Investigation on the net cascade using Ant Colony optimization algorithm

Progress in Nuclear Energy xxx (xxxx) xxx

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Investigation on the net cascade using Ant Colony optimization algorithm Farzaneh Ezazi a, Mohammad Hassan Mallah a, *, Javad Karimi Sabet a, Ali Norouzi b, Aadel Mahmoudian b a b

Materials and Nuclear Fuel Research School, Nuclear Science and Technology Research Institute, Atomic Energy Organization, P.O. Box: 11365-8486, Tehran, Iran Advanced Technologies Company, Atomic Energy Organization, P.O. Box: 143995-5931, Tehran, Iran

A R T I C L E I N F O

A B S T R A C T

Keywords: Net cascade Matched abundance ratio Optimization Ant Colony algorithm

In the present work, the analysis of the net cascades with recirculation has been investigated. Therefore, survey on performance of some configurations of the net cascade has been conducted. It was demonstrated that the lozenge net cascades with recycled flows to different layers indicate the same function. Also, it is found that they perform as well as their equivalent conventional matched abundance ratio cascade and corresponding matched abundance ratio net cascade. Moreover, based on the metaheuristic algorithms, the Ant Colony optimization for continuous domain is used to optimize the net cascade parameters. For this purpose, the code ACORNET is developed and the waste and the product concentrations of a target component and the separation power of single machine were selected as the fitness function main items. It has been revealed that the application of the algorithm to this problem guarantees finding the acceptable solution and from an evolutionary point of view the performance of it is superior.

1. Introduction The use of stable isotopes arising from the demand for fundamental research in physics, medical diagnosis and treatment, and the develop­ ment of new materials is increasing on an ongoing basis, and this has doubled the need to develop methods for the separation of stable iso­ topes. In this regard, multistage separation installations of gas centri­ fuges have been globally identified for the separation of heavy and semiheavy stable isotopes (Armstrong et al., 1968; Kudziev, 1989). From the point of view of stable isotopes separation theory, the first step is to use a model cascade in order to construct a practical cascade (Zeng et al., 2014). The model cascade that is used to separate isotopes of a binary mixture is an ideal cascade in which the concentration of flows is necessarily equal at the confluent points, as well as the enrichment and stripping coefficients are equal at all stages, resulting in the least loss in separation power at stages and thus maximum efficiency of the cascade. Swift Generalization of the non-mixing concept in separating binary isotope mixtures to separating isotopes from multicomponent isotope mixtures is not entirely possible (Borisevich et al., 2014). In other words, the development of the conditions for the equality of concentrations of all isotopes in confluent points is impossible, since the number of gov­ erning equations is more than the number of unknowns, taking into account this condition for all isotopes, and the unique cascade cannot be

deduced. As a result, the consideration of only one condition at each stage of the cascade lead to various types of cascade models in the separation of multicomponent isotope mixtures that all of them act as the ideal cascade in the case of binary separation. To refer to some cases, we can point to the Matched Abundance Ratio Cascade (MARC) pro­ posed by De la Garza et al. (De La Garza et al., 1961) in which, the abundance ratio of the selected two components of the mixture is equal at the confluent points, and this is the condition considered at each stage. Moreover, Matched-X Cascade was also investigated by De la Garza et al. and Murphy. In this cascade, only the concentration of one of the isotopes at the confluent points satisfies the ideal condition (De La Garza et al., 1961; Murphy, 1962). Similarly, the quasi-ideal cascade was proposed by Apelblat and Ilamed-Lehrer (1968) and in 2000 by Sazykin, in which partial cut of one of isotopes, at each stage, embraces the desired condition (Sazykin, 2000). Pseudo-Binary Types 1, 2 and 3 Cascades were also proposed by Zeng et al. (2014). Zeng, in 2012 proposed the ideal net cascade in order to achieve the non-mixing condition at confluent points for all isotopes in the separa­ tion of multicomponent mixtures. This cascade is defined as twodimensional and has different layers. Each layer consists of several stages, like the stages of conventional cascades, and the flows are frac­ tioned in the stages of each layer based on the cut relevant of those stages, entering into the stages of the subsequent layers. In this way, the

* Corresponding author. E-mail address: [email protected] (M.H. Mallah). https://doi.org/10.1016/j.pnucene.2019.103169 Received 18 September 2018; Received in revised form 18 June 2019; Accepted 26 September 2019 0149-1970/© 2019 Elsevier Ltd. All rights reserved.

Please cite this article as: Farzaneh Ezazi, Progress in Nuclear Energy, https://doi.org/10.1016/j.pnucene.2019.103169

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cascades. Zeng et al., in 2015 introduced the net cascades with recir­ culation from the final layer that governs matched abundance ratio conditions (Zeng and Ying, 2015). In this cascade, the upper part is an ideal net cascade, and the last two layers are a recycled cascade that has the MARC conditions of binary components, and the upstream and the downstream output flows of the final layer stages are reversed to the stages of the penultimate layer. This configuration is not unique, and other forms can be considered and analyzed for recycled flows from different layers based on various purposes. For a separation, finding the cascade with best performance is always of practical and theoretical significance. The optimization of a cascade is very important from the economical point of view to make the isotope separation cascade competitive. Even a one-percent difference in the separation performance suggests considerable economic benefit (Bor­ isevich et al., 2014; Mansourzadeh et al., 2019). In this way, instead of imposing the MARC conditions in the lower part of net cascade, it is possible to find optimal cascade parameters using a specific objective function and an appropriate optimization method. Thus, we will benefit from the ideal cascade advantage in the upper part, and the optimal parameters (here, cuts of stages of two last layer of the cascade) can replace the MARC parameters in the lower part (by applying the ob­ tained cuts to the corresponding stages), lacking ideal condition, and more suitable results can be achieved in relation to the specified purpose. The present research was conducted to investigate and analyze some configurations of the net cascades with recirculation. Lozenge net cascade with different situations of recycled flows and matched abun­ dance Ratio net cascade (R-net cascade) are the main forms discussed in the paper. Comparative study of the cascades with the conventional MARC, also, is implemented. The data obtained make it possible to use of the results to making decision in application of various types of the cascades. Additionally, it is desirable to have a net cascade, which can be as efficient as the ideal net cascade and simultaneously supports lack of flow in the late layers with the evident advantage of meeting optimum results for the specific criteria. To determine optimal parameters of lower part of R-net cascade, ACOR for continuous domain algorithm is implemented through development of ACORNET code. The advantage of the ACOR algorithm to obtain optimum cascade, in comparison with the implementation of R-net cascade is explained using better perfor­ mance in regard to the criterion is considered in which in addition to maximizing of single machine separation power, can achieve the con­ centration of a particular component in the product and the waste of the cascade to the desired predetermined values. In this regard, the code ACORNET has been successfully developed and the functionality of the code is well shown.

