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7DEVELOPING A MATHEMATICAL MODEL OF THE POLYMERIZATION REACTOR BASED ON THE INITIAL DATA SET FOR THE PRODUCTION OF POLYVINYL
Sodiqov B.Q. - doctoral student, Navoi State University of Mining and Technologies, E_mail: [email protected], Goziyev O.S. - assistant, Navoi State University of Mining and Technologies E_mail: [email protected], Qudratova M.A - master student, Navoi State Pedagogical Institute, E_mail: [email protected], Erkinov F.F. - student, Navoi State University of Mining and Technologies, E_mail: [email protected], Uzbekistan.
Annotation. In this article, a mathematical model was created in MATLAB based on the heat balance equations of the material balance of the polymerization reactor in the production of polyvinyl chloride. Key words: Database, mathematical model, polymerization reactor, PVC production control system.
Annotatsiya. Ushbu maqolada polivinil xlorid ishlab chiqarishda polimerlash reaktorining issiqlik balans
material balans tenglamalari asosida MATLAB dasturida matematik model yaratilgan.
Kalit so'zlar: Ma'lumotlar bazasi, matematik model, polimerlash reaktori, PVX ishlab chiqarish boshqarish
tizimi
Аннотация. В данной статье в программе MATLAB создана математическая модель на основе уравнений теплового баланса материального баланса реактора полимеризации при производстве поливинилхлорида.
Ключевые слова: База данных, математическая модель, реактор полимеризации, система управления производством ПВХ.
Introduction
The production of polyvinyl chloride (PVC) is a critical process in the chemical industry, owing to its extensive use in various applications such as construction, healthcare, and consumer goods. PVC is synthesized through the polymerization of vinyl chloride monomer (VCM), a process that requires precise control to ensure the desired quality and properties of the final product. This predictive capability is crucial for improving process efficiency, reducing production costs, and ensuring product consistency. In this article, we focus on the development of a mathematical model for the polymerization reactor used in PVC production. The model is based on an initial data set that includes various operational parameters and chemical properties [1].
Steps of developing mathematical model of this process.
Creating a mathematical model of the PVC polymerization process in a batch reactor and finding its transfer function involves several steps. Below, I'll outline a general approach and derive the key equations and transfer function [2]
Step 1: Define the Process and Assumptions Reactants: Vinyl Chloride Monomer (VCM) and initiator. Reactor type: Batch reactor. CO Key assumptions:
CO • The reactor is well-mixed (perfect mixing).
CHLORIDE
Baxtiyor Sodiqov [0009-0007-2346-6670], Olmos G'oziyev [0009-0001-5488-4716], Medina Qudratova [0009-0001-4739-548X], Farxod Erkinov [0009-0005-7478-3401]
m
• Temperature is constant (isothermal conditions).
• The reaction kinetics follow a known rate law.
• The polymerization process is first-order with respect to the initiator and VCM concentrations [3]
Step 2: Write the Governing Equations
The polymerization of VCM can be described by the following kinetic equations: 1. Monomer consumption (1):
® = -M^][/] (1)
where [M] is the monomer concentration, [/] is the initiator concentration, and kp is the rate constant for polymerization [4].
