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\title{Non-Linear Mathematical Modelling of Nylon-6 Polymerization Reactors}
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             \author[1]{Devika, P.  D.}

             \affil[1]{  Ramaiah Institute of Technology}

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\date{\small \em Received: 15 December 2018 Accepted: 3 January 2019 Published: 15 January 2019}

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\begin{abstract}
        


Investigations on Nylon-6 polymerization reactors such as VK (Vereinfacht Kontinuierliches Rohr) tube (vertical column) reactors have been explored to a very limited extent owing to their complex internal structure, which leads to uncertain velocity profile of the reactants inside the reactor. However, usually a flat velocity profile (plug flow profile) of reaction mixture is assumed inside the reactor leading to large errors in prediction of final properties of the product. In the current study, a modified simulation model has been developed for the Nylon-6 polymerization using mono acid stabilizers (acetic acid) in an industrial VK tube reactor. The variation of the velocity profile, which may result in axial changes of concentrations of different reactants along the length of the reactor has been accounted for by considering a small value non-linear time dependent decaying function. The model fetched significantly better results than the earlier model with plug flow profile of the reactants. NonLinearMathematicalModellingofNylon6PolymerizationReactorsStrictly as per the compliance and regulations of:

\end{abstract}


\keywords{Nylon-6 polymerization, VK tube reactors, modelling and simulation, optimization.}

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\let\tabcellsep& 	 	 		 
\section[{Introduction}]{Introduction}\par
K (Vereinfacht Kontinuierliches Rohr) column reactor is one among the most well known reactors used for the hydrolytic polymerization of ?-caprolactam. Vertical tube reactors are designed using the internal gratings to achieve the equal heating and close uniform flow (i.e. flat velocity profile). In literature \hyperref[b24]{24} , raw materials, viz., ?-caprolactam, water and acetic acid are passed freely at the reactor entrance.\par
The velocity profile of the feed seems to be one more important parameter in controlling the rate of reactions in the model in addition to the pressure, temperature, and water concentration. Work on modelling and simulation of such reactors is scanty due to their complicated reactor configuration and lack of industrial data.\par
Many models were developed to study of the significance of H 2 O elimination in the polymer mixture in a tubular reactor by different authors \hyperref[b4]{5,}\hyperref[b10]{11,}\hyperref[b16]{[16]}\hyperref[b17]{[17]}\hyperref[b18]{[18]}\hyperref[b19]{[19]} . Recently, we reported \hyperref[b24]{24} the result of hydrostatic pressure within the reacting system using a plug flow velocity profile. The model predicted M n , NH 2 and COOH concentrations with higher accuracy compared to the other models reported in the literature. However, the non-linear time dependent effects of induced velocity profile due to the internal structure of the reactor could not be incorporated in this model. The industrial VK tube reactors used for nylon-6 polymerization are very complex and it is extremely difficult to include actual velocity profiles of the reactants inside the reactors.\par
In the current study, a simulation model has been developed for Nylon-6 polymerization with mono acid stabilizers in a VK tube reactor. The axial changes of concentrations of various reaction components down the length of the reactor have been simulated by considering the non-linear time dependent decaying factor for each reactant species. 
\section[{II.}]{II.} 
\section[{Model of Vk Tube Reactor With Correction for Velocity Profile}]{Model of Vk Tube Reactor With Correction for Velocity Profile}\par
The VK tube reactor that was modelled has five independent portions . The temperatures in the reaction mixture were measured by the thermocouple wells at six places.\par
Figure  {\ref 1A} represents the schematic representation of the VK tube reactor. Input consisting of molten caprolactam, mono functional acid and water was passed to the entrance of the VK tube under a nitrogen cover. A water seal was kept above the reactor to allow the purged nitrogen to escape. The heat exchanger 1 (HE-1) heated the reaction mixture in zone-1 as it passes through HE-1. The melt in zone-3 upper and lower was heated by the heating jackets and the exothermic reactions. The temperature of the polymer mixture slowly increased from the top layer to the centre portion until the temperature reached the maximum point. Another heat exchanger was used to decrease and stabilize the temperature towards the bottom of the reactor. The end products such as unreacted caprolactam and the polymer were collected from the bottom of reactor.\par
In our earlier work \hyperref[b24]{24} , feed to the VK tube reactors were considered to be flowing normally (that is with no velocity profile), which might not be realistic. V However, in the present work, the flow of reactants was assumed to have a mild and possibly an irregular velocity profile caused by the shear forces exerted in the reactor walls and gratings in the upward direction and the gravitational force in the downward direction. Therefore, our earlier model \hyperref[b24]{24} was modified to incorporate non-linear time dependent effect on the reaction mixture (monomer (m), W, P, and acetic acid (P 1x )). Considering the geometry of the reactor to be symmetrical, an empirical non-linear time dependent exponential quantity given by equation ( 1) has been included in the mass balance equations of the reaction components. These modified mass balance equations which are of first order non-linear ordinary differential equations type (Table \hyperref[tab_1]{2}) have been solved by using the fourth order Runge-Kutta numerical method for the concentrations of all the species in the reaction scheme.\par
Quantity of reaction species changed due to the faster or slower velocity of the species 'i' compared to the average velocity profile of the reaction mixture is given as =? i exp (? i t) moles/kg/h. 
