Search results for: Tracer kinetic modeling

Authors: Bothinah Altaf, Gary Montague, Elaine B. Martin

Abstract:

This study involves the modeling and monitoring of an ammonia synthesis fixed-bed reactor using partial least squares (PLS) and its variants. The process exhibits complex dynamic behavior due to the presence of heat recycling and feed quench. One limitation of static PLS model in this situation is that it does not take account of the process dynamics and hence dynamic PLS was used. Although it showed, superior performance to static PLS in terms of prediction, the monitoring scheme was inappropriate hence adaptive PLS was considered. A limitation of adaptive PLS is that non-conforming observations also contribute to the model, therefore, a new adaptive approach was developed, robust adaptive dynamic PLS. This approach updates a dynamic PLS model and is robust to non-representative data. The developed methodology showed a clear improvement over existing approaches in terms of the modeling of the reactor and the detection of faults.

Keywords: Ammonia synthesis fixed-bed reactor, dynamic partial least squares modeling, recursive partial least squares, robust modeling.

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Authors: Yousef Abu Nahleh, Arun Kumar, Fugen Daver, Reham Al-Hindawi

Abstract:

Despite the availability of natural disaster related time series data for last 110 years, there is no forecasting tool available to humanitarian relief organizations to determine forecasts for emergency logistics planning. This study develops a forecasting tool based on identifying probability of disaster for each country in the world by using decision tree modeling. Further, the determination of aggregate forecasts leads to efficient pre-disaster planning. Based on the research findings, the relief agencies can optimize the various resources allocation in emergency logistics planning.

Keywords: Decision tree modeling, Forecasting, Humanitarian relief, emergency supply chain.

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Authors: Robest Flasza, Milan Rippel, Jan Solc

Abstract:

The main aim of this paper is to develop and calibrate an econometric model for modeling prices of long term electricity futures contracts. The calibration of our model is performed on data from EEX AG allowing us to capture the specific features of German electricity market. The data sample contains several structural breaks which have to be taken into account for modeling. We model the data with an ARIMAX model which reveals high correlation between the price of electricity futures contracts and prices of LT futures contracts of fuels (namely coal, natural gas and crude oil). Besides this, also a share price index of representative electricity companies traded on Xetra, spread between 10Y and 1Y German bonds and exchange rate between EUR and USD appeared to have significant explanatory power over these futures contracts on EEX.

Keywords: electricity futures, EEX, ARIMAX, emissionallowances

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Authors: Sarma Cakula, Maija Sedleniece

Abstract:

Simulation and modeling computer programs are concerned with construction of models for analyzing different perspectives and possibilities in changing conditions environment. The paper presents theoretical justification and evaluation of qualitative e-learning development model in perspective of advancing modern technologies. There have been analyzed principles of qualitative e-learning in higher education, productivity of studying process using modern technologies, different kind of methods and future perspectives of e-learning in formal education. Theoretically grounded and practically tested model of developing e-learning methods using different technologies for different type of classroom, which can be used in professor-s decision making process to choose the most effective e-learning methods has been worked out.

Keywords: E-learning, modeling, E-learning methods development, personal knowledge management

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Authors: M. Sedighizadeh, A. Rezazadeh

Abstract:

A method has been developed for preparing load models for power flow and stability. The load modeling (LOADMOD) computer software transforms data on load class mix, composition, and characteristics into the from required for commonly–used power flow and transient stability simulation programs. Typical default data have been developed for load composition and characteristics. This paper defines LOADMOD software and describes the dynamic and static load modeling techniques used in this software and results of initial testing for BAKHTAR power system.

Keywords: Load Modelling, Static, Power Flow.

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Authors: Tsuyoshi Okita

Abstract:

For a given specific problem an efficient algorithm has been the matter of study. However, an alternative approach orthogonal to this approach comes out, which is called a reduction. In general for a given specific problem this reduction approach studies how to convert an original problem into subproblems. This paper proposes a formal modeling language to support this reduction approach in order to make a solver quickly. We show three examples from the wide area of learning problems. The benefit is a fast prototyping of algorithms for a given new problem. It is noted that our formal modeling language is not intend for providing an efficient notation for data mining application, but for facilitating a designer who develops solvers in machine learning.

Keywords: Formal language, statistical inference problem, reduction.