Initiator consumption:
f = -M/] (2)
where kd is the rate constant for initiator decomposition. Polymer production
^ = kp[M][/] (3)
Where [P] - is the concentration of the polymer
Step 3: Combine the Equations
Assuming the initiator concentration is much smaller than the monomer concentration and remains relatively constant (pseudo-steady-state approximation):
[l]»[l]oe~kdt
The monomer concentration changes over time as:
d[M] . ,
-LJ. =-kp[M][I]0e-k^
If we calculate the integral of this equation over the function [M](t) and exponentiate both sides of the equation, we get the following equation:
-[M](t) = [M]0exp(-(1 - e-kdt)) (4)
Step 4: Transfer Function
To derive the transfer function, we need to linearize the model around an operating point and apply Laplace transforms [5]
1. Linearization:
Let ( t)=Mo+SM(t) and I( t)=/o+SI( t) Linearize around the steady-state concentrations Mo and /o
2. Laplace Transform: Apply Laplace transforms to the linearized equations:
= -kpZ[SM(t)Io + MoSI(t)} (5)
Let A(s)and A I(s) be the Laplace transforms of SM(t)and SI( t), respectively. The Laplace transform of the above equation is:
sAM(s)=-kp(I0AM(s)+M0AI(s))
3.Transfer Function: Solving for AM(s) in terms of AI(s):
A M(s)(s+kpIo)=-kpMoA I(s)
The transfer function is G(s)= AM(s)/ A I(s):
G(s) =
A[M](s) kpM o
A/(s) s + kvl0
Necessary values of parametres
• Rate constant for polymerization (kp): 2*103 L/mol
• Rate constant for initiator decomposition (kd): 5^10-4 s-1
• Initial concentration of monomer ([M]o): 8mol/L
• Initial concentration of initiator ([I]o): 0.05 mol/L
• Reactor temperature: 60°C
• Reactor volume (V): 5 L
From the previous derivation, the transfer function was:
A[M](s) kpM0
G(s) =
M(s) s + kvl0
Substituting the values:
G(s) =
-2 * 1000*8 _ -16000 s + 2 * 1000 * 0.05 = s + 100
This transfer function describes the dynamic response of the monomer concentration to changes in the initiator concentration for the given set of physical parameters [6]
Checking mathematical model on MATLAB
I plot the transient process function of the given transfer function.
Fig.1. Code for finding h(t) function
in
CO
to
CO
Fig.2. Step response of transfer function.
The stability of this system according to the given transfer function is checked by the Nyquist criterion in the MATLAB program.
Fig.3. Amplitude phase characteristic of the transfer function
Looking at this graph, the mathematical model we have built is stable and has the required quality indicators [7]
Conclusion
In this study, we have successfully developed a mathematical model of the polymerization reactor for the production of polyvinyl chloride (PVC) based on initial data sets. By leveraging key parameters and reaction kinetics, we have formulated differential equations that describe the dynamic behavior of monomer, initiator, and polymer concentrations within the reactor. The model was implemented and simulated using MATLAB, providing a comprehensive framework for analyzing the polymerization process.
References
[1.] Botirov, T. V., & Sodiqov, B. Q. (2023). Use of intellectual regulators in the production of polyvinyl chloride. Journal of Advances in Engineering Technology, (4), 31-35.
[2.] Sodiqov, B. Q., & Erkinov. F. F. (2022). classification of information-intellectual systems. Yosh Tadqiqotchi Jurnali, 1(3), 23-27.
[3.] Sain, D., & Mohan, B. M. (2021). A simple approach to mathematical modelling of integer order and fractional order fuzzy PID controllers using one-dimensional input space and their experimental realization. Journal of the Franklin Institute, 358(7), 3726-3756.
[4.] Sodiqov, B. Q., & Jumabayev, E. O. (2024). Advancements in control systems: understanding fopid controllers. Молодые ученые, 2(4), 76-78.
[5.] Timofeeva, S., Smirnov, G., Boboev, A., & Goziev, O. (2023). Use of artificial intelligence to prevent spontaneous combustion of coal. In E3S Web of Conferences (Vol. 417, p. 04006). EDP Sciences. [6.] Mukhitdinov, D., Boybutayev, S., Goziev, O., & Qudratov, J. (2023). Construction of continuous and discrete non-linear prognostic models of the control system for the process of grinding ore materials. In E3S Web of Conferences (Vol. 417, p. 05009). EDP Sciences.
[7.] Kadirov, Y., Samadov, A., & Goziev, O. (2023). Equation of dynamics of greenhouse microclimate parameters. In E3S Web of Conferences (Vol. 390). EDP Sciences.
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