Markov Chain Monte Carlo Analysis of the Variable-Volume Exothermic Model for a Continuously Stirred Tank Reactor

  • J. P. Muhirwa Department of Applied Mathematics and Computational Science (AMCS), Nelson Mandela African Institution of Science and Technology Tanzania (NM-AIST), Tanzania | Department of Mathematics, University of Rwanda, College of Science and Technology (UR-CST), Rwanda
  • S. I. Mbalawata Research Department, African Institute for Mathematical Sciences, Rwanda
  • V. G. Masanja Department of Applied Mathematics and Computational Science (AMCS), Nelson Mandela African Institution of Science and Technology Tanzania (NM-AIST), Tanzania

Abstract

In this paper, a variable-volume Continuously Stirred Tank Reactor (CSTR) deterministic exothermic model has been formulated based on the Reynold Transport Theorem. The numerical analysis of the formulated model and the identifiability of its physical parameters are done by using the least squares and the Delayed-Rejection Adaptive Metropolis (DRAM) method. The least square estimates provide the prior information for the DRAM method. The overall numerical results show that the model gives an insight in describing the dynamics of CSTR processes, and 14 parameters of the CSTR are well identified through DRAM convergence diagnostic tests, such as trace, scatter, autocorrelation, histograms, and marginal density plots. Global sensitivity analysis was further performed, by using the partial rank correlation coefficients obtained from the Latin hypercube sampling method, in order to study and quantify the impact of estimated parameters, uncertainties on the model outputs. The results showed that 7 among the 14 estimated model parameters are very sensitive to the model outcomes and so those parameters need to be handled and treated carefully.

Keywords: parameter identifiability, variable volume, exothermic, CSRT, RTT, MCMC, DRAM

Downloads

Download data is not yet available.

References

A. S. Al-Araji, "Modeling of Continuous Stirred Tank Reactor based on Artificial Neural Network," Al-Nahrain Journal for Engineering Sciences, vol. 18, no. 2, pp. 202-207, 2015.

A. O. Ahmed, G. A. Gasmelseed, A. B. Karama, and A. E. Musa, "Cascade Control of a Continuous Stirred Tank Reactor (CSTR)," Journal of Applied and Industrial Sciences, vol. 1, no. 4, pp. 16-23, 2013.

H. A. Maddah, "Numerical Analysis for the Oxidation of Phenol with TiO2 in Wastewater Photocatalytic Reactors," Engineering, Technology & Applied Science Research, vol. 8, no. 5, pp. 3463-3469, Oct. 2018. https://doi.org/10.48084/etasr.2304

A. Simorgh, A. Razminia, and V. I. Shiryaev, "System identification and control design of a nonlinear continuously stirred tank reactor," Mathematics and Computers in Simulation, vol. 173, pp. 16-31, Jul. 2020. https://doi.org/10.1016/j.matcom.2020.01.010

A. Uppal, W. H. Ray, and A. B. Poore, "On the dynamic behavior of continuous stirred tank reactors," Chemical Engineering Science, vol. 29, no. 4, pp. 967-985, Apr. 1974. https://doi.org/10.1016/0009-2509(74)80089-8

A. S. Ibrehem, "Modified Mathematical Model For Neutralization System In Stirred Tank Reactor," Bulletin of Chemical Reaction Engineering & Catalysis, vol. 6, no. 1, pp. 47-52, May 2011. https://doi.org/10.9767/bcrec.6.1.825.47-52

A. Z. Al-Khazaal, F. Ahmad, and N. Ahmad, "Study on the Removal of Thiosulfate from Wastewater by Catalytic Oxidation," Engineering, Technology & Applied Science Research, vol. 9, no. 2, pp. 4053-4056, Apr. 2019. https://doi.org/10.48084/etasr.2553

