Search results for: fixed biomass

Authors: Sandija Zeverte-Rivza, Irina Pilvere, Baiba Rivza

Abstract:

The use of renewable energy sources incl. biogas has become topical in accordance with the increasing demand for energy, decrease of fossil energy resources and the efforts to reduce greenhouse gas emissions as well as to increase energy independence from the territories where fossil energy resources are available.

As the technologies of biogas production from agricultural biomass develop, risk assessment and risk management become necessary for farms producing such a renewable energy. The need for risk assessments has become particularly topical when discussions on changing the biogas policy in the EU take place, which may influence the development of the sector in the future, as well as the operation of existing biogas facilities and their income level.

The current article describes results of the risk assessment for farms producing biomass from agriculture biomass in Latvia, the risk assessment system included 24 risks, that affect the whole biogas production process and the obtained results showed the high significance of political and production risks.

Keywords: Biogas production, risks, risk assessment.

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Authors: Thidarat Papone, Supaporn Kookkhunthod, Ratanaporn Leesing

Abstract:

Monoculture and mixed cultures of microalgae and the oleaginous yeast for microbial oil productions were investigated using sugarcane juice as carbon substrate. The monoculture of yeast Torulaspora maleeae Y30, Torulaspora globosa YU5/2 grew faster than that of microalgae Chlorella sp. KKU-S2. In monoculture of T. maleeae Y30, a biomass of 8.267g/L with lipid yield of 0.920g/L were obtained, while 8.333g/L of biomass with lipid yield of 1.141g/L were obtained for monoculture of T. globosa YU5/2. A biomass of 1.933g/L with lipid yield of 0.052g/L was found for monoculture of Chlorella sp. KKU-S2. The biomass concentration in the mixed culture of the oleaginous yeast with microalgae increased faster and was higher compared with that in the monocultures. A biomass of 8.733g/L with lipid yield of 1.564g/L was obtained for a mixed culture of T. maleeae Y30 with Chlorella sp. KKU-S2, while 8.010g/L of biomass with lipid yield of 2.424g/L was found for mixed culture of T. globosa YU5/2 with Chlorella sp. KKU-S2. Maximum cell yield coefficient (YX/S, g/L) was found of 0.323 in monoculture of Chlorella sp. KKU-S2 but low level of both specific yield of lipid (YP/X, g lipid/g cells) of 0.027 and volumetric lipid production rate (QP, g/L/d) of 0.003 were observed. While, maximum YP/X (0.303), QP (0.105) and maximum process product yield (YP/S, 0.061) were obtained in mixed culture of T. globosa YU5/2 with Chlorella sp. KKU-S2. The results obtained from the study shows that mixed culture of yeast with microalgae is a desirable cultivation process for microbial oil production.

Keywords: Microbial oil, Chlorella sp. KKU-S2, Torulaspora maleeae Y30, Torulaspora globosa YU5/2, mixed culture, biodiesel.

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Authors: Aki Happonen, Adrian Burian, Erwin Hemming

Abstract:

Fixed-point simulation results are used for the performance measure of inverting matrices using a reconfigurable processing element. Matrices are inverted using the Cholesky decomposition algorithm. The reconfigurable processing element is capable of all required mathematical operations. The fixed-point word length analysis is based on simulations of different condition numbers and different matrix sizes.

Keywords: Cholesky Decomposition, Fixed-point, Matrixinversion, Reconfigurable processing.

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Authors: Menglong Su, Shaoyun Shi, Qing Xu

Abstract:

In this paper, by introducing twice continuously differentiable mappings, we develop an interior path following following method, which enables us to give a constructive proof of the general Brouwer fixed point theorem and thus to solve fixed point problems in a class of non-convex sets. Under suitable conditions, a smooth path can be proven to exist. This can lead to an implementable globally convergent algorithm. Several numerical examples are given to illustrate the results of this paper.

