(55b) A Quadrature-Based Uncertainty Quantification Approach With Reconstruction of the Probability Distribution Function of the System Response in Bubbling Fluidized Beds
In simulations of industrial systems, it is important to have an estimate of the distributions of errors due to uncertainty in the model parameters and input data. This can be accomplished by developing uncertainty quantification tools that can be combined with available CFD codes. Here a non-intrusive, quadrature-based, uncertainty quantification (QBUQ) method is presented. The approach relies on Gaussian quadrature formulae to generate the set of samples of the distribution of the uncertain parameters of the model. A numerical simulation is performed for each sample, and the moments of the system response are directly computed from the simulation results by means of quadrature formulae. These moments are then used to determine the reconstructed distribution function of the values of the system response by means of the extended quadrature method of moments .
The QBUQ procedure is demonstrated by considering a bubbling fluidized bed as an example application. The initial bed height, the particle restitution coefficient and the mean particle size are assumed to be the independent and uniformly distributed uncertain input parameters. The system is simulated with a standard two-fluid model with kinetic theory closures for the particulate phase  implemented into MFIX. The effect of uncertainty on the disperse-phase volume fraction, on the phase velocities and on the pressure drop inside the fluidized bed are examined, and the reconstructed probability density function of the system response is provided for the three quantities studied.
 C. Yuan, F. Laurent, R.O. Fox, An extended quadrature method of moments for population balance equations, J. Aerosol Sci. 51 (2012) 1–23.
 D. Gidaspow, Multiphase flow and fluidization, Academic Press, 1994.
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