sin;j ; p ˛ P; c ˛ C csin ðc; pÞ ¼ (42) j msin ð j; pÞ csout ðc; pÞ ¼
P qws ðp 1Þcsout ðc; p 1Þ þ j msin ð j; pÞcsin ðc; pÞ P qws ðp 1Þ þ j msin ð j; pÞ
cj ˛ J; p ˛ P; p p0; c ˛ C
csout ðc; p1 Þ ¼ CSoout ðcÞ cc ˛ C qws ðpÞ QwU s
cp ˛ P
(43) (44) (45)
Constraint (46) ensures that no water is stored in the storage vessel at the end of the time horizon in order to give a true optimum. Otherwise the resulting minimum effluent could be misleading. qws ðpÞ ¼ 0 cp ¼ jP j (46) The scheduling constraints for the wastewater minimization model are as follows.
Task Scheduling Constraints Task scheduling constraints ensure that each water using operation is integrated with production scheduling. Constraints (47) and (48) together ensure that unit j is washed immediately after a task has produced sout,j. If water is used in the unit, yw(sin,j, p) has a value of 1 causing Constraints (47) and (48) to become active and the start time of washing is forced to coincide with the end time of production. Otherwise, when water is not used in the unit, that is, yw(sin,j, p) has a value of zero, the two constraints become relaxed. Constraint (49) represents the duration of the washing operation performed in unit j. Constraint (50) stipulates that the washing operation can only commence at time point p if the task using s * in,j was active at the previous time point.
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Energy and Water Conservation in Batch Processes twin sin;j ; p tout sout;j ; p MM 1 yw sin;j ; p cj ˛ J; sin;j ˛ Sin;j ; sout;j ˛ Sout;j ; p ˛ P
(47)
twin sin;j ; p tout sout;j ; p þ MM 1 yw sin;j ; p cj ˛ J; sin;j ˛ Sin;j ; sout;j ˛ Sout;j ; p ˛ P
(48)
twout sout;j ; p ¼ twin sin;j ; p 1 þ sw sin;j yw sin;j ; p 1 yw sin;j ; p ¼ y sin;j ; p 1
cj ˛ J; sin;j ˛ Sin;j ; sout;j ˛ Sout;j ; p ˛ P; p > p0
cj ˛ J; sin;j ˛ Sin;j ; sin;j ˛ Sin;j ; p ˛ P; p ˛ p0
(49) (50)
Recycle/Reuse Scheduling Wastewater can only be directly recycled/reused if the unit producing wastewater and the unit receiving wastewater finish operating and begin operating at the same time, respectively. Constraint (51) describes the relationship between usage of water in a unit and the opportunity for recycle and reuse. The constraint states that for a unit j to transfer water to unit j0 , unit j0 should require water at that time point. It does not, however, mean that unit j0 must use water from unit j, it could still obtain water from other sources. Constraints (52) and (53) state that the time at which water recycle/reuse takes place coincides with the time at which the water is produced. Constraints (54) and (55) ensure that the time at which water recycle/reuse takes place coincides with the starting time of the unit receiving the water. ywr sout;j;j0 ; p yw sin;j0 ; p cj; j0 ˛ J; sin;j ˛ Sin;j ; sout;j;j0 ˛ Sout;j;j0 ; p ˛ P (51) twr sout;j;j0 ; p twout sout;j ; p MM 1 ywr sout;j;j0 ; p cj; j0 ˛ J; sout;j ˛ Sout;j ; sout;j;j0 ; p ˛ P twr sout;j;j0 ; p twout sout;j ; p þ MM 1 ywr sout;j;j0 ; p
cj; j0 ˛ J; sout;j ˛ Sout;j ; sout;j;j0 ˛ Sout;j;j0 ; p ˛ P
(52) (53)
twr sout;j;j0 ; p twin sin;j ; p MM 1 ywr sout;j;j0 ; p cj; j0 ˛ J; sin;j ˛ Sin;j ; sout;j ˛ Sout;j ; p ˛ P
(54)
twr sout;j;j0 ; p twin sin;j ; p þ MM 1 ywr sout;j;j0 ; p cj; j0 ˛ J; sin;j ˛ Sin;j ; sout;j ˛ Sout;j ; p ˛ P
(55)
Central Storage Scheduling Constraint (56) relates water usage in a unit and water transfer from storage. It states that water can only be transferred to a unit if it uses water at the same time point. However it is not a prerequisite for the unit to use stored water, the water could be provided from other sources. Constraints (57) and (58) ensure that the time at which water is sent from storage to a unit coincides with the start time of washing of the same unit. (56) ysout ð j; pÞ yw sin;j ; p cj ˛ J; sin;j ˛ Sin;j ; p ˛ P tsout ð j; pÞ twin sin;j ; p MM 2 ysout ð j; pÞ yw sin;j ; p cj ˛ J; sin;j ˛ Sin;j ; p ˛ P
tsout ð j; pÞ twin sin;j ; p þ MM 2 ysout ð j; pÞ yw sin;j ; p
cj ˛ J; sin;j ˛ Sin;j ; p ˛ P
(57) (58)
Constraint (59) relates water usage in a unit and water transfer to storage. It states that water can only be transferred from a unit to storage if the unit used water at the previous time point. However, washing can take place in the unit without discharging water to the storage tank. The water could be discharged to other sinks. Constraints (60) and (61) ensure that the time at which water is sent to storage from a unit must coincide with the finishing time of washing of the same unit. (59) ysin ð j; pÞ yw sin;j ; p 1 cj ˛ J; sin;j ˛ Sin;j ; p ˛ P; p > p1 tsout ð j; pÞ twout sout;j ; p MM 2 ysout ð j; pÞ yw sin;j ; p 1
cj ˛ J; sin;j ˛ Sin;j ; sout;j ˛ Sout;j ; p ˛ P; p > p1
(60)
tsout ð j; pÞ twout sout;j ; p þ MM 2 ysout ð j; pÞ yw sin;j ; p 1
cj ˛ J; sin;j ˛ Sin;j ; sout;j ˛ Sout;j ; p ˛ P; p > p1
(61)
If water is transferred from storage to a unit at a later time point, the time at which this happens must correspond to a later time in the time horizon. This is specified in Constraint (62). Constraint (63) ensures that if water is transferred from a unit to storage at a later time point, the time at which this happens corresponds to a later time in the time horizon. tsout ð j; pÞ tsout ðj0 ; p0 Þ MMð2 ysout ð j; pÞ ysout ðj0 ; p0 ÞÞ
cj; j0 ˛ J; p ˛ P; p p0
(62)
Energy and Water Conservation in Batch Processes tsin ð j; pÞ tsin ðj0 ; p0 Þ MMð2 ysin ð j; pÞ ysin ðj0 ; p0 ÞÞ
cj; j0 ˛ J; p ˛ P; p p0
409 (63)
Constraints (64) and (65) state that if water is transferred to storage from more than one unit at the same time point, the time at which they do so must coincide. Constraints (66) and (67) state that if water is discharged from storage to more than one unit at the same time point, the time at which the water is discharged must coincide. tsin ð j; pÞ tsin ðj0 ; pÞ MMð2 ysin ð j; pÞ ysin ðj0 ; pÞÞ
cj; j0 ˛ J; p ˛ P
(64)
tsin ð j; pÞ tsin ðj0 ; pÞ þ MMð2 ysin ð j; pÞ ysin ðj0 ; pÞÞ
cj; j0 ˛ J; p ˛ P
(65)
tsout ð j; pÞ tsout ðj0 ; pÞ MMð2 ysout ð j; pÞ ysout ðj0 ; pÞÞ
cj; j0 ˛ J; p ˛ P
(66)
tsout ð j; pÞ tsout ðj0 ; pÞ þ MMð2 ysout ð j; pÞ ysout ðj0 ; pÞÞ
cj; j0 ˛ J; p ˛ P
(67)
If water is simultaneously being transferred to and discharged from storage, the time at which this happens should coincide. This is given in Constraints (68) and (69) tsin ð j; pÞ tsout ðj0 ; pÞ MMð2 ysin ð j; pÞ ysout ðj0 ; pÞÞ
cj; j0 ˛ J; p ˛ P
(68)
tsin ð j; pÞ tsout ðj0 ; pÞ þ MMð2 ysin ð j; pÞ ysout ðj0 ; pÞÞ
cj; j0 ˛ J; p ˛ P
(69)
Constraint (70) ensures that if water leaves storage at a later time point compared to water entering the storage, the time at which water leaves the storage must correspond to a later time in the time horizon. tsout ð j; pÞ tsin ðj0 ; p0 Þ MMð2 ysout ð j; pÞ ysin ðj0 ; p0 ÞÞ
cj; j0 ˛ J; p; p0 ˛ P; p p0
(70)
The following feasibility and time horizon constraints also hold. Constraint (71) ensures that if a processing unit j is reusing water from unit j0 at time point p, then unit j0 cannot reuse water from unit j at the same time point. ywr sout;j;j0 ; p þ ywr sout;j0 ;j ; p 1 cj; j0 ˛ J; sout;j;j0 ; sout;j0 ;j ˛ Sout;j;j0 ; p ˛ P (71) Constraints (72)–(76) ensure that each event occurs within the time horizon of interest. twin sin;j ; p H cj ˛ J; sin;j ˛ Sin;j ; p ˛ P twout sout;j ; p H twr sout;j;j0 ; p H tsin sin;j ; p H tsout sout;j ; p H
(72)
cj ˛ J; sout;j ˛ Sout;j ; p ˛ P
(73)
cj ˛ J; sout;j;j0 ˛ Sout;j;j ; p ˛ P
(74)
cj ˛ J; sin;j ˛ Sin;j ; p ˛ P
(75)
cj ˛ J; sout;j ˛ Sout;j ; p ˛ P
(76)
The objective of the formulation is either the maximization of profit or the minimization of makespan. Constraint (77) expresses the profit as the difference between the product revenue and the sum of freshwater, effluent treatment, cooling water, and steam costs. When the objective is the maximization of profit, Constraint (77) is maximized. As a result, the amount of external utilities as well as freshwater consumption is minimized. XX XX XX XX X Profit ¼ CPðsÞdðs; pÞ CF mwf sin;j ; p CE mwe sout;j ; p Costc w cw sin;jh ; p Costs t s
p
X st sin;jc ; p
sin;j
p
sout;j
p
sin;jh
p
sin;jc
(77)
p
Constraint (78) is the objective function for makespan minimization. In Constraint (78), the quotient of profit and time horizon is maximized. As a result, the makespan is minimized and the amount of external utilities as well as freshwater consumed are also minimized using the same reasoning as above. Max
Profit H
(78)
Solution Procedure The overall model, whether for a profit maximization problem or a makespan minimization problem is an MINLP. When solving a profit maximization problem, the model is linearized and solved as MILP, the solution of which is then used as a starting point for
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the exact MINLP model. If the solutions from the two models are equal, the solution is globally optimal, as global optimality can be proven for MILP problems. If the solutions differ, the MINLP solution is locally optimal. The possibility also exists that no feasible starting point is found. This solution procedure was used by Gouws et al. (2008), Adekola and Majozi (2011), and Stamp and Majozi (2011). When solving a makespan minimization problem, the MINLP cannot be linearized completely to an MILP due to the nonlinear objective function, Constraint (78). However, the MINLP was linearized to a relaxed MINLP problem which provides a starting point for the exact MINLP problem. However, the resulting solution to the exact MINLP cannot be guaranteed to be a global optimum. Constraints (6)–(8) have trilinear terms where a binary variable and two continuous variables are multiplied. Constraints (30)– (32), (42), and (43) have bilinear terms where two continuous variables are multiplied. This results in a nonconvex MINLP formulation. The bilinearity resulting from the multiplication of a continuous variable with a binary variable may be handled effectively with the Glover transformation. This is an exact linearization technique and as such will not compromise the accuracy of the model. For the profit maximization problem, the linearization procedure is carried out for each of the nonlinear terms in Constraints (6)–(8), (30)–(32), (42), and (43) to produce a MILP problem, which provides a starting point for the exact MINLP problem. The objective function for the makespan minimization problem was also nonlinear, however, only Constraints (6)–(8), (30)–(32), (42), and (43) were linearized to produce a relaxed MINLP problem, the solution of which provides a starting point for the exact MINLP problem.
