Dynamic Optimization of the Tandem Acetylene Hydrogenation Process

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Dynamic optimization of tandem acetylene hydrogenation process Wenli DU, Chunyu Bao, Xu Chen, Liang Tian, and Da Jiang Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.6b00860 • Publication Date (Web): 21 Oct 2016 Downloaded from http://pubs.acs.org on October 23, 2016

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Industrial & Engineering Chemistry Research

Dynamic optimization of the tandem acetylene hydrogenation process WenliDu1*, Chunyu Bao1, XuChen2, Liang Tian1, Da Jiang1 1

Key Laboratory of Advanced Control and Optimization for Chemical Processes, Ministry of Education, East China University of Science and Technology, Shanghai 200237, PR China

2

School of Electrical and Information Engineering, Jiangsu University, Zhenjiang, 212013, China

ABSTRACT: In ethylene production units, the acetylene side product is converted into additional C 2 H 4 over a palladium-carbon fixed bed reactor, in which the catalyst activity is affected by the formation of green oil impurities. We determined the optimal dynamic operating conditions for two reactors in series undergoing catalyst deactivation while maintaining the selectivity and conversion rate. The optimal switching periods for the reactors were investigated with dynamic optimization of the hydrogen input and inlet temperature with the incremental feature. Control vector parameterization was used to transform this dynamic problem into a finite-dimensional nonlinear programming problem. An incremental-encoding differential evolution with constraint ranking-based mutation operator (IEDE-CRMO) was proposed based on the reaction selectivity and catalyst activity variation to meet certain requirement of the control variables and to solve a complex industrial process. The IEDE-CRMO method was applied in an 1

ACS Paragon Plus Environment


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industrial C2 hydrogenation tandem reaction process, and results showed improvement in overall operation efficiency.

Keywords: acetylene hydrogenation, reactor switching strategy, dynamic optimization, IEDE-CRMO

1. INTRODUCTION Production of polymer-grade ethylene through high-temperature cracking of naphtha or C2–C6 saturated hydrocarbons is an important process in the petrochemical industry.1,2 The C2 hydrogenation process that occurs in the ethylene unit is relatively complicated and sensitive compared with that in cracking furnace and rectifying column. Optimization of the C2 hydrogenation process is one of the bottlenecks for the entire ethylene production process because its location in the middle of the production process stream allows fewer changeable operating conditions.3 The acetylene conversion rate and hydrogenation selectivity of the C2 hydrogenation reactor directly affects the product quality and output, because excess acetylene will poison the downstream ethylene polymerization catalyst; 4 thus, the acetylene content must be less than 10 ppm. Considering the dynamic variation of catalyst activity, simultaneous optimization of the switching periods of series reactors in stage 1 and stage 2 reactors and the operating condition is vital in increasing overall performance.

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Two types of production processes are available for acetylene hydrogenation, namely, front-end and back-end hydrogenation,1,2 which are differentiated by the location of the acetylene hydrogenation reactor in the process flow of the ethylene unit. In the more common back-end hydrogenation process, the acetylene hydrogenation reactor is located after the deethanization unit, and the inlet stream mostly contains C2 hydrocarbons. Acetylene and ethylene hydrogenation are moderately exothermic reactions, and in the back-end reactor, the relatively low hydrogen content in the feed tends to cause the deposition of carbonaceous components on the catalyst, which increases the rate of catalyst passivation. Thus, the operating temperature of reaction must be increased to compensate for passivation to achieve the required acetylene content specifications in the outlet material. Catalyst regeneration is usually performed after a few months of operation. The activity and selectivity are controlled by adjusting the feed, catalyst bed temperature, and hydrogen input, among other factors.5 The control system of the acetylene hydrogenation reactor has been studied for steady hydrogenation processes,6,7 in which steady-state optimization focusing on the increase of product yield was conducted. In practical application by actual plants, the operating conditions of a single reactor are given particular attention, whereas the combined performance of several reactors is neglected. However, in a reaction system, the catalyst performance for the various reactors differs. Hence, optimization of the switching periods of the two reactors to achieve the optimal output while satisfying all the system requirements is necessary. Catalyst deactivation is closely related to temperature, in which higher temperature 3



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increases the possibility of deactivation. Thus, if only catalyst deactivation is considered, then reaction temperature should be kept as low as possible. However, if the temperature is below the critical temperature required for the reaction, the acetylene concentration at the reactor outlet will greatly fluctuate, thus affecting product quality.8 By contrast, an excessively high temperature will facilitate the production of the side product, i.e., ethane.9,10 Thus, the reaction temperature must be controlled within a proper range.11 The catalyst of the stage 1 reactor can generally be used for a dozen months. Catalysts have initially high selectivity and activity, and the conversion of acetylene can be maintained above 90% even with minimum hydrogen input and inlet temperature. However, as the reaction proceeds, the activity and selectivity of the catalyst reduce unceasingly because of the green oil formed from the reaction that covers the catalysts. The greater the amount of green oil generated, the more the inlet temperature and hydrogen input need to be increased to meet the system requirements. In industrial application, most catalytic processes maintain constant conversion by increasing the temperature according to operation experience. However, the method does not consider the interaction between the inlet temperature and the hydrogen input. Optimization of the aforementioned hydrogenation process is a dynamic optimization problem with time-sequence constraints. A set of differential algebraic equations is used to describe the reaction process, wherein dynamic optimization refers to the control of the operating variables in a dynamic model to optimize the process performance indexes.12,13 Solutions for dynamic 4


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optimization problems can be simply categorized into dynamic programming (DP), indirect, and direct methods. Bellman’s principle of optimality is applied in the DP method.14 This method can effectively solve dynamic optimization problems but is subject to the difficulties of dimensionality. To overcome this problem, Luus15 proposed iterative dynamic programming (IDP), which reduces dimension expansion and is an effective global optimization method. However, its computation speed is low and it is only suitable for small-scale problems. The indirect method expands the original problem to a Hamiltonian system by applying the Pontryagin maximum principle,16,17 and is the most accurate method for solving optimal control problems.18,19 However, its application in actual dynamic optimization problems is complex. The direct method has two different parameterization methods such as the simultaneous method and control vector parameterization (CVP). The simultaneous method simultaneously discretizes the state and control variables, whereas CVP only discretizes the control variables. The simultaneous method can better handle path constraint problems and may generate extremely large-scale non-linear programming (NLP) problems for which certain handling methods are needed, but will still result in unstable convergence. The CVP method transforms the original problem into a NLP problem. Even though the accuracy of the results is dependent on the number of discrete points, the CVP method is frequently used in industrial process problems because of its simple and feasible NLP solutions.20–22 Gradient optimization algorithms can also be used in solving such problems, but 5



