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In the last three decades much effort has been devoted in process integration as a way to improve economic and environmental performance of chemical processes. Although the established frameworks have undergone constant refinement toward formulating and solving complicated process integration problems, less attention has been drawn to the problem of sequential applications of mass integration. This work addresses this problem by proposing an algorithm for optimal ordering of the process sinks in direct recycling problems, which is compatible with the typical mass integration formulation. The order consists in selecting the optimal sink at a specific integration step given the selection of the previous steps and the remaining process sources. Such order is identified through a succession of preemptive goal programming problems, namely of optimization problems characterized by more objectives at different priority levels. Indeed, the target for each sink is obtained by maximizing the total flow recycled from the available process sources to this sink and then minimizing the use of pure sources, starting from the purest one; the hierarchy is respected through a succession of linear optimization problems with a single objective function. While the conditional optimality of the algorithm holds always, a thorough statistical analysis including structured to random scenarios of process sources and process sinks shows how frequently the sequential ordering algorithm is outperformed with respect to the total recycled amount by a different selection of process sinks with the same cardinality. Two more case studies proving the usefulness of ordering the process sinks are illustrated. Extensions of the algorithm are also identified to cover more aspects of the process integration framework.

The reduction of the cost and environmental impact of industrial processes is among the most relevant challenges for chemical engineers to design sustainable processes [

Among the various methods and objectives of process integration, mass integration through direct recycling and mass exchange networks is a methodology that allows recovering part of the waste streams of a process (i.e., process sources) and reusing them in appropriate process units (i.e., process sinks), thus reducing the purchase of fresh resources [

More recent methodological development in the problem formulation and solution strategies of mass integration has focused on advanced optimization techniques. In this regard, Bagajewicz and Savelski presented a formulation based on mixed integer linear programing (MILP) to solve the water allocation problem in process plants when a single contaminant is present [

Recent works have also extended the complexity of the mass integration problem formulation. This includes, for instance, multi objective optimization of waste water integration problems with multiple pollutant substances [

In all the approaches mentioned above, the main focus is to find the optimal solution of the respective mass integration problem (e.g., proposing a final optim0al design of a direct recycling or mass exchange network) without proposing a sequential order of actions to reach this optimal solution. However, creating an order of integration actions can be of special interest in practical engineering problems where the optimal overall solution cannot be realized at once but only as a sequence of steps. Some reasons can be significant deviations in the actual performance of the integration steps (e.g., because of modeling uncertainties or simplifications), the fact that it may not be a priori known if all the necessary integration steps will be executed (e.g., because of time and capital expenses limitations considering also the relative significance of the integration step) or, due to process operability reasons, the transition to the optimal integration design has to be practically sequential (e.g., to avoid production loss through extended shut-down). A further more qualitative reason refers to the enhanced interpretability of the proposed solutions that is often critical for the implementation of new designs in industrial practice.

This work addresses this problem by proposing a new algorithm for optimal ordering of the process sinks in direct recycling mass integration problems. The order consists in selecting the optimal sink at a specific integration step when the selection of the previous steps and the remaining process sources are given, ensuring also that the overall optimal direct recycling target is reached. The latter is achieved through a succession of preemptive goal programming (PGP) problems, namely of optimization problems characterized by more objectives at different priority levels [

From the classic direct recycling perspective, a generic chemical process is characterized by the presence of waste streams comprising a target compound which in presence of several impurities can be partially or fully recovered through direct recycling to suitable process units. Such waste streams are referred to as process sources and all the process units that can accept recycled waste streams are named process sinks. Typically, fractions of process sources are mixed together with an external fresh source (e.g., pure stream of the target compound being recycled) to satisfy the required amount of the target compound in a given process sink. At the same time, each process sink can accept a predefined maximum allowable level of impurity.

With this formulation (

s.t:

where _{n} is the amount of fresh resource used in process sink n,

It should be noted that in this particular problem formulation there is one type of fresh source assumed to be 100% pure, all possible impurities are lumped into one, and there is no minimum allowable level of impurity at any process sink. Typically, the values of

loads (

This problem is based on the simplifying assumption that the impurities in the process sources recycled back to the process sinks after integration have a negligible impact on the respective process output variables, at least as long as the maximum allowable impurity constraints are satisfied; in this way, there is no need to include any process model that describes the impact of the recycled impurities on the performance of the process units. This assumption practically allows this kind of mass integration problem to be linear and solved to global optimality. Typically, there may be more than one optimal solutions, characterized by different connections among process sources and sinks.

Solving the direct recycling problem, as presented above, leads to an optimal target in the form of minimum fresh sources or maximum recycling of a target compound and one (of possible many or even infinite) optimum source-to-sink connectivity pattern that corresponds to the optimal target. In large problems (i.e., with many sources and sinks), the structure of the global solution can be quite complicated, impractical to realize in its full extent or all at once, and difficult to interpret. Additionally, significant simplifications may have been made in the problem formulation and this may have an impact on the reliability of the overall solution of the direct recycling problem from industrial practice point of view.

For these reasons, a sequential methodology implying some ordering of recycling actions can be beneficial. In principle, it would be desirable that a sequential method can also lead to the same overall optimum of the superstructure approach, if executed in all its steps. It is also important for such an approach to result in a decreasing gradient of the respective cumulative curve for the recycling loads as a function of the steps of the sequential method; in this way, the user can terminate the computational procedure quite in advance without having to compute many subsequent steps of marginal improvement. It is also important to know if the obtained optimum at any termination point is also globally optimum given the number of steps (i.e., not only sequentially optimum in the sense that an optimum is found at any step given the decisions made at previous steps) or at least to have a list of necessary conditions for this to be true.

One can think of many ways to define the concept of sequential steps in the direct recycling problem. For instance, one step of the sequential method could be defined as one source-to-sink connection (i.e., using the terminology of the direct recycling formulation this corresponds to the variable

Remark-1: In _{j}) is updated at every process sink selection step (j) according to the procedure of box-1 of the algorithmic scheme, which is illustrated in example-1.