Case Study I This case study is a simple sequential process. A single product is produced via three tasks in five units. The case study has been adapted to include water and energy using operations. Fig. 2 shows the recipe representation of the process. Steam is available for heating, cooling water is available for cooling, and freshwater is available for washing. The objective is to maximize profit over a given time horizon, while minimizing freshwater usage and utility requirements. Data required for case study I may be obtained from Tables 1–3. The following should be noted: 1. 2. 3. 4.
The processing duration of a batch is fixed. The mass load of contaminants is dependent on the batch size. A single contaminant is present. The energy requirements of the operations are a function of the batch size and the given initial and final temperatures. Due to this fact, careful consideration must be given to ensure that whenever heat integration occurs between two operations, the minimum temperature difference for heat transfer, 10 C was not violated.
The first case study was solved with the proposed formulation, using Constraints (1), (2), (17)–(22), (27)–(30), and (32)–(37). The objective function was the maximization of Constraint (77). It is important to note that in Constraints (27), (28), (30), and (32), variables associated with wastewater storage are not included. Due to the fact that a single contaminant was present, the formulation for this case study could be reduced to a MILP. The outlet contaminant concentration was fixed to the maximum, Constraint (30) was substituted into Constraint (32) and a Glover transformation was performed on the resulting equation. The computer used to solve the model had an Intel(R) Core(TM) i7-2670QM, 2.2 GHz processor with 4.0 GB RAM. The problem was solved with GAMS using CPLEX as the MIP solver. The results from the first case study as compared to the results achieved by Halim and Srinivasan (2011) may be obtained from Table 4. As can be observed from Table 4, a better profit overall was obtained with the proposed formulation. Although the amounts for steam and cooling water were lower for the model by Halim and Srinivasan (2011), the amount of freshwater required was lower in the proposed formulation, which also results in a lower effluent production. Similar to the method by Halim and Srinivasan (2011), the washing of a unit may not necessarily occur immediately after the end of a processing task, but can be delayed to improve reuse opportunities with other units. To cater for this, Constraint (49) was modified to Constraint (49*) as follows. (49*) twout sout;j ; p ¼ twin sin;j ; p 1 þ sw sin;j yw sin;j ; p 1 cj ˛ J; sin;j ˛ Sin;j ; sout;j ˛ Sout;j ; p ˛ P; p > p0
Fig. 2
Recipe representation of the simple sequential process for case study I.
Energy and Water Conservation in Batch Processes
Table 1
Data pertaining to production, for case study I
Task ( i)
Unit ( j)
Max batch size (kg)
Task 1
Unit 1 Unit 2 Unit 3 Unit 4 Unit 5
100 150 200 100 150
Task 2 Task 3
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Table 2
Table 3
Processing time (h)
Washing time (h)
Material state (m)
1.25 1.70 1.50 0.75 1.20
0.25 0.30 0 0.25 0.30
A B C D Wash water Wastewater Cooling water Steam
Initial inventory (kg)
Max storage (kg)
Revenue/cost ($/kg or $/MJ)
1000 0 0 0
1000 200 250 1000
0 0 0 5 0.1 0.05 0.02 1
Data pertaining to energy requirements, for case study I
Task ( i)
Tin ( C)
Tout ( C)
Unit ( j)
Cp (kJ/kg C)
Task 1
140
60
Task 2 Task 3
60 40
40 80
Unit 1 Unit 2 Unit 3 Unit 4 Unit 5
4.0 4.0 3.5 3.0 3.0
Cooling water Steam
20 170
30 160
Data pertaining to water requirements, for case study I
Task ( i)
Unit ( j)
Max inlet concentration (ppm)
Max outlet concentration (ppm)
Contaminant loading (g contaminant/kg batch)
Task 1
Unit 1 Unit 2 Unit 3 Unit 4 Unit 5
500 50 – 150 300
1000 100 – 300 2000
0.2 0.2 0.2 0.2 0.2
Task 2 Task 3
Table 4
Comparison of results for case study I
Profit ($) Steam (MJ) Cooling water (MJ) Total freshwater (kg) Revenue from product ($)t Cost of steam ($) Cost of cooling water ($) Cost of freshwater ($) Cost of wastewater ($) Number of slots Number of time points Number of iterations CPU time (s)
Halim and Srinivasan (2011)
This formulation
4764.1 43.9 313.9 1238.37 5000 43.9 6.28 123.8 61.9 7 N/A 1500 Not reported
4775.28 66 336 1013.33 5000 66 6.72 101.3 50.7 N/A 15 N/A 28,797
412
Fig. 3
Energy and Water Conservation in Batch Processes
Resulting production schedule for case study I.
Fig. 3 shows the Gantt chart with the resulting production schedule for the first case study. The clear horizontal blocks represent the tasks associated with production processes. The amount of material processed in each unit is labeled within the clear blocks. The dark horizontal blocks represent the washing of a unit. The amount of water associated with washing is labeled underneath the relevant dark blocks. The double-sided arrows (up-down arrows) represent direct heat integration between two tasks. The numbers in round brackets represent the amount of energy associated with heat integration. The numbers in curly brackets represent steam duties while the numbers in square brackets represent cooling water duties.
Case Study II This case study comprises of the popular literature example by Kondili et al. (1993). Halim and Srinivasan (2011) adapted this complex production process to include energy and water using aspects, and it is described here. For the production process, two products, Product 1 and Product 2 are to be produced from three raw materials FeedA, FeedB, and FeedC. A heater, HR is used to heat FeedA. Two reactors, RR1 and RR2 are available to perform three different chemical reactions, Reaction 1, Reaction 2, and Reaction 3. Finally, a separator exists to purify Impure E. Fig. 4 shows the STN representation of the problem. Steam is available for heating, cooling water is available for cooling, and freshwater is available for washing. The production recipe is as follows: 1. Heating: FeedA is heated from 50 C to 70 C inside HR. Steam is required as a heating medium for this reaction. 2. Reaction 1: A mixture of 50% FeedB and 50% FeedC, on a mass basis, is reacted in either unit RR1 or RR2. The product of this reaction is IntBC. The reaction requires cooling from 100 C to 70 C. 3. Reaction 2: A mixture of 40% Hot A and 60% IntBC is reacted to form Prod1 (40%) and IntAB (60%). This reaction can be performed in either unit RR1 or RR2 and requires heating from 70 C to 100 C. 4. Reaction 3: A mixture of 20% FeedC and 80% IntAB is reacted in either units RR1 and RR2. The reaction produces ImpureE and requires heating from 100 C to 130 C. 5. Separation: In SR, ImpureE is purified to produce Prod2 (90%) and intermediate IntAB (10%). The separation requires cooling from 130 C to 100 C. Water is required for washing RR1 and RR2 at the end of any reaction. In this case study, four contaminants, ar, br, cp, and dw are present. Table 5 shows information regarding the production process. Data pertaining to heating and cooling are given in Table 6, while data pertaining to the washing of RR1 and RR2 are given in Table 7. The objective in this case study was to determine the minimum makespan required to produce 200 kg each of Prod1 and Prod2, while also minimizing external utility and freshwater consumption. It is important to note the following from the data in Tables 5–7: 1. The processing duration of a batch is dependent on the batch size. 2. The mass load of contaminants is not fixed, but dependent on the batch size. 3. The energy requirements of the operations are a function of the batch size and the given initial and final temperatures. Due to this fact, careful consideration must be given to ensure that whenever heat integration occurs between two operations, the minimum temperature difference for heat transfer, 10 C, is not violated.
Energy and Water Conservation in Batch Processes
Fig. 4
Table 5
413
STN representation of the complex production facility for case study I.
Data pertaining to production, for case study II
Task ( i)
Unit ( j)
Max batch size (kg)
aij (h)
bij (h)
Washing time (h)
Heating (H) Reaction 1 (R1)
HR RR1 RR2 RR1 RR2 RR1 RR2 SR
100 50 80 50 80 50 80 200
0.667 1.084 1.034 1.09 1.034 0.417 0.367 1.334
0.007 0.027 0.017 0.027 0.017 0.013 0.008 0.007
0 0.25 0.3 0.25 0.3 0.25 0.3 0
Reaction 2 (R2) Reaction 3 (R3) Separation (S)
Table 6
Material state FeedA (s1) FeedB (s2) FeedC (s3/s4) HotA (s5) IntAB (s8) IntBC (s6) ImpureE (s9) Prod1 (s7) Prod2 (s10) Wash water Wastewater Cooling water Steam
Initial inventory (kg)
Max storage (kg)
Revenue/cost ($/kg or $/MJ)
1000 1000 1000 0 0 0 0 0 0
1000 1000 1000 100 200 150 200 1000 1000
10 10 10 0 0 0 0 20 20 0.1 0.05 0.02 1
Data pertaining to energy requirements, for case study II
Task (i)
Tin ( C)
Tout ( C)
Unit ( j)
Cp (kJ/kg C)
Heating (H) Reaction 1 (R1)
50 100
70 70
Reaction 2 (R2)
70
100
Reaction 3 (R3)
100
130
Separation (S) Cooling water Steam
130 20 170
100 30 160
HR RR1 RR2 RR1 RR2 RR1 RR2 SR
2.5 3.5 3.5 3.2 3.2 2.6 2.6 2.8
The second case study was solved using the proposed formulation, using Constraints (1), (2), (17)–(22), (27)–(30), (32)–(37), and (39). The objective function was the minimization of makespan, Constraint (78). It is important to note that in Constraints (27), (28), (30), and (32), variables associated with wastewater storage are not included. The computer used to solve the model had an Intel(R) Core(TM) i7-2670QM, 2.2 GHz processor with 4.0 GB RAM. The problem was solved with GAMS using DICOPT for the MINLP with CPLEX as the MIP solver and MINOS as the NLP solver.
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Energy and Water Conservation in Batch Processes
Data pertaining to water requirements, for case study II
Table 7
Max inlet concentration (ppm) Task ( i) Reaction 1 (R1) Reaction 2 (R2) Reaction 3 (R3)
Fig. 5
Max outlet concentration (ppm)
Unit ( j)
ar
br
cp
dw
ar
br
cp
dw
Contaminant loading (g contaminant/kg batch)
RR1 RR2 RR1 RR2 RR1 RR2
300 300 700 700 500 500
500 500 600 600 200 200
800 800 300 300 400 400
400 400 400 400 300 300
700 700 1200 1200 800 800
800 800 1000 1000 500 500
1200 1200 600 600 700 700
900 900 800 800 900 900
0.2 0.2 0.2 0.2 0.2 0.2
Resulting production schedule for case study II.
