moving_bed.rst

Moving Bed Reactor

The IDAES Moving Bed Reactor (MBR) model represents a unit operation where two material streams – a solid phase and a gas phase – pass through a linear reactor vessel while undergoing chemical reaction(s). The two streams have opposite flow directions (counter-flow). The MBR mathematical model is a 1-D rigorous first-principles model consisting of a set of differential equations obtained by applying the mass, energy (for each phase) and momentum balance equations.

Assumptions:

• The radial concentration and temperature gradients are assumed to be negligible.
• The reactor is assumed to be adiabatic.
• The solid phase is assumed to be moving at a constant velocity determined by the solids feed rate to the reactor.

Requirements:

• Property package contains temperature and pressure variables.
• Property package contains minimum fluidization velocity.

The MBR model is based on:

1. Ostace, A. Lee, C.O. Okoli, A.P. Burgard, D.C. Miller, D. Bhattacharyya, Mathematical modeling of a moving-bed reactor for chemical looping combustion of methane, in: M.R. Eden, M. Ierapetritou, G.P. Towler (Eds.),13th Int. Symp. Process Syst. Eng. (PSE 2018), Computer-Aided Chemical Engineering 2018, pp. 325–330 , San Diego, CA.

Degrees of Freedom

MBRs generally have at least 2 (or more) degrees of freedom, consisting of design and operating variables. The design variables of reactor length and diameter are typically the minimum variables to be fixed.

Model Structure

The core MBR unit model consists of two ControlVolume1DBlock Blocks (named gas_phase and solid_phase), each with one Inlet Port (named gas_inlet and solid_inlet) and one Outlet Port (named gas_outlet and solid_outlet).

Constraints

In the following, the subscripts g and s refer to the gas and solid phases, respectively. In addition to the constraints written by the control_volume Block, MBR units write the following Constraints:

Geometry Constraints

Area of the reactor bed:

$$A_{bed} = \pi \left( \frac{ D_{bed} }{ 2 } \right)^2$$

Area of the gas domain:

A_g, t, x = εA_bed

Area of the solid domain:

A_s, t, x = (1 − ε)A_bed

Length of the gas domain:

L_g = L_bed

Length of the solid domain:

L_s = L_bed

Hydrodynamic Constraints

Superficial velocity of the gas:

$$u_{g,t,x} = \frac{ F_{mol,g,t,x} }{ A_{bed} \rho_{mol,g,t,x} }$$

Superficial velocity of the solids:

$$u_{s,t} = \frac{ F_{mass,s,t,inlet} }{ A_{bed} \rho_{mass,s,t,inlet} }$$

Pressure drop:

The constraints written by the MBR model to compute the pressure drop (if has_pressure_change is 'True') in the reactor depend upon the construction arguments chosen:

If pressure_drop_type is `simple_correlation`:

$$- \frac{ dP_{g,t,x} }{ dx } = 0.2 \left( \rho_{mass,s,t,x} - \rho_{mass,g,t,x} \right) u_{g,t,x}$$

If pressure_drop_type is `ergun_correlation`:

$$- \frac{ dP_{g,t,x} }{ dx } = \frac{ 150 \mu_{g,t,x} {\left( 1 - \varepsilon \right)}^{2} \left( u_{g,t,x} + u_{s,t} \right) }{ \varepsilon^{3} d_{p}^2 } + \frac{ 1.75 \left( 1 - \varepsilon \right) \rho_{mass,g,t,x} \left( u_{g,t,x} + u_{s,t} \right)^{2} }{ \varepsilon^{3} d_{p} }$$

Reaction Constraints

Gas phase reaction extent:

If gas_phase_config.reaction_package is not 'None':

ξ_{g, t, x, r} = r_{g, t, x, r}A_g, t, x

Solid phase reaction extent:

If solid_phase_config.reaction_package is not 'None':

ξ_{s, t, x, r} = r_{s, t, x, r}A_s, t, x

Gas phase heterogeneous rate generation/consumption:

$$M_{g,t,x,p,j} = A_{s,t,x} \sum_{r}^{reactions} {\nu_{s,j,r} r_{s,t,x,r}}$$

Dimensionless numbers, mass and heat transfer coefficients

Particle Reynolds number:

$$Re_{p,t,x} = \frac{ u_{g,t,x} \rho_{mass,g,t,x} }{ \mu_{g,t,x} d_{p}}$$

Prandtl number:

$$Pr_{t,x} = \frac{ c_{p,t,x} \mu_{g,t,x} }{ k_{g,t,x} }$$

Particle Nusselt number:

Nu_p, t, x = 2 + 1.1Pr_t, x^1/3|Re_p, t, x|^0.6

Particle to fluid heat transfer coefficient

h_gs, t, xd_p = Nu_p, t, xk_g, t, x

If energy_balance_type not `EnergyBalanceType.none`:

Gas phase - gas to solid heat transfer:

$$H_{g,t,x} = - \frac{ 6 } { d_{p} } h_{gs,t,x} \left( T_{g,t,x} - T_{s,t,x} \right) A_{s,t,x}$$

Solid phase - gas to solid heat transfer:

$$H_{s,t,x} = \frac{ 6 } { d_{p} } h_{gs,t,x} \left( T_{g,t,x} - T_{s,t,x} \right) A_{s,t,x}$$

List of Variables

Variable	Description	Reference to
Abed	Reactor bed cross-sectional area	bed_area
Ag, t, x	Gas phase area (interstitial cross-sectional area)	gas_phase.area
As, t, x	Solid phase area	solid_phase.area
cp, t, x	Gas phase heat capacity (constant P)	gas_phase.properties.cp_mass
Dbed	Reactor bed diameter	bed_diameter
Fmass, s, t, inlet	Total mass flow rate of solids, at inlet (x = 1)	solid_phase.properties.flow_mass
Fmol, g, t, x	Total molar flow rate of gas	gas_phase.properties.flow_mol
Hg, t, x	Gas to solid heat transfer term, gas phase	gas_phase.heat
Hs, t, x	Gas to solid heat transfer term, solid phase	solid_phase.heat
hgs, t, x	Gas-solid heat transfer coefficient	gas_solid_htc
kg, t, x	Gas thermal conductivity	gas_phase.properties.therm_cond
Lbed	Reactor bed height	bed_height
Lg	Gas domain length	gas_phase.length
Ls	Solid domain length	solid_phase.length
Mg, t, x, p, j	Rate generation/consumption term, gas phase	gas_phase.mass_transfer_term
Nup, t, x	Particle Nusselt number	Nu_particle
dPg, t, x	Total pressure derivative w.r.t. x (axial position)	gas_phase.deltaP
Prt, x	Prandtl number	Pr
rg, t, x, r	Gas phase reaction rate	gas_phase.reactions.reaction_rate
rs, t, x, r	Solid phase reaction rate	solid_phase.reactions.reaction_rate
Rep, t, x	Particle Reynolds number	Re_particle
Tg, t, x	Gas phase temperature	gas_phase.properties.temperature
Ts, t, x	Solid phase temperature	solid_phase.properties.temperature
ug, t, x	Superficial velocity of the gas	velocity_superficial_gas
us, t	Superficial velocity of the solids	velocity_superficial_solid
Greek letters
ε	Reactor bed voidage	bed_voidage
μg, t, x	Dynamic viscosity of gas mixture	gas_phase.properties.visc_d
ξg, t, x, r	Gas phase reaction extent	gas_phase.rate_reaction_extent
ξs, t, x, r	Solid phase reaction extent	solid_phase.rate_reaction_extent
ρmass, g, t, inlet	Density of gas mixture	gas_phase.properties.dens_mass
ρmass, s, t, inlet	Density of solid particles	solid_phase.properties.dens_mass_particle
ρmol, g, t, x	Molar density of the gas	gas_phase.properties.dens_mole

List of Parameters

Parameter	Description	Reference to
dp	Solid particle diameter	solid_phase.properties._params.particle_dia
νs, j, r	Stoichiometric coefficients	solid_phase.reactions.rate_reaction_stoichiometry

Initialization

The initialization method for this model will save the current state of the model before commencing initialization and reloads it afterwards. The state of the model will be the same after initialization, only the initial guesses for unfixed variables will be changed.

The model allows for the passing of a dictionary of values of the state variables of the gas and solid phases that can be used as initial guesses for the state variables throughout the time and spatial domains of the model. This is optional but recommended. A typical guess could be values of the gas and solid inlet port variables at time t = 0.

The model initialization proceeds through a sequential hierarchical method where the model equations are deactivated at the start of the initialization routine, and the complexity of the model is built up through activation and solution of various sub-model blocks and equations at each initialization step. At each step the model variables are updated to better guesses obtained from the model solution at that step.

The initialization routine proceeds in as follows:

• Step 1: Initialize the thermo-physical and transport properties model blocks.
• Step 2: Initialize the hydrodynamic properties.
• Step 3a: Initialize mass balances without reactions and pressure drop.
• Step 3b: Initialize mass balances with reactions and without pressure drop.
• Step 3c: Initialize mass balances with reactions and pressure drop.
• Step 4: Initialize energy balances.

MBR Class

idaes.gas_solid_contactors.unit_models.moving_bed

MBR

MBRData Class

MBRData