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computation difficulties may easily be encountered when the gradient information becomes unavailable.23, 24 Unlike a gradient algorithm, intelligent optimization algorithms are simple, easy to implement, and have strong global searching abilities. Their popularity has been rising rapidly, and they have been widely applied to solve many kinds of optimization problems, including a gradual extension to solving dynamic optimization problems. Intelligent optimization algorithms have been formulated as genetic algorithms (GA), ant colony algorithms (ACA), particle swarm optimizations (PSO), differential evolution (DE), and so on.25–31 The advantages of DE over the other optimization problem solutions include structure simplicity, easy implementation, relatively low space complexity, good performance, and better searching ability and robustness.32 However, its inefficient exploitation is still a problem. To overcome this weakness, many researchers developed new algorithms based on DE, such as modified DE (MDE)[33], trigonometric DE (TDE)[45], composite DE (CoDE)[34], differential evolution with ranking-based mutation operator (DE-RMO)[35], and so on. In this paper, constraint ranking-based mutation operator (CRMO) has been proposed which uses a constraint ranking technique combined with RMO. The incremental-encoding differential evolution with constraint ranking-based mutation operator IEDE-CRMO has better performance in terms of accuracy and convergence speed especially for problem with constraints. The CVP-based IEDE-CRMO can avoid certain flaws of the gradient

6


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algorithms and convergence speed when used to solve increasing parameter dynamic optimization problem with constraints. The acetylene hydrogenation system consists of two reactors connected in series, and its optimization can be described as a constrained non-linear dynamic optimization problem; the acetylene content in the reaction process is minimized by a comprehensive consideration of the catalyst operating life, inlet temperature, hydrogen input, and acetylene removal ratios of stages 1 and 2.11 At the same time, the amount of hydrogenation must satisfy the periodic increment and constant increase of inlet temperature. In this study, an effective dynamic optimization method that can deal with the special requirements of control variables and the optimization of the switching periods for the two-stage tandem C2 hydrogenation process was established based on the dynamic model for the two-stage acetylene hydrogenation system. The in-process operating conditions of the reactors are described. The CVP-based IEDE-CRMO method is able to handle the control variable increment constraints, thus providing solutions for complex dynamic optimization problems in chemical processes. Using dynamic optimization, the catalyst switching and regeneration strategies were rationally designed and the optimal operating conditions were determined under the premise of ensuring product quality to increase ethylene production and economic benefits. The models involved in the process optimization of acetylene hydrogenation are introduced in Section 2, and the optimization problems that require solutions are fully described in Section 3, as 7



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well as solving strategies for the corresponding dynamic optimization methods. The optimization results and analyses of the acetylene hydrogenation system are presented in Section 4. 2. C2 HYDROGENATION PROCESS MODEL Accurate dynamics is the foundation for the successful simulation of reactor performance. It can be used to find the temperature and pressure that will increase the selectivity of ethylene production through acetylene hydrogenation and reduce the use of hydrogen. Regarding the optimization of the hydrogenation process switching periods and the operating conditions in each period that were solved in this study, not only was the model for each reactor separately established but catalyst deactivation was also considered.36,37 The dynamic model for the hydrogenation reactor used in this study is suitable for actual industrial reactors. Heat exchanger Temperature control T

Ethylene product

Preheater Add hydrogen H Stage 1 reactor Stage 2 reactor

Deethanizer

Ethylene distillation Green oil remover

column

Ethylene dryer

Figure 1. Flowchart of acetylene hydrogenation process.

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Figure 1 illustrates the process flow diagram for the main elements in the acetylene gas-phase selective catalytic hydrogenation process. In this process, C2 materials are emitted from the top of the de-ethanizer, passed through the heat exchanger, and exchanged heat with the stream from the reactor outlet. Subsequently, they are mixed with the proper amount of hydrogen and entered the feed preheater. The materials are heated by low-pressure steam to a certain temperature and entered the stage 1 acetylene converter, where the majority of the acetylene in the feed material is hydrogenated to ethylene or ethane. The materials from the stage 1 reactor outlet are mixed with additional hydrogen and then passed through an intercooler to enter the stage 2 acetylene converter, where almost all the residual acetylene in the feed material is removed. After passing through the green oil (oligomeric byproducts) remover, the mixture is sent to the ethylene distillation column to be fractionated for polymer-grade ethylene. Control variables considered in the two-stage tandem acetylene hydrogenation process are the inlet temperature T and hydrogen input H. Since the majority of the acetylene is converted in the stage 1 reactor, the acetylene content in the stage 2 reactor is extremely low, making it impossible to describe the internal mechanism of the stage 2 reactor using either macroscopic or microscopic mechanisms. In this work, a data-driven model for stage 2 reactor was applied by fitting actual production data. 2.1 Principles of catalytic hydrogenation In the presence of a hydrogenation catalyst, the acetylene in the C2 fraction is hydrogenated to ethylene. The following reactions may occur during hydrogenation: 9



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Main reaction: C2 H 2 + H 2 → C2 H 4 + 174.3 kJ mol −1

(1)

Side reactions: C2 H 4 + H 2 → C2 H 6 + 136.7 kJ mol -1

(2)

C2 H 2 + 2 H 2 → C2 H 6 + 311.0 kJ mol −1

(3)

mC2 H 2 + nC2 H 2 → green oil (oligomeric byproducts )

(4)

The following cracking reaction can also occur at high temperature: C2 H 2 → 2C + H 2 + 227.8 kJ mol −1

(5)

Considering only the hydrogenation of ethylene and acetylene,6 reaction rate equations may be written as follows: rC2 H 2 =

rC2 H 4 =

(

k3 pC2 H 2 pH 2

(

1 + K H 2 pH 2

(1 + ( K

)

