Estimation of Mass-Transfer Efficiency for Industrial Distillation Columns

Article pubs.acs.org/IECR

Estimation of Mass-Transfer Efficiency for Industrial Distillation Columns Na Luo, Feng Qian,* Zhen-Cheng Ye, Hui Cheng, and Wei-Min Zhong Key Laboratory of Advanced Control and Optimization for Chemical Processes (East China University of Science and Technology), Ministry of Education, Shanghai 200237, China ABSTRACT: Tray efficiencies are data fundamental to the evaluation of the performance of distillation columns, and many estimation methods have been proposed on the basis of the mass-transfer process. However, determining tray efficiencies in industrial practice remains a challenge. In the current paper, we provide a new method of estimating tray efficiencies of distillation columns by using industrial operating data, in which a multiobjective optimization problem is formulated and a univariate marginal distribution algorithm is used to solve this problem. Using the obtained tray efficiencies, a neural network model is then trained to predict the future efficiencies. The feasibility of the proposed method is verified by an industrial C8 aromatics distillation column. With the prediction errors limited to 5%, the approach shows more accurate generalization than traditional methods.

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INTRODUCTION Distillation is usually the first choice to separate mixtures with different boiling points, and many mathematic models have been developed for implementing the design and optimization of these units. However, complex and various operating conditions often make these models inconsistent with the actual process. Therefore, mass-transfer efficiencies are used to relate theoretical stages with actual plates, and tray efficiency estimation methods have been widely investigated. In general, two kinds of models, namely, empirical and theoretical models,1 are used for tray efficiency estimation. Empirical models are data-driven and have been developed to describe experimental efficiencies in terms of the physical properties, tray geometry, and operating conditions. They express the efficiency either as a functional relationship or as relationships of dimensionless groups. O’Connell2 proposed an empirical correlation for global efficiency, which is widely used in the industry because of its simplicity. The correlation is a function of the liquid viscosity and relative volatility evaluated at the average temperature between the top and bottom column temperatures and feed compositions. MacFarland et al.3 proposed an empirical correlation based on dimensionless groups for the estimation of the Murphree tray efficiency with an absolute mean deviation from experimental data. With the development of intelligent modeling technology, Olivier and Eldridge4 trained and validated a neural network model to predict the sieve tray efficiency. Unlike the empirical or semiempirical approaches, theoretical models are based on phenomenological relationships developed from the analysis of the two-phase characteristics, mass transfer, and the cross-flow hydraulic effects on the tray. The most widely used model is the American Institute of Chemical Engineers (AIChE) approach,5,6 which is based on experiments with mostly air and aqueous liquids on small bubble-cap trays. Given its unreliability in certain situations, Chan and Fair7 modified the AIChE approach and provided a satisfactory fit for a number of larger scale experimental studies. Prado and Fair8 © 2012 American Chemical Society

developed a model from fundamental relationships based on gas−liquid transfer in systems of air and water. This model was then modified and extended by Garcia and Fair9,10 into a mechanistic model for the prediction of the point tray efficiency in aqueous and nonaqueous systems. Recently, the computational fluid dynamics (CFD) method has been applied to simulate hydrodynamics, which is an essential influential factor of mass transfer in both interfacial and bulk diffusions. Through CFD, the effect of the velocity distribution on the concentration profile can be simulated well. Two-dimensional CFD models11 were first developed for the liquid-phase flow simulation on sieve trays. Furthermore, the effect of vapor was considered using three-dimensional CFD models.12 On the basis of the complex model, the turbulent mass-transfer diffusivity, three-dimensional velocity, concentration profiles, and efficiency of the mass-transfer equipment were simultaneously predicted. Despite considerable effort, the evaluation of the tray efficiency in industrial practice remains a challenge. Empirical models can only be used in a narrow range and are difficult to generalize, because they use simple relationships between the tray efficiencies and other factors. Furthermore, the predicted results from empirical models can be inaccurate, because of their limited experiments. As for the theoretical models, numerical simulations are very complex and time-consuming. When these models are applied in industrial processes, many necessary data, such as the tray bubbling area and the fraction of jetting active holes, cannot be measured online. On the other hand, a large amount of operating data are generated in the plant every minute. The data are discarded despite the amount of modeling information they contain. Thus, coping with the Received: Revised: Accepted: Published: 3023

