# 1D Control Volume Class¶

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).

class idaes.core.base.control_volume1d.ControlVolume1DBlock(*args, **kwds)

ControlVolume1DBlock is a specialized Pyomo block for IDAES control volume blocks discretized in one spatial direction, and contains instances of ControlVolume1DBlockData.

ControlVolume1DBlock should be used for any control volume with a defined volume and distinct inlets and outlets where there is a single spatial domain parallel to the material flow direction. This encompases unit operations such as plug flow reactors and pipes.

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, default - useDefault. Valid values: { useDefault - get flag from parent, True - set as a dynamic model, False - set as a steady-state model}

has_holdup

Indicates whether holdup terms should be constructed or not. Must be True if dynamic = True, default - False. Valid values: { True - construct holdup terms, False - do not construct holdup terms}

property_package

Property parameter object used to define property calculations, default - useDefault. Valid values: { useDefault - use default package from parent model or flowsheet, PropertyParameterObject - a PropertyParameterBlock object.}

property_package_args

A ConfigBlock with arguments to be passed to a property block(s) and used when constructing these, default - None. Valid values: { 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.}

auto_construct

If set to True, this argument will trigger the auto_construct method which will attempt to construct a set of material, energy and momentum balance equations based on the parent unit’s config block. The parent unit must have a config block which derives from CONFIG_Base, default - False. Valid values: { True - use automatic construction, False - do not use automatic construciton.}

area_definition

Argument defining whether area variable should be spatially variant or not. default - DistributedVars.uniform. Valid values: { DistributedVars.uniform - area does not vary across spatial domian, DistributedVars.variant - area can vary over the domain and is indexed by time and space.}

transformation_method

Method to use to transform domain. Must be a method recognised by the Pyomo TransformationFactory.

transformation_scheme

Scheme to use when transforming domain. See Pyomo documentation for supported schemes.

finite_elements

Number of finite elements to use in transformation (equivalent to Pyomo nfe argument).

collocation_points

Number of collocation points to use (equivalent to Pyomo ncp argument).

• 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

(ControlVolume1DBlock) New instance

class idaes.core.base.control_volume1d.ControlVolume1DBlockData(component)[source]

1-Dimensional ControlVolume Class

This class forms the core of all 1-D IDAES models. It provides methods to build property and reaction blocks, and add mass, energy and momentum balances. The form of the terms used in these constraints is specified in the chosen property package.

Method to create spatial domain and volume Var in ControlVolume.

Parameters
• - (length_var) – domain for the ControlVolume. If not provided, a new ContinuousSet will be created (default=None). ContinuousSet should be normalized to run between 0 and 1.

• - – a new ContinuousSet if length_domain is not provided (default = [0.0, 1.0]).

• - – the spatial domain,. If a variable is provided, a reference will be made to this in place of the length Var.

• flow (flow_direction - argument indicating direction of material) –

relative to length domain. Valid values:
• FlowDirection.forward (default), flow goes from 0 to 1.

• FlowDirection.backward, flow goes from 1 to 0

Returns

None

add_phase_component_balances(has_rate_reactions=False, has_equilibrium_reactions=False, has_phase_equilibrium=False, has_mass_transfer=False, custom_molar_term=None, custom_mass_term=None)[source]

This method constructs a set of 1D material balances indexed by time, length, phase and component.

Parameters
• has_rate_reactions – whether default generation terms for rate reactions should be included in material balances

• has_equilibrium_reactions – whether generation terms should for chemical equilibrium reactions should be included in material balances

• has_phase_equilibrium – whether generation terms should for phase equilibrium behaviour should be included in material balances

• has_mass_transfer – whether generic mass transfer terms should be included in material balances

• custom_molar_term – a Pyomo Expression representing custom terms to be included in material balances on a molar basis. Expression must be indexed by time, length domain, phase list and component list

• custom_mass_term – a Pyomo Expression representing custom terms to be included in material balances on a mass basis. Expression must be indexed by time, length domain, phase list and component list

Returns

Constraint object representing material balances

Method for adding energy balances (including kinetic energy) indexed by phase to the control volume.

See specific control volume documentation for details.

Method for adding enthalpy balances indexed by phase to the control volume.

See specific control volume documentation for details.

Method for adding momentum balances indexed by phase to the control volume.

See specific control volume documentation for details.

Method for adding pressure balances indexed by phase to the control volume.

See specific control volume documentation for details.

This method constructs the reaction block for the control volume.

Parameters
• has_equilibrium – indicates whether equilibrium calculations will be required in reaction block

• package_arguments – dict-like object of arguments to be passed to reaction block as construction arguments

Returns

None

This method constructs the state blocks for the control volume.

Parameters
• information_flow – a FlowDirection Enum indicating whether information flows from inlet-to-outlet or outlet-to-inlet

• has_phase_equilibrium – indicates whether equilibrium calculations will be required in state blocks

• package_arguments – dict-like object of arguments to be passed to state blocks as construction arguments

Returns

None

add_total_component_balances(has_rate_reactions=False, has_equilibrium_reactions=False, has_phase_equilibrium=False, has_mass_transfer=False, custom_molar_term=None, custom_mass_term=None)[source]

This method constructs a set of 1D material balances indexed by time length and component.