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Authors: Jaewon Jang, Gregory R. Miller

Abstract:

This paper presents work characterizing finite element performance boundaries within which live, interactive finite element modeling is feasible on current and emerging systems. These results are based on wide-ranging tests performed using a prototype finite element program implemented specifically for this study, thereby enabling the unified investigation of numerous direct and iterative solver strategies and implementations in a variety of modeling contexts. The results are intended to be useful for researchers interested in interactive analysis by providing baseline performance estimates, to give guidance in matching solution strategies to problem domains, and to spur further work addressing the challenge of extending the present boundaries.

Keywords: Finite Elements, Interactive Modeling, NumericalAnalysis.

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Authors: Rana Abdul Jabbar Khan, Muhammad Akmal

Abstract:

Personal computers draw non-sinusoidal current with odd harmonics more significantly. Power Quality of distribution networks is severely affected due to the flow of these generated harmonics during the operation of electronic loads. In this paper, mathematical modeling of odd harmonics in current like 3rd, 5th, 7th and 9th influencing the power quality has been presented. Live signals have been captured with the help of power quality analyzer for analysis purpose. The interesting feature is that Total Harmonic Distortion (THD) in current decreases with the increase of nonlinear loads has been verified theoretically. The results obtained using mathematical expressions have been compared with the practical results and exciting results have been found.

Keywords: Harmonic Distortion, Mathematical Modeling, Power Quality.

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Authors: A. Elsayed Ahmed, A. Hafez, A. N. Ouda, H. Eldin Hussein Ahmed, H. Mohamed Abd-Elkader

Abstract:

Unmanned aircraft systems (UAS) are playing increasingly prominent roles in defense programs and defense strategies around the world. Technology advancements have enabled the development of it to do many excellent jobs as reconnaissance, surveillance, battle fighters, and communications relays. Simulating a small unmanned aerial vehicle (SUAV) dynamics and analyzing its behavior at the preflight stage is too important and more efficient. The first step in the UAV design is the mathematical modeling of the nonlinear equations of motion. . In this paper, a survey with a standard method to obtain the full non-linear equations of motion is utilized, and then the linearization of the equations according to a steady state flight condition (trimming) is derived. This modeling technique is applied to an Ultrastick-25e fixed wing UAV to obtain the valued linear longitudinal and lateral models. At the end the model is checked by matching between the behavior of the states of the nonlinear UAV and the resulted linear model with doublet at the control surfaces.

Keywords: Equations of motion, linearization, modeling, nonlinear model, UAV.

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Authors: L. Parisi

Abstract:

In this research work, neural networks were applied to classify two types of hip joint implants based on the relative hip joint implant side speed and three components of each ground reaction force. The condition of walking gait at normal velocity was used and carried out with each of the two hip joint implants assessed. Ground reaction forces’ kinetic temporal changes were considered in the first approach followed but discarded in the second one. Ground reaction force components were obtained from eighteen patients under such gait condition, half of which had a hip implant type I-II, whilst the other half had the hip implant, defined as type III by Orthoload®. After pre-processing raw gait kinetic data and selecting the time frames needed for the analysis, the ground reaction force components were used to train a MLP neural network, which learnt to distinguish the two hip joint implants in the abovementioned condition. Further to training, unknown hip implant side and ground reaction force components were presented to the neural networks, which assigned those features into the right class with a reasonably high accuracy for the hip implant type I-II and the type III. The results suggest that neural networks could be successfully applied in the performance assessment of hip joint implants.

Keywords: Kinemic gait data, Neural networks, Hip joint implant, Hip arthroplasty, Rehabilitation Engineering.

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Authors: Anthony Spiteri Staines

Abstract:

This paper examines the modeling and analysis of a cruise control system using a Petri net based approach, task graphs, invariant analysis and behavioral properties. It shows how the structures used can be verified and optimized.

Keywords: Software Engineering, Real Time Analysis andDesign, Petri Nets, Task Graphs, Parallelism.

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Authors: Farhad Kolahan, Mohammad Bironro

Abstract:

This paper addresses modeling and optimization of process parameters in powder mixed electrical discharge machining (PMEDM). The process output characteristics include metal removal rate (MRR) and electrode wear rate (EWR). Grain size of Aluminum powder (S), concentration of the powder (C), discharge current (I) pulse on time (T) are chosen as control variables to study the process performance. The experimental results are used to develop the regression models based on second order polynomial equations for the different process characteristics. Then, a genetic algorithm (GA) has been employed to determine optimal process parameters for any desired output values of machining characteristics.

