Development of an RTD-Based Flowsheet Modeling Framework for the Assessment of In-Process Control Strategies

Abstract

Continuous manufacturing (CM) is an emerging technology which can improve pharmaceutical manufacturing and reduce drug product quality issues. One challenge that needs to be addressed when adopting CM technology is material traceability through the entire continuous process, which constitutes one key aspect of control strategy. Residence time distribution (RTD) plays an important role in material traceability as it characterizes the material spreading through the process. The propagation of upstream disturbances could be predictively tracked through the entire process by convolution of the disturbance and the RTD. The present study sets up the RTD-based modeling framework in a commonly used process modeling environment, gPROMS, and integrates it with existing modules and built-in tools (e.g., parameter estimation). Concentration calculations based on the convolution integral requires access to historical stream property information, which is not readily available in flowsheet modeling platforms. Thus, a novel approach is taken whereby a partial differential equation is used to propagate and store historical data as the simulation marches forward in time. Other stream properties not modeled by an RTD are determined in auxiliary modules. To illustrate the application of the framework, an integrated RTD-auxiliary model for a continuous direct compression manufacturing line was developed. An excellent agreement was found between the model predictions and experiments. The validated model was subsequently used to assess in-process control strategies for feeder and material traceability through the process. Our simulation results show that the employed modeling approach facilitates risk-based assessment of the continuous line by promoting our understanding on the process.

APPENDIX

To show that by using the advection problem described it is possible to propagate historical state information to the current simulation values. The method of characteristics will be used. The first step to the method of characteristics is to propose that all variables, dependent and independent, are functions of a single characteristic variable s, i.e. f(t, a) = g(s). The problem is then transformed to a system of ordinary differential equations by writing the total derivative of g using the chain rule of differentiation as shown below.

$$ \frac{dg}{ds}=\frac{\partial f}{\partial t}\frac{dt}{ds}+\frac{\partial f}{\partial a}\frac{da}{ds} $$

(7)

By comparing Eqs. (4) to (7), it is possible to obtain the following system of ordinary differential equations for our system:

$$ \frac{dg}{ds}=0;\frac{dt}{ds}=1;\frac{da}{ds}=1 $$

(8)

The initial conditions for this problem need to be chosen carefully. Based on the equation for g it is clear that the solution is that g remains constant with the characteristic value. That is to say that the following equation must hold:

$$ g(s)=g(0) $$

(9)

Based on the original replacement that was performed, the original variable f can now be replaced to obtain the solution.

$$ f\left(t(s),a(s)\right)=f\left(t(0),a(0)\right) $$

(10)

Comparing this result to the initial conditions that are presented in Eq. (5) give us an indication for the choice of initial conditions. Note that if the initial condition f(0, a) = 0 is used, the result is a solution, albeit a very uninteresting one physically. However, the solution that is being sought can be found by examining the boundary condition, f(t, 0) = ΔC_in(t). This would indicate that it is desirable to trace characteristics in time, where t(0) = t₀ and a(0) = 0. The result of this choice of initial conditions is that t(s) = t₀ + s and a(s) = s. Using this solution, the following characteristic curve for time is defined:

$$ t(a)={t}_0+a $$

(11)

This characteristic equation is then used to finally solve the problem fully for f by substituting back into Eq. (10) recognizing that this is equivalent to writing f(t, a) = f(t₀, 0) with the choice of initial conditions that was made previously. The final result is shown below:

$$ f\left(t,a\right)=\Delta {C}_{in}\left(t-a\right) $$

(12)

This result is what is needed in the convolution integral that needs to be calculated in gPROMS.