B. G. Osorio, H. B. Castro, and J. D. S. Torres, "State and unknown input estimation in a CSTR using higher-order sliding mode observer," in IEEE IX Latin American Robotics Symposium and IEEE Colombian Conference on Automatic Control, Bogota, Colombia, Oct. 2011, pp. 1-5. https://doi.org/10.1109/LARC.2011.6086829

C. G. Hill, An Introduction to Chemical Engineering Kinetics and Reactor Design. New York, USA: Wiley, 1977.

J. Du, C. Song, and P. Li, "Modeling and Control of a Continuous Stirred Tank Reactor Based on a Mixed Logical Dynamical Model," Chinese Journal of Chemical Engineering, vol. 15, no. 4, pp. 533-538, Aug. 2007. https://doi.org/10.1016/S1004-9541(07)60120-7

D. P. Karadimou, P. A. Papadopoulos, and N. C. Markatos, "Mathematical modelling and numerical simulation of two-phase gas-liquid flows in stirred-tank reactors," Journal of King Saud University - Science, vol. 31, no. 1, pp. 33-41, Jan. 2019. https://doi.org/10.1016/j.jksus.2017.05.015

D. Ndanguza, J. P. Muhirwa, and A. Uwimana, "Modeling and parameters estimation of a Spatial Predator-Prey distribution," Rwanda Journal of Engineering, Science, Technology and Environment, vol. 2, no. 1, pp. 1-17, Jul. 2019. https://doi.org/10.4314/rjeste.v2i1.5

D. Tamboli and R. Chile, "Multi-model approach for 2-DOF control of nonlinear CSTR process," International Journal of Modelling, Identification and Control, vol. 30, no. 2, pp. 143-161, Jan. 2018. https://doi.org/10.1504/IJMIC.2018.10014979

E. Shakeri, G. Latif-Shabgahi, and A. E. Abharian, "Design of an intelligent stochastic model predictive controller for a continuous stirred tank reactor through a Fokker-Planck observer," Transactions of the Institute of Measurement and Control, vol. 40, no. 10, pp. 3010-3022, Jun. 2018. https://doi.org/10.1177/0142331217712583

E. Vlahakis and G. Halikias, "Temperature and concentration control of exothermic chemical processes in continuous stirred tank reactors," Transactions of the Institute of Measurement and Control, vol. 41, no. 15, pp. 4274-4284, Nov. 2019. https://doi.org/10.1177/0142331219855591

E. A. Buehler, J. A. Paulson, and A. Mesbah, "Lyapunov-based stochastic nonlinear model predictive control: Shaping the state probability distribution functions," in American Control Conference, Boston, MA, USA, Jul. 2016, pp. 5389-5394. https://doi.org/10.1109/ACC.2016.7526514

E. H. Karimi and K. B. McAuley, "A Bayesian Method for Estimating Parameters in Stochastic Differential," IFAC-PapersOnLine, vol. 48, no. 8, pp. 147-152, Jan. 2015. https://doi.org/10.1016/j.ifacol.2015.08.172

F. Remo, L. S. Luboobi, I. S. Mabalawata, and B. K. Nannyonga, "A mathematical model for the dynamics and MCMC analysis of tomato bacterial wilt disease," International Journal of Biomathematics, vol. 11, no. 1, Aug. 2017, Art. no. 1850001. https://doi.org/10.1142/S1793524518500018

G. I. Valderrama-Bahamondez and H. Frohlich, "MCMC Techniques for Parameter Estimation of ODE Based Models in Systems Biology," Frontiers in Applied Mathematics and Statistics, vol. 5, 2019, Art. no. 55. https://doi.org/10.3389/fams.2019.00055

G. L. Foutch and A. H. Johannes, "Reactors in Process Engineering," in Encyclopedia of Physical Science and Technology, 3rd ed., R. A. Meyers, Ed. New York, NY, USA: Academic Press, 2003, pp. 23-43. https://doi.org/10.1016/B0-12-227410-5/00654-2

