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doc/modelling/binding/freundlich_ldf.rst

@@ -3,7 +3,7 @@
 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`.
+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::
@@ -42,6 +42,3 @@ This approximation can be used for any pore phase cocentration :math:`c_p < \eps
 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`.
 
-
-
-

doc/modelling/binding/index.rst

@@ -51,23 +51,28 @@ This can be achieved by a (nonlinear) parameter transform
 
 Linear Driving Force (LDF)
 ---------------------------
+Some authors use the linear driving force (LDF) approximation instead of the native kinetic form of an isotherm.
+All three approaches are equivalent in rapid equilibrium (``IS_KINETIC = 0``) but not equivalent when binding kinetics are considered (``IS_KINETIC = 1``).
 
-A quantitative description of an actual separation process requires the simultaneous solution of the diffusion equation and the bulk material balances, for which no analytical solutions exist. It is therefore computationally convenient to approximate adsorption rate by a time-independent expression given as:
+1. In the native approach, :math:`\frac{dq}{dt}` is an explicit funtion of :math:`c` and :math:`q`. For example :math:`\frac{dq}{dt}=k_a c (q_m - q)-k_d q` in the Langmuir model.
 
-.. math::
-    \begin{aligned}
-        \frac{\mathrm{d} q_i}{\mathrm{d} t} = k_{kin,i}\left(q^*_i - q_i\right) && i = 0, \dots, N_{\text{comp}} - 1.
-    \end{aligned}
+2. A linear driving force approximation is based on the equilibrium concentration :math:`q^*` for given :math:`c`.
+For example :math:`q^*= \frac{q_m k_{eq} c}{1 + k_{eq} c}` in the Langmuir model.
+Here, :math:`\frac{dq}{dt}` is proportional to the actual difference from equilibrium, i.e. :math:`\frac{dq}{dt} = k_{kin}(q^*-q)`.
+Note that, the sign of :math:`\frac{dq}{dt}` causes the resulting flux to act towards the equilibrium.
+In CADET, this approach is denoted by ``LDF``, for example in ``MULTI_COMPONENT_LANGMUIR_LDF``.
 
-where :math:`q` and :math:`q^*` are the adsorbed-phase concentration in equilibrium with the bulk phase concentration and adsorbed-phase concentration averaged over the entire bulk volume respectively. While, :math:`k_{kin}` is the linear driving force coefficient :cite:`Alpay1992`. 
+3. An alterniative linear driving force approximation is based on the equilibrium concentration :math:`c^*` for given :math:`q`.
+For example :math:`c^*=\frac{q}{k_{eq} \left(q_{m}-q\right)}` in the Langmuir model.
+Here, :math:`\frac{dq}{dt}` is proportional to the actual difference from equilibrium concentrations, i.e. :math:`\frac{dq}{dt} = k_{kin}(c-c^*)`.
+Note that, the sign of :math:`\frac{dq}{dt}` causes the resulting flux to act towards the equilibrium.
+In CADET, this approach is denoted by ``LDF_LIQUID_PHASE``, for example in ``MULTI_COMPONENT_LANGMUIR_LDF_LIQUID_PHASE``.
 
-In case of rapid equllibrium :math:`\left(\frac{dq}{dt}=0\right)`, following relationship holds:
-
-.. math::
-    \begin{aligned}
-        q = q^*
-    \end{aligned} 
+In both LDF examples, the original rate constants :math:`k_a` and :math:`k_d` are replaced by the equilibrium contant :math:`k_{eq}=\frac{k_a}{k_d}`.
+The linear driving force approximations are based on a new kinetic constant, :math:`k_{kin}`.
 
+Note that some isotherms, such as the Freundlich model, don't have a native representation in the above sense.
+LDF versions are availabe for some but not all binding models implemented in CADET.
 
