Development of Heat-Integrated Evaporation and Crystallization

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Ind. Eng. Chem. Res. 2001, 40, 2842-2856

Development of Heat-Integrated Evaporation and Crystallization Networks for Ternary Wastewater Systems. 2. Interception Task Identification for the Separation and Allocation Network Gautham Parthasarathy† Saflex Technology, Solutia Inc., 730 Worcester Street, Springfield, Massachusetts 01151

Russell F. Dunn* Nylon Technology, Solutia Inc., 3000 Old Chemstrand Road, Cantonment, Florida 32533

Mahmoud M. El-Halwagi‡ Department of Chemical Engineering, Auburn University, Auburn, Alabama 36849

This work introduces the problem of simultaneous allocation and recovery via crystallization and evaporation for ternary wastewater systems and is a continuation of work proposed in part 1. The problem at hand is solved within the framework of mass integration. Different sources and sinks are identified. Varying sink inlet flow and composition constraints necessitate the use of the path equations for global tracking of all species of interest. Constraint propagation is invoked to determine bounds on allowable sink flow-rate and composition constraints. A twostep solution procedure is proposed to solve the problem. The first step consists of interception task identification for a given stream in the process. This is useful because it enables elimination of infeasible nodes prior to detailed design, thus leading to computational efficiency. The second step consists of the design of a separation network for an identified separation task. This step has been previously described in part 1. Part 2 couples the separation task via evaporation and crystallization to the allocation requirements of various sinks. A case study dealing with the ammonium nitrate manufacturing process is included to demonstrate the broad applicability and value of this work. 1. Introduction Crystallization has been extensively studied as a unit operation and finds broad application in various industries1 including (1) separation (e.g., separation of pxylene from o- and m-xylene), (2) concentration (e.g., concentration of fruit juice), (3) solidification (e.g., modification of the appearance of sugar), (4) purification (e.g., separation of an essential amino acid, L-isoleucine, from a fermentation broth), and (5) analysis (e.g., determination of the molecular structure) Crystallization operations play a vital role in product recovery and are capable of producing very high purity products from impure solutions. Another attractive feature of crystallization is that it generally requires less energy for separation as compared to distillation or other commonly used methods of purification. Another important application of crystallization is in its value in lowering pollution.2 For instance, crystallization can be used to separate salts from aqueous effluent streams so that these streams may be recycled and reused in the process. It has been observed that there * To whom correspondence should be addressed. E-mail: [email protected]. Tel: 850-968-8216. Fax: 850-9688732. † E-mail: [email protected]. Tel: 413-730-2255. Fax: 413-730-3496. ‡ E-mail: [email protected]. Tel: 334-844-2064. Fax: 334-844-2063.

were less established design techniques for crystallization systems as compared to distillation and other separation techniques. In response to this shortcoming, significant work has gone into developing systematic design methodologies for crystallization systems. Melt crystallization manufacturing systems were optimized and mathematically characterized.3 A systematic procedure was formulated for the separation of binary and ternary mixtures via extractive crystallization.4,5 A methodology to generate phase diagrams from solubility product data was developed along with a synthesis procedure for fractional crystallization processes to obtain pure solids from conjugate salt solutions.6 A design technique for suspension crystallization was studied, and a computer algorithm was developed for crystallizer design including mass and population balances, crystallization kinetics, and operating conditions.7 Crystallization paths, analogous to residue curve maps in vapor-liquid equilibrium systems, for melt crystallization processes were introduced.8 A set of systematized procedures for generating equilibriumbased fractional crystallization processes was described.9 A technique for solids mixture separation based on solubility was presented.10 A methodology for the identification of alternative process designs for fractional crystallization from solution was described.11 Guidelines for separation of two and three solute mixtures and bypass regions of multiple saturation via solvent addition/removal and cooling/heating were de-

10.1021/ie0008934 CCC: $20.00 © 2001 American Chemical Society Published on Web 05/26/2001

Ind. Eng. Chem. Res., Vol. 40, No. 13, 2001 2843

scribed.12 A general method for the calculation of various types of phase diagrams for aqueous electrolytic systems was developed and applied toward simulation and optimization of fractional crystallization processes.13 The retrofit of an existing fractional crystallization process to improve product recovery was considered.14 A method to synthesize fractional crystallization process flow sheets was presented and a network flow model was developed from equilibrium data for a candidate set of potential operating point temperatures.15 This flow model represented the potential separation flow sheets. A network flow model for synthesizing crystallization sequences for multicomponent systems was also presented.16 In the following sections, the focus is on separation of species via evaporation coupled with crystallization. Some of the issues not fully addressed in previous crystallization research include the following: 1. The entire focus has been on product recovery, and less attention was paid to the mother liquor. For instance, in applications where discharge to the environment is to be minimized, the mother liquor and concentrations of its constituent species are equally as important as the product separated out following crystallization. The mother liquor has to be tracked for problems dealing with recovery and allocation of streams. 2. Another assumption made previously is that the crystallization system should be designed to operate at the point of multiple saturation (i.e., a eutectic). This aspect allows maximum product separation from the feed stream. However, it may be necessary to operate the system away from the eutectic. The maximum amount separated at the eutectic acts as an upper bound on the amount of pure product that can be separated. The actual amount separated will be decided by the overall recovery and allocation goals of the process. 3. Heat integration of the energy flows in a typical evaporation and crystallization system is not addressed. It has been identified that the major component of the operating cost of an evaporation and crystallization process lies in the heat duty to be added for evaporation and removed in cooling/heating the crystallization block. Although crystallization is a heat-induced separation network (HISEN), the previously developed solution methodologies for the design of volatile organic compound (VOC) condensation HISENs17-20 cannot address all of the unique aspects of the evaporation and crystallization operation. Part 1 defined the problem of the design of an evaporation and crystallization network and described a solution methodology for optimization of the separation task. 4. Most of the previous work on evaporation and crystallization considered the separation operations as stand-alone. In several processes (especially from the environmental perspective), decisions about separation have to be taken in conjunction with allocation requirements. The simultaneous design of a separation and allocation network based on evaporation and crystallization has not been addressed. 2. Allocation and the Mass Integration Framework Separation networks constitute only one aspect of the general recovery and allocation problem. For evaporation and crystallization processes, the mother liquor following salt separation is an important stream that can be potentially recycled and reused. Hence, separa-

Figure 1. Schematic representation of the flow sheet from a species viewpoint.

