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EN
A novel application of building demolition waste for removal benzene from aqueous solutions. In this research, demolition waste from buildings has been studied for possible use as benzene removal adsorbent from aquatic solution. The effect of adsorbent dosage, contact time, initial benzene concentration, and initial pH on benzene adsorption capacity have been investigated in the batch adsorption experiments. The adsorption effects initially happened very rapidly and achieved equilibrium within 180 min. Benzene removal was observed to decrease by an increase in the initial concentration of benzene of 300–700 mg·L–1, an increase in the adsorbent dose of 0.4–2.4 g per 100 mL, where an optimum adsorbent dose equal to 1.2 g per 100 mL was found. The potential of adsorption increases with pH 3.0–7.0 to reach the maximum removal efficiency at pH 6.0. The findings showed that equilibrium data were adequately adapted and correlated with the Freundlich isotherm models. The average percentage of the removal at room temperature was about 98%. Results suggest that building demolition waste can be used effectively in industrial wastewater treatment for the removal of aromatic hydrocarbon, benzene, as a low-cost option.
2
EN
Very fast reactions of forming higher nitrogen oxides set out an equilibrium framework for the course of the reaction of nitrogen monoxide oxidation. The slow course of reaction of nitrogen monoxide with oxygen permanently violates the created equilibria. In particular, the equilibrium of the oxidation reaction of nitrogen monoxide with nitrogen dioxide. The contribution of this reaction to the transformation of nitrogen monoxide in the conditions of nitrogen trioxide removal from the gas phase was estimated.
PL
Przedstawiono problematykę oceny nośności przy wyboczeniu technicznym prętów ściskanych, zwichrzeniu prętów zginanych w płaszczyźnie większej bezwładności przekroju oraz wyboczeniu giętno-skrętnym prętów zginanych w płaszczyźnie większej bezwładności przekroju i ściskanych. Przytoczono analityczne modele eurokodowe oraz dokonano identyfikacji zastępczej imperfekcji geometrycznej do budowy imperfekcyjnych modeli numerycznych wyboczenia technicznego. Na przykładzie oceny nośności przy wyboczeniu słupa stalowego porównano różne podejścia analityczne oraz numeryczne.
EN
Problems related to the technical buckling resistance of compressed members, lateral-torsional buckling of members bent in the plane of section greater moment of inertia and flexural-torsional buckling of members compressed and bent in the plane of section greater moment of inertia are presented. Analytical Eurocode models are quoted and equivalent geometric amplitudes to build imperfect numerical models have been identified. Various analytical and numerical approaches were compared on examples of the buckling resistance assessment of a real steel column.
EN
In this work, we introduce a simple multi-agent simulation model with two roles of agents that correspond to moral and immoral attitudes. The model is given explicitly by a set of mathematical equations with continuous variables and is characterized by four parameters: morality, protection, and two efficiency parameters. Agents are free to adjust their roles to maximize individual gains. The model is analyzed theoretically to find conditions for its stability, i.e., the fractions of agents of both roles that lead to an equilibrium in their gains. A multi-agent simulation is also developed to verify the dynamics of the model for all values of morality and protection parameters, and to identify potential discrepancies with the theoretical analysis.
EN
Moisturizing the intake air by spraying water in the liquid phase significantly lowers the intake air temperature, mainly due to the high value of latent heat of evaporation. The paper presents a methodology for calculating the parameters of the air-fuel mixture after water injection and during subsequent processes of the Otto cycle: compression, combustion and expansion of exhaust gases. For octane as a fuel, exemplary calculations have been carried out to investigate the effect of water injection on the composition of combustion products and selected parameters of the theoretical Otto cycle (temperature, pressure, output power and thermal efficiency).
EN
A method for analyzing the equilibrium of a stack of loaded post pallets is presented. The finite element method was used to investigate the behavior of the bottom pallet in the stack during the addition of successive pallets. The stack was regarded as a self-stable multi-storey structure without bracings which is subjected to the weight of loaded pallets, horizontal forces resulting from sway and bow imperfections, and the impact of a forklift truck. The definite quadratic form of the tangent stiffness matrix after every increment in load was determined by nonlinear analysis to indicate the loss of post stability. An analysis of the stacking process of the evaluated pallets did not reveal a buckling trend in the posts of the bottommost pallet and demonstrated that the loss of equilibrium can lead to the collapse of the entire stack when a critical number of pallets is reached.
