Add linear driving force implementations of certain isotherms

The following isotherms are added:
- Freundlich LDF
- Multi-component-Langmuir LDF
- Multi-component-Langmuir LDF liquid phase
- Multi-component-Bi-Langmuir LDF

For more information on linear driving force models, refer to the documentation.

doc/README.md

@@ -18,5 +18,4 @@ To build the documentation for all releases and the master branch, run:
 
 `sphinx-multiversion ./ ./build/`. 
 
-This documentation is published under https://cadet.github.io
 Any changes to the documentation will automatically be pushed to the github-pages repository (https://github.com/cadet/cadet.github.io) using github actions.

doc/interface/binding/bi_steric_mass_action.rst

@@ -12,9 +12,9 @@ Bi Steric Mass Action
    bound states. Otherwise, the adsorption mode is set for each bound
    state separately.
 
-===================  =========================  =========================================
-**Type:** int        **Range:** {0,1}  		**Length:** 1/NTOTALBND
-===================  =========================  =========================================
+===================  =========================  =======================
+**Type:** int        **Range:** {0,1}           **Length:** 1/NTOTALBND
+===================  =========================  =======================
 
 ``BISMA_KA``
    Adsorption rate constants in state-major ordering

doc/interface/binding/freundlich_ldf.rst

@@ -0,0 +1,52 @@
+.. _freundlich_ldf_config:
+
+Freundlich LDF
+~~~~~~~~~~~~~~~
+
+**Group /input/model/unit_XXX/adsorption – ADSORPTION_MODEL = FREUNDLICH_LDF**
+
+
+``IS_KINETIC``
+   Selects kinetic or quasi-stationary adsorption mode: 1 = kinetic, 0 =
+   quasi-stationary. If a single value is given, the mode is set for all
+   bound states. Otherwise, the adsorption mode is set for each bound
+   state separately.
+
+===================  =========================  ==================================
+**Type:** int        **Range:** {0,1}   	 **Length:** 1/NTOTALBND
+===================  =========================  ==================================  
+
+``FLDF_KKIN``
+   Driving force coefficient for each component
+
+
+**Unit:** :math:`s^{-1}`
+
+===================  =========================  ==================================
+**Type:** double     **Range:** :math:`\ge 0`    **Length:** 1/NTOTALBND
+===================  =========================  ==================================  
+
+
+``FLDF_KF``
+   Freundlich coefficient for each component
+
+**Unit:** :math:`m_{MP}^3~mol^{-1}`
+
+===================  =========================  ==================================
+**Type:** double     **Range:** :math:`\ge 0`    **Length:** 1/NTOTALBND
+===================  =========================  ==================================  
+
+``FLDF_N``
+   Freundlich exponent for each component
+
+**Unit:** :[-]
+
+===================  =========================  ==================================
+**Type:** double     **Range:** :math:`> 0`      **Length:** 1/NTOTALBND
+===================  =========================  ==================================  
+
+
+
+
+
+

doc/interface/binding/multi_component_bi_langmuir.rst

@@ -13,11 +13,11 @@ Multi Component Bi-Langmuir
    state separately.
 
 ===================  =========================  =========================================
-**Type:** int        **Range:** {0,1}  		**Length:** 1/NTOTALBND
+**Type:** int        **Range:** {0,1}           **Length:** 1/NTOTALBND
 ===================  =========================  =========================================
 
 ``MCBL_KA``
-   Adsorption rate constants in state-major ordering
+   Adsorption rate constants in state-major ordering (see :ref:`ordering_multi_dimensional_data`)
 