tion has to be integrated with decisions on how much mother liquor to recover, at what composition, and how the recovered streams will be used. Hence, the more general concept of mass integration is invoked which is a holistic approach that deals with the optimum generation, separation, and routing of streams and species.21-27 A particularly useful framework for representing mass integration strategies is to represent the flow sheet from a species perspective as shown in Figure 1. This representation allows one to look at a process flow sheet from a different viewpoint. First, sources and sinks are defined in the given process. Sources are process streams that carry the targeted species. Sinks are process units that are capable of processing the sources carrying the targeted species. Sinks include reactors, separators, heaters, coolers, pumps, compressors, pollution-control facilities, discharge media, and the like. Streams leaving the sinks, in turn, become sources. Therefore, sinks also are generators of the targeted species. Each sink/generator can be manipulated via design or operating changes to alter the flow rates and compositions that it can accept and that it discharges. It may be necessary to modify compositions of sources to prepare them for the sinks. This is done in a network of separation units referred to as the species interception network (SPIN). For the scope of this work, all interception operations are carried out via the use of evaporation and crystallization networks. Mass integration, therefore, involves a combination of stream segregation, mixing, and interception (both terminal and in-plant); recycle from sources to sinks (with or without interception); and sink/generator manipulation. Segregation is simply avoiding stream mixing. Segregating streams with different compositions may make it unnecessary later to change their compositions. This can result in reduced cost and may allow streams to be recycled directly to sinks without further processing. Mixing can be used to reach appropriate flow rates and compositions. Interception is the use of separation technologies to adjust the compositions of the speciesladen streams to make them acceptable for the sinks. As has been mentioned earlier, these separations may be effected by the use of mass separating agents (MSAs), energy separating agents (ESAs), or membranes. Identifying the right combination for a SPIN can be a large and complex problem because numerous streams typically must be processed, many separation technologies may be applicable, and initially it is not known how much of a species must be removed to make that stream suitable for a sink. Therefore, a systematic technique

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Ind. Eng. Chem. Res., Vol. 40, No. 13, 2001

Figure 2. General problem statement for recovery and allocation.

is needed to screen the candidate separating agents and separation technologies to find the optimal SPIN. Recycle refers to the routing of a source to a sink. Each sink has a number of constraints on the flow rates and compositions of feeds that it can process. If a source satisfies these constraints, it may be recycled directly to the sink. If the source violates these constraints, however, then segregation, mixing, or interception may be used to prepare the stream for recycle. Sink/generator manipulation involves design or operating changes that alter the flow rates and compositions of sources entering or leaving the sinks. These measures can include temperature or pressure changes, unit replacement, catalyst alteration, feedstock, solvent, and product substitution,28,29 and reaction changes.30,31 Part 1 allowed definition of a separation task and designed the evaporation and crystallization network focused on the required reduction of loads of all of the species of interest. Therefore, separation tasks were defined independently of potential usage of the streams generated from evaporation and crystallization. The mother liquor following crystallization and the pure water stream are important candidates to be considered for allocation. In this paper, the task of separating the species of interest is linked to the task of allocating the various streams created to their most appropriate users in the process. In this regard, the mass integration framework offers the following merits: (i) It provides a global tracking of the different species throughout the process and describes the interaction among the various components. (ii) It integrates the usage of the pure water streams and mother liquor streams with the definition of the separation tasks. Although significant advances have been accomplished in developing systematic techniques for mass integration, most of these techniques have been developed to address only single-component ideal systems. A few attempts have targeted mass integration for multicomponent systems, wherein it was generally assumed that each component behaves independently

of the other species. However, in many industrial applications, there is a strong interaction among the species. This nonideal interaction can govern the process performance and must be taken into account. To tackle these challenges, a systematic framework and solution strategy to obtain the global solution was proposed for the separation and allocation of multicomponent nonideal VOC systems.32 However, although condensation is a heat-induced separation process similar to crystallization, there are certain unique features to the solidliquid-phase equilibrium that are different from the vapor-liquid-phase equilibrium for condensation. These aspects result in a need to define a new problem statement and develop a solution procedure to design and optimize an evaporation and crystallization network for recovery and allocation. 3. Problem Statement for Recovery and Allocation The general problem can be stated as follows: Given a process with NSource wastewater sources containing multiple species of interest (i.e., salts) and NSink sinks (with known inlet flow rate and composition constraints for all species of interest) that can accept the potential sources that may be recycled and reused, it is desired to determine minimum cost strategies for segregation, mixing, recycle, and interception (via evaporation and crystallization) that would reduce the discharge of all species of interest from the process to a prespecified level. The general problem can be represented as in Figure 2. The focus of the subsequent work is to solve the following problem, which is a subproblem of the general problem. The system being considered is ternary and consists of water and two salts A and B (which may be pollutants). Interception via evaporation and crystallization is assumed to be implemented one node at a time.33 The rationale behind this assumption is that most chemical plants are open to implementing interception on one stream at a time rather than multiple


Figure 3. Specific recovery and allocation problem representation.

interceptions of multiple streams. Simultaneous interception of multiple nodes may be considered by allowing intercepted compositions of multiple nodes to be treated as optimization variables in the same formulation.22,33 Each stream upon interception results in three product streams, namely, a pure water stream, a salt stream, and a mother liquor stream. The specific problem being solved may be stated as follows: Given a process with NSource wastewater sources containing two salts A and B and NSink sinks (with known inlet flow rate and composition constraints for both salts) that can accept the potential sources that may be recycled and reused, it is desired to determine minimum cost strategies for segregation, mixing, recycle, and interception (via evaporation and crystallization) one node at a time, that would reduce the discharge of water and the two salts from the process to a prespecified level. The problem can be represented as in Figure 3. For the process under consideration, there are several water streams containing the species of interest (i.e., salts A and B). These streams are referred to as sources and are represented by the set WASTE_SOURCES ) {i|i ) 1, 2, ..., NSource}. The total flow rate of each wastewater source, Fi, is known along with the composition of the two salts (zAi and zBi for salts A and B, respectively) and the supply temperature . The initial mass load of the two salts in the ith TSource i wastewater source are given by

of water from all terminal sources is denoted by WNominal, where NTerminal Source Nominal

W

)

∑ j)1

Fj(1 - zAj - zBj )

(3)

The initial discharge of salt A from all terminal sources is denoted by UNominal, where NTerminal Source

U

Nominal

)

