From 8b78fe068c6574ead62fb23af549556a3ac52e14 Mon Sep 17 00:00:00 2001 From: schmoelder Date: Wed, 21 Aug 2024 20:06:25 +0000 Subject: [PATCH] =?UTF-8?q?Deploying=20to=20master=20from=20@=20cadet/CADE?= =?UTF-8?q?T-Website@34444dc9c07c97ff415ebaa770647139e53b86e3=20?= =?UTF-8?q?=F0=9F=9A=80?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- CADET-Match/master/.buildinfo | 2 +- .../master/.doctrees/cli/examples.doctree | Bin 9435 -> 9435 bytes .../master/.doctrees/cli/index.doctree | Bin 11640 -> 11640 bytes .../configuration/basic_config.doctree | Bin 12372 -> 12372 bytes .../.doctrees/configuration/error.doctree | Bin 16419 -> 16419 bytes .../configuration/experiments.doctree | Bin 11625 -> 11625 bytes .../.doctrees/configuration/graphing.doctree | Bin 7220 -> 7220 bytes .../.doctrees/configuration/index.doctree | Bin 7299 -> 7299 bytes .../.doctrees/configuration/misc.doctree | Bin 11833 -> 11833 bytes 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Section :ref:`dependence-on-external-function_react`). +For the sake of brevity, only the standard variant of those reaction models is specified below. +In order to obtain the format for the externally dependent variant, first replace the reaction model name ``XXX`` by ``EXT_XXX``. +Each parameter :math:`p` (except for stoichiometric and exponent matrices) depends on a (possibly distinct) external source in a polynomial way: + +.. math:: + + \begin{aligned} + p(T) &= p_{\texttt{TTT}} T^3 + p_{\texttt{TT}} T^2 + p_{\texttt{T}} T + p. + \end{aligned} + +Thus, a parameter ``XXX_YYY`` of the standard reaction model variant is replaced by the four parameters ``EXT_XXX_YYY``, ``EXT_XXX_YYY_T``, ``EXT_XXX_YYY_TT``, and ``EXT_XXX_YYY_TTT``. +Since each parameter can depend on a different external source, the dataset ``EXTFUN`` (not listed in the standard variants below) should contain a vector of 0-based integer indices of the external source of each parameter. +The ordering of the parameters in ``EXTFUN`` is given by the ordering in the standard variant. +However, if only one index is passed in ``EXTFUN``, this external source is used for all parameters. + +Note that parameter sensitivities with respect to column radius, column length, particle core radius, and particle radius may be wrong when using externally dependent reaction models. +This is caused by not taking into account the derivative of the external profile with respect to column position. + + +.. _multiple-particle-types_reactions: + +Multiple particle types +----------------------- + +The group that contains the parameters of a reaction model in unit operation with index ``XXX`` reads ``/input/model/unit_XXX/reaction_particle``. +This is valid for models with a single particle type. +If a model has multiple particle types, it may have a different reaction model in each type. +The parameters are then placed in the group ``/input/model/unit_XXX/reaction_particle_YYY`` instead, where ``YYY`` denotes the index of the particle type. + +Note that, in any case, ``/input/model/unit_XXX/reaction_particle_000`` contains the parameters of the first (and possibly sole) particle type. +This group also takes precedence over a possibly existing ``/input/model/unit_XXX/adsorption_particle`` group. diff --git a/master/_sources/interface/reactions.rst.txt b/master/_sources/interface/reaction/mass_action_law.rst.txt similarity index 75% rename from master/_sources/interface/reactions.rst.txt rename to master/_sources/interface/reaction/mass_action_law.rst.txt index f874dce2c..26c20b89f 100644 --- a/master/_sources/interface/reactions.rst.txt +++ b/master/_sources/interface/reaction/mass_action_law.rst.txt @@ -1,48 +1,9 @@ -.. _FFReaction: +.. _mass_action_law_config: -Reaction models -=============== +Mass Action Law +~~~~~~~~~~~~~~~ -Externally dependent reaction models ------------------------------------- - -Some reaction models have a variant that can use external sources as specified :ref:`/input/model/external/` (also see Section :ref:`dependence-on-external-function_react`). -For the sake of brevity, only the standard variant of those reaction models is specified below. -In order to obtain the format for the externally dependent variant, first replace the reaction model name ``XXX`` by ``EXT_XXX``. -Each parameter :math:`p` (except for stoichiometric and exponent matrices) depends on a (possibly distinct) external source in a polynomial way: - -.. math:: - - \begin{aligned} - p(T) &= p_{\texttt{TTT}} T^3 + p_{\texttt{TT}} T^2 + p_{\texttt{T}} T + p. - \end{aligned} - -Thus, a parameter ``XXX_YYY`` of the standard reaction model variant is replaced by the four parameters ``EXT_XXX_YYY``, ``EXT_XXX_YYY_T``, ``EXT_XXX_YYY_TT``, and ``EXT_XXX_YYY_TTT``. -Since each parameter can depend on a different external source, the dataset ``EXTFUN`` (not listed in the standard variants below) should contain a vector of 0-based integer indices of the external source of each parameter. -The ordering of the parameters in ``EXTFUN`` is given by the ordering in the standard variant. -However, if only one index is passed in ``EXTFUN``, this external source is used for all parameters. - -Note that parameter sensitivities with respect to column radius, column length, particle core radius, and particle radius may be wrong when using externally dependent reaction models. -This is caused by not taking into account the derivative of the external profile with respect to column position. - - -.. _multiple-particle-types_reactions: - -Multiple particle types ------------------------ - -The group that contains the parameters of a reaction model in unit operation with index ``XXX`` reads ``/input/model/unit_XXX/reaction_particle``. -This is valid for models with a single particle type. -If a model has multiple particle types, it may have a different reaction model in each type. -The parameters are then placed in the group ``/input/model/unit_XXX/reaction_particle_YYY`` instead, where ``YYY`` denotes the index of the particle type. - -Note that, in any case, ``/input/model/unit_XXX/reaction_particle_000`` contains the parameters of the first (and possibly sole) particle type. -This group also takes precedence over a possibly existing ``/input/model/unit_XXX/adsorption_particle`` group. - -.. _FFReactionMassActionLaw: - -Group /input/model/unit_XXX/reaction - REACTION_MODEL = MASS_ACTION_LAW ------------------------------------------------------------------------ +**Group /input/model/unit_XXX/reaction - REACTION_MODEL = MASS_ACTION_LAW** ``MAL_KFWD_BULK`` diff --git a/master/_sources/interface/reaction/michaelis_menten_kinetics.rst.txt b/master/_sources/interface/reaction/michaelis_menten_kinetics.rst.txt