Fig. 5 shows the Gantt chart with the resulting production schedule for the case study. The clear horizontal blocks represent the tasks associated with production processes. The amount of material processed in each unit is labeled within the clear blocks. The dark horizontal blocks represent the washing of a unit. The amount of water associated with washing is labeled underneath the relevant dark blocks. The double-sided arrows (up-down arrows) represent direct heat integration between two tasks. The numbers in round brackets represent the amount of energy associated with heat integration. The numbers in curly brackets represent steam duties while the numbers in square brackets represent cooling water duties. A comparison between the results of this case study obtained by Halim and Srinivasan (2011) and the proposed formulation is described in Table 8. As can be observed from Table 8, improved objectives were obtained using the proposed formulation with the exception of the amount of freshwater used for washing. This was probably due to the fact that the cost associated with using steam was more significant than the cost associated with using freshwater. Hence, to minimize overall costs, promoting opportunities for heat integration took precedence over promoting opportunities for water reuse. This can be observed from the production schedule in Fig. 5. With the operations aligned to promote heat integration, no opportunities for water reuse exist at all. The MINLP model was solved by the aforementioned initialization procedure. The objective value from the relaxed MINLP model was 123.543 $/h and the objective value from the exact MINLP model was 123.369 $/h. As both the relaxed model and exact models were MINLP, due to the nonlinear objective function, the global optimality of the solution could not be guaranteed.
Case Study III This case study was obtained from Majozi and Gouws (2009) and was extended to include heat integration opportunities. The multipurpose batch facility investigated, is similar to that discussed in Case Study II. The heating and separating tasks performed in HR and SR respectively are not to be heat integrated with any other units. Heat integration can only occur between RR1 and
Energy and Water Conservation in Batch Processes
Table 8
415
Comparison of results for case study II
Makespan (h) Steam (MJ) Cooling water (MJ) Total freshwater (kg) Revenue from products ($) Cost of raw materials ($) Cost of steam ($) Cost of cooling water ($) Cost of freshwater ($) Cost of wastewater ($) Profit ($) Number of slots Number of time points Number of iterations CPU time (s) Major iterations
Halim and Srinivasan (2011)
This formulation
19.96 61.36 35.39 275.09 8000 5604.4 61.36 0.7078 27.509 13.75 2292.3 8 N/A 3500 not reported –
19.93 44.88 19.72 341.2 8000 5444.4 44.88 0.3994 34.12 17.06 2459.1 N/A 17 N/A 24532 9
RR2 depending on the tasks they perform. Similarly, water is required for washing RR1 and RR2 at the end of any reaction. In this case study, three contaminants, C1, C2, and C3 are present. Table 9 shows information regarding the production process, while data pertaining to the washing of RR1 and RR2, obtained from Majozi and Gouws (2009) are given in Table 10. The heat integration data are provided in Table 11. In this case study, storage facilities for heat storage and water reuse are available. Different scenarios of the case study were solved to demonstrate the capabilities of the model. These scenarios are as follows: l
Scenario 1: During production, only freshwater is available for washing. Heating and cooling are provided exclusively by steam and cooling water. l Scenario 2: In addition to freshwater for washing, steam, and cooling water for heating and cooling respectively, opportunities for direct water reuse and direct heat integration are explored. l Scenario 3: The effect of the inclusion of storage facilities for heat and wastewater (indirect water reuse and indirect heat integration) to Scenario 2 is explored. Constraints (1)–(16), (21)–(26), (27)–(31), and (34)–(77) were used to solve this case study. The bilinear terms present in the model were linearized and the solution procedure as described previously was used. The capacity of the storage vessel for water was 200 kg. The objective of this case study was to maximize profit while minimizing wastewater production and energy consumption, within a time horizon of 12 h. The storage capacity and the initial storage temperature of the storage vessel were variables to be optimized. Heat losses were not considered. The computer used to solve the model had an Intel(R) Core(TM) i7-2670QM, 2.2 GHz processor with 4.0 GB RAM. The problem was solved with GAMS using DICOPT for the MINLP with CPLEX as the MIP solver and CONOPT as the NLP solver. A
Table 9
Production data for case study III
Task ( i)
Unit ( j)
Max batch size (kg)
Mean processing time (h)
Washing time (h)
Heating (H) Reaction 1 (R1)
HR RR1 RR2 RR1 RR2 RR1 RR2 SR
100 50 80 50 80 50 80 200
1 2 2 2 2 1 1 1 for Prod2, 2 for IntAB
0 0.25 0.3 0.5 0.25 0.25 0.25 0
Reaction 2 (R2) Reaction 3 (R3) Separation (S)
Material state FeedA (s1) FeedB (s2) FeedC (s3/s4) HotA (s5) IntAB (s8) IntBC (s6) ImpureE (s9) Prod1 (s7) Prod2 (s10) Wash water Wastewater Cooling water Steam
Initial inventory (kg)
Max storage (kg)
Revenue/cost (cost units/kg or cost units/kJ)
Unlimited Unlimited Unlimited 0 0 0 0 0 0
Unlimited Unlimited Unlimited 0 0 0 0 0 0
0 0 0 0 0 0 0 100 100 2 3 2 10
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Energy and Water Conservation in Batch Processes
Table 10
Wastewater minimization data for case study III Maximum contaminant concentration (g contaminant/kg water) C1
Reaction 1 (RR1) Reaction 2 (RR1) Reaction 3 (RR1) Reaction 1 (RR2) Reaction 2 (RR2) Reaction 3 (RR2)
Reaction 1
Reaction 3
C3
0.5 0.5 2.3 1 0.9 3 0.01 0.05 0.3 0.2 0.1 1.2 0.15 0.2 0.35 0.3 1 1.2 0.05 0.2 0.05 0.1 1 12 0.03 0.1 0.2 0.075 0.2 1 0.3 0.6 1.5 2 1.5 2.5 Contaminant mass load (g) C1 C2 C3 4 80 10 15 24 358 28.5 7.5 135 9 2 16 15 80 85 22.5 45 36.5
Reactor 1 Reactor 2 Reactor 1 Reactor 2 Reactor 1 Reactor 2
Reaction 2
Table 11
Max. inlet Max. outlet Max. inlet Max. outlet Max. inlet Max. outlet Max. inlet Max. outlet Max. inlet Max. outlet Max. inlet Max. outlet
C2
Heat integration data for case study III.
Reaction
Type
Heating/cooling requirement (kWh)
Operating temperature ( C)
RX1 RX2 RX3
Exothermic Endothermic Exothermic
60 (cooling) 80 (heating) 70 (cooling)
100 60 140
comparison between the three scenarios is provided in Table 12. The results of Scenario 1, Scenario 2, and Scenario 3 are contained in columns 2, 3, and 4 respectively, of Table 12. From Table 12 it can be observed that profit increases from Scenario 1 to Scenario 3, with a decrease in steam and cooling water requirements. However, while cooling water and steam decreased in Scenario 2 compared to Scenario 1, the amount of freshwater increased. The total amount of product in Scenario 1 is 258 kg, while the total amount of product in Scenario 2 is 272.4 kg. This is as a result of additional unit operations being performed in Scenario 2 than are performed in Scenario 1. These
Table 12
Comparison between different scenarios for case study III
Profit (cu) Amount of Prod1 (kg) Amount of Prod2 (kg) Cooling water (kWh) Steam (kWh) Freshwater (kg) Time points CPU time (s) Binary variables Initial storage temperature ( C) Heat storage capacity (ton) Major iterations
Freshwater and utilities
Direct water reuse and direct heat integration
Direct/indirect water reuse and direct/indirect heat integration
18,537 96 162 390 240 816 11 3.1 128
19,836 116 156.4 250 180 1020 11 14.8 508
3
3
22,235 116 188 190 10 896 13 285.2 954 82.9 2.024 4
Energy and Water Conservation in Batch Processes
Fig. 6
417
Resulting process schedule when only direct heat integration and direct water reuse are possible.
additional unit operations contribute to the increased amount of washing water required. Furthermore, no opportunities for direct water reuse were realized due to the cost of steam relative to the cost of freshwater. The unit operations are aligned in such a way as to promote as much heat integration as possible. In so doing, opportunities for direct water reuse are lost. In Scenario 3, storage for water is available and hence a decrease is observed in the amount of freshwater used. The resulting process schedule for the results of Scenario 2 is illustrated in Fig. 6, while the corresponding schedule for Scenario 3 is illustrated in Fig. 7. The clear blocks represent the production operation in the unit while the dark blocks represent the washing operations which take place after the reactions are completed. The numbers above the clear blocks represent the amount of material processed in the unit during production and the numbers below the washing operations represent freshwater. Water transfer to and from storage has been clearly labeled. The up-down arrows represent direct heat integration, while the bent arrows represent indirect heat integration to or from the heat storage unit. The results of Scenario 1 are globally optimal with the objective function of the linearized MILP and exact MINLP being 18,537 c.u. Similarly, the results of Scenario 2 are globally optimal, with the objective function of the linearized MILP and exact MINLP being 19,836 c.u. In Scenario 3, the objective function of the linearized MILP was 25,270 c.u, while the objective function of the exact MINLP was 22,235 c.u. Hence, the results for Scenario 3 are locally optimal.
Fig. 7
Resulting process schedule when both direct/indirect heat integration and wastewater minimization are possible.
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Energy and Water Conservation in Batch Processes
Conclusions A simultaneous method for the optimization of energy and water embedded within a scheduling framework has been developed. Furthermore, opportunities for direct and indirect heat integration as well as direct and indirect water reuse have been explored. The mathematical formulation led to an MINLP problem for which an initialization procedure was employed. The applicability of the method has been demonstrated with three case studies. The developed formulation has proved to effectively solve a complex makespan minimization problem in which duration is a function of batch size and which included multiple contaminants.
See also: Process Integration for Sustainable Industries.
References Adekola, O., Majozi, T., 2011. Wastewater minimization in multipurpose batch plants with a regeneration unit: Multiple contaminants. Computers & Chemical Engineering 35 (12), 2824–2836. Gouws, J.F., Majozi, T., Gadalla, M., 2008. Flexible mass transfer model for water minimization in batch plants. Chemical Engineering and Processing 47, 2323–2335. Halim, I., Srinivasan, R., 2011. Sequential methodology for integrated optimization of energy and water use during batch process scheduling. Computers & Chemical Engineering 35, 1575–1597. Kondili, E., Pantelides, C.C., Sargent, R.W.H., 1993. A general algorithm for short-term scheduling of batch operations – I MILP formulation. Computers & Chemical Engineering 17 (2), 211–227. Majozi, T., Gouws, J.F., 2009. A mathematical optimisation approach for wastewater minimisation in multipurpose batch plants: Multiple contaminants. Computers & Chemical Engineering 33, 1826–1840. Stamp, J.D., Majozi, T., 2011. Optimum heat storage design for heat integrated multipurpose batch plants. Energy 36, 5119–5131.