\section[{a) Determination of water concentration}]{a) Determination of water concentration}\par
As mentioned in our previous work \hyperref[b24]{24} the following vapour liquid equilibrium equation which is selected after testing several vapor liquid equilibrium equations \hyperref[b10]{11,}\hyperref[b20]{20,}\hyperref[b21]{21} was used in the model to find the water quantity in the polymer mass in the vaporizing zone of the reactor.\par
[W]=10 -2 P T exp (8220/T-24.36)\par
The given relation was used to determine the amount of water that was retained in the reaction mixture at each point in the reactor in the vaporizing zone. For the modified model with pressure effect and the nonlinear time dependent adjustment of the concentration, the above equation was modified as given below.\par
[W]=10 -2 P T exp (8220/T -24. 24) +? 4 exp (? 2 t)\par
Where P T is the total pressure in pascals given by P T = P A + 9.81 (? W H W + ? M H P ) (4)\par
Figure  {\ref 1}: VK Tube Reactor\par
Where P A is the atmospheric pressure given by 1.013x10 \hyperref[b4]{5} Pa. The original atmospheric pressure (P A ) at the industrial place was 2\% lower than P A and hence the change from 24.36 to 24.24 in Equation (2). ? 4 and ? 2 are the factors to be adjusted to fit the industrial results and to get the quantity of water consumed/produced in value of [W] was calculated using Equation (3) at all the points in the top portion of the tube. In the non vaporizing zone, i.e, from the LWC point downward, the total water in the reaction mass and in the product was where ? i and ? i are the parameters to be fitted for each species 'i' so that the output polymer properties match well with the industrial data. moles/kg/h, representing the non-linear time dependent influence of the velocity on the water concentration. The assumed to be a constant. Therefore the value of [W] is calculated using a mass balance equation.   The reaction model of nylon-6 polymerization has been presented in Table \hyperref[tab_0]{1}. The previous model rate equations \hyperref[b21]{21} for different species were modified to incorporate the non-linear time dependent factor for the various species and are presented in Table \hyperref[tab_1]{2}. The major reactions included in the reaction model in nylon-6 polymerization were ring opening, poly-condensation and poly-addition. The production of cyclic oligomers were accounted by the ring opening and poly-addition reactions of cyclic dimer (C 2 ) in the process. Higher order cyclic oligomers were neglected to make the analysis easy. The kinetic scheme includes the reactions involving a mono functional acid such as acetic acid and the reactions were catalysed by the carboxyl end groups of the mono functional acid. The rate constants for all the reactions were estimated by the values of kinetic data determined from experimental work \hyperref[b1]{2,}\hyperref[b2]{3,}\hyperref[b4]{5} .) t exp( + p) - ( k + m k - p k + mw -k 1 1 0 ' 3 0 3 ' 1 1 ? ? ? ? = dt dm ) t exp( + p k + c p k - ) p - ( w k + p k - p k + m p k - p) - ( mw k 2 + p 2k - p k - w m k 1 2 ' 5 2 5 1x 0x ' 2 0x 2 ' 3 3 0 ' 2 0 2 ' 1 1 ? ? ? ? ? = ? ? ? ? ? ? dt dp p k - c w k + ) - ( w k + k - ) - ( w k + k - p k - w m k ' 4 2 4 0x 1x ' 2 0x 0 2 0 1 ' 2 2 0 2 ' 1 1 0 ? ? ? ? ? ? ? ? = dt d ? ? ? ? ? = p k 2 - c w k 2 + 2p) - ( k 2 - c k 2 + )/2 - ( w k - k - p) - ( k - m k + p k - w m k ' 4 2 4 0 ' 5 0 2 5 2x 1x ' 2 1 0x 2 0 ' 3 0 3 ' 1 1 1 ? ? ? ? ? ? ? ? ? dt d ? ? ? ? ? ? ? = p k 4 - c w k 4 + p) + - ( k 4 + ) + ( c k 4 + )/6 + 3 - (2 w k + k - p) + 2 - ( k + ) 2 + ( m k + )/3 - ( w k + k 2 + p k - w m k ' 4 2 4 1 0 ' 5 0 1 2 5 1x 2x 3x ' 2 0x 2 2 1 0 ' 3 1 0 3 3 1 ' 2 2 1 2 ' 1 1 2 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? dt d 2 = dt dc -k 4 c 2 w -k 4 ? p -k 5 c 2 ? 0 + k 5 ? (? 0 -2p) = dt d x 1 ? k 2 ? 1 ? 0x -k 2 ? w (? 2x -? 1x )/2 = dt d x 2 ? = k 2 (2 ? 1 ? 1x + ? 2 ? 0x ) -k 2 ? w (4? 3x -3? 2x -? 1x )/6 = dt dp x 1 -k 2 p 1x ? 0 +k 2 ? w (? 0x -p 1x ) + +? 3 exp(?t )\par
