Search results for: Heat equation.

Authors: Imane Khalil, Quinn Pratt

Abstract:

In a nuclear reactor, an array of fuel rods containing stacked uranium dioxide pellets clad with zircalloy is the heat source for a thermodynamic cycle of energy conversion from heat to electricity. After fuel is used in a nuclear reactor, the assemblies are stored underwater in a spent nuclear fuel pool at the nuclear power plant while heat generation and radioactive decay rates decrease before it is placed in packages for dry storage or transportation. A computational model of a Boiling Water Reactor spent fuel assembly is modeled using FLUENT, the computational fluid dynamics package. Heat transfer simulations were performed on the two-dimensional 9x9 spent fuel assembly to predict the maximum cladding temperature for different input to the FLUENT model. Uncertainty quantification is used to predict the heat transfer and the maximum temperature profile inside the assembly.

Keywords: Spent nuclear fuel, conduction, heat transfer, uncertainty quantification.

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Authors: Jae Kyun Bang, Sung Jae Hwang, Chang Yong Ko, Chi Hyun Kim

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Prolonged immobilization leads to significant weakness and atrophy of the skeletal muscle and can also impair the recovery of muscle strength following injury. Therefore, it is important to minimize the period under immobilization and accelerate the return to normal activity. This study examined the effects of heat treatment and rest-inserted exercise on the muscle activity of the lower limb during knee flexion/extension. Twelve healthy subjects were assigned to 4 groups that included: (1) heat treatment + rest-inserted exercise; (2) heat + continuous exercise; (3) no heat + rest-inserted exercise; and (4) no heat + continuous exercise. Heat treatment was applied for 15 mins prior to exercise. Continuous exercise groups performed knee flexion/extension at 0.5 Hz for 300 cycles without rest whereas rest-inserted exercise groups performed the same exercise but with 2 mins rest inserted every 60 cycles of continuous exercise. Changes in the rectus femoris and hamstring muscle activities were assessed at 0, 1, and 2 weeks of treatment by measuring the electromyography signals of isokinetic maximum voluntary contraction. Significant increases in both the rectus femoris and hamstring muscles were observed after 2 weeks of treatment only when both heat treatment and rest-inserted exercise were performed. These results suggest that combination of various treatment techniques, such as heat treatment and rest-inserted exercise, may expedite the recovery of muscle strength following immobilization.

Keywords: Electromyography, Heat Treatment, Muscle, Rest-Inserted Exercise.

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Authors: Ali Dehgan Moroozeh, B. Farhadi Bansouleh

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Evapotranspiration is one of the most important components of the hydrological cycle. Evapotranspiration (ETo) is an important variable in water and energy balances on the earth’s surface, and knowledge of the distribution of ET is a key factor in hydrology, climatology, agronomy and ecology studies. Many researchers have a valid relationship, which is a function of climate factors, to estimate the potential evapotranspiration presented to the plant water stress or water loss, prevent. The FAO-Penman method (PM) had been recommended as a standard method. This method requires many data and these data are not available in every area of world. So, other methods should be evaluated for these conditions. When sufficient or reliable data to solve the PM equation are not available then Hargreaves equation can be used. The Hargreaves equation (HG) requires only daily mean, maximum and minimum air temperature extraterrestrial radiation .In this study, Hargreaves method (HG) were evaluated in 12 stations in the North West region of Iran. Results of HG and M.HG methods were compared with results of PM method. Statistical analysis of this comparison showed that calibration process has had significant effect on efficiency of Hargreaves method.

Keywords: Evapotranspiration, Hargreaves equation, FAOPenman method.

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Authors: Manisha Kankarej

Abstract:

The purpose of this paper is to show a relation between CR structure and F-structure satisfying polynomial equation. In this paper, we have checked the significance of CR structure and F-structure on Integrability conditions and Nijenhuis tensor. It was proved that all the properties of Integrability conditions and Nijenhuis tensor are satisfied by CR structures and F-structure satisfying polynomial equation.

Keywords: CR-submainfolds, CR-structure, Integrability condition & Nijenhuis tensor.