inter stage flow rate decreases in subsequent layers by increasing the number of layers. The principle of the ideal net cascade development is that the cuts of components at each stage must be equal, which imposes the conditions for choosing separation factors (Zeng and Ying, 2012). For example, considering the symmetrical separation for the selected binary components in the cascade can provide equalization of concen­ trations in confluent points. One of the issues associated with these cascades is reduction of the inter stage flow rate in the final layers of cascade as well as the stages leading to the two ends of the cascade in different layers. One of the solutions to cope with this item is the recy­ cled flow of some stages of lower layers to input of the upper layers. This can be a combination of cascades with recirculation and ideal net

2. Theory 2.1. Net cascade scheme and various configuration of Recycled Net Cascade The separating element, centrifuge, is the smallest unit of an isotope separating plant in which the feed is divided to enrich the lighter iso­ topes in the enriching section and to deplete same isotopes in the stripping section. Some centrifuges, connected parallel, are named stage. Feed of all machines of a stage have the same isotope composition, and the same is true for the head and tail streams. The number of cen­ trifuges in a stage is proportional to the feed flow rate of it. The sepa­ ration of isotopes can be achieved by connecting several stages in series called cascade. Fig. 1 shows a conventional counter current cascade. A stage of the cascade is numbered from heavy side consecutively to N: Nf 0 is the feed stage. Ln , L n ; ​ ​ and L00n are the flow rate of input, head and tail 0 streams of arbitrary stage n, respectively. Ci;n ; ​ ​ ​ C i;n ; ​ ​ and ​ C00i;n are the corresponding concentrations of component i, at the streams and the

Fig. 1. Schematic of a conventional counter current cascade. 2

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mixture has been considered to consist of Nc components. F, P and W are the external flows of the cascade and CFi , CPi and CW i are concentrations of the corresponding pipes. The point in which pipes of head and tail of adjacent stages are connected to the feed pipe of the stage is termed confluent point. Unlike conventional cascades, the net cascade considered here is a two-dimensional cascade and is illustrated in Fig. 2. The two l and s indexes are arranged in ordered pair (l, s) to name the position of the stages. The index of l represents the layer in the vertical direction and the index of s refers to the stage number in the horizontal direction. The arrows to the right and left indicate the upstream and downstream flows of each stage, respectively. As shown in the figure, the number of stages in each layer is equal to the number of that layer and the cascade feed is entered into stage (1, 0), which is located on the first layer. Depending on the number of cascade layers and the number of recycled flows into the upper layers, the number of product and waste streams of the cascade will not be constant and unique. As discussed in the introduction, different configurations can be considered in the net cascade with recirculation. Fig. 2 shows the ideal net cascade. There is no recycled flow in this cascade, and thus the nonmixing conditions can be established in all confluent point and for all isotopes. Fig. 3 exhibits three different configurations for the cascade in terms of recycled flow; (a) the net cascade with recirculation from the last layer (R-net cascade), (b) an example of the net cascade with recirculation of different layers and (c) the lozenge net cascade. As stated in the introduction, if there is not the recycled flow in the cascade, the flow rate will be lost. This is also true in the conventional counter current cascades. The method used to reverse the flow of the

Lðl;sÞ ¼ θðl

1;s 1Þ Lðl 1;s

� 1Þ þ 1

stages in a net cascade, operationally, is the same as the one used in the conventional counter current cascades. 2.2. Governing equations of the net cascade The feed, upstream and downstream flows of stage (l, s) are shown 0 with Lðl;sÞ , Lðl;sÞ and L}ðl;sÞ , respectively, and the concentration in these flows as Ci;ðl;sÞ , Ci;ðl;sÞ , and C}i; ​ 0

​ ðl;sÞ . i;F

The ith isotope concentration in the

cascade feed is expressed as C . The general form quoted for a mass balance is the mass that enters a system must, by conservation of mass, either leave the system or accumulate within the system. In the present work, in the stages of the cascade, there is not any accumulation and thus, any mass enters to the stages by the streams, will go out of the stage. Based on this law, the feed flow rate of stage (l, s) is obtained using total mass balance for each stage, and it will be different considering whether the recycled flow of stages of the other layers is entered. For example, feed flow rate of first stage is equal to the cascade feed. Also, in the upper part of the cascade, the flow rate of the first stage of each layer is equal to the downstream flow of the first stage of the previous layer and the flow rate of the last stage of each layer is equal to the upstream flow of the last stage of the previous layer. Equation set 1 represents relevant relations. Moreover, the concentration of the feed flows to stage (l, s) will also be calculated using partial mass balance that is the con­ servation law for the mass of any components. The relations are given in Equation set 2. θðl;sÞ , indicates the cut of stage (l, s) which is the ratio of the upstream flow of the stage to the entering flow of that stage.

� ; l ¼ 1; s ¼ 0 � Lðl;sÞ ¼ Feed Lðl;sÞ ¼ 1 θðl 1;sþ1Þ Lðl 1;sþ1Þ ; *s ¼ ðl 1Þ Lðl;sÞ ¼ θðl� 1;s 1Þ Lðl 1;s �1Þ ; *s ¼ ðl 1Þ Lðl;sÞ ¼ θðl 1;s � 1 θðl 1;sþ1Þ Lðl 1;sþ1Þ ; * ðl 1Þ < s < l 1 � 1Þ Lðl 1;s 1Þ þ Lðl;sÞ ¼ 1 θðl 1;sþ1Þ Lðl 1;sþ1Þ � þ θðlþ1;s 1Þ�Lðlþ1;s 1Þ ; **s ¼ ðl 1Þ �Lðl;sÞ ¼ θðl 1;s� 1Þ Lðl 1;s 1Þ þ� 1 θðlþ1;sþ1Þ Lðlþ1;sþ1Þ ; **s ¼� ðl 1Þ � θðl 1;sþ1Þ Lðl 1;sþ1Þ þ 1 θðl 1;sþ1Þ Lðl 1;sþ1Þ þ θðlþ1;s 1Þ Lðlþ1;s 1Þ þ 1 θðlþ1;sþ1Þ Lðlþ1;sþ1Þ ; ** *ðStage ​ with ​ RecirculationÞ **ðStage ​ with ​ RecirculationÞ

