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## Brazilian Journal of Chemical Engineering

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*Print version* ISSN 0104-6632*On-line version* ISSN 1678-4383

### Braz. J. Chem. Eng. vol.17 n.3 São Paulo Sept. 2000

#### https://doi.org/10.1590/S0104-66322000000300006

**STOCHASTIC SIMULATION OF SUPERCRITICAL FLUID EXTRACTION PROCESSES**

**F. T. Mizutani, A. L. H. Costa and F. L. P. Pessoa **Department of Chemical Engineering, Federal University of Rio de Janeiro (UFRJ)

Escola de Química, Bloco E, Centro de Tecnologia, Cidade Universitária,

CEP 21949-900, Phone: 21-590-3192Rio de Janeiro - RJ, Brazil

E-mail: pessoa@h2o.eq.ufrj.br

*(Received: February 10, 2000 ; Accepted: April 24, 2000)*

Abstract -Process simulation involves the evaluation of output variables by the specification of input variables and process parameters. However, in a real process, input data and parameters cannot be known without uncertainty. This fact may limit the utilization of simulation results to predict plant behavior. In order to achieve a more realistic analysis, the procedure of stochastic simulation can be conducted. This technique is based on a large set of simulation runs where input variables and parameters are randomly selected according to adequate probability density functions. The objective of this work is to illustrate the application of a stochastic simulation procedure to the process of fractionation of orange essential oil, using supercritical carbon dioxide in a multistage extraction column. Analysis of the proposed example demonstrates the importance of the stochastic simulation to develop more reliable designs and operating conditions for a supercritical fluid extraction process.

Keywords:stochastic simulation; supercritical fluid; extraction.

**INTRODUCTION**

The procedure of process simulation consists in the resolution of the set of mathematical relations that describe system behavior, e.g. mass and energy balances, phase equilibrium equations, physical property calculations, etc. The mathematical model contains input variables, process parameters and output variables. The traditional simulation approach involves definition of input variables and process parameters; thus output variables are evaluated by use of the model. However, utilization of the simulation results can be limited due to the uncertainties associated with the variables and parameters in the real process. The solution of the model can be significantly different from the real process response. The usual analysis of uncertainties by a sensitivity study has several drawbacks: it presents possible process scenarios but does not indicate how probable they are, the results are hard to interpret when the number of variables is large and it is difficult to identify the interactions between different variables.

An alternative tool for uncertainty analysis employs the concept of stochastic simulation (Diwekar and Rubin, 1991). This approach is based on a large number of simulation runs; in each run, input variables and process parameters are randomly selected according to adequate probability density functions. The set of output variables evaluated gives important information about the behavior of the statistical process, indicating the most probable variable ranges. This analysis can be found in the literature, for example, in a study of the inaccuracies of thermodynamic evaluations in a liquefied natural gas plant (Melaaen and Owren, 1996) or in the simulation of a pressure relief header network (Costa et al., 1998).

This paper describes the application of a stochastic simulation procedure for the analysis of a supercritical fluid extraction process. The fractionation process of orange essential oil using supercritical carbon dioxide as solvent is studied. An example of how stochastic simulation can guide the process engineer is presented.

**PROCESS MODEL**

The process of fractionation of orange essential oil is carried out in a multistage extraction column. The oil is introduced at the top of the column and the supercritical solvent (carbon dioxide) is introduced at the bottom. The purified oil leaves the column at the bottom.

The orange essential oil is modeled with two key components: limonene and linalool. Limonene represents the terpenes and linalool represents the oxygenated compounds in the oil. The extraction target is to concentrate the oxygenated compounds. This concentration is measured by the fold, that is, how many times linalool composition increases in the oil. On average, the composition of the orange essential oil has a 0.985 mole fraction of terpenes, i.e., it has a 0.015 mole fraction of oxygenated compounds. In the present work, we consider a feed composition with a 0.850 mole fraction of terpenes, which represents a pretreated 10-fold essential oil.

The mathematical model of the fractionation process is based on an equilibrium stage as depicted in Figure 1, where F_{j} is the feed flow rate to stage "j," L_{j-1} is the liquid stream flow rate from stage "j-1" to stage "j," L_{j} is the liquid stream flow rate from stage "j" to stage "j+1," V_{j} is the vapor stream flow rate from stage "j" to stage "j-1," V_{j+1} is the vapor stream flow rate from stage "j+1" to stage "j," U_{j} is the liquid stream flow rate leaving stage "j," W_{j} is the vapor stream flow rate leaving stage "j" and Q_{j} is the heat duty to/from stage "j." Variables x_{ij} and y_{ij} are the liquid and vapor equilibrium composition, respectively, for component "i" in stage "j", and z_{ij} is the composition of component "i" in the feed at stage "j".