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Authors: P. K. Nagarajan, P. Sivashanmugam

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Experimental investigation of heat transfer and friction factor characteristics of circular tube fitted with 300 right-left helical screw inserts with 100 mm spacer of different twist ratio has been presented for laminar and turbulent flow.. The experimental data obtained were compared with those obtained from plain tube published data. The heat transfer coefficient enhancement for 300 RL inserts with 100 mm spacer is quite comparable with for 300 R-L inserts. Performance evaluation analysis has been made and found that the performance ratio increases with increasing Reynolds number and decreasing twist ration with the maximum for the twist ratio 2.93. Also, the performance ratio of more than one indicates that the type of twist inserts can be used effectively for heat transfer augmentation.

Keywords: Heat transfer augmentation, right-left helical screw inserts with spacer, Twist ratio, Heat Transfer

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Authors: Kavita H. Dhanawade, Hanamant S. Dhanawade

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Heat transfer enhancement objects like extended surfaces, fins etc. are chosen for their thermal performance as well as for other design parameters depending on various applications. The present paper is on experimental study to investigate the heat transfer enhancement through wire mesh fin arrays equipped with horizontal base plate. The data used in performance analysis were obtained experimentally for the material (mild steel) for different heat inputs such as 40, 60, 80, 100 and 120 watt, by varying wire mesh diameter, fin height and spacing between two fin arrays. Using the Taguchi experimental design method, optimum design parameters and their levels were investigated. Average heat transfer coefficient was considered as a performance characteristic parameter. An L₉ (3³) orthogonal array was selected as an experimental plan. Optimum results were found by experimenting. It is observed that the wire mesh diameter and fin height have a higher impact on heat transfer coefficient as compared to spacing between two fin arrays.

Keywords: Heat transfer enhancement, finned surface, wire mesh diameter, natural convection.

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Authors: Mohammad Reza Soltani, Mohammad Farahani, Javad Sepahi Younsi

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Heat source addition to the axisymmetric supersonic inlet may improve the performance parameters, which will increase the inlet efficiency. In this investigation the heat has been added to the flow field at some distance ahead of an axisymmetric inlet by adding an imaginary thermal source upstream of cowl lip. The effect of heat addition on the drag coefficient, mass flow rate and the overall efficiency of the inlet have been investigated. The results show that heat addition causes flow separation, hence to prevent this phenomena, roughness has been added on the spike surface. However, heat addition reduces the drag coefficient and the inlet mass flow rate considerably. Furthermore, the effects of position, size, and shape on the inlet performance were studied. It is found that the thermal source deflects the flow streamlines. By improper location of the thermal source, the optimum condition has been obtained. For the optimum condition, the drag coefficient is considerably reduced and the inlet mass flow rate and its efficiency have been increased slightly. The optimum shape of the heat source is obtained too.

Keywords: Drag coefficient, heat source, performanceparameters, supersonic inlet.

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Authors: Wisam K. Hussam, Ali Alfeeli, Gergory J. Sheard

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Solar panels that use photovoltaic (PV) cells are popular for converting solar radiation into electricity. One of the major problems impacting the performance of PV panels is the overheating caused by excessive solar radiation and high ambient temperatures, which degrades the efficiency of the PV panels remarkably. To overcome this issue, an aluminum heat sink was used to dissipate unwanted heat from PV cells. The dimensions of the heat sink were determined considering the optimal fin spacing that fulfils hot climatic conditions. In this study, the effects of cooling on the efficiency and power output of a PV panel were studied experimentally. Two PV modules were used: one without and one with a heat sink. The experiments ran for 11 hours from 6:00 a.m. to 5:30 p.m. where temperature readings in the rear and front of both PV modules were recorded at an interval of 15 minutes using sensors and an Arduino microprocessor. Results are recorded for both panels simultaneously for analysis, temperate comparison, and for power and efficiency calculations. A maximum increase in the solar to electrical conversion efficiency of 35% and almost 55% in the power output were achieved with the use of a heat sink, while temperatures at the front and back of the panel were reduced by 9% and 11%, respectively.

Keywords: Photovoltaic cell, natural convection, heat sink, efficiency.

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Authors: Mohamed M. Mousa, Aidarkhan Kaltayev

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In this paper, He-s homotopy perturbation method (HPM) is applied to spatial one and three spatial dimensional inhomogeneous wave equation Cauchy problems for obtaining exact solutions. HPM is used for analytic handling of these equations. The results reveal that the HPM is a very effective, convenient and quite accurate to such types of partial differential equations (PDEs).