�� Ci;ðl;sÞ ¼ FeedCi;F Lðl;sÞ ; l ¼ 1; s ¼ 0 �. � θðl 1;sþ1Þ Lðl 1;sþ1Þ C}i;ðl 1;sþ1Þ Lðl;sÞ ; *s ¼ ðl 1Þ Ci;ðl;sÞ �. � Ci;ðl;sÞ ¼ θðl 1;s 1Þ Lðl 1;s 1Þ C’i;ðl 1;s 1Þ Lðl;sÞ ; *s ¼ ðl 1Þ �. � � � Ci;ðl;sÞ ¼ θðl 1;s 1Þ Lðl 1;s 1Þ C’i;ðl 1;s 1Þ þ 1 θðl 1;sþ1Þ Lðl 1;sþ1Þ C}i;ðl 1;sþ1Þ Lðl;sÞ ; * ðl 1Þ < s < l 1 �. �� � Ci;ðl;sÞ ¼ 1 θðl 1;sþ1Þ Lðl 1;sþ1Þ C}i;ðl 1;sþ1Þ þ θðlþ1;s 1Þ Lðlþ1;s 1Þ C’i;ðlþ1;s 1Þ Lðl;sÞ ; **s ¼ ðl 1Þ �. � � � Ci;ðl;sÞ ¼ θðl 1;s 1Þ Lðl 1;s 1Þ C’i;ðl 1;s 1Þ þ 1 θðlþ1;sþ1Þ Lðlþ1;sþ1Þ C}i;ðlþ1;sþ1Þ Lðl;sÞ ; **s ¼ ðl 1Þ � � � � � Ci;ðl;sÞ ¼ θðl 1;s 1Þ Lðl 1;s 1Þ C’i;ðl 1;s 1Þ þ 1 θðl 1;sþ1Þ Lðl 1;sþ1Þ C}i;ðl 1;sþ1Þ þ 1 θðl 1;sþ1Þ Lðl 1;sþ1Þ C}i;ðl �. � � þ 1 θðlþ1;sþ1Þ Lðlþ1;sþ1Þ C}i;ðlþ1;sþ1Þ Lðl;sÞ ; ** ðl 1Þ < s < l 1

(1) ðl

1Þ < s < l

1

�� ¼ 1

*ðStage ​ with ​ RecirculationÞ **ðStage ​ with ​ RecirculationÞ

3

(2)

1;sþ1Þ

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help of equations (3)–(5). The concentration of the upstream flows at stages of that layer and the corresponding cut of those stages are the unknowns and they are calculated using a suitable solving method such as trust-region dogleg. Ci ¼ θC ’i þ ð1 C’k1 C’k2 Nc X

1=2

¼ αk1 k2

C’ θÞ PNc i

’ j¼1 ij C j

α

ði ¼ 1; 2; …; Nc Þ

Ck 1 Ck 2

C’i ¼ 1

(3)

(4)

(5)

i¼1

Afterwards, the tail concentrations of the stages of the current layer are also calculated as follow: C’i;ðl;sÞ C}i;ðl;sÞ ¼ PNc ’ j¼1 αij C j;ðl;sÞ

(6)

For each stage, the separation factor is defined as follow (Ying et al., 1996): . 0 0 Ci;n Cj;n M M αij;n ¼ . ¼ α0;nj i ði ¼ j 1; j ¼ 2; …; Nc Þ (7) C00i;n C00j;n

Fig. 2. Schematic diagram of a 4-layer ideal net cascade.

After calculating the inter stage flow rate and the concentration of these flows in the desired layer using the equation sets 1 and 2, a system of nonlinear equations is formed for each stage of that layer with the

Where, αi;j is the separation factor of components i and j, Mi and Mj are the molecular weights of the components i and j, respectively. The components are arranged as if i < j, then Mi
Fig. 3. Schematic of a Recycled Net Cascade: a) from last layer b) from different layers c) lozenge net cascade. 4

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separation factor for the unit mass difference (Mj -Mi ¼ 1). Thus, to achieve an ideal net cascade, all parameters are individually calculated layer by layer to reach the end of the cascade. If there is recycled flow in the cascade, the calculation procedure will be the same, with the difference that it is necessary to consider an iteration loop for layers with recirculation to converge the feed flows of the stages in the layers involved recirculation. The convergence criteria for flow rates in layers with recycled flow are satisfied using Equation (8), where m shows the number of itera­ tions, N indicates the sum of the total number of stages in the cascade and ε is the convergence parameter considered 10 8 in this article. sffiffiffiffiffiffiffiffiffiffiffiffiffiffi�ffiffiffiffiffiffiffiffiffiffiffi XX� 1 Lmðl;sÞ j2 ≪ε ​ (8) �Lmþ1 ðl;sÞ N S l

and Gambardella, 1997; Dorigo and Stutzle, 2004). In order to generalize the ACO algorithm to the continuous problems, some approaches were introduced until 2008, when they were not widely publicized. Contrary to those approaches, Dorigo and Socha presented a very interesting idea this year. Their idea was related to the inspiration from the pheromone nature in tracking, on the one hand and integration of it with estimation of distribution algorithms (EDAs) in order to use it for tackling continuous issues. Dorigo and Socha have tested the performance of this method, called the Ant Colony optimization for continuous domain algorithm, against the substantial number of other algorithms and approaches. ACOR when compared to other probability-learning methods, proved to the best on four out of 10 test problems. On the others, the quality of the solutions found was not significantly worse than the state-of-the-art. Also, ACOR is a clear winner when compared to other ant-related algorithms for continuous optimization that were proposed in the past. When compared to these methods, ACOR was better by almost two orders of magnitude. Finally, when compared to other metaheuristic adapted to continuous optimization, ACOR was the winner in one-third of the test problems and perform not much worse than the others (Socha and Dorigo, 2008). Given the issues raised, we decided to use this method in our problem. To the best of our knowledge, this is the first time that an optimization method is used to find the optimal parameters of a net cascade. This method, ACOR, uses the bases of the ACO algorithm and EDAs to generate a solution archive and update it based on a given objective function. The basic idea underlying ACOR is the switch from a discrete probability distribution to using a continuous one, that is, a Probability Density Function (PDF). In this case, the Gaussian kernel PDF can be used, so that it is generated on the basis of each of the solutions in the archive, and based on the better or worse of that answer, it is given a probability of choice. In order to implement this algorithm, it is assumed that there exist n continuous variables in the problem. Thus, each of the solutions ob­ tained includes n variables. The number of solutions stored in the so­ lution archives is also considered to be k. The jth solution with the Sol symbol is represented as follow: � � Solj ¼ S1j ; ​ S2j ; ​ …; ​ Sij ; ​ …; ​ Snj (9)

