control_volume_1d.rst

1D Control Volume Class

Contents

The ControlVolume1DBlock block is used for systems with one spatial dimension where material flows parallel to the spatial domain. Examples of these types of unit operations include plug flow reactors and pipes. ControlVolume1DBlock blocks are discretized along the length domain and contain one StateBlock and one ReactionBlock (if applicable) at each point in the domain (including the inlet and outlet).

idaes.core.control_volume1d

ControlVolume1DBlock

ControlVolume1DBlockData

ControlVolume1DBlock Equations

This section documents the variables and constraints created by each of the methods provided by the ControlVolume0DBlock class.

• t indicates time index
• x indicates spatial (length) index
• p indicates phase index
• j indicates component index
• e indicates element index
• r indicates reaction name index

Most terms within the balance equations written by ControlVolume1DBlock are on a basis of per unit length (e.g. mol/m ⋅ s).

add_geometry

The add_geometry method creates the normalized length domain for the control volume (or a reference to an external domain). All constraints in ControlVolume1DBlock assume a normalized length domain, with values between 0 and 1.

This method also adds variables and constraints to describe the geometry of the control volume. ControlVolume1DBlock does not support varying dimensions of the control volume with time at this stage.

Variables

Variable Name	Symbol	Indices	Conditions
length_domain	x	None	None
volume	V	None	None
area	A	None	None
length	L	None	If length_var argument is provided, a reference to the provided component is made in place of creating a new variable

Constraints

`geometry_constraint`:

V = A × L

add_phase_component_balances

Material balances are written for each component in each phase (e.g. separate balances for liquid water and steam). Physical property packages may include information to indicate that certain species do not appear in all phases, and material balances will not be written in these cases (if has_holdup is True holdup terms will still appear for these species, however these will be set to 0).

Variables

Variable Name	Symbol	Indices	Conditions
material_holdup	Mt, x, p, j	t, x, p, j	has_holdup = True
phase_fraction	ϕt, x, p	t, x, p	has_holdup = True
material_accumulation	$\frac{\partial M_{t,x,p,j}}{\partial t}$	t, x, p, j	dynamic = True
_flow_terms	Ft, x, p, j	t, x, p, j	None
material_flow_dx	$\frac{\partial F_{t,x,p,j}}{\partial x}$	t, x, p, j	None
rate_reaction_generation	Nkinetic, t, x, p, j	t, x, p ,j	has_rate_reactions = True
rate_reaction_extent	Xkinetic, t, x, r	t, x, r	has_rate_reactions = True
equilibrium_reaction_generation	Nequilibrium, t, x, p, j	t, x, p ,j	has_equilibrium_reactions = True
equilibrium_reaction_extent	Xequilibrium, t, x, r	t, x, r	has_equilibrium_reactions = True
phase_equilibrium_generation	Npe, t, x, p, j	t, x, p ,j	has_phase_equilibrium = True
mass_transfer_term	Ntransfer, t, x, p, j	t, x, p ,j	has_mass_transfer = True

Constraints

`material_balances(t, x, p, j)`:

$$L \times \frac{\partial M_{t, x, p, j}}{\partial t} = fd \times \frac{\partial F_{t, x, p, j}}{\partial x} + L \times N_{kinetic, t, x, p, j} + L \times N_{equilibrium, t, x, p, j} + L \times N_{pe, t, x, p, j} + L \times N_{transfer, t, x, p, j} + L \times N_{custom, t, x, p, j}$$

fd is a flow direction term, which allows for material flow to be defined in either direction. If material flow is defined as forward, fd =  − 1, otherwise fd = 1.

The N_{custom, t, x, p, j} term allows the user to provide custom terms (variables or expressions) in both mass and molar basis which will be added into the material balances, which will be converted as necessary to the same basis as the material balance (by multiplying or dividing by the component molecular weight). The basis of the material balance is determined by the physical property package, and if undefined (or not mass or mole basis), an Exception will be returned.

`material_flow_linking_constraints(t, x, p, j)`:

This constraint is an internal constraint used to link the extensive material flow terms in the StateBlocks into a single indexed variable. This is required as Pyomo.DAE requires a single indexed variable to create the associated DerivativeVars and their numerical expansions.

If has_holdup is True, `material_holdup_calculation(t, x, p, j)`:

M_{t, x, p, j} = ρ_{t, x, p, j} × A × ϕ_t, x, p

where ρ_{t, x, p, j} is the density of component j in phase p at time t and location x.

If dynamic is True:

Numerical discretization of the derivative terms, $\frac{\partial M_{t,x,p,j}}{\partial t}$, will be performed by Pyomo.DAE.

If has_rate_reactions is True, `rate_reaction_stoichiometry_constraint(t, x, p, j)`:

N_{kinetic, t, x, p, j} = α_r, p, j × X_{kinetic, t, x, r}

where α_r, p.j is the stoichiometric coefficient of component j in phase p for reaction r (as defined in the PhysicalParameterBlock).