The constants, ? and ?,, which are magnitudes and amplitudes of the nonlinear time dependent functions incorporated in the mass balance equations of the reactive species are expressed by the second order polynomial equations as a function of total polymerization time (T) in Table \hyperref[tab_2]{3}. Since the velocity profile of the reaction mass is expected to be highly dependent on the total reaction time (obtained from total volume of reactor/feed velocity), it is reasonable to relate the two using empirical relations given in the table. The best fit values of constants obtained for the three experimental cases are also given in the table. Even though the three data sets have different termperature profiles in the zones, the changes in values from one data set to another data set are insignificant in most cases. The equations given in the table can be used to predict the constants for an unknown set of reaction conditions.   Three sets of experimental data measured from a nylon-6 polymerization VK tube were used for the modelling and simulation of the process. At any point of time, the total pressure in the reactor was calculated by using water seal length above the reactor, the atmospheric pressure and the pressure exerted by the mass of polymer above the interested point in the reactor. The reactor design parameters were used to determine the time of the reactions and the hydrostatic pressures of the polymer mass at any point along the length of the reactor. The temperatures of the polymer mixture were measured at different poin ts in the reactor . Straight line assumption has been used to interpolate the temperature values between any two closely measured points. The temperatures and the properties of the polymer mixture have been listed in Table \hyperref[tab_2]{3} in all the zones for three sets of experimental data d) Starting parameters used in the simulation of the model\par
The modified mathematical model represented by the set of non-linear ordinary differential equations were solved simultaneously by using Runge-Kutta IV order method, with an iteration step size of 0.01h. To break the hierarchy of equations closure conditions \hyperref[b21]{21} were used. The starting parameters for all the three data  a , b, c as defined elsewhere \hyperref[b24]{24} The properties of polymer such as numberaverage molecular weight (M n ), [COOH], [NH 2 ], polydispersity index (excluding the effect of caprolactam and cyclic dimers) and the quantity of extractables have been determined. 
\section[{III. Data from Industrial Observations}]{III. Data from Industrial Observations}\par
The change in pressure down the length of the reactor and the temperature of the polymer mixture in various zones were measured by finding the average of 10-12 values, found after every two hours and are presented in Table \hyperref[tab_3]{4}. The resident time of polymer mass in each zone was measured by using the mechanical dimensions and the internal structure of the tube. The polymer properties were measured from the polymer samples collected at the bottom of the reactor in the raw form before the monomer was extracted at every two hours. The relative viscosity was calculated by the methods given in literature \hyperref[b22]{22} and the average values were listed in Table \hyperref[tab_2]{3}. The following equations were used to find M n , relative viscosity (RV C ) and extractables wt\% (E) from the experimental raw chips and were presented in Table \hyperref[tab_2]{3}.Intrinsic viscosity 
\section[{Global}]{Global}[?] (dL/g)=\{[1 + 4k H (RV C -1)] 1/2 -1\}/2K H (1 -0.01E) (6)\par
Relative viscosity (extracted chips)RV 1 = 1 + [?] + K H [?] 2 (7)\par
Number-average molecular weight M n (g/mol)=113.16F(RV 1 -1) (8)   where k H is the Huggins constant (0.286 for nylon-6 polymer ) and F is an empirical relation constant \hyperref[b23]{23} determined for the industrial reactor (95.7). The refractive index was measured by an immersion refractometer. The percent of extractables can be measured by refluxing 15.0g of raw chips for 8 hours in 80ml of distilled water. The refractive indices calibration table was used for various concentrations of caprolactam to obtain the percentage of extractables.\par
IV. 
