Search results for: entropy and exergy analysis

Authors: Sheng-An Yang, Ren-Yi Hung, Ying-Yi Ho

Abstract:

This paper aims to perform the second law analysis of thermodynamics on the laminar film condensation of pure saturated vapor flowing in the direction of gravity on an ellipsoid with variable wall temperature. The analysis provides us understanding how the geometric parameter- ellipticity and non-isothermal wall temperature variation amplitude “A." affect entropy generation during film-wise condensation heat transfer process. To understand of which irreversibility involved in this condensation process, we derived an expression for the entropy generation number in terms of ellipticity and A. The result indicates that entropy generation increases with ellipticity. Furthermore, the irreversibility due to finite temperature difference heat transfer dominates over that due to condensate film flow friction and the local entropy generation rate decreases with increasing A in the upper half of ellipsoid. Meanwhile, the local entropy generation rate enhances with A around the rear lower half of ellipsoid.

Keywords: Free convection; Non-isothermal; Thermodynamic second law; Entropy, Ellipsoid.

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Authors: Ali Mchirgui, Nejib Hidouri, Mourad Magherbi, Ammar Ben Brahim

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The paper provides a numerical investigation of the entropy generation analysis due to natural convection in an inclined square porous cavity. The coupled equations of mass, momentum, energy and species conservation are solved using the Control Volume Finite-Element Method. Effect of medium permeability and inclination angle on entropy generation is analysed. It was found that according to the Darcy number and the porous thermal Raleigh number values, the entropy generation could be mainly due to heat transfer or to fluid friction irreversibility and that entropy generation reaches extremum values for specific inclination angles.

Keywords: Porous media, entropy generation, convection, numerical method.

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Authors: David Sherwell, Vivien Visaya

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We consider the topological entropy of maps that in general, cannot be described by one-dimensional dynamics. In particular, we show that for a multivalued map F generated by singlevalued maps, the topological entropy of any of the single-value map bounds the topological entropy of F from below.

Keywords: Multivalued maps, Topological entropy, Selectors

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Authors: Young-Seok Choi

Abstract:

This paper provides a quantitative measure of the time-varying multiunit neuronal spiking activity using an entropy based approach. To verify the status embedded in the neuronal activity of a population of neurons, the discrete wavelet transform (DWT) is used to isolate the inherent spiking activity of MUA. Due to the de-correlating property of DWT, the spiking activity would be preserved while reducing the non-spiking component. By evaluating the entropy of the wavelet coefficients of the de-noised MUA, a multiresolution Shannon entropy (MRSE) of the MUA signal is developed. The proposed entropy was tested in the analysis of both simulated noisy MUA and actual MUA recorded from cortex in rodent model. Simulation and experimental results demonstrate that the dynamics of a population can be quantified by using the proposed entropy.

Keywords: Discrete wavelet transform, Entropy, Multiresolution, Multiunit activity.

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Authors: Saria Abed, Tahar Khir, Ammar Ben Brahim

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A thermo-exergy optimization of a gas turbine cycle with two different regenerator designs is established. A comparison was made between the performance of the two regenerators and their roles in improving the cycle efficiencies. The effect of operational parameters (the pressure ratio of the compressor, the ambient temperature, excess of air, geometric parameters of the regenerators, etc.) on thermal efficiencies, the exergy efficiencies, and irreversibilities were studied using thermal balances and quantitative exegetic equilibrium for each component and for the whole system. The results are given graphically by using the EES software, and an appropriate discussion and conclusion was made.

Keywords: Exergy efficiency, gas turbine, heat transfer, irreversibility, optimization, regenerator, thermal efficiency.

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Authors: Anabel Fernandez, Leandro Rodriguez-Ortiz, Rosa Rodríguez

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Due to the negative impact of fossil fuels, renewable energies are promising sources to limit global temperature rise and damage to the environment. Also, the development of technology is focused on obtaining energetic products from renewable sources. In this study, a thermodynamic model including exergy balance and a subsequent Life Cycle Assessment (LCA) were carried out for four subsystems of the integrated gasification-combustion of pinewood. Results of exergy analysis and LCA showed the process feasibility in terms of exergy efficiency and global energy efficiency of the life cycle (GEELC). Moreover, the energy return on investment (EROI) index was calculated. The global exergy efficiency resulted in 67%. For pretreatment, reaction, cleaning, and electric generation subsystems, the results were 85%, 59%, 87%, and 29%, respectively. Results of LCA indicated that the emissions from the electric generation caused the most damage to the atmosphere, water, and soil. GEELC resulted in 31.09% for the global process. This result suggested the environmental feasibility of an integrated gasification-combustion system. EROI resulted in 3.15, which determines the sustainability of the process.

