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Concept# Reaction–diffusion system

Summary

Reaction–diffusion systems are mathematical models which correspond to several physical phenomena. The most common is the change in space and time of the concentration of one or more chemical substances: local chemical reactions in which the substances are transformed into each other, and diffusion which causes the substances to spread out over a surface in space.
Reaction–diffusion systems are naturally applied in chemistry. However, the system can also describe dynamical processes of non-chemical nature. Examples are found in biology, geology and physics (neutron diffusion theory) and ecology. Mathematically, reaction–diffusion systems take the form of semi-linear parabolic partial differential equations. They can be represented in the general form
:\partial_t \boldsymbol{q} = \underline{\underline{\boldsymbol{D}}} ,\nabla^2 \boldsymbol{q} + \boldsymbol{R}(\boldsymbol{q}),
where q(x, t) represents the unknown vector function, is a diagonal matrix of diffusion coef

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The concept of reaction variants and invariants for lumped reaction systems has been known for several decades. Its applications encompass model identification, data reconciliation, state estimation and control using kinetic models. In this thesis, the concept of variants and invariants is extended to distributed reaction systems and used to develop new applications to estimation, control and optimization.
The thesis starts by reviewing the material and heat balances and the concept of variants and invariants for several lumped reaction systems. Different definitions of variants and invariants, in particular the vessel extents, are presented for the case of homogeneous reaction systems, and transformations to variants and invariants are obtained. The extension to systems with heat balance and mass transfer is also reviewed.
The concept of extents is generalized to distributed reaction systems, which include many processes involving reactions and described by partial differential equations. The concept of extents and the transformation to extents are detailed for various configurations of tubular reactors and reactive separation columns, as well as for a more generic framework that is independent of the configuration.
New developments of the extent-based incremental approach for model identification are presented. The approach, which compares experimental and modeled extents, results in maximum-likelihood parameter estimation if the experimental extents are uncorrelated and the modeled extents are unbiased. Furthermore, the identification problem can be reformulated as a convex optimization problem that is solved efficiently to global optimality.
The estimation of unknown rates without the knowledge or the identification of the rate models is described. This method exploits the fact that the variants computed from the available measurements allow isolating the different rates. Upon using a Savitzky-Golay filter for differentiation of variants, one can show that the resulting rate estimator is optimal and obtain the error and variance of the rate estimates.
The use of variants and invariants for reactor control is also considered. Firstly, offset-free control via feedback linearization is implemented using kinetic models. Then, it is shown how rate estimation can be used for control via feedback linearization without kinetic models. By designing an outer-loop feedback controller, the expected values of the controlled variables converge exponentially to their setpoints.
This thesis presents an approach to speed up steady-state optimization, which takes advantage of rate estimation without rate models to speed up the estimation of steady state for imperfectly known dynamic systems with fast and slow states. Since one can use feedback control to speed up convergence of the fast part, rate estimation allows estimating the steady state of the slow part during transient operation.
The application to dynamic optimization is also shown. Adjoint-free optimal control laws are computed for all the types of arcs in the solution. In the case of reactors, the concept of extents allows the symbolic computation of optimal control laws in a systematic way. A parsimonious input parameterization is presented, which approximates the optimal inputs well with few parameters. For each arc sequence, the optimal parameter values are computed via numerical optimization.
The theoretical results are illustrated by simulated examples of reaction systems.

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This thesis presents a systematic study on the merits and limitations on using pin-by-pin resolution and transport theory based approaches for nuclear core design calculations. Starting from the lattice codes and an optimal cross section generation scheme, it compares different methods, transport approximations, and spatial discretizations used in pin-by-pin homogenized codes.
It is in the interest of nuclear power plant operators to employ more heterogeneous core loadings in order to improve the fuel utilization and decrease the amount of spent fuel. This necessarily increases the requirements on the accuracy of the computation tools used for the core design and safety analysis. One possibility is employing 3D core solvers with higher spatial resolution, e.g. pin-cell wise.
The comparison of several lattice codes indicates that already the proper generation of diffusion coefficients and higher-order scattering moments for pin-cell geometry is not straightforward. Out of three available lattice codes, two generated unphysical diffusion coefficient when using the inscatter approximation, while the last code was not able to provide the higher scattering moments.
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A study on spatial discretization indicated that the finite difference method applied on pin-cells does not properly capture the big flux changes between MOX and uranium fuel, while the nodal expansion method is more accurate but too slow. It was suggested to use the finite difference method with finer mesh in the outer assembly pin-cells, which increases the required computation time by only 50 % and decreases the pin power errors below 1 % with respect to lattice code results.
Due to some problems which were observed with the available diffusion/SP3 solvers, a new SP3 solver was implemented in the DORT-TD platform. Several core tests showed that the SP3 pin-by-pin solver can significantly outperform the state-of-the-art nodal solver SIMULATE-5, in particular for reactor cores with inserted control rods.
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