(245u) Kinetic Parameter Estimation Including Uncertainty Under Mass Transfer Limited Conditions

Kinetic models are commonly used to describe chemical
processes (1) to allow gaining insight in the underlying reactions, and (2) for
process optimisation. In the former application, the model needs to be
calibrated using experimental data. The main aim of such a calibration exercise
is to obtain kinetic parameter values which are independent from process
settings such as the mixing speed, the reactor geometry, etc. This can only be
achieved if the reactor is not suffering from mass transfer limitations and,
hence, no major spatial heterogeneities exist. This “homogeneous” reactor
behaviour is aimed for by mixing. However, it is often not possible to have
non-limiting mass transfer, and thus it becomes impossible to use kinetic
models to estimate the intrinsic parameter values using only a kinetic model
and assuming a CSTR. However, by coupling computational fluid dynamics (CFD)
with the kinetic model, it is possible to account for these spatial
heterogeneities and calibrate the kinetic model.

Recently, Verbruggen et al.
(2016) [1] coupled a Langmuir model with CFD, and stated that they were able to
accurately retrieve the intrinsic kinetic parameter values. This approach
proved to be very powerful, but requires the evaluation of the CFD model at
multiple parameter values to find the minimum of the objective function, and
thus is computationally very demanding. When overcoming the computational
burden, however, it might seem that the problem of estimating the intrinsic
parameter values has been solved. But this is not the case as one can only prove
that the obtained parameter values really represent the intrinsic parameter
values when the measurements are not noise corrupted and an unlimited amount of
such data is available (the so-called theoretical identifiability). In reality,
only a limited amount of noise-corrupted measurements is available, and thus
the reliability of the estimated parameter value will depend on the measurement
uncertainty and the experimental design (e.g. the sampling locations and the initial
substrate concentrations). As a consequence, only parameters with a
sufficiently important (and uncorrelated) impact on the model predictions will
be estimated properly. Therefore, it is important to assess the uncertainty of
the parameter estimates in such a calibration context. This is often referred
to as practical identifiability analysis [2].

It is clear that even for parameter estimations
performed with CFD, assessing the parameter uncertainty remains an important
aspect, since uncertain parameter estimates will yield models
with low predictive power. The application of uncertainty analysis for
CFD-based kinetic model calibration has not yet been performed to our
knowledge. A new methodology is developed, and is applied in silico to a second-order reaction (Eq. (1)) for a microreactor
where the enzyme is immobilised at the wall (Figure 1).

Figure
1: Microreactor (W=200 µm, L=0.10 m) with enzyme immobilised at the walls.

The corresponding kinetic model is given in Eq. (2),
and was implemented in OpenFOAM 2.2.2:

The uncertainty of k_cat
was estimated as a function of the “optimal” k_cat value, using the likelihood confidence region
method [3]. Since the parameter uncertainty is affected by the mass transfer,
the sampling locations, and measurement noise, a Monte Carlo strategy was used
to determine the impact of these degrees of freedom on the parameter
uncertainty. The use of the developed methodology showed that in this case k_cat, could be properly estimated
when using the CFD-based model calibration, even when mass transfer limitations
were high. Moreover, it allows estimating the uncertainty of the parameter
values.

References

[1] Verbruggen, S.W., Keulemans, M., van Walsem, J., Tytgat, T.,  Lenaerts, S., Denys, S., (2016) CFD modeling
of transient adsorption/desorption behavior in a gas
phase photocatalytic fiber
reactor, Chem Eng J, 292:42-50

[2] Dochain, D.,
Vanrolleghem, P.A., (2001) Dynamical Modelling and Estimation in Wastewater
Treatment Processes

[3] Seber G.A.F., Wild C.J., (1989) Nonlinear
Regression