Keywords: Homotopy perturbation method, Exact solution, Cauchy problem, inhomogeneous wave equation

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Authors: Sh. Hosseini, M. B. Limooei, M. Hossein Zade, E. Askarnia, Z. Asadi

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Manganese steel (Hadfield) is one of the important alloys in industry due to its special properties. High work hardening ability with appropriate toughness and ductility are the properties that caused this alloy to be used in wear resistance parts and in high strength condition. Heat treatment is the main process through which the desired mechanical properties and microstructures are obtained in Hadfield steel. In this study various heat treatment cycles, differing in austenising temperature, time and quenching solution are applied. For this purpose, the same samples of manganese steel was heat treated in 9 different cycles, and then the mechanical properties and microstructures were investigated. Based on the results of the study, the optimum heat treatment cycle was obtained.

Keywords: Manganese steel (Hadfield), heat treatment, austenising temperature, austenising time, quenching solution, mechanical properties.

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Authors: Kew Lee Ming, Norhashidah Hj. Mohd. Ali

Abstract:

In a previous work, we presented the numerical solution of the two dimensional second order telegraph partial differential equation discretized by the centred and rotated five-point finite difference discretizations, namely the explicit group (EG) and explicit decoupled group (EDG) iterative methods, respectively. In this paper, we utilize a domain decomposition algorithm on these group schemes to divide the tasks involved in solving the same equation. The objective of this study is to describe the development of the parallel group iterative schemes under OpenMP programming environment as a way to reduce the computational costs of the solution processes using multicore technologies. A detailed performance analysis of the parallel implementations of points and group iterative schemes will be reported and discussed.

Keywords: Telegraph equation, explicit group iterative scheme, domain decomposition algorithm, parallelization.

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Authors: Dalvir Singh, Somnath Chattopadhyaya

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During welding or flame cutting of metals, the prediction of heat affected zone (HAZ) is critical. There is need to develop a simple mathematical model to calculate the temperature variation in HAZ and derivative analysis can be used for this purpose. This study presents analytical solution for heat transfer through conduction in mild steel plate. The homogeneous and nonhomogeneous boundary conditions are single variables. The full field analytical solutions of temperature measurement, subjected to local heating source, are derived first by method of separation of variables followed with the experimental visualization using infrared imaging. Based on the present work, it is suggested that appropriate heat input characteristics controls the temperature distribution in and around HAZ.

Keywords: Conduction Heat Transfer, Heat Affected Zone (HAZ), Infra-Red Imaging, Numerical Method, Orthogonal Function, Plasma Arc Cutting, Separation of Variables, Temperature Measurement.

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Authors: Costa, E.S., Borges, E.N.M., Afonso, M.M.

Abstract:

The modeling of sound radiation is of fundamental importance for understanding the propagation of acoustic waves and, consequently, develop mechanisms for reducing acoustic noise. The propagation of acoustic waves, are involved in various phenomena such as radiation, absorption, transmission and reflection. The radiation is studied through the linear equation of the acoustic wave that is obtained through the equation for the Conservation of Momentum, equation of State and Continuity. From these equations, is the Helmholtz differential equation that describes the problem of acoustic radiation. In this paper we obtained the solution of the Helmholtz differential equation for an infinite cylinder in a pulsating through free and homogeneous. The analytical solution is implemented and the results are compared with the literature. A numerical formulation for this problem is obtained using the Boundary Element Method (BEM). This method has great power for solving certain acoustical problems in open field, compared to differential methods. BEM reduces the size of the problem, thereby simplifying the input data to be worked and reducing the computational time used.

Keywords: Acoustic radiation, boundary element

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Authors: Se Kyung Oh, Ary Bachtiar Krishna Putra, Soo Whan Ahn

Abstract:

A numerical study on the turbulent flow and heat transfer characteristics in the rectangular channel with different types of baffles is carried out. The inclined baffles have the width of 19.8 cm, the square diamond type hole having one side length of 2.55 cm, and the inclination angle of 5o. Reynolds number is varied between 23,000 and 57,000. The SST turbulence model is applied in the calculation. The validity of the numerical results is examined by the experimental data. The numerical results of the flow field depict that the flow patterns around the different baffle type are entirely different and these significantly affect the local heat transfer characteristics. The heat transfer and friction factor characteristics are significantly affected by the perforation density of the baffle plate. It is found that the heat transfer enhancement of baffle type II (3 hole baffle) has the best values.

Keywords: Turbulent flow, rectangular channel, inclined baffle, heat transfer, friction factor.