0.5

+ K C2 H 2 pC2 H 2 + K C2 H 4 pC2 H 4 + K C2 H 6 pC2 H 6 k2 pC2 H 4 pH 2

H2

pH 2

)

0.5

+ K C2 H 2 pC2 H 2 + K C2 H 4 pC2 H 4 + K C2 H 6 pC2 H 6

) )

3

(6)

3

(7)

Where,  E  ki = k0,i exp  − i   RT 

(8)

2.2 Reactor model For industrial hydrogenation, temperature and concentration gradients between the gas-phase substrates and the solid catalyst particles are eliminated by the high speed of the gas flow inside the reactor. Hence, in practice, a one-dimensional pseudo-homogeneous model can sufficiently reflect the main features of the hydrogenation reactor and meet the requirements for industrial

10


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application. In this study, an isobaric, adiabatic, pseudo-homogeneous one-dimensional plug flow model was used to simulate the hydrogenation process. The following equations were applied: Material balance equation: dFi = ∑ ( ρ B rj ,d ) S dz

(9)

Energy balance equation: dT − S ρ B ∑ ( H i rj ,d ) = dz ∑ ( FiC p,i )

(10)

2.3 Deactivation kinetics Acetylene

molecules

adsorbed

on

the

catalyst

will

undergo

slow

hydrogenation–oligomerization to generate high-molecular-weight materials containing even numbers of carbon atoms. These oligomers may block the micropores of the catalyst, reducing the catalyst’s specific surface area and consequently, its activity.38 Catalyst deactivation is a complex phenomenon, with mechanisms and patterns that are dependent on the specific actual process.39,40 According to various research studies, hydrogenation processes with different mechanisms have different deactivation coefficients. The deactivation coefficient at a given time and temperature, ai ( t , T ) , can be expressed as in Eq. (11): dai ( t , T )  E  = −ka , j exp  − a ,i  * ain dt  RT 

(11)

where n is the deactivation order. An apparent rate can be expressed using Eq. (12):

11



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ri ,d ( t, T ) = ri 0 (T ) ai ( t , T )

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(12)

where ri ,d ( t, T ) is the reaction rate for the reactant on the catalyst at a given time, and ri00 (T ) is the reaction rate of reactant on fresh catalyst 2.4 Stage 2 reactor model The main purpose of stage 2 reactor in the acetylene hydrogenation system is to ensure that output acetylene content is reduced to 10 ppm. Since acetylene content in stage 2 reactor is extremely low and is in a mesoscopic state, the stage 2 reaction process cannot be described using any reaction mechanisms. The stage 2 reactor model used in this study was established by fitting parameters obtained from production process. The outlet acetylene concentration of stage 2 reactor was able to meet the system requirements, although there was a redundancy in real production requirements. Given that the stage 2 reactor’s catalyst has relatively short life cycle and strong catalytic activity, the ethylene increment and hydrogen/acetylene ratio obtained by fitting the actual production data are approximately linear descriptions. The stage 2 reactor hydrogen/acetylene ratio can be described by the following equation: R 2 = s * ( d R 2 ) + s1* d R 2 + s 2 2

(13)

where s, s1, and s2 are fitted parameter values. The stage 2 reactor ethylene increment can be described by the following equation: x R 2 = (2 − R 2) * x R1C2 H 2

(14)

where (2-R2) is the acetylene selectivity of stage 2 reactor. 12


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3 C2 HYDROGENATION PROCESS OPTIMIZATION PROBLEM AND SOLUTION 3.1 Description of acetylene hydrogenation optimization problem According to the above descriptions and performance indexes of dynamic optimization problem, the optimization problem of two-stage acetylene hydrogenation system can be described as follows: Set the objective function as the ethylene content of the maximized catalyst operation cycle, and the equation is: Di ,R1

Di ,R1 , D j ,R 2

d R1 =1

dR1 ,d R 2 =1

max obj = {( ∑ ( xR1C2 H4 ( dR1 )) +

∑

( xR2 ( d R1 , d R2 )))*24*28* PC2 H4

(15)

− Preg (1 +  Di,R1 / D j,R2 )}/ Di ,R1

Descriptions of constraints: dxR1 ( d R1 ) = f 1 ( x0 , H ( d R 1 ) , T ( d R 1 ) , r , d R 1 ) dz xR 2 (d R1 , d R 2 ) = f 2( xR1 ( d R1 ) , d R 2 ) x ( t0 ) = x0 g ( x R1 ( d R1 ) , d R1 ) ≤ 0

(16)

H ( d R 2 ) ≥ H ( d R 2 − 1) T ( d R1 ) ≥ T ( d R1 − 1) u L ≤ u ≤ uU

Eq. (15) represents the maximized average daily revenue under certain switching period of the stage 1 and stage 2 reactor conditions, where obj is the average daily revenue during operation. The x0 = ( x0C H , x0C H , x0C H ) represents the flow rates of ethane, ethylene, and acetylene at the inlet, 2

6

2

4

2

2

respectively; x R1 = ( x R1C H , x R1C H , x R1C H , x R1H , x R1T ) represents the flow rates of ethane, ethylene, 2

6

2

4

2

2

2

13



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acetylene, and hydrogen, as well as the thermodynamic temperature, at the stage 1 reactor outlet. The term ∆xR1C H ( d R1 ) = xR1C H ( d R1 ) − x0C H is the ethylene flow rate increment in the stage 1 reactor, 2