EN
In this study, ANN (artificial neural network) model was applied to estimate the Ni(II) removal efficiency of peanut shell based on batch adsorption tests. The effects of initial pH, metal concentrations, temperature, contact time and sorbent dosage were determined. Also, COD (chemical oxygen demand) was measured to evaluate the possible adverse effects of the sorbent during the tests performed with varying temperature, pH and sorbent dosage. COD was found as 96.21 mg/dm3 at pH 2 and 54.72 mg/dm3 at pH 7. Also, a significant increase in COD value was observed with increasing dosage of the used sorbent. COD was found as 12.48 mg/dm3 after use of 0.05 g sorbent and as 282.78 mg/dm3 after use of 1 g sorbent. During isotherm studies, the highest regression coefficient (R 2) value was obtained with Freundlich isotherm (R 2 = 0.97) for initial concentration and with Temkin isotherm for sorbent dosage. High pseudo-second order kinetic model regression constants were observed (R 2 = 0.95-0.99) during kinetic studies with varying pH values. In addition, Ni(II) ion adsorption on peanut shell was further defined with pseudo-second order kinetic model, since qe values in the second order kinetic equation were very close to the experimental values. The relation between the estimated results of the built ANN model and the experimental results were used to evaluate the success of ANN modeling. Consequently, experimental results of the study were found to be in good agreement with the estimated results of the model.
8
Content available remote Preparation of bentonite/nAg nanocomposites
EN
In this work results of the preparation of bentonite/nAg nanocomposites were presented. In the first stage, the bentonite sorption properties were determined, including the equilibrium and kinetics parameters of the sorption process of silver ions on the bentonite. The study analyzed the filler sorption properties for different concentrations of silver ions in solution. The equilibrium sorption data were analyzed using Freundlich, Langmuir and Temkin equations. It was found that the best fit is given by the Freundlich equation. Analysis of the kinetics of the sorption process showed that the pseudo-second-order equation was characterized by the best fit for the experimental data, suggesting the chemical character of the adsorption process. In order to obtain a nanocomposite, silver ions contained in the composite were subjected to a reduction process using tannic acid with stabilizing and reducing properties. The obtained bentonite/nAg nanocomposites contained silver nanoparticles in the range of 162÷266 mg/g. The structures of the nanomaterials were studied by XRD and SEM methods.
PL
Celem niniejszej pracy była analiza równowagi i kinetyki procesu sorpcji jonów srebra na powierzchni napełniacza oraz otrzymanie nanokompozytu bentonit/nAg. W badaniu analizowano właściwości sorpcyjne napełniacza dla różnych stężeń jonów srebra w roztworze. Do opisu równowagi sorpcji analizowanych jonów zastosowano równania Langmuira, Freundlicha oraz Temkina. Stwierdzono, że najlepsze dopasowanie daje równanie Freundlicha. Celem badania było również określenie kinetyki sorpcji. Równanie pseudodrugiego rzędu uzyskało lepsze dopasowanie do danych doświadczalnych, co świadczy o chemicznym charakterze sorpcji. W celu otrzymania nanokompozytu jony srebra zawarte w kompozycie poddano procesowi redukcji przy użyciu kwasu taninowego, charakteryzującego się właściwościami stabilizująco-redukujących. Otrzymano nanokompozyty bentonit/nAg o zawartości nanocząstek srebra 162÷266 mg/g.
DE
Die traditionelle Interpretation des ökonomischen Gleichgewichts ist nicht erfolgreich bei der Realisierung von Marktgleichgewichten. Das Hauptargument gegen die traditionelle Interpretation des ökonomischen Gleichgewichts ist die normalerweise getroffene Annahme einer unrealistischen reinen Konkurrenz. Eine realistische Idee ist die moderne Interpretation des ökonomischen Gleichgewichts. Diese Interpretation benötigt keine unrealistischen Annahmen und kein hohes mathematisches Verständnis. Die moderne Interpretation des ökonomischen Gleichgewichts führt zu totalen quantitativen und wertorientierten Marktgleichgewichten. Alle drei Teilmärkte (Waren, Geld und Arbeitsmarkt) befinden sich im Gleichgewicht, wenn alle auf diesen Märkten tätigen Wirtschaftseinheiten im Gleichgewicht sind. Der Vorteil der modernen Interpretation des ökonomischen Gleichgewichts ist, dass es keine Utopie, sondern eine „Topia“ ist. Die Erkenntnis führt zur Überwindung von Wirtschaftskrisen und zu einem stetig wachsenden Wohlstand trotz weltweit geringer Ressourcen. Die einzige Bedingung, um die moderne Interpretation des wirtschaftlichen Gleichgewichts zu realisieren, ist der Wille, etwas zu verändern. Es ist möglich, die moderne Interpretation des Gleichgewichts für Transport- und Logistikmärkte zu verwenden.
