Some authors use the linear driving force (LDF) approximation instead of the native kinetic form of an isotherm. The binding model can be combined with different transport models, e.g. the lumped rate model with pores in a recent publication by Moreno-Gonalez et al. (2020). This post raises the question if the native and LDF approaches to modeling binding kinetics are equivalent.

They are not. For a given bound phase concentration q, the equilibrium liquid phase concentration c* can be computed. The quasi stationary approach forces c=c^* (`IS_KINETIC = 0`

in CADET). The native approch is chosen by setting `IS_KINETIC = 1`

. In contrast, the LDF approach forces the adsorptive flux to be proportional to the difference c-c^*. This is currently not supported by CADET, but we might add it in the future. Unfortunately both approaches yield different results. For example, the kinetic form of the Langmuir model reads:

\frac{dq}{dt}=k_a c(q_m-q)-k_d q \tag{I}

In equilibrium (\frac{dq}{dt}=0) we get:

c^*=\frac{k_d}{k_a}\frac{q}{q_m-q}

Using the equilibrium constant we get:

c^*=\frac{1}{k_{eq}}\frac{q}{q_m-q}\tag{II}

As descried above, CADET can either use eq. I (`IS_KINETIC = 1`

) or eq. II with c=c^* (`IS_KINETIC = 0`

).

The LDF approach uses the following kinetic equation:

\frac{dq}{dt}=k_{kin}(c-c^*) = k_{kin}(c-\frac{1}{k_{eq}}\frac{q}{q_m-q})

Setting k_{kin}=k_a q_m is the closest we can get to eq. I:

\frac{dq}{dt}=k_a q_m (c-\frac{1}{k_{eq}}\frac{q}{q_m-q}) = k_a q_m c-k_d q \frac{q_m}{q_m-q}\tag{III}

This is actually not too bad. If we replace the q in the denominator (and only that one) in eq. III by the equilibrium concentration q^* = q_m \frac{k_{eq}c}{1+k_{eq}c}, we get:

\frac{dq}{dt}=k_a q_m c - k_d q \frac{qm}{q_m- q_m k_{eq}\frac{c}{1+k_{eq}c}} = k_a c (q_m-q)-k_d q

However, since c and q are not really in equilibrium, eq. III will never be exactly identical with eq. I. The difference between both equations decreases with increasing k_{kin}.

Note that there is a second LDF approach by setting

\frac{dq}{dt} = k_{kin}(q^*-q)

which is also not equivalent to the native approach.

Another possible tweak is to use the lumped rate model with pores with quasi-stationary approach (`IS_KINETIC=0`

), k_a=k_{eq}, k_d=1 and k_f = k_{kin}. Because the pore phase concentration is now the equilibrium concentration, c_p = c^*, (a factor times) k_{kin}(c-c^*) is substracted from the the interstitial phase concentration in the respective partial differential equation. However, this is not exactly equal to \frac{dq}{dt}, due to the particle porosity, \varepsilon_p>0. The difference decreases with \varepsilon_p approaching zero (\varepsilon_p=0 would cause a division by zero). In the limit \varepsilon_p \rightarrow 0, the lumped rate model with pores reduces to the lumped rate model without pores, but also eliminates film diffusion.

For completeness, I am adding both LDF approximations of the multi-component Langmuir model:

\frac{dq_i}{dt}=k_{a,i} c_i q_{m,i} (1-\sum_{j=1}^{N_{comp}}{{\frac{q_j}{q_{m,j}}}})-k_{d,i} q_i (`MULTI_COMPONENT_LANGMUIR`

)

\frac{dq_i}{dt} = k_{kin,i}(q_i^*-q_i) with q_i^*=\frac{q_{m,i} k_{eq,i} c_i}{1 + \sum_{j=1}^{n_{comp}}{k_{eq,j} c_j}} (`MULTI_COMPONENT_LANGMUIR_LDF`

)

\frac{dq_i}{dt} = k_{kin,i}(c_i-c_i^*) with c_i^*=\frac{q_i}{k_{eq,i} q_{m,i}\left(1-\sum_{j=1}^{N_{comp}}{\frac{q_j}{q_{m,j}}}\right)} (`MULTI_COMPONENT_LANGMUIR_LDF_LIQUID_PHASE`

)

In rapid equiibrium (`IS_KINETIC = 0`

), all three versions provide identical results.