If has_equilibrium_reactions argument is True, `equilibrium_reaction_stoichiometry_constraint(t, x, p, j)`:

N_{equilibrium, t, x, p, j} = α_r, p, j × X_{equilibrium, t, x, r}

where α_r, p.j is the stoichiometric coefficient of component j in phase p for reaction r (as defined in the PhysicalParameterBlock).

add_total_component_balances

Material balances are written for each component across all phases (e.g. one balance for both liquid water and steam). Physical property packages may include information to indicate that certain species do not appear in all phases, and material balances will not be written in these cases (if has_holdup is True holdup terms will still appear for these species, however these will be set to 0).

Variables

Variable Name	Symbol	Indices	Conditions
material_holdup	Mt, x, p, j	t, x, p, j	has_holdup = True
phase_fraction	ϕt, x, p	t, x, p	has_holdup = True
material_accumulation	$\frac{\partial M_{t,x,p,j}}{\partial t}$	t, x, p, j	dynamic = True
_flow_terms	Ft, x, p, j	t, x, p, j	None
material_flow_dx	$\frac{\partial F_{t,x,p,j}}{\partial x}$	t, x, p, j	None
rate_reaction_generation	Nkinetic, t, x, p, j	t, x, p ,j	has_rate_reactions = True
rate_reaction_extent	Xkinetic, t, x, r	t, x, r	has_rate_reactions = True
equilibrium_reaction_generation	Nequilibrium, t, x, p, j	t, x, p ,j	has_equilibrium_reactions = True
equilibrium_reaction_extent	Xequilibrium, t, x, r	t, x, r	has_equilibrium_reactions = True
mass_transfer_term	Ntransfer, t, x, p, j	t, x, p ,j	has_mass_transfer = True

Constraints

`material_balances(t, x, p, j)`:

$$L \times \sum_p{\frac{\partial M_{t, x, p, j}}{\partial t}} = fd \times \sum{\frac{\partial F_{t, x, p, j}}{\partial x}} + L \times \sum_p{N_{kinetic, t, x, p, j}} + L \times \sum_p{N_{equilibrium, t, x, p, j}} + L \times \sum_p{N_{transfer, t, x, p, j}} + L \times N_{custom, t, x, j}$$

fd is a flow direction term, which allows for material flow to be defined in either direction. If material flow is defined as forward, fd =  − 1, otherwise fd = 1.

The N_{custom, t, x, j} term allows the user to provide custom terms (variables or expressions) in both mass and molar basis which will be added into the material balances, which will be converted as necessary to the same basis as the material balance (by multiplying or dividing by the component molecular weight). The basis of the material balance is determined by the physical property package, and if undefined (or not mass or mole basis), an Exception will be returned.

`material_flow_linking_constraints(t, x, p, j)`:

This constraint is an internal constraint used to link the extensive material flow terms in the StateBlocks into a single indexed variable. This is required as Pyomo.DAE requires a single indexed variable to create the associated DerivativeVars and their numerical expansions.

If has_holdup is True, `material_holdup_calculation(t, x, p, j)`:

M_{t, x, p, j} = ρ_{t, x, p, j} × A × ϕ_t, x, p

where ρ_{t, x, p, j} is the density of component j in phase p at time t and location x.

If dynamic is True:

Numerical discretization of the derivative terms, $\frac{\partial M_{t,x,p,j}}{\partial t}$, will be performed by Pyomo.DAE.

If has_rate_reactions is True, `rate_reaction_stoichiometry_constraint(t, x, p, j)`:

N_{kinetic, t, x, p, j} = α_r, p, j × X_{kinetic, t, x, r}

where α_r, p.j is the stoichiometric coefficient of component j in phase p for reaction r (as defined in the PhysicalParameterBlock).

If has_equilibrium_reactions argument is True, `equilibrium_reaction_stoichiometry_constraint(t, x, p, j)`:

N_{equilibrium, t, x, p, j} = α_r, p, j × X_{equilibrium, t, x, r}

where α_r, p.j is the stoichiometric coefficient of component j in phase p for reaction r (as defined in the PhysicalParameterBlock).

add_total_element_balances

Material balances are written for each element in the mixture.

Variables

Variable Name	Symbol	Indices	Conditions
element_holdup	Mt, x, e	t, x, e	has_holdup = True
phase_fraction	ϕt, x, p	t, x, p	has_holdup = True
element_accumulation	$\frac{\partial M_{t,x,e}}{\partial t}$	t, x, e	dynamic = True
elemental_mass_transfer_term	Ntransfer, t, x, e	t, x, e	has_mass_transfer = True
elemental_flow_term	Ft, x, e	t, x, e	None

Constraints

`elemental_flow_constraint(t, x, e)`:

F_t, x, e = ∑_p∑_jF_{t, x, p, j} × n_j, e

where n_j, e is the number of moles of element e in component j.