Example-1

At some process sink selection step j of the algorithm there are three available process sources, namely SR_{1}

(50 kg/sec), SR_{2} (40 kg/sec), and SR_{3 }(60 kg/sec), the values in parenthesis denoting the loads of the sources. This means that_{max}) to which the process sources in SRav_{j} recycle the maximum possible load and impurity. For instance, this could be a process sink with _{1}, SR_{2}, SR_{3} respectively and the maximum allowable impurity level of the sink, 20 kg/sec and 30 kg/sec are recycled from SR_{1} and SR_{2}, respectively, and no recycle from SR_{3}. This means that SRsel and SRnew in box-8 of _{1} and SRnew_{2} have now remaining loads of 30 kg/sec and 10 kg/sec, respectively. Moving now to the next step (j = j + 1) and updating the available process sources set in box-1 of _{1}, SRnew_{2}, SR_{3}}.

Remark-2: The impurity concentrations in the process sources are assumed not to change after closing the recycle loops. However, this is not a particularity of the sequential ordering algorithm presented here, since the same assumption is implicitly made in the general formulation of direct recycling problems (i.e., in its LP form), when it is stated that the recycled impurities do not affect the performance of the process sinks and, consequently, the composition of the process sources does not change.

Remark-3: As shown in box-8 of _{j} for every process sink selection step (i.e., box-2 of

Remark-4: From box-3 to box-5 there is an internal loop at every process sink selection step. In this loop the maximum recycled amount to each process sink in SKav_{j} is independently calculated (i.e., by solving a LP such as the one described by Equations (1) to (8) with N_{sinks} = 1) considering the process sources in SRav_{j}. It should be reminded here that the solution of each LP problem may not be unique, since more than one (or even infinite) combinations of the process sources may be maximizing the recycled amount (_{n}.

Remark-5: After finding SK_{max} (box-6 of

Remark-6: In case that in a process sink selection sept j, more than one SK_{max} are found in box-6 of

Remark-7: Since the availability of process sources at SRav_{j} is more limited than at SRav_{j−1}, it follows that

Remark-8: At the end of the ordering algorithm, when all the sinks have been selected for recycling, the total amount recovered from the waste streams corresponds necessarily to the one calculated in the general LP problem (Equations (1) to (8) considering all the sinks simultaneously). Actually, it is already known that starting from any sink and following a similar procedure like the one described in

In the same way, a subgroup of process sinks is characterized by a fixed target for maximum recycle, given a set of available process sources. Therefore, at a general step of the process sink ordering algorithm, the total amount of recycle corresponds necessarily to the target quantities calculated in the general LP problem (Equations (1)-(8)) considering that group of n process sinks. The ordering algorithm is further illustrated in example-2.

Example-2

The sets of process sinks and process sources in _{1} sets, respectively. If no mass integration was performed, the process sinks would require 280 kg/sec of the fresh target compound, while 310 kg/sec of waste

Process sinks | |||||
---|---|---|---|---|---|

Required amount of target compound (kg/sec) | Maximum allowable impurity (% w/w) | Maximum allowable impurity (kg/sec) | |||

SK1 | 50 | 2 | 1 | ||

SK2 | 40 | 5 | 2 | ||

SK3 | 60 | 10 | 6 | ||

SK4 | 30 | 20 | 6 | ||

SK5 | 100 | 25 | 25 | ||

Process sources | |||||

Waste load kg/sec | Impurity content (% w/w) | Impurity content (kg/sec) | |||

SR1 | 30 | 1 | 0.3 | ||

SR2 | 40 | 5 | 2 | ||

SR3 | 50 | 10 | 5 | ||

SR4 | 50 | 15 | 7.5 | ||

SR5 | 40 | 20 | 8 | ||

SR6 | 100 | 30 | 30 | ||

should be disposed.

In this case, for the first process selection step (j = 1), one may heuristically start from SK5 in box-3, namely the sink with the maximum required amount of target compound; if this amount could be recycled from the available process sources, there would be clearly no further need to check the rest of the sinks at this process selection step. One solution of the respective LP maximization problem could be to recycle the waste loads of SR3 and SR4. This would satisfy recycling 100 kg/sec to SK5, but only 12.5 kg/sec of impurities would be recycled, while SK5 can take up to 25 kg/sec of impurities. According to the PGP problem in box-7, the goal is to successively replace purer with more impure sources to achieve the goal of maximizing the impurity (or equivalently minimizing the use the purer sources at earlier steps of the process sink selection). It can be easily verified that solving the respective PGP problem results in using SR4 (6.7 kg/sec), SR5 (40 kg/sec) and SR6 (53.3 kg/sec). This recycles 100 kg/sec to SK5 and 25 kg/sec of impurity, which is its maximum allowable impurity level. Thus, in box-8 of

Using the same heuristic like before for the next process selection step (j = 2), it now makes sense to start from SK3 in box-3. Doing so, one can easily find that the maximum required amount of target compound (60 kg/sec) can be indeed recycled to SK3 and the impurities can be maximized and reach again the maximum allowable impurity level with the following solution: SR2 (5 kg/sec), SR3 (50 kg/sec) and SR4new (5 kg/sec). Thus, in box-8 of

Following this procedure until all sinks are ordered (j = 5, not presented here because of space limitations) results in the following ordering of process sinks

recycled from the process sources to the process sinks, reducing the process requirements for fresh target compound to only 18.7 kg/sec and the respective waste streams to 48.7 kg/sec. It can be easily verified that the same targets are calculated by the general LP problem of Equations (1) to (8), when all process sinks are considered simultaneously.

Comparing the recycling cumulative curve with the cumulative curve of the required amount of target compound, it can be seen that no fresh compound is required in process sinks SK5 and SK3. It can also be seen that the cumulative recycling curve has a decreasing slope, which, however, in this case does not lead to marginally increasing recycling steps, and, therefore, it would not make sense to terminate the algorithm before considering all process sinks.