H. Ballesteros-Moncada, E. J. Herrera-Lopez, and J. Anzurez-Marin, "Fuzzy model-based observers for fault detection in CSTR," ISA Transactions, vol. 59, pp. 325-333, Nov. 2015. https://doi.org/10.1016/j.isatra.2015.10.006

H. Haario, E. Saksman, and J. Tamminen, "Adaptive proposal distribution for random walk Metropolis algorithm," Computational Statistics, vol. 14, no. 3, pp. 375-395, Sep. 1999. https://doi.org/10.1007/s001800050022

H. Haario, E. Saksman, and J. Tamminen, "An adaptive Metropolis algorithm," Bernoulli, vol. 7, no. 2, pp. 223-242, Apr. 2001. A. Stepanov, "Exact Calculation of the Internal Energy of the Ideal Gas in Statistical Mechanics," Physical Science International Journal, vol. 14, no. 5, pp. 1-5, Apr. 2017. https://doi.org/10.2307/3318737

I. S. Mbalawata, S. Sarkka, and H. Haario, "Parameter estimation in stochastic differential equations with Markov chain Monte Carlo and non-linear Kalman filtering," Computational Statistics, vol. 28, no. 3, pp. 1195-1223, Jun. 2013. https://doi.org/10.1007/s00180-012-0352-y

I. S. Mbalawata, S. Sarkka, M. Vihola, and H. Haario, "Adaptive Metropolis algorithm using variational Bayesian adaptive Kalman filter," Computational Statistics & Data Analysis, vol. 83, pp. 101-115, Mar. 2015. https://doi.org/10.1016/j.csda.2014.10.006

J. C. Etchells, "Process Intensification: Safety Pros and Cons," Process Safety and Environmental Protection, vol. 83, no. 2, pp. 85-89, Mar. 2005. https://doi.org/10.1205/psep.04241

J. Jiang, J. Wu, S. Poncin, and H. Z. Li, "Effect of hydrodynamic shear on biogas production and granule characteristics in a continuous stirred tank reactor," Process Biochemistry, vol. 51, no. 3, pp. 345-351, Mar. 2016. https://doi.org/10.1016/j.procbio.2015.12.014

J. Vojtesek and P. Dostal, "Simulation analysis of continuous stirred tank reactor," in 22nd European Conference on Modelling and Simulation, Nicosia, Cyprus, Jun. 2008, pp. 1-6.

J. Vojtesek and P. Dostal, "Simulation of adaptive control of continuous stirred tank reactor," International Journal of Simulation Modelling, vol. 8, no. 3, pp. 133-144, 2009. https://doi.org/10.2507/IJSIMM08(3)1.129

J. P. Muhirwa and D. Ndanguza, "Effect of random noise, quasi random noise and systematic random noise on unknown continuous stirred tank reactor (cstr)," Applied Mathematical Sciences, vol. 11, no. 62, pp. 3051-3071, 2017. https://doi.org/10.12988/ams.2017.79283

K. Lopez Buritica, S. Casanova Trujillo, C. D. Acosta, and H. A. Granada Diaz, "Dynamical Analysis of a Continuous Stirred-Tank Reactor with the Formation of Biofilms for Wastewater Treatment," Mathematical Problems in Engineering, vol. 2015, Jun. 2015, Art. no. e512404. https://doi.org/10.1155/2015/512404

K. Cahyari, Sarto, S. Syamsiah, and A. Prasetya, "Performance of continuous stirred tank reactor (CSTR) on fermentative biohydrogen production from melon waste," IOP Conference Series: Materials Science and Engineering, vol. 162, no. 1, Nov. 2016, Art. no. 012013. https://doi.org/10.1088/1757-899X/162/1/012013

K. J. Keesman et al., "Aquaponics Systems Modelling," in Aquaponics Food Production Systems: Combined Aquaculture and Hydroponic Production Technologies for the Future, S. Goddek, A. Joyce, B. Kotzen, and G. M. Burnell, Eds. Cambridge, England: Springer International Publishing, 2019, pp. 267-299. https://doi.org/10.1007/978-3-030-15943-6_11