\section[{Results and Discussion}]{Results and Discussion}\par
Total pressure and temperature along the length of the reactor were calculated using the procedure explained elsewhere \hyperref[b24]{24} . The pressure of the polymer melt varies from 1.023x10 5 Pa to 2.081x10 \hyperref[b4]{5} Pa down the length of the reactor. Moreover, the viscous polymer mixture flowing down the VK tube decreases the pressure. However these changes are very small to be neglected. The temperature at any point was measured by linearly interpolating the two measured points. The temperature and pressure profiles along the length of the system were plotted in Figure \hyperref[fig_3]{2} for data 3. The volumes of the various portions and the mass flow rate into the VK tube were used to calculate the reaction times. The temperature and pressure values were shown with respect to the estimated polymerization time at all points from the top to the bottom of the reactor.    The conversion of caprolactam plotted versus polymerization time (Figure  {\ref 4}) revealed increased faster conversion in the first 3-8 hours and attained equilibrium thereafter. The rates of the conversion in the present model are slower than in the previous model because of the slightly controlled vaporization of water in the top zone of the reactor. The present model determined a monomer (m) concentration of approximately 8.2 \% when polymerization was completed. The result has been compared with the experimental extractables concentration of 9.66\% in Table \hyperref[tab_2]{3}. The difference is about 14\%. For the other two data sets also similar results were obtained. The estimation was not so good because the experimental extractables contained around 15-20 wt\% of cyclic and linear oligomers along with the remaining monomer (m). d) Similar results were obtained in the earlier model \hyperref[b24]{24} as well.\par
This is because both the models are not capable of calculating the concentrations of higher oligomers and hence a better comparison could not be made. Figures  {\ref 5-7}  C groups and number average molecular weight for old model \hyperref[b24]{24} and present model. The comparison has also been presented in Table \hyperref[tab_2]{3}. In the present model, the end group concentrations increase in the middle zone and in the bottom zone of the reactor better than the old model because of more water retained due to the controlled vaporization by the induced velocity.However in the bottom zone M n values decrease in the present model because of the increased quantity of water due to the controlled vaporization in the top and the middle zones. This happens because of the assumed non plug velocity profile where water moves down the reactor at a faster rate than when there is flat velocity profile for the flow of the reactants. The error percentages reduced considerably for the carboxylic end groups and the M n values. The predicted values for the present model incorporating the non-linear time dependent effect for the various species namely, caprolactam, polymer, water and acetic acid gives the better comparison for carboxylic end groups and M n values with the experimental values. The errors in amino end groups were also improved considerably in some cases. The improved results are due to adjustment of concentrations of various species induced due to nonflat velocity profile, which controls the reaction rates of the polymerization process in the top zone and in the middle zone thereby maintaining the required quantity of water concentration for better results. This helps to control the end polymer properties and the M n value of the end polymer.\par
In Figure 8, the influence of head pressure,that is the pressure caused by the water seal at the top of the reactor can also be observed to affect the M n value of polymer. For instance, when the water seal length is increased from 0.1 -5m, rates of reactions improve significantly because of the retention of huge quantity of water inside the reactor. However M n decreases by about 10.7 \% (Figure 8) because of the higher quantity of water at equilibrium. Similar results were obtained in the old model \hyperref[b24]{24} . As discussed in the previous section, down pressure in the reactor was observed to be a controlling parameter. This is observed from Figures 9-12 as the discrepancies between the experimental values and the predicted model values were significant. Therefore, the reaction rates and the end product values at the bottom of the reactor can be controlled using the pressure in the reactor.  VK tube reactor's performance can be optimized by another important parameter, the polymer melt temperature profile down the reactor. To get the proper initiation of the reactions water should be retained at the beginning by slowly increasing the temperature to the highest temperature point. Thereafter, water should be allowed to evaporate to the required quantity at the maximum temperature and then reduced to the expected equilibrium value. The properties of the polymer at the end of the reaction are determined by the water content at the highest temperature point. If the highest temperature point is increased there will be a decrease in [W] in the nonvaporizing zone. This will increase the average molecular weight as shown in Figure 13. Therefore, the temperature at the maximum temperature position (T h ) affects the final polymer properties. When the maximum temperature (T h ) has been changed nominally by a few degrees (10K), M n changes by about 21.6\%. Similar behaviour was observed in the previous model \hyperref[b24]{24}    The predicted net reaction values and the equilibrium values of the three major reactions have been depicted in Figures  {\ref 15 and 16}. The pattern and the performance are similar to those of earlier model \hyperref[b24]{24} V. 
