# Evaluation of the adsorption isotherm function in infinite dimensions using Gateaux differential

## Jozef Minár

### Comenius University, Bratislava

Abstract:We consider fully and partially saturated flow of contaminated water into an initially dry porous sample together with contaminant adsorption in sample. At the one side of the sample is inflow chamber, from which contaminated water originates. At the opposite side is outflow chamber, where, after some time, contaminated water begins to flow out. We consider a model where adsorbed and dissolved contaminant does not affect flow of water. We test applicability of an infinite dimensional gradient method (using Gateaux differential) for determination of adsorption isotherm function in a contaminant model. The differential is applied to a function measuring a norm of difference between contaminant fluxes into outflow chamber. Here, one flux is “objective” flux given at the beginning of the optimization and second flux is computed using argument of adsorption isotherm function.

# Mathematical Modeling of Biomass Co-Firing in Circulating Fluidized Bed

### Czech Technical University in Prague

Abstract:This contribution presents the work in progress on the design and numerical solution of a quasi-1D CFD model of multiphase flow and combustion dynamics in industrial scale circulating fluidized bed boilers. First, a slight introduction to fluidized bed reactors and their advantages over traditional pulverized fuel boilers is given. Afterward, the current status of the development of the corresponding mathematical model is described. The basis of the model consists in equations for multiphase flow in a variable cross section duct, describing the mechanical interaction of gas and multiple granular solid phases - bed material, coal, and biomass. In addition, attrition of fuel particles together with natural devolatilization and burnout due to their expositon to hot oxidizing environment are considered. The most important part of the model is the correct equation for energy balance which accounts for heat production, heat transfer in terms of diffusion, convection, and radiation together with mixture flow effects. The talk is concluded by demonstrating the results of some preliminary numerical simulations.

# Mathematical Model of Coke Combustion in Moving Bed

## Robert Straka

### AGH University of Science and Technology, Krakow

Abstract:Large coke particles combustion and gasification in rock-melting furnaces are described by complex chemical reactions and non-linear heat and mass transfer which occurs inside the reactor. The tricky part of the modeling is the setup of parameters whose values cannot be measured directly and are of essential meaning together with great influence on the results obtained from the simulation. The one dimensional model of coke-fired moving beds, its numerical solution by FVM and results' validation will be presented.

# A dissection solver with kernel detection for finite element matrices on shared memory computers

## Atsushi Suzuki

### Universite Pierre et Marie Curie, Paris

Abstract:We have developed a factorization algorithm with kernel detection for sparse matrices from finite element methods. Obtaining solution in the image space and construction of the kernel vectors are important especially for sub-domain solver in the FETI domain decomposition method. In the developed solver, the matrix is decomposed into sub-blocks by a bisection algorithm, which is realized by SCOTCH software package. Sub-matrices in the first level consist of sparse matrices solved by a skyline solver and ones in the other levels consist of dense matrices solved with BLAS libraries. In higher levels, factorization of dense sub-matrices is parallelized by introducing sub-blocks and rank-n updates for Schur complement. Parallel implementation is done on multi-core architecture by using POSIX threads library and asynchronous task management, and the code can achieve competitive performance to Intel Pardiso solver. The kernel detection algorithm is based on measurement of residuals with orthogonal projections, which is free from computation of eigenvalues, then it is robust with large condition numbers.

# Thermal calculations of rock-melting cupola furnace

### AGH University of Science and Technology, Krakow

Abstract:Preliminary calculations of heat and mass balance for the process of lava melting have been carried out. On the basis of the calculation process will be analyzed in energetic point of view. Performance characteristics of the furnace indicated cost estimates including cost loss due to emissions.

# Numerical Model of Fluidized Bed in Stokes Regime

## Michal Beneš

### Czech Technical University in Prague

Abstract:We discuss the one-dimensional model of fluidized bed formed by slowly moving dense fluid carrying solid-phase particles. This model originates in the conservation laws for the two-phase vertical flow which are simplified due to low values of the Reynolds number and leeds to a free-boundary problem for bed height. The bed height varies according to the superficial velocity changes which is demonstrated in the presented numerical results.

Abstract:

# On an inverse Wulff problem

## Daniel Ševčovič

### Comenius University, Bratislava

Abstract:In this presentation we propose a constrained optimization method for solving the inverse Wulff problem. Given a closed Jordan curve in the plane, our aim is to find the optimal anisotropy function minimizing the isoperimetric ratio in the relative geometry. This can be viewed as an inverse Wulff problem. Our approach is based on a finite Fourier mode discretization of the optimization problem. We also show how the discretized problem can be solved by means of a semidefinite relaxation method applied to certain non-convex quadratic problem with conic constraints formulated in terms of a cone consisting of positive semidefinite matrices. It is furthermore shown that such a relaxed optimization problem can be effectively solved by means of the Matlab solver SeDuMi. We also show that, in the limit when the number of Fourier modes tends to infinity, the approximate solution converges to the minimizaer of the isoperimetric ratio in the relative geometry. Practical examples of construction of the optimal anisotropy function will be also presented. In particular, these examples include closed curves corresponding to snowflakes from Bentley's list of snowflakes. In these examples we show that the optimal anisotropy function corresponds to the hexagonal type of anisotropy, as expected. This is a joint work with Mária Trnovská.