L. P. Russo and B. W. Bequette, "Cstr performance limitations due to cooling jacket dynamics," in Dynamics and Control of Chemical Reactors, Distillation Columns and Batch Processes, J. G. Balchen, Ed. Oxford, UK: Pergamon, 1993, pp. 149-154. https://doi.org/10.1016/B978-0-08-041711-0.50025-8

M. Danish, M. K. Al Mesfer, and M. M. Rashid, "Effect of operating conditions on cstr performance: an experimental study," International journal of engineering research and applications, vol. 5, no. 2, pp. 74-78, 2015.

M. Laine, Adaptive MCMC methods with applications in environmental and geophysical models. Helsinki, Finland: Finnish Meteorological Institute, 2008.

D. F. A. M. N. Esmaeel, "Fuzzy logic Control of Continuous Stirred Tank Reactor," Tikrit Journal of Engineering Sciences, vol. 20, no. 2, pp. 70-80, 2013.

M. A. S. Aboelela and R. H. M. Hennas, "Development of a fractional order pid controller using adaptive weighted pso and genetic algorithms with applications," in Fractional Order Systems, A. T. Azar, A. G. Radwan, and S. Vaidyanathan, Eds. Cambridge, Massachusetts, MA, USA: Academic Press, 2018, pp. 511-551. https://doi.org/10.1016/B978-0-12-816152-4.00017-0

M.-H. Cui, D. Cui, L. Gao, H.-Y. Cheng, and A.-J. Wang, "Efficient azo dye decolorization in a continuous stirred tank reactor (CSTR) with built-in bioelectrochemical system," Bioresource Technology, vol. 218, pp. 1307-1311, Oct. 2016. https://doi.org/10.1016/j.biortech.2016.07.135

N. M. Ramli and M. S. Mohamad, "Modelling for Temperature Non-Isothermal Continuous Stirred Tank Reactor Using Fuzzy Logic," International Journal of Environmental and Ecological Engineering, vol. 11, no. 2, pp. 169-175, Jan. 2017.

R. B. Bird, W. E. Stewart, and E. N. Lightfoot, Transport Phenomena, 2nd edition. New York, NY, USA: Wiley, 2009.

R. Kandiyoti, Fundamentals of Reaction Engineering. London, UK: Bookboon, 2009.

R. C. S. Dias and M. R. P. F. N. Costa, "Transient Behavior and Gelation of Free Radical Polymerizations in Continuous Stirred Tank Reactors," Macromolecular Theory and Simulations, vol. 14, no. 4, pp. 243-255, 2005. https://doi.org/10.1002/mats.200400086

R. M. Sudhanan and D. P. Poongodi, "Comparative Analysis of Various Control Strategies for a Nonlinear CSTR System," International Journal of Nonlinear Sciences and Numerical Simulation, vol. 18, no. 3-4, pp. 269-276, Jun. 2017. https://doi.org/10.1515/ijnsns-2015-0125

S. Brooks, "Markov chain Monte Carlo method and its application," Journal of the Royal Statistical Society: Series D (The Statistician), vol. 47, no. 1, pp. 69-100, 1998. https://doi.org/10.1111/1467-9884.00117

S. S. Jang and R. B. Gopaluni, "Parameter estimation in nonlinear chemical and biological processes with unmeasured variables from small data sets," Chemical Engineering Science, vol. 66, no. 12, pp. 2774-2787, Jun. 2011. https://doi.org/10.1016/j.ces.2011.03.029

S. Masoumi, T. A. Duever, and P. M. Reilly, "Sequential Markov Chain Monte Carlo (MCMC) model discrimination," The Canadian Journal of Chemical Engineering, vol. 91, no. 5, pp. 862-869, 2013. https://doi.org/10.1002/cjce.21711

S. Nanda, "Reactors and Fundamentals of Reactors Design for Chemical Reaction," Ph.D. dissertation, Maharshi Dayanand University, Haryana, India, 2008.