The ordering algorithm assures the conditional optimality of the process sink selection at a given algorithmic step (i.e., given the previously selected process sinks) and also that the recycled amount to it cannot be superior to any previous one with the available set of process sources. However, it does not guarantee that for a given cardinality of the process sink selection (i.e., ordering the first n sinks out of the totality of N_{sinks}) the sequential procedure leads to the maximum cumulative recycling among any other group of n sinks. Nevertheless, one trivial condition can point out whether another group of n sinks has the potential to lead to a higher recycled amount or, instead, the first n sinks of the ordering algorithm are globally optimal. The condition requires that the cumulative recycling to the first n ordered process sinks is superior to the cumulative required amount of the target compound of any other group of n sinks. This is expressed by Equation (9):

where

Thus, we can make the following statements:

Statement-1: Any group of process sinks that violates condition (9) may be characterized by a higher target

for maximum recycling (i.e., such a group is a candidate for outperforming the ordered group of process sinks and thus it should be further tested, for example, by solving the respective linear optimization problem, expressed by Equations (1) to (8), for this specific candidate group of process sinks.

Statement-2: If no group violates the condition (9), then the ordering algorithm has identified the globally maximum recycling for the cardinality of sinks at the specific step of the process sink selection. Therefore, the condition (9) is a necessary condition.

For instance, in the previously presented example-2, at the third process sink selection step, where

In this particular example,

Since the ordering algorithm does not always guarantee global optimality for a given cardinality of ordered pro- cess sinks (i.e., other than the first and the last step of the algorithm, namely when only the first sink or all the sinks of the direct recycling problem are ordered), it is interesting to test the frequency of such cases under different conditions. Thus, a thorough computational screening of direct recycling problems was performed includ- ing various factors differentiating the scenarios tested.

The first factor refers to the target compound for recycling with three levels of differentiation, namely inert, generated or consumed compounds in the process system. The tested levels of depletion or excess of generation for the last two cases range from 10% to 90% with respect to the fresh source required before any recycling. The second factor refers to cases with either higher number of sources or sinks, the number of sources and sinks ranging each from 1 to 15. The third factor refers to the type of sampling with respect to the loads and the impurity concentrations of the process sources and the required amount of target compound and maximum allowable level of impurity of the process sinks. Three types of sampling were considered: latin-hypercube without space filling, allowing repetitions and totally randomized.

Combinations of these factors have been used to define the different scenario categories (e.g., consumed compound with 90% depletion, higher number of process sources than sinks and totally randomized sampling, may refer to one combination of factors defining one scenario). Multiple samples were created (i.e., 1000 for each sub-problem) based on the size of the problem (i.e., number of sinks) and the loads and impurity combinations.

For the evaluation of the scenarios, three indices were defined (Equations (11)-(13)):

Factors of differentiation | Levels of differentiation | |||||
---|---|---|---|---|---|---|

Type of target compound to be recycled | Inert (P1) | Depleted (P2) 10 to 90% | Generated (P3) 10 to 90% | |||

Number of process sources and sinks | Sources > sinks | Sinks < sources | ||||

Type of sampling | Latin hypercube (LH) | Allowing repetitions (Rand) | Totally random (Totrand) | |||

where

The number of sinks obviously defines the steps of the process sink selection and, thus, also the potential violation points for condition (9). Therefore,

The results of the statistical evaluation on the basis of the three indices and the lumped cases of

A more detailed look into sub-cases of the lumped P2 case is provided in

trends as the level of depletion is decreasing (i.e., P2_10 means that only 10% of the required amount of the target compound in the process sinks exists in the process sources, while the rest 90% is depleted in the intermediate process units): the values of

Other sub-cases of P2, such as those presented in

One particular case of interest for the application of the sequential ordering algorithm is when the realization of mass integration through direct recycling can be carried out only by a gradual retrofitting, namely reducing the fresh source of the target compound to one process sink at a time. In this case study, a pharmaceutical process producing an intermediate has to undergo mass integration by recycling of its most relevant solvent, the N-methylpyrrolidone, whose price is 2500 $/tonne.

The recycling of the process sources to the sinks can be realized without serious problems that would interrupt the production of the main chemical, if one sink is disconnected from the rest of the process at a time, keeping all the other sinks in normal operation. Before any process sink is disconnected from the rest of the process, it discharges enough amount to a buffer tank to keep the process operating with the same load discharged into the process sources. Then, it is linked with the pipes of the process sources to receive the recycling load and, subsequently, reconnected with the rest of the process. The buffer tank is then used for the same purpose in the next process sink, and so on. Disconnection of the sink and reconnection with the process sources requires in total 3 days. This means that realizing any direct recycling strategy for all 5 process sinks requires 15 days.

In this kind of problem, the sequential ordering algorithm can provide the optimal order of the sinks so that the sinks with the highest recycling are disconnected and reconnected first. In this way, fresh resources are saved during the intermediate period of realizing the mass integration strategy, although at the end of the procedure other sink orders would also recycle the same total amount. Therefore, it is convenient to integrate the process sinks following the cumulative curve to obtain the highest economic saving from the first step.

The application of the algorithm results in the process sink ordering presented in

Process sinks | ||||
---|---|---|---|---|

Required amount of target compound (kg/hr) | Maximum allowable impurity (% w/w) | Maximum allowable impurity (kg/hr) | ||

SK1 | 300 | 1 | 3 | |

SK2 | 250 | 2 | 5 | |

SK3 | 280 | 10 | 28 | |

SK4 | 150 | 20 | 30 | |

SK5 | 100 | 30 | 30 | |

Process sources | ||||

Waste load (kg/hr) | Impurity content (% w/w) | Impurity content (kg/hr) | ||

SR1 | 100 | 5 | 5 | |

SR2 | 120 | 10 | 12 | |

SR3 | 180 | 20 | 36 | |

SR4 | 120 | 30 | 36 | |

SR5 | 150 | 50 | 75 | |

SR6 | 150 | 70 | 105 | |

Ordered sinks | Recycled amount per sink (kg/hr) | Cumulative recycled amount (kg/hr) | Cumulative solvent savings ($) | Sources connected to sink (kg/hr) | |||||
---|---|---|---|---|---|---|---|---|---|

