I discovered that I had not closed the energy balance for the dense bed...the catalyst temperature from the reactor wasn't updated in the energy balance. When this oversight was corrected, I had to remove this energy balance from the system in order to converge the rest of the system.
So now there are two variables to converge outside the system...the regenerated catalyst coke level and temperature. Let's see how these variables affect the errors in the carbon balance. In the plot below, the temperatures for the two curves are 50 K apart.
It appears that the bed temperature doesn't greatly affect the coke level. Thus, my convergence program first estimates the coke level. A bisection method is used to obtain this estimate.
The program then converges the bed temperature, again using a bisection routine. Finally, the carbon balance is reconverged using the updated bed temperature. The entire process takes about 10 s. I have checked the energy balance after these steps and it still is within the convergence tolerance.
The new results
If all goes well, the next post will include some parametric studies using the integrated model.