2.3. Optimization algorithm As already mentioned, optimizing a cascade and finding its optimal parameters is very important in terms of achieving a higher economic efficiency, and many scientists have done many researches in this re­ gard. Different approaches are used to find optimal parameters in solving constrained multi-objective optimization problems, depending on the type of the problem and the user requests. Some of these methods operate on the basis of nonlinear programming. For example, in the work of Palkin at 2002, the sorting procedure is combined with a random search and in the other, direct search solution is available for optimization. In addition, Palkin at 1998 and 2012 minimize the total number of machines in the case of variable separation factor with cut and feed flow rate of the centrifuge (Hooke and Jeeves, 1961; Palkin, 1998, 2013a,b; Palkin et al., 2002; Palkin and Igoshin, 2012; Palkin, 2013b). One of the most powerful methods that have recently been taken the interest of many scholars and researchers is metaheuristic methods based on taking inspiration from nature and living beings, for example, using foraging techniques of honey bees or locating methods of birds, Elephants and fish, and so on (Correia et al., 2018; Deb, 2015; Safdari et al., 2017; Wang et al., 2015). The metaheuristic algorithms, unlike classical methods, often use a population of solutions in their own process rather than using one answer. Because in these methods, unlike the one-way methods, the search space is explored comprehensively, there is less chance of getting into a locally optimal point. The meta­ heuristic algorithms are route-independent methods. This distinguishes them from the old ones, so they can effectively explore the entire search space. In the separation of stable isotopes and the design of conventional counter current separation cascades, the use of these methods is becoming more widespread. Some of the metaheuristic algorithms used to optimize the multicomponent mixture separation in conventional cascades are genetic algorithms (Norouzi and Zolfaghari, 2011), simu­ lated annealing (Kirkpatrick et al., 1983; Song et al., 2010), particle swarm optimization (Safdari et al., 2017) and so on. One of these intelligent optimization methods inspired by the col­ lective intelligence of ants in routing from the nest to the food source is the ACO algorithm, which has been designed for use in combinatorial optimization problems or permutation problems. The ACO algorithm is one of the most successful congestion-based algorithms, which was presented by Dorigo and Di Caro in 1999. The most interesting joint part in the behavior of the ants is their ability to find alternate routes be­ tween the nest and food sources by traversing the pheromone traces. Then the ants, with a possible pheromone level-based decision making, will choose a route to continue; the stronger pheromone trace, the greater the utility of that route. This algorithm is an obvious example of collective intelligence, in which, factors that are not very potent, can work together to achieve great results. This algorithm is used to solve a wide range of optimization problems (Dorigo et al., 1996, 1999; Dorigo

The fitness function (or the objective function) of this solution is displayed by fj ¼ FðSolj Þ. Gaussian kernel Gi can be created using Equation (10) with the help of the ith variables of all solutions in the solution archive, which can regenerate these variables with the proba­ bility of producing answers around them. � � X 2 (10) Gi ¼ Pj N Sij ; σij j

Equation (11) is used to calculate the parameter of Pj that determines the probability of selecting any of the variables based on their fitness. Pj ¼ P

wj m wm

α

(11)

Where, α is used to increase or reduce probabilities and wj is the weighting coefficients for each of the variables, computed as follow: � �2 ! 1 1 j 1 wj ¼ pffiffiffiffiffi exp (12) 2 qk qk 2π Where, q is the selection pressure factor and q12 can be used instead of

α in Equation (11), that in this case, the parameter q can be eliminated in Equation (12). Another needed parameter is σ ij that specifies the adjacent of Sij with

which scatter is searched. The longer the average distance of others than one solution, the larger the area should to be searched around that so­ lution. Equation (13) is used to calculate this parameter.

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2.4. Objective function of optimization If there is recycled flow from the stages of the last layer of the net cascade to the penultimate layer stages, it will be possible to create the R conditions in the lower part of the cascade. In this case, like the ideal upper part of the cascade, the symmetrical separation is imposed for the selected binary components of k1 and k2 in the cascade by equation (14). The definition of βðk1;k2Þ and γ ðk1;k2Þ is given in Equation (15). Where, ​ βðk1;k2Þ , γðk1;k2Þ and αðk1;k2Þ , respectively, are the enriching and stripping coefficients as well as separation factor of the selected binary compo­ nents of k1 and k2 . Thus the matched abundance ratio condition for components of k1 and k2 , given in equations (16) and (17), are established. pffiffiffiffiffiffiffiffiffiffiffiffi βðk1;k2Þ ¼ γ ðk1;k2Þ ¼ αðk1;k2Þ (14) βðk1;k2Þ ¼ R’ðk1; k2Þ;

0 � 0 Ck1 Ck2 ; Ck1 =Ck2

n 1

γðk1;k2Þ ¼

¼ R}ðk1; k2Þ;

Ck1 =Ck2 � C00k1 C00k2

nþ1 ¼ Rðk1; k2Þ; n

(15) ðk1 < k2 ; n ¼ 2; 3; …; N

1Þ (16)

0

R’ðk1; k2Þ ¼

σ ¼δ

P �� i r �Sj k

;

R}ðk1; k2Þ ¼

C00k1 C00k2

(17)

The idea here is that to improve the performance of this cascade in the lower part, which is not ideal. It is possible to use a powerful opti­ mization method taking into account a suitable objective function rather than the matched-abundance ratio conditions for achieving optimal parameters of this domain, providing a more efficient function for the cascade. Here, Equation (4) is necessary to be deleted in the equation system consisting of Equations of 3, 4 and 5, and consider the cuts of stages in the last two layers as optimization variables instead of imposing symmetrical separation in the two last layers of the cascade. These cut values should be used whenever the cascade is solved. Finally, appropriate cut values must be found in order to obtain an extremum for the objective function. Thus, if the number of layers in the cascade is considered NL, then jth solution will be defined as follow.

