Yiqing Luo*,Liang Kong,Xigang Yuan

Chemical Engineering Research Center,School of Chemical Engineering and Technology,Tianjin University,Tianjin 300072,China

Keywords:Energy System engineering Low-temperature distillation system Process synthesis Distillation sequence Shaft work target

A B S T R A C T In this paper,by combining a stochastic optimization method with a refrigeration shaft work targeting method,an approach for the synthesis of a heat integrated complex distillation system in a low-temperature process is presented.The synthesis problem is formulated as a mixed-integer nonlinear programming(MINLP)problem,which is solved by simulated annealing algorithm under a random procedure to explore the optimal operating parameters and the distillation sequence structure.The shaft work targeting method is used to evaluate the minimum energy cost of the corresponding separation system during the optimization without any need for a detailed design for the heat exchanger network(HEN)and the refrigeration system(RS).The method presented in the paper can dramatically reduce the scale and complexity of the problem.A case study of ethylene cold-end separation isused to illustrate the application of the approach.Compared with the original industrial scheme,the result is encouraging.

1.Introduction

Many chemical separation processes,such as gas separation and ethylene production that operate completely or partially in low temperature circumstance consume significant quantities of energy.The design and operation of low-temperature distillation processes generally include three sections:the core process(the separation sequence),the heat exchanger network(HEN)and the refrigeration system(RS).The traditional approach in practice for the design of low-temperature processes begins with the design of the separation process;after that,the HEN is designed to recover heat,and finally,the refrigeration system is designed to support the former systems.Because the three parts of the system are interrelated,a separation sequence optimized with the consideration of the heat recovery in the HEN may not necessarily correspond to the minimal energy cost of the global system without considering the RS simultaneously because the energy cost for a low-temperature separation system depends on the shaft work of the compressors.As a result,it is difficult for the traditional method to obtain an optimal design of the entire system with three decoupled parts.Thus,a simultaneous design method is necessary to obtain the optimal total cost for the entire system.

Most of the existing research studies regarding distillation systems in low-temperature processes focused on the optimization of the separation sequences and HEN simultaneously,while the refrigeration system was designed separately[1–4].Because the refrigeration system accounts for a large proportion of the total energy consumption for a low temperature process,the minimum total energy consumption cannot be guaranteed if the refrigeration system is not simultaneously considered in the optimization problem.Wang and Smith[5]attempted to build a mathematic model covering the separation sequence,integration and refrigeration simultaneously,and Tahouni et al.[6]improved the model.How ever,their model is complicated because of the consideration of the design of both HEN and RS,which requires a considerable effort to solve.

Linnhoff and Dhole[7]proposed a shaft work targeting method by extending the pinch analysis to yield the shaft work targets directly from basic process data.This approach simplifies the identification of the most energy-efficient system for a low-temperature process by not requiring know ledge of the HEN and RS structures.Therefore,in the present paper,by combining stochastic optimization with the shaft work targeting method,an approach for the synthesis of a heat integrated complex distillation system in a low-temperature process is presented.This approach involves the use of an improved simulated annealing algorithm[8]to explore the optimal operating parameters and the distillation sequence structure by a random procedure,a shaft work targeting procedure to evaluate the energy consumption of the separation processes that by-passes the detailed design of HEN and RS,and the shortcut method based on Fenske–Underwood–Gillil and to evaluate the capital cost of the distillation columns.The method presented in this paper can dramatically reduce the problem of the scale and complexity caused by searching a large number of optimal integer variables corresponding to the HEN and RS structures.The optimal distillation separation system that has the minimum total annual cost can be obtained with the consideration of the separation sequence,HEN and RS simultaneously.

2.Model for the Separation Processes

2.1.Problem statement and assumptions

The problem in this paper can be described as follow s.

Given an N-component mixture and the available utilities, fi nd the distillation sequence that can separate the mixture into N products with a specified separation request and the minimum total annual cost,including the energy cost and the equipment depreciation cost.The separation structure can consist of simple columns,complex columns with side rectifying or side stripping,pre-fractionators and thermally coupled columns.

To simplify the problem,the following assumptions are adopted:

(1)Constant molar flow inside the distillation columns;

(2)Constant relative volatility for the components in the mixture;

(3)No azeotrope exists in the mixture;

(4)All the streams are saturated;

(5)Only one middle(distributing)component exists in each non-sharp separator;

(6)N product streams exist.

Assumptions(1)–(4)are widely used in the synthesis of a distillation sequence to simplify the problem so that the shortcut method for a distillation column design can be used.For non-sharp separation,the complexity of the problem will increase with an increase of the number of distributing components,and thus assumption(5)ensures that the problem is not too complicated.

