Fixed Bed 1D Reactor¶
The IDAES Fixed Bed 1D Reactor (FixedBed1D) model represents a unit operation where a gas stream passes through a solid phase bed in a linear reactor vessel. The FixedBed1D mathematical model is a 1D time variant model with two phases (gas and solid). The model captures the gassolid interaction between both phases through reaction/adsorption, mass and heat transfer.
Assumptions:
The radial concentration and temperature gradients are assumed to be negligible.
The reactor is assumed to be adiabatic.
Requirements:
Property package contains temperature and pressure variables.
Degrees of Freedom¶
FixedBed1D Reactors 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 FixedBed1D unit model consists of one ControlVolume1DBlock Block named gas_phase, which has one Inlet Port (named gas_inlet) and one Outlet Port (named gas_outlet). As there are no flow variables in the solid_phase a ControlVolume1DBlock Block is not used, instead indexed state blocks are used to access the solid thermophysical properties whilst the mass and energy balance equations of the solid are written in the core FixedBed1D unit model.
Construction Arguments¶
The IDAES FixedBed1D model has construction arguments specific to the whole unit and to the individual regions.
Arguments that are applicable to the FixedBed1D unit as a whole are:
finite_elements  sets the number of finite elements to use when discretizing the spatial domains (default = 10).
length_domain_set  sets the list of point to use to initialize a new ContinuousSet (default = [0.0, 1.0]).
transformation_method  sets the discretization method to use by the Pyomo TransformationFactory to transform the spatial domain (default = dae.finite_difference):
dae.finite_difference  finite difference method.
dae.collocation  orthogonal collocation method.
transformation_scheme  sets the scheme to use when transforming a domain. Selected schemes should be compatible with the transformation_method chosen (default = None):
None  defaults to “BACKWARD” for finite difference transformation method and to “LAGRANGERADAU” for collocation transformation method
BACKWARD  use a finite difference transformation method.
FORWARD  use a finite difference transformation method.
LAGRANGERADAU  use a collocation transformation method.
collocation_points  sets the number of collocation points to use when discretizing the spatial domains (default = 3, collocation methods only).
flow_type  indicates the flow arrangement within the unit to be modeled. Options are:
‘forward_flow’  (default) gas flows in the forward direction (from x=0 to x=1)
‘reverse_flow’  gas flows in the reverse direction (from x=1 to x=0).
material_balance_type  indicates what type of energy balance should be constructed (default = MaterialBalanceType.componentTotal).
MaterialBalanceType.componentTotal  use total component balances.
MaterialBalanceType.total  use total material balance.
energy_balance_type  indicates what type of energy balance should be constructed (default = EnergyBalanceType.enthalpyTotal).
EnergyBalanceType.none  excludes energy balances.
EnergyBalanceType.enthalpyTotal  single enthalpy balance for material.
momentum_balance_type  indicates what type of momentum balance should be constructed (default = MomentumBalanceType.pressureTotal).
MomentumBalanceType.none  exclude momentum balances.
MomentumBalanceType.pressureTotal  single pressure balance for material.
has_pressure_change  indicates whether terms for pressure change should be constructed (default = True).
True  include pressure change terms.
False  exclude pressure change terms.
pressure_drop_type  indicates what type of pressure drop correlation should be used (default = “ergun_correlation”).
“ergun_correlation”  use the Ergun equation.
“simple_correlation”  use a simplified pressure drop correlation.
Arguments that are applicable to the gas phase:
property_package  property package to use when constructing gas phase Property Blocks (default = ‘use_parent_value’). This is provided as a Physical Parameter Block by the Flowsheet when creating the model. If a value is not provided, the ControlVolume Block will try to use the default property package if one is defined.
property_package_args  set of arguments to be passed to the gas phase Property Blocks when they are created (default = ‘use_parent_value’).
reaction_package  reaction package to use when constructing gas phase Reaction Blocks (default = None). This is provided as a Reaction Parameter Block by the Flowsheet when creating the model. If a value is not provided, the ControlVolume Block will try to use the default property package if one is defined.
reaction_package_args  set of arguments to be passed to the gas phase Reaction Blocks when they are created (default = None).
has_equilibrium_reactions  sets flag to indicate if terms of equilibrium controlled reactions should be constructed (default = False).
Arguments that are applicable to the solid phase:
property_package  property package to use when constructing solid phase Property Blocks (default = ‘use_parent_value’). This is provided as a Physical Parameter Block by the Flowsheet when creating the model. If a value is not provided, the Indexed State Blocks will try to use the default property package if one is defined.
property_package_args  set of arguments to be passed to the solid phase Property Blocks when they are created (default = ‘use_parent_value’).
reaction_package  reaction package to use when constructing solid phase Reaction Blocks (default = None). This is provided as a Reaction Parameter Block by the Flowsheet when creating the model. If a value is not provided, the Indexed State Blocks will try to use the default property package if one is defined.
reaction_package_args  set of arguments to be passed to the solid phase Reaction Blocks when they are created (default = None).
has_equilibrium_reactions  sets flag to indicate if terms of equilibrium controlled reactions should be constructed (default = False).
Additionally, FixedBed1D units have the following construction arguments which are passed to all the ControlVolume1DBlock and state Blocks and are always specified to True.
Argument 
Value 

