A 2D Gas-Solid Fluidized Bed - Multiphase Reactor Modeling using PyMRM

In a gas-fluidized bed (with diameter $D = 2.0~\mathrm{m}$ and expanded bed height $L = 3.0~\mathrm{m}$ ), heterogeneously catalyzed chemical reactions take place involving the species $A$ , $B$ , and $C$ . The reaction scheme and associated kinetics are as follows:

A \overset{K_{r,1}}{\rightarrow} B \overset{K_{r,2}}{\rightarrow} C, \quad r_{A} = -K_{r,1} c_{A}, \quad r_{B} = K_{r,1} c_{A} - K_{r,2} c_{B}, \quad r_{C} = K_{r,2} c_{B}.

(1)

The catalyst material consists of Geldart A type particles with a particle size of $50~\mathrm{\mu m}$ and a particle density of $1500~\mathrm{kg~m^{-3}}$ . Additional data are given in Table 1. In the extended Kunii and Levenspiel fine-particle model, the radial gas (bubble) velocity profile is taken into account as well as the radial gas dispersion. For the radial profile of the bubble rise velocity, we assume a parabolic profile given by:

U_{0}(r) = 2 \langle U_{0} \rangle \left[ 1 - \left( \frac{r}{R} \right)^{2} \right],

(2)

where $\langle U_{0} \rangle$ is the average bubble rise velocity. This expression approximately accounts for the fact that the preferential pathway of the bubbles is located in the central part of the gas-fluidized bed. Due to the relatively high value of the average superficial gas velocity, the axial gas dispersion can be neglected. Moreover, it can be assumed that the reactor is isothermal. The bubble-to-cloud mass transfer coefficients $K_{bc}$ and the cloud-to-emulsion mass transfer coefficients $K_{ce}$ can be taken as the same for the components $A$ , $B$ , and $C$ (see Table 1).

Table 1: Additional Data

Parameter	Value
Average bubble rise velocity $\langle U_{0} \rangle$	$0.50~\mathrm{m~s^{-1}}$
Reaction rate constant $K_{r,1}$	$0.080~\mathrm{s^{-1}}$
Reaction rate constant $K_{r,2}$	$0.010~\mathrm{s^{-1}}$
Bubble-to-cloud mass transfer coefficient $K_{bc}$	$2.5~\mathrm{s^{-1}}$
Cloud-to-emulsion mass transfer coefficient $K_{ce}$	$1.5~\mathrm{s^{-1}}$
Radial gas dispersion coefficient $D_{i,r}$ (i = A, B, C)	$0.01~\mathrm{m^{2}~s^{-1}}$
Bubble phase solids holdup parameter $\gamma_{b}$	$0.005~(-)$
Cloud phase solids holdup parameter $\gamma_{c}$	$0.200~(-)$
Emulsion phase solids holdup parameter $\gamma_{e}$	$5.000~(-)$
Inlet concentration of species $A$	$5.0~\mathrm{mol~m^{-3}}$
Inlet concentration of species $B$	$0.0~\mathrm{mol~m^{-3}}$
Inlet concentration of species $C$	$0.0~\mathrm{mol~m^{-3}}$

Questions:

Formulate the steady-state model equations and the associated boundary conditions according to the extended Kunii and Levenspiel Fine Particle Model, taking into account the radial profile of bubble rise velocity and the radial gas dispersion.
Implement the formulated model equations in Python.
Compute the steady-state concentration profiles at the outlet of the gas-fluidized bed using the implemented Python model. Inspect the computed concentration profiles carefully and explain the qualitative shape of these profiles. Use 100 radial grid points and 200 axial grid points.
Compute the degree of chemical conversion of species $A$ and the selectivity towards the desired product $B$ . Carefully account for the radial gas (bubble) velocity profiles.
Consider the case where the bubble rise velocity is uniform and radial gas dispersion can be neglected (assuming uniform inlet conditions). Solve the resulting Ordinary Differential Equations (ODEs) and compute the degree of chemical conversion of species $A$ and the selectivity towards the desired product $B$ .

Note that all numerical implementations should be consistent with the provided equations and parameter values.