Keywords: Exergy analysis, Life Cycle Assessment, LCA, renewability, sustainability.

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Authors: Young-Seok Choi

Abstract:

This work assesses the cortical and the sub-cortical neural activity recorded from rodents using entropy and mutual information based approaches to study how hypothermia affects neural activity. By applying the multi-scale entropy and Shannon entropy, we quantify the degree of the regularity embedded in the cortical and sub-cortical neurons and characterize the dependency of entropy of these regions on temperature. We study also the degree of the mutual information on thalamocortical pathway depending on temperature. The latter is most likely an indicator of coupling between these highly connected structures in response to temperature manipulation leading to arousal after global cerebral ischemia.

Keywords: Spiking activity, entropy, mutual information, temperature modulation.

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Authors: Arvinder Kaur, Deepti Chopra

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Software Entropy Metrics for bug prediction have been validated on various software systems by different researchers. In our previous research, we have validated that Software Entropy Metrics calculated for Mozilla subsystem’s predict the future bugs reasonably well. In this study, the Software Entropy metrics are calculated for a subsystem of Android and it is noticed that these metrics are not suitable for bug prediction. The results are compared with a subsystem of Mozilla and a comparison is made between the two software systems to determine the reasons why Software Entropy metrics are not applicable for Android.

Keywords: Android, bug prediction, mining software repositories, Software Entropy.

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Authors: Youssef Khmou, Said Safi, Miloud Frikel

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The Maximum entropy principle in spectral analysis was used as an estimator of Direction of Arrival (DoA) of electromagnetic or acoustic sources impinging on an array of sensors, indeed the maximum entropy operator is very efficient when the signals of the radiating sources are ergodic and complex zero mean random processes which is the case for cosmic sources. In this paper, we present basic review of the maximum entropy method (MEM) which consists of rank one operator but not a projector, and we elaborate a new operator which is full rank and sum of all possible projectors. Two dimensional Simulation results based on Monte Carlo trials prove the resolution power of the new operator where the MEM presents some erroneous fluctuations.

Keywords: Maximum entropy, Cosmic source, Localization, operator, projector, azimuth, elevation, DoA, circular array.

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Authors: R. K. Tuli

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In present communication, we have developed the suitable constraints for the given the mean codeword length and the measures of entropy. This development has proved that Renyi-s entropy gives the minimum value of the log of the harmonic mean and the log of power mean. We have also developed an important relation between best 1:1 code and the uniquely decipherable code by using different measures of entropy.

Keywords: Codeword, Instantaneous code, Prefix code, Uniquely decipherable code, Best one-one code, Mean codewordlength

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Authors: Nurdan Yildirim, Arif Hepbasli

Abstract:

Heating, cooling and lighting appliances in buildings account for more than one third of the world’s primary energy demand. Therefore, main components of the building heating systems play an essential role in terms of energy consumption. In this context, efficient energy and exergy utilization in HVAC-R systems has been very essential, especially in developing energy policies towards increasing efficiencies. The main objective of the present study is to assess the performance of a family house with a volume of 326.7 m3 and a net floor area of 121 m2, located in the city of Izmir, Turkey in terms of energetic, exergetic and sustainability aspects. The indoor and exterior air temperatures are taken as 20°C and 1°C, respectively. In the analysis and assessment, various metrics (indices or indicators) such as exergetic efficiency, exergy flexibility ratio and sustainability index are utilized. Two heating options (Case 1: condensing boiler and Case 2: air heat pump) are considered for comparison purposes. The total heat loss rate of the family house is determined to be 3770.72 W. The overall energy efficiencies of the studied cases are calculated to be 49.4% for Case 1 and 54.7% for Case 2. The overall exergy efficiencies, the flexibility factor and the sustainability index of Cases 1 and 2 are computed to be around 3.3%, 0.17 and 1.034, respectively.

Keywords: Buildings, exergy, low exergy, sustainability, efficiency, heating, renewable energy.