dynamic 
True 
has_holdup 
True 
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, FixedBed1D units write the following Constraints:
Geometry Constraints¶
Area of the reactor bed:
Area of the gas domain:
Area of the solid domain:
Hydrodynamic Constraints¶
Superficial velocity of the gas:
Pressure drop:
The constraints written by the FixedBed1D 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:
If pressure_drop_type is ergun_correlation:
Reaction Constraints¶
If gas_phase_config.reaction_package is not ‘None’:
Gas phase reaction extent:
\[\xi_{g,t,x,r} = r_{g,t,x,r} A_{g,t,x}\]
If solid_phase_config.reaction_package is not ‘None’:
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¶
Sum of component mass fractions in solid phase:
Solid material holdup:
If solid_phase_config.reaction_package is not ‘None’:
Solid material accumulation:
\[{\dot{J}}_{mass,s,t,x,j} = A_{s,t,x} {MW}_{s,j} {\Sigma}_{r} {r_{s,t,x,r} \nu_{s,j,r}}\]
If solid_phase_config.reaction_package is ‘None’:
Solid material accumulation:
\[{\dot{J}}_{mass,s,t,x,j} = 0\]
If energy_balance_type not EnergyBalanceType.none:
Solid energy holdup:
\[q_{energy,s,t,x} = A_{s,t,x} \rho_{mass,s,t,x} H_{mass,s,t,x}\]Solid energy accumulation:
\[{\dot{q}}_{energy,s,t,x} = H_{s,t,x}  A_{s,t,x} {\Sigma}_{r} {r_{s,t,x,r} H_{rxn,s,t,x,r}}\]
If energy_balance_type is EnergyBalanceType.none:
Isothermal solid phase:
\[T_{s,t,x} = T_{s,t=0,x}\]Isothermal gas phase:
\[T_{g,t,x} = T_{s,t=0,x}\]
If energy_balance_type not EnergyBalanceType.none:
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:
\[Nu_{p,t,x} = 2 + 1.1 Pr_{t,x}^{1/3} \left Re_{p,t,x} \right^{0.6}\]Particle to fluid heat transfer coefficient
\[h_{gs,t,x} d_{p} = Nu_{p,t,x} k_{g,t,x}\]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 

\(A_{bed}\) 
Reactor bed crosssectional area 

\(A_{g,t,x}\) 
Gas phase area (interstitial crosssectional area) 

\(A_{s,t,x}\) 
Solid phase area 

\(c_{p,t,x}\) 
Gas phase heat capacity (constant \(P\)) 

