Search results for: heat equation

Authors: Chuanyun Gu

Abstract:

By using fixed point theorems for a class of generalized concave and convex operators, the positive solution of nonlinear fractional differential equation with integral boundary conditions is studied, where n ≥ 3 is an integer, μ is a parameter and 0 ≤ μ < α. Its existence and uniqueness is proved, and an iterative scheme is constructed to approximate it. Finally, two examples are given to illustrate our results.

Keywords: Fractional differential equation, positive solution, existence and uniqueness, fixed point theorem, generalized concave and convex operator, integral boundary conditions.

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Authors: N. M. Nusi, M. Othman, M. Suleiman, F. Ismail, N. Alias

Abstract:

In this paper, an alternating implicit block method for solving two dimensional scalar wave equation is presented. The new method consist of two stages for each time step implemented in alternating directions which are very simple in computation. To increase the speed of computation, a group of adjacent points is computed simultaneously. It is shown that the presented method increase the maximum time step size and more accurate than the conventional finite difference time domain (FDTD) method and other existing method of natural ordering.

Keywords: FDTD, Scalar wave equation, alternating direction implicit (ADI), alternating group explicit (AGE), asymmetric approximation.

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Authors: Adeosun S. Oluropo, Sekunowo O. Israel, Talabi S. Isaac

Abstract:

This study investigated the combined effect of cold rolling and heat treatment on the mechanical properties of Al-Ti alloy. Samples of the alloy are cast in metal mould to obtain 0.94-2.19wt% mixes of titanium. These samples are grouped into untreated (as-cast) and those that are cold rolled to fifty percent reduction, homogenized at 500⁰C and soaked for one hour. The cold rolled and heat treated samples are normalized (RT_n) and quench-tempered (RT_q-t) at 100⁰C. All these samples are subjected to tensile, micro-hardness and microstructural evaluation. Results show remarkable improvement in the mechanical properties of the cold rolled and heat treated samples compared to the as-cast. In particular, the RT_q-t samples containing titanium in the range of 1.7-2.2% demonstrates improve tensile strength by 24.7%, yield strength, 28%, elastic modulus, 38.3% and micro-hardness, 20.5%. The Al₃Ti phase being the most stable precipitate in the α-Al matrix appears to have been responsible for the significant improvement in the alloy’s mechanical properties. It is concluded that quench and temper heat treatment is an effective method of improving the strength-strain ratio of cold rolled Al-.0.9-2.2%Ti alloy.

Keywords: Aluminum-titanium alloy, heat treatment, mechanical properties, precipitate.

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Authors: Nik Mohd Asri Nik Long, Koo Lee Feng, Zainidin K. Eshkuvatov, A. A. Khaldjigitov

Abstract:

This paper study the behavior of the solution at the crack edges for an elliptical crack with developing cusps, Ω in the plane elasticity subjected to shear loading. The problem of finding the resulting shear stress can be formulated as a hypersingular integral equation over Ω and it is then transformed into a similar equation over a circular region, D, using conformal mapping. An appropriate collocation points are chosen on the region D to reduce the hypersingular integral equation into a system of linear equations with (2N+1)(N+1) unknown coefficients, which will later be used in the determination of shear stress intensity factors and maximum shear stress intensity. Numerical solution for the considered problem are compared with the existing asymptotic solution, and displayed graphically. Our results give a very good agreement to the existing asymptotic solutions.

Keywords: Elliptical crack, stress intensity factors, hyper singular integral equation, shear loading, conformal mapping.

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Authors: Elhameh Narimani, Mikhail Sorin, Philippe Micheau, Hakim Nesreddine

Abstract:

Numerical models of the heat exchangers in ejector refrigeration system (ERS) were developed and validated with the experimental data. The models were based on the switched heat exchangers model using the moving boundary method, which were capable of estimating the zones’ lengths, the outlet temperatures of both sides and the heat loads at various experimental points. The developed models were utilized to investigate the influence of the primary flow pressure on the performance of an R245fa ERS based on its coefficient of performance (COP) and exergy efficiency. It was illustrated numerically and proved experimentally that increasing the primary flow pressure slightly reduces the COP while the exergy efficiency goes through a maximum before decreasing.