4

2

4

2

4

and xR 2 is the ethylene flow rate increment at the outlet of the stage 2 reactor. The unit for xR1C2 H4 ( d R1 ) and that for xR 2 is kmol h −1 ; there are 24 h per day, representing the daily ethylene

increment, given that the molecular weight of ethylene is unit of weight. Moreover, and Preg

is

the

cost

PC2 H 4

of

28 kg kmol −1

and can be converted to a

is the price of the ethylene (7,690 Chinese Yuan (CNY)/t), catalyst

regeneration

(300,000CNY). Di,R1 ∈[300,440]

and

D j , R 2 ∈[100,140] represent the switching periods of the stage 1 and stage 2 reactors, respectively;

where i and j enumerate different stage 1 and stage 2 reactor-switching periods for identification based on the number of days in a discrete phase. The term  Di ,R1 / D j ,R 2  rounding represents the number of catalyst switches performed for the two-stage reactors in series. Furthermore, z ∈ [0,3.16] is the height of the reactor, and the d R1 and d R 2 are the numbers of days spent in the stage 1 and 2 reactors during the operation, respectively, where d R1 ∈ 0, Di ,R1  and d R 2 ∈ 0, D j ,R 2  . The xR1 ( ) can be solved using Eqs. (9) and (10); r is the reaction rate of the reactant on the catalyst at a given time and can be solved using Eqs. (11) and (12), and Eqs. (6) to (8). In addition, xR 2 ( ) represents the ethylene increment in stage 2 reactor using Eqs. (13) and (14). The inequality constraint g ( ) means the flow rates of acetylene and hydrogen at the stage 1 reactor outlet are below the standard values. The inequality constraints of control variables H ( d R 2 ) ≥ H ( d R 2 − 1) and T ( d R1 ) ≥ T ( d R1 − 1) mean that the hydrogen input must satisfy the periodic increment and the constant increase of inlet 14


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temperature. The hydrogen input is decided by the periodically increasing D j , R 2 and the increasing D j , R 2 hydrogen input. The inlet temperature within the D j , R1 increases gradually. The foregoing optimization problem aims to maximize the performance index obj in a certain sense by controlling the hydrogen feed rate and material inlet temperature (control variable u ) of the stage 1 reactor and selecting appropriate operation cycles (discrete optimization variables Di , R1 , D j , R 2 ), under the premise of satisfying material and energy balances while meeting the constraints for the acetylene and hydrogen flow rates at the outlet of stage 1 reactor and the outlet acetylene content of stage 2 reactor. Considering the significant decays of catalyst activity overtime in both hydrogenation reactors, as well as the constraint conditions in which process variables also change with time, the corresponding control variables need to be dynamically adjusted and must meet the constraint conditions to achieve the optimal operating conditions while meeting product quality requirements. Handling industrial complex problems not only need to consider the internal relations and the extreme conditions but also need more effective methods. 3.2 Dynamic optimization solution based on the DE algorithm For intelligent algorithm-based solutions of dynamic optimization problems, discrete methods must be used to transform the original dynamic optimization problem into a finite-dimensional nonlinear programming problem. Then, the original problem’s solution can be approximated by solving the NLP problem. In this case, the starting and ending times of the dynamic optimization problem are deterministic: discretize the problem’s time domain d R1 ∈ 0, Di ,R1  into N segments, 15



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and set the search range for the decision variable as [ rmin , rmax ] , with dimension D. If the control variable in stage j can be expressed as rj , and the decision variable set of the entire time domain can be expressed as R = ( r1 , r2 ,..., rD ) , then the DE algorithm can be used to solve the decision variables and optimize the objective function value. DE is an intelligent algorithm proposed by Price and Storn41 which is highly efficient, converges rapidly, and is robust among other advantages. The most important factor is the generation of trial vectors. Mutation and crossover are used to generate trial vectors, then the optimal individual of the next generation will be selected. The processes are as follows: 1. Initialization: DE randomly initializes NP individuals in the searching space, each of which can be expressed as xig =  xig,1 , xig,2 ,..., xig, D  , i = 1, 2,..., NP , where g is the number of current generations and D is the number of optimization variables. 2. Evaluation: Each individual is evaluated to locate the current optimal individual. 3. Mutation operation: The mutation vector is denoted as Vi g , and the most commonly used mutation method DE/rand/1 is given by: DE / rand / 1: Vi g = X rg1 + F ( X rg2 − X rg3 )

(17)

In Eq. (17), r1 , r2 and r3 are random integers in the interval [1, NP] that are not equal to either each other or i. Parameter F is known as the zoom factor, and generally has a value of [0, 1]. 4.Repair: If vig, j ∉  rj min , rj max  , use a repair operator to make

V rg

feasible：

16


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 min {rj max , 2* rj min − vig, j } if vig, j < r j min  vig, j =  g g max {rj min , 2* rj max − vi , j } if vi , j > r j max

(18)

5. Crossover operation: This operation is used to increase the diversity of DE populations. Binomial restructuring is applied to generate a trial vector denoted as Rig : g v , if rand ≤ CR or j=jrand , j=1,2,...,D rijg =  ijg  xij , otherwise

where rijg , vijg , xijg are the jth elements of

(19)

Rig ,Vi g , X ig

, respectively; CR is the crossover rate whose

value usually falls in [0, 1]; rand is a random number in [0, 1]; and jrand is a random integer in [1, NP]. 6. Selection operation: The winner between the target vector

X ig

and trial vector

Rig

is

selected to enter the population of the next generation.  R g , if Rig is better than X ig X ig +1 =  i g  X i , otherwise

(20)

7. If the termination criterion is satisfied, the algorithm is terminated. Otherwise, the DE algorithm repeats the operations of mutation, repair, crossover, and selection. The DE algorithm has been applied in dynamic optimization problems with excellent results.42–44 Among the main factors that affect its optimization efficiency are the variable discretization method, expression of variable constraints, and selection criteria for the optimal solution. In this study, the approach to arriving at a solution was proposed within the framework of the DE algorithm. 3.3 Control vector parameterization 17



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The initial step in solving dynamic optimization problems is to discretize the original problem. The discretization method CVP is used to segment the time intervals, and the control variables are divided into a limited number of subcomponents in the time domain. A primary function containing a limited number of parameters is used to approximate each subcomponent. The implementation is described in the following statements. The time interval [t 0 , t f

] is divided into

N stages by the CVP, where the primary function of each stage can be expressed as a constant function, linear function, wavelet function, and so on.45 The constant function method is the most commonly used. The segmented constant function can be used to approximate the control variable u: c , t ∈ [ti −1 , ti ] ,i=1,2,...,N r (t ) =  i 0 , else

As such, each set of

( c1 , c2 ,..., cN )

(21)

values will correspond to one control trajectory. Then, the DE

algorithm can be used to obtain the control trajectory that optimizes the objective function. 3.4 Incremental encoding The preceding discussion showed that, as the reaction progresses, the catalyst activity gradually decreases, although the system needs to satisfy these requirements: the outlet acetylene and hydrogen flow rates of stage 1 reactor are below the maximum constraint values, and the acetylene level at stage 2 reactor outlet meets the system specifications. To accomplish this, the control variables must continuously increase during the reaction. The degree of constraint violations was calculated to minimize the violations of constraints of control variables in the 18