new file mode 100644 index 000000000..76884ea2e --- /dev/null +++ b/master/_sources/interface/reaction/michaelis_menten_kinetics.rst.txt @@ -0,0 +1,41 @@ +.. _michaelis_menten_kinetics_config: + +Michaelis Menten kinetics +~~~~~~~~~~~~~~~~~~~~~~~~~ + +**Group /input/model/unit_XXX/reaction - REACTION_MODEL = MICHAELIS_MENTEN** + +``MM_STOICHIOMETRY_BULK`` + + Stochiometric matrix :math:`S`. + The substrate component :math:`c_S` is identified by the index of the first negative entry in the stoichiometry of the corresponding reaction. + Input as reaction index major. + + ================ ============================= ======================================================== + **Type:** double **Range:** :math:`\mathbb{R}` **Length:** :math:`\texttt{NREACT} \cdot \texttt{NCOMP}` + ================ ============================= ======================================================== + +``MM_VMAX`` + + Limiting rate :math:`\mu_{\mathrm{max},j}` at saturation. + + ================ ============================= =================================== + **Type:** double **Range:** :math:`\mathbb{R}` **Length:** :math:`\texttt{NREACT}` + ================ ============================= =================================== + +``MM_KMM`` + + Michaelis constant :math:`k_{\mathrm{MM},j}`. + + ================ ============================= =================================== + **Type:** double **Range:** :math:`\mathbb{R}` **Length:** :math:`\texttt{NREACT}` + ================ ============================= =================================== + +``MM_KI`` + + Inhibition constant :math:`k_{\mathrm{I},j,i}` w.r.t component :math:`i` and reaction :math:`j`. If :math:`k_{\mathrm{I},j,i} <= 0`, the component does not inhibit the reaction. + Input as reaction index major. + + ================ ============================= ======================================================== + **Type:** double **Range:** :math:`\mathbb{R}` **Length:** :math:`\texttt{NREACT} \cdot \texttt{NCOMP}` + ================ ============================= ======================================================== \ No newline at end of file diff --git a/master/_sources/modelling/index.rst.txt b/master/_sources/modelling/index.rst.txt index 5741194d2..283c9f3a9 100644 --- a/master/_sources/modelling/index.rst.txt +++ b/master/_sources/modelling/index.rst.txt @@ -8,5 +8,5 @@ This section gives complete information about the supported unit operations, bin .. toctree:: unit_operations/index binding/index - reactions + reaction/index networks diff --git a/master/_sources/modelling/reaction/index.rst.txt b/master/_sources/modelling/reaction/index.rst.txt new file mode 100644 index 000000000..172908182 --- /dev/null +++ b/master/_sources/modelling/reaction/index.rst.txt @@ -0,0 +1,41 @@ +.. _reaction_models: + +Reaction models +=============== + + +Reaction models describe the (net) fluxes :math:`f_{\mathrm{react}}` of a +reaction mechanism. +CADET features one reaction type: + + - :ref:`mass_action_law_model` (most commonly used) + - :ref:`michaelis_menten_kinetics_model` + + +.. _dependence-on-external-function_react: + +Dependence on external function +------------------------------- + +A reaction model may depend on an external function or profile :math:`T\colon \left[ 0, T_{\mathrm{end}}\right] \times [0, L] \to \mathbb{R}`, where :math:`L` denotes the physical length of the unit operation, or :math:`T\colon \left[0, T_{\mathrm{end}}\right] \to \mathbb{R}` if the unit operation model has no axial length. +By using an external profile, it is possible to account for effects that are not directly modeled in CADET (e.g., temperature). +The dependence of each parameter is modeled by a polynomial of third degree. +For example, the forward rate constant :math:`k_{\mathrm{fwd}}` is really given by + +.. math:: + + \begin{aligned} + k_{\mathrm{fwd}}(T) &= k_{\mathrm{fwd},3} T^3 + k_{\mathrm{fwd},2} T^2 + k_{\mathrm{fwd},1} T + k_{\mathrm{fwd},0}. + \end{aligned} + +While :math:`k_{\mathrm{fwd},0}` is set by the original parameter ``XXX_KFWD`` of the file format (``XXX`` being a placeholder for the reaction model), the parameters :math:`k_{\mathrm{fwd},3}`, :math:`k_{\mathrm{fwd},2}`, and :math:`k_{\mathrm{fwd},1}` are given by ``XXX_KFWD_TTT``, ``XXX_KFWD_TT``, and ``XXX_KFWD_T``, respectively. +The identifier of the externally dependent reaction model is constructed from the original identifier by prepending ``EXT_`` (e.g., ``MASS_ACTION_LAW`` is changed into ``EXT_MASS_ACTION_LAW``). +This pattern applies to all parameters and supporting reaction models. +Note that the parameter units have to be adapted to the unit of the external profile by dividing with an appropriate power. + +Each parameter of the externally dependent reaction model can depend on a different external source. +The 0-based indices of the external source for each parameter is given in the dataset ``EXTFUN``. +By assigning only one index to ``EXTFUN``, all parameters use the same source. +The ordering of the parameters in ``EXTFUN`` is given by the ordering in the file format specification. + +For the configuration of external function dependence and more information on model parameters required to define in CADET file format, see section :ref:`FFReaction`. \ No newline at end of file diff --git a/master/_sources/modelling/reactions.rst.txt b/master/_sources/modelling/reaction/mass_action_law.rst.txt similarity index 70% rename from master/_sources/modelling/reactions.rst.txt rename to master/_sources/modelling/reaction/mass_action_law.rst.txt index 578801021..79b25a400 100644 --- a/master/_sources/modelling/reactions.rst.txt +++ b/master/_sources/modelling/reaction/mass_action_law.rst.txt @@ -1,13 +1,4 @@ -.. _reaction_models: - -Reaction models -=============== - - -Reaction models describe the (net) fluxes :math:`f_{\mathrm{react}}` of a -reaction mechanism. The most common mechanism is the mass action law. - -.. _MRMassActionLaw: +.. _mass_action_law_model: Mass action law --------------- @@ -59,7 +50,7 @@ where The forward and backward rates of the liquid phase particle reactions can be modified by a power of every bound state in the solid phase of the particle. The exponents of these powers are given by the matrices :math:`E^{ps}_{\mathrm{fwd}} = (e^{ps}_{\mathrm{fwd},m,j})` and :math:`E^{ps}_{\mathrm{bwd}} = (e^{ps}_{\mathrm{bwd},m,j})`, which are both of size :math:`(\sum_i N_{\mathrm{bnd},i}) \times N_{\mathrm{react}}`. -Whereas the exponent matrices :math:`E^{p}_{\mathrm{fwd}}, E^{p}_{\mathrm{bwd}} \in \mathbb{R}^{N_{\mathrm{comp}} \times N_{\mathrm{react}}}` are initialized based on the stoichiometric matrix :math:`S^{p} \in \mathbb{R}^{N_{\mathrm{comp}} \times N_{\mathrm{react}}}`, see Eq. :eq:`MRMassActionLawExpMatDefault`, the exponent matrices :math:`E^{ps}_{\mathrm{fwd}}, E^{ps}_{\mathrm{bwd}}` of the modifier terms default to :math:`0`. +Whereas the exponent matrices :math:`E^{p}_{\mathrm{fwd}}, E^{p}_{\mathrm{bwd}} \in \mathbb{R}^{N_{\mathrm{comp}} \times N_{\mathrm{react}}}` are initialized based on the stoichiometric matrix :math:`S^{p} \in \mathbb{R}^{N_{\mathrm{comp}} \times N_{\mathrm{react}}}`, see Eq. :eq:`MRMassActionLawExpMatDefault`, the exponent matrices :math:`E^{ps}_{\mathrm{fwd}}, E^{ps}_{\mathrm{bwd}}` of the modifier terms default to :math:`0`. Vice versa, the rates of solid phase reactions can be modified by liquid phase concentrations. The corresponding exponent matrices :math:`E^{sp}_{\mathrm{fwd}} = (e^{sp}_{\mathrm{fwd},\ell,j})` and :math:`E^{sp}_{\mathrm{bwd}} = (e^{sp}_{\mathrm{bwd},\ell,j})` are both of size :math:`N_{\mathrm{comp}} \times N_{\mathrm{react}}`. @@ -79,7 +70,7 @@ where - k^p_{\mathrm{bwd},j} &\left[\prod_{m=0}^{\sum_{i=0}^{N_{\mathrm{comp}}-1} N_{\mathrm{bnd},i}-1} \left(c^s_{m}\right)^{e^{s}_{\mathrm{bwd},m,j}}\right] \left[\prod_{\ell=0}^{N_{\mathrm{comp}}-1} \left(c^p_{\ell}\right)^{e^{sp}_{\mathrm{bwd},\ell,j}}\right]. \end{split} -Whereas the exponent matrices :math:`E^{s}_{\mathrm{fwd}}, E^{s}_{\mathrm{bwd}} \in \mathbb{R}^{(\sum_i N_{\mathrm{bnd},i}) \times N_{\mathrm{react}}}` are initialized based on the stoichiometric matrix :math:`S^{s} \in \mathbb{R}^{(\sum_i N_{\mathrm{bnd},i}) \times N_{\mathrm{react}}}`, see Eq. :eq:`MRMassActionLawExpMatDefault`, the exponent matrices :math:`E^{sp}_{\mathrm{fwd}}, E^{sp}_{\mathrm{bwd}}` of the modifier terms default to :math:`0`. +Whereas the exponent matrices :math:`E^{s}_{\mathrm{fwd}}, E^{s}_{\mathrm{bwd}} \in \mathbb{R}^{(\sum_i N_{\mathrm{bnd},i}) \times N_{\mathrm{react}}}` are initialized based on the stoichiometric matrix :math:`S^{s} \in \mathbb{R}^{(\sum_i N_{\mathrm{bnd},i}) \times N_{\mathrm{react}}}`, see Eq. :eq:`MRMassActionLawExpMatDefault`, the exponent matrices :math:`E^{sp}_{\mathrm{fwd}}, E^{sp}_{\mathrm{bwd}}` of the modifier terms default to :math:`0`. Correlation of forward- and backward rate constants @@ -115,33 +106,3 @@ This can be achieved by a (nonlinear) parameter transform F\left( k_{\mathrm{eq},i}, k_{\mathrm{bwd},i} \right) &= \begin{pmatrix} k_{\mathrm{eq},i} k_{\mathrm{bwd},i} \\ k_{\mathrm{bwd},i} \end{pmatrix} \\ \text{ with Jacobian } J_F\left( k_{\mathrm{eq},i}, k_{\mathrm{bwd},i} \right) &= \begin{pmatrix} k_{\mathrm{bwd},i} & k_{\mathrm{eq},i} \\ 0 & 1 \end{pmatrix}. \end{aligned} - - -.. _dependence-on-external-function_react: - -Dependence on external function -------------------------------- - -A reaction model may depend on an external function or profile :math:`T\colon \left[ 0, T_{\mathrm{end}}\right] \times [0, L] \to \mathbb{R}`, where :math:`L` denotes the physical length of the unit operation, or :math:`T\colon \left[0, T_{\mathrm{end}}\right] \to \mathbb{R}` if the unit operation model has no axial length. -By using an external profile, it is possible to account for effects that are not directly modeled in CADET (e.g., temperature). -The dependence of each parameter is modeled by a polynomial of third degree. -For example, the forward rate constant :math:`k_{\mathrm{fwd}}` is really given by - -.. math:: - - \begin{aligned} - k_{\mathrm{fwd}}(T) &= k_{\mathrm{fwd},3} T^3 + k_{\mathrm{fwd},2} T^2 + k_{\mathrm{fwd},1} T + k_{\mathrm{fwd},0}. - \end{aligned} - -While :math:`k_{\mathrm{fwd},0}` is set by the original parameter ``XXX_KFWD`` of the file format (``XXX`` being a placeholder for the reaction model), the parameters :math:`k_{\mathrm{fwd},3}`, :math:`k_{\mathrm{fwd},2}`, and :math:`k_{\mathrm{fwd},1}` are given by ``XXX_KFWD_TTT``, ``XXX_KFWD_TT``, and ``XXX_KFWD_T``, respectively. -The identifier of the externally dependent reaction model is constructed from the original identifier by prepending ``EXT_`` (e.g., ``MASS_ACTION_LAW`` is changed into ``EXT_MASS_ACTION_LAW``). -This pattern applies to all parameters and supporting reaction models. -Note that the parameter units have to be adapted to the unit of the external profile by dividing with an appropriate power. - -Each parameter of the externally dependent reaction model can depend on a different external source. -The 0-based indices of the external source for each parameter is given in the dataset ``EXTFUN``. -By assigning only one index to ``EXTFUN``, all parameters use the same source. -The ordering of the parameters in ``EXTFUN`` is given by the ordering in the file format specification in Section :ref:`FFReaction`. - -The layout of the matrices in the file format is presented in Table :ref:`FFReactionMassActionLaw`. - diff --git a/master/_sources/modelling/reaction/michaelis_menten_kinetics.rst.txt b/master/_sources/modelling/reaction/michaelis_menten_kinetics.rst.txt new file mode 100644 index 000000000..2efe7cc40 --- /dev/null +++ b/master/_sources/modelling/reaction/michaelis_menten_kinetics.rst.txt @@ -0,0 +1,40 @@ +.. _michaelis_menten_kinetics_model: + +Michaelis Menten kinetics +------------------------- + +Implements liquid phase Michaelis-Menten reaction kinetics of the form + +.. math:: + + \begin{aligned} + f_\text{react} = S \mathbf{\nu}, + \end{aligned} + +where :math:`S` is the stoichiometric matrix and the fluxes are given by + +.. math:: + + \begin{aligned} + \nu_j = \frac{\mu_{\mathrm{max},j} \, c_S}{k_{\mathrm{MM},j} + c_S}, + \end{aligned} + +where + +- :math:`j` is the reaction index, +- :math:`c_S` is the substrate component, +- :math:`\mu_{\mathrm{max},j}`, is the limiting rate approached by the system at saturation, +- :math:`k_{\mathrm{MM},j}` is the Michaelis constant, which is defined as the concentration of substrate at which the reaction rate is half ov :math:`\mu_{\mathrm{max},j}`. + + +In addition, the reaction might be inhibited by other components. +In this case, the flux has the form + +.. math:: + + \begin{aligned} + \nu_j = \frac{\mu_{\mathrm{max},j} \, c_S}{k_{\mathrm{MM},j} + c_S} \prod_i \frac{k_{\mathrm{I},j,i}}{k_{\mathrm{I},j,i} + c_{\mathrm{I},i}}. + \end{aligned} + +Here, :math:`k_{\mathrm{I},j,i}` is the inhibition constant w.r.t component :math:`i` and reaction :math:`j`. +If :math:`k_{\mathrm{I},j,i} <= 0`, component :math:`i` does not inhibit the reaction. diff --git a/master/developer_guide/cadet_core_architecture.html b/master/developer_guide/cadet_core_architecture.html index 58676a06e..49d78b4de 100644 --- a/master/developer_guide/cadet_core_architecture.html +++ b/master/developer_guide/cadet_core_architecture.html @@ -176,8 +176,8 @@