Problem Statement The problem addressed in this article can be stated as follows: Given: Scheduling data i. Production recipe for each product ii. Available units and their capacities iii. Task durations iv. Time horizon of interest or predetermined production v. Costs of raw materials vi. Selling price of final products Heat integration data (i) Hot duties for tasks requiring heating and cold duties for tasks that require cooling (ii) Operating temperatures of heat sources and heat sinks (iii) Minimum allowable temperature differences (iv) Costs of hot and cold utilities (v) Design limits on heat storage Wastewater minimization data (i) Mass load of contaminants (ii) Maximum inlet and outlet concentrations of contaminants (iii) Washing time for each unit (iv) Maximum storage available for reuse (v) Cost of freshwater (vi) Cost of effluent treatment Determine an optimal production schedule that achieves a maximum profit or minimum makespan, requiring the minimum amount of external utilities and freshwater use.
Mathematical Formulation The mathematical formulation consists of three modules, that is, production scheduling, heat integration, and wastewater minimization modules. For production scheduling, the objective is either profit maximization or minimization of makespan. During production, certain tasks require cooling or heating, for instance, exothermic reactions that require cooling or endothermic reactions that require heating. The requirements for heating or cooling afford opportunities for heat integration. In multipurpose batch plants, particularly at the completion of a processing task in a unit, the equipment unit is washed before performing subsequent tasks. This is to ensure product integrity through prevention of contamination. The washing operations present an opportunity for wastewater minimization. Sets C {c|c ¼ contaminant} J {j|j ¼ processing unit} Jc {jc|jc ¼ processing unit which may conduct tasks requiring heating} 4J Jh {jh|jh ¼ processing unit which may conduct tasks requiring cooling} 4J P {p|p ¼ time point} S {s|s ¼ any state} Sin,j {sin,j|sin,j ¼ input state to a processing unit} 4S Sin {Sin|Sin ¼ input state into any unit} Sout {sout|s out ¼ output state from any unit} Sin;j {sin;j sin;j ¼ effective state into processing unit} 4Sin Sout,j {Sout, j|Sout,j ¼ output stream from a processing unit} 4S U {u|u ¼ heat storage unit}
Encyclopedia of Sustainable Technologies, Volume 4
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Continuous variables B(sin,j) Batch size, either fixed or variable cin(sin,j,c,p) Inlet concentration of contaminant c, to unit j at time point p cout(sout,j,c,p) Outlet concentration of contaminant c, from unit j at time point p CL sin;jh Cooling load for hot state csin(c, p) Inlet concentration of contaminant c, to storage at time point p csout(c, p) Outlet concentration of contaminant c, from storage at time point p cw sin;jh ; p External cooling required by unit jh conducting the task corresponding to state sin;jh at time point p d(s, p) Amount of state delivered to customers at time point p dur(sin,j, p) Duration of task, either fixed or dependent on batch size H Time horizon of interest, optimization variable for makespan minimization problem HL sin;jc Heating load for cold state MB(sin,j, c, p) Mass load of contaminant c in unit j at time point p after processing sin,j that is added to the water stream mwin(sin,j, p) Mass of water into unit j for cleaning state sout at time point p mwout(sout,j, p) Mass of water produced at time point p from unit j mwf (sin,j, p) Mass of freshwater into unit j at time point p mwe(sout,j, p) Mass of effluent water from unit j at time point p mwr sout;j;j0 ; p Mass of water recycled to unit j0 from j at time point p msin(j, p) Mass of water transferred from unit j to storage at time point p msout(j, p) Mass of water transferred from storage to unit j at time point p Q(sin,j, u, p) Heat exchanged with heat storage unit u at time point p q sin;jh ; sin;jc ; p Amount of heat exchanged during direct heat integration qws(p) Amount of water stored in storage at time point p st sin;jc ; p External heating required by unit jc conducting the task corresponding to state sin;jc at time point p t0(u, p) Initial temperature in heat storage unit u at time point p tf (u, p) Final temperature in heat storage unit u at time point p t0(sin,j, u, p) Time at which heat storage unit commences activity tf (sin,j, u, p) Time at which heat storage unit ends activity tu(sin,j, p) Time at which a state is used in unit j tout(sout,j, p) Time at which a state is produced from unit j at time point p twin(sin,j, p) Time at which water is used at time point p in unit j twout(sout,j, p) Time at which water is produced at time point p from unit j twr sout;j;j0 ; p Time at which water is recycled from unit j to unit j0 at time point p tsin( j, p) Time at which water is transferred from unit j to storage at time point p tsout( j, p) Time at which water is transferred from storage to unit j at time point p W(u) Capacity of heat storage unit u G(sin,j, u, p) Glover transformation variable J(sin,j, u, p) Reformulation-Linearization variable Binary variables x sin;jc ; sin;jh ; p Binary variable associated with heat integration between unit jc conducting the task corresponding to state sin;jc and unit jh conducting the task corresponding to state sin;jh , at time point p y(s * in,j, p) Binary variable associated with usage of state s in unit j at time point p ywr sout;j;j0 ; p Binary variable showing usage of recycle from unit j to unit j0 at time point p yw(sin,j, p) Binary variable showing usage of water in unit j at time point p ysin(j, p) Binary variable showing transfer of water from unit j to storage at time point p ysout(j, p) Binary variable showing transfer of water from storage to unit j at time point p z(sin,j, u, p) Binary variable associated with heat integration between unit j conducting the task corresponding to state sin,j with heat storage unit u at time point p Parameters A(c) Contaminant loading (g contaminant/kg batch) a Constant coefficient of processing time b Variable coefficient of processing time cp Specific heat capacity of heat storage fluid CPstate(sin,j) Specific heat capacity of state CU in(sin,j, c) Maximum inlet concentration of contaminant c in unit j
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CU out(sout,j, c) Maximum outlet concentration of contaminant c from unit j CE Cost of effluent water treatment (c.u./kg water) CF Cost of freshwater (c.u./kg water) Cost_cw Cost of cooling water Cost_st Cost of steam CP(s) Selling price of product s, s ¼ product CSoout(c) Initial concentration of contaminant in storage E(sin,j) Fixed amount of heat required by or removed from unit j conducting the task corresponding to state sin,j H Time horizon of interest, for profit maximization problem M(sin,j, c) Mass load of contaminant c in unit j after processing sin,j that is added to the water stream MM Any large number MwU(sin,j) Maximum inlet water mass of unit j Qwos Initial amount of water in storage QwU s Maximum capacity of storage T(sin,j) Operating temperature for processing unit j conducting the task corresponding to state sin,j, for constant temperature processes Tin(sin,j) Inlet temperature for state sin,j Tout(sin,j) Outlet temperature for state sin,j T L Lower bound for heat storage temperature T U Upper bound for heat storage temperature DTmin Minimum allowable thermal driving force s(sin,j) Duration of the task corresponding to state sin,j conducted in unit j sw(sout,j) Duration of washing for unit j W L Lower bound for heat storage capacity W U Upper bound for heat storage capacity The necessary heat integration and wastewater minimization constraints are embedded in the scheduling framework. The heat integration constraints are presented first, followed by the necessary wastewater minimization constraints. The scheduling framework as used in the models by Adekola and Majozi (2011) and Stamp and Majozi (2011), was used in the proposed model. The model uses the state sequence network recipe representation and an uneven discretization of the time horizon. This has proven to result in fewer binary variables compared to models based on other representations. The mathematical model is based on the superstructure in Fig. 1. Fig. 1A is the superstructure for heat integration while Fig. 1B is the superstructure for water optimization.
Heat Integration Constraints The heat integration constraints are based on the superstructure in Fig. 1A. In Fig. 1A, each processing unit may operate using either direct or indirect heat integration. Direct heat integration refers to the use of heat generated from a processing unit to supply a processing unit requiring heat without use of storage. Indirect heat integration refers to the use of heat previously stored in a heat storage vessel to supply a processing unit requiring heat. The heat is stored through a heat transfer fluid medium, for example, water. Processing units may also operate in standalone mode, using only external utilities. This may be required for control reasons or when thermal driving forces or time do not allow for heat integration. If either direct or indirect heat integration is not sufficient to satisfy the required duty, external utilities may make up for any deficit. Constraints (1)–(26) constitute the heat integration model, useful for multipurpose batch processes. The formulation is based on the model by Stamp and Majozi (2011) and includes direct and indirect heat integration. Constraints (1) and (2) are active simultaneously and ensure that one hot unit will be integrated with one cold unit when direct heat integration takes place, in order to simplify operation of the process. Also, if two units are to be heat integrated at a given time point, they must both be active at that time point. However, if a unit is active, it may operate in either integrated or standalone mode. X x sin;jc ; sin;jh ; p y sin;jh ; p ; cp ˛ P; sin;jh ˛ sin;j (1) Sin;jc
X x sin;jc ; sin;jh ; p y sin;jc ; p ;
cp ˛ P; sin;jc ˛ sin;j
(2)
Sin;jh
Constraint (3) ensures that only one hot or cold unit is heat integrated with one heat storage unit at any point in time. This is to simplify and improve operational efficiency in the plant. X X x sin;jc ; u; p þ x sin;jh ; u; p 1; cp ˛ P; u ˛ U (3) Sin;jc
Sin;jh
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Fig. 1
Energy and Water Conservation in Batch Processes
Superstructure for mathematical formulation: (A) when units perform heating/cooling tasks and (B) when units perform washing tasks.
Constraints (4) and (5) ensure that a unit cannot simultaneously undergo direct and indirect heat integration. This condition simplifies the operation of the process. X x sin;jc ; sin;jh ; p þ z sin;jc ; u; p 1; cp ˛ P; sin;jc ˛ Sin;j ; u ˛ U (4) Sin;jh
X x sin;jc ; sin;jh ; p þ z sin;jh ; u; p 1;
cp ˛ P; sin;jh ˛ Sin;j ; u ˛ U
(5)
Sin;jc
Constraints (6) and (7) quantify the amount of heat received from or transferred to the heat storage unit, respectively. There will be no heat received or transferred if the binary variable signifying use of the heat storage vessel, z(sin,j, u, p), is zero. These constraints are active over the entire time horizon, where p is the current time point and p 1 is the previous time point. Q sin;jc ; u; p 1 ¼ W ðuÞcp T0 ðu; p 1Þ Tf ðu; pÞ z sin;jc ; u; p 1 ; cp ˛ P; p > p0; sin;jc ˛ Sin;j ; u ˛ U (6) Q sin;jh ; u; p 1 ¼ W ðuÞcp Tf ðu; pÞ T0 ðu; p 1Þ z sin;jh ; u; p 1 ;
cp ˛ P; p > p0; sin;jh ˛ Sin;j ; u ˛ U
(7)
Constraint (8) quantifies the heat transferred to the heat storage vessel at the beginning of the time horizon. The initial temperature of the heat storage fluid is T0(u, p1). Q sin;jh ; u; p0 ¼ W ðuÞcp Tf ðu; p1Þ T0 ðu; p1Þ z sin;jh ; u; p0 ; csin;jh ˛ Sin;j ; u ˛ U (8) Constraint (9) ensures that the final temperature of the heat storage fluid at any time point becomes the initial temperature of the heat storage fluid at the next time point. This condition will hold regardless of whether or not there was heat integration at the previous time point. T0 ðu; pÞ ¼ Tf ðu; p 1Þ;
cp ˛ P; u ˛ U
(9)
Constraints (10) and (11) ensure that temperature of heat storage does not change if there is no heat integration with the heat storage unit, unless there unit. MM is any large number, thereby resulting in an overall “Big M” is heat loss from the heat storage formulation. If either z sin;jc ; u; p 1 or z sin;jh ; u; p 1 is equal to one, Constraints (10) and (11) will be redundant. However, if these two binary variables are both zero, the initial temperature at the previous time point will be equal to the final temperature at the current time.