Keywords: Coefficient of performance, ejector refrigeration system, exergy efficiency, heat exchangers modeling, moving boundary method.

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Authors: Gopal Nandan, P. K. Sahooa, Ravi Kumara, B Chatterjeeb, D. Mukhopadhyayb, H. G. Leleb

Abstract:

In a nuclear reactor Loss of Coolant accident (LOCA) considers wide range of postulated damage or rupture of pipe in the heat transport piping system. In the case of LOCA with/without failure of emergency core cooling system in a Pressurised Heavy water Reactor, the Pressure Tube (PT) temperature could rise significantly due to fuel heat up and gross mismatch of the heat generation and heat removal in the affected channel. The extent and nature of deformation is important from reactor safety point of view. Experimental set-ups have been designed and fabricated to simulate ballooning (radial deformation) of PT for 220 MWe IPHWRs. Experiments have been conducted by covering the CT by ceramic fibers and then by submerging CT in water of voided PTs. In both the experiments, it is observed that ballooning initiates at a temperature around 665´┐¢C and complete contact between PT and Caldaria Tube (CT) occurs at around 700´┐¢C approximately. The strain rate is found to be 0.116% per second. The structural integrity of PT is retained (no breach) for all the experiments. The PT heatup is found to be arrested after the contact between PT and CT, thus establishing moderator acting as an efficient heat sink for IPHWRs.

Keywords: Pressure Tube, Calandria Tube, Thermo-mechanicaldeformation, Boiling heat transfer, Reactor safety

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Authors: A. Suleng, T. Jelstad Olsen, J. Šindler, P. Bárta

Abstract:

A dead leg is a typical subsea production system component. CFD is required to model heat transfer within the dead leg. Unfortunately its solution is time demanding and thus not suitable for fast prediction or repeated simulations. Therefore there is a need to create a thermal FEA model, mimicking the heat flows and temperatures seen in CFD cool down simulations. This paper describes the conventional way of tuning and a new automated way using parametric model order reduction (PMOR) together with an optimization algorithm. The tuned FE analyses replicate the steady state CFD parameters within a maximum error in heat flow of 6 % and 3 % using manual and PMOR method respectively. During cool down, the relative error of the tuned FEA models with respect to temperature is below 5% comparing to the CFD. In addition, the PMOR method obtained the correct FEA setup five times faster than the manually tuned FEA.

Keywords: CFD, convective heat, FEA, model tuning, subseaproduction

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Authors: Jahedul Islam Chowdhury, Bao Kha Nguyen, David Thornhill, Roy Douglas, Stephen Glover

Abstract:

Organic Rankine Cycle (ORC) is the most commonly used method for recovering energy from small sources of heat. The investigation of the ORC in supercritical condition is a new research area as it has a potential to generate high power and thermal efficiency in a waste heat recovery system. This paper presents a steady state ORC model in supercritical condition and its simulations with a real engine’s exhaust data. The key component of ORC, evaporator, is modelled using finite volume method, modelling of all other components of the waste heat recovery system such as pump, expander and condenser are also presented. The aim of this paper is to investigate the effects of mass flow rate and evaporator outlet temperature on the efficiency of the waste heat recovery process. Additionally, the necessity of maintaining an optimum evaporator outlet temperature is also investigated. Simulation results show that modification of mass flow rate is the key to changing the operating temperature at the evaporator outlet.

Keywords: Organic Rankine cycle, supercritical condition, steady state model, waste heat recovery.