Implementation Details\(f_\text{ads,j}\left(c^p_j, c^s_j\right)\) (as used in, e.g., the General rate model (GRM)), according to some Binding models. This interface fully encapsulates a binding model. If supported, binding model parameters may have a Dependence on external function.

IDynamicReactionModel -Defines functions for evaluating the reaction rates \(f_\text{react,i}\left( c \right)\) (as used in, e.g., the General rate model (GRM)), according to some Reaction models. -If supported, reaction model parameters may have a Dependence on external function.

+Defines functions for evaluating the reaction rates \(f_\text{react,i}\left( c \right)\) (as used in, e.g., the General rate model (GRM)), according to some Reaction models. +If supported, reaction model parameters may have a Dependence on external function.

Unit Operation submodel assumption¶

Setup connections and switches¶

diff --git a/master/interface/binding/bi_steric_mass_action.html b/master/interface/binding/bi_steric_mass_action.html index 16098eb1c..0bac8431d 100644 --- a/master/interface/binding/bi_steric_mass_action.html +++ b/master/interface/binding/bi_steric_mass_action.html @@ -52,7 +52,7 @@

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    • @@ -115,10 +115,8 @@

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          • Specification of network connections
          • Valve switches
            • @@ -124,13 +124,13 @@

            Branches

            The network of unit operations uses “connectionâ€-variables \(c_{\text{con}}\) to connect the different unit operation ports with each other. The inlet port variables \(c_{\text{in},n,k}\) of unit operation \(n\) are attached to \(c_{\text{con},n}\) via

            -(23)¶\[\begin{aligned} +(22)¶\[\begin{aligned} c_{\text{in},n,k,i} &= c_{\text{con},n,k,i}, \qquad k = 1, \dots, N_{\text{port},\text{in},n},\quad i = 1, \dots, N_{\text{comp},n}. \end{aligned}\]

            While \(N_{\text{port},\text{in},n}\) denotes the number of inlet ports of unit operation \(n\), the number of outlet ports is given by \(N_{\text{port},\text{out},n}\). The connection variables \(c_{\text{con},n,k,i}\) collect all inflows of component \(i\) into port \(k\) of unit operation \(n\):