Energy and Water Conservation in Batch Processes 1 X X T0 ðu; p 1Þ Tf ðu; pÞ þ MM@ z sin;jc ; u; p 1 þ z sin;jh ; u; p 1 A;
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0
sin;jc
cp ˛ P; p > p0; u ˛ U
(10)
cp ˛ P; p > p0; u ˛ U
(11)
sin;jh
0
1 X X T0 ðu; p 1Þ Tf ðu; pÞ þ MM@ z sin;jc ; u; p 1 þ z sin;jh ; u; p 1 A; sin;jc
sin;jh
Constraint (12) ensures that minimum thermal driving forces are obeyed when there is direct heat integration between a hot and a cold unit. This constraint holds when both hot and cold units operate at constant temperature, which is commonly encountered in practice. An example is when there is heat integration between an exothermic and an endothermic reaction. T sin;jh T sin;jc DT min MM 1 x sin;jc ; sin;jh ; p 1 ; cp ˛ P; p > p0; sin;jc ; sin;jh ˛ Sin;j (12) Constraints (13) and (14) ensure that minimum thermal driving forces are obeyed when there is heat integration with the heat storage unit. Constraint (13) applies for heat integration between heat storage and a heat sink, while Constraint (14) applies for heat integration between heat storage and a heat source. In Constraints (13) and (14), the units operate at fixed temperatures. Tf ðu; pÞ T sin;jc DT min MM 1 z sin;jc ; u; p 1 ; cp ˛ P; p > p0; sin;jh ˛ Sin;j ; u ˛ U (13) T sin;jh Tf ðu; pÞ DT min MM 1 z sin;jh ; u; p 1 ;
cp ˛ P; p > p0; sin;jh ˛ Sin;j ; u ˛ U
(14)
Constraint (15) states that the cooling of a fixed heat source will be satisfied by either direct or indirect heat integration as well as external utility if required. X min E sin;jc ; E sin;jh x sin;jc ; sin;jh ; p ; cp ˛ P; sin;jh ˛ Sin;j ; u ˛ U (15) E sin;jh y sin;jh ; p ¼ Q sin;jh ; u; p þ cw sin;jh ; p þ sin;jc
sin;jc ;sin;jh
Constraint (16) ensures that the heating of a fixed heat sink will be satisfied by either direct or indirect heat integration as well as external utility if required. X E sin;jc y sin;jc ; p ¼ Q sin;jc ; u; p þ st sin;jc ; p þ min E sin;jc ; E sin;jh x sin;jc ; sin;jh ; p ; cp ˛ P; sin;jc ˛ sin;j ; u ˛ U (16) sin;jh
sin;jc ;sin;jh
Constraints (17) and (18) give the heating load for a cold state and cooling load for a hot state, respectively, for situations where the batch size of material processed in a unit is variable. HL sin;jc ; p ¼ B sin;jc ; p CPstate sin;jc Tout sin;jc Tin sin;jc ; cp ˛ P; sin;jc ˛ Sin;j (17) CL sin;jh ; p ¼ B sin;jh ; p CPstate sin;jh Tin sin;jh Tout sin;jh ;
cp ˛ P; sin;jh ˛ Sin;j
(18)
Constraints (19) and (20) give the heating load for a cold state and cooling load for a hot state, respectively. The heating load and cooling load are made up using heat from other units as well as external utility if required. X HL sin;jc ; p ¼ st sin;jc ; p þ q sin;jc ; sin;jh ; p ; cp ˛ P; sin;jc ˛ Sin;j (19) sin;jh
X q sin;jh ; sin;jc ; p ; CL sin;jh ; p ¼ cw sin;jh ; p þ
cp ˛ P; sin;jh ˛ Sin;j
(20)
sin;jh
Constraints (21) and (22) ensure that the times at which units are active are synchronized when direct heat integration takes place. Starting times for the tasks in the integrated units are the same. This constraint may be relaxed for operations requiring preheating or precooling and is dependent on the process. tu sin;jh ; p tu sin;jc ; p MM 1 x sin;jc ; sin;jh ; p cp ˛ P; sin;jc ; sin;jh ˛ Sin;j (21) tu sin;jh ; p tu sin;jc ; p þ MM 1 x sin;jc ; sin;jh ; p
cp ˛ P; sin;jc ; sin;jh ˛ Sin;j
(22)
Constraints (23) and (24) ensure that if indirect heat integration takes place, the time a unit is active will be equal to the time a heat storage unit starts either to transfer or receive heat. tu sin;j ; p tu sin;j ; p MM y sin;j ; p z sin;j ; u; p cp ˛ P; u ˛ U; sin;j ˛ Sin;j (23) tu sin;j ; p t0 sin;j ; u; p þ MM y sin;j ; p z sin;j ; u; p
cp ˛ P; u ˛ U; sin;j ˛ Sin;j
(24)
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Constraints (25) and (26) state that the time when heat transfer to or from a heat storage unit is finished will coincide with the time the task transferring or receiving heat has finished processing. tu sin;j ; p 1 þ s sin;j y sin;j ; p 1 tf sin;j ; u; p MM y sin;j ; p 1 z sin;j ; u; p 1 cp ˛ P; p > p0; u ˛ U; sin;j ˛ Sin;j (25) cp ˛ P; p > p0; u ˛ U; sin;j ˛ Sin;j tu sin;j ; p 1 þ s sin;j y sin;j ; p 1 tf sin;j ; u; p þ MM y sin;j ; p 1 z sin;j ; u; p 1 (26)
Wastewater Minimization Constraints The wastewater minimization constraints are based on the superstructure in Fig. 1B. Unit j represents a water using operation in which the water used consists of freshwater, stored water, or recycled/reused water. Water from unit j can be recycled into the same unit, reused by other units, or sent to storage. Direct water reuse refers to the use of an outlet wastewater stream from a processing unit in another processing unit while indirect water reuse refers to the use of previously stored wastewater in a processing unit. Constraints (27)–(77) constitute the wastewater minimization model useful for multipurpose batch processes, which involve multiple contaminants. The formulation is based on the model by Adekola and Majozi (2011) and includes both direct water reuse and indirect water reuse due to the presence of a central storage vessel. Mass balances around each processing unit and the central storage vessel are formulated as follows.
Mass Balance Around a Unit Constraint (27) is the water balance over the inlet to a unit. Water entering the unit is a combination of reuse/recycle streams from other units, j0 , freshwater and water from storage. Constraint (28) states that the water leaving a unit could be recycled/reused, sent to storage, or discarded as effluent. Constraint (29) states that the amount of water exiting a unit must equal the amount of water entering the unit at the previous time point. This constraint captures the fact that water is neither produced nor lost in the unit during the washing operation. X mwin sin;j ; p ¼ mwr sout;j0;j ; p þ mwf sin;j ; p þ msout ð j; pÞ cj; j0 ˛ J; sin;j ˛ Sin;j ; sout;j0 ;j ˛ Sout;j0;j ; p ˛ P (27) sout;j0;j
X mwout sout;j ; p ¼ mwr sout;j;j0 ; p þ mwe sout;j ; p þ msin ð j; pÞ
cj; j0 ˛ J; sout;j;j0 ˛ Sout;j;j0 ; p ˛ P
(28)
sout;j;j0
mwin sin;j ; p 1 ¼ mwout sout;j ; p
cj ˛ J; sin;j ˛ Sin;j sout;j ; ˛ Sout;j ; p ˛ P; p > p0
(29)
Constraint (30) represents the inlet contaminant mass balance. The contaminant mass load in the inlet stream is the sum of the contaminant mass load in recycle/reuse water and that in water from storage. Constraint (31) represents the outlet contaminant mass as the mass of contaminant that entered the unit at the previous time point and the mass load of contaminant picked up in the unit during its operation. In Constraint (31), the mass load of contaminant in a unit is a given parameter. X mwin sin;j ; p cin sin;j ; c; p ¼ mwr sout;j0 ;j ; p cout sout;j ; c; p þ msout ð j; pÞcsout ðc; pÞ cj; j0 ˛ J; sin;j ˛ Sin;j ; sout;j0 ;j ˛ Sout;j0 ;j ; p ˛ P; c ˛ C sout;j0 ;j
(30) mwout sout;j ; p cout sout;j ; c; p ¼ M sout;j ; c yw sout;j ; p 1 þ mwin sin;j ; p 1 cin sin;j ; c; p 1 cj ˛ J; sout;j ˛ Sout;j ; p ˛ P; p > p1 ; c ˛ C (31) In the case where mass load of contaminant in a unit is defined as a function of batch size of material processed in the unit, Constraint (32) represents the outlet contaminant mass. Constraint (33) defines the variable contaminant mass load as the product of the contaminant loading and the batch size of material processed. mwout sout;j ; p cout sout;j ; c; p ¼ MB sin;j ; c yw sin;j ; p 1 þ mwin sin;j ; p 1 cin sin;j ; c; p 1 cj ˛ J; sin;j ˛ Sin;j ; sout;j ˛ Sout;j ; p ˛ P; p > p1 ; c ˛ C (32) MB sin;j ; c; p ¼ AðcÞB sin;j ; p
cj ˛ J; sin;j ˛ Sin;j ; p ˛ P; c ˛ C
(33)
Constraints (34) and (35) ensure that the inlet and outlet contaminant concentrations do not exceed the allowed maximum. Similarly, the maximum allowable water in a unit must not be exceeded. This is governed by Constraint (36). Constraint (37)
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restricts the mass of water entering the unit from recycle/reuse, to the maximum allowable for the unit. Likewise, Constraint (38) restricts the mass of water entering the unit from storage, to the maximum allowable for the unit. cin sin;j ; c; p CU cj ˛ J; sin;j ˛ Sin;j ; p ˛ P; c ˛ C (34) in sin;j ; c yw sin;j ; p cout sout;j ; c; p CU out sout;j ; c yw sin;j ; p 1
cj ˛ J; sin;j ˛ Sin;j ; sout;j ˛ Sout;j ; p ˛ P; p > p1 ; c ˛ C
mwin sin;j ; p MW U sin;j yw sin;j ; p
cj ˛ J; sin;j ˛ Sin;j ; p ˛ P
(35) (36)
mwr sout;j0 ;j ; p MwU sin;j ywr sout;j0 ;j ; p cj; j0 ˛ J; sin;j ˛ Sin;j ; sout;j0 ;j ˛ Sout;j0 ;j ; p ˛ P
(37)
msout sout;j ; p MwU sin;j ysout sout;j ; p cj ˛ J; sin;j ˛ Sin;j ; sout;j ˛ Sout;j ; p ˛ P
(38)
The maximum quantity of water into a unit is represented by Constraint (39). It is important to note that for multicontaminant wastewater the outlet concentration of the individual components cannot all be set to the maximum, since the contaminants are not limiting simultaneously. The limiting contaminant(s) will always be at the maximum outlet concentration and the nonlimiting contaminants will be below their respective maximum outlet concentrations. ( ) M sin;j ; c (39) MwU sin;j ¼ max U cj ˛ J; sin;j ˛ Sin;j ; sout;j ˛ Sout;j c ˛ C c˛C Cout sout;j ; c CU in sin;j ; c
Mass Balance Around Central Storage Constraint (40) is the water mass balance around the storage tank. The amount of water stored in the storage tank consists of water stored at the previous time point and the difference between water entering the storage tank from a processing unit and water leaving the storage tank to a processing unit. Constraint (41) defines the initial amount of water in the tank. X X msin ð j; pÞ msout ð j; pÞ cj ˛ J; p ˛ P; p > p0 (40) qws ðpÞ ¼ qws ðp 1Þ þ j
j
qws ðp0Þ ¼ Qwos
X
msout ð j; p0Þ
cj ˛ J
(41)
j
The definition of the inlet contaminant concentration to the storage tank is given in Constraint (42). The concentration of water exiting the storage tank is assumed to be equal to the concentration of water in the tank as given in Constraint (43). This condition is true in the case of perfect mixing. The initial concentration in the storage tank is given in Constraint (44). Constraint (45) ensures that the maximum capacity of the tank is not exceeded. P j msin ð j; pÞcout sin;j ; c; p P cj ˛ J; sin;j ˛ Sin;j ; p ˛ P; c ˛ C csin ðc; pÞ ¼ (42) j msin ð j; pÞ csout ðc; pÞ ¼
P qws ðp 1Þcsout ðc; p 1Þ þ j msin ð j; pÞcsin ðc; pÞ P qws ðp 1Þ þ j msin ð j; pÞ
cj ˛ J; p ˛ P; p p0; c ˛ C
csout ðc; p1 Þ ¼ CSoout ðcÞ cc ˛ C qws ðpÞ QwU s
cp ˛ P
(43) (44) (45)
Constraint (46) ensures that no water is stored in the storage vessel at the end of the time horizon in order to give a true optimum. Otherwise the resulting minimum effluent could be misleading. qws ðpÞ ¼ 0 cp ¼ jP j (46) The scheduling constraints for the wastewater minimization model are as follows.