∑ j)1

FjzAj

(4)

The initial discharge of salt B from all terminal sources is denoted by VNominal, where NTerminal Source

V

Nominal

)

∑ j)1

FjzBj

(5)

The targeted final discharges from the process for water, salt A, and salt B are given as

WFinal e RWNominal

(6)

UFinal e βUNominal

(7)

VFinal e γVNominal

(8)

where

MassAi MassBi

)

FizAi

(1)

0eRe1

(9)

)

FizBi

(2)

0eβe1

(10)

0eγe1

(11)

The set WASTE_SOURCES consist of two subsets, namely, TERMINAL_SOURCES ) {j|j ) 1, 2, ..., NTerminal Source } and INTERNAL_SOURCES ) {k|k ) 1, 2, ..., NInternal Source }. TERMINAL_SOURCES refers to all source streams that are discharged from the process, while INTERNAL_SOURCES refers to all source streams that are used internally in the process. The initial discharge

This reduction in mass load is based on process considerations that can be technical (e.g., to meet composition requirements of units accepting resultant created sources), environmental (e.g., to satisfy discharge standards), and operational (e.g., to stay away from plugging and corrosion limits). The reduction in mass load is to be accomplished via invoking the mass

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integration strategies of segregation, mixing, recycle, and interception (via evaporation and crystallization). Besides the reduction in discharge loads of water and the two salts, there are allocation requirements to be satisfied by the sources present. The wastewater sources that are a part of the set WASTE_SOURCES are processed through a set PROCESS_SINKS ) {p|p ) 1, 2, ..., NTotal Sink } of process equipment that is a subset of the entire flow sheet. A subset of the set PROCESS_SINKS is the set SINKS ) {m|m ) 1, 2, ..., NSink}, whichs consists of those sinks in the process requiring freshwater and is considered as part of the overall allocation problem. For each sink, m, bounds on the required inlet flow rate, LSink m , and composition of Sink Sink both salts, zm,A and zm,B , are known. These values are determined a priori using the principles of constraint propagation as described in a subsequent section. Also available for service is freshwater with flow rate LFresh i that can be used to meet the allocation demands of the sinks in the set SINKS. The sources out of the evaporation and crystallization network and external freshwater are to be mixed and allocated to meet the flow rate and composition requirements of each sink. The separation of the species of interest from the wastewater streams is accomplished via evaporation and crystallization using a set ESA {e|e ) 1, 2, ..., NE} consisting of two sets COOLANT {C|C ) 1, 2, ..., NC} and HOT {H|H ) 1, 2, ..., NH} of candidate coolants and heating media, respectively. The temperature of each coolant, tCool C , and the cost of the coolant (given in $/kJ removed), COOLANT_COSTC, is known. The temperature of each heating media, tHot H , and the cost of the heating medium (given in $/kJ removed), HOT_COSTH, is known. The objective of the design task is to minimize the total cost of freshwater and evaporation and crystallization and considers two tasks, namely, (1) to identify optimum strategies for evaporation and crystallization of the wastewater sources to meet prespecified separation tasks and (2) to develop strategies for mixing sources from evaporation and crystallization with external freshwater to meet allocation tasks. 4. Design Challenges The aforementioned problem poses several design challenges. Because of the interactions among the water and salts, the solid-liquid equilibrium governing the evaporation and crystallization of the species of interest is nonideal and possibly nonlinear, thereby complicating the computational aspects of the design problem. Thus, the final design must account for the nonideal interactions existing between the species of interest. Simultaneous global tracking of all of the species of interest is required. Because of the linkage between separation and allocation, decisions on separation have to be integrated with allocation requirements. The flow rates and compositions of streams exiting separation are unknown. This poses a challenge, because the allocation scheme can be determined only after fixing these unknown values. To meet the allocation requirements of the sinks, the amount of external freshwater required should be traded off with the water available in the sources created from evaporation and crystallization. Besides these challenges, there exist several questions that must be addressed.

(i) Which wastewater sources are to be selected for recycle and reuse? (ii) How much of each wastewater source must be recycled? (iii) Where should the recycled sources be used? (iv) Should interception (via evaporation and crystallization) be implemented? If yes, which wastewater source should be targeted for interception? (v) Should both salts be separated from the source targeted for interception? If no, which salt should be targeted for removal? (vi) How much of each or both salts should be removed via interception? The final design should incorporate both separation and allocation constraints and incur minimum cost. 5. Path Equations Some of the critical tools needed to solve the aforementioned problem are the path equations.21,22,25,34 The path equations allow global tracking of all species of interest throughout the process. The path equations represent the load of the targeted species (e.g., salts) throughout the process as a function of its composition in carrying streams. While developing the path equations, one needs to keep track of only those units involving the targeted species instead of considering the entire flow sheet. Overall and component material balances and unit modeling equations can be used to derive the mathematical expressions for the path equations. In the path equations, a node is used to represent each salt-laden water source. These nodes are connected with composition profiles of the streams within the units. For more details on the graphical representation of the path equation, the reader is referred to the literature.22,33,34 Each node is related to the other nodes via process models. These mathematical expressions of the path equations linking input and output nodes can be developed as described subsequently. Consider a process unit, p, which involves one or both salts (i.e., p ∈ PROCESS_SINKS). The input water sources to this unit are described by the set INPUTp ) {i|i is an input stream to unit p}. The output water sources from this unit are described by the set OUTPUTp ) {i|i is an output stream from unit p}.

Overall Material Balance

∑

i∈INPUTp

Fi )

∑

Fi

p ∈ PROCESS_SINKS

i∈OUTPUTp

(12)

If the process unit p involves streams that are not laden with the salts, these should be included in the material balance.

Component Balance for Salt A FizAi ) FizAi + NetAp ∑ ∑ i∈INPUT i∈OUTPUT p

p

p ∈ PROCESS_SINKS (13) where the term NetAp accounts for the net loss of salt A within the process unit p (e.g., losses in fugitive emissions + depletion by chemical reaction - generation by chemical reaction, etc).


Component Balance for Salt B FizBi ) FizBi + NetBp ∑ ∑ i∈INPUT i∈OUTPUT p

p

p ∈ PROCESS_SINKS (14) NetBp

where the term accounts for the net loss of salt B within the process unit p (e.g., losses in fugitive emissions + depletion by chemical reaction - generation by chemical reaction, etc).