S. Sharma, "Markov Chain Monte Carlo Methods for Bayesian Data Analysis in Astronomy," Annual Review of Astronomy and Astrophysics, vol. 55, pp. 213-259, Aug. 2017. https://doi.org/10.1146/annurev-astro-082214-122339

S. Sinharay, "Assessing Convergence of the Markov Chain Monte Carlo Algorithms: A Review," ETS Research Report Series, vol. 2003, no. 1, pp. i-52, 2003. https://doi.org/10.1002/j.2333-8504.2003.tb01899.x

S. Tronci, M. Grosso, J. Alvarez, and R. Baratti, "Stochastic dynamical nonlinear behavior analysis of a class of single-state CSTRs," IFAC Proceedings Volumes, vol. 42, no. 11, pp. 697-702, Jan. 2009. https://doi.org/10.3182/20090712-4-TR-2008.00113

S. Zhang, D. Muller, H. Arellano-Garcia, and G. Wozny, "CFD simulation of the fluid hydrodynamics in a continuous stirred-tank reactor," Chemical Engineering Transactions, vol. 32, pp. 1441-1446, Jun. 2013.

S. N. Naikwad and S. V. Dudul, "Identification of a Typical CSTR Using Optimal Focused Time Lagged Recurrent Neural Network Model with Gamma Memory Filter," Applied Computational Intelligence and Soft Computing, vol. 2009, Jan. 2010, Art. no. 385757. https://doi.org/10.1155/2009/385757

S. R. Tofighi, F. Bayat, and F. Merrikh-Bayat, "Robust feedback linearization of an isothermal continuous stirred tank reactor: H∞ mixed-sensitivity synthesis and DK-iteration approaches," Transactions of the Institute of Measurement and Control, vol. 39, no. 3, pp. 344-351, Mar. 2017. https://doi.org/10.1177/0142331215603446

T. Niederberger, "Markov Chain Monte Carlo Methods for Parameter Identification in Systems Biology Models," Ph.D. dissertation, Ludwig Maximilian University of Munich, Bad Reichenhall, Germany, 2012.

T. Rajagopalan and V. Seshadri, "Analysis of continuous stirred tank reactor as a multivariable process and algorithms for computer determination of the equilibrium states," International Journal of Control, vol. 15, no. 3, pp. 497-507, Mar. 1972. https://doi.org/10.1080/00207177208932165

V. Nicoulaud-Gouin, L. Garcia-Sanchez, M. Giacalone, J. C. Attard, A. Martin-Garin, and F. Y. Bois, "Identifiability of sorption parameters in stirred flow-through reactor experiments and their identification with a Bayesian approach," Journal of Environmental Radioactivity, vol. 162-163, pp. 328-339, Oct. 2016. https://doi.org/10.1016/j.jenvrad.2016.06.008

V. Roy, "Convergence Diagnostics for Markov Chain Monte Carlo," Annual Review of Statistics and Its Application, vol. 7, no. 1, pp. 387-412, 2020. https://doi.org/10.1146/annurev-statistics-031219-041300

Y. Lu, Z. Fang, and C. Gao, "Stabilization of (state, input)-disturbed CSTRs through the port-Hamiltonian systems approach," arXiv:1707.01560 [math], Jun. 2017, Accessed: Feb. 20, 2021. [Online]. Available: http://arxiv.org/abs/1707.01560.

Z. Prokopova and R. Prokop, "Modelling and Simulation of Chemical Industrial Reactors," in 23rd European Conference on Modelling and Simulation, Madrid, Spain, Jun. 2009, pp. 378-383. https://doi.org/10.7148/2009-0378-0383

Metrics

Abstract Views: 63
PDF Downloads: 56

Metrics Information
Bookmark and Share