The LDF approximations will be added to CADET in the future.

The LDF approximations are straightforfard to apply to each bound state of the bi-Langmuir model. We should consider adding these models to CADET, too. The formulation of, e.g., Kaspereit et al. (2011) can be extended by allowing for more than two bound states and different k_{kin} for each of them.

Let q_{i,j} denote the concentration of component i at binding site j. Then \frac{dq_i}{dt} = \sum_{j=1}^{n_{site}}{\frac{dq_{i,j}}{dt}} and

\frac{dq_{i,j}}{dt}=k_{a,i,j} c_i q_{m,i,j} (1-\sum_{k=1}^{N_{comp}}{{\frac{q_{k,j}}{q_{m,k,j}}}})-k_{d,i,j} q_{i,j} (`MULTI_COMPONENT_BILANGMUIR`

)

\frac{dq_{i,j}}{dt} = k_{kin,i,j}(q_{i,j}^*-q_{i,j}) with 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}} (`MULTI_COMPONENT_BILANGMUIR_LDF`

)

Let the binding sites of each component have the same k_{kin,i}. Then

\frac{dq_i}{dt} = k_{kin,i} (q_i^* - q_i) with q_i^*=\sum_{j=1}^{n_{site}}{\frac{q_{m,i,j} k_{eq,i,j} c_i}{1 + \sum_{k=1}^{n_{comp}}{k_{eq,k,j} c_k}}} and q_i = \sum_{j=1}^{n_{site}}{q_{i,j}} (`MULTI_COMPONENT_BILANGMUIR_LDF`

)

The residual of `MULTI_COMPONENT_LANGMUIR_LDF`

reads

F_i = \frac{dq_i}{dt} - k_{kin,i}\left(q_i^*-q_i\right) with q_i^*=\frac{q_{m,i} k_{eq,i} c_i}{1 + \sum_{k=1}^{n_{comp}}{k_{eq,k} c_k}}

with derivatives

\frac{\partial F_i}{\partial c_i} = k_{kin,i} \left(\frac{q_{m,i} k_{eq,i}^2 c_i}{\left(1 + \sum_{k=1}^{n_{comp}}{k_{eq,k} c_k}\right)^2} - \frac{q_{m,i} k_{eq,i}}{1 + \sum_{k=1}^{n_{comp}}{k_{eq,k} c_k}}\right)

\frac{\partial F_i}{\partial c_j} = k_{kin,i} \frac{q_{m,i} k_{eq,i} k_{eq,j} c_i}{\left(1 + \sum_{k=1}^{n_{comp}}{k_{eq,k} c_k}\right)^2} for i \neq j

\frac{\partial F_i}{\partial q_i} = k_{kin,i}

\frac{\partial F_i}{\partial q_j} = 0 for i \neq j

In analogy to the first answer, for the second LDF approach we get:

\frac{dq}{dt} = k_{kin}(q^*-q) = k_{kin}(\frac{q_m k_{eq} c}{1 + k_{eq} c}-q)\tag{IV}

Here, setting k_{kin}=k_d is the closest we can get to eq. I:

\frac{dq}{dt} = k_d\frac{q_m k_{eq} c}{1 + k_{eq} c}-k_d q = k_a c \frac{q_m}{1 + k_{eq} c}-k_d q \tag{V}

If we replace the c in the denominator (and only that one) in eq. V by the equilibrium concentration c^*=\frac{1}{k_{eq}}\frac{q}{q_m-q}, we get:

\frac{dq}{dt} = k_a c \frac{q_m}{1 + k_{eq} \frac{1}{k_{eq}}\frac{q}{q_m-q}}-k_d q = k_a c \left(q_m-q\right)-k_d q\tag{VI}

Since c and q are not really in equilibrium, eq. VI will never be exactly identical with eq. I. The difference between both equations decreases with increasing k_{kin}.