\section[{Conclusions}]{Conclusions}\par
Following conclusions could be drawn from the present work:\par
? The Nylon-6 polymerization was simulated in an industrial VK tube reactor using a modified nonlinear model incorporating a time dependent modification function, T. ? Non-linear time dependent decay functions were introduced in mass balance equations for calculating the concentrations of various species at any position in the reactor to account for the change in concentration of the species due to non-flat velocity profile. This reduces the errors compared to models that use a strict plug flow model for a VK Tube. Since the internal design of the VK Tube is not ideally suited to achieve a flat velocity profile, the incorporated functions tries to accommodate small differences from the ideal condition of plug flow. ? A modified empirical VLE relation was used in the vaporizing zone to predict the water profile inside the reactor incorporating the non-linear time dependent functions as mentioned above. ? A lowest water content (LWC) point was noted, which defined the end of the vaporizing zone, due to the changing temperature and pressure down the axis of the reactor. The location and [H 2 O] at the LWC point were observed to be very important in determining the end product results. ? The non-flat velocity profiles of the various reactants in the VK tube reactor controlled the reaction rates, controlled the vaporization of water to the required Quantity in the vaporizing zone and in the nonvaporizing zone that favoured simulation of polymer properties with exceptional accuracy. ? The calculated values of M n for data 3 by the present model evaluates the industrial data with an error of 0.03, as compared to the previous model with an error of 1.9\%. ? Amino end group concentrations were predicted by the present model for data 3 with an error of 0.8\% as against the prediction error of 2.7\% in the earlier model. ? Carboxyl end groups predicted by the present model for data 3 match with the industrial data with an error of 4\% over the error of 7.2\% in the previous model. ? The present model was found to be in conformity with the experimental values, superior over the earlier model \hyperref[b24]{24} , where the reactants were assumed to flow in a perfect flat velocity profile.\par
? Hence it can be concluded that the present nonlinear explicit time dependent simulation model is useful in estimating the nylon-6 end group results in a VK tube reactor accurately, in improving the system operation and provides insights for the improvement of such reactor designs.\par
Literature Cited\begin{figure}[htbp]
\noindent\textbf{}\includegraphics[]{image-2.png}
\caption{\label{fig_0}}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{}\includegraphics[]{image-3.png}
\caption{\label{fig_1}}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{}\includegraphics[]{image-4.png}
\caption{\label{fig_2}}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{2}\includegraphics[]{image-5.png}
\caption{\label{fig_3}Figure 2 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{0}\includegraphics[]{image-6.png}
\caption{\label{fig_4}[m 0}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{}\includegraphics[]{image-7.png}
\caption{\label{fig_5}}\end{figure}
          \begin{figure}[htbp]
\noindent\textbf{1} \par 
\begin{longtable}{P{0.7999999999999999\textwidth}P{0.049999999999999996\textwidth}}
6 Polymerization\tabcellsep -\end{longtable} \par
 
\caption{\label{tab_0}Table 1 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{2} \par 
\begin{longtable}{}
\end{longtable} \par
 
\caption{\label{tab_1}Table 2 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{3} \par 
\begin{longtable}{}
\end{longtable} \par
 
\caption{\label{tab_2}Table 3 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{4} \par 
\begin{longtable}{P{0.3267992424242424\textwidth}P{0.12556818181818183\textwidth}P{0.045075757575757575\textwidth}P{0.1271780303030303\textwidth}P{0.051515151515151514\textwidth}P{0.10303030303030303\textwidth}P{0.07083333333333333\textwidth}}
Property a\tabcellsep \multicolumn{2}{l}{Data 1}\tabcellsep \multicolumn{2}{l}{Data 2}\tabcellsep \multicolumn{2}{l}{Data 3}\\