The mathematical model comprises material and energy balances associated with equilibrium relationships for each stage and component. The mathematical model (MESH equations) is composed of equations 1 to 4:

Equilibrium equations for each component at each stage:

(1) |

Material balance for each component at each stage:

(2) |

Energy balance equation for each stage:** **

(3) |

Mole fraction summation for each phase at each stage:

(4) |

The thermodynamic system is modeled with the Peng-Robinson cubic equation of state (Peng and Robinson, 1976) with the Van der Waals mixing rule and the Aznar-Telles approach to the temperature dependence of the attractive term (Almeida et al., 1991). The mathematical model is solved by the Newton-Raphson method, using a special algorithm for initial guess generation (Mizutani, 1999).

**STOCHASTIC ANALYSIS**

The procedure of stochastic simulation is applied to the analysis of an extraction column with 10 stages where the process engineer must select the feed temperature and pressure to reach a 25-fold oil. The uncertainty associated with the fractionation process is inserted in the real values of temperature and pressure in relation to the values set by the process engineer. The exact values of these variables in the industrial plant cannot be established, but their statistical behavior is known. Other uncertainty sources could be easily included, using the same procedure presented here.

Each stochastic analysis is composed of a group of 1000 simulator runs. In each run, feed temperature and pressure are randomly selected according to normal distributions; their means correspond to the values set by the process engineer and the standard deviations are 4 K for temperature and 2 atm for pressure (1atm = 1.01325.10^{5}Pa).

The first trial values of feed temperature and pressure to attain the process specification are 333 K and 82 atm. The deterministic simulation results obtained from these values predict a fold of F = 25.9 and a yield of Y = 30.9%. At first, the isolated analysis of the deterministic simulation results indicates that the temperature and pressure selected are the solution to the problem, since the column operates according to the specification, i.e., F = 25.9 ³ 25.

In order to check the feasibility of the proposed solution, a stochastic analysis is applied. Histograms of the pressure and temperature of the group of runs are presented in Figures 2 and 3.

The stochastic simulation generated a set of values of fold and yield that represents the statistical process response. The mean and standard deviation of the set of results are shown in Table 1.

Figures 4 and 5 show the histogram and a graph of the cumulative probability of the fold values, respectively.

Analysis of Figure 5 indicates that there is a considerable probability, of about 40 %, that the proposed process conditions will not attain the process specification. This information alerts the process engineer to the need to propose new operating conditions. The stochastic analysis served as a warning that deterministic results are not reliable due to variable uncertainty.

In this context, new operating conditions with higher pressure are established: 333 K and 83.8 atm. A histogram of the new feed pressure is presented in Figure 6.

The mean and standard deviation of the fold and yield under this new operating condition are presented in Table 2.

A histogram and the cumulative probability distribution of the fold values are presented in Figures 7 and 8.

In Figure 8, it is possible to observe that the probability of violating the specification was reduced to about 10 %. Thus, drawing on his experience, the process engineer must decide if this value is acceptable; if not, new operating conditions must be tested and the stochastic analysis repeated.

**CONCLUSIONS**

This paper presents an application of the concept of stochastic simulation for the analysis of uncertainties in an extraction process involving a supercritical fluid. Analysis of the example of the fractionation of orange essential oil indicates that this approach can be an important tool to estimate more reliable process conditions.

**ACKNOWLEDGEMENTS**

The authors are grateful to the Research Support Foundation of the State of Rio de Janeiro (FAPERJ) and the Brazilian National Council for Scientific and Technological Development (CNPq) for their financial support.

**REFERENCES**

Almeida, G.S., Aznar, M. and Telles, A.S., Uma Nova Forma de Dependência com a Temperatura do Termo Atrativo de Equações de Estado Cúbicas, RBE, Caderno de Eng. Química, 8, No. 2, p. 95 (1991). [ Links ]

Costa, A.L.H., de Medeiros, J.L. and Pessoa, F.L.P., Steady-State Modeling and Simulation of Pipeline Networks for Compressible Fluids, Brazilian Journal of Chemical Engineering, 15, No. 4, p. 344 (1998). [ Links ]

Diwekar, U.M. and Rubin, E.S., Stochastic Modeling of Chemical Processes, Computers and Chemical Engineering, 15, No. 2, p. 105 (1991). [ Links ]

Melaaen, I.S. and Owren, G., How Do the Inaccuracies of Enthalpy and Vapour-Liquid Equilibrium Calculations Influence Baseload LNG Plant Design?, Computers and Chemical Engineering, 20, No. 1, p. 1 (1996). [ Links ]

Mizutani, F.T., Design and Analysis of Supercritical Extraction Processes, M.Sc. thesis (in Portuguese), COPPE, Federal University of Rio de Janeiro/UFRJ (1999). [ Links ]

Peng, D.Y. and Robinson, D.B., A New Two Constants Equation of State, Ind. Eng. Chem. Fundam., 15, No. 1, p. 58 (1976). [ Links ]