I am starting with a mixed model that is often used to represent fluidized beds. A diagram of the model appears below.
The model contains a dense bed and a freeboard region above it.
The dense bed
The height of the dense bed is assumed known. If not measured, it may be estimated [McFarlane et al.].
Within the bed are two flow branches through two volumes. One branch for the bubble phase is modeled as a plug flow reactor. The other branch for the catalyst emulsion phase is modeled as a CSTR. In this simple model, no gas is exchanged between the two branches. The cross flow feature will be added later.
The plug flow reactor is actually modeled as a series of tanks. This will make it easy to add exchange between the two regions in a later model. With the tanks-in-series approach, a set of ordinary differential equations for the dynamic material balances of all regions is readily developed.
I have assumed 15 tanks for the plug flow model. This allows for some axial dispersion, but it is not great.
Himmelblau and Bischoff have an extensive discussion of mixed models with branching. They give the formula below for the number of parameters needed to describe a branched model.
The third parameter, besides f and alpha, is the number of tanks in the plug flow region. This has been set at 15, as mentioned. The parameter estimation scheme will not be able to vary that parameter, so if it needs to be adjusted to better match the axial dispersion of the plug model, then a trial-and-error manual approach is needed.
Himmelblau and Bischoff presented age distribution curves for a number of examples with various values of f, alpha and the number of tanks in both branches. However, no parameter estimation scheme was presented.
This region is modeled as a pure, plug flow which means it is just a delay of the mixed output from the dense bed regions. There are no parameters to estimate for this submodel.
A versatile model
This model is capable of modeling the bed as a nearly total plug flow reactor or a nearly total CSTR. The "nearly total" is due to the fact that f cannot be 0 or 1 without error. However, I have used it for 0.01 < f < 0.99 with success.
The model is also versatile in application. It may be used for any gas or liquid flow within a known confined volume, such as tanks and reactors, and waste treatment ponds. A single entrance and a single exit are required. It might still be possible to do tracer studies with multiple sources and exits, but a more complicated model would be needed.
R. C. McFarlane, R. C. Reineman, J. F. Bartere and C. Georgakis, “Dynamic Simulator for a Model IV Fluid Catalytic Cracking Unit,” Computers & Chemical Engineering, Vol. 17, 1993, pp. 275-300.
Himmelblau, D. M. and K. B. Bischoff, Process Analysis and Simulation: Deterministic Systems, pp 73-80, Wiley, 1968.