February 26, 2011 December 31, 2011 January 11, 2012 January 11, 2012 dx.doi.org/10.1021/ie2008407 | Ind. Eng. Chem. Res. 2012, 51, 3023−3031

Industrial & Engineering Chemistry Research

Article

the tray efficiency is transformed into a solution of an MOP as follows:

contradiction provides a new idea for the estimation of the tray efficiency. The main goal of the current study is to establish an appropriate method of estimating the tray efficiencies of industrial distillation columns. Unlike previous empirical and theoretical models, a new method called MOP-NN (MOP = multiobjective optimization problem, NN = neural network) is developed on the basis of a large amount of industrial operating data. The tray efficiency is obtained by solving a multiobjective problem, and a new multiobjective estimation of the distribution algorithm is proposed in the current paper to prevent failure of general optimization algorithms. A neural network model is trained using the obtained tray efficiencies to predict the future efficiencies. The predicted tray efficiencies are compared with the efficiencies evaluated from plant data of industrial columns to test the feasibility and generalization of the proposed method.

m ⎧ ⎪ obj 1 = ∑ (Ti − T ′i )2 ⎪ i=1 ⎪ ⎨ n l ⎪ ⎪ obj 2 = ∑ ∑ (Mk , j − M′k , j )2 ⎪ k=1 j=1 ⎩

where m is the number of online temperature measurement points, l denotes the number of key components in the products, n is the number of product streams, Ti and T′i are the predicted and online measurement temperatures on the ith stage, respectively, and Mk,j and M′k,j are the prediction and measured values of the jth component fraction in the kth product stream, respectively. The first objective is used to make the temperature profile obtained via simulation agree with the online measurements of the industrial column. At each stage of the distillation column, the component fractions depend on both the temperature and tray efficiency when the pressure is fixed. The temperatures and component fractions are adjusted to be close to the real data so that the tray efficiencies are near the actual value. The second objective is used to reduce the error between the simulation and the measurement mass fraction of the key components in the bottom and overhead products. This makes the tray efficiencies in the upper and bottom parts of the column consistent with the actual separation conditions. To realize the aforementioned idea, the MOP-NN method is used to estimate the tray efficiency in industrial columns. Unlike empirical or theoretical models, the mechanistic model of the distillation column is first developed. The thermodynamic package and the equation of state are chosen according to the mixture properties in the column. Commercial software such as Aspen Plus or HYSYS or custom programming can be used in establishing such a model. Online operating data are acquired from a real plant. The obtained data contain industrial noise acquired from the industrial environment. Thus, the data should undergo steady-state identification and gross error detection. Afterward, the tray efficiencies for the industrial distillation column are estimated. At the beginning of the estimation, the tray efficiencies are randomly initialized and set for the column simulation. The temperature profile and the overhead and bottom compositions obtained via simulation clearly do not agree with the corresponding actual measurements of the industrial column. With the evolution of the optimization algorithm, the tray efficiencies can be obtained to make the simulation results consistent with the industrial data. This process runs under various operating conditions to obtain accurate values. With these tray efficiency data, a neural network model is trained to predict the future tray efficiency under a new operating condition. The flow diagram for the tray efficiency estimation is shown in Figure 1. The estimation of the tray efficiencies is transformed into an MOP in which both the objectives are in conflict with each other and no single solution can optimize all the objectives at the same time. Classical optimization methods often convert the MOP to a single-objective optimization problem by emphasizing one particular Pareto optimal solution at a time. By contrast, multiobjective evolutionary algorithms (MOEAs) work with a population of candidate solutions and thus can produce a set of Pareto optimal solutions to approximate the

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TRAY EFFICIENCY ESTIMATION BASED ON INDUSTRIAL OPERATING DATA The overall tray efficiency is defined as the number of ideal equilibrium trays divided by the actual number of trays as follows: Eo =

Nideal Nreal

(1)

In practice, the Murphree efficiency is usually used to relate the theoretical stages with the actual trays instead of the overall efficiency. For component i on stage j in the column, the Murphree efficiency is defined as Ei , j =

yi , j − yi , j + 1 Kxi , j − yi , j + 1

(3)