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Authors: Hossein Shokouhmand , Koohyar Vahidkhah, Mohammad A. Esmaeili

Abstract:

A numerical investigation of surface heat transfer characteristics of turbulent air flows in different parallel plate grooved channels is performed using CFD code. The results are obtained for Reynolds number ranging from 10,000 to 30,000 and for arc-shaped and rectangular grooved channels. The influence of different geometric parameters of dimples as well as the number of them and the geometric and thermophysical properties of channel walls are studied. It is found that there exists an optimum value for depth of dimples in which the largest wall heat flux can be achieved. Also, the results show a critical value for the ratio of wall thermal conductivity to the one of fluid in which the dependence of wall heat flux to this ratio almost vanishes. In most cases examined, heat transfer enhancement is larger for arc-shaped grooved channels than rectangular ones.

Keywords: dimple, heat transfer enhancement, Numerical, optimum value, turbulent air flow.

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Authors: Mohamed Essahafi, Mustapha Ait Lafkih

Abstract:

In industrial environments, the heat exchanger is a necessary component to any strategy of energy conversion. Much of thermal energy used in industrial processes passes at least one times by a heat exchanger, and methods systems recovering thermal energy. This survey paper tries to presents in a systemic way an sample control of a heat exchanger by comparison between three controllers LQR (linear quadratic regulator), PID (proportional, integrator and derivate) and Pole Placement. All of these controllers are used mainly in industrial sectors (chemicals, petrochemicals, steel, food processing, energy production, etc…) of transportation (automotive, aeronautics), but also in the residential sector and tertiary (heating, air conditioning, etc...) The choice of a heat exchanger, for a given application depends on many parameters: field temperature and pressure of fluids, and physical properties of aggressive fluids, maintenance and space. It is clear that the fact of having an exchanger appropriate, well-sized, well made and well used allows gain efficiency and energy processes.

Keywords: LQR linear-quadratic regulator, PID control, Pole Placement, Heat exchanger.

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Authors: Esther T. Akinlabi, Stephen A. Akinlabi

Abstract:

This paper reports on the effects of heat treatment on 3CR12 and AISI 316 stainless steel grades. Heat treatment was conducted on the steel grades and cooled using two different media; air and water in order to study the effect of each medium on the evolving properties of the samples. The heat treated samples were characterized through the evolving microstructure and hardness. It was found that there was a significant grain size reduction in both the heat treated stainless steel specimens compared to the parent materials. The finer grain sizes were achieved as a result of impediment to growth of one phase by the other. The Vickers microhardness values of the heat treated samples were higher compared to the parent materials due to the fact that each of the steel grades had a proportion of martensitic structures in their microstructures thereby improving the integrity of the material.

Keywords: Austenite, Ferrite, Grain size, Hardness, Martensite, Microstructure and stainless steel.

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Authors: A. D. Parekh, P. R. Tailor

Abstract:

Capacity and efficiency of any refrigerating system diminish rapidly as the difference between the evaporating and condensing temperature is increased by reduction in the evaporator temperature. The single stage vapour compression refrigeration system is limited to an evaporator temperature of -40 0C. Below temperature of -40 0C the either cascade refrigeration system or multi stage vapour compression system is employed. Present work describes thermal design of main three heat exchangers namely condenser (HTS), cascade condenser and evaporator (LTS) of R404A-R508B and R410A-R23 cascade refrigeration system. Heat transfer area of condenser (HTS), cascade condenser and evaporator (LTS) for both systems have been compared and the effect of condensing and evaporating temperature on heat-transfer area for both systems have been studied under same operating condition. The results shows that the required heat-transfer area of condenser and cascade condenser for R410A-R23 cascade system is lower than the R404A-R508B cascade system but heat transfer area of evaporator is similar for both the system. The heat transfer area of condenser and cascade condenser decreases with increase in condensing temperature (Tc), whereas the heat transfer area of cascade condenser and evaporator increases with increase in evaporating temperature (Te).