Fig. 4. Flowchart of the ACOR algorithm.

i j

Ck1 0 Ck2

� � Sir �

(13)

1

Solj ¼ θNL; θNL

ðNL 1Þ ; θNL 1; ðNL 2Þ ; θNL; ðNL 3Þ ; θNL 1; ðNL 4Þ ; …;

1;ðNL 4Þ ; θNL;ðNL 3Þ ; θNL 1;ðNL 2Þ ; θNL;ðNL 1Þ

​

(18)

�

Where value of δ is chosen to be greater than one; the larger the value of this parameter, the greater the exploration and vice versa; the smaller the parameter, the greater the exploitation. In general, the steps to implement the ACOR algorithm are as follows:

3. Results and discussion

1. Generating initial answers, evaluating them, sorting out and placing in the archive 2. Calculating weight coefficients, wj and probabilities, Pj 3. Creating the probable model for each variable of i separately (Gi ) 4. Creating a specific number of random samples using the probable models obtained in the previous step (xeG ¼ GðG1 ; …; Gn ) 5. Integrating new samples with those in the archives, sorting out and deleting additional members 6. Updating the best solution found so far 7. If the termination condition is not met, the iteration resumes from step 3, otherwise, the algorithm will end (Socha and Dorigo, 2008). The flowchart of the algorithm implementation is shown in Fig. 4.

3.1. Performance comparison of matched abundance ratio net cascade and conventional matched abundance ratio cascade

In order to analyze the contents of the previous sections, it is necessary to make calculations to determine their accuracy. The calcu­ lations are performed for separation of Tellurium isotopes as Tellurium hexafluoride gas with the natural composition shown in Table 1 (Suvorov and Tcheltsov, 1993).

The function of these two cascades was compared by evaluating the R-net cascade (with the specifications: NL ¼ 10, α0​ ¼ 1:2, k1​ ¼ 2, k2​ ¼ 3 and Feed ¼ 20) and the conventional MARC cascade (with the speci­ fications: N ¼ 19, NF ¼ 10, α0​ ¼ 1:2, k1​ ¼ 2, k2​ ¼ 3 and Feed ¼ 20).

Table 1 Specification of natural TeF6 as the cascade feed. Feed Composition Isotope

120

Molar Mass Concentration

234 0.00089

TeF6

122

TeF6

236 0.0246

123

TeF6

237 0.0087

124

TeF6

238 0.0461

6

125

TeF6

239 0.0699

126

TeF6

240 0.1872

128

TeF6

242 0.3179

130

TeF6

244 0.3447

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Table 2 Inter stage flows of the conventional MARC cascade and R-net cascade. MARC cascade

L1

L2

27.3714 L11

R-net cascade

28.4346 P Lð:; 9Þ

L3

38.9150 L12

L4

44.5846 L13

16.8910 P Lð:; 8Þ

11.2215 P Lð:; 7Þ

L5

47.8169 L14 7.9892 P Lð:;

5.8814 P Lð:;

6Þ

Isotope

2.0046 P Lð:;

3Þ

2Þ

0.9914 P Lð:;

51.4674 P Lð:; 6Þ

52.7127 P Lð:; 7Þ

53.7981 P Lð:; 8Þ

54.8139 P Lð:; 9Þ

28.4341

16.8909

11.2214

7.9891

5.8814

4.3381

3.0928

2.0046

0.9915

Product Concentration

Waste Concentration

MARC cascade

R-net cascade

MARC cascade

R-net cascade

0.03553

0.03553

9.4669E-06

9.4648E-06

0.70791

0.70790

7.2313E-03

7.2313E-03

0.10061

0.10061

6.3637E-03

6.3637E-03

TeF6

3

TeF6

123

4

TeF6

124

TeF6

0.11334

0.11335

0.04439

0.04439

5

125

TeF6

0.02925

0.02925

0.07093

0.07093

6

126

TeF6

0.01276

0.01276

0.19161

0.19161

7

128

TeF6

5.6620E-04

5.6620E-04

0.32596

0.32596

8

130

TeF6

1.6017E-05

1.5990E-05

0.35349

0.35349

Parameter

Total Number of Machine Cascade Separation Power Separation Power of single Machine Number of Recycled Streams

R-net cascade (NL ¼ 7)

Conventional MARC cascade (N ¼ 13)

109

104

101

2.69019

2.69018

2.69021

0.02468

0.02586

0.02663

12

12

12

558.061 P Lð:; 0Þ

1Þ

55.8054 P LR Net 558.054

3.2. Performance comparison of 11-layer lozenge net cascade and equivalent 7-layar R-net cascade This section compares the function of the lozenge net cascade having 11 layers shown in Fig. 3(c) with its equivalent R-net cascade having seven layers (which are called cascades 1 and 2). Both cascades deliver the final product from the last stage of the seventh layer and the waste from the first stage of the seventh layer. As shown in Table 4, the sep­ aration power of the three cascades is roughly equal, and the number of machines in the lozenge net cascade will be larger and the singlemachine separation power will be lower. This result was foreseeable given that the number of stages is greater in the lozenge net cascade. In all three cascades, the number of recycled streams onto the upper layers is equal to each other (¼12). Table 5 shows that the hydraulic performance of these two cascades, as well as their equivalent conventional MARC cascade having 13 layers (which is called cascade 3) is similar and the inter stage flow rates in the three cascades have the same profile. As can be deduced from Table 6, the separation performance of these three cascades is very similar to each other and the product and the waste concentrations in the three cascades will be the same.

Cascades Lozenge net cascade (NL ¼ 11)

55.8059 P LMARC

P and 3. The symbol Lð:; sÞ is the sum of the input flow rates of the stages with the number s in the various layers. As shown in Tables 2 and 3, the separation performance of these two cascades is the same and the total flow rates in the stages with identical numbers in different layers in the R-net cascade is equal to the inter stage flow rate of the equivalent stage in the conventional MARC cascade.

Table 4 Total number of machine, cascade separation power and separation power of single machine of cascades 1, 2 and 3.