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production. However, minimizing the degree of violation does not guarantee that control variables must meet the requirements. In addition, calculating the violation degree is difficult because of the periodic increase in hydrogen input. The idea of the incremental encoding came from the original non-uniform discretization-based CVP (ndCVP)46. In this paper, we introduced the incremental encoding (IE) into DE evaluation operator, which can be satisfied by the indirect method in the DE algorithm by using IE, which is simple and easy to implement. At the same time, this method can make the control variable meet the system requirements. This method encodes parameters ri ∈[ ri min , ri max ] = [0,1] ,i = 1,..., N + 1 into parameter vector U = [u1, u2 ,..., uN ] , which needs to N +1

be optimized. IE must satisfy the following conditions: ri ∈ [0,1] , and r1 / ∑ ri must equal the ratio i =1

of the difference of the ith interval [ui −1 , ui ] over the entire range [umin , umax ] of the control variable. r

i N +1

∑r

=

( ui − ui −1 )

( umax − umin )

, i = 1, 2,..., N

(22)

i

i =1

A fractional parameter, R = [ r1 , r2 ,..., rN +1 ] that corresponds to parameter U = [u1, u2 ,..., uN ] must also be optimized; the mapping relationship is shown below: N +1  u1 = umin + ( umax − umin ) r1 / ∑ ri i =1  ⋅⋅⋅  k N +1  uk = umin + ( umax − umin ) ∑ ri / ∑ ri i =1 i =1  ⋅⋅⋅  N N +1 u = u + ( u − u ) N min max min ∑ ri / ∑ ri  i =1 i =1

(23)

The requirements for the parameters in the acetylene hydrogenation system can be simply satisfied by using IE. 19



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3.5 Constraint handling mechanism based on Deb’s rule In the constrained DE algorithm, the objective function value obji and the constraint violation degree vioi corresponding to each individual U i can be calculated. The constraint handling mechanism will be considered next. Constraint handling methods usually include a penalty function method that transforms a single-objective constrained optimization problem into a single-objective optimization problem, and a multi-objective method that discriminates between feasible and infeasible solutions to generate a multi-objective optimization problem. In this study, the constraints were handled using the priority rule of feasible solutions proposed by Deb,47 which is a multi-objective method. Compared with a penalty function, the advantage of the priority rule of feasible solutions is that no parameters need to be introduced. Moreover, Deb’s rule is a simple and effective constraint handling method. This rule compares the advantages and disadvantages of two solutions using the following method: For two individuals U i and

Uj

that satisfy i ≠ j , the corresponding objective function value

and constraint violation degree are ( obji , vioi ) and ( obj j , vio j ) , respectively, and if U i is superior to Uj

, then one of the following 3 rules must be satisfied: Rule 1: vioi >0 && vio j > 0 , and vioi < vio j ; Rule 2: vioi =0 && vio j = 0 , and

obji > obj j ;

Rule 3: vioi =0 && vio j > 0 . 3.6 Constraint ranking-based mutation operator 20


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DE algorithm may suffer from low convergence speed. Introducing the CRMO into DE is helpful for enhancing its performance.45 Dynamic optimization of tandem acetylene hydrogenation is the problem with time-increasing parameter constraints. Here, the CRMO is used to deal with this problem. The constraint ranking method is used to sort the DE population from best to worst based on the rule described in Figure 2. Input:

Evaluate the obj and vio for each individual in the population

1:

vio=0 and the fitness =obj

2:

select the minimum obj(minobj)

3:

vio≠0 and the fitness = minobj-vio

4:

sort the population based on the fitness of each individual

Figure 2. Constraint ranking technique First, evaluation operation with DEB’s approach is done, then a quick sort algorithm can be used to sort the whole population from best to worst. Subsequently, the ranking Ri of ith individual is assigned as: Ri = NP − i , i = 1, 2,..., NP

(24)

The selection probability of ith individual is calculated as: pi =

Ri , i = 1, 2,..., NP NP

(25)

According to Eq. (25), the best individual in the current population will give the highest ranking. The individuals in the RMO are finally selected based on the selection probability

21



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pi .The ranking-based individual selection in RMO is illustrated in Figure 3. The individuals will

be chosen in the mutation operator with high selection probability. Input：：

The target vector index

Output:

The selected vector indexes r1, r2, r3

1:

Randomly select r1 ϵ [1,NP] While rand > pr1 or r1 ==i

2:

i; selection probability

p1 , p2 ,..., p NP

Randomly select r1 ϵ [1,NP]

3: 4:

END

5:

Randomly select r2

6: 7:

While rand > pr or r2 ==i Randomly select r2 ϵ [1,NP]

8:

END

9:

Randomly select r3 ϵ[1,NP]

10: 11:

While rand > pr or r2 ==i Randomly select r3 ϵ [1,NP]

12:

END

ϵ [1,NP]

2

3

Figure 3. Ranking-based individual selection 3.7 Implementation steps of hydrogenation process optimization The regulation of the acetylene hydrogenation process mainly includes controlling the reactor feed temperature and hydrogen feed flow rate. The average daily revenue (Eq. (15)) was optimized by applying the aforementioned calculations. In this study, the number of days, k, of the discrete phase was 20, and an enumeration method was used to determine the switching periods, Di , R1 and D j , R 2 .

The prediction accuracy can be improved by refining k, although the computation

time increases accordingly. However, the value of k can be practically selected based on the actual need.

22


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Hydrogen input H needs to satisfy a gradual increase or periodic increment due to the catalyst activity of the stage 2 reactor. The temperature T needs to meet the time-increasing constraint caused by the catalyst activity of the stage 1 reactor. Hydrogen input H with the incremental encoding parameter is expressed as RH : m1 =  Di , R1 / Di , R 2  m2 = Di , R 2 / k

(26)

m3 = ( Di , R1 − m1* Di , R 2 ) / k RH = ( rH 1,1 , rH 1,2 ,..., rH 1,( m 2 +1) ,..., rHm1,1 , rHm1,2 ,

(27)

..., rHm1,( m 2 +1) , rH ( m1+1),1 ,..., rH ( m1+1),( m 3+1) )

H will experience an

( m1 + 1)

times incremental encoding process，which can be expressed as

Eq. (28): RH = ( rH 1 , rH 2 ,..., rH ( N + m1+1) )

(28)

Temperature T with the incremental encoding parameter is expressed as RT :

(

RT = rT 1 , rT 2 ,..., rT ( N +1)

)

(29)

3.7.1. Implementations of IEDE-CRMO algorithms Step 1: The respective catalyst switching periods for both acetylene hydrogenation reactors, Di , R1 and D j , R 2 ,

and the number of days k for each stage based on the actual need were determined.