2.2.Problem formulation

For a certain distillation system,the total annual cost can be the sum of the utility cost required by the separation and the capital costs of all the columns.The costs depend on the variables of the separation sequence(represented by as et of integer variables),the thermal coupling structure(a set of binary variables),the pressure of each column,the reflux ratio of each column and the recovery of the key components in non-sharp separation columns.Therefore,the synthesis problem can be formulated as the following optimization problem:

where the integer number series{si}and{?i}represent the separation sequence structure and the thermal coupling scheme,respectively.SP and Φ are the sets of all possible separation sequences of{si}and thermal coupling structures{?i},respectively.P is a set of the feasible ranges of operating pressure p.The bound ranges of the recovery of light and heavy key components in non-sharp separation column ξLKand ξHK,respectively,are assumed based on operation specifications,and the ratio of the actual reflux ratio to minimum reflux ratio r=R/Rmin,is fixed to 1.2 which is assumed as a relatively better value and will not be optimized in the model for simplification.

An improved simulated annealing algorithm presented by An and Yuan[9]is used to solve the problem.The algorithm combines simulated annealing with the simplex method,in which the simplex method is used to search for continuous variables,and the discrete variables are generated and evolve via the random method.Because of its characteristic of stochastic nature,the algorithm does not require an explicit definition of the search space through a superstructure,which includes all the potential solutions and the necessary constraint equations,and the convexity that is necessary for a traditional mathematical MINLP approach.Furthermore,because the detailed design of HEN and RS can be by-passed via the shaft work target method,there is no need for any other variables to describe HEN and RS.Once the above variables are fixed on a set of feasible values,the temperature of each stream and the heat duty of each condenser or reboiler corresponding to a certain separation sequence will be determined by the shortcut method.The capital cost is estimated using methods recommended by H.Silla[10].Because the exergy of cold utilities in a low-temperature process comes from the shaftwork of refrigeration compressors,the energy cost of the refrigerant utilities is calculated by electric cost of the refrigeration compressors.

2.3.Representation of the separation sequence and the thermal couple structure

Based on An and Yuan[11],either sharp separation or non-sharp separation of the distillation tasks in a separation sequence can be represented by a set of integer number series.As is shown in Fig.1,C1–CNare N components arranged in volatility descending order.An arrow with an odd number that points between two adjacent components indicates a sharp separation task,while an arrow with an even number that points at a component indicates a non-sharp separation task.For example,the arrow with number 2 indicates the non-sharp separation in a pre-fractionator in which the light key component is C1,the heavy key component is C3,and the distributing component is C2.In this w ay,the integer number series{si}(i=1,2,…,T)could be a T-dimensional ordered array,which consists of the ordinals of the separation tasks.

Fig.1.Codes for splitting tasks.

As has been assumed before,the feed of each column will either be saturated liquid or be saturated vapor.Thus,a T-dimensional binary array{?i}(i=1,2,…,T,?i={0,1},?1=0)can be used to describe the feed condition of each column,namely,whether the interconnections between the separation tasks are a thermal couple or not,with a value of 0 showing the saturated liquid,and a value of 1 indicating the thermal couple stream.

2.4.Decomposition of the problem

To apply the shortcut method to the distillation column design,acomplex distillation column flowsheet is converted into its thermodynamic equivalent simple column flow sheet.For example,in Fig.2,the flowsheet(b)is the thermodynamic equivalent simple column flow sheet of the complex distillation column configuration(c).Each simple column has one stripping section and one rectifying section and finishes a single separation task(M).The number of column sections(S)and the separation tasks(M)in the thermodynamic equivalent simple column flow sheet meet the following inequalities according to An and Yuan[11]:

Therefore,when a separation sequence contains only sharp separation to separate N-components' mixture,(N?1)sharp separation tasks with 2(N?1)column sections are required.The number of separation tasks will increase when non-sharp separation is introduced.Each additional non-sharp separation increases a separation task,and thus,the maximum number of separation tasks(M)for N components is the sum of(N?1)sharp separation tasks and(N?2)non-sharp separation tasks,namely(2N?3)separation tasks.

Fig.2.Merging the product streams.

For example in Fig.2(a),if the mixture containing A,B and C is separated into three products by sharp separation in a directive separation sequence,2 separation tasks with 4 column sections are required,while 3 separation tasks with 6 column sections are needed when one non-sharp separation exists in the first column in Fig.2(b).Therefore,according to An and Yuan(2006),a decomposition strategy can be used to divide a large multi-component separation problem into a series of small sub-problems according to the number of non-sharp separation tasks.Because the number of non-sharp separation tasks ranges from 0 to N?2,the original problem will be divided to N?1 sub-problems.Each sub-problem only contains simple distillation columns with different numbers of column sections.