`element_balances(t, x, e)`:

$$L \times \frac{\partial M_{t, x, e}}{\partial t} = fd \times \frac{\partial F_{t, x, e}}{\partial x} + L \times N_{transfer, t, p, j} + L \times N_{custom, t, e}$$

fd is a flow direction term, which allows for material flow to be defined in either direction. If material flow is defined as forward, fd =  − 1, otherwise fd = 1.

The N_{custom, t, x, e} term allows the user to provide custom terms (variables or expressions) which will be added into the material balances.

If has_holdup is True, `elemental_holdup_calculation(t, x, e)`:

M_t, x, e = ρ_{t, x, p, j} × A × ϕ_t, x, p

where ρ_{t, x, p, j} is the density of component j in phase p at time t and location x.

If dynamic is True:

Numerical discretization of the derivative terms, $\frac{\partial M_{t,x,p,j}}{\partial t}$, will be performed by Pyomo.DAE.

add_total_enthalpy_balances

A single enthalpy balance is written for the entire mixture at each point in the spatial domain.

Variables

Variable Name	Symbol	Indices	Conditions
energy_holdup	Et, x, p	t, x, p	has_holdup = True
phase_fraction	ϕt, x, p	t, x, p	has_holdup = True
energy_accumulation	$\frac{\partial E_{t,x,p}}{\partial t}$	t, x, p	dynamic = True
_enthalpy_flow	Ht, x, p	t, x, p	None
enthalpy_flow_dx	$\frac{\partial H_{t,x,p}}{\partial x}$	t, x, p	None
heat	Qt, x	t, x	has_heat_transfer = True
work	Wt, x	t, x	has_work_transfer = True
enthalpy_transfer	Htransfer, t, x	t, x	has_enthalpy_transfer = True

Expressions

`heat_of_reaction(t, x)`:

Q_{rxn, t, x} = sum_rX_{kinetic, t, x, r} × ΔH_rxn, r + sum_rX_{equilibrium, t, x, r} × ΔH_rxn, r

where Q_{rxn, t, x} is the total enthalpy released by both kinetic and equilibrium reactions, and ΔH_rxn, r is the specific heat of reaction for reaction r.

Parameters

Parameter Name	Symbol	Default Value
scaling_factor_energy	senergy	1E-6

Constraints

`enthalpy_balance(t)`:

$$s_{energy} \times L \times \sum_p{\frac{\partial E_{t, x, p}}{\partial t}} = s_{energy} \times fd \ times \sum_p{\frac{\partial H_{t, x, p}}{\partial x}} + s_{energy} \times L \times Q_{t,x} + s_{energy} \times L \times W_{t,x} + s_{energy} \times L \times H_{transfer,t,x} + s_{energy} \times L \times Q_{rxn, t, x} + s_{energy} \times L \times E_{custom, t, x}$$

fd is a flow direction term, which allows for material flow to be defined in either direction. If material flow is defined as forward, fd =  − 1, otherwise fd = 1.

The E_{custom, t, x} term allows the user to provide custom terms which will be added into the energy balance.

`enthalpy_flow_linking_constraints(t, x, p)`:

This constraint is an internal constraint used to link the extensive enthalpy flow terms in the StateBlocks into a single indexed variable. This is required as Pyomo.DAE requires a single indexed variable to create the associated DerivativeVars and their numerical expansions.

If has_holdup is True, `enthalpy_holdup_calculation(t, x, p)`:

E_t, x, p = u_t, x, p × A × ϕ_t, x, p

where u_t, x, p is the internal density (specific internal energy) of phase p at time t and location x.

If dynamic is True:

Numerical discretization of the derivative terms, $\frac{\partial E_{t,x,p}}{\partial t}$, will be performed by Pyomo.DAE.

add_total_pressure_balances

A single pressure balance is written for the entire mixture at all points in the spatial domain.

Variables

Variable Name	Symbol	Indices	Conditions
pressure	Pt, x	t, x	None
pressure_dx	$\frac{\partial P_{t,x}}{\partial x}$	t, x	None
deltaP	ΔPt, x	t, x	has_pressure_change = True

Parameters

Parameter Name	Symbol	Default Value
scaling_factor_pressure	spressure	1E-4

Constraints

`pressure_balance(t, x)`:

$$0 = s_{pressure} \times fd \times \frac{\partial P_{t,x}}{\partial x} + s_{pressure} \times L \times \Delta P_{t,x} + s_{pressure} \times L \times \Delta P_{custom, t, x}$$

fd is a flow direction term, which allows for material flow to be defined in either direction. If material flow is defined as forward, fd =  − 1, otherwise fd = 1.

The ΔP_{custom, t, x} term allows the user to provide custom terms which will be added into the pressure balance.

`pressure_linking_constraint(t, x)`:

This constraint is an internal constraint used to link the pressure terms in the StateBlocks into a single indexed variable. This is required as Pyomo.DAE requires a single indexed variable to create the associated DerivativeVars and their numerical expansions.