The two-stage-combined acetylene hydrogenation dynamics was discretized using the CVP method, as well as the discretization dimension N according to the k, in which the control variables include the hydrogen input and temperature U = [ H T] . DE maintains a population of NP

23



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individual members, where NP is the population size. The incremental encoding parameter is expressed as R = [ RH , RT ] = ( r1, r2 ,..., r2*N +m1+2 ) = (r1, r2 ,..., rD ) . 0  , was initialized, where individuals Ri0 =  ri0,1 , ri0,2 ,..., ri0,D  in Step 2: A population R0 =  R10 , R20 ,..., RNP

the population are used to represent solutions, and D is the dimension of the optimization problem. IE was used to solve the corresponding U i0 = ui0,1 , ui0,2 ,..., ui0, N  , so as to solve the objective function values f (U i0 ) of all the populations, as well as the degrees of constraint violation vio (Ui0 ) . Then, Deb’s rule was used to find the optimal solution. Step 3: The DE algorithm was implemented, where F=0.5, CR=0.9, NP=3*D and MaxGen=200048,49. CRMO: For each individual, the population was sorted according to the feasibility-based ranking technique, and the ranking-based mutation operator was performed. g  . Mutation: The mutation operation was performed, solving R g =  R1g , R2g ,..., RNP

Repair: If vig, j ∉ [0,1] , a repair operator was used to make

vig, j ∈ [ 0,1]

Crossover: The crossover operation was performed based on the population parameters that are obtained. g  , the incremental encoding method was used Evaluation: Based on the solved R g =  R1g , R2g ,..., RNP

g  . Then, the population’s objective function to solve for the corresponding U g = U1g ,U 2g ,...,U NP

value and the degree of constraint violation were solved. Finally, Deb’s rule was used to find the optimal solution. 24


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Selection: The DE selection operation was performed. Step 4: Termination. If the population satisfies the termination criterion or reaches the maximum number of iterations MaxGen, then output the optimal solution. Otherwise, return to Step 3.

Figure 4. Flow diagram of the IEDE-CRMO algorithm.

25



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4. EXPERIMENTAL RESULTS FOR ACETYLENE HYDROGENATION SYSTEM For the optimization problem given by Eq. (15), which was aimed at the production conditions for actual industrial units, the discrete time (k = 20days) and the range of switching periods Di , R1 ∈ [300, 440] and D j, R 2 ∈[100,140] were set according to actual production experience. To reduce the computational complexity, the switching period was then enumerated for stage 1 reactor Di, R1 ∈{300,320,340,360,380,400,420, 440} and stage 2 reactor D j ,R 2 ∈{100,120,140} . Different combinations of stage 1 and stage 2 switching periods are listed in Table 1, and optimal objective function values that met the system requirements were found for different switching period conditions. The roles of the two reactors in series are different; the stage 1 reactor converts the majority of the acetylene, whereas the stage 2 reactor serves as a protective auxiliary for converting the residual acetylene. In our experiment, industrial data were used for the stage 1 reactor, and a genetic algorithm was used to estimate the reaction dynamics and deactivation dynamic parameters.11,47 The parameters of stage 2 reactor in Eq. (13) was also fitted using industrial data, in which the parameter values fitted were s = −6 × 10−6 , s1 = 6.3 × 10−3 , s 2 = 1.3 . By adjusting the control variables of the system, optimal objective function values were found through the obtained stage 1 and stage 2 dynamic models that satisfied all constraint conditions. Here, we selected the hydrogen input and the reactor temperature as the control variables. In the case of excess hydrogen input, the occurrence of side reactions will intensify, and excess ethylene will react with hydrogen to produce ethane, thus reducing the ethylene yield. In the case of 26


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insufficient hydrogen addition, the large amount of residual acetylene will affect product quality. Hence, there is an optimal value for the hydrogen input during hydrogenation that optimizes the reaction selectivity.51 Temperature is closely related to the reaction rate.52 As the temperature increases, the increments in the main and side reactions’ rates are different, thus affecting the selectivity of the hydrogenation. For example, hydrogenation cannot proceed if the temperature is too low, but the catalyst activity will increase if the temperature is too high. In that case, the intensity of the reaction may easily cause run away temperatures, as well as increase the production of green oil that would affect catalyst activity. Therefore, as control variables, the hydrogen input H and temperature T of the hydrogenation reactor must be controlled within certain ranges. In this study, based on actual industrial conditions, by controlling the H and T of the stage 1 reactor, the boundary conditions H ∈[ 40,80] and T ∈ [298.15, 308.15] were satisfied, in which it was strictly specified that stage 1 reactor’s outlet acetylene flow rate should be less than 15kmol h −1 and the hydrogen flow rate should be less than 2 kmol h −1 , so that stage 2 reactor can be operated with an ideal hydrogen/acetylene ratio, and Eq. (15) is optimized. The variations of some switching periods’ objective functions with evaluation times are shown in Figure 5. Among these, all stage 2 reactions were carried out for 120 days. When the number of days of the stage 1 reaction was greater than 400 days, the obtained optimal values became infeasible solutions. Figure 6 shows the variation of the objective function value with the evaluation times under the optimal switching periods D2, R1 = 320 and D2, R 2 = 120 . Here, the objective function value remained nearly unchanged in the later phase of the evaluation, so that the function 27