Keywords: interior path following method, general Brouwer fixed point theorem, non-convex sets, globally convergent algorithm

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Authors: Li Yongfeng, Liu Yixing, Li Jian, Li Jun

Abstract:

A novel biomass composite inspired from wood porous structure was manufactured by impregnating vinyl monomer into wood cellular structure under vacuum conditions, and initiating the monomer for in situ polymerization through a thermal treatment. The vacuum condition was studied, and the mechanical properties of the composite were also tested. SEM observation shows that polymer generated in the wood porous structure, and strongly interacted with wood matrix; and the polymer content increased with vacuum value increasing. FTIR indicates that polymer grafted onto wood matrix, resulting chemical complex between them. The rate of monomer loading increased with increasing vacuum value and time, accordance with rate of polymer loading. The compression strength and modulus of elasticity linearly increased with the increasing rate of polymer loading. Results indicate that the novel biomass composite possesses good mechanical properties capable of applying in the fields of construction, traffic and so forth.

Keywords: Biomass composite, manufacture, vinyl monomer, wood cellular structure.

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Authors: C. Sinescu, M. Negrutiu, M. Romînu, C. Haiduc, E. Petrescu, M. Leretter, A.G. Podoleanu

Abstract:

Optical Coherence Tomography (OCT) combined with the Confocal Microscopy, as a noninvasive method, permits the determinations of materials defects in the ceramic layers depth. For this study 256 anterior and posterior metal and integral ceramic fixed partial dentures were used, made with Empress (Ivoclar), Wollceram and CAD/CAM (Wieland) technology. For each investigate area 350 slices were obtain and a 3D reconstruction was perform from each stuck. The Optical Coherent Tomography, as a noninvasive method, can be used as a control technique in integral ceramic technology, before placing those fixed partial dentures in the oral cavity. The purpose of this study is to evaluate the capability of En face Optical Coherence Tomography (OCT) combined with a fluorescent method in detection and analysis of possible material defects in metalceramic and integral ceramic fixed partial dentures. As a conclusion, it is important to have a non invasive method to investigate fixed partial prostheses before their insertion in the oral cavity in order to satisfy the high stress requirements and the esthetic function.

Keywords: Ceramic Fixed Partial Dentures, Material Defects, En face Optical Coherence Tomography, Fluorescence.

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Authors: Jun Li, Wenze Wang, Zhanggui Hu, Yongwei Zhu, Dunwen Zuo

Abstract:

Non-water based fixed abrasive polishing was adopted to manufacture LBO crystal for nano precision surface quality because of its deliquescent. Ethyl alcohol was selected as the non-water based slurry solvent and ethanediamine, lactic acid, hydrogen peroxide was added in the slurry as a chemical additive, respectively. Effect of different additives with non-water based slurry on material removal rate, surface topography, microscopic appearances, and surface roughness were investigated in fixed abrasive polishing of LBO crystal. The results show the best surface quality of LBO crystal with surface roughness Sa 8.2 nm and small damages was obtained by non-water based slurry with lactic acid. Non-water based fixed abrasive polishing can achieve nano precision surface quality of LBO crystal with high material removal.

Keywords: Non-water based slurry, LBO crystal, Fixed abrasive polishing, Surface roughness.

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Authors: Phuong T. Dang, Hy G. Le, Giang T. Pham, Hong T. M. Vu, Kien T, Nguyen, Canh D. Dao, Giang H. Le, Hoa T. K. Tran, Quang K. Nguyen, Tuan A. Vu

Abstract:

Nanostructured catalysts were successfully prepared by acidification of diatomite and regeneration of FCC spent catalysts. The obtained samples were characterized by IR, XRD, SEM, EDX, MAS-NMR (27Al and 29Si), NH3-TPD and tested in catalytic pyrolysis of biomass (rice straw). The results showed that the similar bio-oil yield of 41.4% can be obtained by pyrolysis with catalysts at 450oC as compared to that of the pyrolysis without catalyst at 550oC. The bio-oil yield reached a maximum of 42.55% at the pyrolysis temperature of 500oC with catalytic content of 20%. Moreover, by catalytic pyrolysis, bio-oil quality was better as reflected in higher ratio of H/C, lower ratio of O/C. This clearly indicated high application potential of these new nanostructured catalysts in the production of bio-oil with low oxygenated compounds.