SR1 | SR2 | SR3 | SR4 | SR5 | SR6 | ||||

SK3 | 275 | 275 | 49500 | 100 | 120 | 55 | 0 | 0 | 0 |

SK4 | 141.7 | 416.7 | 124506 | 0 | 0 | 125 | 16.7 | 0 | 0 |

SK5 | 100 | 516.7 | 217512 | 0 | 0 | 0 | 100 | 0 | 0 |

SK2 | 11.3 | 528 | 312552 | 0 | 0 | 0 | 3.3 | 8 | 0 |

SK1 | 6 | 534 | 408672 | 0 | 0 | 0 | 0 | 6 | 0 |

Ordered sinks | Recycled amount per sink (kg/hr) | Cumulative recycled amount (kg/hr) | Cumulative solvent savings ($) | Sources connected to sink (kg/hr) | |||||
---|---|---|---|---|---|---|---|---|---|

SR1 | SR2 | SR3 | SR4 | SR5 | SR6 | ||||

SK1 | 60 | 60 | 10800 | 60 | 0 | 0 | 0 | 0 | 0 |

SK3 | 230 | 290 | 63000 | 40 | 120 | 70 | 0 | 0 | 0 |

SK2 | 25 | 315 | 119700 | 0 | 0 | 25 | 0 | 0 | 0 |

SK4 | 128.3 | 443.3 | 199494 | 0 | 0 | 85 | 43.3 | 0 | 0 |

SK5 | 90.7 | 534 | 295614 | 0 | 0 | 0 | 76.7 | 14 | 0 |

presents an alternative solution (i.e., not obtained by the sequential ordering algorithm), resulting in the same overall recycled amount. It is worth noticing that, although the two solutions have different structure (i.e., source-to-sink connectivity), the same total amount is recycled from each process source. However, these two solutions have significantly different economic performance. The solution based on the sequential ordering algorithm saves 113058 $ more in this intermediate period for realizing the full source-to-sink connectivity of the direct recycling solution. The reason for the better performance of the sequential ordering algorithm lies in the property of the monotonically decreasing gradient of its respective cumulative curve of the recycled amount.

Another possible application of the methodology is when the hypothesis of negligible impact on the process sink performance by the recycled impurities is not anymore valid. From modelling point of view, this means that a detailed model of the process sink is available, which can describe the impact of the impurities on the process sink output streams, this impact being nonlinear in the general case. In such cases, the calculation of the target for mass integration through direct recycling requires to solve a NLP optimization problem, whose solution can be hard to calculate. Besides the computational difficulties, another practical issue is related to the frequent use of advanced process simulators for rigorous modelling of the process units, which are not typically designed to handle NLP superstructure optimization. Moreover, in industrial practice, it is often required to verify and understand the impact of any process change (e.g., such as recycling impure process streams), even when rigorous process models are available.

To tackle this problem, the sequential ordering algorithm offers an interesting alternative; the optimal ordering of process sinks obtained by solving the respective linear representation of the problem can be considered as the basis for a subsequent sequential procedure to calculate the optimum recycling target of the nonlinear representation of the process. Thus, the steps of this new sequential procedure follow those of the ordering algorithm applied to the linear representation of the problem. However, because of the nonlinear relations among process streams and units, the impurity content of the process sources may change significantly when the recycling is realized; this in turn can cause the violation of the maximum allowable impurity constraints of the process sinks, if the recycling loads are not properly adjusted. This adjustment of the recycling loads is performed sequentially according to the order of the process sinks. This has the advantage of studying the nonlinear performance of only one sink at a time, starting from the sinks with the highest recycling loads, which are more likely to have a greater impact on the overall process performance.

As an example, the process depicted in ^{3} reactor, the following reaction takes place at 150˚C and 1 bar:

The kinetics follow the power law with respect to compound concentrations, with Arrhenius expressions for the kinetic constants both for the direct and reverse reactions. The evaporation at flash-1 takes place at 150˚C and 1 bar, the decanter operates at 50˚C and 1 bar, and the evaporation at flash-2 at 70˚C and 1 bar. The compositions of the streams for both the vapor-liquid and liquid-liquid phase separation at thermodynamic equilibrium are calculated with the Non-Random-Two-Liquid (NRTL) method. These kinetic expressions and thermodynamic models introduce strong nonlinearities in the performance of the respective process units. A full stream table of the process is provided in ESI (Part 3).

In this case study, it is desired to minimize the use of the fresh amount of compound A in the process sinks 1, 2, 3 and 4 through the recycling of the process stream 16 (“process source”). The relevant process sinks and source data are provided in

First, the recycling target according to the linear representation of the problem is calculated resulting in 7.11 kmol/hr. The respective cumulative curve is presented in

Process sinks | ||||
---|---|---|---|---|

Required amount of target compound (kmol/hr) | Maximum allowable impurity (% mol/mol) | Maximum allowable impurity (kmol/hr) | ||

SK1 | 13 | 20 | 2.6 | |

SK2 | 15 | 15 | 2.25 | |

SK3 | 10 | 5 | 0.5 | |

SK4 | 10 | 3 | 0.3 | |

Process sources | ||||

Waste load (kmol/hr) | Impurity content (% mol/mol) | Impurity content (kmol/hr) | ||

SR1 | 100 | 79.5 | 79.5 | |

process sources and sinks. This case study is only used to highlight a potential way to decompose a complicated NLP direct recycling problem with the help of the proposed sequential ordering algorithm for the process sinks.

The design of optimal mass integration networks often results in complicated superstructure formulations and solutions, which can be cumbersome to interpret and realize in practice. Therefore, in many cases a more structured construction of the solution based on sequential approaches is advantageous. This work addresses this problem by proposing an algorithm for optimal ordering of the process sinks in direct recycling problems, which is compatible with the typical mass integration formulation and reaches the same recycling target in the case of the linear representation of the problem. The obtained order consists in selecting the optimal sink at a specific integration step given the selection of the previous steps and the remaining process sources. Such order is identified through a succession of preemptive goal programming problems, namely of optimization problems characterized by more objectives at different priority levels. Indeed, the target for each sink is obtained by maximizing the total flow recycled from the available process sources to this sink and then minimizing the use of pure sources, starting from the purest one; the hierarchy is respected through a succession of linear optimization problems with a single objective function.