Reaction Time(H)\tabcellsep 11.71\tabcellsep \tabcellsep \multicolumn{2}{l}{14.5}\tabcellsep \multicolumn{2}{l}{15.17}\\
Acetic Acid Fed, (Wt\%)\tabcellsep 0.07\tabcellsep \tabcellsep \multicolumn{2}{l}{0.07}\tabcellsep \multicolumn{2}{l}{0.06}\\
(Mol/Kg)\tabcellsep \multicolumn{2}{l}{0.0117}\tabcellsep \multicolumn{2}{l}{0.0117}\tabcellsep \multicolumn{2}{l}{0.0100}\\
\tabcellsep Melt b\tabcellsep Dow c\tabcellsep Melt b\tabcellsep Dow c\tabcellsep Melt b\tabcellsep Dow c\\
\tabcellsep Temp\tabcellsep Temp\tabcellsep Temp\tabcellsep Temp\tabcellsep \multicolumn{2}{l}{Temp Temp}\\
\tabcellsep (K)\tabcellsep (K)\tabcellsep (K)\tabcellsep (K)\tabcellsep (K)\tabcellsep (K)\\
T 1 (Zone I)\tabcellsep 517\tabcellsep -\tabcellsep 518\tabcellsep -\tabcellsep 521\tabcellsep -\\
T 2 (Zone II)\tabcellsep 521\tabcellsep 546\tabcellsep 520.5\tabcellsep 538\tabcellsep \tabcellsep 521 550\\
T 3 (Zone III Upper)\tabcellsep 524\tabcellsep 555\tabcellsep 522\tabcellsep 549\tabcellsep \tabcellsep 527 550\\
T 4 (Zone III Lower)\tabcellsep 548\tabcellsep 555\tabcellsep 546.5\tabcellsep 549\tabcellsep \tabcellsep 547 550\\
T 5 (Zone IV)\tabcellsep 531.5\tabcellsep 497\tabcellsep 529\tabcellsep 501.5\tabcellsep \tabcellsep 531 504\\
T 6 (Zone V)\tabcellsep 526.5\tabcellsep 496\tabcellsep 523.5\tabcellsep 501.5\tabcellsep \tabcellsep 527 505\\
Extractables(\%)\tabcellsep 10.56\tabcellsep \tabcellsep \multicolumn{2}{l}{9.62}\tabcellsep \multicolumn{2}{l}{9.66}\\
M n (G/Mol)\tabcellsep \multicolumn{2}{l}{17084}\tabcellsep \multicolumn{2}{l}{16.484}\tabcellsep \multicolumn{2}{l}{16.384}\\
{}[COOH](Mol/Kg)\tabcellsep \multicolumn{2}{l}{0.0517}\tabcellsep \multicolumn{2}{l}{0.0548}\tabcellsep \multicolumn{2}{l}{0.0590}\\
{}[NH 2 ](Mol/Kg)\tabcellsep \multicolumn{2}{l}{0.0460}\tabcellsep \multicolumn{2}{l}{0.0452}\tabcellsep \multicolumn{2}{l}{0.0480}\end{longtable} \par
 
\caption{\label{tab_3}Table 4 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{5} \par 
\begin{longtable}{P{0.85\textwidth}}
Sets\end{longtable} \par
 
\caption{\label{tab_4}Table 5 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{5} \par 
\begin{longtable}{P{0.1335304054054054\textwidth}P{0.5240709459459459\textwidth}P{0.08327702702702702\textwidth}P{0.10768581081081081\textwidth}P{0.0014358108108108109\textwidth}}
\tabcellsep \multicolumn{4}{l}{Non-Linear Mathematical Modelling of Nylon-6 Polymerization Reactors}\\
Year 2019\tabcellsep \tabcellsep \tabcellsep \tabcellsep \\
48\tabcellsep \tabcellsep \tabcellsep \tabcellsep \\
Volume XIX Issue I Version I\tabcellsep Polydispersity index\tabcellsep 1 1.5 2 2.5\tabcellsep B\tabcellsep A\\
D D D D ) C\tabcellsep \tabcellsep 0.5\tabcellsep \tabcellsep \\
(\tabcellsep \tabcellsep \tabcellsep \tabcellsep \\
Journal of Researches in Engineering\tabcellsep \multicolumn{4}{l}{Time(H) 0.0 1.0 2.0 3.0 2 4 Figure 14: Predicted Polydispersity Index Values Comparison For Data 3: (A) With The Influence Of Pressure M w /M n 1.0003 1.4014 1.5364 1.7220 0 0 6 8 10 12 14 16 Variation In The Vk Tube And (B) At Normal Atmospheric Pressure}\\
Global\tabcellsep \tabcellsep 4.0 5.0 6.0\tabcellsep 1.9035 2.0215 2.0782\tabcellsep \\
\tabcellsep \tabcellsep 7.0\tabcellsep 2.0907\tabcellsep \\
\tabcellsep \tabcellsep 8.0\tabcellsep 2.0851\tabcellsep \\
\tabcellsep \tabcellsep 9.0\tabcellsep 2.0735\tabcellsep \\
\tabcellsep \tabcellsep 10.0\tabcellsep 2.0615\tabcellsep \\
\tabcellsep \tabcellsep 11.0\tabcellsep 2.0696\tabcellsep \\
\tabcellsep \tabcellsep 12.0\tabcellsep 2.1002\tabcellsep \\
\tabcellsep \tabcellsep 13.0\tabcellsep 2.1130\tabcellsep \\
\tabcellsep \tabcellsep 14.0\tabcellsep 2.1192\tabcellsep \\
\tabcellsep \tabcellsep 15.0\tabcellsep 2.1232\tabcellsep \\
\tabcellsep © 2019 Global Journals\tabcellsep \tabcellsep \tabcellsep \end{longtable} \par
 
\caption{\label{tab_5}Table 5 :}\end{figure}
 			\footnote{Non-Linear Mathematical Modelling of Nylon-6 Polymerization Reactors} 			\footnote{© 2019 Global JournalsNon-Linear Mathematical Modelling of Nylon-6 Polymerization Reactors} 		 		\backmatter  			  				\begin{bibitemlist}{1}
\bibitem[Ahn ()]{b19}\label{b19} 	 		‘Effects of Diffusional Water Removal and Heat Transfer in Nylon-6 Reactors’.  		 			Y.-C Ahn 		.  	 	 		\textit{Polym. Eng. Sci}  		1997. 37  (2)  p. .  	 