EN
The traditional interpretation of economic equilibrium is not successful in realizing market equilibria. The main argument against the traditional interpretation of economic equilibrium is the normally made assumption of unrealistic pure competition. A realistic idea is the modern interpretation of economic equilibrium. This interpretation needs no unrealistic assumptions and no high level of mathematic understanding. The modern interpretation of economic equilibrium leads to total quantitative- and valuebased market equilibria. All three partial markets (the goods, the money and the labor market) are in equilibrium, when all economic units acting on these markets are in equilibrium. The advantage of the modern interpretation of economic equilibrium is that it is no utopia but a “topia”. The realization leads to overcome economic crises and to steady growing welfare in spite of worldwide low resources. The only condition to realize the modern interpretation of economic equilibrium is the will to change something. It is possible to use the modern interpretation of equilibrium for transport and logistic markets.
EN
The adsorbent of bone char (BC), produced from the pyrolysis of crushed animal bones, was dominated by the mesopores of the Brunauer Emmett Teller (BET) surface area. The optimal condition for defluoridation with BC was a pH level near 5.0. Chloride and nitrate ions could increase fluoride adsorption capacity in contrast with the effect of sulfate and carbonate ions. The interchangeability between fluoride and hydroxyl groups on BC sorbent was proved by the Fourier transform infrared spectroscopy. Langmuir equation had a better correlation coefficient than the Freundlich equation at various temperatures. Thermodynamic parameters such as Delta G degrees, Delta H degrees, Delta S degrees, Ea and S*, have been calculated to describe the nature of fluoride adsorption onto BC. Negative Delta G degrees and Delta H degrees values at various temperatures indicate a spontaneous process, and its exothermic effect, respectively. However, a positive Delta S degrees value represents an increasing process for entropy. The E-a and S* values ranging from 5 to 40 kj.mol-1 and 0 to 1, respectively, demonstrated that the adsorption is dominated by physical process, although the adsorption kinetic process was involved external diffusion, intraparticle diffusion and chemical reaction equilibrium stage. A high concentration of NaOH solution increases efficiency of removing adsorbed F- ions from the BC surface.
EN
The paper introduces the concept of a strict local equilibrium of order k in the Gale economic model. We obtain higher-order necessary and sufficient conditions for such equilibria without assuming continuity of the utility functions. These conditions are formulated in terms of generalized lower and upper directional derivatives, introduced by Studniarski (1986). A stability theorem for strict local equilibria of order k is also included.
12
Content available remote Ergodicity of non-Hamiltonian Equilibrium Systems
EN
It is well known that ergodic theory can be used to formally prove a form of relaxation to microcanonical equilibrium for finite, mixing Hamiltonian systems. In this manuscript we substantially modify this proof using an approach similar to that used in umbrella sampling, and use this approach to consider relaxation in both Hamiltonian and non- Hamiltonian systems. In doing so, we demonstrate the need for a form of ergodic consistency of the initial and final distribution. The approach only applies to relaxation of averages of physical properties and low order probability distribution functions. It does not provide any information about whether the full 6N-dimensional phase space distribution relaxes towards the equilibrium distribution or how long the relaxation of physical averages takes.