 **Unit:** :math:`m_{MP}^3~mol^{-1}~s^{-1}`
 

doc/interface/binding/multi_component_bi_langmuir_ldf.rst

@@ -0,0 +1,44 @@
+.. _multi_component_bi_langmuir_ldf_config:
+
+Multi Component Bi-Langmuir LDF
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+**Group /input/model/unit_XXX/adsorption – ADSORPTION_MODEL = MULTI_COMPONENT_BILANGMUIR_LDF**
+
+
+``IS_KINETIC``
+   Selects kinetic or quasi-stationary adsorption mode: 1 = kinetic, 0 =
+   quasi-stationary. If a single value is given, the mode is set for all
+   bound states. Otherwise, the adsorption mode is set for each bound
+   state separately.
+
+===================  =========================  =========================================
+**Type:** int        **Range:** {0,1}           **Length:** 1/NTOTALBND
+===================  =========================  =========================================
+
+``MCBLLDF_KEQ``
+   Equillibrium loading constants in state-major ordering (see :ref:`ordering_multi_dimensional_data`)
+
+**Unit:** :math:`m_{MP}^3~mol^{-1}`
+
+===================  =========================  =========================================
+**Type:** double     **Range:** :math:`\ge 0`   **Length:** NCOMP :math:`\cdot` NSTATES
+===================  =========================  =========================================
+
+``MCBLLDF_KKIN``
+   Linear driving force coefficients in state-major ordering
+
+**Unit:** :math:`s^{-1}`
+
+===================  =========================  =========================================
+**Type:** double     **Range:** :math:`\ge 0`   **Length:** NCOMP :math:`\cdot` NSTATES
+===================  =========================  =========================================
+
+``MCBLLDF_QMAX``
+   Maximum adsorption capacities in state-major ordering
+
+**Unit:** :math:`mol~m_{SP}^{-3}`
+
+===================  =========================  =========================================
+**Type:** double     **Range:** :math:`\gt 0`   **Length:** NCOMP :math:`\cdot` NSTATES
+===================  =========================  =========================================

doc/interface/binding/multi_component_langmuir_ldf.rst

@@ -0,0 +1,44 @@
+.. _multi_component_langmuir_ldf_config:
+
+Multi Component Langmuir LDF
+============================
+
+**Group /input/model/unit_XXX/adsorption – ADSORPTION_MODEL = MULTI_COMPONENT_LANGMUIR_LDF**
+
+
+``IS_KINETIC``
+   Selects kinetic or quasi-stationary adsorption mode: 1 = kinetic, 0 =
+   quasi-stationary. If a single value is given, the mode is set for all
+   bound states. Otherwise, the adsorption mode is set for each bound
+   state separately.
+
+===================  =========================  =========================================
+**Type:** int        **Range:** {0,1}  		    **Length:** 1/NTOTALBND
+===================  =========================  =========================================
+
+``MCLLDF_KEQ``
+   Equillibrium loading constants
+
+**Unit:** :math:`m_{MP}^3~mol^{-1}`
+
+===================  =========================  =========================================
+**Type:** double     **Range:** :math:`\ge 0`   **Length:** NCOMP
+===================  =========================  =========================================
+
+``MCLLDF_KKIN``
+   Linear driving force coefficients
+
+**Unit:** :math:`s^{-1}`
+
+===================  =========================  ==================================
+**Type:** double     **Range:** :math:`\ge 0`   **Length:** NCOMP
+===================  =========================  ================================== 
+
+``MCLLDF_QMAX``
+   Maximum adsorption capacities
+
+**Unit:** :math:`mol~m_{SP}^{-3}`
+
+===================  =========================  ==================================
+**Type:** double     **Range:** :math:`\gt 0`   **Length:** NCOMP
+===================  =========================  ================================== 

doc/interface/binding/multi_component_langmuir_ldf_liquid_phase.rst

@@ -0,0 +1,44 @@
+.. _multi_component_langmuir_ldf_liquid_phase_config:
+
+Multi Component Langmuir LDF Liquid Phase
+==========================================
+
+**Group /input/model/unit_XXX/adsorption – ADSORPTION_MODEL = MULTI_COMPONENT_LANGMUIR_LDF_LIQUID_PHASE**
+
+
+``IS_KINETIC``
+   Selects kinetic or quasi-stationary adsorption mode: 1 = kinetic, 0 =
+   quasi-stationary. If a single value is given, the mode is set for all
+   bound states. Otherwise, the adsorption mode is set for each bound
+   state separately.
+
+===================  =========================  =========================================
+**Type:** int        **Range:** {0,1}  		    **Length:** 1/NTOTALBND
+===================  =========================  =========================================
+
+``MCLLDFC_KEQ``
+   Equillibrium loading constants
+
+**Unit:** :math:`m_{MP}^3~mol^{-1}`
+
+===================  =========================  =========================================
+**Type:** double     **Range:** :math:`\ge 0`   **Length:** NCOMP
+===================  =========================  =========================================
+
+``MCLLDFC_KKIN``
+   Linear driving force coefficients
+
+**Unit:** :math:`s^{-1}`
+
+===================  =========================  ==================================
+**Type:** double     **Range:** :math:`\ge 0`   **Length:** NCOMP
+===================  =========================  ================================== 
+
+``MCLLDFC_QMAX``
+   Maximum adsorption capacities
+
+**Unit:** :math:`mol~m_{SP}^{-3}`
+
+===================  =========================  ==================================
+**Type:** double     **Range:** :math:`\gt 0`   **Length:** NCOMP
+===================  =========================  ================================== 