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converged to a global optimal solution. With these switching periods, after subtracting the catalyst regeneration cost from the ethylene production revenue of the two stages, the average daily revenue was 110,353.1 CNY. For the optimal switching periods ( D2, R1 = 320 , D2, R 2 = 120 ), the optimal control curves for the control variables H and T over time are shown in Figures 7 and 8, respectively. In the first case, the demand for hydrogen was relatively low for the first 200 days and gradually increased thereafter (Figures 5). As the reaction progresses, the catalyst activity gradually drops and the amount of hydrogen required to satisfy the constraint conditions increases. The control of T during the reaction process is closely related to the hydrogenation. The rates of the main and side reactions change with the change of temperature, affecting the selectivity of the hydrogenation. Figure 8 shows the optimal control set for T under the optimal switching period conditions, which shows a tendency to increase. Based on the operation optimization, an optimized objective function was achieved by adjusting the switching periods and controlling H and T, and the overall economic benefit was optimized. An hourly ethylene increment can be obtained through the dynamic model of acetylene hydrogenation. The hourly ethylene increments for the stage 1( ∆x R1C 2 H 4 ) and 2( xR 2 C 2 H 4 ) reactors are shown in Figures 9 and 10, respectively. In Figure 9, the ethylene production gradually decreases with the change in catalyst activity. The decrease in ethylene production is relatively small in the early phase then gradually becomes larger after 200 days. Figures 5 and 7 show that, although the hydrogen input is greatly increased, ethylene production still gradually declines, 28


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which is indicative of the decline in the stage 1 reactor’s catalyst activity. Thus, the cost for catalyst switching needs to be evaluated to ensure the optimal benefit to the entire system. Figure 10 shows that the catalyst of the stage 2 reactor has a relatively short life cycle and strong catalytic activity. The fact that the peak is gradually increasing shows that the catalytic activity of the stage 1 reactor declined gradually, with an increase of acetylene at the outlet of the stage 1 reactor. In Figures 7, 11, and 12, with the number of days fixed for the stage 1 reactor at D2, R1 = 320 and varied for the stage 2 reactor ( D 2, R 2

= 120 , D1, R 2 = 100

and

D 3, R 2 = 140 ),

the given H value for each

stage was obtained using the algorithm. As shown in the figures, with relatively high catalyst activities early on for the stage 1 and 2 reactors, adding only a small amount of hydrogen meets the system requirements. However, as the reactions progress, the catalyst switches for the stage 2 reactor result in a decreased amount of hydrogen required because of the change in catalyst activity. Different values of H are observed due to the different switching periods for the stage 2 reactor. After switching, the stage 2 reactor’s catalyst exhibits relatively good catalytic activity, since the catalyst for this stage is changed periodically within the stage 1 reactor’s switching period. Concomitantly, the hydrogen input also demonstrates a certain periodicity under the premise of meeting system requirements. Figure 13 shows the control value of H in each stage with switching periods of D8, R1 = 440, D2, R 2 = 140 , under which Eq. (15) would be optimal. It is evident that the hydrogen input is quite low under good catalytic conditions in the early phase then gradually increases as the reaction proceeds. 29



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Figure 5. Variation of objective function with time for different switching periods

Figure 6. Variation of objective function value with time for the optimal switching periods

30


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Figure 7. Hydrogen input over time with switching periods 320 d for stage 1 and 120 d for stage 2

Figure 8. Thermodynamic temperature over time with switching periods of 320 d for stage 1 and 120 d for stage 2

Figure 9. Daily ethylene increment of stage 1 reactor over time

31



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Figure 10. Daily ethylene increment of stage 2 reactor over time

Figure 11. Hydrogen input over time with switching periods of 320 d for stage 1 and 100 d for stage 2

32


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For different switching periods Di , R1 and D j , R 2 , the final objective function values, constraint violation degrees, and corresponding H and T control values are shown in Table 1. Here, the value of 0 for the constraint violation degrees in all cases suggests that the constraints are eventually satisfied, and only the objective functions need to be compared to determine the optimal switching periods under a given discrete phase k condition. Table 1. Optimization results for different switching periods for the stage 1 and stage 2 reactors 33



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Stage 1 switching period

Stage 2 switching period

obj

vio

300

100 120 140 100 120 140 100 120 140 100 120 140 100 120 140 100 120 140 100 120 140 100 120 140

103830.438 103535.158 99113.379 101449.59 110353.092 96462.5914 97831.5281 94604.3899 92718.7666 93118.63 88251.869 88047.908 87547.3311 84254.2032 82506.8491 80506.5183 79319.8002 76075.5011 74602.7543 73072.793 68704.4362 68792.3292 66677.2885 63263.0399

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

320

340

360

380

400

420

440

34


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Figure 14. Relationship between reactor life cycle and economic benefits

Figure 14 can be generated based on the results in Table 1, which shows the influence of both reactors’ switching periods on the overall economic benefit, and also enumerates all switching regeneration strategies. It is apparent that the maximum overall economic benefit was achieved when the stage 1 reactor was used for 320 days and the stage 2 reactor was used for 120 days. After the stage 2 reactor had been running for 120 days, it began to have difficulty in converting trace amounts of acetylene, and more hydrogen was needed to reduce the acetylene content to the system’s required level. Since the cost for catalyst regeneration is relatively high, the regeneration operational costs and the increasing overall selectivity must be considered to increase and

35



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maximize the overall revenue. In this way, safe production by the system can be ensured while the maximum economic benefit can be obtained. By comparing the objective function values of the switching periods, considerable economic benefit can be obtained by optimizing the switching strategy and the corresponding hydrogen input and temperature parameters. Different methods of the objective function value are compared to prove the validity of the method of dynamic optimization, as shown in Table 2. The boundaries of control variables are H ∈[ 40,80] and T ∈ [298.15, 308.15] . The first case (C-HT) is to increase the hydrogen input H and

temperature T with constant, i.e., the interval of H is 2.5, the interval of T is 0.625, thus the objective function can be derived as 19,196.00 CNY. The second case (C-T) is to fix the H at the maximum hydrogen input value (H = 80 kmol/h) and only increase the T constantly, the best objective function is 62,339.71 CNY. For the third case (Min T), the minimum thermodynamic temperature of T = 298.15 K (min T) and a maximum hydrogen input of H = 80 kmol/h are used, and the best objective function is 45,453.45 CNY. For the fourth case (IEDE-CRMO), the dynamic optimization (IEDE-CRMO) is used and compared, and the objective function is 110,353.1 CNY. Here, the constraint violations in all cases are satisfied, and only the objective function needs to be compared. From table 2, the dynamic optimization method shows evident advantages compared with others. For the optimal switching periods ( D2, R1 = 320，D2, R 2 = 120 ), the control curves for control variables H and T over time with different methods are shown in Figures 16 and 17, respectively. 36