Task Scheduling Constraints Task scheduling constraints ensure that each water using operation is integrated with production scheduling. Constraints (47) and (48) together ensure that unit j is washed immediately after a task has produced sout,j. If water is used in the unit, yw(sin,j, p) has a value of 1 causing Constraints (47) and (48) to become active and the start time of washing is forced to coincide with the end time of production. Otherwise, when water is not used in the unit, that is, yw(sin,j, p) has a value of zero, the two constraints become relaxed. Constraint (49) represents the duration of the washing operation performed in unit j. Constraint (50) stipulates that the washing operation can only commence at time point p if the task using s * in,j was active at the previous time point.
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Energy and Water Conservation in Batch Processes twin sin;j ; p tout sout;j ; p MM 1 yw sin;j ; p cj ˛ J; sin;j ˛ Sin;j ; sout;j ˛ Sout;j ; p ˛ P
(47)
twin sin;j ; p tout sout;j ; p þ MM 1 yw sin;j ; p cj ˛ J; sin;j ˛ Sin;j ; sout;j ˛ Sout;j ; p ˛ P
(48)
twout sout;j ; p ¼ twin sin;j ; p 1 þ sw sin;j yw sin;j ; p 1 yw sin;j ; p ¼ y sin;j ; p 1
cj ˛ J; sin;j ˛ Sin;j ; sout;j ˛ Sout;j ; p ˛ P; p > p0
cj ˛ J; sin;j ˛ Sin;j ; sin;j ˛ Sin;j ; p ˛ P; p ˛ p0
(49) (50)
Recycle/Reuse Scheduling Wastewater can only be directly recycled/reused if the unit producing wastewater and the unit receiving wastewater finish operating and begin operating at the same time, respectively. Constraint (51) describes the relationship between usage of water in a unit and the opportunity for recycle and reuse. The constraint states that for a unit j to transfer water to unit j0 , unit j0 should require water at that time point. It does not, however, mean that unit j0 must use water from unit j, it could still obtain water from other sources. Constraints (52) and (53) state that the time at which water recycle/reuse takes place coincides with the time at which the water is produced. Constraints (54) and (55) ensure that the time at which water recycle/reuse takes place coincides with the starting time of the unit receiving the water. ywr sout;j;j0 ; p yw sin;j0 ; p cj; j0 ˛ J; sin;j ˛ Sin;j ; sout;j;j0 ˛ Sout;j;j0 ; p ˛ P (51) twr sout;j;j0 ; p twout sout;j ; p MM 1 ywr sout;j;j0 ; p cj; j0 ˛ J; sout;j ˛ Sout;j ; sout;j;j0 ; p ˛ P twr sout;j;j0 ; p twout sout;j ; p þ MM 1 ywr sout;j;j0 ; p
cj; j0 ˛ J; sout;j ˛ Sout;j ; sout;j;j0 ˛ Sout;j;j0 ; p ˛ P
(52) (53)
twr sout;j;j0 ; p twin sin;j ; p MM 1 ywr sout;j;j0 ; p cj; j0 ˛ J; sin;j ˛ Sin;j ; sout;j ˛ Sout;j ; p ˛ P
(54)
twr sout;j;j0 ; p twin sin;j ; p þ MM 1 ywr sout;j;j0 ; p cj; j0 ˛ J; sin;j ˛ Sin;j ; sout;j ˛ Sout;j ; p ˛ P
(55)
Central Storage Scheduling Constraint (56) relates water usage in a unit and water transfer from storage. It states that water can only be transferred to a unit if it uses water at the same time point. However it is not a prerequisite for the unit to use stored water, the water could be provided from other sources. Constraints (57) and (58) ensure that the time at which water is sent from storage to a unit coincides with the start time of washing of the same unit. (56) ysout ð j; pÞ yw sin;j ; p cj ˛ J; sin;j ˛ Sin;j ; p ˛ P tsout ð j; pÞ twin sin;j ; p MM 2 ysout ð j; pÞ yw sin;j ; p cj ˛ J; sin;j ˛ Sin;j ; p ˛ P
tsout ð j; pÞ twin sin;j ; p þ MM 2 ysout ð j; pÞ yw sin;j ; p
cj ˛ J; sin;j ˛ Sin;j ; p ˛ P
(57) (58)
Constraint (59) relates water usage in a unit and water transfer to storage. It states that water can only be transferred from a unit to storage if the unit used water at the previous time point. However, washing can take place in the unit without discharging water to the storage tank. The water could be discharged to other sinks. Constraints (60) and (61) ensure that the time at which water is sent to storage from a unit must coincide with the finishing time of washing of the same unit. (59) ysin ð j; pÞ yw sin;j ; p 1 cj ˛ J; sin;j ˛ Sin;j ; p ˛ P; p > p1 tsout ð j; pÞ twout sout;j ; p MM 2 ysout ð j; pÞ yw sin;j ; p 1
cj ˛ J; sin;j ˛ Sin;j ; sout;j ˛ Sout;j ; p ˛ P; p > p1
(60)
tsout ð j; pÞ twout sout;j ; p þ MM 2 ysout ð j; pÞ yw sin;j ; p 1
cj ˛ J; sin;j ˛ Sin;j ; sout;j ˛ Sout;j ; p ˛ P; p > p1
(61)
If water is transferred from storage to a unit at a later time point, the time at which this happens must correspond to a later time in the time horizon. This is specified in Constraint (62). Constraint (63) ensures that if water is transferred from a unit to storage at a later time point, the time at which this happens corresponds to a later time in the time horizon. tsout ð j; pÞ tsout ðj0 ; p0 Þ MMð2 ysout ð j; pÞ ysout ðj0 ; p0 ÞÞ
cj; j0 ˛ J; p ˛ P; p p0
(62)
Energy and Water Conservation in Batch Processes tsin ð j; pÞ tsin ðj0 ; p0 Þ MMð2 ysin ð j; pÞ ysin ðj0 ; p0 ÞÞ
cj; j0 ˛ J; p ˛ P; p p0
409 (63)
Constraints (64) and (65) state that if water is transferred to storage from more than one unit at the same time point, the time at which they do so must coincide. Constraints (66) and (67) state that if water is discharged from storage to more than one unit at the same time point, the time at which the water is discharged must coincide. tsin ð j; pÞ tsin ðj0 ; pÞ MMð2 ysin ð j; pÞ ysin ðj0 ; pÞÞ
cj; j0 ˛ J; p ˛ P
(64)
tsin ð j; pÞ tsin ðj0 ; pÞ þ MMð2 ysin ð j; pÞ ysin ðj0 ; pÞÞ
cj; j0 ˛ J; p ˛ P
(65)
tsout ð j; pÞ tsout ðj0 ; pÞ MMð2 ysout ð j; pÞ ysout ðj0 ; pÞÞ
cj; j0 ˛ J; p ˛ P
(66)
tsout ð j; pÞ tsout ðj0 ; pÞ þ MMð2 ysout ð j; pÞ ysout ðj0 ; pÞÞ
cj; j0 ˛ J; p ˛ P
(67)
If water is simultaneously being transferred to and discharged from storage, the time at which this happens should coincide. This is given in Constraints (68) and (69) tsin ð j; pÞ tsout ðj0 ; pÞ MMð2 ysin ð j; pÞ ysout ðj0 ; pÞÞ
cj; j0 ˛ J; p ˛ P
(68)
tsin ð j; pÞ tsout ðj0 ; pÞ þ MMð2 ysin ð j; pÞ ysout ðj0 ; pÞÞ
cj; j0 ˛ J; p ˛ P
(69)
Constraint (70) ensures that if water leaves storage at a later time point compared to water entering the storage, the time at which water leaves the storage must correspond to a later time in the time horizon. tsout ð j; pÞ tsin ðj0 ; p0 Þ MMð2 ysout ð j; pÞ ysin ðj0 ; p0 ÞÞ
cj; j0 ˛ J; p; p0 ˛ P; p p0
(70)
The following feasibility and time horizon constraints also hold. Constraint (71) ensures that if a processing unit j is reusing water from unit j0 at time point p, then unit j0 cannot reuse water from unit j at the same time point. ywr sout;j;j0 ; p þ ywr sout;j0 ;j ; p 1 cj; j0 ˛ J; sout;j;j0 ; sout;j0 ;j ˛ Sout;j;j0 ; p ˛ P (71) Constraints (72)–(76) ensure that each event occurs within the time horizon of interest. twin sin;j ; p H cj ˛ J; sin;j ˛ Sin;j ; p ˛ P twout sout;j ; p H twr sout;j;j0 ; p H tsin sin;j ; p H tsout sout;j ; p H
(72)
cj ˛ J; sout;j ˛ Sout;j ; p ˛ P
(73)
cj ˛ J; sout;j;j0 ˛ Sout;j;j ; p ˛ P
(74)
cj ˛ J; sin;j ˛ Sin;j ; p ˛ P
(75)
cj ˛ J; sout;j ˛ Sout;j ; p ˛ P
(76)
The objective of the formulation is either the maximization of profit or the minimization of makespan. Constraint (77) expresses the profit as the difference between the product revenue and the sum of freshwater, effluent treatment, cooling water, and steam costs. When the objective is the maximization of profit, Constraint (77) is maximized. As a result, the amount of external utilities as well as freshwater consumption is minimized. XX XX XX XX X Profit ¼ CPðsÞdðs; pÞ CF mwf sin;j ; p CE mwe sout;j ; p Costc w cw sin;jh ; p Costs t s
p
X st sin;jc ; p
sin;j
p
sout;j
p
sin;jh
p
sin;jc
(77)
p
Constraint (78) is the objective function for makespan minimization. In Constraint (78), the quotient of profit and time horizon is maximized. As a result, the makespan is minimized and the amount of external utilities as well as freshwater consumed are also minimized using the same reasoning as above. Max
Profit H
(78)
Solution Procedure The overall model, whether for a profit maximization problem or a makespan minimization problem is an MINLP. When solving a profit maximization problem, the model is linearized and solved as MILP, the solution of which is then used as a starting point for
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Energy and Water Conservation in Batch Processes
the exact MINLP model. If the solutions from the two models are equal, the solution is globally optimal, as global optimality can be proven for MILP problems. If the solutions differ, the MINLP solution is locally optimal. The possibility also exists that no feasible starting point is found. This solution procedure was used by Gouws et al. (2008), Adekola and Majozi (2011), and Stamp and Majozi (2011). When solving a makespan minimization problem, the MINLP cannot be linearized completely to an MILP due to the nonlinear objective function, Constraint (78). However, the MINLP was linearized to a relaxed MINLP problem which provides a starting point for the exact MINLP problem. However, the resulting solution to the exact MINLP cannot be guaranteed to be a global optimum. Constraints (6)–(8) have trilinear terms where a binary variable and two continuous variables are multiplied. Constraints (30)– (32), (42), and (43) have bilinear terms where two continuous variables are multiplied. This results in a nonconvex MINLP formulation. The bilinearity resulting from the multiplication of a continuous variable with a binary variable may be handled effectively with the Glover transformation. This is an exact linearization technique and as such will not compromise the accuracy of the model. For the profit maximization problem, the linearization procedure is carried out for each of the nonlinear terms in Constraints (6)–(8), (30)–(32), (42), and (43) to produce a MILP problem, which provides a starting point for the exact MINLP problem. The objective function for the makespan minimization problem was also nonlinear, however, only Constraints (6)–(8), (30)–(32), (42), and (43) were linearized to produce a relaxed MINLP problem, the solution of which provides a starting point for the exact MINLP problem.