Parameters
• has_rate_reactions – whether default generation terms for rate reactions should be included in material balances

• has_equilibrium_reactions – whether generation terms should for chemical equilibrium reactions should be included in material balances

• has_phase_equilibrium – whether generation terms should for phase equilibrium behaviour should be included in material balances

• has_mass_transfer – whether generic mass transfer terms should be included in material balances

• custom_molar_term – a Pyomo Expression representing custom terms to be included in material balances on a molar basis. Expression must be indexed by time, length domain and component list

• custom_mass_term – a Pyomo Expression representing custom terms to be included in material balances on a mass basis. Expression must be indexed by time, length domain and component list

Returns

Constraint object representing material balances

This method constructs a set of 1D element balances indexed by time and length.

Parameters
• rate (has_rate_reactions - whether default generation terms for) – reactions should be included in material balances

• for (has_equilibrium_reactions - whether generation terms should) – chemical equilibrium reactions should be included in material balances

• phase (has_phase_equilibrium - whether generation terms should for) – equilibrium behaviour should be included in material balances

• be (has_mass_transfer - whether generic mass transfer terms should) – included in material balances

• custom (custom_elemental_term - a Pyomo Expression representing) – terms to be included in material balances on a molar elemental basis. Expression must be indexed by time, length and element list

Returns

Constraint object representing material balances

Method for adding a total energy balance (including kinetic energy) to the control volume.

See specific control volume documentation for details.

This method constructs a set of 1D enthalpy balances indexed by time and phase.

Parameters
• should (has_heat_of_reaction - whether terms for heat of reaction) – be included in enthalpy balance

• be (has_work_transfer - whether terms for work transfer should) – included in enthalpy balances

• be – included in enthalpy balances

• to (has_enthalpy_transfer - whether terms for enthalpy transfer due) – mass transfer should be included in enthalpy balance. This should generally be the same as the has_mass_transfer argument in the material balance methods

• representing (custom_term - a Python method which returns Pyomo expressions) – custom terms to be included in enthalpy balances. Method should accept time, length and phase list as arguments.

Returns

Constraint object representing enthalpy balances

Method for adding a total material balance to the control volume.

See specific control volume documentation for details.

Method for adding a total momentum balance to the control volume.

See specific control volume documentation for details.

This method constructs a set of 1D pressure balances indexed by time.

Parameters
• be (has_pressure_change - whether terms for pressure change should) – included in enthalpy balances

• to (custom_term - a Pyomo Expression representing custom terms) – be included in pressure balances. Expression must be indexed by time and length domain

• representing (custom_term - a Python method which returns Pyomo expressions) – custom terms to be included in enthalpy balances. Method should accept time and length as arguments.

Returns

Constraint object representing pressure balances

apply_transformation()[source]

Method to apply DAE transformation to the Control Volume length domain. Transformation applied will be based on the Control Volume configuration arguments.

build()[source]

Build method for ControlVolume1DBlock blocks.

Returns

None

initialize(state_args=None, outlvl=0, optarg=None, solver=None, hold_state=True)[source]

Initialization routine for 1D control volume.

Keyword Arguments
• 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 = {}).

• 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)

• hold_state – flag indicating whether the initialization routine should unfix any state variables fixed during initialization, default - True. Valid values: True - states variables are not unfixed, and a dict of returned containing flags for which states were fixed during initialization, False - state variables are unfixed after initialization by calling the release_state method.

Returns

If hold_states is True, returns a dict containing flags for which states were fixed during initialization else the release state is triggered.

model_check()[source]

This method executes the model_check methods on the associated state blocks (if they exist). This method is generally called by a unit model as part of the unit’s model_check method.

Parameters

None

Returns

None

release_state(flags, outlvl=0)[source]

Method to release state variables fixed during initialization.

Keyword Arguments
• flags – dict containing information of which state variables were fixed during initialization, and should now be unfixed. This dict is returned by initialize if hold_state = True.

• outlvl – sets output level of logging

Returns

None

report(time_point=0, dof=False, ostream=None, prefix='')[source]

No report method defined for ControlVolume1D class. This is due to the difficulty of presenting spatially discretized data in a readable form without plotting.

## 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. $$mol/m \cdot s$$).

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:

$V = A \times L$

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

$$M_{t,x,p,j}$$

t, x, p, j

has_holdup = True

phase_fraction

$$\phi_{t,x,p}$$

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

$$F_{t, x, p, j}$$

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

$$N_{kinetic,t,x,p,j}$$

t, x, p ,j

has_rate_reactions = True

rate_reaction_extent

$$X_{kinetic,t,x,r}$$

t, x, r

has_rate_reactions = True

equilibrium_reaction_generation

$$N_{equilibrium,t,x,p,j}$$

t, x, p ,j

has_equilibrium_reactions = True

equilibrium_reaction_extent

$$X_{equilibrium,t,x,r}$$

t, x, r

has_equilibrium_reactions = True

phase_equilibrium_generation

$$N_{pe,t,x,p,j}$$

t, x, p ,j

has_phase_equilibrium = True

mass_transfer_term

$$N_{transfer,t,x,p,j}$$

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.