Keywords: Acidified diatomite, biomass, catalytic pyrolysis, bio-oil, nanostructured catalysts, regenerated FCC catalyst.

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Authors: Shinya Yokoyama, Katsunari Jonouchi, Kenji Imou

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This paper discusses the utilization of marine biomass as an energy resource in Japan. A marine biomass energy system in Japan was proposed consisting of seaweed cultivation (Laminaria japonica) at offshore marine farms, biogas production via methane fermentation of the seaweeds, and fuel cell power generation driven by the generated biogas. We estimated energy output, energy supply potential, and CO2 mitigation in Japan on the basis of the proposed system. As a result, annual energy production was estimated to be 1.02-109 kWh/yr at nine available sites. Total CO2 mitigation was estimated to be 1.04-106 tonnes per annum at the nine sites. However, the CO2 emission for the construction of relevant facilities is not taken into account in this paper. The estimated CO2 mitigation is equivalent to about 0.9% of the required CO2 mitigation for Japan per annum under the Kyoto Protocol framework.

Keywords: CO2 mitigation, Fuel cell power generation, Laminaria japonica, Marine biomass, Seaweed.

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Authors: Lukáš Plošek, Jaroslav Hynšt, Jaroslav Záhora, Jakub Elbl, Antonín Kintl, Ivana Charousová, Silvia Kovácsová

Abstract:

Compost can influence soil fertility and plant health. At the same time compost can play an important role in the nitrogen cycle and it can influence leaching of mineral nitrogen from soil to underground water.

This paper deals with the influence of compost addition and mineral nitrogen fertilizer on leaching of mineral nitrogen, nitrogen availability in microbial biomass and plant biomass production in the lysimetric experiment. Twenty one lysimeters were filed with topsoil and subsoil collected in the area of protection zone of underground source of drinking water - Březová nad Svitavou. The highest leaching of mineral nitrogen was detected in the variant fertilized only mineral nitrogen fertilizer (624.58 mg m^-2), the lowest leaching was recorded in the variant with high addition of compost (315.51 mg m^-2). On the other hand, losses of mineral nitrogen are not in connection with the losses of available form of nitrogen in microbial biomass. Because lost of mineral nitrogen was detected in variant with the least change in the availability of N in microbial biomass.

The leaching of mineral nitrogen, yields as well as the results concerning nitrogen availability from the first year of long term experiment suggest that compost can positive influence the leaching of nitrogen into underground water.

Keywords: Nitrogen, Compost, Biomass production, Lysimeter.

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Authors: Alemayehu Geremew Negash

Abstract:

In this paper, we prove some common fixed point theorems for co-cyclic weak contractions in compact metric spaces.

Keywords: Cyclic weak contraction, Co-cyclic weak contraction, Co-cyclic representation, Common fixed point.

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Authors: Okolo Chimaobi Valentine

Abstract:

Commercial banks in Nigeria adopted many strategies to attract fresh deposits including the use of high deposit rate. However, pricing of banking services moved in favor of the banks at the expense of customers, resulting in their seeking other investment alternatives rather than saving their money in the bank. Both deposit and lending rates were greatly influenced by the Central Bank of Nigeria (CBN) decision on interest rate. Therefore, commercial bank effort to attract deposits via manipulation of her rates was greatly limited, otherwise the banks will be giving out more than it earned. The study aimed at examining the relationship between interest rate and fixed fund deposit of commercial banks, how policy-controlled interest rate affected commercial bank’s fixed fund deposit The researcher employed ordinary least square technique, using, multiple linear regression, unrestricted vector auto-regression, correlation matrix test, granger causality and impulse response graph in the analysis. Commercial bank’s interest rates affected commercial bank’s fixed fund deposit significantly while policy-controlled interest rate did not significantly transmit through the commercial bank’s interest rates to affect fixed fund deposit. While commercial banks seek creative ways to expand their fixed fund deposit, policy authorities in Nigeria should better coordinate interest rate fluctuation and induce competition in the entire financial sector.