 .. _reference_concentrations:
 

doc/modelling/binding/multi_component_bi_langmuir.rst

@@ -3,7 +3,7 @@
 Multi Component Bi-Langmuir
 ~~~~~~~~~~~~~~~~~~~~~~~~~~~
 
-The multi component Bi-Langmuir model :cite:`Guiochon2006` adds :math:`M - 1` *additional* types of binding sites :math:`q_{i,j}` (:math:`0 \leq j \leq M - 1`) to the Langmuir model (see Section :ref:`multi_component_langmuir_model`) without allowing an exchange between the different sites :math:`q_{i,j}` and :math:`q_{i,k}` (:math:`k \neq j`).
+The multi component Bi-Langmuir model :cite:`Guiochon2006` adds :math:`M - 1` additional types of binding sites :math:`q_{i,j}` (:math:`0 \leq j \leq M - 1`) to the Langmuir model (see Section :ref:`multi_component_langmuir_model`) without allowing an exchange between the different sites :math:`q_{i,j}` and :math:`q_{i,k}` (:math:`k \neq j`).
 Therefore, there are no competitivity effects between the different types of binding sites and they have independent capacities.
 
 .. math::
@@ -18,5 +18,4 @@ See the Section :ref:`multi_component_langmuir_model`.
 Originally, the Bi-Langmuir model is limited to two different binding site types.
 Here, the model has been extended to arbitrary many binding site types.
 
-
 For more information on model parameters required to define in CADET file format, see :ref:`multi_component_bi_langmuir_config`.

doc/modelling/binding/multi_component_bi_langmuir_ldf.rst

@@ -3,20 +3,13 @@
 Multi Component Bi-Langmuir LDF
 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 
-This model is a :ref:`ldf_model` approximation of  multi component Bi-Langmuir model :cite:`Guiochon2006`, that adds :math:`M - 1` *additional* types of binding sites :math:`q_{i,j}` (:math:`0 \leq j \leq M - 1`) to the LDF based Langmuir model (see Section :ref:`multi_component_langmuir_model_ldf`). The implementation follows the same principle as followed in :ref:`multi_component_bi_langmuir_model`.
-Adsorbed phase concnetration averaged over the entire bulk volume is given as:
+This a linear driving force model variant of the :ref:`multi_component_bi_langmuir_model` model.
+It is based on the equilibrium concentration :math:`q^*` for a given liquid phase concentration :math:`c` (see also :ref:`ldf_model`).
 
 .. math::
     \begin{aligned}
         q_{i,j}^*=\frac{q_{m,i,j} k_{eq,i,j} c_i}{1 + \sum_{k=1}^{N_{comp}}{k_{eq,k,j} c_k}} && i = 0, \dots, N_{\text{comp}} - 1, \: j = 0, \dots, M - 1.% 	           (0 \leq i \leq N_{\text{comp}} - 1, \: 0 \leq j \leq M - 1). 
     \end{aligned}
 
 
-Note that all binding components must have exactly the same number of binding site types :math:`M \geq 1`.
-See the Section :ref:`multi_component_langmuir_model_ldf`.
-
-Originally, the Bi-Langmuir model is limited to two different binding site types.
-Here, the model has been extended to arbitrary many binding site types.
-
-
 For more information on model parameters required to define in CADET file format, see :ref:`multi_component_bi_langmuir_ldf_config`.

doc/modelling/binding/multi_component_langmuir_ldf.rst

@@ -3,14 +3,13 @@
 Multi Component Langmuir LDF
 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 
-The Langmuir :ref:`ldf_model` binding model also includes a saturation term and takes into account the capacity of the resin :cite:`Langmuir1916,Guiochon2006`. 
-All components compete for the same binding sites.
+This a linear driving force model variant of the :ref:`multi_component_langmuir_model` model.
+It is based on the equilibrium concentration :math:`q^*` for a given liquid phase concentration :math:`c` (see also :ref:`ldf_model`).
 
 .. math::
 
     \begin{aligned}
         q_i^*=\frac{q_{m,i} k_{eq,i} c_i}{1 + \sum_{j=1}^{n_{comp}}{k_{eq,j} c_j}} && i = 0, \dots, N_{\text{comp}} - 1.
     \end{aligned}
 
-
 For more information on model parameters required to define in CADET file format, see :ref:`multi_component_langmuir_ldf_config`.

doc/modelling/binding/multi_component_langmuir_ldf_liquid_phase.rst

@@ -3,7 +3,8 @@
 Multi Component Langmuir LDF Liquid Phase
 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 
-The liquid phase based variation of the Langmuir :ref:`ldf_model` binding model is given as:
+This a linear driving force model variant of the :ref:`multi_component_langmuir_model` model.
+It is based on the equilibrium concentration :math:`c^*` for a given solid phase concentration :math:`q` (see also :ref:`ldf_model`).
 
 .. math::