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Authors: Antonino Greco, Nadia Mammone, Francesco Carlo Morabito, Mario Versaci

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Artifact rejection plays a key role in many signal processing applications. The artifacts are disturbance that can occur during the signal acquisition and that can alter the analysis of the signals themselves. Our aim is to automatically remove the artifacts, in particular from the Electroencephalographic (EEG) recordings. A technique for the automatic artifact rejection, based on the Independent Component Analysis (ICA) for the artifact extraction and on some high order statistics such as kurtosis and Shannon-s entropy, was proposed some years ago in literature. In this paper we try to enhance this technique proposing a new method based on the Renyi-s entropy. The performance of our method was tested and compared to the performance of the method in literature and the former proved to outperform the latter.

Keywords: Artifact, EEG, Renyi's entropy, kurtosis, independent component analysis.

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Authors: Mohsen Nikaein

Abstract:

Because of importance of energy, optimization of power generation systems is necessary. Gas turbine cycles are suitable manner for fast power generation, but their efficiency is partly low. In order to achieving higher efficiencies, some propositions are preferred such as recovery of heat from exhaust gases in a regenerator, utilization of intercooler in a multistage compressor, steam injection to combustion chamber and etc. However thermodynamic optimization of gas turbine cycle, even with above components, is necessary. In this article multi-objective genetic algorithms are employed for Pareto approach optimization of Regenerative-Intercooling-Gas Turbine (RIGT) cycle. In the multiobjective optimization a number of conflicting objective functions are to be optimized simultaneously. The important objective functions that have been considered for optimization are entropy generation of RIGT cycle (Ns) derives using Exergy Analysis and Gouy-Stodola theorem, thermal efficiency and the net output power of RIGT Cycle. These objectives are usually conflicting with each other. The design variables consist of thermodynamic parameters such as compressor pressure ratio (Rp), excess air in combustion (EA), turbine inlet temperature (TIT) and inlet air temperature (T0). At the first stage single objective optimization has been investigated and the method of Non-dominated Sorting Genetic Algorithm (NSGA-II) has been used for multi-objective optimization. Optimization procedures are performed for two and three objective functions and the results are compared for RIGT Cycle. In order to investigate the optimal thermodynamic behavior of two objectives, different set, each including two objectives of output parameters, are considered individually. For each set Pareto front are depicted. The sets of selected decision variables based on this Pareto front, will cause the best possible combination of corresponding objective functions. There is no superiority for the points on the Pareto front figure, but they are superior to any other point. In the case of three objective optimization the results are given in tables.

Keywords: Exergy, Entropy Generation, Brayton Cycle, DesignParameters, Optimization, Genetic Algorithm, Multi-Objective.

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Authors: Stoyan Nedeltchev, Markus Schubert

Abstract:

By means of the ultrafast X-ray tomography facility, data were obtained at different superficial gas velocities U_G in a bubble column (0.1 m in ID) operated with an air-deionized water system at ambient conditions. Raw reconstructed images were treated by both the information entropy (IE) and the reconstruction entropy (RE) algorithms in order to identify the main transition velocities in a bubble column. The IE values exhibited two well-pronounced minima at U_G=0.025 m/s and U_G=0.085 m/s identifying the boundaries of the homogeneous, transition and heterogeneous regimes. The RE extracted from the central region of the column’s cross-section exhibited only one characteristic peak at U_G=0.03 m/s, which was attributed to the transition from the homogeneous to the heterogeneous flow regime. This result implies that the transition regime is non-existent in the core of the column.

Keywords: Bubble column, ultrafast X-ray tomography, information entropy, reconstruction entropy.

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Authors: Antonino Greco, Nadia Mammone, Francesco Carlo Morabito, Mario Versaci

Abstract:

The goal of this work is to improve the efficiency and the reliability of the automatic artifact rejection, in particular from the Electroencephalographic (EEG) recordings. Artifact rejection is a key topic in signal processing. The artifacts are unwelcome signals that may occur during the signal acquisition and that may alter the analysis of the signals themselves. A technique for the automatic artifact rejection, based on the Independent Component Analysis (ICA) for the artifact extraction and on some high order statistics such as kurtosis and Shannon-s entropy, was proposed some years ago in literature. In this paper we enhance this technique introducing the Renyi-s entropy. The performance of our method was tested exploiting the Independent Component scalp maps and it was compared to the performance of the method in literature and it showed to outperform it.