In this study, the sequential ordering algorithm has undergone a thorough statistical test (case study-1) to identify the frequency of conditions under which, at a certain step, the algorithm has selected the group of sinks with the highest target for recycling (global optimality) among all the groups with the same cardinality. It has been shown that in the vast majority of the cases the algorithm identifies a globally optimum group of process sinks for each step of process sink selection: the most challenging cases are those where the process sinks are more than the process sources in inert or slightly decreasing target compound scenarios. Of course, the conditional optimality is always guaranteed at each process selection step, while ensuring that at the end of the process sink ordering procedure the maximum amount of the target compound is recycled. It would be, however, useful to define simple, yet stricter algebraic conditions than those proposed herein, to efficiently identify these groups of process sinks that need to be tested for outperforming the process sinks of the same cardinality selected by the sequential ordering algorithm.

Two more case studies were conducted to demonstrate the usefulness of ordering the process sinks. In the case study of gradual retrofitting (case study-2), ordering the process sinks resulted in significant economic savings, compared to a solution with an equivalent recycling target but with different ordering strategy. In the case study considering the nonlinear characteristics of the process units with respect to the recycled impurity amounts (case study-3), the sequential ordering algorithm provides an interesting alternative for decomposing the overall NLP problem. The solutions may be by definition suboptimal compared to the overall NLP problem, but are significantly easier to compute, especially in practical applications where commercial process simulators are used, which are cumbersome to introduce into the NLP superstructure formulation. Although the potential is demonstrated in a simple case study, it is evident that further work is needed to optimally tune the use of the sequential ordering algorithm in this kind of problems of larger size. In this direction, one should try to minimize the optimality gap compared to the overall NLP problem and interpret the deviation from the solutions obtained by the linear problem representation.

The sequential ordering algorithm can be properly adapted and extended to cover more aspects of the process integration framework, including heat integration, property integration, and mass exchange networks. This will result in a unified framework for sequential process integration, which can be used in parallel with the superstructure problem formulations to enhance the realization and interpretability of the obtained solutions in complicated integration problems.

Filippo Marchione,Stavros Papadokonstantakis,Konrad Hungerbuehler, (2016) Sequential Ordering Algorithm for Mass Integration: The Case of Direct Recycling. Advances in Chemical Engineering and Science,06,158-182. doi: 10.4236/aces.2016.62018

As mentioned in the manuscript (Section 2.3), the sequential ordering algorithm does not guarantee global optimality for a given cardinality of ordered process sinks (i.e., with the exception of the first and last step of the algorithm, namely when only the first sink or all sinks are ordered). In the manuscript, we stated one simple condition (Equations (9)) that can be used to identify groups of process sinks that may outperform, from a global optimality perspective, those selected by the sequential ordering algorithm for the same cardinality. We present the following two examples to illustrate the application of this condition.

Example-1

Let us consider the sets of process sinks and sources in

The sum of the required amounts of the target compound at the process sinks SK2 and SK3 exceeds the corresponding recycled amount at the ordered sinks SK1 and SK3 of the sequential algorithm, namely:

Therefore, the couple of process sinks {SK2, SK3} is a potential group that outperforms the selected group of sinks by the ordering algorithm (i.e.,

Example-2

Let us consider the sets of process sinks and sources in

The sum of the required amounts of the target compound at the process sinks SK3 and SK1 exceeds the corresponding recycled amount at the ordered sinks SK1 and SK3 of the sequential algorithm, namely:

Therefore, the couple of sinks {SK3, SK1} is a potential group that outperforms the selected group of sinks by the ordering algorithm (i.e.,

Process sinks | ||||
---|---|---|---|---|

Required amount of target compound (mol/sec) | Maximum allowable impurity (% mol/mol) | Maximum allowable impurity (mol/sec) | ||

SK1 | 150 | 10 | 15.0 | |

SK2 | 95 | 17 | 16.2 | |

SK3 | 90 | 19 | 17.1 | |

SK4 | 80 | 2.5 | 2.0 | |

Process sources | ||||

Waste load mol/sec | Impurity content (% mol/mol) | Impurity content (mol/sec) | ||

SR1 | 100 | 15 | 15 | |

SR2 | 90 | 40 | 36 | |

SR3 | 120 | 50 | 60 | |

Process sinks | |||
---|---|---|---|

Required amount of target compound (mol/sec) | Maximum allowable impurity (% mol/mol) | Maximum allowable impurity (mol/sec) | |

SK1 | 150 | 10 | 15.0 |

SK2 | 95 | 17 | 16.2 |

SK3 | 90 | 19 | 17.1 |

SK4 | 80 | 2.5 | 2.0 |

Process sources | |||

Waste load mol/sec | Impurity content (% mol/mol) | Impurity content (mol/sec) | |

SR1 | 180 | 15 | 27 |

SR2 | 30 | 40 | 12 |

SR3 | 30 | 50 | 15 |

Tables S3-S5 present the statistical evaluation of the indices Ind1, Ind2, and Ind3, respectively, according to their definition in the manuscript for each category of scenarios. Diverse categories of scenarios were identified based on factors and their levels of differentiation described in

For instance, the number of scenarios for the P2_10 problems refers to the case of a 10% remaining (i.e., 90% depleting) target compound in the process sources, the P2_30 problems refers to the case of a 30% remaining (i.e., 70% depleting) target compound in the process sources, and so on. Thus, summing up the number of scenarios for the problems for P2_10 up to P2_90 (i.e., 6900 for each type of these problems) results in the total number of scenarios for the P2 (all problems) case, namely 34500 scenarios.