(2)

where Ei,j is the Murphree efficiency, K denotes the equilibrium constant, x and y are the mean vapor and liquid fractions, respectively, i is the component number, and j is the stage number. In the current work, the Murphree efficiency was calculated on the basis of the vapor composition of a particular stage under specific temperature and pressure conditions. Its value changes with the thermodynamic equilibrium between the vapor and liquid phases. For commercial distillation columns, the estimation of the overall efficiency is too coarse. Murphree efficiencies are widely used, and their values are often assumed the same for all components on the jth stage in ideal systems. On the other hand, in nonideal systems, the estimated Murphree efficiencies of the ith composition on the jth stage can be different. For simplification, the Murphree efficiencies on a number of continuous trays can be considered of the same value. Hence, the estimated tray efficiencies are defined as a vector of variables. In the current study, the tray efficiencies are chosen by checking the temperature profile and overhead and bottom compositions obtained via simulation against the corresponding measurements of the industrial columns. The same comparative data are selected as in a previous study13 because the experimental data of Murphree efficiencies are difficult to directly obtain without collection of samples from the stage outlets during the industrial process. Hence, the estimation of 3024

dx.doi.org/10.1021/ie2008407 | Ind. Eng. Chem. Res. 2012, 51, 3023−3031


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algorithm (EDA) is proposed to estimate the probability distributions of the parent individuals over the search space and generate offspring individuals from the current distribution through sampling. EDAs were originally used in discrete (binary) optimization problems. The EDAs can be categorized into three groups according to the complexity of variable dependency as follows: (i) univariate dependency, such as PBIL,16 CGA,17 and UMDA,18 (ii) bivariate dependencies, such as MIMIC,19 COMIT,20 and BMDA,21 and (iii) multivariate dependencies, such as ECGA,22 BOA,23 and FDA.24 In the continuous domain, Larrañage et al.25 extended the UMDA to UMDAC by constructing Gaussian networks as probabilistic graphic models. PBILC,26 MIMICC,25 and EGNAee25 are extensions of different EDAs in the continuous domain. In the current study, UMDAC was used as the basis of the new multiobjective estimation of distribution algorithm. A general outline of UMDAC is as follows: (1) A population of individuals is randomly initialized. (2) Individuals are selected into the selection set, and the elitism set is chosen. (3) The probability density is estimated using the selection set. (4) New individuals are sampled from the probability model as the new population. (5) The new population is partially replaced by the elitism set. (6) The process is stopped if the stopping criterion is reached; otherwise, step 2 is repeated. In UMDAC, the probability model of each dimension i is assumed as a Gaussian distribution as follows:

Figure 1. Estimation approach for the tray efficiencies based on operating data.

Pareto front in a single run. A multiobjective univariate marginal distribution algorithm is proposed by considering the interrelationship between the tray efficiencies and avoiding the failure of general optimization algorithms.

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MULTIOBJECTIVE UNIVARIATE MARGINAL DISTRIBUTION ALGORITHM (MUMDA) The MOP is usually defined as yi = fi (x),

2πσi 2

2 2 e(−1/2σi )(xi −μi)

(5)

where μi and σi2 refer to the mean and variance values, respectively. Equation 5 is denoted as P(xi) ≈ N(μi,σi2). Using the maximum likelihood method, the parameters μi and σi2 are estimated as

i = 1, ..., D

minimize y = f (x) = (f1 (x), ..., fD (x)) μ̂i =

subject to e(x) = (e1(x), ..., eJ (x)) ≥ 0 x∈S

1

P(xi) =

1 n

(4)

where D is the number of objective functions f i:Rp→R and x = [x1, ..., xp]T is the vector of decision variables with the feasible domain. The particular set of values x*1, ..., x*p should be determined to yield the optimum values of all objective functions from the set S that satisfy the J constraints ej(x) ≥ 0, j = 1, ..., J. Given that more than one best possible solution is available, the Pareto optimal solution set is introduced as nondominated solutions in MOP. A number of MOEAs have been suggested over the past decade to obtain the Pareto front; a detailed review for this can be found elsewhere.14 The most popular multiobjective optimization algorithm is the nondominated sorting genetic algorithm II (NSGA-II),15 which is an extension of the genetic algorithm (GA) in MOP. It uses crossover and mutation strategies similar to those in GA. These strategies fail when the problem is complex and interactive. Instead of applying genetic operators such as mutations and crossovers, an estimation of distribution