Case Study I This case study is a simple sequential process. A single product is produced via three tasks in five units. The case study has been adapted to include water and energy using operations. Fig. 2 shows the recipe representation of the process. Steam is available for heating, cooling water is available for cooling, and freshwater is available for washing. The objective is to maximize profit over a given time horizon, while minimizing freshwater usage and utility requirements. Data required for case study I may be obtained from Tables 1–3. The following should be noted: 1. 2. 3. 4.
The processing duration of a batch is fixed. The mass load of contaminants is dependent on the batch size. A single contaminant is present. The energy requirements of the operations are a function of the batch size and the given initial and final temperatures. Due to this fact, careful consideration must be given to ensure that whenever heat integration occurs between two operations, the minimum temperature difference for heat transfer, 10 C was not violated.
The first case study was solved with the proposed formulation, using Constraints (1), (2), (17)–(22), (27)–(30), and (32)–(37). The objective function was the maximization of Constraint (77). It is important to note that in Constraints (27), (28), (30), and (32), variables associated with wastewater storage are not included. Due to the fact that a single contaminant was present, the formulation for this case study could be reduced to a MILP. The outlet contaminant concentration was fixed to the maximum, Constraint (30) was substituted into Constraint (32) and a Glover transformation was performed on the resulting equation. The computer used to solve the model had an Intel(R) Core(TM) i7-2670QM, 2.2 GHz processor with 4.0 GB RAM. The problem was solved with GAMS using CPLEX as the MIP solver. The results from the first case study as compared to the results achieved by Halim and Srinivasan (2011) may be obtained from Table 4. As can be observed from Table 4, a better profit overall was obtained with the proposed formulation. Although the amounts for steam and cooling water were lower for the model by Halim and Srinivasan (2011), the amount of freshwater required was lower in the proposed formulation, which also results in a lower effluent production. Similar to the method by Halim and Srinivasan (2011), the washing of a unit may not necessarily occur immediately after the end of a processing task, but can be delayed to improve reuse opportunities with other units. To cater for this, Constraint (49) was modified to Constraint (49*) as follows. (49*) twout sout;j ; p ¼ twin sin;j ; p 1 þ sw sin;j yw sin;j ; p 1 cj ˛ J; sin;j ˛ Sin;j ; sout;j ˛ Sout;j ; p ˛ P; p > p0
Fig. 2
Recipe representation of the simple sequential process for case study I.
Energy and Water Conservation in Batch Processes
Table 1
Data pertaining to production, for case study I
Task ( i)
Unit ( j)
Max batch size (kg)
Task 1
Unit 1 Unit 2 Unit 3 Unit 4 Unit 5
100 150 200 100 150
Task 2 Task 3
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Table 2
Table 3
Processing time (h)
Washing time (h)
Material state (m)
1.25 1.70 1.50 0.75 1.20
0.25 0.30 0 0.25 0.30
A B C D Wash water Wastewater Cooling water Steam
Initial inventory (kg)
Max storage (kg)
Revenue/cost ($/kg or $/MJ)
1000 0 0 0
1000 200 250 1000
0 0 0 5 0.1 0.05 0.02 1
Data pertaining to energy requirements, for case study I
Task ( i)
Tin ( C)
Tout ( C)
Unit ( j)
Cp (kJ/kg C)
Task 1
140
60
Task 2 Task 3
60 40
40 80
Unit 1 Unit 2 Unit 3 Unit 4 Unit 5
4.0 4.0 3.5 3.0 3.0
Cooling water Steam
20 170
30 160
Data pertaining to water requirements, for case study I
Task ( i)
Unit ( j)
Max inlet concentration (ppm)
Max outlet concentration (ppm)
Contaminant loading (g contaminant/kg batch)
Task 1
Unit 1 Unit 2 Unit 3 Unit 4 Unit 5
500 50 – 150 300
1000 100 – 300 2000
0.2 0.2 0.2 0.2 0.2
Task 2 Task 3
Table 4
Comparison of results for case study I
Profit ($) Steam (MJ) Cooling water (MJ) Total freshwater (kg) Revenue from product ($)t Cost of steam ($) Cost of cooling water ($) Cost of freshwater ($) Cost of wastewater ($) Number of slots Number of time points Number of iterations CPU time (s)
Halim and Srinivasan (2011)
This formulation
4764.1 43.9 313.9 1238.37 5000 43.9 6.28 123.8 61.9 7 N/A 1500 Not reported
4775.28 66 336 1013.33 5000 66 6.72 101.3 50.7 N/A 15 N/A 28,797
412
Fig. 3
Energy and Water Conservation in Batch Processes
Resulting production schedule for case study I.
Fig. 3 shows the Gantt chart with the resulting production schedule for the first case study. The clear horizontal blocks represent the tasks associated with production processes. The amount of material processed in each unit is labeled within the clear blocks. The dark horizontal blocks represent the washing of a unit. The amount of water associated with washing is labeled underneath the relevant dark blocks. The double-sided arrows (up-down arrows) represent direct heat integration between two tasks. The numbers in round brackets represent the amount of energy associated with heat integration. The numbers in curly brackets represent steam duties while the numbers in square brackets represent cooling water duties.
Case Study II This case study comprises of the popular literature example by Kondili et al. (1993). Halim and Srinivasan (2011) adapted this complex production process to include energy and water using aspects, and it is described here. For the production process, two products, Product 1 and Product 2 are to be produced from three raw materials FeedA, FeedB, and FeedC. A heater, HR is used to heat FeedA. Two reactors, RR1 and RR2 are available to perform three different chemical reactions, Reaction 1, Reaction 2, and Reaction 3. Finally, a separator exists to purify Impure E. Fig. 4 shows the STN representation of the problem. Steam is available for heating, cooling water is available for cooling, and freshwater is available for washing. The production recipe is as follows: 1. Heating: FeedA is heated from 50 C to 70 C inside HR. Steam is required as a heating medium for this reaction. 2. Reaction 1: A mixture of 50% FeedB and 50% FeedC, on a mass basis, is reacted in either unit RR1 or RR2. The product of this reaction is IntBC. The reaction requires cooling from 100 C to 70 C. 3. Reaction 2: A mixture of 40% Hot A and 60% IntBC is reacted to form Prod1 (40%) and IntAB (60%). This reaction can be performed in either unit RR1 or RR2 and requires heating from 70 C to 100 C. 4. Reaction 3: A mixture of 20% FeedC and 80% IntAB is reacted in either units RR1 and RR2. The reaction produces ImpureE and requires heating from 100 C to 130 C. 5. Separation: In SR, ImpureE is purified to produce Prod2 (90%) and intermediate IntAB (10%). The separation requires cooling from 130 C to 100 C. Water is required for washing RR1 and RR2 at the end of any reaction. In this case study, four contaminants, ar, br, cp, and dw are present. Table 5 shows information regarding the production process. Data pertaining to heating and cooling are given in Table 6, while data pertaining to the washing of RR1 and RR2 are given in Table 7. The objective in this case study was to determine the minimum makespan required to produce 200 kg each of Prod1 and Prod2, while also minimizing external utility and freshwater consumption. It is important to note the following from the data in Tables 5–7: 1. The processing duration of a batch is dependent on the batch size. 2. The mass load of contaminants is not fixed, but dependent on the batch size. 3. The energy requirements of the operations are a function of the batch size and the given initial and final temperatures. Due to this fact, careful consideration must be given to ensure that whenever heat integration occurs between two operations, the minimum temperature difference for heat transfer, 10 C, is not violated.
Energy and Water Conservation in Batch Processes
Fig. 4
Table 5
413
STN representation of the complex production facility for case study I.
Data pertaining to production, for case study II
Task ( i)
Unit ( j)
Max batch size (kg)
aij (h)
bij (h)
Washing time (h)
Heating (H) Reaction 1 (R1)
HR RR1 RR2 RR1 RR2 RR1 RR2 SR
100 50 80 50 80 50 80 200
0.667 1.084 1.034 1.09 1.034 0.417 0.367 1.334
0.007 0.027 0.017 0.027 0.017 0.013 0.008 0.007
0 0.25 0.3 0.25 0.3 0.25 0.3 0
Reaction 2 (R2) Reaction 3 (R3) Separation (S)
Table 6
Material state FeedA (s1) FeedB (s2) FeedC (s3/s4) HotA (s5) IntAB (s8) IntBC (s6) ImpureE (s9) Prod1 (s7) Prod2 (s10) Wash water Wastewater Cooling water Steam
Initial inventory (kg)
Max storage (kg)
Revenue/cost ($/kg or $/MJ)
1000 1000 1000 0 0 0 0 0 0
1000 1000 1000 100 200 150 200 1000 1000
10 10 10 0 0 0 0 20 20 0.1 0.05 0.02 1
Data pertaining to energy requirements, for case study II
Task (i)
Tin ( C)
Tout ( C)
Unit ( j)
Cp (kJ/kg C)
Heating (H) Reaction 1 (R1)
50 100
70 70
Reaction 2 (R2)
70
100
Reaction 3 (R3)
100
130
Separation (S) Cooling water Steam
130 20 170
100 30 160
HR RR1 RR2 RR1 RR2 RR1 RR2 SR
2.5 3.5 3.5 3.2 3.2 2.6 2.6 2.8
The second case study was solved using the proposed formulation, using Constraints (1), (2), (17)–(22), (27)–(30), (32)–(37), and (39). The objective function was the minimization of makespan, Constraint (78). It is important to note that in Constraints (27), (28), (30), and (32), variables associated with wastewater storage are not included. The computer used to solve the model had an Intel(R) Core(TM) i7-2670QM, 2.2 GHz processor with 4.0 GB RAM. The problem was solved with GAMS using DICOPT for the MINLP with CPLEX as the MIP solver and MINOS as the NLP solver.