            -(24)¶\[\begin{aligned} +(23)¶\[\begin{aligned} c_{\text{con},n,k,i} &= \frac{\sum_{m=1}^{N_{\text{units}}} \sum_{\ell = 1}^{N_{\text{port},\text{out},n}} \sum_{j = 1}^{N_{\text{comp},m}} S_{(n,k,i),(m,\ell,j)} Q_{m,\ell} c_{\text{out},m,\ell,j}}{\sum_{m=1}^{N_{\text{units}}} \sum_{\ell=1}^{N_{\text{port},\text{out},m}} \hat{S}_{(n,k),(m,\ell)} Q_{m,\ell} }, \end{aligned}\]

            where \(F_{m,\ell}\) denotes the volumetric flow rate from outlet port \(\ell\) of unit operation \(m\), \(S_{(n,k,i),(m,\ell,j)} \in \{0, 1\}\) is a connection matrix indicating whether component \(i\) at outlet port \(k\) of unit operation \(n\) is connected to component \(j\) at inlet port \(\ell\) of unit operation \(m\), and \(\hat{S}_{(n,k),(m,\ell)} \in \{0, 1\}\) is another connection matrix indicating whether outlet port \(k\) of unit operation \(n\) is connected to inlet port \(\ell\) of unit operation \(m\), that is

            @@ -187,13 +187,13 @@

            Branches

            \[Q = Q_0 + Q_1(t - t_s) + Q_2(t-t_s)^2 + Q_3(t-t_s)^3,\]

            where \(t_s\) is the beginning of the time section that triggers the valve switch.

            -

            Note that the denominator in Eq. 24 must always be positive. +

            Note that the denominator in Eq. 23 must always be positive. That is, the flow rate coefficients have to be chosen such that the flow into every connected inlet port is strictly positive at all times.

    • Solution of the linear systems¶

    -

    Each time step in the simulation requires the solution of a nonlinear system Eq. 25 (see Sec. Time stepping). -The nonlinear problem is solved by a Newton iteration, which, in turn, requires the solution of a linear system that essentially consists of the Jacobians of the unit operations and some coupling matrices from Eqs. 23 and 24.

    +

    Each time step in the simulation requires the solution of a nonlinear system Eq. 24 (see Sec. Time stepping). +The nonlinear problem is solved by a Newton iteration, which, in turn, requires the solution of a linear system that essentially consists of the Jacobians of the unit operations and some coupling matrices from Eqs. 22 and 23.

    These linear systems are either solved in parallel or sequentially. The parallel method first solves each unit operation (in parallel) to compute the solution at its outlet. Using these values, the inlets are adjusted and the unit operations are solved again. This is iterated until the system is fully solved.

    diff --git a/master/modelling/reaction/index.html b/master/modelling/reaction/index.html new file mode 100644 index 000000000..04fdc01fc --- /dev/null +++ b/master/modelling/reaction/index.html @@ -0,0 +1,160 @@ + + + + + + + + Reaction models — CADET + + + + + + + + + + + + + + + + + + + + +
    + + +
    +
    + + +
    + + +

    + + + You're reading the documentation for a development version. + For the latest released version, please have a look at v4.4.0. + + +

    + + +
    +

    Reaction models¶

    +

    Reaction models describe the (net) fluxes \(f_{\mathrm{react}}\) of a +reaction mechanism. +CADET features one reaction type:

    +
    +
    +
    +
    +

    Dependence on external function¶

    +

    A reaction model may depend on an external function or profile \(T\colon \left[ 0, T_{\mathrm{end}}\right] \times [0, L] \to \mathbb{R}\), where \(L\) denotes the physical length of the unit operation, or \(T\colon \left[0, T_{\mathrm{end}}\right] \to \mathbb{R}\) if the unit operation model has no axial length. +By using an external profile, it is possible to account for effects that are not directly modeled in CADET (e.g., temperature). +The dependence of each parameter is modeled by a polynomial of third degree. +For example, the forward rate constant \(k_{\mathrm{fwd}}\) is really given by

    +
    +\[\begin{aligned} + k_{\mathrm{fwd}}(T) &= k_{\mathrm{fwd},3} T^3 + k_{\mathrm{fwd},2} T^2 + k_{\mathrm{fwd},1} T + k_{\mathrm{fwd},0}. +\end{aligned}\]
    +

    While \(k_{\mathrm{fwd},0}\) is set by the original parameter XXX_KFWD of the file format (XXX being a placeholder for the reaction model), the parameters \(k_{\mathrm{fwd},3}\), \(k_{\mathrm{fwd},2}\), and \(k_{\mathrm{fwd},1}\) are given by XXX_KFWD_TTT, XXX_KFWD_TT, and XXX_KFWD_T, respectively. +The identifier of the externally dependent reaction model is constructed from the original identifier by prepending EXT_ (e.g., MASS_ACTION_LAW is changed into EXT_MASS_ACTION_LAW). +This pattern applies to all parameters and supporting reaction models. +Note that the parameter units have to be adapted to the unit of the external profile by dividing with an appropriate power.

    +

    Each parameter of the externally dependent reaction model can depend on a different external source. +The 0-based indices of the external source for each parameter is given in the dataset EXTFUN. +By assigning only one index to EXTFUN, all parameters use the same source. +The ordering of the parameters in EXTFUN is given by the ordering in the file format specification.

    +

    For the configuration of external function dependence and more information on model parameters required to define in CADET file format, see section Reaction models.

    +
    +
    + + + +
    + +
    +
    +
    +
    +
    + ©2008-2024. + + | + Page source +
    + + + + + + \ No newline at end of file diff --git a/master/modelling/reactions.html b/master/modelling/reaction/mass_action_law.html similarity index 54% rename from master/modelling/reactions.html rename to master/modelling/reaction/mass_action_law.html index 7fef63d88..58a3641d7 100644 --- a/master/modelling/reactions.html +++ b/master/modelling/reaction/mass_action_law.html @@ -1,25 +1,23 @@ - + - Reaction models — CADET - - - - - + Mass action law — CADET + + + + + - - - - - - + + + + - + @@ -30,7 +28,7 @@
    -

    Correlation of forward- and backward rate constants¶

    +

    Correlation of forward- and backward rate constants¶

    Note that forward rate constant \(k_{\mathrm{fwd},i}\) and backward rate constant \(k_{\mathrm{bwd},i}\) of reaction \(i\) are linearly correlated due to the form of the equilibrium constant @@ -184,27 +167,6 @@