Keywords: Heat-transfer area, R410A, R404A, R508B, R23, Refrigeration system, Thermal design

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Authors: Huaizhi Han, Bingxi Li, Yurong He, Rushan Bie, Zhao Wu

Abstract:

The flow and heat transfer mechanism in convex corrugated tubes have been investigated through numerical simulations in this paper. Two kinds of tube types named as symmetric corrugated tube (SCT) and asymmetric corrugated tube (ACT) are modeled and studied numerically based on the RST model. The predictive capability of RST model is examined in the corrugation wall in order to check the reliability of RST model under the corrugation wall condition. We propose a comparison between the RST modelling the corrugation wall with existing direct numerical simulation of Maaß C and Schumann U [14]. The numerical results pressure coefficient at different profiles between RST and DNS are well matched. The influences of large corrugation tough radii to heat transfer and flow characteristic had been considered. Flow and heat transfer comparison between SCT and ACT had been discussed. The numerical results show that ACT exhibits higher overall heat transfer performance than SCT.

Keywords: Asymmetric corrugated tube, RST, DNS, flow and heat transfer mechanism.

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Authors: Ou Xie, Zhenyu Zhao

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In this paper, we consider the problem for identifying the unknown source in the Poisson equation. A modified Tikhonov regularization method is presented to deal with illposedness of the problem and error estimates are obtained with an a priori strategy and an a posteriori choice rule to find the regularization parameter. Numerical examples show that the proposed method is effective and stable.

Keywords: Ill-posed problem, Unknown source, Poisson equation, Tikhonov regularization method, Discrepancy principle

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Authors: N. Kumaresan, J. Kavikumar, M. Kumudthaa, Kuru Ratnavelu

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In this paper, solution of fuzzy differential equation under general differentiability is obtained by genetic programming (GP). The obtained solution in this method is equivalent or very close to the exact solution of the problem. Accuracy of the solution to this problem is qualitatively better. An illustrative numerical example is presented for the proposed method.

Keywords: Fuzzy differential equation, Generalized differentiability, Genetic programming and H-difference.

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Authors: Yiqin Lin, Liang Bao, Qinghua Wu, Liping Zhou

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In this paper, we propose a direct method based on the real Schur factorization for solving the projected Sylvester equation with relatively small size. The algebraic formula of the solution of the projected continuous-time Sylvester equation is presented. The computational cost of the direct method is estimated. Numerical experiments show that this direct method has high accuracy.

Keywords: Projected Sylvester equation, Schur factorization, Spectral projection, Direct method.

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Authors: Kelong Zheng, Jinsong Hu,

Abstract:

In this paper, numerical solution for the generalized Rosenau-Burgers equation is considered and Crank-Nicolson finite difference scheme is proposed. Existence of the solutions for the difference scheme has been shown. Stability, convergence and priori error estimate of the scheme are proved. Numerical results demonstrate that the scheme is efficient and reliable.

Keywords: Generalized Rosenau-Burgers equation, difference scheme, stability, convergence.

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Authors: N. Dizadji, P. Entezar

Abstract:

In this paper, the ice melting in rectangular, cylindrical and conical forms, which are erected vertically against air flow, are experimentally studied in the free convection regime.The results obtained are: Nusslet Number, heat transfer coefficient andGrashof Number, and the variations of the said numbers in relation to the time. The variations of ice slab area and volume are measured, too.

Keywords: Nusselt Number, Heat Transfer, Grashof Number, Heat Transfer Coefficient.

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Authors: Nguyen Thi Thanh Truc, Chi-Hyeon Lee, Byeong-Kyu Lee, Srinivasa Reddy Mallampati

Abstract:

This study evaluated to facilitate separation of ABS plastics from other waste plastics by froth flotation after surface hydrophilization of ABS with heat treatment. The mild heat treatment at 100oC for 60s could selectively increase the hydrophilicity of the ABS plastics surface (i.e., ABS contact angle decreased from 79o to 65.8o) among other plastics mixture. The SEM and XPS results of plastic samples sufficiently supported the increase in hydrophilic functional groups and decrease contact angle on ABS surface, after heat treatment. As a result of the froth flotation (at mixing speed 150 rpm and airflow rate 0.3 L/min) after heat treatment, about 85% of ABS was selectively separated from other heavy plastics with 100% of purity. The effect of optimum treatment condition and detailed mechanism onto separation efficiency in the froth floatation was also investigated. This research is successful in giving a simple, effective, and inexpensive method for ABS separation from waste plastics.

Keywords: ABS, hydrophilic, heat treatment, froth flotation, contact angle.