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Energy and Water Conservation in Batch Processes
Data pertaining to water requirements, for case study II
Table 7
Max inlet concentration (ppm) Task ( i) Reaction 1 (R1) Reaction 2 (R2) Reaction 3 (R3)
Fig. 5
Max outlet concentration (ppm)
Unit ( j)
ar
br
cp
dw
ar
br
cp
dw
Contaminant loading (g contaminant/kg batch)
RR1 RR2 RR1 RR2 RR1 RR2
300 300 700 700 500 500
500 500 600 600 200 200
800 800 300 300 400 400
400 400 400 400 300 300
700 700 1200 1200 800 800
800 800 1000 1000 500 500
1200 1200 600 600 700 700
900 900 800 800 900 900
0.2 0.2 0.2 0.2 0.2 0.2
Resulting production schedule for case study II.
Fig. 5 shows the Gantt chart with the resulting production schedule for the case study. The clear horizontal blocks represent the tasks associated with production processes. The amount of material processed in each unit is labeled within the clear blocks. The dark horizontal blocks represent the washing of a unit. The amount of water associated with washing is labeled underneath the relevant dark blocks. The double-sided arrows (up-down arrows) represent direct heat integration between two tasks. The numbers in round brackets represent the amount of energy associated with heat integration. The numbers in curly brackets represent steam duties while the numbers in square brackets represent cooling water duties. A comparison between the results of this case study obtained by Halim and Srinivasan (2011) and the proposed formulation is described in Table 8. As can be observed from Table 8, improved objectives were obtained using the proposed formulation with the exception of the amount of freshwater used for washing. This was probably due to the fact that the cost associated with using steam was more significant than the cost associated with using freshwater. Hence, to minimize overall costs, promoting opportunities for heat integration took precedence over promoting opportunities for water reuse. This can be observed from the production schedule in Fig. 5. With the operations aligned to promote heat integration, no opportunities for water reuse exist at all. The MINLP model was solved by the aforementioned initialization procedure. The objective value from the relaxed MINLP model was 123.543 $/h and the objective value from the exact MINLP model was 123.369 $/h. As both the relaxed model and exact models were MINLP, due to the nonlinear objective function, the global optimality of the solution could not be guaranteed.
Case Study III This case study was obtained from Majozi and Gouws (2009) and was extended to include heat integration opportunities. The multipurpose batch facility investigated, is similar to that discussed in Case Study II. The heating and separating tasks performed in HR and SR respectively are not to be heat integrated with any other units. Heat integration can only occur between RR1 and
Energy and Water Conservation in Batch Processes
Table 8
415
Comparison of results for case study II
Makespan (h) Steam (MJ) Cooling water (MJ) Total freshwater (kg) Revenue from products ($) Cost of raw materials ($) Cost of steam ($) Cost of cooling water ($) Cost of freshwater ($) Cost of wastewater ($) Profit ($) Number of slots Number of time points Number of iterations CPU time (s) Major iterations
Halim and Srinivasan (2011)
This formulation
19.96 61.36 35.39 275.09 8000 5604.4 61.36 0.7078 27.509 13.75 2292.3 8 N/A 3500 not reported –
19.93 44.88 19.72 341.2 8000 5444.4 44.88 0.3994 34.12 17.06 2459.1 N/A 17 N/A 24532 9
RR2 depending on the tasks they perform. Similarly, water is required for washing RR1 and RR2 at the end of any reaction. In this case study, three contaminants, C1, C2, and C3 are present. Table 9 shows information regarding the production process, while data pertaining to the washing of RR1 and RR2, obtained from Majozi and Gouws (2009) are given in Table 10. The heat integration data are provided in Table 11. In this case study, storage facilities for heat storage and water reuse are available. Different scenarios of the case study were solved to demonstrate the capabilities of the model. These scenarios are as follows: l
Scenario 1: During production, only freshwater is available for washing. Heating and cooling are provided exclusively by steam and cooling water. l Scenario 2: In addition to freshwater for washing, steam, and cooling water for heating and cooling respectively, opportunities for direct water reuse and direct heat integration are explored. l Scenario 3: The effect of the inclusion of storage facilities for heat and wastewater (indirect water reuse and indirect heat integration) to Scenario 2 is explored. Constraints (1)–(16), (21)–(26), (27)–(31), and (34)–(77) were used to solve this case study. The bilinear terms present in the model were linearized and the solution procedure as described previously was used. The capacity of the storage vessel for water was 200 kg. The objective of this case study was to maximize profit while minimizing wastewater production and energy consumption, within a time horizon of 12 h. The storage capacity and the initial storage temperature of the storage vessel were variables to be optimized. Heat losses were not considered. The computer used to solve the model had an Intel(R) Core(TM) i7-2670QM, 2.2 GHz processor with 4.0 GB RAM. The problem was solved with GAMS using DICOPT for the MINLP with CPLEX as the MIP solver and CONOPT as the NLP solver. A
Table 9
Production data for case study III
Task ( i)
Unit ( j)
Max batch size (kg)
Mean processing time (h)
Washing time (h)
Heating (H) Reaction 1 (R1)
HR RR1 RR2 RR1 RR2 RR1 RR2 SR
100 50 80 50 80 50 80 200
1 2 2 2 2 1 1 1 for Prod2, 2 for IntAB
0 0.25 0.3 0.5 0.25 0.25 0.25 0
Reaction 2 (R2) Reaction 3 (R3) Separation (S)
Material state FeedA (s1) FeedB (s2) FeedC (s3/s4) HotA (s5) IntAB (s8) IntBC (s6) ImpureE (s9) Prod1 (s7) Prod2 (s10) Wash water Wastewater Cooling water Steam
Initial inventory (kg)
Max storage (kg)
Revenue/cost (cost units/kg or cost units/kJ)
Unlimited Unlimited Unlimited 0 0 0 0 0 0
Unlimited Unlimited Unlimited 0 0 0 0 0 0
0 0 0 0 0 0 0 100 100 2 3 2 10
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Energy and Water Conservation in Batch Processes
Table 10
Wastewater minimization data for case study III Maximum contaminant concentration (g contaminant/kg water) C1
Reaction 1 (RR1) Reaction 2 (RR1) Reaction 3 (RR1) Reaction 1 (RR2) Reaction 2 (RR2) Reaction 3 (RR2)
Reaction 1
Reaction 3
C3
0.5 0.5 2.3 1 0.9 3 0.01 0.05 0.3 0.2 0.1 1.2 0.15 0.2 0.35 0.3 1 1.2 0.05 0.2 0.05 0.1 1 12 0.03 0.1 0.2 0.075 0.2 1 0.3 0.6 1.5 2 1.5 2.5 Contaminant mass load (g) C1 C2 C3 4 80 10 15 24 358 28.5 7.5 135 9 2 16 15 80 85 22.5 45 36.5
Reactor 1 Reactor 2 Reactor 1 Reactor 2 Reactor 1 Reactor 2
Reaction 2
Table 11
Max. inlet Max. outlet Max. inlet Max. outlet Max. inlet Max. outlet Max. inlet Max. outlet Max. inlet Max. outlet Max. inlet Max. outlet
C2
Heat integration data for case study III.
Reaction
Type
Heating/cooling requirement (kWh)
Operating temperature ( C)
RX1 RX2 RX3
Exothermic Endothermic Exothermic
60 (cooling) 80 (heating) 70 (cooling)
100 60 140
comparison between the three scenarios is provided in Table 12. The results of Scenario 1, Scenario 2, and Scenario 3 are contained in columns 2, 3, and 4 respectively, of Table 12. From Table 12 it can be observed that profit increases from Scenario 1 to Scenario 3, with a decrease in steam and cooling water requirements. However, while cooling water and steam decreased in Scenario 2 compared to Scenario 1, the amount of freshwater increased. The total amount of product in Scenario 1 is 258 kg, while the total amount of product in Scenario 2 is 272.4 kg. This is as a result of additional unit operations being performed in Scenario 2 than are performed in Scenario 1. These
Table 12
Comparison between different scenarios for case study III
Profit (cu) Amount of Prod1 (kg) Amount of Prod2 (kg) Cooling water (kWh) Steam (kWh) Freshwater (kg) Time points CPU time (s) Binary variables Initial storage temperature ( C) Heat storage capacity (ton) Major iterations
Freshwater and utilities
Direct water reuse and direct heat integration
Direct/indirect water reuse and direct/indirect heat integration
18,537 96 162 390 240 816 11 3.1 128
19,836 116 156.4 250 180 1020 11 14.8 508
3
3
22,235 116 188 190 10 896 13 285.2 954 82.9 2.024 4
Energy and Water Conservation in Batch Processes
Fig. 6
417
Resulting process schedule when only direct heat integration and direct water reuse are possible.
additional unit operations contribute to the increased amount of washing water required. Furthermore, no opportunities for direct water reuse were realized due to the cost of steam relative to the cost of freshwater. The unit operations are aligned in such a way as to promote as much heat integration as possible. In so doing, opportunities for direct water reuse are lost. In Scenario 3, storage for water is available and hence a decrease is observed in the amount of freshwater used. The resulting process schedule for the results of Scenario 2 is illustrated in Fig. 6, while the corresponding schedule for Scenario 3 is illustrated in Fig. 7. The clear blocks represent the production operation in the unit while the dark blocks represent the washing operations which take place after the reactions are completed. The numbers above the clear blocks represent the amount of material processed in the unit during production and the numbers below the washing operations represent freshwater. Water transfer to and from storage has been clearly labeled. The up-down arrows represent direct heat integration, while the bent arrows represent indirect heat integration to or from the heat storage unit. The results of Scenario 1 are globally optimal with the objective function of the linearized MILP and exact MINLP being 18,537 c.u. Similarly, the results of Scenario 2 are globally optimal, with the objective function of the linearized MILP and exact MINLP being 19,836 c.u. In Scenario 3, the objective function of the linearized MILP was 25,270 c.u, while the objective function of the exact MINLP was 22,235 c.u. Hence, the results for Scenario 3 are locally optimal.
Fig. 7
Resulting process schedule when both direct/indirect heat integration and wastewater minimization are possible.
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Energy and Water Conservation in Batch Processes
Conclusions A simultaneous method for the optimization of energy and water embedded within a scheduling framework has been developed. Furthermore, opportunities for direct and indirect heat integration as well as direct and indirect water reuse have been explored. The mathematical formulation led to an MINLP problem for which an initialization procedure was employed. The applicability of the method has been demonstrated with three case studies. The developed formulation has proved to effectively solve a complex makespan minimization problem in which duration is a function of batch size and which included multiple contaminants.
See also: Process Integration for Sustainable Industries.
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