Keywords: Artifact, EEG, Renyi's entropy, independent component analysis, kurtosis.

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Authors: Young-Seok Choi

Abstract:

We present a refined multiscale Shannon entropy for analyzing electroencephalogram (EEG), which reflects the underlying dynamics of EEG over multiple scales. The rationale behind this method is that neurological signals such as EEG possess distinct dynamics over different spectral modes. To deal with the nonlinear and nonstationary nature of EEG, the recently developed empirical mode decomposition (EMD) is incorporated, allowing a decomposition of EEG into its inherent spectral components, referred to as intrinsic mode functions (IMFs). By calculating the Shannon entropy of IMFs in a time-dependent manner and summing them over adaptive multiple scales, it results in an adaptive subscale entropy measure of EEG. Simulation and experimental results show that the proposed entropy properly reveals the dynamical changes over multiple scales.

Keywords: EEG, subscale entropy, Empirical mode decomposition, Intrinsic mode function.

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Authors: Fahad Alsharari, Mohd Salmi Md Noorani

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A Motzkin shift is a mathematical model for constraints on genetic sequences. In terms of the theory of symbolic dynamics, the Motzkin shift is nonsofic, and therefore, we cannot use the Perron- Frobenius theory to calculate its topological entropy. The Motzkin shift M(M,N) which comes from language theory, is defined to be the shift system over an alphabet A that consists of N negative symbols, N positive symbols and M neutral symbols. For an x in the full shift, x will be in the Motzkin subshift M(M,N) if and only if every finite block appearing in x has a non-zero reduced form. Therefore, the constraint for x cannot be bounded in length. K. Inoue has shown that the entropy of the Motzkin shift M(M,N) is log(M + N + 1). In this paper, a new direct method of calculating the topological entropy of the Motzkin shift is given without any measure theoretical discussion.

Keywords: Motzkin shift, topological entropy.

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Authors: Francis Chinweuba Eboh, Peter Ahlström, Tobias Richards

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Entropy, as an outcome of the second law of thermodynamics, measures the level of irreversibility associated with any process. The identification and reduction of irreversibility in the energy conversion process helps to improve the efficiency of the system. The entropy of pure substances known as absolute entropy is determined at an absolute reference point and is useful in the thermodynamic analysis of chemical reactions; however, municipal solid waste (MSW) is a structurally complicated material with unknown absolute entropy. In this work, an empirical model to calculate the absolute entropy of MSW based on the content of carbon, hydrogen, oxygen, nitrogen, sulphur, and chlorine on a dry ash free basis (daf) is presented. The proposed model was derived from 117 relevant organic substances which represent the main constituents in MSW with known standard entropies using statistical analysis. The substances were divided into different waste fractions; namely, food, wood/paper, textiles/rubber and plastics waste and the standard entropies of each waste fraction and for the complete mixture were calculated. The correlation of the standard entropy of the complete waste mixture derived was found to be s^o_msw= 0.0101C + 0.0630H + 0.0106O + 0.0108N + 0.0155S + 0.0084Cl (kJ.K^-1.kg) and the present correlation can be used for estimating the absolute entropy of MSW by using the elemental compositions of the fuel within the range of 10.3% ≤ C ≤ 95.1%, 0.0% ≤ H ≤ 14.3%, 0.0% ≤ O ≤ 71.1%, 0.0 ≤ N ≤ 66.7%, 0.0% ≤ S ≤ 42.1%, 0.0% ≤ Cl ≤ 89.7%. The model is also applicable for the efficient modelling of a combustion system in a waste-to-energy plant.

Keywords: Absolute entropy, irreversibility, municipal solid waste, waste-to-energy.

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Authors: Sui Yan Wong, Keat Ping Yeoh, Chi Wai Hui

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In this study, supercritical steam is introduced to Combined Cycle Power Plant (CCPP) in an attempt to further optimize energy recovery. Subcritical steam is commonly used in the CCPP, operating at maximum pressures around 150-160 bar. Supercritical steam is an alternative to increase heat recovery during vaporization period of water. The idea of improvement using supercritical steam is further examined with the use of exergy, pinch analysis and Aspen Plus simulation.

Keywords: Exergy, pinch, combined cycle power plant, CCPP, supercritical steam.