    Correlation of forward- and backward rate constants

    -
    -

    Dependence on external function¶

    -

    A reaction model may depend on an external function or profile \(T\colon \left[ 0, T_{\mathrm{end}}\right] \times [0, L] \to \mathbb{R}\), where \(L\) denotes the physical length of the unit operation, or \(T\colon \left[0, T_{\mathrm{end}}\right] \to \mathbb{R}\) if the unit operation model has no axial length. -By using an external profile, it is possible to account for effects that are not directly modeled in CADET (e.g., temperature). -The dependence of each parameter is modeled by a polynomial of third degree. -For example, the forward rate constant \(k_{\mathrm{fwd}}\) is really given by

    -
    -\[\begin{aligned} - k_{\mathrm{fwd}}(T) &= k_{\mathrm{fwd},3} T^3 + k_{\mathrm{fwd},2} T^2 + k_{\mathrm{fwd},1} T + k_{\mathrm{fwd},0}. -\end{aligned}\]
    -

    While \(k_{\mathrm{fwd},0}\) is set by the original parameter XXX_KFWD of the file format (XXX being a placeholder for the reaction model), the parameters \(k_{\mathrm{fwd},3}\), \(k_{\mathrm{fwd},2}\), and \(k_{\mathrm{fwd},1}\) are given by XXX_KFWD_TTT, XXX_KFWD_TT, and XXX_KFWD_T, respectively. -The identifier of the externally dependent reaction model is constructed from the original identifier by prepending EXT_ (e.g., MASS_ACTION_LAW is changed into EXT_MASS_ACTION_LAW). -This pattern applies to all parameters and supporting reaction models. -Note that the parameter units have to be adapted to the unit of the external profile by dividing with an appropriate power.

    -

    Each parameter of the externally dependent reaction model can depend on a different external source. -The 0-based indices of the external source for each parameter is given in the dataset EXTFUN. -By assigning only one index to EXTFUN, all parameters use the same source. -The ordering of the parameters in EXTFUN is given by the ordering in the file format specification in Section Reaction models.

    -

    The layout of the matrices in the file format is presented in Table Group /input/model/unit_XXX/reaction - REACTION_MODEL = MASS_ACTION_LAW.

    -
    - @@ -218,7 +180,7 @@

    Correlation of forward- and backward rate constantsPage source diff --git a/master/modelling/reaction/michaelis_menten_kinetics.html b/master/modelling/reaction/michaelis_menten_kinetics.html new file mode 100644 index 000000000..e6115cad2 --- /dev/null +++ b/master/modelling/reaction/michaelis_menten_kinetics.html @@ -0,0 +1,145 @@ + + + + + + + + Michaelis Menten kinetics — CADET + + + + + + + + + + + + + + + + + + +
    + + +
    +
    + + +
    + + +

    + + + You're reading the documentation for a development version. + For the latest released version, please have a look at v4.4.0. + + +

    + + +
    +

    Michaelis Menten kinetics¶

    +

    Implements liquid phase Michaelis-Menten reaction kinetics of the form

    +
    +\[\begin{aligned} + f_\text{react} = S \mathbf{\nu}, +\end{aligned}\]
    +

    where \(S\) is the stoichiometric matrix and the fluxes are given by

    +
    +\[\begin{aligned} + \nu_j = \frac{\mu_{\mathrm{max},j} \, c_S}{k_{\mathrm{MM},j} + c_S}, +\end{aligned}\]
    +

    where

    +
      +

    • \(j\) is the reaction index,

      • +

    • \(c_S\) is the substrate component,

      • +

    • \(\mu_{\mathrm{max},j}\), is the limiting rate approached by the system at saturation,

      • +

    • \(k_{\mathrm{MM},j}\) is the Michaelis constant, which is defined as the concentration of substrate at which the reaction rate is half ov \(\mu_{\mathrm{max},j}\).

      • +
    +

    In addition, the reaction might be inhibited by other components. +In this case, the flux has the form

    +
    +\[\begin{aligned} + \nu_j = \frac{\mu_{\mathrm{max},j} \, c_S}{k_{\mathrm{MM},j} + c_S} \prod_i \frac{k_{\mathrm{I},j,i}}{k_{\mathrm{I},j,i} + c_{\mathrm{I},i}}. +\end{aligned}\]
    +

    Here, \(k_{\mathrm{I},j,i}\) is the inhibition constant w.r.t component \(i\) and reaction \(j\). +If \(k_{\mathrm{I},j,i} <= 0\), component \(i\) does not inhibit the reaction.

    +
    + + + +
    + +
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This ansatz is inserted into the DAE and the resulting nonlinear algebraic equation system is solved for \(y_{\tau} = y\left( t_\tau \right)\) using a Newton method.

    -(25)¶\[\begin{aligned} +(24)¶\[\begin{aligned} H(y_\tau) = F\left(t_\tau,y_\tau,\frac{1}{\Delta t_\tau} \,\sum_{i=0}^{q_\tau}{\alpha_{\tau,i} \, y_{\tau-i}}, p\right) = 0 \end{aligned}\]

    This requires (possibly many) solutions of linear equation systems involving the Jacobian of \(H\) given by

    @@ -221,11 +221,11 @@

    Branches

    Adaptivity¶

    The BDF order \(q_{\tau}\) and size of the time step \(t_\tau\) are chosen such that the error bounds are satisfied with minimal work. -Equation 25 is solved with a Newton iteration using very few iterations (MAX_NEWTON_ITER from Table Group /solver/time_integrator) since the starting point is assumed to be close to the solution as the time steps are small. +Equation 24 is solved with a Newton iteration using very few iterations (MAX_NEWTON_ITER from Table Group /solver/time_integrator) since the starting point is assumed to be close to the solution as the time steps are small. If the solution of the Newton iteration does not pass the convergence test, the step size \(t_\tau\) is reduced and the Newton iteration is tried again. This may happen at most MAX_CONVTEST_FAIL times, otherwise time integration is aborted with failure. Having a solution of the Newton iteration at hand, a local truncation error test is performed which controls the error of the BDF method. -If this test fails, step size \(t_\tau\) and order \(q_{\tau}\) are adapted and the process restarts with solving the nonlinear system Eq. 25. +If this test fails, step size \(t_\tau\) and order \(q_{\tau}\) are adapted and the process restarts with solving the nonlinear system Eq. 24. The error test may be failed at most MAX_ERRTEST_FAIL times before time integration is aborted with failure.