Process Unit Modeling Equation(s) (MassAi , MassBi , zAi , zBi : i ∈ OUTPUTp) ) fp(MassAi ,MassBi ,zAi ,zBi :i∈INPUTp) p ∈ PROCESS_SINKS (15) where fp is a vector of several performance equations relating outputs to inputs for process unit p. The path equations can be utilized to determine the effect of manipulating any node on the rest of the process. Thus, the effects of intercepting a particular node (e.g., a wastewater source) can be followed via the path equations. Let the ith water source be intercepted. Upon interception, the total flow, mass load, and composition of both salts of the ith source are altered A,Int , from Fi, MassAi , MassBi , zAi , and zBi to FInt i , Massi B,Int A,Int B,Int Massi , zi , and zi , respectively. For the process unit p, the output flow, mass loads, and compositions are now functions of the intercepted flow, mass loads, and compositions of the input streams. This can be represented as A,Int FInt ) FizAi + NetAp ∑ ∑ i zi i∈INPUT i∈OUTPUT p

p

p ∈ PROCESS_SINKS (16)

∑

B,Int FInt ) i zi

i∈INPUTp

∑

FizBi + NetBp

i∈OUTPUTp

p ∈ PROCESS_SINKS (17) (MassAi , MassBi , zAi , zBi : i ∈ OUTPUTp) ) ,MassB,Int ,zA,Int ,zB,Int :i∈INPUTp) fp(MassA,Int i i i i p ∈ PROCESS_SINKS (18) The path equations developed previously can also be used to relate the terminal discharge of water and the two salts to the stream being intercepted. Thus, assuming the ith wastewater source is being intercepted, the following is applicable to the case of independent species of interest A,Int B,Int ,zi ) WFinal ) φi(FInt i ,zi

U

Final

)

A,Int B,Int φi(FInt ,zi ) i ,zi

A,Int B,Int VFinal ) µi(FInt ,zi ) i ,zi

i ) 1, 2, ..., NSource (19) i ) 1, 2, ..., NSource (20) i ) 1, 2, ..., NSource (21)

When water and the two salts are interacting with each other, A,Int B,Int (WFinal, UFinal, VFinal) ) θi(FInt ,zi ) i ,zi i ) 1, 2, ..., NSource (22)

6. Constraint Propagation In the general problem, the concentrations of water and the two salts in various process streams would

Figure 4. Constraint propagation to determine sink inlet flow rate and composition constraints.

depend on each other through the sinks present. A tool that can be used to track the propagation of mass throughout the process is the path diagram.22,34 The path diagram is critical in determining the allowable sink inlet flow rates and compositions of both salts for sinks requiring freshwater. In the general problem, constraints on each sink requiring water are affected by constraints existing on all other sinks in the process. For the specific problem being considered, it is assumed that only one sink in the process affects each sink requiring water. The maximum allowable inlet flow rates and/or compositions of both salts are determined by the limits tolerated by the other sink. These maximum allowable values are determined by backpropagation of constraints existing on the other sink outside the problem formulation. Figure 4 is an example of constraint propagation. Sink I requires freshwater. Downstream from sink I, there are several units in the process, namely, sinks II-V. Sink V has tight constraints on the maximum amount of the two salts in streams that it can accept. Therefore, the maximum salt content of a water stream going into sink I is determined by the allowable compositions for sink V. Backpropagation of the constraints for sink V will result in the determination of allowable compositions for the two salts in inlet water streams to sink I. Thus, the problem at hand will have known maximum allowable sink inlet flow rate and composition constraints for all sinks, requiring freshwater, that are a part of the overall allocation problem. 7. Mathematical Formulation In this section, the design problem is formulated as an optimization program whose objective is to minimize cost subject to constraints of materials and energy balances, thermodynamic limitations, and technical and environmental requirements. 7.1. Evaporation and Crystallization System. In the evaporation and crystallization network, one node is intercepted at a time. The ith wastewater source is initially heated and sent to the evaporation block, leading to the creation of a pure water stream. The residual stream from evaporation is cooled and sent to the crystallization block to reduce the mass load of either or both salts to the target mass loads of both salts. The pure water stream created as a result of evaporation has a total flow rate Soli. The residual stream from

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Figure 5. Determination of mixing and routing strategies.

the evaporation block has a total flow rate Ri and Resi Resi and zi,B of salts A and B, respeccompositions zi,A tively. The result of crystallization is the creation of a salt stream with respective total flow rates Salti and Salt for salt A. Crystallization also results composition zi,A in a mother liquor stream with respective total flow rate ML ML and zi,B of salts A and B, MLi and compositions zi,A respectively. The typical structure and design methodology for an evaporation and crystallization network was described in part 1. The overall and component material balances, solid-liquid equilibrium, enthalpy balance, pinch equations for heat integration, and unit sizing and cost equations are given in part 1. 7.2. Constraints for Mixing and Allocation. In addition to constraints describing the evaporation and crystallization network, constraints characterizing optimal strategies for mixing and routing the streams created from the separation network must be included. To account for all mixing possibilities, a mixing box is defined (Figure 5) which can accept freshwater as well as streams from the evaporation and crystallization network. Once these streams are mixed, they are routed to the sinks utilizing them. Each stream getting intercepted results in the creation of a pure water stream, a salt stream, and a mother liquor stream. The pure water stream and the mother liquor stream can act as potential sources. The following constraints describe the mixing and allocation problem.

Source Material Balance NSink

Soli )

Pure ∑ li,m

i)1

(23)

m)1

NSink

MLi )

ML li,m ∑ m)1

i)1

(24)

NSink

Fi )

∑ li,m

i ) 2, ..., NSource

(25)

m)1

NSink

) LFresh i

Fresh li,m ∑ m)1

i ) NSource + 1

(26)

Sink Material Balance NSource Pure ML Fresh LSink ) l1,m + l1,m + l1,m + m

∑ i)2

li,m

m ) 1, 2, ..., NSink (27)

Sink Inlet Composition Balance for Salt A NSource Sink LSink m zm,A

)

ML Sink l1,m zm,A

+

∑ i)2

li,mzAi m ) 1, 2, ..., NSink (28)

Sink Inlet Composition Balance for Salt B NSource Sink ML Sink LSink m zm,B ) l1,mzm,B +

∑ i)2

li,mzBi m ) 1, 2, ..., NSink (29)

Having listed the problem constraints, the objective function can be described. The task at hand is to converge to the minimum cost from among the various cost values obtained by minimizing the total annualized cost of evaporation and crystallization and the annual cost of purchasing freshwater for each source i, i.e.