σ̂i 2 =

N

∑ xik k=1

1 N

(6)

N

∑ (xik − μ̂i)2 k=1

(7)

where i is the dimension of the variant, xik is the sample of variant xi, N is the number of samples, and μ̂i and σ̂i2 are the estimated values of μi and σi2, respectively. In general, the MOP gives rise to a set of optimal solutions (known as Pareto-optimal solutions) instead of a single optimal solution. To solve this kind of problem, UMDAc was extended into MOPs, called MUMDA in the current work. The framework of MUMDA is as follows: (1) A population of individuals is randomly initialized, and the fitnesses of the individuals are calculated. (2) Individuals Ds are selected according to the Pareto domination selection method. (3) Elitism individuals De are selected from the current population. 3025



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Table 1. Test Problems Used in the Current Study

which any two solutions share each other’s fitness. It is defined as

(4) The probability density model p̂(x) is estimated using eqs 5−7. (5) N individuals are sampled from p̂(x). (6) The fitnesses of the new population are calculated, and the e worst individuals are replaced using De. (7) The process is stopped if the stopping criterion is reached; otherwise, step 2 is repeated. The Pareto domination selection method and diversity maintenance are discussed in detail in the following subsections. Pareto Domination Selection Method. To identify the solutions of the Pareto front in the population, each solution should be compared with every other solution in the population to determine if it is dominated. The computational complexity is high. The Pareto domination tournament method is used to select the Pareto front. Two candidates are randomly selected from the current generation, and a comparison set of individuals are randomly picked from the elitism set. The two candidates are compared against each other, and the nondominated candidate is then compared against each individual in the comparison set. If the candidate is not dominated by the comparison set, it is selected and placed into the elitism set, and vice versa. Diversity Maintenance. Niche fitness sharing is used to select the fixed elitism set and preserve the diversity. The sharing function method involves a sharing parameter, which sets the extent of the sharing desired in a problem. This parameter is related to the distance used to calculate the proximity measure between two population members. The parameter represents the largest value of that distance, within

F(Pop(i)) =

1 ∑Pop(j) ∈ P s(d(Pop(i), Pop(j)))

(8)

where P is the elitism set and s(d(Pop(i),Pop(j))) is the sharing function of individuals Pop(i) and Pop(j) s(d(Pop(i), Pop(j))) ⎧ ⎪1 if (Pop(i) ⎪ ⎛ d(Pop(i), Pop(j)) ⎞α , Pop(j)) =⎨ −⎜ ⎟ ⎪ σshare < σshare ⎝ ⎠ ⎪ ⎩0 otherwise

(9)

where d(Pop(i),Pop(j)) is the distance between individuals Pop(i) and Pop(j). Equation 9 is calculated as the Euclidean distance. and σshare is the niche radius, which is fixed by the average distance of the individuals in a population. The fitness of individuals within the σshare distance is reduced because they are in the same niche. Thus, convergence occurs within a niche, but the convergence of the full population is avoided. As one niche fills up, its niche count increases to the point that its shared fitness is lower than that of the other niches.27 Five multiobjective benchmark functions are adopted to test the performance of the proposed method in the MOP and compare it with those of the related studies. These functions are Schaffer, FON, KUR, ZDT4, and ZDT6 (Table 1) and include characteristics that are suitable for the 3026



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Figure 2. Obtained nondominated solutions to the Schaffer, FON, KUR, ZDT4, and ZDT6 problems.

determination of the effectiveness of MOP approaches in maintaining the population diversity as well as the convergence to the Pareto front. All problems have two objective functions, and none of the problems have any constraint. The performances of MUMDA are compared with those of NSGA-II15 and multiobjective particle swarm optimization (MPSO).28

The results of the five benchmark functions are shown in Figure 2. For NSGA-II, the selection parameter is 0.3, the crossover operator is 5, and the mutation operator is 5. For MPSO, the inertial weight is 0.729, the first acceleration coefficient is 1.495, and the second acceleration coefficient is 1.495. For MUMDA, the truncation ratio is set at 0.3 and the elitism population is 5. In all algorithms, the population size is 3027



