This section describes the mathematical approach to modeling the fast pyrolysis process in fluidized-bed reactors and its computational implementation. Along with this, the chemical kinetic models adopted and the setup for each test case are described.
2.1. Multifluid Model
Fast pyrolysis processes involve the presence of multiple phases of different natures (fluid and particulate phases) and different morphological and chemical properties (sand, pelletized organic materials, gases, etc.). These phases are composed of different species that react with each other in thermally active chemical processes. In other words, fast pyrolysis is a multiphase hydrodynamic process with thermochemical reactions.
Therefore, a multifluid computational model is required, consisting of a gas phase and one or more solid phases, each containing an arbitrary number of species. In this model, the phases are treated as interpenetrating continuous media, and their balance equations undergo an averaging process to derive the local balance equations for the thermally and chemically reactive coupled multifluid system [
28,
29].
The following mass and momentum balances are presented for phase
i:
where
Here, is the volume-phase fraction for phase i, is the i-phase density field, is the i-phase velocity field, p is the shared pressure field, is the particle pressure field (non-zero only for granular phases), is the gravitational acceleration, is the drag coefficient between phases i and j, is the i-phase dynamic viscosity, and is the i-phase bulk viscosity.
Momentum exchange between phases occurs primarily through drag forces, and many suitable correlations can be found in the literature for this type of system [
30,
31,
32,
33,
34]. In this work, the Syamlal–O’Brien drag model is adopted [
31], which has been widely used for simulating the fluidization of Geldart B particles over the years. This is mainly due to its accuracy in predicting solid distributions [
35] and flexibility in its formulation, allowing the adjustment of coefficients in order to reproduce fluidization patterns under different experimental conditions. The drag coefficient is written as
where
is the Reynolds number, and
is the terminal velocity for the granular phase
i given by the following expression:
with
where
j represents the continuous phase,
and
are modeling coefficients, and
The rheology of the granular phases is modeled based on the kinetic theory of granular flow (KTGD) [
30,
36,
37] and frictional theory [
38,
39].
For a low concentration of particles, KTGF models based on the corresponding granular temperature are used. This field is computed based on an energy balance equation, which needs to be solved prior to the modeling of the solid stress tensor. This equation was developed considering perfectly spherical particles and assuming that only binary collisions may occur, based on the work of [
36,
40]:
The parameters involved are defined as [
36,
41,
42]
where
is the solid pressure, which represents normal stress contributions due to particle collisions;
is the bulk viscosity of the granular compound;
represents granular energy dissipation due to inelastic collisions;
and
are the rates of granular energy transfer between the continuous phase and the particles (see [
30]); and
is the radial distribution, which represents the dimensionless distance between particles.
For high concentrations, the frictional theory [
39,
43,
44] is adopted following the additive approach proposed by Johnson and Jackson [
39]. For these conditions, the solid pressure is modeled by introducing frictional pressure, given by
while the solid viscosity is computed following the work of Schaeffer [
38]:
This parameter represents the viscosity of the solid phase for highly packed conditions, and its definition is based on the theories of soil mechanics and plasticity. Usually,
, and for solid concentrations higher than this value, the solid pressure and solid viscosity are given by the sum of the kinetic and frictional contribution (i.e., Equations (
10) and (
13) and Equations (
11) and (
14)).
The thermal energy equation for phase
i is given by
where
,
, and
are the i-phase temperature, heat capacity, and thermal conductivity. Moreover,
represents the heat given by chemical reactions involving the i-phase.
The heat transfer coefficient between phases
i and
j can be written based on different correlations. Among the most popular for multiphase flow, the Ranz–Marshall correlation [
45] is given by
For gas-particle heat transfer, the Gunn correlation [
46] has been extensively adopted:
where
j represents the continuous phase.
Finally, the species conservation is considered, which can be written for species
m of phase
i as
where
is the mass fraction of species m in phase
i, and
is the source of species
m in phase
i given by chemical reactions.
2.2. Fast Pyrolysis Kinetics
The kinetics of the process depend on the composition of the biomass material to be pyrolyzed. There are several ways of establishing the sequence of chemical reactions and the main components to be considered in a way that is computationally affordable [
7,
47,
48,
49]. In this work, bagasse and red oak are considered as the pyrolyzing material, and the kinetics previously reported by Bradbury et al. [
50], and subsequently adapted by Di Blasi [
7], Miller and Bellan [
47], are adopted as a first approach. In this context, the virgin biomass, composed of cellulose (C), hemicellulose (H), and lignine (L), reacts until it becomes active biomass. Then, the active biomass is converted to bio-oil (TAR) and biochar (CHAR). At the same time, the TAR (considered a gas phase) can react, forming bio-gas (GAS). This whole process takes place within the reactor, where the TAR and GAS phases can leave the reactor before completing the reaction process, while CHAR, in the solid state, accumulates at the bottom. The chemical reaction sequence is described in
Figure 1.