Categories of scenarios | Ind1 | |||
---|---|---|---|---|

Mean | Median | Standard deviation | Trimmed mean over [5% - 95%] percentiles | |

All problems (75,900 scenarios) | 69 | 2 | 244 | 22 |

P1 (all problems, 6,900 scenarios) | 47 | 2 | 176 | 16 |

P2 (all problems, 34,500 scenarios) | 113 | 4 | 319 | 53 |

P3 (all problems, 34,500 scenarios) | 28 | 1 | 139 | 7 |

Sinks > sources (33,000 scenarios) | 119 | 9 | 320 | 59 |

Sinks < sources (42,900 scenarios) | 30 | 1 | 153 | 5 |

LH problems (25,300 scenarios) | 60 | 2 | 229 | 17 |

Rand problems (25,300 scenarios) | 70 | 2 | 248 | 23 |

Totrand problems (25,300 scenarios) | 76 | 3 | 253 | 28 |

P2_10 problems (6900 scenarios) | 179 | 9 | 433 | 103 |

P2_30 problems (6900 scenarios) | 154 | 7 | 385 | 85 |

P2_50 problems (6900 scenarios) | 111 | 5 | 288 | 58 |

P2_70 problems (6900 scenarios) | 72 | 4 | 211 | 33 |
---|---|---|---|---|

P2_90 problems (6900 scenarios) | 51 | 2 | 181 | 19 |

P3_10 problems (6900 scenarios) | 20 | 1 | 125 | 4 |

P3_30 problems (6900 scenarios) | 22 | 1 | 122 | 4 |

P3_50 problems (6900 scenarios) | 28 | 1 | 146 | 6 |

P3_70 problems (6900 scenarios) | 32 | 1 | 147 | 8 |

P3_90 problems (6900 scenarios) | 39 | 2 | 154 | 13 |

P1, sinks > sources (3000 scenarios) | 86 | 8 | 249 | 39 |

P1, sinks < sources (3900 scenarios) | 17 | 1 | 72 | 5 |

P2, sinks > sources (3000 scenarios) | 189 | 24 | 403 | 119 |

P2, sinks < sources (3900 scenarios) | 56 | 2 | 217 | 15 |

P3, sinks > sources (3000 scenarios) | 56 | 4 | 202 | 20 |

P3, sinks < sources (3900 scenarios) | 7 | 1 | 43 | 2 |

P1, LH problems (2300 scenarios) | 33 | 2 | 120 | 13 |

P1, rand problems (2300 scenarios) | 47 | 2 | 180 | 15 |

P1, totrand problems (2300 scenarios) | 61 | 3 | 215 | 21 |

P2, LH problems (2300 scenarios) | 111 | 4 | 316 | 51 |

P2, rand problems (2300 scenarios) | 113 | 4 | 320 | 52 |

P2, totrand problems (2300 scenarios) | 116 | 5 | 320 | 55 |

P3, LH problems (2300 scenarios) | 14 | 1 | 90 | 3 |

P3, rand problems (2300 scenarios) | 32 | 1 | 153 | 8 |

P3, totrand problems (2300 scenarios) | 39 | 1 | 163 | 11 |

P2_10, sinks > sources (3000 scenarios) | 297 | 54 | 533 | 212 |

P2_10, sinks < sources (3900 scenarios) | 88 | 2 | 308 | 28 |

P2_30, sinks > sources (3000 scenarios) | 250 | 30 | 480 | 170 |

P2_30, sinks < sources (3900 scenarios) | 81 | 2 | 269 | 28 |

P2_50, sinks > sources (3000 scenarios) | 190 | 28 | 366 | 130 |

P2_50, sinks < sources (3900 scenarios) | 51 | 2 | 189 | 15 |

P2_70, sinks > sources (3000 scenarios) | 120 | 15 | 274 | 72 |

P2_70, sinks < sources (3900 scenarios) | 35 | 1 | 135 | 10 |

P2_90, sinks > sources (3000 scenarios) | 86 | 9 | 243 | 42 |

P2_90, sinks < sources (3900 scenarios) | 23 | 1 | 103 | 6 |

P3_10, sinks > sources (3000 scenarios) | 43 | 2 | 186 | 11 |

P3_10, sinks < sources (3900 scenarios) | 3 | 0 | 20 | 1 |

P3_30, sinks > sources (3000 scenarios) | 45 | 3 | 178 | 14 |

P3_30, sinks < sources (3900 scenarios) | 4 | 1 | 33 | 1 |

P3_50, sinks > sources (3000 scenarios) | 57 | 4 | 214 | 18 |

P3_50, sinks < sources (3900 scenarios) | 6 | 1 | 34 | 1 |

P3_70, sinks > sources (3000 scenarios) | 64 | 4 | 212 | 25 |

P3_70, sinks < sources (3900 scenarios) | 8 | 1 | 48 | 2 |

P3_90, sinks > sources (3000 scenarios) | 74 | 7 | 215 | 34 |

P3_90, sinks < sources (3900 scenarios) | 13 | 1 | 66 | 4 |

Categories of scenarios | Ind2 | |||
---|---|---|---|---|

Mean | Median | Standard deviation | Trimmed mean over [5% - 95%] percentiles | |