Keywords: Regression modeling, PMEDM, GeneticAlgorithm, Optimization.

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Authors: Ahmed S. Al-Abdulwahab

Abstract:

Generation system reliability assessment is an important task which can be performed using deterministic or probabilistic techniques. The probabilistic approaches have significant advantages over the deterministic methods. However, more complicated modeling is required by the probabilistic approaches. Power generation model is a basic requirement for this assessment. One form of the generation models is the well known capacity outage probability table (COPT). Different analytical techniques have been used to construct the COPT. These approaches require considerable mathematical modeling of the generating units. The unit-s models are combined to build the COPT which will add more burdens on the process of creating the COPT. Decimal to Binary Conversion (DBC) technique is widely and commonly applied in electronic systems and computing This paper proposes a novel utilization of the DBC to create the COPT without engaging in analytical modeling or time consuming simulations. The simple binary representation , “0 " and “1 " is used to model the states o f generating units. The proposed technique is proven to be an effective approach to build the generation model.

Keywords: Decimal to Binary, generation, reliability.

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Authors: H. C. O. Reis, A. S. Cossolin, B. A. P. Santos, K. C. Castro, G. M. Pereira, V. C. Silva, P. T. Sousa Jr, E. L. Dall’Oglio, L. G. Vasconcelos, E. B. Morais

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In this study, malt bagasse, a low-cost waste biomass, was tested as a biosorbent to remove the cationic dye Malachite green (MG) from aqueous solution. Batch biosorption experiments were investigated as functions of different experimental parameters such as initial pH, salt (NaCl) concentration, contact time, temperature and initial dye concentration. Higher removal rates of MG were obtained at pH 8 and 10. The equilibrium and kinetic studies suggest that the biosorption follows Langmuir isotherm and the pseudo-second-order model. The maximum monolayer adsorption capacity was estimated at 117.65 mg/g (at 45 °C). According to Dubinin–Radushkevich (D-R) isotherm model, biosorption of MG onto malt bagasse occurs physically. The thermodynamic parameters such as Gibbs free energy, enthalpy and entropy indicated that the MG biosorption onto malt bagasse is spontaneous and endothermic. The results of the ionic strength effect indicated that the biosorption process under study had a strong tolerance under high salt concentrations. It can be concluded that malt bagasse waste has potential for application as biosorbent for removal of MG from aqueous solution.

Keywords: Color removal, kinetic and isotherm studies, thermodynamic parameters, FTIR.

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Authors: Phumelele P. Kubheka, Pius A. Owolawi, Gbolahan Aiyetoro

Abstract:

Twitter is one of the most popular social media platforms where users share their opinions on different subjects. Twitter can be considered a great source for mining text due to the high volumes of data generated through the platform daily. Many industries such as telecommunication companies can leverage the availability of Twitter data to better understand their markets and make an appropriate business decision. This study performs topic modeling on Twitter data using Latent Dirichlet Allocation (LDA). The obtained results are benchmarked with another topic modeling technique, Latent Semantic Indexing (LSI). The study aims to retrieve topics on a Twitter dataset containing user tweets on South African Telcos. Results from this study show that LSI is much faster than LDA. However, LDA yields better results with higher topic coherence by 8% for the best-performing model in this experiment. A higher topic coherence score indicates better performance of the model.

Keywords: Big data, latent Dirichlet allocation, latent semantic indexing, Telco, topic modeling, Twitter.

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Authors: Gionata Carmignani, Mario G. C. A. Cimino, Franco Failli

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Business Processes (BPs) are the key instrument to understand how companies operate at an organizational level, taking an as-is view of the workflow, and how to address their issues by identifying a to-be model. In last year’s, the BP Model and Notation (BPMN) has become a de-facto standard for modeling processes. However, this standard does not incorporate explicitly the Problem- Solving (PS) knowledge in the Process Modeling (PM) results. Thus, such knowledge cannot be shared or reused. To narrow this gap is today a challenging research area. In this paper we present a framework able to capture the PS knowledge and to improve a workflow. This framework extends the BPMN specification by incorporating new general-purpose elements. A pilot scenario is also presented and discussed.