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Authors: Xiaoshu Lu, Martti Viljanen

Abstract:

A new analytical model is developed which provides close-formed solutions for both transient indoor and envelope temperature changes in buildings. Time-dependent boundary temperature is presented as Fourier series which can approximate real weather conditions. The final close-formed solutions are simple, concise, and comprehensive. The model was compared with numerical results and good accuracy was obtained. The model can be used as design and control guidelines in engineering applications for analysing mechanical heat transfer properties for buildings.

Keywords: Analytical model, heat transfer, whole building.

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Authors: D. Bala Bhaskar

Abstract:

An algorithm is proposed for the order reduction of large scale linear dynamic multi variable systems where the reduced order model denominator is obtained by using Stability equation method and numerator coefficients are obtained by using SRAM. The proposed algorithm produces a lower order model for an original stable high order multivariable system. The reduction procedure is easy to understand, efficient and computer oriented. To highlight the advantages of the approach, the algorithm is illustrated with the help of a numerical example and the results are compared with the other existing techniques in literature.

Keywords: Multi variable systems, order reduction, stability equation method, SRAM, time domain characteristics, ISE.

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Authors: Maria Neagu

Abstract:

This paper presents the heat and mass driven natural convection succession in a Darcy thermally stratified porous medium that embeds a vertical semi-infinite impermeable wall of constant heat flux and concentration. The scale analysis of the system determines the two possible maps of the heat and mass driven natural convection sequence along the wall as a function of the process parameters. These results are verified using the finite differences method applied to the conservation equations.

Keywords: Finite difference method, natural convection, porous medium, scale analysis, thermal stratification.

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Authors: Ziya Uddin

Abstract:

This paper deals with the theoretical and numerical investigation of magneto hydrodynamic boundary layer flow of a nanofluid past a wedge shaped wick in heat pipe used for the cooling of electronic components and different type of machines. To incorporate the effect of nanoparticle diameter, concentration of nanoparticles in the pure fluid, nanothermal layer formed around the nanoparticle and Brownian motion of nanoparticles etc., appropriate models are used for the effective thermal and physical properties of nanofluids. To model the rotation of nanoparticles inside the base fluid, microfluidics theory is used. In this investigation ethylene glycol (EG) based nanofluids, are taken into account. The non-linear equations governing the flow and heat transfer are solved by using a very effective particle swarm optimization technique along with Runge-Kutta method. The values of heat transfer coefficient are found for different parameters involved in the formulation viz. nanoparticle concentration, nanoparticle size, magnetic field and wedge angle etc. It is found that, the wedge angle, presence of magnetic field, nanoparticle size and nanoparticle concentration etc. have prominent effects on fluid flow and heat transfer characteristics for the considered configuration.

Keywords: Heat transfer, Heat pipe, numerical modeling, nanofluid applications, particle swarm optimization, wedge shaped wick.

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Authors: Mohammed A. Elhaj, Jamal S. Yassin

Abstract:

Owing to the lack of synchronization between the solar energy availability and the heat demands in a specific application, the energy storing sub-system is necessary to maintain the continuity of thermal process. The present work is dealing with an active solar heating storing system in which an air solar collector is connected to storing unit where this energy is distributed and provided to the heated space in a controlled manner. The solar collector is a box type absorber where the air flows between a number of vanes attached between the collector absorber and the bottom plate. This design can improve the efficiency due to increasing the heat transfer area exposed to the flowing air, as well as the heat conduction through the metal vanes from the top absorbing surface. The storing unit is a packed bed type where the air is coming from the air collector and circulated through the bed in order to add/remove the energy through the charging / discharging processes, respectively. The major advantage of the packed bed storage is its high degree of thermal stratification. Numerical solution of the packed bed energy storage is considered through dividing the bed into a number of equal segments for the bed particles and solved the energy equation for each segment depending on the neighbor ones. The studied design and performance parameters in the developed simulation model including, particle size, void fraction, etc. The final results showed that the collector efficiency was fluctuated between 55%-61% in winter season (January) under the climatic conditions of Misurata in Libya. Maximum temperature of 52ºC is attained at the top of the bed while the lower one is 25ºC at the end of the charging process of hot air into the bed. This distribution can satisfy the required load for the most house heating in Libya.