}] min [min {TACi + CFreshLFresh i for i ) 1, 2, ..., NSource (30) where TACi is the total annualized cost of the evapora-


tion and crystallization network to reduce the mass load of all of the species of interest to the predefined discharge loads in the ith stream being intercepted. This problem cannot be solved easily using commercial optimization packages. In addition to the nonconvexity of the problem, it has two inner minimization programs. The pinch formulation for heat-induced separation has an inner minimization problem besides the one posed by the overall objective function. These complications call for the development of a two-step solution strategy that is described subsequently.

Path Equations.

Overall Material Balance

∑

∑

Fi )

i∈INPUTp

(33)

Component Balance for Salt A FizAi ) FizAi + NetAp ∑ ∑ i∈INPUT i∈OUTPUT p

p

p ∈ PROCESS_SINKS (34)

8. Solution Strategy Before describing the solution procedure, the following aspects of the problem can be used to reduce complexity without losing rigor. In general, the cost of freshwater is lower than the cost of separation via evaporation and crystallization. The cost of piping and rerouting streams, required to implement segregation, recycling, and mixing strategies, can also be neglected as compared to the cost of separation. Thus, the objective function for this problem can be rewritten as

min [min {TACi}]

p ∈ PROCESS_SINKS

Fi

i∈OUTPUTp

for i ) 1, 2, ..., NSource

Component Balance for Salt B

∑

FizBi + NetBp

i∈OUTPUTp

p ∈ PROCESS_SINKS (35) Process Unit Modeling Equation(s) (MassAi , MassBi , zAi , zBi : i ∈ OUTPUTp) ) fp(MassAi ,MassBi ,zAi ,zBi :i∈INPUTp) p ∈ PROCESS_SINKS (36)

(31)

Generally, in most discharge load reduction problems, the reduction in discharge of salts (or pollutants) is more critical than the water reduction requirement. In other words, the problem may be driven by the required reduction in species discharge. As a consequence of reduction in species discharge and recycling of effluents, there will be a corresponding reduction in the net discharge of water from the process as well. Thus, the minimization of the cost of separation will result in an overall minimum cost solution. It may be noted that there is a direct relationship between the load of species separated out and the corresponding cost of separation. Because only one node is being intercepted at a time, the cost of separation is reduced upon separation of lower amounts of the two salts. Therefore, minimization of the cost of separation of salts from a given node occurs at minimum separation of both salts to achieve the predefined separation target. A two-step procedure is proposed to solve this problem. The first step involves interception task identification for the various available nodes. The second step involves designing the optimal heat-integrated evaporation and crystallization network that can deliver the identified task at minimum cost. 8.1. Step I: Task Identification. The problem statement has listed required minimum reductions in discharge of the two salts from the process in terms of β and γ, respectively. The objective of this step is the identification of the minimum separation load of each salt from a given node in order to meet the overall discharge reduction targets. This load removal is the separation task for the evaporation and crystallization network for the node under consideration. The subsequent mathematical formulation is executed twice, once for each salt present. The solution for the case where the minimum separation loads for both salts are satisfied is chosen as the final solution strategy. The mathematical formulation may be given as follows for the ith node being intercepted:

∑

FizBi )

i∈INPUTp

Modification of Path Equations due to Interception. For the ith source being intercepted,

∑

A,Int FInt ) i zi

i∈INPUTp

∑

FizAi + NetAp

i∈OUTPUTp

p ∈ PROCESS_SINKS (37) B,Int FInt ) FizBi + NetBp ∑ ∑ i zi i∈INPUT i∈OUTPUT p

p

p ∈ PROCESS_SINKS (38) (MassAi , MassBi , zAi , zBi : i ∈ OUTPUTp) ) ,MassB,Int ,zA,Int ,zB,Int :i∈INPUTp) fp(MassA,Int i i i i p ∈ PROCESS_SINKS (39) Interception Block Constraints. For the ith source being intercepted,

+ FInt Fi ) FPure i i +

Fui ∑ u)A,B

(40a)

A,Int FizAi ) FInt + FAi i zi

(40b)

B,Int + FBi FizBi ) FInt i zi

(40c)

Allocation Constraints.

Source Material Balance NSink

Soli )

Pure li,m ∑ m)1

i)1

(41)

i)1

(42)

NSink

MLi )

ML ∑ li,m

m)1

min {Fui }

for i ) 1, 2, ..., NSource, u ) A, B

subject to the following constraints.

(32)

NSink

Fi )

∑ li,m

m)1

i ) 2, ..., NSource

(43)

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Ind. Eng. Chem. Res., Vol. 40, No. 13, 2001 NSink

∑

LFresh ) i

Fresh li,m

i)1

(44)

to be followed to converge to the optimal design at minimum cost.

m)1

9. Solution Algorithm

Sink Material Balance NSource

LSink m

)

Pure l1,m

+

ML l1,m

+

Fresh l1,m

+

∑ i)2

li,m

m ) 1, 2, ..., NSink (45)

Sink Inlet Composition Balance for Salt A NSource Sink ML Sink LSink m zm,A ) l1,mzm,A +

∑ i)2

li,mzAi m ) 1, 2, ..., NSink (46)

Sink Inlet Composition Balance for Salt B NSource Sink LSink m zm,B

)

ML Sink l1,m zm,B

+

∑ i)2

li,mzBi m ) 1, 2, ..., NSink (47)

Predefined Discharge Reduction Constraints. NTerminal Source

W

Nominal

)

∑ j)1

Fj(1 - zAj - zBj )

(48)

NTerminal Source

U

Nominal

)

∑ j)1

FjzAj

(49)

NTerminal Source

VNominal )

∑ j)1

FjzBj

(50)

WFinal e RWNominal

(51)

UFinal e βUNominal

(52)

VFinal e γVNominal

(53)

0eRe1

(54)

0eβe1

(55)

0eγe1

(56)

where

A,Int B,Int ,zi ) WFinal ) φi(FInt i ,zi

i ) 1, 2, ..., NSource (57)

A,Int B,Int ,zi ) UFinal ) φi(FInt i ,zi

i ) 1, 2, ..., NSource (58)

A,Int B,Int VFinal ) µi(FInt ,zi ) i ,zi

i ) 1, 2, ..., NSource (59)