All the reactions are assumed to be irreversible first-order, while the dependence of the specific reaction rate on temperature is expressed by the Arrhenius equation:
where
represents the rate constant of reaction
m,
is the activation energy of reaction
m, and
R is the gas constant. These coefficients for each reaction are detailed in
Table 1. The link between the reaction rate
of Equation (
18) and the reaction coefficients
follows the structure presented by Xue et al. [
12,
13].
The ratios of CHAR in relation to the products CHAR+GAS () for reaction 3 for , , and are , and , respectively.
It is worth mentioning that the product yield depends on several factors, such as the geometry, the operating conditions, the material proportions, the physical and numerical modeling approaches, and, very importantly, the kinetic model adopted. This work seeks to contrast the relative importance of these variables, including the simplified kinetics adopted.
2.3. Numerical Approach
The previous set of equations forming the multifluid model for thermal reactive and particulate flow is addressed in the framework of the Finite Volume Method (FVM). The chemical kinetics and thermal fluid dynamics equations are solved in a segregated manner by a time-splitting procedure. To solve the fluid dynamics, SIMPLE-based algorithms are adopted for pressure–velocity coupling [
52,
53], and the partial elimination algorithm is used for multiphase momentum coupling [
54].
In this work, 3 distinct phases are addressed (gas, sand, and biomass), which interact with each other by exchanging mass, momentum, and energy, each one consisting of multiple species, allowing chemical reactions between them. For the simulations, the suites OpenFOAM(R) v11 [
16] and ANSYS Fluent(R) v18.1 [
27] were used. General conclusions from the use of both are drawn later on in the Results Section, and, in general, both codes preserve the following structure:
Establish the initial conditions for each phase variable.
Compute the phase fractions based on phase-continuity equations but for the continuous phase (gas). Then, calculate the gas-phase fraction by subtracting the solid-phase fraction from unity.
Obtain the drag and heat transfer coefficients between phases based on the stored values of the variables.
Compute the granular viscosity of each phase and granular pressure.
Compute the temperature field of each phase based on each phase’s energy balance.
Compute the species fraction based on each species’ transport equation considering the reaction rates obtained from the new temperature fields.
Compute the velocity field prediction of each phase based on the momentum equation.
Construct and compute the shared pressure field based on the mass and momentum equations.
Update the velocity field of each phase based on the new pressure field and flux reconstruction. This step can be iterated with the previous one in OpenFOAM following the PISO method.
Iterate from 2 to favor the coupling between equations until a convergence criterion is reached and proceed with the next time step.
In the following tests, the results shown will be those given by OpenFOAM by default unless otherwise specified.
2.4. Test Cases
The described model was used to simulate lab-scale and pilot plant bubbling fluidized-bed reactors for the fast pyrolysis of biomass. Two experimental setups were considered for the simulations: Setup 1 [
13], a lab-scale experiment, and Setup 2 [
55], a pilot plant experiment. The scheme of the arrangement is shown in
Figure 2, and the dimensions are detailed in
Table 2. For both cases, three phases are considered: sand (solid), biomass (solid), and gas. The biomass (with a density of
kg/m
3 and a mean particle diameter of
μm) is fed at 300 K into the reactor from the side at a fixed mass flow rate (
), with differences in the initial composition (using red oak for Setup 1 and bagasse for Setup 2). At the same time, a bed of silica sand particles (with a density of
kg/m
3 and a mean particle diameter of
μm) at an initial packing of 0.58 is fluidized by nitrogen injection at 773 K from the bottom of the bed at
superficial velocity. The heated walls of the reactor (with a height
from the bottom) are kept at a fixed temperature of 773 K to allow the fast pyrolysis reaction. The rest of the material properties are detailed in
Table 3.
Regarding the numerical setup, the outlet conditions are specified at the top of the reactor; Johnson–Jackson partial slip boundary conditions for the solids [
39] are adopted at the walls with a null gradient for the temperature, except for part of the heated wall (up to
). The inlet conditions for the gas (pure nitrogen) are imposed at the bottom, and the inlet conditions for biomass and nitrogen are dense packing conditions through the lateral feed. Most simulations were run until 100 s to allow statistical stationary conditions for averaging, and quasi-hexahedral cells were adopted. A mesh sensitivity analysis was performed to define the grid refinement in the following section.