\bibitem[Fukumoto ()]{b20}\label{b20} 	 		‘Equilibria between Polycapramide and Water’.  		 			O Fukumoto 		.  	 	 		\textit{I. J. Polym. Sci}  		1956. 22 p. .  	 
\bibitem[Gupta and Kothari ()]{b23}\label{b23} 	 		\textit{Manufactured Fibre Technol-ogy},  		 			V B Gupta 		,  		 			V K Kothari 		.  		1997. Chapman \&Hall: London. 363.  	 
\bibitem[Jacobs and Schweigman]{b10}\label{b10} 	 		\textit{Mathematical Model for the Polymerization of Caprolactam to Nylon-6},  		 			H Jacobs 		,  		 			C Schweigman 		.  		 	 
\bibitem[Gupta and Gandhi ()]{b16}\label{b16} 	 		‘Modeling of Backmixing in Continuous Polymerization of Caprolactam in VK Column Reac-tors’.  		 			A Gupta 		,  		 			K S Gandhi 		.  	 	 		\textit{Ind. Eng. Chem. Prod. Res. Dev}  		1985. 24 p. .  	 
\bibitem[Kumar and Gupta ()]{b7}\label{b7} 	 		‘Modeling of Higher Cyclic Oligomer Formation in Nylon-Polymerization’.  		 			V S Kumar 		,  		 			S K Gupta 		.  	 	 		\textit{Ind. Eng. Chem. Res}  		1997. 36 p. .  	 
\bibitem[Gupta ()]{b17}\label{b17} 	 		‘Modeling of VK Column Reactors for Manufac-turing of Nylon-6’.  		 			A Gupta 		.  	 	 		\textit{Ind. J. Fibre Text. Res}  		1995. 20 p. .  	 
\bibitem[Ramesh and Gupta ()]{b15}\label{b15} 	 		‘Modelling of an Industrial Autothermal Nylon-6 Flow Reactor’.  		 			G M Ramesh 		,  		 			S K Gupta 		.  	 	 		\textit{Polymer}  		1993. 34  (8)  p. .  	 
\bibitem[Gupta et al. ()]{b6}\label{b6} 	 		‘Molecular Weight Distribution for Reversible Nylon-6 Polymerizations in Batch Reactors’.  		 			S K Gupta 		,  		 			A Kumar 		,  		 			P Tandon 		,  		 			C D Naik 		.  	 	 		\textit{Polymer}  		1981. 22 p. .  	 
\bibitem[Sareen and Gupta ()]{b8}\label{b8} 	 		‘Multiobjective Optimization of an Industrial Semibatch Nylon-6 Reactor’.  		 			R Sareen 		,  		 			S K Gupta 		.  	 	 		\textit{J. Appl. Polym. Sci}  		1995. 58 p. .  	 
\bibitem[Reimschuessel ()]{b0}\label{b0} 	 		‘Nylon-6 Chemistry and Mechanism’.  		 			H K Reimschuessel 		.  	 	 		\textit{J. Polym. Sci.: Macromolecular Reviews}  		1977. 12 p. .  	 
\bibitem[Reimschuessel and Nagasubramanian ()]{b3}\label{b3} 	 		‘On the Opti-mization of Caprolactam Polymerization’.  		 			H K Reimschuessel 		,  		 			K Nagasubramanian 		.  	 	 		\textit{Chem. Eng. Sci}  		1972. 27 p. .  	 
