Modelling polyurethane synthesis rates on the example of heteropolyaddition of 2,4toluene diisocyanate (TDI) and 1,4butanediol (BD)
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Abstract
This paper investigates the effects of increase of reaction mixture viscosity on the kinetics of linear polymer creation in a bulk heteropolyaddition process of 2,4toluene diisocyanate (TDI) and 1,4butanediol (BD). The paper presents a method for solving a system of bulk polyaddition of 2,4toluene diisocyanate and 1,4butanediol process balance equations, allowing the determination of the process kinetic parameters. Determination of polymerisation reaction kinetic parameters was also made possible by the use of the socalled partial reaction rate constant. Such an approach enabled a significant simplification of the mathematical expressions describing the heteropolyaddition process and provided an opportunity to associate kinetic parameters with the average molar mass of the mixture and, thus, with the viscosity. The method presented herein facilitates an analysis of the linear polymers heteropolyaddition process.
Keywords
Kinetics of polymerisations Reaction rate constant Linear polymer Viscosity Heteropolyaddition Modelling of polymerisationsList of symbols
 E_{i}
Experimental value (literature), [g/mol]
 k_{1}
Partial reaction rate constant of component 1, [dm^{3}/(mol s)^{1/2}]
 k_{A}, k_{B}
Partial reaction rate constants of reagents assigned to reagents A and B, respectively, [dm^{3}/(mol s)^{1/2}]
 k_{AB}
Reaction rate constant A + B = AB, [dm^{3}/(mol s)]
 k_{j}
Partial reaction rate constant j of this component, [dm^{3}/(mol s)^{1/2}]
 M_{0}
Molar mass of monomer A or B, [g/mol]
 M_{n}
Polymer numberaverage molar mass, [g/mol]
 O_{i}
Calculated value, [g/mol]
 t
Time [min]
 w_{i}
Part by weight
 Z_{i}
Correction coefficient of component i
 β(η)
Correlation coefficient, β(η) ∈ (0.1)
 γ
Constant coefficient characteristic for a particular polymerisation process
Introduction
Very often during the course of polymerisation processes, the viscosity of the reaction medium changes. When these changes significantly influence the polymerisation process rate, it becomes necessary to take into account these variables in balance equations describing this process. The need to take into account changes in viscosity means that systems of mass balance equations describing the polymerisation process are usually quite complex and often difficult to solve due to the large number of unknown kinetic parameters. In connection with difficulties in determining polymerisation process kinetic parameters, literature often assumes the same value for all constant rates of these reactions (k = const), and the effect of viscosity changes on the numerical values of the k constants is ignored. Used method is appropriate for systems, where viscosity is practically constant.
This paper attempts to take into account increasing viscosity of the reaction medium on the kinetics of the linear polymer creation process occurring during heteropolyaddition process. The increase in average molar mass of the mixture and, thus, the viscosity caused by the course of this process makes the constant reaction rate numerical values decrease and as such may have a significant effect on the results.
Due to the complexity of the mass balance equation system and the necessity to take into account viscosity changes in numerical calculations, this paper uses the concept of partial reaction rate constants, the introduction of which into the balance equations allows a meaningful simplification of the mathematical model describing the polyaddition process being considered here [1]. The influence of viscosity on the polymerisation rate has been taken into account by associating that rate with the numberaverage molecular mass M_{n} of the reaction mixture, which provided an opportunity to associate kinetic parameters k with the numberaverage molar mass M_{n}. Using the derived mass balance system of equations, which takes into account numberaverage molar mass changes on the reaction rate constant numerical values, kinetic parameter values for the analysed processes were determined.
The presented method facilitates determination of partial reaction rate constant numerical values and, thus, it assists the progress of polymerisation kinetic parameters definition, which facilitates the system of balance equations to be solved.
The used model
Literature data [1] show that in real systems, very often during the course of polymerisation processes, the viscosity of the reaction medium changes. When the viscosity changes significantly influence the polymerisation process rate, it becomes necessary to take into account these variations in rate equations describing this process. A change in the viscosity of a reaction system, most often exhibiting itself by a change in the numeric kinetic value (which determine rate of reaction), causes a reduction in the reaction rate numerical constants, which should be taken into consideration in numeric calculations. Changes in viscosity, during the course of a polymerisation process, cause a change in the process kinetic parameters, resulting in adding significant complexity to the mass balance equation system. As reaction rate constant values change with the progress of the polymerisation process, it becomes necessary, from the point of view of numeric calculations, to determine the reaction rate values at each step of the said calculations.
The mass ratios and numberaverage molar masses M_{n} [g/mol] of the selected fractions of the reaction mixture in the heteropolyaddition process in 2,4TDI (0.172 mol) and 1,4BD (0.172 mol) in the solution of chlorobenzene and tetrahydrofuran at a temperature of t_{1} = 86 °C [7]
t [min]  M_{n} [g/mol]  F1 + F2  F3 + F4  F5 + F6 

Mass proportion  
20  309  0.73  0.252  0.016 
40  332  0.574  0.352  0.074 
60  356  0.46  0.385  0.155 
90  412  0.32  0.354  0.225 
120  456  0.252  0.314  0.267 
180  537  0.164  0.234  0.269 
240  667  0.087  0.157  0.218 
300  801  0.067  0.112  0.166 

F1—fraction 1 is equal to the sum of the concentration of components A and B.

F2—fraction 2 is equal to the concentration of component AB.

F3—fraction 3 is equal to the sum of the concentration of components A_{1}B_{2} and A_{2}B_{1}.