Keywords: Commercial bank, fixed fund deposit, fluctuation effects, interest rate.

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Authors: S. J. Mitchual, K. Frimpong-Mensah, N. A. Darkwa

Abstract:

The fuel potential of six tropical hardwood species namely: Triplochiton scleroxylon, Ceiba pentandra, Aningeria robusta, Terminalia superba, Celtis mildbreadii and Piptadenia africana were studied. Properties studied included species density, gross calorific value, volatile matter, ash content, organic carbon and elemental composition. Fuel properties were determined using standard laboratory methods. The result indicates that the gross calorific value (GCV) of the species ranged from 20.16 to 22.22 MJ/kg and they slightly varied from each other. Additionally, the GCV of the biomass materials were higher than that of other biomass materials like; wheat straw, rice straw, maize straw and sugar cane. The ash and volatile matter content varied from 0.6075 to 5.0407%, and 75.23% to 83.70% respectively. The overall rating of the properties of the six biomass materials suggested that Piptadenia africana has the best fuel property to be used as briquettes and Aningeria robusta the worse. This study therefore suggests that a holistic assessment of a biomass material needs to be done before selecting it for fuel purpose.

Keywords: Ash content, Briquette, Calorific value, Elemental composition, Species, Volatile matter.

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Authors: František Včelař, Zuzana Pátíková

Abstract:

Recently a new type of very general relational structures, the so called (L-)complete propelattices, was introduced. These significantly generalize complete lattices and completely lattice L-ordered sets, because they do not assume the technically very strong property of transitivity. For these structures also the main part of the original Tarski’s fixed point theorem holds for (L-fuzzy) isotone maps, i.e., the part which concerns the existence of fixed points and the structure of their set. In this paper, fundamental properties of (L-)complete propelattices are recalled and the so called L-fuzzy relatively isotone maps are introduced. For these maps it is proved that they also have fixed points in L-complete propelattices, even if their set does not have to be of an awaited analogous structure of a complete propelattice.

Keywords: Fixed point, L-complete propelattice, L-fuzzy (relatively) isotone map, residuated lattice, transitivity.

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Authors: Lv Yuhua

Abstract:

In this paper, by constructing a special cone and using fixed point theorem and fixed point index theorem of cone, we get the existence of positive solution for a class of singular eigenvalue value problems with p-Laplace operator, which improved and generalized the result of related paper.

Keywords: Cone, fixed point index, eigenvalue problem, p-Laplace operator, positive solutions.

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Authors: C. Sinescu, M. Negrutiu, R. Negru, M. Romînu, A.G. Podoleanu

Abstract:

The fixed partial dentures are mainly used in the frontal part of the dental arch because of their great esthetics. There are several factors that are associated with the stress state created in ceramic restorations, including: thickness of ceramic layers, mechanical properties of the materials, elastic modulus of the supporting substrate material, direction, magnitude and frequency of applied load, size and location of occlusal contact areas, residual stresses induced by processing or pores, restoration-cement interfacial defects and environmental defects. The purpose of this study is to evaluate the capability of Polarization Sensitive Optical Coherence Tomography (PSOCT) in detection and analysis of possible material defects in metal-ceramic and integral ceramic fixed partial dentures. As a conclusion, it is important to have a non invasive method to investigate fixed partial prostheses before their insertion in the oral cavity in order to satisfy the high stress requirements and the esthetic function.