2.5.Strategy of merging columns

As shown in Fig.2,the flow sheet(b)is the thermodynamic equivalent simple column flow sheet when the shortcut method is used to design configuration(c).The first column finishes a non-sharp separation task,in which the middle component B is distributed in both of the distillate AB and bottoms product BC of the column,and then is separated by two downstream columns in flow sheet(b).To obtain the equivalent result of flow sheet(c),larger values of pressure and vapor flow rate in the two downstream columns will be selected as the operating parameters for the evaluation of both of the two columns.

2.6.Heat integration and shaft work calculation

In low-temperature distillation processes,the driving force primarily comes from refrigerants,more specifically,and it belongs to the shaftwork of refrigeration compressors.First,the compressor shaftwork is transformed to the exergy of the refrigerants.Next,the exergy will be transferred to the separation sequence via the HEN,and part of the exergy will be lost in this procedure.Therefore,if a separation sequence is determined,the minimum exergy requirement for the separation process is also determined,and the minimum compressor shaftwork depends on the exergy loss in the HEN,which could be determined from the exergy grand composite curve(EGCC)according to the research of Umeda[12]and Linnhoff[7].

Fig.3(a)shows a process grand composite curve(GCC)below ambient,which is widely used in the design of a HEN.The curve reflects the net heat flow against the temperature of a system.In Fig.3(b),the process exergy grand composite curve(EGCC)is obtained with the Carnot Factor(ηc=1? T0/T)as the vertical axis instead of the temperature T in Fig.3(a).The are a surrounded by utility lines and EGCC,i.e.,the area in the shade represents the exergy loss(EL)HENin the HEN.Consequently,any change in the HEN and the RS is easily assessed in terms of the consequent reduction of the exergy loss and therefore of the exergy supplied by the refrigeration system.The reduction in the overall shaftwork is equal to that of the shaded area/ηex,where ηexis the exergy efficiency of the refrigeration system,which is approximately constant.Hence,only setting the utilities for the separation tasks is required,and then the shaftwork target can be determined without the need for the HEN and RS designs.

3.Algorithm and Optimization Procedure

Fig.3.Exergy loss during heat exchange.

The low-temperature distillation process synthesis problem is a MINLP problem.An improved simulated annealing(SSA)algorithm presented by An[9]is used to solve the problem.The algorithm combines the simulated annealing with the simplex method,in which the simplex method is used to search for continuous variables and the discrete variables are generated via the random method.Fig.4 show s the optimization flow chart of the algorithm.In the algorithm,x represents continuous variables and y represents discrete variables.New continuous variables will be generated in each circle via the simplex method.If the new solution is accepted,a new discrete variable will be generated and the annealing temperature will decrease.Otherwise,random numbers are used to determine whether and how to change the discrete variables,and then the simplex search is performed again.The algorithm will end when the current annealing temperature is lower than the terminal temperature,and then the optimal solution is output.

4.Case Study

An actual ethylene plant cold end separation process is taken as an example.The feed composition and separation specifications are listed in Table 1.The utility price is listed in Table 2,of which,the refrigerant cost resulting from the shaftwork is indicated in the electricity cost of the refrigeration compressor.The objective of the example is to determine the separation sequence with a minimum total annual cost.

4.1.Problem decomposition

There are five components in this case.The separation requires at least four sharp separation tasks and at most three non-sharp separation tasks.Thus,the original problem can be divided into four subproblems depending on the number of non-sharp separation tasks,as listed in Table 3.Every sub-problem will be solved independently and the best solution of the four sub-problems is the optimum solution of the example.

4.2.Solutions for the sub-problems

Table 4 lists the optimal solution for each sub-problem.From the result,the direct separation sequence is determined to be the most economical for all of the sub-problems regardless of whether they contain non-sharp separations.When non-sharp separation tasks are introduced,the total annual cost decreases compared with the cost of the completely sharp separation scheme.How ever,the cost increases when more non-sharp separators are added.This cost increases may be caused by the cost increasing when merging simple column sections that have the same product into a complex column.In conclusion,the separation scheme using one non-sharp separation task with component C distributing is the best scheme for the case.