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):

$M_{t, x, p, j} = \rho_{t, x, p, j} \times A \times \phi_{t, x, p}$

where $$\rho_{t, x, p ,j}$$ 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):

$N_{kinetic, t, x, p, j} = \alpha_{r, p, j} \times X_{kinetic, t, x, r}$

where $$\alpha_{r, p. j}$$ 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):

$N_{equilibrium, t, x, p, j} = \alpha_{r, p, j} \times X_{equilibrium, t, x, r}$

where $$\alpha_{r, p. j}$$ is the stoichiometric coefficient of component $$j$$ in phase $$p$$ for reaction $$r$$ (as defined in the PhysicalParameterBlock).

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

$$M_{t,x,p,j}$$

t, x, p, j

has_holdup = True

phase_fraction

$$\phi_{t,x,p}$$

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

$$F_{t, x, p, j}$$

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

$$N_{kinetic,t,x,p,j}$$

t, x, p ,j

has_rate_reactions = True

rate_reaction_extent

$$X_{kinetic,t,x,r}$$

t, x, r

has_rate_reactions = True

equilibrium_reaction_generation

$$N_{equilibrium,t,x,p,j}$$

t, x, p ,j

has_equilibrium_reactions = True

equilibrium_reaction_extent

$$X_{equilibrium,t,x,r}$$

t, x, r

has_equilibrium_reactions = True

mass_transfer_term

$$N_{transfer,t,x,p,j}$$

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.

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):

$M_{t, x, p, j} = \rho_{t, x, p, j} \times A \times \phi_{t, x, p}$

where $$\rho_{t, x, p ,j}$$ 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):

$N_{kinetic, t, x, p, j} = \alpha_{r, p, j} \times X_{kinetic, t, x, r}$

where $$\alpha_{r, p. j}$$ 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):

$N_{equilibrium, t, x, p, j} = \alpha_{r, p, j} \times X_{equilibrium, t, x, r}$

where $$\alpha_{r, p. j}$$ is the stoichiometric coefficient of component $$j$$ in phase $$p$$ for reaction $$r$$ (as defined in the PhysicalParameterBlock).

Material balances are written for each element in the mixture.

Variables

Variable Name

Symbol

Indices

Conditions

element_holdup

$$M_{t,x,e}$$

t, x, e

has_holdup = True

phase_fraction

$$\phi_{t,x,p}$$

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

$$N_{transfer,t,x,e}$$

t, x, e

has_mass_transfer = True

elemental_flow_term

$$F_{t,x,e}$$

t, x, e

None

Constraints

elemental_flow_constraint(t, x, e):

$F_{t,x,e} = \sum_p{\sum_j{F_{t,x,p,j} \times n_{j, e}}}$

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):

$M_{t, x, e} = \rho_{t, x, p, j} \times A \times \phi_{t, x, p}$

where $$\rho_{t, x, p ,j}$$ 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.

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

$$E_{t,x,p}$$

t, x, p

has_holdup = True

phase_fraction

$$\phi_{t,x,p}$$

t, x, p

has_holdup = True

energy_accumulation

$$\frac{\partial E_{t,x,p}}{\partial t}$$

t, x, p

dynamic = True

_enthalpy_flow

$$H_{t,x,p}$$

t, x, p

None

enthalpy_flow_dx

$$\frac{\partial H_{t,x,p}}{\partial x}$$

t, x, p

None

heat

$$Q_{t,x}$$

t, x

has_heat_transfer = True

work

$$W_{t,x}$$

t, x

has_work_transfer = True

enthalpy_transfer

$$H_{transfer,t,x}$$

t, x

has_enthalpy_transfer = True

Expressions

heat_of_reaction(t, x):

$Q_{rxn, t, x} = sum_r{X_{kinetic, t, x, r} \times \Delta H_{rxn, r}} + sum_r{X_{equilibrium, t, x, r} \times \Delta H_{rxn, r}}$

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

Parameters

Parameter Name

Symbol

Default Value

scaling_factor_energy

$$s_{energy}$$

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.

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):

$E_{t, x, p} = u_{t, x, p} \times A \times \phi_{t, x, p}$

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.

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

Variables

Variable Name

Symbol

Indices

Conditions

pressure

$$P_{t,x}$$

t, x

None

pressure_dx

$$\frac{\partial P_{t,x}}{\partial x}$$

t, x

None

deltaP

$$\Delta P_{t,x}$$

t, x

has_pressure_change = True

Parameters

Parameter Name

Symbol

Default Value

scaling_factor_pressure

$$s_{pressure}$$

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 $$\Delta P_{custom, t, x}$$ term allows the user to provide custom terms which will be added into the pressure balance.