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Authors: Tomoaki Hashimoto

Abstract:

Recent technological advance has prompted significant interest in developing the control theory of quantum systems. Following the increasing interest in the control of quantum dynamics, this paper examines the control problem of Schrödinger equation because quantum dynamics is basically governed by Schrödinger equation. From the practical point of view, stochastic disturbances cannot be avoided in the implementation of control method for quantum systems. Thus, we consider here the robust stabilization problem of Schrödinger equation against stochastic disturbances. In this paper, we adopt model predictive control method in which control performance over a finite future is optimized with a performance index that has a moving initial and terminal time. The objective of this study is to derive the stability criterion for model predictive control of Schrödinger equation under stochastic disturbances.

Keywords: Optimal control, stochastic systems, quantum systems, stabilization.

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Authors: H.Y. Lai, S. C. Chang, W. L. Chen

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The hydrodynamic and thermal lattice Boltzmann methods are applied to investigate the turbulent convective heat transfer in the wavy channel flows. In this study, the turbulent phenomena are modeling by large-eddy simulations with the Smagorinsky model. As a benchmark, the laminar and turbulent backward-facing step flows are simulated first. The results give good agreement with other numerical and experimental data. For wavy channel flows, the distribution of Nusselt number and the skin-friction coefficients are calculated to evaluate the heat transfer effect and the drag force. It indicates that the vortices at the trough would affect the magnitude of drag and weaken the heat convection effects on the wavy surface. In turbulent cases, if the amplitude of the wavy boundary is large enough, the secondary vortices would be generated at troughs and contribute to the heat convection. Finally, the effects of different Re on the turbulent transport phenomena are discussed.

Keywords: Heat transfer, lattice Boltzmann method, turbulence, wavy channel.

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Authors: Chen-Yu Wang, Tsung-Chien Chen, Ming-Hui Lee, Jen-Feng Huang

Abstract:

In this paper, the innovative intelligent fuzzy weighted input estimation method (FWIEM) can be applied to the inverse heat transfer conduction problem (IHCP) to estimate the unknown time-varying heat flux efficiently as presented. The feasibility of this method can be verified by adopting the temperature measurement experiment. We would like to focus attention on the heat flux estimation to three kinds of samples (Copper, Iron and Steel/AISI 304) with the same 3mm thickness. The temperature measurements are then regarded as the inputs into the FWIEM to estimate the heat flux. The experiment results show that the proposed algorithm can estimate the unknown time-varying heat flux on-line.

Keywords: Fuzzy Weighted Input Estimation Method, IHCP andHeat Flux.

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Authors: Chang Su Woo, Hyun Sung Park

Abstract:

Useful lifetime evaluation of railpads were very important in design procedure to assure the safety and reliability. It is, therefore, necessary to establish a suitable criterion for the replacement period of rail pads. In this study, we performed properties and accelerated heat aging tests of rail pads considering degradation factors and all environmental conditions including operation, and then derived a lifetime prediction equation according to changes in hardness, thickness, and static spring constants in the Arrhenius plot to establish how to estimate the aging of rail pads. With the useful lifetime prediction equation, the lifetime of e-clip pads was 2.5 years when the change in hardness was 10% at 25°C; and that of f-clip pads was 1.7 years. When the change in thickness was 10%, the lifetime of e-clip pads and f-clip pads is 2.6 years respectively. The results obtained in this study to estimate the useful lifetime of rail pads for high speed trains can be used for determining the maintenance and replacement schedule for rail pads.

Keywords: Rail pads, accelerated test, Arrhenius plot, useful lifetime prediction.

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Authors: Bouchmel Mliki, Chaouki Ali, Mohamed Ammar Abbassi

Abstract:

In this numerical study, natural convection of Cu–water nanofluid in a cavity submitted to different heating modes on its vertical walls is analyzed. Maxwell-Garnetts (MG) and Brinkman models have been utilized for calculating the effective thermal conductivity and dynamic viscosity of nanofluid, respectively. Influences of Rayleigh number (Ra = 10³−10⁶), nanoparticle volume concentration (f = 0-0.04) and Hartmann number (Ha = 0-90) on the flow and heat transfer characteristics have been examined. The results indicate that the Hartmann number influences the heat transfer at Ra = 10⁶ more than other Raleigh numbers, as the least effect is observed at Ra = 10³. Moreover, the results show that the solid volume fraction has a significant influence on heat transfer, depending on the value of Hartmann, heat generation or absorption coefficient and Rayleigh numbers.

Keywords: Heat transfer, linearly/sinusoidally heated, Lattice Boltzmann Method, natural convection, nanofluid.

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