Figs.5 to 8 show the respective flow charts of each sub-problem.The solid arrows in the figures represent material flow and the dotted lines inside columns are used to indicate column sections.Note that the solutions obtained in the paper should be the separation sequences and thermal couple structure that result in the minimum total annual cost with the optimal HEN and RS energy targets and that the HEN and RS designs are not included in the model.How ever,the heat integration structure is easier to match manually considering the limited number of streams in this example.Thus,the heat integration is also indicated by two heat exchangers connected with a dotted line in these figures.

Fig.4.Flowchart of the SSA algorithm.

Table 1Feed compositions and product speci fi cations

Table 2Specifi cation of available utilities

Table 3Four sub-problems of the problem

Table 4Optimal schemes for sub-problems

Fig.6.Optimal design for sub-problem 2.

Fig.7.Optimal design for sub-problem 3.

Fig.5.Optimal design for sub-problem 1.

Fig.8.Optimal design for sub-problem 4.

Fig.5 show s the optimal design obtained by solving sub-problem 1,w hich consists of only sharp separation tasks.The scheme is similar to simple column sequential separation,except that the bottom flow of columns 1 and 3 is thermal coupled streams without reboilers,and there is only one pair of heat integration,in which the condenser of column 4 rejects heat to column 2.

Fig.6 show s one of the flowsheets for sub-problem 2 after thermodynamic equivalent simple column mergence.In the flowsheet,there is only one non-sharp separation task.Columns 2 and 4 only have a rectifying section each.Note that there are different thermodynamic equivalent flow sheets for the solution,but the number of column sections and energy cost will remain the same.Flow sheets of subproblems 3 and 4 are similar to that of sub-problem 2 with the only difference being in the non-sharp separation tasks,column sections and thermal couple conditions.

“Is it possible?” said the lord-in-waiting, “I never imagined it would be a little, plain, simple thing like that. She has certainly changed color at seeing so many grand people around her.”

4.3.Comparison with the original industrial scheme

Fig.9.Original separation sequence.

Fig.9 shows the separation scheme of an actual industry plant consisting of five columns.Among these columns,the first column is a non-sharp separation column with two middle distributing components,B and C,while the other columns are all sharp separation columns.The scheme is simulated and its total annual cost is also calculated using the method of this paper to enable a fair comparison with the optimal result obtained in this paper.As shown in Table 5,the optimal complex column scheme obtained using the method described in this paper can save17.9%of the total annual cost compared with the industrial scheme,which is an inspiring result.

Table 5Comparison between the actual scheme and the optimal scheme

5.Conclusions

In this study,a systematic approach for synthesizing a heat integrated complex distillation system of a low-temperature process using refrigeration shaftwork targeting and the stochastic optimization approach is presented.In the proposed method,the optimal operating parameters and the distillation sequence structure of the method are obtained through an improved simulated annealing algorithm under a random procedure,a shaft work targeting procedure is used to calculate the energy cost of the separation processes that by-passes the detailed design of HEN and RS,and the shortcut method based on Fenske–Underwood–Gilliland is used to evaluate the capital cost of the distillation columns.The method presented in the paper can dramatically reduce the scale and complexity of the problem regarding the HEN and RS structures,and the optimal total annual cost can be obtained with the consideration of the separation sequence,HEN and RS simultaneously.The cold end separation process in an industrial ethylene plant is used as an example of the application of the method.The result achieved using the proposed method provides a significant improvement in the total annual cost over the practical plant.

The method does not involve the design of the HEN and RS.Only the optimal separation sequence and the thermal couple structure that result in the minimum total annual cost with the optimal HEN and RS energy targets are obtained,and an additional design procedure is required to obtain the HEN and RS.How ever,the significance of the proposed method is that a fairly good separation sequence can be obtained easily at the initial design procedure,and the solution can be used as a criterion to value current schemes and show the energy savings potential.

Currently,the model only considers distillation separation,but there are some other widely used separation methods that are low-temperature processes.In addition,the assumption of only one middle component limits some potential schemes of non-sharp separations.Future work is warranted to improve the model by considering more separation methods and reducing the restrictions on the non-sharp separations.

Nomenclature

M number of separation tasks

N number of feed components

p pressure,kPa

R actual reflux ratio

Rminminimum re fl ux ratio

r R/Rmin

S number of column sections

SP the sets of all possible separation sequences of{si}

siseparation task in column i

T current annealing temperature,K

Tendterminal annealing temperature,K

T0initial annealing temperature,K

x continuous variables

y discrete variables

Φ the sets of all thermal coupling structure{?i}

?ifeed condition of column i

ξ recovery of the key component in non-sharp separation

Subscripts

HK heavy key component

LK light key component