EN
The survival of social groups depends on internal factors (the size of a group, its socio-diversity, inner organization, coherence and synergy of actions for the common good), external factors, mostly on a safe natural and social environment, and on sustainable interactions with this environment. In addition, the survival and development of groups is determined by their stability, which in turn depends on the homeostatic mechanisms that maintain a state of balance within groups and in their environments. People have an influence on the stability of social systems; their actions may lead to strengthening or weakening of this homeostasis. The implementation of the concept of sustainable development serves, among others, to strengthen the homeostasis of social systems and consequently, to prolong the existence of mankind. However, paradoxically enough, the more the system tends to equilibrium, the less stable it becomes reducing its chance of survival. But still, striving to achieve a state of equilibrium has become an imperative nowadays in view of the concept of sustainable development. Moreover, since the beginning of the Anthropocene era, people's interference in the homeostasis of natural and social systems has been growing, helped by the progress of science and technology. However, only a handful of the world's population, the financial elite, benefit from this. Driven by economic interests and ignoring ecological criteria, they weaken this homeostasis carelessly and irresponsibly. Focused on their own benefits here and now, they do not care much about the fate of future generations.
PL
Przetrwanie grup społecznych zależy od czynników wewnętrznych (liczebności, socjodywergencji, organizacji wewnętrznej, koherencji i synergii działań na rzecz dobra wspólnego), zewnętrznych – przede wszystkim od bezpiecznego środowiska przyrodniczego i społecznego oraz od zrównoważonych interakcji z otoczeniem. Oprócz tego o przetrwaniu i rozwoju decyduje ich stabilność, która zależy od mechanizmów homeostazy zachowujących równowagę w grupach i w środowisku, w jakim przebywają. O homeostazie systemów społecznych decydują ludzie. Ich działania mogą prowadzić do wzmacniania albo do osłabiania homeostazy. Wzmacnianiu homeostazy systemów społecznych, a w konsekwencji ekstensji czasu istnienia ludzkości, służy – między innymi – urzeczywistnianie idei rozwoju zrównoważonego. Tu jednak pojawia się paradoks: im bardziej system zmierza do równowagi, tym mniejszą osiąga stabilność i tym samym zmniejszą swoją szansę na przetrwanie. Nie zważając na to, dążenie do równowagi stało się nakazem chwili za sprawą koncepcji rozwoju zrównoważonego. Poza tym, od początku epoki antropocenu postępuje ingerencja ludzi w homeostazę systemów przyrodniczych i społecznych. A postęp wiedzy i techniki pomaga im w majsterkowaniu przy homeostazie. Korzysta z tego tylko garstka populacji świata – elity finansowe. Nie kierując się kryteriami ekologicznymi, lecz ekonomicznymi, beztrosko i nieodpowiedzialnie osłabiają homeostazę. Mają one na uwadze wyłącznie swoje korzyści osiągane teraz. Dlatego nie obchodzi ich, jaki los szykują wskutek tego przyszłym pokoleniom.
EN
The paper briefly presents a theory of games with finitely many infinite populations (types) each of whom has finitely many available strategies; the payoff of an individual player depends on the distribution of choices of strategies in all populations and his own particular choice. We give specific examples of applications of the theory in several areas: spatial allocation (of species), economic models – household economy and transportation networks. We also briefly discuss questions of computation of equilibria and relations of large games, as understood in the present paper, to ordinary matrix games, games with continuum of players and evolutionary game theory.
PL
Praca zawiera skrótowy opis gier ze skończoną liczbą nieskończonych populacji (typów), z których każda dysponuje skończoną liczbą strategii; wypłata pojedynczego gracza zależy od rozkładu wyborów strategii wszystkich populacji i jego własnego wyboru strategii. Podajemy konkretne przykłady zastosowań w kilku dziedzinach: alokacja przestrzenna (gatunków), modele ekonomiczne – model gospodarki drobnotowarowej i model ruchu drogowego. Dyskutujemy też krótko zagadnienia obliczeniowe i związki dużych gier, jak przedstawiono je w tym artykule, ze zwykłymi grami macierzowymi, grami z continuum graczy i z ewolucyją teorią gier.
EN
Based on the minimum Gibbs Free Enthalpy algorithm (FEM), model of the ferrosilicon smelting process has been presented. It is a system of two closed isothermal reactors: an upper one with a lower temperature T1, and a lower one with a higher temperature T2. Between the reactors and the environment as well as between the reactors inside the system, a periodical exchange of mass occurs at the moments when the equilibrium state is reached. The condensed products of chemical reactions move from the top to the bottom, and the gas phase components move in the opposite direction. It can be assumed that in the model, the Reactor 1 corresponds to the charge zone of submerged arc furnace where heat is released as a result of resistive heating, and the Reactor 2 corresponds to the zones of the furnace where heat is produced by electric arc. Using the model, a series of calculations was performed for the Fe-Si-O-C system and was determined the influence of temperatures T1, T2 on the process. The calculation results show a good agreement model with the real ferrosilicon process. It allows for the determination of the effects of temperature conditions in charge zones and arc zones of the ferrosilicon furnace on the carbothermic silica reduction process. This allows for an explanation of many characteristic states in the ferrosilicon smelting process.