doc/interface/introduction.rst

@@ -77,6 +77,7 @@ Common notation and identifiers that are used in the subsequent description are
    * - PARAM_VALUE
      - Value of a generic unspecified parameter
 
+.. _ordering_multi_dimensional_data:
 
 Ordering of multi dimensional data
 ----------------------------------
@@ -85,14 +86,21 @@ Some model parameters, especially in certain binding models, require multi dimen
 Since CADET only reads one dimensional arrays, the layout of the data has to be specified (i.e., the way how the data is linearized in memory).
 The term “*xyz*-major” means that the index corresponding to *xyz* changes the slowest.
 
-For instance, suppose a model with :math:`2` components and :math:`3` bound states has a “component-major” dataset.
-Then, the requested matrix is stored in memory such that all bound states are listed for each component (i.e., the component index changes the slowest and the bound state index the fastest):
+For instance, suppose a model with :math:`2` components and :math:`3` bound states has a “state-major” dataset.
+Then, the requested matrix is stored in memory such that all components are listed for each bound state (i.e., the bound state index changes the slowest and the component index the fastest):
 
 ::
 
-     comp0bnd0, comp0bnd1, comp0bnd2, comp1bnd0, comp1bnd1, comp1bnd2.
+     comp0bnd0, comp1bnd0, comp0bnd1, comp1bnd1, comp0bnd2, comp1bnd2
 
-This linear array can be represented as a :math:`2 \times 3` matrix in “row-major” storage format, or a :math:`3 \times 2` matrix in “column-major” ordering.
+
+This linear array can also be represented as a :math:`3 \times 2` matrix in “row-major” storage format:
+
+::
+
+     comp0bnd0, comp1bnd0
+     comp0bnd1, comp1bnd1
+     comp0bnd2, comp1bnd2
 
 
 .. _section_dependent_parameters:

doc/literature.bib

@@ -374,4 +374,43 @@ @article{Huuk2017
 url = {http://doi.wiley.com/10.1002/biot.201600336},
 volume = {12},
 year = {2017}
+}
+@article{Alpay1992,
+title = {The linear driving force model for fast-cycle adsorption and desorption in a spherical particle},
+journal = {Chemical Engineering Science},
+volume = {47},
+number = {2},
+pages = {499-502},
+year = {1992},
+issn = {0009-2509},
+doi = {https://doi.org/10.1016/0009-2509(92)80041-A},
+url = {https://www.sciencedirect.com/science/article/pii/000925099280041A},
+author = {E. Alpay and D.M. Scott}
+}
+@incollection{Singh2016,
+title = {Chapter 8 - Nanoparticle Ecotoxicology},
+editor = {Ashok K. Singh},
+booktitle = {Engineered Nanoparticles},
+publisher = {Academic Press},
+address = {Boston},
+pages = {343-450},
+year = {2016},
+isbn = {978-0-12-801406-6},
+doi = {https://doi.org/10.1016/B978-0-12-801406-6.00008-X},
+url = {https://www.sciencedirect.com/science/article/pii/B978012801406600008X},
+author = {Ashok K. Singh},
+keywords = {Anthropogenic sphere, Atmosphere, Biosphere, Climate, Environment, Fate of nanoparticles, Human activity, Hydrosphere, Lithosphere, Man made, Nanoparticles, Nanotoxicology, Natural},
+abstract = {Although the physicochemical and beneficial properties of nanoparticles have drawn widespread attention in recent years, their potential for causing environmental damage and ecological toxicity (ecotoxicity) is not fully understood. This chapter provides contemporary information regarding characteristics of the environment (atmosphere, hydrosphere, lithosphere, and biosphere), distribution of bulk and nanoparticles in different spheres, fate of nanoparticles (in air, water, and soil), and adverse effects of different groups of nanoparticles on various types of ecosystems. Although many nanoparticles have been shown to cause potent ecological damage, the results from studies often differ, possibly due to heterogeneity in the physical properties of nanoparticles and the environment, the species and habitat differences, nanoparticles' unique electronic and chemical properties, differences in biomarkers used, and presence of toxins and other contaminants associated with the nanoparticle that may modulate its toxicity. Elucidating these issues may represent a critical future research direction to establish the safe use of nanoparticles.}
+}
+@article{Benedikt2019,
+author = {Aumeier, Benedikt M. and Dang, Anh H. Q. and Ohs, Burkhard and Yüce, Süleyman and Wessling, Matthias},
+title = {Aqueous-Phase Temperature Swing Adsorption for Pesticide Removal},
+journal = {Environmental Science \& Technology},
+volume = {53},
+number = {2},
+pages = {919-927},
+year = {2019},
+doi = {10.1021/acs.est.8b05873},
+URL = { https://doi.org/10.1021/acs.est.8b05873},
+eprint = { https://doi.org/10.1021/acs.est.8b05873}
 }