All problems (75,900 scenarios) | 0.037 | 0.000 | 0.489 | 0.002 |

P1 (all problems, 6,900 scenarios) | 0.049 | 0.000 | 0.391 | 0.006 |

P2 (all problems, 34,500 scenarios) | 0.056 | 0.000 | 0.670 | 0.006 |

P3 (all problems, 34,500 scenarios) | 0.016 | 0.000 | 0.212 | 0.000 |

Sinks > sources (33,000 scenarios) | 0.063 | 0.000 | 0.718 | 0.006 |

Sinks < sources (42,900 scenarios) | 0.018 | 0.000 | 0.160 | 0.000 |

LH problems (25,300 scenarios) | 0.034 | 0.000 | 0.249 | 0.002 |

Rand problems (25,300 scenarios) | 0.055 | 0.000 | 0.711 | 0.005 |

Totrand problems (25,300 scenarios) | 0.023 | 0.000 | 0.386 | 0.000 |

P2_10 problems (6900 scenarios) | 0.007 | 0.000 | 0.073 | 0.000 |

P2_30 problems (6900 scenarios) | 0.049 | 0.000 | 0.597 | 0.004 |

P2_50 problems (6900 scenarios) | 0.096 | 0.000 | 1.118 | 0.011 |

P2_70 problems (6900 scenarios) | 0.072 | 0.000 | 0.646 | 0.012 |

P2_90 problems (6900 scenarios) | 0.057 | 0.000 | 0.463 | 0.009 |

P3_10 problems (6900 scenarios) | 0.001 | 0.000 | 0.020 | 0.000 |

P3_30 problems (6900 scenarios) | 0.008 | 0.000 | 0.091 | 0.000 |

P3_50 problems (6900 scenarios) | 0.011 | 0.000 | 0.087 | 0.000 |

P3_70 problems (6900 scenarios) | 0.025 | 0.000 | 0.371 | 0.001 |

P3_90 problems (6900 scenarios) | 0.037 | 0.000 | 0.265 | 0.004 |

P1, sinks > sources (3000 scenarios) | 0.079 | 0.000 | 0.552 | 0.013 |

P1, sinks < sources (3900 scenarios) | 0.026 | 0.000 | 0.187 | 0.002 |

P2, sinks > sources (3000 scenarios) | 0.097 | 0.000 | 0.990 | 0.013 |

P2, sinks < sources (3900 scenarios) | 0.025 | 0.000 | 0.197 | 0.001 |

P3, sinks > sources (3000 scenarios) | 0.025 | 0.000 | 0.300 | 0.001 |

P3, sinks < sources (3900 scenarios) | 0.009 | 0.000 | 0.101 | 0.000 |

P1, LH problems (2300 scenarios) | 0.048 | 0.000 | 0.246 | 0.008 |

P1, rand problems (2300 scenarios) | 0.060 | 0.000 | 0.355 | 0.010 |

P1, totrand problems (2300 scenarios) | 0.039 | 0.000 | 0.522 | 0.001 |

P2, LH problems (2300 scenarios) | 0.050 | 0.000 | 0.327 | 0.007 |

P2, rand problems (2300 scenarios) | 0.084 | 0.000 | 0.991 | 0.009 |

P2, totrand problems (2300 scenarios) | 0.035 | 0.000 | 0.508 | 0.002 |

P3, LH problems (2300 scenarios) | 0.015 | 0.000 | 0.131 | 0.000 |

P3, rand problems (2300 scenarios) | 0.026 | 0.000 | 0.322 | 0.002 |

P3, totrand problems (2300 scenarios) | 0.008 | 0.000 | 0.119 | 0.000 |

P2_10, sinks > sources (3000 scenarios) | 0.009 | 0.000 | 0.095 | 0.000 |
---|---|---|---|---|

P2_10, sinks < sources (3900 scenarios) | 0.006 | 0.000 | 0.049 | 0.000 |

P2_30, sinks > sources (3000 scenarios) | 0.087 | 0.000 | 0.888 | 0.009 |

P2_30, sinks < sources (3900 scenarios) | 0.020 | 0.000 | 0.143 | 0.001 |

P2_50, sinks > sources (3000 scenarios) | 0.177 | 0.000 | 1.664 | 0.030 |

P2_50, sinks < sources (3900 scenarios) | 0.034 | 0.000 | 0.268 | 0.003 |

P2_70, sinks > sources (3000 scenarios) | 0.124 | 0.000 | 0.942 | 0.030 |

P2_70, sinks < sources (3900 scenarios) | 0.032 | 0.000 | 0.229 | 0.003 |

P2_90, sinks > sources (3000 scenarios) | 0.090 | 0.000 | 0.657 | 0.019 |

P2_90, sinks < sources (3900 scenarios) | 0.032 | 0.000 | 0.214 | 0.003 |

P3_10, sinks > sources (3000 scenarios) | 0.000 | 0.000 | 0.004 | 0.000 |

P3_10, sinks < sources (3900 scenarios) | 0.001 | 0.000 | 0.026 | 0.000 |

P3_30, sinks > sources (3000 scenarios) | 0.013 | 0.000 | 0.131 | 0.000 |

P3_30, sinks < sources (3900 scenarios) | 0.004 | 0.000 | 0.035 | 0.000 |

P3_50, sinks > sources (3000 scenarios) | 0.016 | 0.000 | 0.113 | 0.000 |

P3_50, sinks < sources (3900 scenarios) | 0.007 | 0.000 | 0.060 | 0.000 |

P3_70, sinks > sources (3000 scenarios) | 0.040 | 0.000 | 0.552 | 0.004 |

P3_70, sinks < sources (3900 scenarios) | 0.012 | 0.000 | 0.095 | 0.000 |

P3_90, sinks > sources (3000 scenarios) | 0.057 | 0.000 | 0.337 | 0.010 |

P3_90, sinks < sources (3900 scenarios) | 0.022 | 0.000 | 0.190 | 0.001 |

Categories of scenarios | Ind3 | |||
---|---|---|---|---|

Mean | Median | Standard deviation | Trimmed mean over [5% - 95%] percentiles | |

All problems (75,900 scenarios) | 0.002 | 0.000 | 0.019 | 0.0000 |

P1 (all problems, 6,900 scenarios) | 0.003 | 0.000 | 0.025 | 0.0001 |

P2 (all problems, 34,500 scenarios) | 0.002 | 0.000 | 0.016 | 0.0000 |

P3 (all problems, 34,500 scenarios) | 0.002 | 0.000 | 0.021 | 0.0000 |

Sinks > sources (33,000 scenarios) | 0.002 | 0.000 | 0.019 | 0.0001 |

Sinks < sources (42,900 scenarios) | 0.002 | 0.000 | 0.020 | 0.0000 |

LH problems (25,300 scenarios) | 0.001 | 0.000 | 0.011 | 0.0000 |

Rand problems (25,300 scenarios) | 0.004 | 0.000 | 0.030 | 0.0001 |

Totrand problems (25,300 scenarios) | 0.001 | 0.000 | 0.011 | 0.0000 |

P2_10 problems (6900 scenarios) | 0.000 | 0.000 | 0.006 | 0.0000 |

P2_30 problems (6900 scenarios) | 0.001 | 0.000 | 0.014 | 0.0000 |

P2_50 problems (6900 scenarios) | 0.002 | 0.000 | 0.018 | 0.0001 |

P2_70 problems (6900 scenarios) | 0.002 | 0.000 | 0.014 | 0.0002 |

P2_90 problems (6900 scenarios) | 0.003 | 0.000 | 0.022 | 0.0002 |
---|---|---|---|---|

