(672h) Analytical Solution for Temperature for Free Radical Polymerization Mathematical Model

Any equation for temperature has to come from energy balance equation in a given mathematical model. Solving for temperature from energy balance equation in Ordinary Differential Equations (ODE) form is always a tedious task as generally, the equation is highly non-linear in terms of temperature. This is valid even for the energy balance equation of the moments based mathematical model of free radical polymerization (FRP). Some researchers have tried to obtain analytical solution for this by applying several assumptions, but the results were not very promising [1]. The solution did not match well with numerical results and also did not give physically correct results. Garg et al. 2014 [2] have shown it is impossible to integrate this equation in expanded form (explicit form). This is because it is highly non-linear equation for temperature in that form. In my current work, I have tried to approach the problem in a bit different way. I have tried to keep various terms of the energy balance equation intact without expressing them as the function of temperature. This was the approach of early researchers which failed to give acceptable and useful results. I rearranged the energy balance equation and replaced the expressions with variables which could be suitably integrated. Because of this approach, the resultant analytical solution for temperature needs to be solved along with analytical solution of other variables as obtained by Garg et al. [2]. So, first we need to solve for other variables using their analytical solution for a given time step assuming the isothermal conditions in that time step. After that, values of those variables are passed on the analytical solution of temperature. The new temperature thus calculated is the temperature at the end of the time step. This new temperature thus becomes the base temperature for next time step where whole analytical solution is solved based on it and new temperature is calculated at the end of the time step. This approach is mathematically consistent (Similar to Euler Method of Numerical Integration) and helped to retain the non-linearity arising out of volume variation as well as temperature variation.

Thus, with this result for temperature, all the eleven ODEs of Free Radical Polymerization have now been solved and a complete analytical solution has been obtained. To confirm the correctness and usefulness of the results, the solution through analytical solution of temperature and monomer conversion under non-isothermal situation has been validated against the solution obtained from ODEs models and found a good agreement (Fig.1). The two ODE models used are complete model without assumption (FRP_Full) and with quasi steady state assumption for live radical polymer chains (FRP_QSSA). This work is important given the fact that the temperature equation is a highly stiff problem. So, now Complete analytical solution of FRP can be used to simulate any desirable situation without worrying about stiff or non-stiff solvers. A simple excel file will also do the work. Thus hardware and software requirements are greatly reduced to implement the analytical solution. The usefulness of results is not limited to predicting monomer conversion alone but also to predicting number averaged chain length and polydispersity index. This has now practical applications for using it in improved process control for isothermal and non-isothermal conditions (batch and semi-batch), improved simulations for better design, optimization and understanding etc.

Figure 1: The results obtained through FRP_Full, FRP_QSSA and Analytical Solution are compared for Temperature and conversion, under non-isothermal condition.

References

1. Venkateshwaran, G., et al., Journal of Applied Polymer Science, Vol. 45, 1992.
2. Garg, D.K., et al. Macromolecules, 47(14), 2014.