    Error bounds for the local truncation error test are specified by an absolute tolerance (ABSTOL) and a relative tolerance (RELTOL). Note that the relative tolerance only works for non-zero values, whereas zero values are accounted for by the absolute tolerance. @@ -255,7 +255,7 @@

    Branches

    \end{aligned}\end{split}\]

    These linear DAEs depend on the solution \(y\), \(\dot{y}\) of the original DAE system. Consequently, the sensitivity systems are solved together with the original DAE system in a staggered approach [25].

    -

    After the nonlinear Eq. 25 has been successfully solved using Newton iteration (i.e., it has passed the convergence test), each sensitivity is solved using the same Newton iteration. +

    After the nonlinear Eq. 24 has been successfully solved using Newton iteration (i.e., it has passed the convergence test), each sensitivity is solved using the same Newton iteration. If direct linear solvers were used, this iteration would convergence with a single iteration as there is no nonlinearity. The Newton method for the sensitivities performs at most MAX_NEWTON_ITER_SENS (see Table Group /solver/time_integrator) iterations. 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    diff --git a/v4.4.0/.buildinfo b/v4.4.0/.buildinfo index 0b99b06cf..13fa63416 100644 --- a/v4.4.0/.buildinfo +++ b/v4.4.0/.buildinfo @@ -1,4 +1,4 @@ # Sphinx build info version 1 # This file hashes the configuration used when building these files. When it is not found, a full rebuild will be done. -config: 37c0dc83db4a7fb58ddc6415554c6072 +config: 71c134945c27b716584f9cfa1da3e5ea tags: 645f666f9bcd5a90fca523b33c5a78b7 diff --git a/v4.4.0/.doctrees/README.doctree b/v4.4.0/.doctrees/README.doctree index 4ded5507c2de96de8d52cec93b442e6e9ecd8bea..adb0bfbbded95cc657f3ec8d4c1c7e401056ddfb 100644 GIT binary patch delta 21 ccmcbnaZO`GA|ppeYPn^qNn!EkOvblD09_XdEdT%j delta 21 ccmcbnaZO`GA|pqESz2O7N?FGj|UCnVrRaGL=2sZDh}+3c2jKku6P|%MO-P znMA)M>4E8O&zZh=daF4+wBWh_-nmPhij zF#}a>7a37%^>_>|kcdQ~r9x_?dn7edGR{t$E9CFS%b8N@j8UM%(A@EJ#S*YxC<2v+ z*_w%E$I1XaZO$_&JClpYGDZ|wqKcUz5id^1P{mxi5I3SmHl9nQvS@BwxilKNHd3Uf zFUf+e#i%F~qz>f^xwH{Sw`gHH0r|0VDVYQP1mVKH97Thc%b!N$x%|08YAlJ5+IDW= zer;su_MJd}0QTJMXPa&5p-P?VSm3QwwpwYe4W_Zjf0imU6ER)GiL`Tc# z&!3By3bAZ46Du(RC(R9xcofYUD;ULMG@gkSiy+ygxoA9AEE$C;bs4iIl}!+VmP&*c z%cG;Ivph}~bQg+f{W5%BDwN|T)HOk6qtkbn=+!zNE1XW`CP1qB-Q$Vr;HK6q($H@Mq~tB2ONIUV5(RG+9{`OZXA!D zO^u^)@SALmhDJ1#%Hm;U$55Fevja6qMp2;=j5G{ZA{HkZcLVL{Z(K#Q^&asP;Aye- zg=nN5EgE2}bGuQ4Ql=OM!%2;v0}>WfERHT(C>4olW$@4_cqfG4kh!=B;*{<#qet@~ zaYUmriIDks_o2Bfomq1178!MI7Vlx+J1-qWpKN8J2KEhtK#+p6Ud=Uow|T&=(jqc&|=Oj#u-lX^=2!_%PBCX*UK{EIg~!VO~%HMXCT(jKg4ig4$RFJ z<>YhJ<-wIXtvy*9&qU+krFN32rN#2>W9w+J#absn-jmZ1o>Uy`XUmWhq99GNydREEvV zIvr_>ep}7O&Z?NVus9A!!oArr5RnZNN2IY4#v`1{mW;6iSY#p$b`*yc3sbc+XzN%w zmQ93<7=l9Xf%g)^4_g+CA}dv6LI{Mi2s=n+498!zf$AkicvI0v5>sz7rAUKgZfbSL@6t{ zLP7vge@lTvgkw(58(?rC)M-juS&_uJK{Y@A0nm-}iup_mC}Uk@Ql^on6%}fU4TI?? z(g--xSBvji9i*hF)L<+nMbwgp1a&acV$#q=xin<0FNS2A6UvZmC6}vIzYFGCQ7q{c z0w$o0lSD+7WRp0LrzS`feT9Y`X;8qPzzN8cFO%kCsj)4r#)AJbd`N0&Ay6wIdkB$& z7((umV6oaduRCl(l(n&#H@d07fJ|3oQ5Ipbv3%XOe9-&`iva~@q0>h7lD2vE!