My original plan was to convert the dynamic model used for the tracer study into a dynamic model of the FCC regenerator. Instead, I have built a steady-state model for a couple of reasons. First, the ODE solvers in Mathcad can't have a matrix as the dependent variable: only vectors are allowed. That limitation would make model development difficult, but still not impossible to achieve. Second, the main purpose of the model is for optimization which doesn't require a dynamic model. A dynamic model would greatly increase the computation time required.
Why was a matrix needed?
The model consists of a number of "tanks", each with a molar outflow vector. The elements of the vectors are the molar rates of each chemical component in the gas. Thus, a matrix of the number of components by the number of tanks was needed.
For the example I used 5 components and 16 tanks. Ten tanks were used for the bubble phase, one for the emulsion phase, and 5 for the freeboard phase. Thus, 80 molar flow rate variables were needed. In the model, these variables are represented by one matrix variable.
In addition, equations for carbon on coke were needed for the emulsion phase and the 5 freeboard tanks, i.e. 6 more equations. Temperatures were solved for the dense bed (bubble phase and emulsion phase assumed at same temperature) and the 5 freeboard tanks. Finally, 5 more equations were needed to compute the dry flue gas composition and 1 equation was used to determine the amount of excess molar flow in the emulsion phase, explained below. A total of 98 equations were required.
Incorporating the tracer study results
The tracer study produced parameters that are used to compute the volumes of the various zones and the flow rates entering each zone. According to theory, once a bubble phase appears, the flow to the emulsion phase remains constant and all gas above that rate goes to the bubble phase. Accordingly, the parameters from the tracer study determine the flow to the emulsion phase and the volumes of the emulsion and bubble phases.
Tracer studies at more than one air flow rate could be used to check the assumption about maximum flow to the emulsion phase. If the assumption is shown to be incorrect, then the tracer studies can be used to develop correlations for emulsion flow rate and volumes.
Excess flow produced by reaction
The burning of coke in the emulsion phase produces additional gas flows. Since the emulsion phase volumetric flow rate is assumed constant, the excess flow is sent to the bubble phase. I assumed that this excess was partitioned equally to the tanks in the bubble phase. Note that I said "volumetric flow rate is constant." The volumetric rate determines the gas velocity which in turn determines the particle drag and fluidization behavior. Thus, changes in bed temperature and pressure can also affect the amount of excess flow.
Using matrix notation greatly simplified the problem
The 98 equations were expressed by 30 vector-matrix equations. An example is shown at the end of this post.
Simultaneous equation solver
The use of the simultaneous equation solver in today's advanced math programs (e.g. Mathcad, Maple, Matlab, Mathematica) makes programming so much easier because I don't have to devise the solution and iteration strategy. I can concentrate on writing the equations and providing good initial guesses and variable constraints.
The model has been written with air flow rate, air temperature, catalyst flow rate, and the parameters from the tracer study as variable input parameters. The first three mentioned are variables that can be controlled for a given reactor. The tracer parameters can't be directly controlled, but their affect on reactor performance might be useful in diagnosing a problem.
Example vector equation
The angle bracket superscripts indicate a column in the "n" matrix. In this example, NR-1 refers to the last tank in the bubble phase, NR is the emulsion phase, and NR+1 is the first tank in the freeboard. The freeboard coke on catalyst is a vector, Cfb, and the freeboard temperature vector is Tfb. The subscript "0" for these variables refers to the first tank in the freeboard region.
Speed for optimization
The model takes about 0.4 s to solve when examining the operating variables. If we assume that the addition of the riser cracking reactor model doubles the time, the integrated model should be fast enough for optimization.