Keywords: Ceramic Fixed Partial Dentures, Material Defects, Polarization Sensitive Optical Coherence Tomography, Numerical Simulation

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Authors: M.F. Mohamad, Anita Ramli, S.E.E Misi, S. Yusup

Abstract:

This work presents the hydrogen production from steam gasification of palm kernel shell (PKS) at 700 oC in the presence of 5% Ni/BEA and 5% Fe/BEA as catalysts. The steam gasification was performed in two-staged reactors to evaluate the effect of calcinations temperature and the steam to biomass ratio on the product gas composition. The catalytic activity of Ni/BEA catalyst decreases with increasing calcinations temperatures from 500 to 700 oC. The highest H2 concentration is produced by Fe/BEA (600) with more than 71 vol%. The catalytic activity of the catalysts tested is found to correspond to its physicochemical properties. The optimum range for steam to biomass ratio if found to be between 2 to 4. Excess steam content results in temperature drop in the gasifier which is undesirable for the gasification reactions.

Keywords: Hydrogen, Palm Kernel Shell, Steam gasification, Ni/BEA, Fe/BEA

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Authors: M. Chafi, S. Akazdam, C. Asrir, L. Sebbahi, B. Gourich, N. Barka, M. Essahli

Abstract:

Fixed bed adsorption has become a frequently used industrial application in wastewater treatment processes. Various low cost adsorbents have been studied for their applicability in treatment of different types of effluents. In this work, the intention of the study was to explore the efficacy and feasibility for azo dye, Acid Orange 7 (AO7) adsorption onto fixed bed column of NaOH Treated eggshell (TES). The effect of various parameters like flow rate, initial dye concentration, and bed height were exploited in this study. The studies confirmed that the breakthrough curves were dependent on flow rate, initial dye concentration solution of AO7 and bed depth. The Thomas, Yoon–Nelson, and Adams and Bohart models were analysed to evaluate the column adsorption performance. The adsorption capacity, rate constant and correlation coefficient associated to each model for column adsorption was calculated and mentioned. The column experimental data were fitted well with Thomas model with coefficients of correlation R2 ≥0.93 at different conditions but the Yoon–Nelson, BDST and Bohart–Adams model (R2=0.911), predicted poor performance of fixed-bed column. The (TES) was shown to be suitable adsorbent for adsorption of AO7 using fixed-bed adsorption column.

Keywords: Adsorption models, acid orange 7, bed depth, breakthrough, dye adsorption, fixed-bed column, treated eggshell.

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Authors: N. P. Dhull, K. Gupta, R. Soni, D. K. Rahi, S. K. Soni

Abstract:

The aim of the present work was to statistically design an autotrophic medium for maximum biomass production by Chlorella pyrenoidosa using response surface methodology. After evaluating one factor at a time approach, K2HPO4, KNO3, MgSO4.7H2O and NaHCO3 were preferred over the other components of the fog’s medium as most critical autotrophic medium components. The study showed that the maximum biomass yield was achieved while the concentrations of MgSO4.7H2O, K2HPO4, KNO3 and NaHCO3 were 0.409 g/L, 0.24 g/L, 1.033 g/L, and 3.265 g/L, respectively. The study reported that the biomass productivity of C. pyrenoidosa improved from 0.14 g/L in defined fog’s medium to 1.40 g/L in modified fog’s medium resulting 10 fold increase. The biochemical composition biosynthesis of C. pyrenoidosa was altered using nitrogen limiting stress bringing about 5.23 fold increase in lipid content than control (cell without stress), as analyzed by FTIR integration method.

Keywords: Autotrophic condition, Chlorella pyrenoidosa, FTIR, Response Surface Methodology, Optimization.

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Authors: Adil AL-Rammahi

Abstract:

In this paper, a new method for solution of second order linear Fredholm integral equation in power series form was studied. The result is obtained by using Banach fixed point theorem.

Keywords: Fredholm integral equation, power series, Banach fixed point theorem, Linear Systems.