F4—fraction 4 is equal to the concentration of component A_{2}B_{2}.

F5—fraction 5 is equal to the sum of the concentration of components A_{2}B_{3} and A_{3}B_{2}.

F6—fraction 6 is equal to the concentration of component A_{3}B_{3}.
The numerical calculations of the presented heteropolyaddition process were based on model (2), on the basis of which a calculation algorithm was built in the Mathcad Prime 2.0 software that allows for approximation of kinetic parameters of the process and consequently for determining the kinetics of the polyurethaneforming reaction.
The calculations were based on the results of experimental studies in Table 1 and the model of the heteropolyaddition process (2). All calculations were done starting with determining the reaction rate constant k_{AB} = 5 × 10^{−4} dm^{3}/(mol s) with the assumption k_{i} = const.
Equation (4) defines the correction coefficient, which is a dimensionless number, with two parameters γ_{1} and p characteristic for a particular polymerisation process. Due to this fact, the partial reaction rate constant of k_{i} decreases with the increase in the numberaverage molar mass of the forming polymer.
Selected values of the partial reaction rate constant k_{i} [(dm^{3}/(mol s))^{1/2}] in the heteropolyaddition process in 2,4TDI (0.172 mol) and 1,4BD (0.172 mol) in chlorobenzene and tetrahydrofuran solution at temperature t_{1} = 86 °C are calculated with model (2)
t [min]  k_{i} [(dm^{3}/(mol s))^{1/2}] 

0  0.02020 
20  0.02015 
40  0.02010 
60  0.02006 
90  0.02001 
120  0.01997 
180  0.01991 
240  0.01986 
300  0.01982 
Determination of kinetic parameters for two different k_{i} values, k_{i} = const [8] and k_{i} = k_{1}Z_{i}, allowed to determine how the mass ratios of respective polymer fractions change with time, and compare them with the results of experimental research, as shown in the figures below.
Comparison of experimental data and the data calculated with model (2) in the form of mass ratios and numberaverage molar mass M_{n} [g/mol] of the selected fractions of the reaction mixture in the heteropolyaddition process in 2,4TDI (0.172 mol) and 1,4BD (0.172 mol) in the solution of chlorobenzene and tetrahydrofuran at a temperature of t_{1} = 86 °C [7]
t [min]  Data from literature [7]  Calculated with model (2)  

M_{n} [g/mol]  w _{ i}  M_{n} [g/mol]  w _{ i}  
F1 + F2  F3 + F4  F5 + F6  F1 + F2  F3 + F4  F5 + F6  
20  309  0.730  0.252  0.016  166  0.818  0.162  0.018 
40  332  0.574  0.352  0.074  207  0.590  0.294  0.087 
60  356  0.460  0.385  0.155  250  0.430  0.326  0.150 
90  412  0.32  0.354  0.225  316  0.284  0.300  0.195 
120  456  0.252  0.314  0.267  383  0.200  0.253  0.200 
180  537  0.164  0.234  0.269  519  0.114  0.176  0.171 
240  667  0.087  0.157  0.218  655  0.074  0.126  0.137 
300  801  0.067  0.112  0.166  792  0.052  0.094  0.110 
The value of objective function S in accordance with Eq. (5)  
\(S_{{M_{n} }}\)  S _{ F1+ F2}  S _{ F3+ F4}  S _{ F5+ F6}  \(S_{{M_{n} }}\)  S _{ F1+ F2}  S _{ F3+ F4}  S _{ F5+ F6}  
S  185,000  0.0480  0.0451  0.0457  62,370  0.0188  0.0274  0.0201 
ΣS  –  0.1388  –  0.0623 

at the same value of reaction rate constant k_{i} = const

with a reaction rate constant dependent on the average molecular mass of the polymer k_{i} = k_{1}Z_{i}.
On the basis of Figs. 1, 2, 3, 4, 5, 6, 7 and 8 and data in Table 3, it can be concluded that if the increase in viscosity of the reaction mixture is omitted and the same reaction rate is assumed during the whole process of heteropolyaddition of 2,4TDI with 1,4BD, the result obtained from the sum of objective function (5) for all analysed fractions S_{min} = 0.1388 is twice higher than when including the changes in the average molecular mass of the forming polymer (and thus viscosity) S_{min} = 0.0623.
In the above equation, the following symbols were used: S_{c}—results of calculations without changes in viscosity; S_{η}—results of calculations with changes in viscosity; and β(η)—numerical value of correlation coefficient included in the range (0,1): β(η) ∈ (0,1).
The measure of the influence of viscosity on the polymerisation process in this case is the value of the correlation coefficient β(η) which increases with the increase in viscosity influence on the process. The value of this coefficient was β_{t}(η) = 0.551. The figure below shows the dynamics of changes in the correction coefficient Z_{i} in relation to the average molar mass.
It is worth noting that this case is an example of a polymerisation in diluted solutions, as evidenced by a small change in the Z_{i} correction coefficient, varying between 0.9530 and 0.9550.
Conclusions
This paper deliberately analyses a process in a solution in which the viscosity is not likely to affect the speed of the polymerisation process or its influence is negligible. The aim was to examine how model (2) behaves, assuming a variable value of kinetic parameters. As we can see, despite slight changes in viscosity of the system resulting from a small increase in average molar mass of polyurethane, the proposed method of kinetic parameters calculation gives a better approximation of the course of polyaddition process of 2,4TDI with 1,4BD than the model based on the same kinetic parameters for the whole polymerisation process.
The results obtained show that the applied approach proposed in model (2) is universal, and prove that kinetic parameters change during the heteropolyaddition process.
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