8.2. Step II: Design of the Evaporation and Crystallization Network. Step I results in the identification of the mass loads of salts A and B to be removed via the separation network to achieve the overall reductions in discharge of the salts from the process. This step is focused on the design of the evaporation and crystallization network to deliver the required separation for the node under consideration. It may be noted that part 1 describes the methodology

The following solution algorithm is proposed to solve this problem. 1. Select a node from among all of the available wastewater sources. 2. For the node under consideration, implement step I described previously. This involves formulation and solution of an optimization program to obtain the minimum separation loads of salts A and B, which would result in the overall discharge targets being satisfied. 3. If the above optimization program does not result in a feasible solution for either salt being considered, discount this node from further consideration. Select a new node and repeat the above procedure. 4. For feasible nodes, implement step II described in part 1. 5. For the node under consideration, determine the total annualized cost of designing an evaporation and crystallization network that can deliver the separation requirement for both salts A and B. 6. Select a new node and repeat the above procedure until all of the available wastewater sources have been considered. 7. Choose the feasible design with minimum total annualized cost as the final design. 10. Case Study The case study considers an ammonium nitrate manufacturing process35 depicted in Figure. 6. Ammonium nitrate is widely used as a fertilizer. It is manufactured by reaction of ammonia with nitric acid. Nitric acid is neutralized with anhydrous gaseous ammonia in an exothermic reaction. Some nitric acid laden gas streams are scrubbed in scrubber 2 with freshwater. The acid-laden wastewater stream from scrubber 2 is sent to neutralization. Following reaction, the ammonium nitrate solution contains between 78% and 84% of the salt. The solution is concentrated in a concentration step. Following addition of appropriate additives, the ammonium nitrate melt is sent to the prilling tower. The melt is sprayed against a hot air stream, resulting in the formation of prills. The exiting gas streams, containing water, ammonium nitrate, and sodium nitrate, are sent to scrubbing before being discharged to the atmosphere. The scrubbing is carried out in two scrubbers operating in series. The first scrubber (scrubber 3) uses the water stream from neutralization of the acid-laden wastewater stream from scrubber 2 while the second scrubber (scrubber 1) requires freshwater. The effluent from scrubber 3 is sent to the concentration step to recover any ammonium nitrate present in the stream. The species of interest in this problem are water, ammonium nitrate (i.e., salt A), and sodium nitrate (i.e., salt B). There are two main wastewater streams being discharged from the process, namely, the effluent from scrubber 1 and the condensed water stream from the concentration step. There are four gaseous sources, three in-plant liquid sources, and two terminal liquid sources in the flow sheet (refer to Figure 6). It may be noted that one of the in-plant liquid streams (namely, stream 2) has nitric acid present in it. This will eliminate it from being considered for


Figure 6. Manufacturing process for ammonium nitrate used in the case study.

potential interception. The nominal material balance for the process is given in Table 1. The path equations describing the process are described in Appendix 1. The two sinks in the process requiring freshwater have the following bounds on acceptable flow rates (kilograms per hour) and compositions.

Scrubber 1 40 e LSink e 50 1 Sink 0 e z1,NH e 0.05 4NO3 Sink e 0.1 0 e z1,NaNO 3

Scrubber 2

Table 1. Nominal Material Balance for the Case Study stream

phase

flow (kg/h)

water (%)

salt A (%)

salt B (%)

acid (%)

air (%)

1 2 3 4 5 6 7 8 9 10

liquid liquid liquid liquid liquid gas gas gas liquid liquid

44.7 63.2 75 100 68 141 116 108 92 100

100 70.7 66.7 50 66.1 3.5 4.3 4.6 100 92

0 0 0 20 2.9 19.1 6.03 1.85 0 5

0 0 33.3 30 30.9 6.4 3.45 0.92 0 3

0 29.3 0 0 0 0 0 0 0 0

0 0 0 0 0 70.9 86.2 92.6 0 0

sources may be considered for interception using evaporation and crystallization. 11. Solution of the Case Study

90 e

LSink 2

e 100

Sink e 0.05 0 e z2,NH 4NO3 Sink e 0.05 0 e z2,NaNO 3

It may be noted that these allowable bounds can be established using constraint propagation as described previously. It is desired to lower the discharge of water and the two salts by the following water ) 75%, NH4NO3 ) 65%, and NaNO3 ) 50%. This reduction in discharge is to be achieved via invoking the mass integration strategies of segregation, mixing, recycle, and interception (via evaporation and crystallization) at minimum cost. Both in-plant and terminal liquid

The case study is solved via the two-step procedure outlined previously. The path equations are formulated based on overall material and component balances (refer to Appendix 1). From the problem statement, the values of discharge reduction required are given by R ) 0.25, β ) 0.35, and γ ) 0.50. The nonlinear optimization problems were solved using LINGO on a 450 MHz Dell Windows NT workstation to converge onto the optimal solution. It may be noted that the nonlinearity present in the problem formulation may cause difficulties in converging to the global solution. The first program was run to identify the degree of reduction that could be achieved via use of the various allocation strategies (i.e., segregation, mixing, and recycling) without any interception. The problem formulation will consist of eqs 32-59, with all

2852


Figure 7. Final separation and allocation solution strategy for the case study.

interception loads being set to zero. The results were reductions in discharge of various species as given by water ) 60% (i.e., R ) 0.4), NH4NO3 ) 50% (i.e., β ) 0.5), and NaNO3 ) 7% (i.e., γ ) 0.93). Stream flow rates and compositions for the allocation solution strategy with no interception are provided in Table 2. It may be noted that use of these strategies by themselves did not allow the desired target to be achieved. Hence, interception is considered in the analysis. The mathematical formulation to be used for interception is given in eqs 32-59. There are two inplant sources (namely, streams 3 and 4) and two terminal sources (namely, streams 5 and 10). Following the solution algorithm described previously, one node is selected at a time. It was observed that sources 3 and 10 gave infeasible solutions. In other words, these sources cannot be intercepted so as to achieve the overall required reductions in discharges of the species of interest. Sources 4 and 5 are both feasible. Source 4 was intercepted such that the minimum loads of water and both salts to be removed via evaporation and crystallization in kilogram per hour are water ) 21.7, NH4NO3 ) 1.125, and NaNO3 ) 10.5. Source 5 gave a feasible solution with salt removal loads equal to source 4. However, this solution required removal of all ammonium nitrate present in the stream via interception, resulting in an ammonium nitrate composition of near zero in the exit stream. Besides, the initial concentration of ammonium nitrate present in stream 5 is low enough to render the solution economically unfavorable with the solution for stream 4. It may be noted that the benefits of in-plant interception, as compared to traditional end-of-pipe treatment, are ap-