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Authors: Nan Li, Ying Wang

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Using entropy weight and TOPSIS method, a comprehensive evaluation is done on the development level of Chinese regional service industry in this paper. Firstly, based on existing research results, an evaluation index system is constructed from the scale of development, the industrial structure and the economic benefits. An evaluation model is then built up based on entropy weight and TOPSIS, and an empirical analysis is conducted on the development level of service industries in 31 Chinese provinces during 2006 and 2009 from the two dimensions or time series and cross section, which provides new idea for assessing regional service industry. Furthermore, the 31 provinces are classified into four categories based on the evaluation results, and deep analysis is carried out on the evaluation results.

Keywords: Chinese regional service industry, Development level, Entropy weight, TOPSIS Evaluation Method

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Authors: Sheema Shuja Khattak, Gule Saman, Imran Khan, Abdus Salam

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Image segmentation plays an important role in medical imaging applications. Therefore, accurate methods are needed for the successful segmentation of medical images for diagnosis and detection of various diseases. In this paper, we have used maximum entropy to achieve image segmentation. Maximum entropy has been calculated using Shannon, Renyi and Tsallis entropies. This work has novelty based on the detection of skin lesion caused by the bite of a parasite called Sand Fly causing the disease is called Cutaneous Leishmaniasis.

Keywords: Shannon, Maximum entropy, Renyi, Tsallis entropy.

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Authors: Young-Seok Choi

Abstract:

This paper presents a multiscale information measure of Electroencephalogram (EEG) for analysis with a short data length. A multiscale extension of permutation entropy (MPE) is capable of fully reflecting the dynamical characteristics of EEG across different temporal scales. However, MPE yields an imprecise estimation due to coarse-grained procedure at large scales. We present an improved MPE measure to estimate entropy more accurately with a short time series. By computing entropies of all coarse-grained time series and averaging those at each scale, it leads to the modified MPE (MMPE) which provides an enhanced accuracy as compared to MPE. Simulation and experimental studies confirmed that MMPE has proved its capability over MPE in terms of accuracy.

Keywords: Multiscale entropy, permutation entropy, EEG, seizure.

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Authors: Wafa Halfaoui Mbarek, Khir Tahar, Ben Brahim Ammar

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This study reports a comparison from an exergetic point of view between three configurations of vapor compression industrial refrigeration systems operating with R134a as working fluid. The performances of the different cycles are analyzed as function of several operating parameters such as condensing temperature and inter stage pressure. In addition, the contributions of component exergy destruction to the total exergy destruction are obtained for each system. The results are estimated to be used in the selection of the most advantageous configuration from an exergetic view point.

Keywords: Vapor compression, exergy, destruction, efficiency, R134a.

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Authors: Hyung Jong Ko, Kyoung Hoon Kim

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In this study thermodynamic performance analysis of a combined organic Rankine cycle and ejector refrigeration cycle is carried out for use of low-grade heat source in the form of sensible energy. Special attention is paid to the effects of system parameters including the turbine inlet temperature and turbine inlet pressure on the characteristics of the system such as ratios of mass flow rate, net work production, and refrigeration capacity as well as the coefficient of performance and exergy efficiency of the system. Results show that for a given source the coefficient of performance increases with increasing of the turbine inlet pressure. However, the exergy efficiency has an optimal condition with respect to the turbine inlet pressure.

Keywords: Coefficient of performance, ejector refrigeration cycle, exergy efficiency, low-grade energy, organic rankine cycle.

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Authors: Anil Kumar, C. Nagaraja Kumar

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The position and momentum space information entropies of hydrogen atom are exactly evaluated. Using isospectral Hamiltonian approach, a family of isospectral potentials is constructed having same energy eigenvalues as that of the original potential. The information entropy content is obtained in position space as well as in momentum space. It is shown that the information entropy content in each level can be re-arranged as a function of deformation parameter.

Keywords: Information Entropy, BBM inequality, Isospectral Potential.

