I've added a riser reactor model to the regenerator model. It's currently not well suited for parametric studies or optimization, but it will obtain a solution.
The riser model
I've assumed plug flow for both the gases and the catalyst with no slip between the two phases. Ten tanks in series are used to simulate this plug flow behavior.
A decay function based on coke level instead of catalyst residence time was assumed. The parameters in this function were adjusted to get results similar to the plant data cited by Dasila, et al. This function will vary with the catalyst, so further work would be needed to determine the proper function. A coke based decay function should provide a better integration of the regenerator and reactor models than a residence time function. Without a decay function based on coke, the regenerated catalyst temperature is the only influence the regenerator performance has on the reactor performance.
Reaction rates and heats of reaction were those used by Dasila.
The oil is assumed to completely evaporate in the first tank. For gas oils such as this example, that assumption may be adequate. For heavy oils, the vaporization process may need to be modeled throughout the riser. this would require more description of the oil and more thermodynamic information.
A bump in the road
I had to remove the carbon balance around the regenerator bed from the system of equations in order to obtain solutions. Without that equation, the coke on the regenerated catalyst is specified as a constant. This results in an error in the carbon balance because the coke removed in the regenerator is not rectified with the coke produced in the riser.
The error in the coke balance is a function of the guessed value of the regenerated coke level as shown below.
The intersection of the two curves gives the solution value for the coke on the regenerated catalyst.
Finding the solution using a minimization routine
The error in the carbon balance can be found using the Levenberg-Marquardt option in Mathcad's Minerr routine. However, it seems like it should take less time to find the solution. I'm going to use a bisection routine next to see if I can speed up this step.
The base case results
The results of this integrated model for the verification case found in Dasila are below. I've made several changes to the problem, so these results will not compare with the Dasila results. The following were changed:
Dasila, Prabha K, Indranil Choudhury, Deoki Saraf, Sawaran Chopra, and Ajay Dalai. “Parametric Sensitivity Studies in a Commercial FCC Unit.” Adv. in Chem. Eng. Sci. 2012, no. January (2012): 136–49.