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steady-state period as well as the difference Δ between the averages of two consecutive steady-state periods as the criteria in a recursive fashion. If σ̂ is larger than the minimum allowed standard deviation, or Δ is larger than the minimum allowed difference between the averages of two consecutive steady-state periods, the current data are in the unsteady state and the identification turns to the next data. After the steady-state was ascertained, the associated gross and random errors were further removed using a gross error

Figure 3. Moving windows of n data points around the kth time.

set to 100 and each run continues until a maximum of 10 000 function evaluations is reached. The results of NSGA-II, MPSO, and MUMDA are illustrated in Figure 2 (subfigures, from left to right). For the Schaffer problem, all the algorithms can converge to the Pareto front and the results of NSGA-II are the best distributed. For the FON problem, all algorithms can obtain solutions close to the Pareto front except for MPSO. MUMDA shows good performance in the KUR, ZDT4, and ZDT6 test functions, compared with NSGA-II and MPSO. From the test performances, MUMDA can find solutions that closely resemble the Pareto front solutions and the uniformity of the solutions is satisfactory.

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CASE STUDY The tray efficiencies can be estimated using the MOP-NN method using industrial operating data. However, case studies from the literature are not suitable to test the proposed method. Meanwhile, actual operating data are usually numerous and confidential. Therefore, an opportunity to work on a raffinate column in an aromatics plant in China was presented; thus, this column was used in a case study for the proposed method. In the raffinate column, a mixture of o-xylene (OX), m-xylene (MX), and the absorbent (p-diethylbenzene, PDEB) is separated. The tray efficiencies cannot be accurately identified; hence, the mechanistic model of the raffinate column is not consistent with the actual conditions. In reality, the reconstruction of the column increases the difficulty of the simulation. The original sieve trays in the rectification section were replaced by high-performance trays, whereas the original trays of the stripping section were kept. Thus, the tray efficiencies in both sections vary considerably. With a large amount of operating data, MOP-NN is suitable for the estimation of the tray efficiencies of the column. According to Figure 1, the industrial operation data should be first acquired and disposed. Industrial Operating Data Processing. The performance analysis of the column was first conducted to obtain industrial data. The compositions of the feed and product streams were analyzed; at the same time the operating data were acquired from the automation system, including the flow rates, mass fraction of the components, temperature, and pressure. To ensure that the column runs in a steady state,29 the state of the column was observed for a few days. In addition, the operating data were processed in turn using steady-state identification and gross error detection. Steady-state identification30 can be performed using various techniques. We used the simplest and most common steadystate detector to analyze the data over a predefined moving window, as shown in Figure 3. The moving window replaces each data point within the 250 min time interval, and the moving window average is equivalent to a low-pass filter. The steady-state detector uses the standard deviation σ̂ within a

Figure 4. Pareto solutions of tray efficiencies under the design capacity.

detection method. For industrial operating data, a simple comparison strategy is always useful to eliminate the gross error. The selected data were sorted, and whether the minimum and maximum values were within the 3 standard deviation limit was tested. If the limit was violated, the gross error was identified, and the magnitude of the gross error was taken as the difference between the measured and reference values. Data with gross errors were discarded, and the average values for the steady state were calculated. Through the identification and detection method, the effective industrial data in the steady state were obtained to estimate the tray efficiencies. Table 2. Comparison between the Objectives of the Predefined and Estimated Values

predefined values estimated values 1 estimated values 2

efficiency in rectification section

efficiency in stripping section

objective 1

objective 2

0.9397

0.2009

853.05

1.0 × 10−4

0.9619

0.2380

562.54

6.2 × 10−5

0.9644

0.2275

85.26

6.4 × 10−5

Tray Efficiency Estimation. The distillation column for the case study contains 64 trays, with the feed flow varying from 120 to 140 m3/h. The feed flow goes into the column at the 32nd stage (from the top) and includes the mass fraction of 55% PDEB, 23% MX, 11% OX, and 11% other components at 179 °C and 0.28 MPa. In practice, MX and OX accumulate on the fifth stage, which is located at the outlet of the distillation product. The pressure profile in the column is assumed to be linearly distributed. The overhead stream has a small flow rate of only 1 kg/h, which can be ignored. 3028



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and Bot′PDEB are the prediction and analytical data of the PDEB mass fraction in the bottom product. The mechanistic model of the raffinate column was first developed using Aspen Plus. Meanwhile, the multiobjective algorithms were programmed in Matlab. A software interface

Figure 5. Relationship between the capacities and tray efficiencies.