\(D_{bed}\) 
Reactor bed diameter 

\(F_{mol,g,t,x}\) 
Total molar flow rate of gas 

\(H_{g,t,x}\) 
Gas to solid heat transfer term, gas phase 

\(H_{s,t,x}\) 
Gas to solid heat transfer term, solid phase 

\(H_{rxn,s,t,x,r}\) 
Solid phase heat of reaction 

\(h_{gs,t,x}\) 
Gassolid heat transfer coefficient 

\(J_{mass,s,t,x,j}\) 
Material holdup, solid phase 

\({\dot{J}}_{mass,s,t,x,j}\) 
Material accumulation, solid phase 

\(k_{g,t,x}\) 
Gas thermal conductivity 

\(L_{bed}\) 
Reactor bed height 

\(M_{g,t,x,p,j}\) 
Rate generation/consumption term, gas phase 

\(Nu_{p,t,x}\) 
Particle Nusselt number 

\(dP_{g,t,x}\) 
Total pressure derivative w.r.t. \(x\) (axial position) 

\(Pr_{t,x}\) 
Prandtl number 

\(q_{energy,s,t,x}\) 
Energy holdup, solid phase 

\({\dot{q}}_{energy,s,t,x}\) 
Energy accumulation, solid phase 

\(r_{g,t,x,r}\) 
Gas phase reaction rate 

\(r_{s,t,x,r}\) 
Solid phase reaction rate 

\(Re_{p,t,x}\) 
Particle Reynolds number 

\(T_{g,t,x}\) 
Gas phase temperature 

\(T_{s,t,x}\) 
Solid phase temperature 

\(u_{g,t,x}\) 
Superficial velocity of the gas 

Greek letters 

\(\varepsilon\) 
Reactor bed voidage 

\(\mu_{g,t,x}\) 
Dynamic viscosity of gas mixture 

\(\xi_{g,t,x,r}\) 
Gas phase reaction extent 

\(\rho_{mass,g,t,inlet}\) 
Density of gas mixture 

\(\rho_{mass,s,t,inlet}\) 
Density of solid particles 

\(\rho_{mol,g,t,x}\) 
Molar density of the gas 

List of Parameters¶
Parameter 
Description 
Reference to 

\(d_{p}\) 
Solid particle diameter 

\(\nu_{s,j,r}\) 
Stoichiometric coefficients 

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 submodel 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 as follows:
Step 1: Initialize the thermophysical 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.
Block Triangularization Initialization¶
The Block Triangularization initialization routine initializes the 1DFixedBed model in the following steps:
 Step 1: Convert the system of equations from a partial differential set of equations (PDAE) to an algebraic set of equations (AE) by deactivating
the discretization equations and sum_component_equations (if present), and fixing the state variables to a guess (typically inlet conditions).
Step 2: Decompose the AE into strongly connected components via block triangularization and solve individual blocks.
Step 3: Revert the fixed variables and deactivated constraints to their original states.
The routine 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 state variables at time \(t=0\).
FixedBed1D Class¶
 class idaes.models_extra.gas_solid_contactors.unit_models.fixed_bed_1D.FixedBed1D(*args, **kwds)¶
 Parameters
rule (function) – A rule function or None. Default rule calls build().
concrete (bool) – If True, make this a toplevel model. Default  False.
ctype (class) –
Pyomo ctype of the block. Default  pyomo.environ.Block
 Config args
 dynamic
Indicates whether this model will be dynamic or not, default = useDefault. Valid values: { useDefault  get flag from parent (default = False), True  set as a dynamic model, False  set as a steadystate model.}
 has_holdup
Indicates whether holdup terms should be constructed or not. Must be True if dynamic = True, default  False. Valid values: { useDefault  get flag from parent (default = False), True  construct holdup terms, False  do not construct holdup terms}
 finite_elements
Number of finite elements to use when discretizing length domain (default=20)
 length_domain_set
length_domain_set  (optional) list of point to use to initialize a new ContinuousSet if length_domain is not provided (default = [0.0, 1.0])
 transformation_method
Method to use to transform domain. Must be a method recognized by the Pyomo TransformationFactory, default  “dae.finite_difference”. Valid values: { “dae.finite_difference”  Use a finite difference transformation method, “dae.collocation”  use a collocation transformation method}
 transformation_scheme
Scheme to use when transforming domain. See Pyomo documentation for supported schemes, default  None. Valid values: { None  defaults to “BACKWARD” for finite difference transformation method, and to “LAGRANGE RADAU” for collocation transformation method, “BACKWARD”  Use a finite difference transformation method, “FORWARD””  use a finite difference transformation method, “LAGRANGERADAU””  use a collocation transformation method}