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Authors: B. Silva, H. Figueiredo, I. C. Neves, T. Tavares

Abstract:

Arthrobacter viscosus biomass was used for Cr(VI) biosorption. The effect of pH on Cr(VI) reduction and removal from aqueous solution was studied in the range of 1-4. The Cr(VI) removal involves both redox reaction and adsorption of metal ions on biomass surface. The removal rate of Cr(VI) was enhanced by very acid conditions, while higher solution pH values favored the removal of total chromium. The best removal efficiency and uptake were reached at pH 4, 72.5 % and 12.6 mgCr/gbiomass, respectively.

Keywords: Biosorption, chromium, pH, reduction.

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Authors: M. Moosavi, H. Khatibzadeh

Abstract:

In this article, we study demiclosed and strongly quasi-nonexpansive of a sequence generated by the proximal point algorithm for a finite family of quasi-nonexpansive mappings in Hadamard spaces. Δ-convergence of iterations for the sequence of strongly quasi-nonexpansive mappings as well as the strong convergence of the Halpern type regularization of them to a common fixed point of sequence are also established. Our results generalize and improve several previously known results of the existing literature.

Keywords: Fixed point, Hadamard space, proximal point algorithm, quasi-nonexpansive sequence of mappings, resolvent.

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Authors: Safeer Hussain Khan

Abstract:

In this paper, we introduce a two-step iterative algorithm to prove a strong convergence result for approximating common fixed points of three contractive-like operators. Our algorithm basically generalizes an existing algorithm..Our iterative algorithm also contains two famous iterative algorithms: Mann iterative algorithm and Ishikawa iterative algorithm. Thus our result generalizes the corresponding results proved for the above three iterative algorithms to a class of more general operators. At the end, we remark that nothing prevents us to extend our result to the case of the iterative algorithm with error terms.

Keywords: Contractive-like operator, iterative algorithm, fixed point, strong convergence.

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Authors: Yasuhito Tanaka

Abstract:

We present a constructive proof of Tychonoff’s fixed point theorem in a locally convex space for uniformly continuous and sequentially locally non-constant functions.

Keywords: sequentially locally non-constant functions, Tychonoff’s fixed point theorem, constructive mathematics.

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Authors: Sh. Shahhosseini, S. Alinia, M. Irani

Abstract:

In this paper 2D Simulation of catalytic Fixed Bed Reactor in Fischer-Tropsch Synthesis of GTL technology has been performed utilizing computational fluid dynamics (CFD). Synthesis gas (a mixture of carbon monoxide and hydrogen) has been used as feedstock. The reactor was modeled and the model equations were solved employing finite volume method. The model was validated against the experimental data reported in literature. The comparison showed a good agreement between simulation results and the experimental data. In addition, the model was applied to predict the concentration contours of the reactants and products along the length of reactor.

Keywords: GTL, Fischer–Tropsch synthesis, Fixed Bed Reactor, CFD simulation.

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Authors: Clement M. Matasane, Mohamed T. Kahn

Abstract:

Biomass resources such as animal waste, agricultural and acro-industrial residues, forestry and woodland waste, and industrial and municipal solid wastes provide alternative means to utilize its untapped potential for biomass/biofuel renewable energy systems. In addition, crop residues (i.e., grain, starch, and energy crops) are commonly available in the district and play an essential role in community farming activities. The remote sensing technology (mappings) and geographic information systems tool will be used to determine the biomass potential in the Vhembe District Municipality. The detailed assessment, estimation, and modeling in quantifying their distribution, abundance, and quality yield an effective and efficient use of their potential. This paper aims to examine the potential and prospects of deploying bioenergy systems in small or micro-systems in the district for community use and applications. This deployment of the biofuels/biomass systems will help communities for sustainable energy supply from their traditional energy use into innovative and suitable methods that improve their livelihood. The study demonstrates the potential applications of Geographical Information Systems (GIS) in spatial mapping analysis, evaluation, modeling, and decision support for easy access to renewable energy systems.

Keywords: Agricultural crops, waste materials, biomass potentials, bioenergy potentials, GIS mappings, environmental data, renewable energy deployment, sustainable energy supply.