4

3.0928 P Lð:;

49.9241 P Lð:; 5Þ

122

3

4Þ

L10

54.8145 L19

47.8164 P Lð:; 4Þ

2

2

4.3381 P Lð:;

L8

53.8014 L18

44.5836 P Lð:; 3Þ

120

1

5Þ

L8

52.7132 L17

38.9147 P Lð:; 2Þ

1

Row

L7

51.4079 L16

27.3713 P Lð:; 1Þ

Table 3 Concentration of components in the waste and the product streams of the con­ ventional MARC cascade and R-net cascade. Row

L6

49.9247 L15

3.3. Comparison of the performance of 11-layer lozenge net cascades with recycled flows to different layers

Feed is the flow rate of the cascade feed. The compared R-net cascade has the configuration of Fig. 2 (a). Obviously, the sum of the stages in the last two layers of the R-net cascade with ten layers is equal to the number of stages in the conventional MARC cascade. Comparison of flow rates in these two cascades as well as the product and the waste concentrations of both cascades are presented in Tables 2

This section discusses 11-layer lozenge net cascade in different configurations. In these configurations, the recycled flows entered to the stages of different upper layers (1 up, 2 up, 3 up, 4 up and 5 up). The Characteristics of the studied lozenge net cascade is NL ¼ 11, α0​ ¼ 1:2, k1​ ¼ 2, k2​ ¼ 3 and Feed ¼ 1. As shown in Table 7, when the recycled flows enter into stages of

Table 5 Inter stage flows of cascades 1, 2 and 3. Lozenge net cascade (NL ¼ 11)

P

Lð:;

6Þ

1.3587 P Lð:; 1Þ R-net cascade (NL ¼ 7)

Conventional MARC cascade (N ¼ 13)

1.2681 P Lð:;

P

Lð:;

5Þ

1.9306 P Lð:; 2Þ 6Þ

0.6963 P Lð:;

P

Lð:;

4Þ

2.2117 P Lð:; 3Þ 5Þ

0.4151 P Lð:;

P

Lð:;

3Þ

2.3731 P Lð:; 4Þ 4Þ

0.2537 P Lð:;

P

Lð:;

2Þ

2.4800 P Lð:; 5Þ 3Þ

0.1468 P Lð:;

P

Lð:;

1Þ

2.5602 P Lð:; 6Þ 2Þ

0.0666 P Lð:;

P

Lð:; 0Þ

2.6268 P LLozenge ​ Net 1Þ

18.38779 P Lð:; 0Þ

1.3587 P Lð:; 1Þ

1.9305 P Lð:; 2Þ

2.2117 P Lð:; 3Þ

2.3732 P Lð:; 4Þ

2.4800 P Lð:; 5Þ

2.5602 P Lð:; 6Þ

2.6268 P LR Net

1.2680 L1

0.6962 L2

0.4151 L3

0.2537 L4

0.1468 L5

0.0666 L6

18.38757 L7

1.2680

0.6963

0.4151

0.2537

0.1468

0.0666

18.38755

1.3587 L8

1.9305 L9

2.2117 L10

2.3731 L11

7

2.4799 L12

2.5602 L13

2.6268 P LMARC

F. Ezazi et al.

Progress in Nuclear Energy xxx (xxxx) xxx

Table 6 Concentration of components in the waste and product streams of the cascades 1, 2 and 3. Row

Isotope

Product Concentration

Waste Concentration

Lozenge net cascade (NL ¼ 11)

R-net cascade (NL ¼ 7)

Conventional MARC cascade (N ¼ 13)

Lozenge net cascade (NL ¼ 11)

R-net cascade (NL ¼ 7)

Conventional MARC cascade (N ¼ 13)

1

120

TeF6

0.02750

0.02750

0.02750

3.6300E-05

3.6301E-05

3.6298E-05

2

122

TeF6

0.51794

0.51794

0.51794

0.00877

0.00877

0.00877

3

123

TeF6

0.09676

0.09676

0.09676

0.00587

0.00587

0.00587

4

124

TeF6

0.19060

0.19060

0.19060

0.04146

0.04146

0.04146

5

125

0.08889

0.08888

0.08888

0.06929

0.06929

0.06929

6

TeF6

126

0.06837

0.06837

0.06837

0.19099

0.19099

0.19099

7

TeF6

128

0.00914

0.00914

0.00914

0.32780

0.32780

0.32780

8

TeF6

130

TeF6

7.7265E-04

7.7262E-04

7.7264E-04

0.35576

0.35576

0.35576

Table 7 Total number of machine, cascade separation power and Separation Power of single machine of lozenge net cascades. Row

1 2 3 4

Parameter

Cascades

Total Number of Machine Cascade Separation Power Separation Power of single Machine Number of Recycled Streams

Lozenge net cascade 1 up

Lozenge net cascade 2 up

Lozenge net cascade 3 up

Lozenge net cascade 4 up

Lozenge net cascade 5 up

109 2.69019 0.02468

109 2.69023 0.02468

108 2.68995 0.02490

107 2.69029 0.02514

108 2.69028 0.02491

12

12

12

12

12

Table 8 Inter stage flows of lozenge net cascades. Cascades

P

Lozenge Lozenge Lozenge Lozenge Lozenge

1.358715 1.35870 1.358713 1.358714 1.358713 P Lð:; 1Þ

1.930541 1.930539 1.930537 1.930539 1.930538 P Lð:; 2Þ

2.211677 2.211674 2.211672 2.211674 2.211673 P Lð:; 3Þ

2.373113 2.37311 2.373107 2.373109 2.373108 P Lð:; 4Þ

2.480016 2.480012 2.480009 2.480011 2.48001 P Lð:; 5Þ

2.560151 2.560147 2.560143 2.560145 2.560145 P Lð:; 6Þ

2.626793 2.626789 2.626785 2.626787 2.626787 P LLozenge ​ Net

1.268078 1.268074 1.268070 1.268071 1.268071

0.696252 0.696247 0.696245 0.696245 0.696245

0.415116 0.415111 0.415110 0.415109 0.415109

0.253679 0.253676 0.253675 0.253674 0.253674

0.146777 0.146774 0.146774 0.146773 0.146773

0.066642 0.066641 0.066641 0.066640 0.066640

18.38755 18.38751 18.38748 18.38749 18.38749

net cascade 1 up net cascade 2 up Net Cascade 3 up net cascade 4 up net cascade 5 up