 collocation_points
Number of collocation points to use per finite element when discretizing length domain (default=3)
 flow_type
Flow configuration of Fixed Bed default  “forward_flow”. Valid values: { “forward_flow”  gas flows from 0 to 1, “reverse_flow”  gas flows from 1 to 0.}
 material_balance_type
Indicates what type of mass balance should be constructed, default  MaterialBalanceType.componentTotal. Valid values: { MaterialBalanceType.none  exclude material balances, MaterialBalanceType.componentPhase  use phase component balances, MaterialBalanceType.componentTotal  use total component balances, MaterialBalanceType.elementTotal  use total element balances, MaterialBalanceType.total  use total material balance.}
 energy_balance_type
Indicates what type of energy balance should be constructed, default  EnergyBalanceType.enthalpyTotal. Valid values: { EnergyBalanceType.none  exclude energy balances, EnergyBalanceType.enthalpyTotal  single enthalpy balance for material, EnergyBalanceType.enthalpyPhase  enthalpy balances for each phase, EnergyBalanceType.energyTotal  single energy balance for material, EnergyBalanceType.energyPhase  energy balances for each phase.}
 momentum_balance_type
Indicates what type of momentum balance should be constructed, default  MomentumBalanceType.pressureTotal. Valid values: { MomentumBalanceType.none  exclude momentum balances, MomentumBalanceType.pressureTotal  single pressure balance for material, MomentumBalanceType.pressurePhase  pressure balances for each phase, MomentumBalanceType.momentumTotal  single momentum balance for material, MomentumBalanceType.momentumPhase  momentum balances for each phase.}
 has_pressure_change
Indicates whether terms for pressure change should be constructed, default  False. Valid values: { True  include pressure change terms, False  exclude pressure change terms.}
 pressure_drop_type
Indicates what type of pressure drop correlation should be used, default  “ergun_correlation”. Valid values: { “ergun_correlation”  Use the Ergun equation, “simple_correlation”  Use a simplified pressure drop correlation.}
 gas_phase_config
gas phase config arguments
 dynamic
Indicates whether this model will be dynamic or not, default = useDefault. Valid values: { useDefault  get flag from parent (default = False), True  set as a dynamic model, False  set as a steadystate model.}
 has_holdup
Indicates whether holdup terms should be constructed or not. Must be True if dynamic = True, default  False. Valid values: { useDefault  get flag from parent (default = False), True  construct holdup terms, False  do not construct holdup terms}
 property_package
Property parameter object used to define property calculations (default = ‘use_parent_value’)  ‘use_parent_value’  get package from parent (default = None)  a ParameterBlock object
 property_package_args
A dict of arguments to be passed to the PropertyBlockData and used when constructing these (default = ‘use_parent_value’)  ‘use_parent_value’  get package from parent (default = None)  a dict (see property package for documentation)
 reaction_package
Reaction parameter object used to define reaction calculations, default  None. Valid values: { None  no reaction package, ReactionParameterBlock  a ReactionParameterBlock object.}
 reaction_package_args
A ConfigBlock with arguments to be passed to a reaction block(s) and used when constructing these, default  None. Valid values: { see reaction package for documentation.}
 has_equilibrium_reactions
Indicates whether terms for equilibrium controlled reactions should be constructed, default  True. Valid values: { True  include equilibrium reaction terms, False  exclude equilibrium reaction terms.}
 solid_phase_config
solid phase config arguments
 dynamic
Indicates whether this model will be dynamic or not, default = useDefault. Valid values: { useDefault  get flag from parent (default = False), True  set as a dynamic model, False  set as a steadystate model.}
 has_holdup
Indicates whether holdup terms should be constructed or not. Must be True if dynamic = True, default  False. Valid values: { useDefault  get flag from parent (default = False), True  construct holdup terms, False  do not construct holdup terms}
 property_package