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Table 2. Results of the different method under the same condition Stage 1

Stage 2

switching

switching

period

period

320

C-HT

C-T

min T

IEDE-CRMO

100

19196

60076.44

44088.75

101242.64

120

18197.84

60700.39

44266.42

110353.09

140

16236.54

62339.71

45453.46

96328.54

Figure15. Hydrogen input over time with switching periods 320 d for stage 1 and 100 d for stage 2 of the different methods

37



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Figure16. Thermodynamic temperature over time with switching periods of 320 d for stage 1 and 100 d for stage 2 of the different methods

Most of the existing acetylene hydrogenation model research is based on a static optimization method to solve the hydrogen input and the inlet temperature, which in the process of the whole production has a constant value. However, the acetylene hydrogenation model is a dynamic model according to the characteristics of the dynamic model given the optimal control values of the whole system using IEDE-RMO dynamic optimization to solve.

5. Conclusions CVP-based IEDE-CRMO was applied to optimize a two-stage tandem acetylene hydrogenation model. The method was used to determine the switching periods and corresponding hydrogen input and temperature parameters for maximizing the economic benefit of the system, while simultaneously satisfying the system constraints of the control variables. The acetylene conversion rate for each reactor could be rationally assigned through computational modeling. 38


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The hydrogen input must also be periodically changed throughout the reaction process given that the switching periods and catalyst activities of the stage 1 and 2 reactors are different. Dynamic optimization can represent this periodically changing process and the corresponding values of the control variables. As a result, more accurate optimization values can be obtained compared with previous methods. Based on different switching periods, the optimal switching period for the stage 2 reactor was found to be 120 days, whereas that for the stage 1 reactor was 320 days. These values were obtained by considering the regeneration cost and product price and by comparing the objective function values. Simultaneously, the hydrogen input and temperature parameters within the switching periods could be optimized and controlled to maximize performance and the objective function. The application of dynamic optimization is a desirable approach for solving complicated and challenging processes in chemical industries. AUTHOR INFORMATION Corresponding Authors *E-mail: [email protected] ACKNOWLEDGMENTS This research was supported by National Natural Science Foundation of China (21276078, 61333010), National Program on Key Basic Research Project of China (973 Program) (2012CB720500), and “Shu Guang” project of Shanghai Municipal Education Commission. Nomenclature k = the number of days of the discrete phase H =the hydrogen input of the stage 1 reactor 39



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T = the inlet temperature of the stage 1 reactor obj = Objective value vio = Violation degree

N=discretization dimension CVP = control vector parameterization NLP = non-linear programming

DE = Differential evolution D = The dimension of optimization problem NP = population size

Variables ai = Deactivation coefficient C p ,i = Specific heat capacity at constant pressure of gas i ( kJ kg −1 K −1 )

d R1 = Operation days of stage 1 reactor d R2 = Operation days of stage 2 reactor Di , R1 = Switching periods of stage 1 reactor D j , R 2 = Switching periods of stage 2 reactor E a ,i = Activation energy of gas i( kJ kmol −1 )

Fi = Molar flow rate of gas i(kmol h−1 )

H = Hydrogen feeding flow rate of stage 1 reactor(kmol h −1 ) H = Enthalpy change of reaction ( kJ kmol −1 ) 40


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k 0 ,i = Pre-exponential factor of reaction i( km ol kg − 1 hr − 1 kPa − 3 ) K H 2 = Adsorption constant of hydro gen ( kPa −1 ) K C 2 H 2 = Adsorption constant of acetylene ( kPa − 1 ) K C 2 H 4 = Adsorption constant of ethylene ( kPa − 1 ) K C 2 H 4 = Adsorption constant of eth ane ( kPa − 1 )

pi = Partial pressure of gas i(kPa)

ρ B = Catalyst density (kg m−3 ) R = Gas constant (kJ kmol −1 K −1 ) R 2 = Hydrogen/acetylene ratio of the stage 2 reactor ri 0 = Rate of reaction i(kmol kg −1 hr −1 ) r j , d = Rate of reactionjon catalyst ( kmol kg −1 K −1 )

S = Cross-sectional area of reactor(m 2 )

T = Thermodynamic temperature ( K ) xR1i = Concentration of gas i at stage 1 outlet (kmol h−1 ) xR 2 = Ethylene concentration at stage 2 outlet (kmol h −1 )

z = Reactor length ( m) References (1) Weirauch,W.; Naphtha. Gasoline demand is forecast[J]. Hydrocarbon Process, 1998, 77(9): 25-27 (2) Zehnder, S. What are Western Europe’s petrochemical feedstock options?[J]．Hydrocarbon Process, 1998, 77(2): 59-60 41



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TABLE OF CONTENTS Figure 1. Flowchart of acetylene hydrogenation process Figure 2. Constraint ranking technique Figure 4. Flow diagram of the IEDE-CRMO algorithm Figure 5. Variation of objective function with over time for different switching periods Figure 6. Objective function value variations time for the optimal switching periods Figure 7. Hydrogen input over time with switching periods 320 d for stage 1 and 120 d for stage 2 Figure 8. Thermodynamic temperature over time with switching periods of 320 d for stage 1 and 120 d for stage 2 Figure 9. Daily ethylene increment of stage1 reactor over time Figure 10. Daily ethylene increment of stage 2 reactor over time Figure 11. Hydrogen input over time with switching periods of 320 d for stage 1 and 100 d for stage 2 Figure 12. Hydrogen input over time with switching periods of 320 d for stage 1 and 140 d for stage 2 Figure 13. Hydrogen input over time with switching periods of 440 d for stage 1 and 120 d for stage 2 Figure 14. Relationship between reactor life cycle and economic benefits Figure 15. Hydrogen input over time with switching periods 320 d for stage 1 and 100 d for stage 2 of the different methods Figure 16. Thermodynamic temperature over time with switching periods of 320 d for stage 1 and 100 d for stage 2 of the different methods

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Dynamic Optimization of the Tandem Acetylene Hydrogenation Process

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