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Authors: Gabriela-Alina Dumitrel, Teodor Todinca, Carmen Holotescu, Cosmina-Mariana Militaru

Abstract:

The paper presents a computational tool developed for the evaluation of technical and economic advantages of an innovative cleaning and conditioning technology of fluidized bed steam/oxygen gasifiers outlet product gas. This technology integrates into a single unit the steam gasification of biomass and the hot gas cleaning and conditioning system. Both components of the computational tool, process flowsheet and economic evaluator, have been developed under IPSEpro software. The economic model provides information that can help potential users, especially small and medium size enterprises acting in the regenerable energy field, to decide the optimal scale of a plant and to better understand both potentiality and limits of the system when applied to a wide range of conditions.

Keywords: biomass, CHP units, economic evaluation, gasification.

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Authors: Martin Lisý, Marek Baláš, Michal Špiláček, Zdeněk Skála

Abstract:

This paper deals with the issue of biomass and sorted municipal waste gasification and cogeneration using hot-air turbo-set. It brings description of designed pilot plant with electrical output 80 kWe. The generated gas is burned in secondary combustion chamber located beyond the gas generator. Flue gas flows through the heat exchanger where the compressed air is heated and consequently brought to a micro turbine. Except description, this paper brings our basic experiences from operating of pilot plant (operating parameters, contributions, problems during operating, etc.). The principal advantage of the given cycle is the fact that there is no contact between the generated gas and the turbine. So there is no need for costly and complicated gas cleaning which is the main source of operating problems in direct use in combustion engines because the content of impurities in the gas causes operation problems to the units due to clogging and tarring of working surfaces of engines and turbines, which may lead as far as serious damage to the equipment under operation. Another merit is the compact container package making installation of the facility easier or making it relatively more mobile. We imagine, this solution of cogeneration from biomass or waste can be suitable for small industrial or communal applications, for low output cogeneration.

Keywords: Biomass, combustion, gasification, microcogeneration.

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Authors: Ismat Beg, Asma Rashid Butt

Abstract:

Let (X,) be a partially ordered set and d be a metric on X such that (X, d) is a complete metric space. Assume that X satisfies; if a non-decreasing sequence xn → x in X, then xn  x, for all n. Let F be a set valued mapping from X into X with nonempty closed bounded values satisfying; (i) there exists κ ∈ (0, 1) with D(F(x), F(y)) ≤ κd(x, y), for all x  y, (ii) if d(x, y) < ε < 1 for some y ∈ F(x) then x  y, (iii) there exists x0 ∈ X, and some x1 ∈ F(x0) with x0  x1 such that d(x0, x1) < 1. It is shown that F has a fixed point. Several consequences are also obtained.

Keywords: Fixed point, partially ordered set, metric space, set valued mapping.

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Authors: Manisha Chaudhary, Joydip Dhar, Govind Prasad Sahu

Abstract:

A mathematical model is proposed considering the forest biomass density B(t), density of wood based industries W(t) and density of synthetic industries S(t). It is assumed that the forest biomass grows logistically in the absence of wood based industries, but depletion of forestry biomass is due to presence of wood based industries. The growth of wood based industries depends on B(t), while S(t) grows at a constant rate, independent of B(t). Further there is a competition between W(t) and S(t) according to market demand. The proposed model has four ecologically feasible steady states, namely, E1: forest biomass free and wood industries free equilibrium; E2: wood industries free equilibrium and two coexisting equilibria E∗1 , E∗2 . Behavior of the system near all feasible equilibria is analyzed using the stability theory of differential equations. In the proposed model, the natural depletion rate h1 is a crucial parameter and system exhibits Hopf-bifurcation about the non-trivial equilibrium with respect to h1. The analytical results are verified using numerical simulation.

Keywords: A mathematical model, Competition between wood based and synthetic industries, Hopf-bifurcation, Stability analysis.

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