P3_10 problems (6900 scenarios) | 0.000 | 0.000 | 0.011 | 0.0000 |

P3_30 problems (6900 scenarios) | 0.002 | 0.000 | 0.022 | 0.0000 |

P3_50 problems (6900 scenarios) | 0.002 | 0.000 | 0.023 | 0.0000 |

P3_70 problems (6900 scenarios) | 0.003 | 0.000 | 0.024 | 0.0000 |

P3_90 problems (6900 scenarios) | 0.003 | 0.000 | 0.024 | 0.0001 |

P1, sinks > sources (3000 scenarios) | 0.004 | 0.000 | 0.025 | 0.0003 |

P1, sinks < sources (3900 scenarios) | 0.003 | 0.000 | 0.024 | 0.0000 |

P2, sinks > sources (3000 scenarios) | 0.002 | 0.000 | 0.014 | 0.0001 |

P2, sinks < sources (3900 scenarios) | 0.002 | 0.000 | 0.017 | 0.0000 |

P3, sinks > sources (3000 scenarios) | 0.002 | 0.000 | 0.022 | 0.0000 |

P3, sinks < sources (3900 scenarios) | 0.002 | 0.000 | 0.021 | 0.0000 |

P1, LH problems (2300 scenarios) | 0.002 | 0.000 | 0.016 | 0.0001 |

P1, rand problems (2300 scenarios) | 0.006 | 0.000 | 0.037 | 0.0004 |

P1, totrand problems (2300 scenarios) | 0.002 | 0.000 | 0.012 | 0.0000 |

P2, LH problems (2300 scenarios) | 0.001 | 0.000 | 0.010 | 0.0000 |

P2, rand problems (2300 scenarios) | 0.003 | 0.000 | 0.022 | 0.0001 |

P2, totrand problems (2300 scenarios) | 0.001 | 0.000 | 0.013 | 0.0000 |

P3, LH problems (2300 scenarios) | 0.001 | 0.000 | 0.011 | 0.0000 |

P3, rand problems (2300 scenarios) | 0.005 | 0.000 | 0.034 | 0.0000 |

P3, totrand problems (2300 scenarios) | 0.001 | 0.000 | 0.008 | 0.0000 |

P2_10, sinks > sources (3000 scenarios) | 0.000 | 0.000 | 0.004 | 0.0000 |

P2_10, sinks < sources (3900 scenarios) | 0.000 | 0.000 | 0.007 | 0.0000 |

P2_30, sinks > sources (3000 scenarios) | 0.001 | 0.000 | 0.008 | 0.0000 |

P2_30, sinks < sources (3900 scenarios) | 0.001 | 0.000 | 0.017 | 0.0000 |

P2_50, sinks > sources (3000 scenarios) | 0.002 | 0.000 | 0.012 | 0.0003 |

P2_50, sinks < sources (3900 scenarios) | 0.002 | 0.000 | 0.022 | 0.0000 |

P2_70, sinks > sources (3000 scenarios) | 0.002 | 0.000 | 0.013 | 0.0004 |

P2_70, sinks < sources (3900 scenarios) | 0.002 | 0.000 | 0.015 | 0.0000 |

P2_90, sinks > sources (3000 scenarios) | 0.004 | 0.000 | 0.025 | 0.0004 |

P2_90, sinks < sources (3900 scenarios) | 0.002 | 0.000 | 0.020 | 0.0001 |

P3_10, sinks > sources (3000 scenarios) | 0.000 | 0.000 | 0.007 | 0.0000 |

P3_10, sinks < sources (3900 scenarios) | 0.001 | 0.000 | 0.013 | 0.0000 |

P3_30, sinks > sources (3000 scenarios) | 0.001 | 0.000 | 0.018 | 0.0000 |

P3_30, sinks < sources (3900 scenarios) | 0.002 | 0.000 | 0.025 | 0.0000 |

P3_50, sinks > sources (3000 scenarios) | 0.003 | 0.000 | 0.027 | 0.0000 |

P3_50, sinks < sources (3900 scenarios) | 0.002 | 0.000 | 0.019 | 0.0000 |

P3_70, sinks > sources (3000 scenarios) | 0..4 | 0.000 | 0.027 | 0.0001 |

P3_70, sinks < sources (3900 scenarios) | 0.003 | 0.000 | 0.022 | 0.0000 |

P3_90, sinks > sources (3000 scenarios) | 0.003 | 0.000 | 0.024 | 0.0003 |

P3_90, sinks < sources (3900 scenarios) | 0.003 | 0.000 | 0.024 | 0.0000 |

The stream flows and temperatures for the case study-3 of the manuscript are presented in

Streams | Temperature (˚C) | A kmol/hr | B kmol/hr | C kmol/hr | D kmol/hr | Sol1 kmol/hr | Sol2 kmol/hr |
---|---|---|---|---|---|---|---|

1 | 20 | 13.0 | |||||

2 | 20 | 15.0 | 10.0 | 10.0 | |||

3 | 20 | 10.0 | 5.0 | 10.0 | 30.0 | 10.0 | |

4 | 20 | 10.0 | 20.0 | 10.0 | 10.0 | ||

5 | 20 | 10.0 | |||||

6 | 20 | 13.0 | 10.0 | ||||

7 | 20 | 20.5 | 15.0 | 5.0 | |||

8 | 150 | 17.3 | 11.8 | 3.2 | 11.3 | ||

9 | 132 | 19.3 | 13.0 | 3.2 | 16.1 | 1.0 | 1.0 |

10 | 150 | 0.1 | 0.0 | 0.0 | 7.0 | 0.9 | 0.6 |

11 | 150 | 19.3 | 13.0 | 3.2 | 9.1 | 0.2 | 0.4 |

12 | 20 | 7.5 | 5.0 | 5.0 | |||

13 | 20 | 7.5 | 5.0 | 5.0 | |||

14 | 20 | 22.5 | 12.5 | 25.0 | 35.0 | 15.0 | |

15 | 50 | 2.0 | 1.1 | 4.8 | 1.0 | 1.0 | |

16 | 50 | 20.5 | 11.4 | 20.2 | 34.0 | 14.0 | |

17 | 20 | 5.0 | 2.5 | 10.0 | 5.0 | 5.0 | |

18 | 20 | 5.0 | 2.5 | 10.0 | 5.0 | 5.0 | |

19 | 70 | 0.0 | 0.0 | 1.0 | 3.8 | 1.1 | |

20 | 70 | 5.0 | 2.5 | 9.0 | 1.2 | 3.9 |