Cascades Lozenge Lozenge Lozenge Lozenge Lozenge

net cascade net cascade net cascade net cascade net cascade

1 up 2 up 3 up 4 up 5 up

Lð:;

6Þ

P

Lð:;

5Þ

P

Lð:;

4Þ

upper layers rather than entering the stages of previous layer, it will not have much effect on the cascade separation power and the cascade performance in terms of the total number of machines and the separa­ tion power of single machine will be roughly constant. Table 8 shows that the total flow rate in the stages having the same number of different layers has always been constant and is equal to the flow rates of equivalent R-net cascade given in Table 5 with the difference that as the recycled flows enters the upper layers, the flow expansion in the stages with the same number will be even more uniform and the flow rate in these stages will be closer to each other, resulting in number of machines becoming more uniform in stages. Concerning the cut values of the stages with the same number, it should be noted that the recycled flow entry to the upper layers also brings the cuts of the stages with the same number closer to each other, which results in a more uniform flow expansion in these stages. Ac­ cording to Table 9, the recycled flow entry to the upper layers has no effect on the separation performance of the lozenge net cascade, and the product and the waste concentrations of the cascades are very close to each other in different modes. It can be seen from Table 10 that the matched-abundance ratio condition is still present throughout the cascade by applying symmet­ rical separation in the cascade with different configurations. Symbols cp

P

Lð:;

3Þ

P

Lð:;

2Þ

P

Lð:;

1Þ

P

Lð:; 0Þ

and cz indicate the head and tail concentrations of the stage, respectively. 3.4. Comparison of performance of R-net cascade with optimized net cascade It has been shown that the cut values can be found for the stages of the last two layers of the net cascade, with the help of the ACOR opti­ mization algorithm, so that the concentration of Te-122 isotope, that is considered to be the target, in the cascade product and the waste, rea­ ches a predetermined values and separation power per single centrifuge in the cascade is also comparable with the value obtained in the R-net cascade. For this purpose, the objective function used is defined as follow: objective Function ¼ PF * ðða = absðcpð15; 15; 2Þ cp � ð15; 15; 2ÞÞÞÞ � � � � � ����� 1 ΔUcascade þ * abs czð15; 1; 2Þ cz � ð15; 1; 2Þ þ c * (19) b N Where, cpð15; 15; 2Þ and czð15; 1; 2Þ are the concentrations of Te-122 isotope in the product and the waste of the cascade. cp� ð15; 15; 2Þ and cz� ð15; 1; 2Þ are the predetermined concentrations for Te-122 isotope in 8

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Progress in Nuclear Energy xxx (xxxx) xxx

Table 9 Concentration of components in the waste and the product streams of lozenge net cascades of different configurations. Row

Isotopes

Lozenge net cascade 1 up

Product Concentration 120 0.027506 1 TeF6

Lozenge net cascade 2 up

Lozenge net cascade 3 up

Lozenge net cascade 4 up

Lozenge net cascade 5 up

0.027507

0.027521

0.027505

0.027506

0.517940

0.517933

0.517922

0.517931

0.517929

0.096767

0.096766

0.096764

0.096765

0.096765

0.190603

0.190607

0.190606

0.190609

0.190609

0.088890

0.088892

0.088891

0.088893

0.088893

0.068378

0.068380

0.068379

0.068380

0.068381

0.009140

0.009140

0.0019140

0.009141

0.009141

7.7265E-04

7.7265E-04

7.7267E-04

7.7267E-04

7.7267E-04

Waste Concentration 120 1 TeF6

3.6300E-05

3.6301E-05

3.6330E-05

3.6296E-05

3.62987E-05

2

122

TeF6

0.008776

0.008775

0.008775

0.008775

0.008775

3

123

TeF6

0.005875

0.005874

0.005874

0.005874

0.005874

4

124

TeF6

0.041465

0.041465

0.041465

0.041465

0.041465

5

125

0.069291

0.069291

0.069291

0.069291

0.069291

6

TeF6

126

0.190990

0.190990

0.190990

0.190991

0.190991

7

TeF6

128

0.327803

0.327803

0.327803

0.327804

0.327803

8

TeF6

130

TeF6

0.355762

0.355762

0.355762

0.355761

0.355761

2

122

3

TeF6

123

4

TeF6

124

5

TeF6

125

6

TeF6

126

7

TeF6

128

8

TeF6

130

TeF6

must be as large as possible. Since the items are not of the same order, coefficients are used to gradually align the items so that they all participate equally in the optimization process. Thus, by maximizing the function, we can reach to predetermined concentration of desired component as well as the highest possible amount for separation power of the single machine. Equation (20) defines the separation power of a cascade. V is the value function. In order to separation of isotopes in a multicomponent mixture, various relations have been proposed for the definition of the value function. Some of them are the Palkin equation (Palkin, 2013a,b), Filippov equation (Filippov and Sosnin, 1974), Yamamoto equation (Yamamoto and Kanagawa, 1979), Kolokoltsov equation (Kolokoltsov et al., 1970), Wood equation (Wood et al., 1999), Ilamed equation (Lehrer-Ilamed, 1969), etc. In this paper, the latest relation provided by Palkin, which is most frequently used, will be applied as Equation (21). Also, ΔMij is the difference of molecular weights of isotopes i and j. It should be noted that although the values obtained by each of the value functions are different, but their trends are all the same and it can be used each of them optionally in the objective function. For example, Norouzi et al. had used Filippov Equation (Norouzi et al., 2017). � � � � � � W W ΔU ¼ PV CP1 ; CP2 ; …; CPNc þ WV CW FV CF1 ; CF2 ; …; CFNc 1 ; C2 ; …; CNc

Table 10 Satisfaction of MARC condition in the lozenge net cascades. Cascades

Concentrations

Isotopes k1 ¼ 122

Lozenge net cascade 1 up Lozenge net cascade 2 up Lozenge net cascade 3 up Lozenge net cascade 4 up Lozenge net cascade 5 up

cp(6,-5) cz(6,-3) cz(8,-1) cp(5,-4) cz(5,-2) cz(9,2) cp(4,-3) cz(4,-1) cz(10,-1) cp(3,0) cz(3,2) cp(11,0) cp(1,0) cp(11,0)

TeF6

0.034134 0.006105 0.033084 0.040861 0.009341 0.049184 0.047002 0.014138 0.0718483 0.0549238 0.112122 0.184738 0.045184 0.014587