Table 2. Allocation (with No Interception) Solution Strategy of the Case Study stream

phase

flow (kg/h)

water (%)

salt A (%)

salt B (%)

acid (%)

air (%)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

liquid liquid liquid liquid liquid gas gas gas liquid liquid liquid liquid liquid liquid liquid liquid liquid liquid

50 68.5 80.3 107.06 65.1 150.9 124.2 116.4 90 97.83 24.2 27.7 45.9 41.5 2.6 54.1 8.3 17.4

85 62 59.5 46.3 60.6 6.7 8.1 7.3 90 82.6 82.6 82.6 82.6 100 60.6 60.6 60.6 100

5 3.6 3.1 21 1.7 20.1 8.3 4.6 5 9.7 9.7 9.7 9.7 0 1.7 1.7 1.7 0

10 7.3 37.3 32.7 37.6 6.9 4.4 2.1 5 7.7 7.7 7.7 7.7 0 37.6 37.6 37.6 0

0 27 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 66.2 79.1 85.9 0 0 0 0 0 0 0 0 0 0

parent. Generally, although the loads being separated may be the same, in-plant interception offers the significant benefit of higher stream concentrations of the species of interest, thus leading to a cheaper interception. Thus, the optimal solution lies in intercepting stream 4 using evaporation and crystallization. The solution strategy for separation and allocation is given in Figure 7. Various stream flow rates and compositions for interception of stream 4 are listed in Table 3. Overall required reductions in discharge targets were satisfied. It may be noted that stream 5 is discharged with zero composition of ammonium nitrate. The in-plant interception resulted in the reduction of undesirable dis-

Ind. Eng. Chem. Res., Vol. 40, No. 13, 2001 2853 Table 3. Allocation with Interception (of Source 4) Solution Strategy stream

phase

flow (kg/h)

water (%)

salt A (%)

salt B (%)

acid (%)

air (%)

1 2 3 4 4 int 5 6 7 8 9 10 11 12 13 14 15 16 17 18

liquid liquid liquid liquid liquid gas gas gas liquid liquid liquid liquid liquid liquid liquid liquid liquid liquid liquid

50 68.5 80.3 107.07 73.74 31.78 150.96 124.2 116.4 90 97.83 25.74 25.74 46.34 41.5 2.15 22.76 6.87 17.39

85 62 3.11 46.3 37.79 55.9 6.68 8.12 7.3 90 82.6 82.6 82.6 82.6 100 55.9 55.9 55.9 100

5 3.65 37.36 21 28.99 0 20.1 8.35 4.6 5 9.71 9.71 9.71 9.71 0 0 0 0 0

10 7.3 59.5 32.7 33.22 44.06 6.95 4.43 2.15 5 7.67 7.67 7.67 7.67 0 44.06 44.06 44.06 0

0 27 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 66.2 79.1 85.9 0 0 0 0 0 0 0 0 0 0 0

charge (and loss) of product to the environment. Another useful consequence is the reduction in freshwater demand. The initial freshwater requirement of 136 kg/h was reduced to 59 kg/h, signifying a reduction of about 57%. Also, interception of stream 4 will result in a pure water stream (as described in part 1). This pure water stream can be used to replace freshwater and reduce the demand of freshwater further. Step II of the design procedure was described in part 1 for a ternary wastewater source with two salts. The case study solved in part 1 is for interception of stream 4 to its target compositions and flow rates to achieve the desired separation loads of water and both salts.

representing flow rates and compositions of intermediate streams. The initial values used for freshwater flow rate (kg/h) and compositions are as follows: F1 ) 44.7, zA1 ) 0.0, zB1 ) 0.0; F9 ) 92, zA9 ) 0.0, zB9 ) 0.0. Scrubber 2. The scrubber operates such that 18.5 kg/h of nitric acid is transferred continuously to the water stream.

F1 + 18.5 ) F2 F1zA1 ) F2zA2 , F1zB1 ) F2zB2 F1(1.0 - zA1 - zB1 ) ) F2zW 2 Acid 1.0 - zA2 - zB2 - zW 2 ) z2

1.0 - zA1 - zB1 ) zW 1 Neutralization. It is assumed that the nitric acid present in the wastewater stream entering neutralization consumes a stoichiometric amount of caustic. Thus, for 18.5 kg/h of nitric acid, 11.8 kg/h of caustic is to be added, producing 25 kg/h of sodium nitrate and 5.3 kg/h of water.

F2 + 11.8 ) F3 F2zA2 ) F3zA3 F2zB2 + 25 ) F3zB3

12. Conclusions This paper has introduced the novel problem of recovery (via evaporation and crystallization) and allocation of sources to meet various process constraints. The unique aspects of evaporation and crystallization are included. The problem has to consider varying sink inlet flow rate and composition requirements that necessitate the use of path equations for global tracking of all species of interest. Constraint propagation is invoked to determine bounds on allowable sink flow rate and composition constraints. A two-step solution procedure is proposed. The first step consists of interception task identification for a given stream in the process. This step is useful as it enables elimination of infeasible nodes prior to detailed design. The second step (previously described in part 1) consists of design of a separation network for the identified separation task. This paper couples the separation task (via evaporation and crystallization) to the allocation requirements of various sinks. A case study is included to demonstrate the broad applicability and value of this approach. Acknowledgment

1.0 - zA3 - zB3 ) zW 3 Scrubber 3. It is assumed that 5 kg/h of sodium nitrate and 20 kg/h of ammonium nitrate are transferred from the gas stream (exiting the prilling tower) to the water stream from neutralization (i.e., stream 3). Also, the air content of all gas sources (i.e., streams 6-8) is constant at 100 kg/h.

F3zB3 + 5 ) F4zB4 F3zA3 + 20 ) F4zA4 F3 + G6 ) F4 + G7 75F4 ) 100F3 141G7 ) 116G6

The authors acknowledge the support of Solutia Inc. toward this research.