&QhF`@0L|y!3U)aVggl=<7#)5 zSP|lo8}x%T_#i7v=5=kAzWq7KOTt@9f7Eyv zxRI4)Y3`Dy)K9D=n%A(Ff_`G9l}T_-nw?f03TQ4W=9CxAOYBrNE=#cjQdv^1v9g(4 zSknQT?yNg}+u&e$sP|ZZ`1<~1y+;Rc?(OS4**`pdp@>wm4rdvRFZsN~CAN%Im)7>}KXq!3+^ED{Li!0j6|i_StZg#KJA zk@)L~M^lL?rvESim1zQk?$;Ry^9GBrNM06?3MJ>XaSoj+LBHK3i|_^gz=Fvl13&EF_uk0?}C5>Ls!WO;|xvrdAiXQ_rj$lZ`QKssB@*ro*yOQy-mTuT<3{TWDlXR=dt z!OE7%VN3`s9z|G+aw?aW%Vq>Gq~96Do^6BFq}fW=VTtliX(^^?RL<4SRczE?$`w479O) z@|={%q=Ijf*~I^A{A9KKiOuk31JiIJJ)~guxR#bGVmu*`6E2pH+0lknsTCCy zCUz3Y?Z(*{iQQt4n4z1SY&%R3w1yH*t7t1ucPkmhpUVI8=2Fq{YiL~;;njBYX+U!vkv&xlUKyS!p zVK$7$U^-+w{dk!=!A3|OBSQ^{vOTbeQED?kdx7L?Ri|2zi%@}LVl`?_bv-2Nx)dDD z4Phq5(i>SmU|xX`J}>5h=4$mT(K(`Hg&Z^kbGyoLSEDvIvHD`sqL|B&0o1&zx_Xsp zA_IcJ(Aw%Blgb#Q5u`v@5(<;KJk3jp#tZ=VOQ;+$c7R@_Z*EcF0&P6SqmFF#0+46| z5=s4tScrv%9V*$41yTbiFLnfQDyPIv|CT60t{*z!v$C(<)A^gC_^>hp&(n;sKm4b zSAsf@-e7%GG%}-@m=$yJlpGZv-L`i7>Y!JCazzkp zlPbNk8wggugzN>F?+WXKxvsKGJafTJ42EuHp05%M82A|j8pj${oJD8Zrj6c}oIp0& zRh+|RCT-qm%dC-DDV~hRlZ712d{WthZL+esR2H2XO9yO-Grc%WZiKAG%n-yfG$|Yo zQR3C_{l3|&C|QYq?@?&6PY(%Gj1^BqHQB3Bd~qNh!n5+jor$Q|pz+{}NeFWiY$eE9 zOah4Q5pzG2f^ERuqFQTFzj~A(sQwX6g1|Uw_1}jEmsNl4rD5z+II=226&Ljdn@pZd z6mt0_N{eGA#fl8NphxGGIA`0blOf7Z9q}%wCW1QD?AgLsZHKrU`RU@NPl3u1D zTP^lN%w~nZ%Hm{&oazL&?-;NUcdC!Ta!6?fvNl&T3jPrUP#!uP&++%E&nnB3%(Rx8 zcV#7PvE)W;7*Sf%!gPlBtIUY4JJhyS3Q5AC6y-^am+o*8>g<)OofhgL|Gcp8C_j~@ zg(0c+IT))2VYFb76$MOR2$+qk>Z6(RSxlC=lhMWR7$K0&Df*wR(|0S z<1n?^9Jf%y#<$ffD%CrzozvXFLrbZIYspRs+G^a@`mw-+qn(wVssn3e{bKhehgC8_8P4Sp^MSFDFkrs-+IXgk11e1%2nvw4z_1Cv$DruQ7OSow3PvR zbFpUxeDE|Zzq#^Q(#5umw9k>H5v$+6-raIIok|AzY6v3V`n#N&$07hCU zUt&5(){kV4tQ0etL^_nrHI-bBwqUWMbwCwq)qCb5a&0BvXj9CYlrb@lO+2(+br-g) z!cLQZ3pUu=3F)^YmOPMtn|S>(UKi4TLcds0qCH{hcTkuNFFPVTwnujC3h&r;VEeTP zwr`h)m-IX72h}zgY>eD)>>J&^&oJVN*p9vN-TOxO?ccF?_xA0(cU>FXyL0F6JtKQZ zjp_8e2|%^6Wu^ew*phxP0nxIuNX(?)--v2_5J3JrOe+o@&+0jqqq0H_SUW^#(03h@ z)I&X$!{rGBh%|uIux)tg)Rh!+iCqu7zzW5qEQJuv=Xcavu%()pRnG~UIr9%RJ9cPh zl>*dK&pl=hawzysd7qfW!hT$d*80<82Y;*lu!j2Qm_1n>vAX}w#+=kci=T?s zo&Nl;PpXNI!WLC54=^A1Xw`MhISWc$9z&fVc1`wr~dbztW%pLxfO57*V2 zcgz$(1Lhr!h#K>bi>t;H^A4yK|GO7nYs@>mRy^pmCT)WI-B?ropFY#^S|9Sm@PGU4 z=%QCTpO{WL6m~C?`z>KTu_Rcv@IM>Vum7)>1RFMk```etow zloq>&tiCmIqWHgN(XDwi(N%PM$>^zAds7qB8u=dc)Jd^E)-KK*6nn5ZiG5Ozl``I< zykk#z$DRYb_hYflXU4jBWRI~YzB9gK`$*#2y~c=f?f#KH*Y4cEJ-*A>v2Wk#$mr;f z-Me<}s<~1&Qvi+`>rtKk$le&NwQzQEwfgUx!0uCavr5CVT%M`5S;b+Jwv3>a3py1` zZcpq<7F)(E(WJJ8IJQ$-IJNdx1lg1PKXz*DSUYImcH{q1%ZgqzuiyE}_6!y|9Q#8q zdTL&4J8_^X`|ryB;3pPKXJN6_V{^h28-Do&gq$FFY?`c#p#R?W*_qDnpV-T59ZmaR zvX^%jSBamnF*L)*@Wc+^(i1y;?c;fl9lm8ajdUEymd*$^RUKiLXYq@0nncdcR?tWB za)JDD+AjQ*bK6eJCO0|oDBX4_$1lToZ50Q`pXq5=*k*kxT{}3wY`sgET+}bb0di_I4nv?>?RSw!pdQ-6 z+|zxP^*ij`!{G}Yq_+JuF1iQ5>!uUYa7(a~-PU4R6kas3KkIZ0_hgR>Epd zfSUr#k%M%y`x4-jvn(eM#dC16$6hYia5~PRVLv>ar`hfwaOQnIXStK3@ZS=J44M*c7IvD208ZMcG9_|fH-D6J;IfrJMEj&OPja&$sxiurP~jT~ zs36ZbXN+l7UF1nWFyy-;%HF+%qXmXlB4>JVSYf#|fGJ z11Bit{S{Sc7oWh{1$z%bYjDnp`~{Pr0ru5egqsogn_25l

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     diff --git a/v4.4.0/modelling/reactions.html b/v4.4.0/modelling/reactions.html index 16c03fc31..b01e07436 100644 --- a/v4.4.0/modelling/reactions.html +++ b/v4.4.0/modelling/reactions.html @@ -82,7 +82,7 @@

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