Keywords: Business Process Management, BPMN, Problem Solving, Process mapping.

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Authors: Seyed Ali Jafari, Mohammad Ghomi Avili, Emad Benhelal

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In this study, a mathematical model was proposed and the accuracy of this model was assessed to predict the growth of Pseudomonas aeruginosa and rhamnolipid production under nitrogen limiting (sodium nitrate) fed-batch fermentation. All of the parameters used in this model were achieved individually without using any data from the literature. The overall growth kinetic of the strain was evaluated using a dual-parallel substrate Monod equation which was described by several batch experimental data. Fed-batch data under different glycerol (as the sole carbon source, C/N=10) concentrations and feed flow rates were used to describe the proposed fed-batch model and other parameters. In order to verify the accuracy of the proposed model several verification experiments were performed in a vast range of initial glycerol concentrations. While the results showed an acceptable prediction for rhamnolipid production (less than 10% error), in case of biomass prediction the errors were less than 23%. It was also found that the rhamnolipid production by P. aeruginosa was more sensitive at low glycerol concentrations. Based on the findings of this work, it was concluded that the proposed model could effectively be employed for rhamnolipid production by this strain under fed-batch fermentation on up to 80 g l- 1 glycerol.

Keywords: Fed-batch culture, glycerol, kinetic parameters, modelling, Pseudomonas aeruginosa, rhamnolipid.

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Authors: Mohamed A. Khlifi, Badr M. Alshammari

Abstract:

This paper discusses the general methods to saturation in the steady-state, two axis (d & q) frame models of synchronous machines. In particular, the important role of the magnetic coupling between the d-q axes (cross-magnetizing phenomenon), is demonstrated. For that purpose, distinct methods of saturation modeling of dumper synchronous machine with cross-saturation are identified, and detailed models synthesis in d-q axes. A number of models are given in the final developed form. The procedure and the novel models are verified by a critical application to prove the validity of the method and the equivalence between all developed models is reported. Advantages of some of the models over the existing ones and their applicability are discussed.

Keywords: Cross-magnetizing, models synthesis, synchronous machine, saturated modeling, state-space vectors.

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Authors: Abhishek Soni, A. Kumaraswamy, M. S. Mahesh

Abstract:

Bullet penetration in steel plate is investigated with the help of three-dimensional, non-linear, transient, dynamic, finite elements analysis using explicit time integration code LSDYNA. The effect of large strain, strain-rate and temperature at very high velocity regime was studied from number of simulations of semi-spherical nose shape bullet penetration through single layered circular plate with 2 mm thickness at impact velocities of 500, 1000, and 1500 m/s with the help of Johnson Cook material model. Mie-Gruneisen equation of state is used in conjunction with Johnson Cook material model to determine pressure-volume relationship at various points of interests. Two material models viz. Plastic-Kinematic and Johnson- Cook resulted in different deformation patterns in steel plate. It is observed from the simulation results that the velocity drop and loss of kinetic energy occurred very quickly up to perforation of plate, after that the change in velocity and changes in kinetic energy are negligibly small. The physics behind this kind of behaviour is presented in the paper.

Keywords: AISI 4340 steel, ballistic impact simulation, bullet penetration, non-linear FEM.

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Authors: Yasuo Obikane

Abstract:

This work is to study a roll of the fluctuating density gradient in the compressible flows for the computational fluid dynamics (CFD). A new anisotropy tensor with the fluctuating density gradient is introduced, and is used for an invariant modeling technique to model the turbulent density gradient correlation equation derived from the continuity equation. The modeling equation is decomposed into three groups: group proportional to the mean velocity, and that proportional to the mean strain rate, and that proportional to the mean density. The characteristics of the correlation in a wake are extracted from the results by the two dimensional direct simulation, and shows the strong correlation with the vorticity in the wake near the body. Thus, it can be concluded that the correlation of the density gradient is a significant parameter to describe the quick generation of the turbulent property in the compressible flows.

Keywords: Turbulence Modeling , Density Gradient Correlation, Compressible

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Authors: Anwar Jarndal

Abstract:

In this paper, a genetic-neural-network (GNN) based large-signal model for GaN HEMTs is presented along with its parameters extraction procedure. The model is easy to construct and implement in CAD software and requires only DC and S-parameter measurements. An improved decomposition technique is used to model self-heating effect. Two GNN models are constructed to simulate isothermal drain current and power dissipation, respectively. The two model are then composed to simulate the drain current. The modeling procedure was applied to a packaged GaN-on-Si HEMT and the developed model is validated by comparing its large-signal simulation with measured data. A very good agreement between the simulation and measurement is obtained.