27F4zA4 ) 20G6yA6

Appendix 1

9F4zB4 ) 30G6yB6

The following equations have been used to model the various units in the ammonium nitrate process described in the case study. Figure 6 lists the variables

W G7yW 7 ) G6y6

2854


G7yA7 ) G6yA6 - 20 G7yB7 ) G6yB6 - 5 G6(1.0 - yA6 - yB6 - yW 6 ) ) 100 Air 1.0 - yA6 - yB6 - yW 6 ) y6 Air 1.0 - yA7 - yB7 - yW 7 ) y7

1.0 - zA4 - zB4 ) zW 4 Concentration/Prilling. Stream 5 is sent to the concentration block. This will affect the gas composition coming off the prilling tower. It is assumed that 9 kg/h of ammonium nitrate generated in the reaction block will enter the gas stream from the prilling tower (i.e., stream 6), in addition to the amount already present.

F4 + 9 + 100 ) F5 + G6 F4zA4 + 9 ) F5zA5 + G6yA6 F4zB4 ) F5zB5 + G6yB6 1.0 - zA5 - zB5 ) zW 5 Scrubber 1. It is assumed that 3 kg/h of sodium nitrate and 5 kg/h of ammonium nitrate are transferred from the gas stream, exiting scrubber 3, to the freshwater stream.

92F10 ) 100F9 F9zB9 + 3 ) F10zB10 F9zA9 + 5 ) F10zA10 F9 + G7 ) F10 + G8 F9zA9 + G7yA7 ) F10zA10 + G8yA8 F9zB9 + G7yB7 ) F10zB10 + G8yB8 G8(1.0 - yA8 - yB8 - yW 8 ) ) 100 Air 1.0 - yA8 - yB8 - yW 8 ) y8

1.0 - zA9 - zB9 ) zW 9 1.0 - zA10 - zB10 ) zW 10 Nomenclature Cfresh ) $/kg of freshwater e ) index denoting ESA fp ) vector of performance equations relating outputs to inputs for unit p Fi ) total flow rate of the ith source, kg/h ) total flow rate of the ith intercepted source, kg/h FInt i i ) index denoting sources in set WASTE_SOURCES j ) index denoting sources in set TERMINAL_SOURCES k ) index denoting sources in set INTERNAL_SOURCES LSink ) total inlet flow rate of the mth sink, kg/h m

) total flow rate of freshwater, kg/h LFresh i li,m ) individual flow rate from the ith source to the mth sink, kg/h Fresh li,m ) individual flow rate of freshwater from the ith source to the mth sink, kg/h ML ) individual flow rate of mother liquor from the ith li,m source to the mth sink, kg/h Pure li,m ) individual flow rate of pure water from the ith source to the mth sink, kg/h m ) index denoting sinks MLi ) total flow rate of mother liquor from the ith source, kg/h MassAi ) initial mass load of salt A in the ith source, kg/h MassBi ) initial mass load of salt B in the ith source, kg/h NetAp ) net loss of salt A within process unit p, kg/h NetBp ) net loss of salt A within process unit p, kg/h NSource ) total number of sources in set WASTE_SOURCES NSink ) total number of sinks in set SINKS Terminal ) total number of sources in set NSource TERMINAL_SOURCES Internal NSource ) total number of sources in set INTERNAL_SOURCES Total NSink ) total number of sinks in set PROCESS_SINKS NC ) total number of coolants NH ) total number of heating media NE ) total number of ESAs p ) index denoting sinks in set PROCESS_SINKS Ri ) total flow rate of the residual liquor for the ith intercepted source, kg/h Salti ) total flow rate of the salt stream for the ith intercepted source, kg/h Soli ) total flow rate of pure water for the ith intercepted source, kg/h TACi ) total annualized cost of the separation network, $/year TSource ) initial temperature of the ith source, °C i tCool ) operating temperature of the Cth coolant, °C C tHot ) operating temperature of the Hth heating media, H °C UNominal ) initial discharge of salt A from all terminal sources, kg/h UFinal ) final discharge of salt A from all terminal sources, kg/h u ) index denoting either salt A or B VNominal ) initial discharge of salt B from all terminal sources, kg/h VFinal ) final discharge of salt B from all terminal sources, kg/h WNominal ) initial discharge of water from all terminal sources, kg/h WFinal ) final discharge of water from all terminal sources, kg/h zAi ) initial weight fraction of salt A in the ith source zBi ) initial weight fraction of salt B in the ith source zA,Int ) weight fraction of salt A following interception of i the ith source zB,Int ) weight fraction of salt B following interception of i the ith source Resi zi,A ) weight fraction of salt A in the residual liquor of the ith intercepted source Resi zi,B ) weight fraction of salt B in the residual liquor of the ith intercepted source ML zi,A ) weight fraction of salt A in the mother liquor of the ith intercepted source ML zi,B ) weight fraction of salt B in the mother liquor of the ith intercepted source zui ) weight fraction of salt u in the ith source

Ind. Eng. Chem. Res., Vol. 40, No. 13, 2001 2855 zu,Int ) weight fraction of salt u following interception of i the ith source Sink zm,A ) inlet weight fraction of salt A for the mth sink Sink ) inlet weight fraction of salt B for the mth sink zm,B Sets COOLANT ) set of coolants COOLANT_COST ) set of cost of coolants ESA ) set of ESAs HOT ) set of heating media HOT_COST ) set of cost of heating media INPUT ) set of input streams to unit INTERNAL_SOURCES ) set of internal sources OUTPUT ) set of output streams to unit PROCESS_SINKS ) set of process sinks SINKS ) set of sinks TERMINAL_SOURCES ) set of terminal sources WASTE_SOURCES ) set of waste sources Greek Letters R ) factor indicating required reduction in water discharge β ) factor indicating required reduction in discharge of salt A γ ) factor indicating required reduction in discharge of salt B φ ) function relating final water discharge to intercepted source variables φ ) function relating final salt A discharge to intercepted source variables µ ) function relating final salt B discharge to intercepted source variables θ ) function relating final discharge to intercepted source variables for the dependent case Subscripts A ) salt A B ) salt B C ) coolants e ) ESAs H ) heating media i ) ith wastewater source j ) jth terminal wastewater source m ) mth sink Sink ) sink Source ) source Superscripts A ) salt A B ) salt B Cool ) temperature of the coolant Final ) final discharge Hot ) temperature of heating media Int ) integrated ML ) mother liquor Nominal ) nominal discharge Pure ) pure water Resi ) residual liquor Sink ) sink Source ) source Terminal ) terminal Total ) total u ) uth salt

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Received for review October 16, 2000 Accepted April 13, 2001 IE0008934

Development of Heat-Integrated Evaporation and Crystallization

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