Property parameter object used to define property calculations (default = ‘use_parent_value’)  ‘use_parent_value’  get package from parent (default = None)  a ParameterBlock object
 property_package_args
A dict of arguments to be passed to the PropertyBlockData and used when constructing these (default = ‘use_parent_value’)  ‘use_parent_value’  get package from parent (default = None)  a dict (see property package for documentation)
 reaction_package
Reaction parameter object used to define reaction calculations, default  None. Valid values: { None  no reaction package, ReactionParameterBlock  a ReactionParameterBlock object.}
 reaction_package_args
A ConfigBlock with arguments to be passed to a reaction block(s) and used when constructing these, default  None. Valid values: { see reaction package for documentation.}
 has_equilibrium_reactions
Indicates whether terms for equilibrium controlled reactions should be constructed, default  True. Valid values: { True  include equilibrium reaction terms, False  exclude equilibrium reaction terms.}
initialize (dict) – ProcessBlockData config for individual elements. Keys are BlockData indexes and values are dictionaries with config arguments as keys.
idx_map (function) – Function to take the index of a BlockData element and return the index in the initialize dict from which to read arguments. This can be provided to override the default behavior of matching the BlockData index exactly to the index in initialize.
 Returns
(FixedBed1D) New instance
FixedBed1DData Class¶
 class idaes.models_extra.gas_solid_contactors.unit_models.fixed_bed_1D.FixedBed1DData(component)[source]¶
1D Fixed Bed Reactor Model Class
 block_triangularization_initialize(gas_phase_state_args=None, solid_phase_state_args=None, outlvl=0, solver=None, calc_var_kwds=None)[source]¶
Block triangularization (BT) initialization routine for 1D FixedBed unit.
The BT initialization routine initializes the unit model in the following steps:
 Step 1: Convert the system of equations from a partial differential set of
equations (PDAE) to an algebraic set of equations (AE) by deactivating the discretization equations and sum_component_equations (if present), and fixing the state variables to a guess (typically inlet conditions).
 Step 2: Decompose the AE into strongly connected components
via block triangularization and solve individual blocks
 Step 3: revert the fixed variables and deactivated constraints to their
original states.
The initialization procedure is wrapped in the TemporarySubsystemManager utility to ensure that the deactivation of constraints and fixing of variables are only temporary and the final model structure is unchanged after initialization.
 Keyword Arguments
gas_phase_state_args – a dict of arguments to be passed to the property package(s) to provide an initial state for initialization (see documentation of the specific property package) (default = None).
solid_phase_state_args – a dict of arguments to be passed to the property package(s) to provide an initial state for initialization (see documentation of the specific property package) (default = None).
outlvl – sets output level of initialization routine
optarg – solver options dictionary object (default=None, use default solver options)
solver – str indicating which solver to use during initialization (default = None, use default solver)
calc_var_kwds – Dictionary of keyword arguments for calculate_variable_from_constraint
 Returns
None
 initialize_build(gas_phase_state_args=None, solid_phase_state_args=None, outlvl=0, solver=None, optarg=None)[source]¶
Initialization routine for 1DFixedBed unit.
 Keyword Arguments
gas_phase_state_args – a dict of arguments to be passed to the property package(s) to provide an initial state for initialization (see documentation of the specific property package) (default = None).
solid_phase_state_args – a dict of arguments to be passed to the property package(s) to provide an initial state for initialization (see documentation of the specific property package) (default = None).
outlvl – sets output level of initialization routine
optarg – solver options dictionary object (default=None, use default solver options)
solver – str indicating which solver to use during initialization (default = None, use default solver)
 Returns
None