PL
Bazując na algorytmie minimalizacji Gibbsa przedstawiono model fizykochemiczny procesu wytopu żelazokrzemu. Modelem procesu jest układ dwóch reaktorów izotermicznych: górnego o temperaturze niższej T1, oraz dolnego o temperaturze wyższej T2 . Pomiędzy reaktorami i otoczeniem, oraz pomiędzy reaktorami wewnątrz układu zachodzi cyklicznie wymiana masy w chwilach, gdy w reaktorach osiągnięty jest stan równowagi. Skondensowane produkty reakcji przechodzą z góry w dół, a składniki fazy gazowej w kierunku przeciwnym. Można założyć, że reaktor 1 odpowiada strefom pieca rezystancyjno-łukowego w których ciepło wydziela się na zasadzie nagrzewania rezystancyjnego, a strefa 2 odpowiada strefom łuku, w których ciepło wydziela się na zasadzie nagrzewania łukowego. Wykorzystując pakiet do obliczeń termochemicznych HSC 7.1 przeprowadzono serię obliczeń symulacyjnych dla układu Fe-Si-O-C oraz określono wpływ temperatury T1, T2 na efektywność procesu redukcji krzemionki węglem.
EN
Based on the minimum Gibbs Free Enthalpy algorithm, model of the ferrosilicon smelting process has been presented. It is a system of two closed isothermal reactors: an upper one with a lower temperature T1, and a lower one with a higher temperature T2. Between the reactors and the environment as well as between the reactors inside the system, a periodical exchange of mass occurs at the moments when the equilibrium state is reached. The condensed products of chemical reactions move from the top to the bottom, and the gas phase components move in the opposite direction. It can be assumed that in the model, the Reactor 1 corresponds to the charge zone of submerged arc furnace where heat is released as a result of resistive heating, and the Reactor 2 corresponds to the zones of the furnace where heat is produced by electric arc. Using the model, a series of calculations was performed for the Fe-Si-O-C system and was determined the influence of temperatures T1, T2 on the efficiency of the carbothermic silica reduction process.
PL
W pracy poddano analizie pojazd ratowniczy przystosowany do pracy w zubożonym powietrzu, którego układ zasilania powietrzem został zaopatrzony w zbiornik tlenu i mieszalnik zapewniający utrzymanie koncentracji tlenu 21% w gazowej mieszaninie utleniającej doprowadzanej do silnika. Została podana metodyka wyznaczania składu równowagowego spalin w przemianie ekspansji powstałych ze spalania paliwa o składzie CαHβOγNδ w gazowej mieszaninie utleniającej o znanym składzie. Przedstawiono wyniki przykładowych obliczeń wybranych parametrów teoretycznego obiegu Otto dla metanu jako paliwa, zadanego współczynnika stechiometrycznego w zależności od składu mieszaniny utleniającej.
EN
The study analyzed the rescue vehicle designed to temporary work in depleted air, which the air supply system was equipped with an oxygen pressure vessel and mixer to maintain an oxygen concentration of 21% in the oxidizing gas mixture supplied to the engine. Was given the methodology for determining the equilibrium combustion products composition (in the expansion stroke) resulting from the combustion of fuel CαHβOγNδ in the gas oxidation mixture of known composition. Paper presents the results of sample calculations of selected parameters of the theoretical Otto cycle for the methane as fuel, depending on the composition of the oxidation mixture and the given equivalence ratio.