Keywords: GaN HEMT, computer-aided design & modeling, neural networks, genetic optimization.

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Authors: P. C. Tewari, Rajiv Kumar, Dinesh Khanduja

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This paper discusses the performance modeling and availability analysis of Yarn Dyeing System of a Textile Industry. The Textile Industry is a complex and repairable engineering system. Yarn Dyeing System of Textile Industry consists of five subsystems arranged in series configuration. For performance modeling and analysis of availability, a performance evaluating model has been developed with the help of mathematical formulation based on Markov-Birth-Death Process. The differential equations have been developed on the basis of Probabilistic Approach using a Transition Diagram. These equations have further been solved using normalizing condition in order to develop the steady state availability, a performance measure of the system concerned. The system performance has been further analyzed with the help of decision matrices. These matrices provide various availability levels for different combinations of failure and repair rates for various subsystems. The findings of this paper are therefore, considered to be useful for the analysis of availability and determination of the best possible maintenance strategies which can be implemented in future to enhance the system performance.

Keywords: Availability Analysis, Markov Process, Performance Modeling, Steady State Availability.

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Authors: Yun-Pang Flötteröd, Jakob Erdmann

Abstract:

The human behaviors during evacuations are quite complex. One of the critical behaviors which affect the efficiency of evacuation is route choice. Therefore, the respective simulation modeling work needs to function properly. In this paper, Simulation of Urban Mobility’s (SUMO) current dynamic route modeling during evacuation, i.e. the rerouting functions, is examined with a real case study. The result consistency of the simulation and the reality is checked as well. Four influence factors (1) time to get information, (2) probability to cancel a trip, (3) probability to use navigation equipment, and (4) rerouting and information updating period are considered to analyze possible traffic impacts during the evacuation and to examine the rerouting functions in SUMO. Furthermore, some behavioral characters of the case study are analyzed with use of the corresponding detector data and applied in the simulation. The experiment results show that the dynamic route modeling in SUMO can deal with the proposed scenarios properly. Some issues and function needs related to route choice are discussed and further improvements are suggested.

Keywords: Evacuation, microscopic traffic simulation, rerouting, SUMO.

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Authors: A. Ghaffari, A. S. Mostafavi

Abstract:

Complex assemblies of interacting proteins carry out most of the interesting jobs in a cell, such as metabolism, DNA synthesis, mitosis and cell division. These physiological properties play out as a subtle molecular dance, choreographed by underlying regulatory networks that control the activities of cyclin-dependent kinases (CDK). The network can be modeled by a set of nonlinear differential equations and its behavior predicted by numerical simulation. In this paper, an innovative approach has been proposed that uses genetic algorithms to mine a set of behavior data output by a biological system in order to determine the kinetic parameters of the system. In our approach, the machine learning method is integrated with the framework of existent biological information in a wiring diagram so that its findings are expressed in a form of system dynamic behavior. By numerical simulations it has been illustrated that the model is consistent with experiments and successfully shown that such application of genetic algorithms will highly improve the performance of mathematical model of the cell division cycle to simulate such a complicated bio-system.

Keywords: Cell cycle, Cyclin-dependent kinase, Fission yeast, Genetic algorithms, Mathematical modeling, Wiring diagram

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Authors: Lambros V. Skarlas, Grigorios N. Beligiannis, Efstratios F. Georgopoulos, Adam V. Adamopoulos

Abstract:

The aim of this contribution is to present a new approach in modeling the electrical activity of the human heart. A recurrent artificial neural network is being used in order to exhibit a subset of the dynamics of the electrical behavior of the human heart. The proposed model can also be used, when integrated, as a diagnostic tool of the human heart system. What makes this approach unique is the fact that every model is being developed from physiological measurements of an individual. This kind of approach is very difficult to apply successfully in many modeling problems, because of the complexity and entropy of the free variables describing the complex system. Differences between the modeled variables and the variables of an individual, measured at specific moments, can be used for diagnostic purposes. The sensor fusion used in order to optimize the utilization of biomedical sensors is another point that this paper focuses on. Sensor fusion has been known for its advantages in applications such as control and diagnostics of mechanical and chemical processes.

