The solving of ordinary differential equations by means of the Laplace transform method

Laplaceova transformace je velmi silným matematickým nástrojem pro řešení obyčejných lineárních diferenciálních rovnic s konstantními koeficienty. Její využití je široké - lze ji použít na lineární rovnice prvního i vyšších řádů, velmi vhodná je pro řešení diferenciálních rovnic s více pravými stranami (a to i nespojitými) a v neposlední řadě ji lze také aplikovat na soustavy ODR. Laplaceova transformace se intenzivně využívá především v teorii řízení, kde transformace odpovídající diferenciální rovnice regulační soustavy umožňuje analyzovat chování této soustavy, např. reakce (odezvy) systému na vstupní veličinu. Cílem práce bylo uvést základy teorie Laplaceovy transformace a demonstrovat tento silný matematický aparát při řešení konkrétních úloh, včetně využití software pro symbolickou matematiku Maple.The Laplace transform is a very powerful mathematical tool for solving of ordinary linear differential equations with constant coefficients. Its usage is wide - it can be applied to first order and also to higher order equations, it is very convenient for solving of differential equations with several forcing terms (including noncontinuous terms) and of course, it can be used for solving of systems of ordinary differential equations. The Laplace transform plays the key role in control theory, where the transformation of the differential equation of the control system enables to analyse the behavior of this system, e. g. its reaction to input values. Our aim was to present essentials of the Laplace transform theory and demonstrate this strong mathematical tool in the solving of concrete problems, including the usage of the software Maple.

Stochastic Programming Algorithms

Stochastické programování a optimalizace jsou mocnými nástroji pro řešení široké škály inženýrských problémů zahrnujících neurčitost. Algoritmus progressive hedging je efektivní dekompoziční metoda určená pro řešení scénářových stochastických úloh. Z důvodu vertikální dekompozice je možno tento algoritmus implementovat paralelně, čímž lze významně ušetřit výpočetní čas a ostatní prostředky. Teoretická část této diplomové práce se zabývá matematickým a zejména pak stochastickým programováním a detailně popisuje algoritmus progressive hedging. V praktické části je navržena a diskutována původní paralelní implementace algoritmu progressive hedging, která je pak otestována na jednoduchých úlohách. Dále je uvedená paralelní implementace použita pro řešení inženýrského problému plynulého odlévání ocelové bramy a na závěr jsou získané výsledky zhodnoceny.Stochastic programming and optimization are powerful tools for solving a wide variety of engineering problems including uncertainty. The progressive hedging algorithm is an effective decomposition method for solving scenario-based stochastic programmes. Due to the vertical decomposition, this algorithm can be implemented in parallel thereby the computing time and other resources could be considerably spared. The theoretical part of this master's thesis deals with mathematical and especially with stochastic programming. Further, the progressive hedging algorithm is presented and discussed in detail. In the practical part, the original parallel implementation of the progressive hedging algorithm is suggested, fruitfully discussed and tested to simple problems. Furthermore, the presented parallel implementation is used for solving the continuous casting process of steel slabs and the results are appraised.

Optimization of Secondary Cooling Parameters of Continuous Steel Casting

Plynulé odlévání je dominantní způsob výroby oceli, pomocí kterého je v současné době vyráběno více než 95 % veškeré celosvětové produkce oceli. Matematické modelování a optimální řízení provozu licího stroje patří mezi klíčové úlohy při plynulém odlévání oceli, které významným způsobem ovlivňují produktivitu a kvalitu vyráběné oceli, konkurenceschopnost ocelárny, bezpečnost při provozu licího stroje a jeho dopad na životní prostředí. Tato práce se zabývá vývojem a implementací numerického modelu teplotního pole plynule odlévaného sochoru a jeho využitím při optimálním řízení dynamického provozu licího stroje. Počítačový model byl vytvořen a implementován v MATLABu. Z důvodu vysoké výpočetní náročnosti byl model paralelizován pomocí výpočtu na grafických kartách NVIDIA s využitím architektury CUDA. Ověření modelu bylo provedeno na základě provozních dat z Třineckých železáren. Vyvinutý model byl následně použit jako základ prediktivního řídícího systému pro řízení dynamických změn při provozu licího stroje. Činnost vyvinutého řídícího systému byla ověřena na modelových dynamických situacích, které potvrdily schopnost navrženého řídícího systému optimálně řídit dynamický provoz licího stroje. Počítačový model teplotního pole a prediktivní řídící systém byly vytvořeny tak, že je lze modifikovat pro libovolný licí stroj, což umožňuje jejich případné komerční použití.Continuous casting is a dominant production technology of steelmaking which is currently used for more that 95% of the world steel production. Mathematical modelling and optimal control of casting machine are crucial tasks in continuous steel casting which directly influence productivity and quality of produced steel, competitiveness of steelworks, safety of casting machine operation and its impact on the environment. This thesis concerns with the development and implementation of the numerical model of temperature field for continuously cast steel billets and its use for optimal control of the casting machine. The numerical model was developed and implemented in MATLAB. Due to computational demands the model was parallelized by means of the computation on graphics processing units NVIDIA with the computational architecture CUDA. Validation and verification of the model were performed with the use of operational data from Trinecke zelezarny steelworks. The model was then utilized as a part of the developed model-based predictive control system for the optimal control of dynamic situations in the casting machine operation. The behaviour of the developed control system was examined by means of dynamic model situations that have confirmed the ability of the implemented system to optimally control dynamic operations of the continuous casting machine. Both the numerical model of the temperature field and the model-based predictive control system have been implemented so that they can be modified for any casting machine and this allows for their prospective commercial applications.