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Authors: Hsing-Kuo Pao, John Case

Abstract:

Biological sequences from different species are called or-thologs if they evolved from a sequence of a common ancestor species and they have the same biological function. Approximations of Kolmogorov complexity or entropy of biological sequences are already well known to be useful in extracting similarity information between such sequences -in the interest, for example, of ortholog detection. As is well known, the exact Kolmogorov complexity is not algorithmically computable. In prac-tice one can approximate it by computable compression methods. How-ever, such compression methods do not provide a good approximation to Kolmogorov complexity for short sequences. Herein is suggested a new ap-proach to overcome the problem that compression approximations may notwork well on short sequences. This approach is inspired by new, conditional computations of Kolmogorov entropy. A main contribution of the empir-ical work described shows the new set of entropy-based machine learning attributes provides good separation between positive (ortholog) and nega-tive (non-ortholog) data - better than with good, previously known alter-natives (which do not employ some means to handle short sequences well).Also empirically compared are the new entropy based attribute set and a number of other, more standard similarity attributes sets commonly used in genomic analysis. The various similarity attributes are evaluated by cross validation, through boosted decision tree induction C5.0, and by Receiver Operating Characteristic (ROC) analysis. The results point to the conclu-sion: the new, entropy based attribute set by itself is not the one giving the best prediction; however, it is the best attribute set for use in improving the other, standard attribute sets when conjoined with them.

Keywords: compression, decision tree, entropy, ortholog, ROC.

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Authors: I. O. Nascimento, J. T. Manzi

Abstract:

The drying process is an important operation in the chemical industry and it is widely used in the food, grain industry and fertilizer industry. However, for demanding a considerable consumption of energy, such a process requires a deep energetic analysis in order to reduce operating costs. This paper deals with thermodynamic optimization applied to rotary dryers based on the entropy production minimization, aiming at to reduce the energy consumption. To do this, the mass, energy and entropy balance was used for developing a relationship that represents the rate of entropy production. The use of the Second Law of Thermodynamics is essential because it takes into account constraints of nature. Since the entropy production rate is minimized, optimals conditions of operations can be established and the process can obtain a substantial gain in energy saving. The minimization strategy had been led using classical methods such as Lagrange multipliers and implemented in the MATLAB platform. As expected, the preliminary results reveal a significant energy saving by the application of the optimal parameters found by the procedure of the entropy minimization It is important to say that this method has shown easy implementation and low cost.

Keywords: Drying, entropy minimization, modeling dryers, thermodynamic optimization.

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Authors: Sabri Mrayed, David Maccioni, Greg Leslie

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Seawater desalination has been accepted as one of the most effective solutions to the growing problem of a diminishing clean drinking water supply. Currently two desalination technologies dominate the market – the thermally driven multi-stage flash distillation (MSF) and the membrane based reverse osmosis (RO). However, in recent years membrane distillation (MD) has emerged as a potential alternative to the established means of desalination. This research project intended to determine the viability of MD as an alternative process to MSF and RO for seawater desalination. Specifically the project involves conducting thermodynamic analysis of the process based on the second law of thermodynamics to determine the efficiency of the MD. Data was obtained from experiments carried out on a laboratory rig. To determine exergy values required for the exergy analysis, two separate models were built in Engineering Equation Solver – the ’Minimum Separation Work Model’ and the ‘Stream Exergy Model’. The efficiency of MD process was found to be 17.3 % and the energy consumption was determined to be 4.5 kWh to produce one cubic meter of fresh water. The results indicate MD has potential as a technique for seawater desalination compared to RO and MSF. However it was shown that this was only the case if an alternate energy source such as green or waste energy was available to provide the thermal energy input to the process. If the process was required to power itself, it was shown to be highly inefficient and in no way thermodynamically viable as a commercial desalination process.

Keywords: Desalination, Exergy, Membrane distillation, Second law efficiency.

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Authors: S. El Safty, A. El-Zonkoly

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The ability to detect and classify the type of fault plays a great role in the protection of power system. This procedure is required to be precise with no time consumption. In this paper detection of fault type has been implemented using wavelet analysis together with wavelet entropy principle. The simulation of power system is carried out using PSCAD/EMTDC. Different types of faults were studied obtaining various current waveforms. These current waveforms were decomposed using wavelet analysis into different approximation and details. The wavelet entropy of such decompositions is analyzed reaching a successful methodology for fault classification. The suggested approach is tested using different fault types and proven successful identification for the type of fault.

Keywords: Fault classification, wavelet transform, waveletentropy.

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Authors: K.C. Deshmukh, P.G. Khot, Nikhil

Abstract:

In the present communication, we have proposed some new generalized measure of fuzzy entropy based upon real parameters, discussed their and desirable properties, and presented these measures graphically. An important property, that is, monotonicity of the proposed measures has also been studied.

Keywords: Fuzzy numbers, Fuzzy entropy, Characteristicfunction, Crisp set, Monotonicity.

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