\bibitem[Srivastava and Gupta ()]{b13}\label{b13} 	 		‘Optimization of a Tubular Nylon-6 Reactor with Radial Gradients’.  		 			D Srivastava 		,  		 			S K Gupta 		.  	 	 		\textit{Polym. Eng. Sci}  		1991.  (8)  p. .  	 
\bibitem[Proc. 5th Eur./2nd Intl ()]{b11}\label{b11} 	 	 	 		\textit{Proc. 5th Eur./2nd Intl},  				 (5th Eur./2nd Intl)  		1972. 7 p. .  	 
\bibitem[Vaidya]{b22}\label{b22} 	 		\textit{Production of Synthetic Fibres},  		 			A A Vaidya 		.  		 	 
\bibitem[Gupta and Kumar ()]{b21}\label{b21} 	 		\textit{Reaction Engineering of Step Growth Polymerization},  		 			S K Gupta 		,  		 			A Kumar 		.  		1987. New York: Plenum Press.  	 
\bibitem[Gupta and Tjahjadi ()]{b14}\label{b14} 	 		‘Simulation of an Industrial Nylon-6 Tubular Reactor’.  		 			S K Gupta 		,  		 			M Tjahjadi 		.  	 	 		\textit{J. Appl. Polym. Sci}  		1987. 33 p. .  	 
\bibitem[Wajge et al. ()]{b9}\label{b9} 	 		‘Simulation of an Industrial Semibatch Nylon-6 Reactor:Optimal Parameter Esti-mation’.  		 			R M Wajge 		,  		 			S S Rao 		,  		 			S K Gupta 		.  	 	 		\textit{Polymer}  		1994. 35  (17)  p. .  	 
\bibitem[Ashwini et al. ()]{b24}\label{b24} 	 		‘Simulation of Hydrolytic Polymerization of Nylon-6 in Industrial Reactors: Part I. Mono-Acid-Stabilized Systems in VK Tube Reactors’.  		 			K Ashwini 		,  		 			Kotilingam Agrawal 		,  		 			Tomonobu Devika 		,  		 			Manabe 		.  	 	 		\textit{Ind. Eng. Chem. Res}  		2001. 40 p. .  	 
\bibitem[Gupta et al. ()]{b5}\label{b5} 	 		‘Simulation of Molecular Weight Distribution and Cyclic Oligomer Formation in the Polymerization of Nylon-6’.  		 			S K Gupta 		,  		 			C D Naik 		,  		 			P Tandon 		,  		 			A Kumar 		.  	 	 		\textit{J. Appl. Polym. Sci}  		1981. 26 p. .  	 
\bibitem[Pal and Gupta ()]{b12}\label{b12} 	 		‘Simulation of Non-Vaporizing Tubular Nylon-6 Reactors with Radial Gradients: Finite-Difference Computations’.  		 			D Pal 		,  		 			S K Gupta 		.  	 	 		\textit{Polymer}  		1989. 1918-1930. 30.  	 
\bibitem[Gupta et al. ()]{b18}\label{b18} 	 		‘Simulation of Three-Stage Nylon-6 Reactors with Intermediate Mass Transfer at Finite Rates’.  		 			S K Gupta 		,  		 			A Kumar 		,  		 			K K Agrawal 		.  	 	 		\textit{J. Appl. Polym. Sci}  		1982. 27 p. .  	 
\bibitem[Tai et al. ()]{b2}\label{b2} 	 		‘The Kinetics of Hydrolytic Polymerization of ?-Caprolactam. II. Determination of the Kinetic and Thermodynamic Constants by Least Squares Curve Fitting’.  		 			K Tai 		,  		 			H Teranishi 		,  		 			Y Arai 		,  		 			T Tagawa 		.  	 	 		\textit{J. Appl. Polym. Sci}  		1980. 25 p. .  	 
\bibitem[Tai et al. ()]{b1}\label{b1} 	 		‘The Kinetics of Hydrolytic Polymerization of is ?-Caprolactam’.  		 			K Tai 		,  		 			H Teranishi 		,  		 			Y Arai 		,  		 			T Tagawa 		.  	 	 		\textit{J. Appl. Polym.Sci}  		1979. 24 p. .  	 
\bibitem[Tai et al. ()]{b4}\label{b4} 	 		‘The Simulation of Hydrolytic Polymerization of-Caprolactam in Various Reactors’.  		 			K Tai 		,  		 			Y Arai 		,  		 			T Tagawa 		.  	 	 		\textit{J. Appl. Polym. Sci}  		1982. 27 p. .  	 
\end{bibitemlist}
 			 		 	 
\end{document}