Rheological models of high viscosity silicone fluids

High viscosity silicone fluids are utilized in multiple technical applications. Viscous type dampers of torsional vibrations are widely used in the field of internal combustion engines. Thin layers of high viscosity silicone oils, exposed to an oscillating shear stress, show both damping and elastic properties which are frequency and temperature dependent. This paper focuses on the issues to determine rheological models of the silicone fluids which may be utilized to create complex dynamic models of powertrains with dynamic torsional vibration dampers

Comparison of the energy conversion efficiency of a solar chimney and a solar PV-powered fan for ventilation applications

A study into the performance of a solar chimney and a solar photovoltaic (PV)-powered fan for ventilation applications was carried out using numerical simulations. The performance of the solar chimney was compared with that of a direct current (DC) fan powered by a solar PV panel. The comparison was carried out using the same area of the irradiated surface—the area of the solar absorber plate in the case of the solar chimney and the area of the solar panel in the case of the photovoltaic-powered fan. The two studied cases were compared under various solar radiation intensities of incident solar radiation. The results indicate that the PV-powered fans significantly outperform solar chimneys in terms of converting solar energy into the kinetic energy of air motion. Moreover, ventilation with PV-powered fans offers more flexibility in the arrangement of the ventilation system and also better control of the air flow rates in the case of battery storag

Utilization of an Air-PCM Heat Exchanger in Passive Cooling of Buildings: A Simulation Study on the Energy Saving Potential in Different European Climates

The energy saving potential (ESP) of passive cooling of buildings with the use of an air-PCM heat exchanger (cold storage unit) was investigated through numerical simulations. One of the goals of the study was to identify the phase change temperature of a PCM that would provide the highest energy saving potential under the specific climate and operating conditions. The considered air-PCM heat exchanger contained 100 aluminum panels filled with a PCM. The PCM had a thermal storage capacity of 200 kJ/kg in the phase change temperature range of 4 °C. The air-PCM heat exchanger was used to cool down the outdoor air supplied to a building during the day, and the heat accumulated in the PCM was rejected to the outdoors at night. The simulations were conducted for 16 locations in Europe with the investigated time period from 1 May–30 September. The outdoor temperature set point of 20 °C was used for the utilization of stored cold. In the case of the location with the highest ESP, the scenarios with the temperature set point and without the set point (which provides maximum theoretical ESP) were compared under various air flow rates. The average utilization rate of the heat of fusion did not exceed 50% in any of the investigated scenarios

Optimal design of structure in rheological models: an automotive application to dampers with high viscosity silicone fluids

Dynamic torsional vibration dampers are for a long time inherent integral components of internal combustion engines. One of the most common types of the dynamic dampers is a silicone damper. It has been, for many years, perceived as an exclusively viscous damper, thus it has been constructed and designed according to this perception. When compared to other types of dynamic dampers of the similar size with flexible components used for their construction, the standard iscous damper has a lower damping effect. Moreover, this damper type has been a significantly cheaper and simpler solution. Current silicone oils with high nominal viscosity, having not only the expected damping properties, but also significant elastic characteristics under alternate shear stress, enable construction of dynamic dampers with a higher damping effect than a viscous damper. Frequency and temperature dependent complicated rheological properties of high viscosity silicone fluids can only be identified experimentally using a suitable dynamic viscometer. However, the measured frequency dependencies of both components of the complex shear modulus are only defined for harmonic loading while internal combustion engine load is periodic and contains several tens harmonics. The key to the solution is therefore to find suitable multiparameter rheological models comprised of linear elastic and damping elements that would approximate in the specified frequency range both components of the complex shear modulus. Such a complicated task can be solved using efficient optimization algorithms. This article focuses on the mathematical description of convolute rheological properties of high viscosity silicone liquids and also contains an example of the application of created rheological models in the complex dynamic model of a V10 diesel engine. A computational tool for the determination of stiffness and damping coefficients of the multi-parameter rheological model was created and solved in the optimization software GAMS by means of the CONOPT solver. The possibility of these modern technologies is shown by the comparison of computation models and experimentally set torsional vibration spectres with standard viscous damper and damper utilizing a high viscosity silicone oil

Importance of the experimental investigation of a concasting technology

The solidification and cooling of a continuously cast billet, slab cylinder, generally of a concasting and the simultaneous heating of the mold is a very complicated problem of three-dimensional (3D) transient heat and mass transfer. The solving of such a problem is impossible without numerical models of the temperature field of the concasting itself hich it is being processed through the concasting machine (caster). The application of the numerical model requires systematic experimentation and measurement of operational parameters on a real caster as well as in the laboratory. The measurement results, especially temperatures, serve not only for the verification of the exactness of the model, but mainly for optímization of the process procedure: real process input data numerical analyses optimization correction of real process. The most important part of the investigation is the measurement of the temperatures in the walls of the mold and the surface of the slab in the zones of secondary and tertiary cooling

Investigation of a Temperature Field of the Steel Billet 150x150 mm Continuously Cast

The solidification and cooling of a continuously cast billet and the simultaneous heating of the mold is a very complicated problem of three-dimensional (3D) transient heat and mass transfer. The solving ofuch a problem is impossible without numerical models of the temperature field of the concasting itself which it is being processed through the concasting machine (caster). The application of the numerical model requires systematic experimentation and measurement of operational parameters on a real caster as well as in the laboratory. The measurement results, especially temperatures, serve not only for the verification of the exactness of the model, but mainly for optimization of the process procedure. The most important part of the investigation is the measurement of the temperatures in the walls of the mold and the surface of the slab in the zones of secondary and tertiary cooling