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}.