As previously mentioned, the trays in the stripping section have lower efficiencies than those in the rectification section equipped with high-performance trays. Hence, the column is divided into two sections. In the first section (from the 1st to the 32nd stage), the range of the efficiency is set, [0.85, 1], and is obtained by testing the model under the typical operating conditions. In the other section, the range is [0.2, 0.4]. In the industrial column, eight temperature measurement points, namely, the 1st, 5th, 10th, 18th, 30th, 40th, 60th, and 64th stages (from the top to the bottom), were selected. Furthermore, the mass fractions of MX and OX in the side stream and PDEB in the bottom product were analyzed every 8 h. The multiobjective problem for estimating the tray efficiencies is defined as ⎧ obj 1 = (T − T ′ )2 + (T − T ′ )2 1 1 5 5 ⎪ + (T10 − T ′10 )2 + (T18 − T ′18 )2 ⎪ ⎪ (T30 − T ′30 )2 + (T40 − T ′40 )2 ⎪ + (T60 − T ′60 )2 + (T64 − T ′64 )2 ⎨ ⎪ ⎪ obj 2 = (SidMX − Sid′MX )2 ⎪ + (Sid′OX − Sid′OX )2 ⎪ + (BotPDEB − Bot′PDEB )2 ⎩

Figure 6. Parity plots of the observed and predicted tray efficiencies.

connected the mechanistic model to Matlab to realize the optimization, which used the ActiveX server as a handle in the Matlab environment. After all these steps were completed, different methods, including NSGA-II, MPSO, and MUMDA, were used to estimate the tray efficiencies using operating data. The population size was set to 100, with 200 evolved generations, and the other parameters were set as previously done. All these algorithms are heuristic methods; thus, we tested every algorithm 30 times. The converged Pareto solutions under the design capacity condition are shown in Figure 4. As shown in Figure 4, the different Pareto solutions were obtained using these three algorithms. The Pareto solutions of MUMDA are better than those of NSGA-II and MPSO because MUMDA adopts the same strategies used by NSGA-II and MPSO to solve the MOP; meanwhile, the evolutionary mechanism allows MUMDA to diverge from the local optimum. From these solutions, one point on the Pareto front was chosen as the final solution considering the balance between the two objectives. To date, the estimation of the tray efficiencies under one working condition was completed. The objectives using one group of predefined values were compared with those using the estimated values for validation (Table 2). The results show that the new values simultaneously reduce both objectives after the optimization of the predefined tray efficiencies. The estimation of tray efficiencies was repeated using data obtained under different operating conditions. Other operating conditions were investigated to enrich the database and obtain

(10)

where Ti and T′i are the predicted and online measurement temperatures on the ith stage, respectively, SidMX, Sid′MX, SidOX, and Sid′OX are the prediction and analytical data of the MX and OX mass fractions in the side stream, and BotPDEB 3029



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Figure 7. Comparison of the temperature profiles of the simulation using the proposed method, the AIChE model, and online measurements.

Table 3. Component Fractions in the Side and Bottom Streams 85% design capacity

115% design capacity

mass fraction

simulation using MOP-NN, %

analytical data, %

error, %

simulation using MOP-NN, %

analytical data, %

error, %

MX (side) OX (side) PDEB (bottom)