doc/modelling/binding/freundlich_ldf.rst

@@ -0,0 +1,44 @@
+.. _freundlich_ldf_model:
+
+Freundlich LDF
+~~~~~~~~~~~~~~~
+
+The Freundlich isotherm model is an empirical model that considers changes in the equilibrium constant of the binding process due to the heterogeneity of the surface and the variation of the interaction strength :cite:`Benedikt2019,Singh2016`.
+This variant of the model is based on the linear driving force approximation (see section :ref:`ldf_model`) and is given as
+
+.. math::
+	\begin{aligned} 
+		q^*_i= k_{F,i}c_{p,i}^{\frac{1}{n_{i}}}  && i = 0, \dots, N_{\text{comp}} - 1.
+	\end{aligned}
+
+No interaction between the components is considered when the model has multiple components. 
+One of the limitation of this isotherm is the first order Jacobian :math:`\left(\frac{dq^*}{dc_p}\right)` tends to infinity as :math:`c_{p} \rightarrow 0` for :math:`n>1`.
+To address this issue an approximation of isotherm is considered near the origin.
+This approximation matches the isotherm in such a way that  :math:`q=0` at :math:`c_p=0` and also matches the first derivative of the istotherm at :math:`c_p = \epsilon`, where :math:`\epsilon` is a very small number, for example :math:`1e-14`.
+The form of approximation and its derivative is given below for :math:`c_p < \varepsilon` and :math:`n>1`:
+
+.. math::
+
+	\begin{aligned} 
+		q^* = \alpha_0+\alpha_1 c+\alpha_2 c_p^2  
+	\end{aligned}
+	
+	\begin{aligned} 
+		\frac{dq^*}{dc_p} = \alpha_1+ 2 \alpha_2 c_p 
+	\end{aligned}
+
+where :math:`\alpha_0=0` and :math:`\alpha_1` and :math:`\alpha_2` are determined from :math:`\alpha_1 \varepsilon+\alpha_2 \varepsilon^2 = k_f \varepsilon^{1/n}` and :math:`\alpha_1 + 2 \alpha_2 \varepsilon = \frac{1}{n}k_f \varepsilon^{\frac{1-n}{n}}`.
+
+.. math::
+	\begin{aligned}
+		\alpha_1 = \frac{2 n-1}{n}k_f \varepsilon^{\frac{1-n}{n}}
+	\end{aligned}
+.. math::
+	\begin{aligned}
+		\alpha_2 = \frac{1-n}{n}k_f \varepsilon^{\frac{1-2 n}{n}}
+	\end{aligned}
+
+This approximation can be used for any pore phase cocentration :math:`c_p < \epsilon` given :math:`n>1`.
+For the case, when :math:`n \le 1` no special treament near the origin is required.
+For more information on model parameters required to define in CADET file format, see :ref:`freundlich_ldf_config`.
+