Keywords: Artificial Neural Networks, Diagnostic System, Health Condition Modeling Tool, Heart Diagnostics Model, Heart Electricity Model.

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Authors: Sema Kurtuluş, Kemal Kurtuluş

Abstract:

The objective of this paper is to review and assess the methodological issues and problems in marketing research, data and knowledge mining in Turkey. As a summary, academic marketing research publications in Turkey have significant problems. The most vital problem seems to be related with modeling. Most of the publications had major weaknesses in modeling. There were also, serious problems regarding measurement and scaling, sampling and analyses. Analyses myopia seems to be the most important problem for young academia in Turkey. Another very important finding is the lack of publications on data and knowledge mining in the academic world.

Keywords: Marketing research, data mining, knowledge mining, research modeling, analyses.

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Authors: F. Hosseinpour, H. Hajihosseini

Abstract:

Simulation is a very helpful and valuable work tool in manufacturing. It can be used in industrial field allowing the system`s behavior to be learnt and tested. Simulation provides a low cost, secure and fast analysis tool. It also provides benefits, which can be reached with many different system configurations. Topics to be discussed include: Applications, Modeling, Validating, Software and benefits of simulation. This paper provides a comprehensive literature review on research efforts in simulation.

Keywords: Manufacturing, modeling, simulation, training.

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Authors: Jatinder Kumar, Ajay Bansal

Abstract:

The present study deals with the modeling and simulation of flow through an annular reactor at different hydrodynamic conditions using computational fluid dynamics (CFD) to investigate the flow behavior. CFD modeling was utilized to predict velocity distribution and average velocity in the annular geometry. The results of CFD simulations were compared with the mathematically derived equations and already developed correlations for validation purposes. CFD modeling was found suitable for predicting the flow characteristics in annular geometry under laminar flow conditions. It was observed that CFD also provides local values of the parameters of interest in addition to the average values for the simulated geometry.

Keywords: Annular reactor, computational fluid dynamics (CFD), hydrodynamics, Rhodamine B

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Authors: R. Kongnuy, E. Naowanich

Abstract:

In this paper we develop and analyze the model for the spread of Leptospirosis by age group in Thailand, between 1997 and 2010 by using mathematical modeling and computer simulation. Leptospirosis is caused by pathogenic spirochetes of the genus Leptospira. It is a zoonotic disease of global importance and an emerging health problem in Thailand. In Thailand, leptospirosis is a reportable disease, the top three age groups are 23.31% in 35-44 years olds group, 22.76% in 25-34 year olds group, 17.60% in 45-54 year olds group from reported leptospirosis between 1997 and 2010, with a peak in 35-44 year olds group. Our paper, the Leptosipirosis transmission by age group in Thailand is studied on the mathematical model. Some analytical and simulation results are presented.

Keywords: Age Group, Equilibrium State, Leptospirosis, Mathematical Modeling.

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Authors: Nguyen Thu Huong, Nguyen Quang Bau

Abstract:

The Hall Coefficient (HC) and the Magnetoresistance (MR) have been studied in two-dimensional systems. The HC and the MR in Rectangular Quantum Wire (RQW) subjected to a crossed DC electric field and magnetic field in the presence of a Strong Electromagnetic Wave (EMW) characterized by electric field are studied in this work. Using the quantum kinetic equation for electrons interacting with optical phonons, we obtain the analytic expressions for the HC and the MR with a dependence on magnetic field, EMW frequency, temperatures of systems and the length characteristic parameters of RQW. These expressions are different from those obtained for bulk semiconductors and cylindrical quantum wires. The analytical results are applied to GaAs/GaAs/Al. For this material, MR depends on the ratio of the EMW frequency to the cyclotron frequency. Indeed, MR reaches a minimum at the ratio 5/4, and when this ratio increases, it tends towards a saturation value. The HC can take negative or positive values. Each curve has one maximum and one minimum. When magnetic field increases, the HC is negative, achieves a minimum value and then increases suddenly to a maximum with a positive value. This phenomenon differs from the one observed in cylindrical quantum wire, which does not have maximum and minimum values.

Keywords: Hall coefficient, rectangular quantum wires, electron-optical phonon interaction, quantum kinetic equation.

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