53.76 24.20 98.26

55.09 23.13 99.54

2.41 4.63 1.29

54.36 25.78 98.68

54.37 25.64 99.30

0.02 0.55 0.62

additional information on the column. The tray efficiencies were found to widely vary in a nonlinear way under different capacities. The estimated tray efficiencies for the distillation and stripping sections related to the unit capacity are shown in Figure 5. In Figure 5, the tray efficiencies decrease with increasing product capacities, because the internal liquid distribution and liquid-to-vapor ratio in each section of the unit change with the changing feed flow. With an increase in capacity, the tray efficiencies in the rectification section are reduced, similar to those in the stripping section. In the actual process, the required heat at the bottom of the column should be increased as the fractionation efficiency is reduced and the reflux ratio should also be increased to obtain qualified productions. Therefore, the appropriate mechanistic model is the basis for the optimization of the operating conditions of the column to guarantee that the commercial operating conditions are beneficial to the plant. Prediction of Tray Efficiencies. The tray efficiencies under different operating conditions were obtained with MUMDA, and a neural network was used to predict the tray efficiencies. Unlike the neural network model developed by Olivier and Eldridge,4 the input variables needed in the current study can easily be measured in the industrial process, including the flow rate, the mass fractions of MX and OX in the bottom stream, the pressure on the top and bottom of the column, the liquid rate of the flux, and the reboiler heat. The outputs are the tray efficiencies of the rectification and stripping sections. All input variables were normalized to a similar magnitude, which accelerated the training and reduced the chances of getting stuck in the local optima. Two feed-forward networks were trained with only one hidden layer, a hyperbolic tangent sigmoid transfer function as the activation function, and a linear transfer function as the output function. The number of epochs is 200, and the number of hidden nodes is 9. To illustrate the generalized ability of the neural network, the data were divided into training and test sets, with the neural network trained on 90 points of the data and tested on

45 points. Parity plots of the predicted and observed tray efficiencies are shown in Figure 6. Figure 6 shows that the predicted tray efficiencies in the stripping and rectification sections are close to the observed values. The maximum absolute error is less than 5%, indicating that the predicted value using the current model is accurate for the investigated case. Discussion of the Simulation Results. Two cases under 85% and 115% design capacities were chosen to validate the proposed method. We simulated these two cases using the predicted tray efficiencies from the neural networks and obtained the temperature profiles. Figure 7 shows the simulated temperature profiles compared with the online measurements. The simulation using the AIChE model is also included in the figure. In Figure 7, the temperature profiles under 85% and 115% design capacities are shown in the left and right images, respectively. The points on the simulated temperature profile using MOP-NN are close to the online measurements on the 1st, 5th, 10th, 18th, 30th, 40th, 60th, and 64th stages, whereas the deviations between the simulated temperature profiles using the AIChE model and the actual measurements are relatively large. This finding indicates that the efficiencies calculated by the AIChE model are not consistent with those of the actual process conditions; whereas the MOP-NN method can give a good simulation of industrial data. The simulated component fractions were compared with the analytical data to validate the performance of MOP-NN; the results are shown in Table 3. The errors between the simulation and analytical data are within 5%. Note that the two cases chosen in the current study are typical in industrial productions and thus are suitable for the proposed method.

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CONCLUSIONS In the current study, an MOP-NN method, which could be used for ideal or nonideal systems, was proposed to estimate the tray efficiencies of industrial distillation columns on the basis of operating data. The Murphree efficiencies were deter3030



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mined by optimizing a multiobjective problem. A multiobjective univariate marginal distribution algorithm was introduced to solve this problem, and its performance was outstanding compared with those of NSGA-II and MPSO. The algorithm was used in the estimation of the tray efficiencies under different operating conditions. A neural network model was trained on the basis of the obtained efficiency data and was used to predict the efficiencies under other operating conditions. An industrial C8 aromatics raffinate distillation column was used in a case study to validate this approach. The predicted efficiencies are in good agreement with those of industrial practice and better than those obtained from other conventional methods. Furthermore, the proposed method avoids the drawbacks of conventional methods, which are usually dependent on the experiments. However, MOP-NN is time-consuming especially when numerous operating data need to be disposed. To improve the efficiencies of MOP-NN, some clustering methods should be used in the disposal of operating conditions to reduce the number of redundant samples and increasing attention should be paid to the selection of the objectives.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Tel.: +86-21-64252060. Fax: +86-21-64252060.

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ACKNOWLEDGMENTS We are grateful to the reviewers for their detailed comments and suggestions. Thanks are also given to Dr. Wangli He and Jinlong Li from the laboratory for their professional editing. This work is funded by the Major State Basic Research Development Program of China (Grant 2009CB320603), 111 Project (Grant B08021), Shanghai Leading Academic Discipline Project (Grant B504), and Major State Basic Research Development Program of Shanghai (Grant 10JC1403400).

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REFERENCES

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Estimation of Mass-Transfer Efficiency for Industrial Distillation Columns

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