EN
Purpose: Body balance, as one of the coordination abilities,is a desirable variable for basketball players as regards the necessity of efficient responses in constantly changing situations on a basketball court. The aim of this study was to check whether physical activity in the form of running and jumping influences variables characterizing the process of keeping body balance of a basketball player in the standing position. Methods: The research was conducted on 11 young basketball players. The measurements were taken with a Kistler force plate. Apart from commonly registered COP displacements, an additional variable describing the process of keeping body balance by a basketball player was ankle joint stiffness on the basis of which an “Index of Balance–Stiffness” (IB-S) was created. Results: Statistically significant differences were obtained for the maximum COP displacements and ankle joint stiffness between measurements of balance in the standing position before and after the employed movement tasks whereas there were no statistically significant differences for the aforementioned variables describing the process of keeping balance between measurements after running and after jumping. Conclusions: The research results indicate that the employed movement activities brought about significant changes in the process of keeping balance of basketball player in the standing position which, after the run performed, remain on a similar level to the series of jumps being performed. The authors attempted to establish an index based on the stiffness which yields a possibility to perceive each basketball player as an individual person in the process of keeping balance.
19
Content available Zastosowanie tanich adsorbentów do uzdatniania wody
PL
W ciągu ostatnich kilku lat poszukiwanie nowych tanich sorbentów stało się ważnym, szeroko analizowanym zagadnieniem przez wiodące światowe ośrodki badawcze. Tematyka obejmuje adsorpcję: jonów metali ciężkich, barwników, fenoli i ich pochodnych, węglowodorów itp. Istotą problemu jest zastąpienie drogich tradycyjnych sorbentów tanimi, które często są produkowane z materiałów odpadowych. Ocena skuteczności naturalnych sorbentów wymaga przeprowadzania eksperymentów na konkretnej grupie związków i na wybranym sorbencie. Tanie naturalne sorbenty nie wymagają regeneracji, na przykład mogą być po procesie adsorpcji spalone, zeszkliwione, kompostowane etc. W artykule przedstawiono przegląd wykorzystania tanich sorbentów w zastosowaniu do usuwania zanieczyszczeń z wody. Zaprezentowano usuwanie jonów metali ciężkich, barwników oraz węglowodorów przy wykorzystaniu naturalnych sorbentów. W pracy omówiono równowagę, kinetykę oraz dynamikę sorpcji na podstawie danych literaturowych i wyników badań własnych.
EN
The search for new low-cost sorbents for sorption is an important issue extensively analyzed by the world leading research centers in the last few years. This includes heavy metal ions, dyes, phenols and their derivatives, hydrocarbons, etc. The essence of the problem is to replace expensive traditional sorbents by means of low-cost ones which are often produced from waste materials. Evaluation of the effectiveness of these natural sorbents requires carrying out experiments on specific group of compounds and a selected sorbent. Low-cost sorbents do not require regeneration of sorbents as it may be, for example burnt in incinerators where exhaust gases mast must be treated at appropriate level. This paper presents an overview of the application of low cost sorbents to remove contaminants from water. The manuscript includes the removal of heavy metal ions, dyes, hydrocarbons, etc using low cost adsorbents. In the paper equilibrium, kinetics and dynamics of sorption are evaluated based on literature data and our own research.
EN
Purpose: Coal gasification is complex technology, which results depend on many variables, connected among others with fuel, the converting agent and the process itself. The paper aims to analyze, by means of simulations, the influence of the following factors – temperature, pressure, characteristic parameters of coal and gasifying agent, on the composition and heating value of the obtained syngas. The other aim of this paper is the determination of gasification efficiency (based on the definition of cold gas efficiency) for various process conditions. Methods: Computer simulations were used as the research method for the work presented. An equilibrium model, based on the stoi-chiometric method with four independent reactions, was formulated and used in this paper. This model was implemented in Mathematica software. The influence of temperature (in a range from 500 to 1500°C), pressure (changed from atmospheric to 35 atm), three types of gasifying agent (mixtures of air, pure oxygen and steam) and the composition of four Polish coals (lignite and three hard-coals) on syngas parameters were analyzed in this paper. Results: Concentrations of CO2, CO, CH4, H2O, H2, N2 in the equilibrium syngas, for the chosen temperature, pressure and parame-ters of the fuel and converting agent were the results of the simulations carried out. Subsequently, the lower heating value and process efficiency for each syngas composition was calculated. Practical implications: The simulations indicated the thermodynamic limits of gasification and allowed for the formulation of the general principles ruling this process. Results presented in this paper may be useful in the preliminary optimization and analysis of coal gasification. They also can be a point of reference for more advanced simulations. Originality/ value: This paper presents own results obtained from equilibrium simulations of coal gasification. The author implemented a mathematical model, based on the method of Deringer and Traustel, presented earlier in literature, to carry out the calculations.
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