R’k1k2 ¼ k2 ¼ 123

R}k1k2

TeF6

0.017383 0.003109 0.016849 0.018996 0.004342 0.022865 0.019947 0.006000 0.0304917 0.017731 0.036198 0.059641 0.182379 0.0588801

1.963601 1.963601 1.963601 2.151017 2.151017 2.151017 2.356321 2.356321 2.356321 3.097465 3.097465 3.097465 0.247752 0.247752

the mentioned flows. ΔUCascade is the total separation power of the cascade. N is the total number of machines in the cascade and a, b and c are the objective function coefficients for each of the objective function items proved after a number of trial and error processes. PF is a penalty factor commonly is considered 1. If the target isotope concentration in the product and the waste of the cascade, respectively, is less or more than the predetermined values, the PF will be active and will eliminate the answer from the population. As can be seen, the objective function consists of three items. The purpose is maximizing them in order to maximize the function. In this way, the value of product and waste concentrations of desired component need to be close to predetermined values so that the denominators become minimum and the ratio of the total separation power of the cascade to the total number of machines

(20) � � V C1 ; C2 ; …; CNc ¼

�2 Nc � X ΔMN 1

1 Nc

c

1

i;j¼1

ΔMij

Ci

� � � Ci Cj ln Cj

(21)

i
In the test, the cascade Characteristics is: NL ¼ 15, α0​ ¼ 1:078, k1​ ¼ 2, k2​ ¼ 3 and Feed ¼ 1. This test has been performed to reach the Te-122 isotope concentration to 0.4488 in the cascade product and 0.0092 in the cascade waste and simultaneously to the maximum separation

Table 11 Concentration of isotopes in the waste and product streams of R-net and optimized net cascades. Product concentration (R- net cascade) Waste concentration (R- net cascade) Product concentration (Optimized net cascade) Waste concentration (Optimized net cascade)

120TeF6

122TeF6

123TeF6

124TeF6

125TeF6

126TeF6

128TeF6

130TeF6

0.0239 5.2058E-05 0.0229 4.1800E-5

0.4468 0.0092 0.4470 0.0079

0.0900 0.0057 0.0934 0.0053

0.2048 0.0403 0.2131 0.0396

0.1125 0.0684 0.1117 0.0685

0.1015 0.1903 0.946 0.1911

0.0184 0.3288 0.0155 0.3296

0.0024 0.3572 0.0017 0.3579

9

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Progress in Nuclear Energy xxx (xxxx) xxx

Table 12 Cuts of the two last layers of R-net and optimized net cascades. Cut of R-net cascade stages θð14; 11Þ θð14; θð14; 13Þ

9Þ

θð14;

7Þ

0.4053 θð15; 14Þ

0.4093 θð15; 12Þ

0.4127 θð15; 10Þ

0.4159 θð15; 8Þ

0.4651 θð15; 14Þ

0.5195 θð15; 12Þ

0.4525 θð15; 10Þ

0.4077 θð15; 8Þ

0.4030 0.4075 0.4111 0.4145 Cut of optimized net cascade stages θð14; 13Þ θð14; 11Þ θð14; 9Þ θð14; 7Þ

0.3041

0.4153

0.2702

0.3711

θð14;

5Þ

0.4197 θð15; 6Þ

θð14;

3Þ

0.4244 θð15; 4Þ

θð14;

1Þ

0.4304 θð15; 2Þ

θð14;1Þ

θð14;3Þ

θð14;5Þ

θð14;7Þ

θð14;9Þ

θð14;11Þ

θð14;13Þ

–

0.4377 θð15;0Þ

0.4458 θð15;2Þ

0.4543 θð15;4Þ

0.4625 θð15;6Þ

0.4696 θð15;8Þ

0.4755 θð15;10Þ

0.4805 θð15;12Þ

– θð15;14Þ

0.4181

0.4224

0.4279

0.4345

0.4422

0.4505

0.4587

0.4662

0.4727

0.4781

0.4826

θð14;

θð14;

θð14;

θð14;1Þ

θð14;3Þ

θð14;5Þ

θð14;7Þ

θð14;9Þ

θð14;11Þ

θð14;13Þ

–

0.3318 θð15;0Þ

0.5437 θð15;2Þ

0.5740 θð15;4Þ

0.4721 θð15;6Þ

0.3688 θð15;8Þ

0.3251 θð15;10Þ

0.5011 θð15;12Þ

– θð15;14Þ

0.2884

0.4287

0.4128

0.5332

0.3899

0.5324

0.5638

0.4575

5Þ

0.2987 θð15; 6Þ

0.4680

3Þ

0.4873 θð15; 4Þ

0.5418

1Þ

0.5299 θð15; 2Þ

0.5276

power of single machine in the cascade. The coefficients of a, b and c in the objective function were obtained equal to 1000, 100 and 1,000,000, respectively. The results show that the ACOR algorithm is well able to find optimal cascade parameters. Table 11 exhibit the concentration of Tellurium isotopes in the product and the waste of the two cascades. It is observed that: - The product concentration of the second isotope is 0.4468 and 0.4470 in the R-net and optimal cascades, respectively, as well as 0.0092 and 0.0079 in the R-net and optimal cascade for the waste concentra­ tion. Clearly, the separation performance results from the cascade optimized by the optimization algorithm are more appropriate. - The single-machine separation power (based on Palkin equation, as mentioned above) are 0.00479 and 0.004800, respectively, in the two Rnet and optimal cascades. It is known that the single-machine separation powers in the two cascades are comparable. In order to compare the behavior of the different relations of sepa­ ration powers, the single-machine separation power based on the Fili­ ppov equation, for example, is also reported that are 0.0000463 and 0.0000465, respectively, in the two R-net and optimal cascades. As a result, despite the different values, the two equations have the same behavior. Table 12 shows the cuts of the last two layers of the two cascades. The cuts of R-net cascade obtained around 0.4 while cuts of optimized cascades have been distributed in wider range. The reason is that the lower and upper bound for the cuts in the optimization algorithm had been considered 0.25 and 0.65 respectively. Thus, the algorithm can explore a larger space to find optimal parameters. For the performance comparison, also, the cascade cuts (the ratio of product and feed of the cascade) are 0.035 and 0.037 respectively, for the two R-net and optimal cascades. This result shows the functionality of the optimization method again.

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