Keywords: Solar energy, thermal process, performance, collector, packed bed, numerical analysis, simulation.

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Authors: Amit Goyal, Alka, Rama Gupta, C. Nagaraja Kumar

Abstract:

We have solved the Burgers-Fisher (BF) type equations, with time-dependent coefficients of convection and reaction terms, by using the auxiliary equation method. A class of solitary wave solutions are obtained, and some of which are derived for the first time. We have studied the effect of variable coefficients on physical parameters (amplitude and velocity) of solitary wave solutions. In some cases, the BF equations could be solved for arbitrary timedependent coefficient of convection term.

Keywords: Solitary wave solution, Variable coefficient Burgers- Fisher equation, Auxiliary equation method.

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Authors: S. Ben Saad, F. Ayad, H. Damotte

Abstract:

This study deals with Computational Fluid Dynamics (CFD) studies of the interactions between the air flow and louvered fins which equipped the automotive heat exchangers. 3D numerical simulation results are obtained by using the ANSYS Fluent 13.0 code and compared to experimental data. The paper studies the effect of louver angle and louver pitch geometrical parameters, on overall thermal hydraulic performances of louvered fins. The comparison between CFD simulations and experimental data show that established 3-D CFD model gives a good agreement. The validation agrees, with about 7% of deviation respectively of friction and Colburn factors to experimental results. As first, it is found that the louver angle has a strong influence on the heat transfer rate. Then, louver angle and louver pitch variation of the louvers and their effects on thermal hydraulic performances are studied. In addition to this study, it is shown that the second half of the fin takes has a significant contribution on pressure drop increase without any increase in heat transfer.

Keywords: CFD simulations, automotive heat exchanger, performances.

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Authors: Marzieh Dosti, Alireza Nazemi

Abstract:

In this paper, the telegraph equation is solved numerically by cubic B-spline quasi-interpolation .We obtain the numerical scheme, by using the derivative of the quasi-interpolation to approximate the spatial derivative of the dependent variable and a low order forward difference to approximate the temporal derivative of the dependent variable. The advantage of the resulting scheme is that the algorithm is very simple so it is very easy to implement. The results of numerical experiments are presented, and are compared with analytical solutions by calculating errors L2 and L∞ norms to confirm the good accuracy of the presented scheme.

Keywords: Cubic B-spline, quasi-interpolation, collocation method, second-order hyperbolic telegraph equation.

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Authors: Anuar Ishak

Abstract:

The characteristics of fluid flow and heat transfer over a permeable shrinking sheet is studied. The governing partial differential equations are transformed into a set of ordinary differential equations, which are then solved numerically using MATLAB routine boundary value problem solver bvp4c. Numerical results show that dual solutions are possible for a certain range of the suction parameter. A stability analysis is performed to determine which solution is linearly stable and physically realizable.

Keywords: Dual solutions, heat transfer, shrinking sheet, stability analysis.

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Authors: P. O. Angienda, B. O. Aketch, E. N. Waindi

Abstract:

Juvenile Nile tilapia subjected to heat treatment at temperatures ranging from 260C to 370C showed positive correlation (P<0.01) between treatment temperatures and resultant sex ratios, while, survival rate of the fry showed a negative correlation against temperature (P<0.01). The optimal temperature for both sex shift towards males and survival rates was 36±0.5°C, producing male percentage of 86.31 and a fry survival of 65.25. To determine the genetic basis of temperature sex-determination in Nile tilapia, we employed three microsatellite markers (Abur36, Abur100 and UNH846). Abur36 predicted the sex of 95% of the heat induced individuals, suggesting that the locus influence sex ratio and its interaction with temperature result in male biased sex ratio. This locus could turn out to be the major sex determining gene operating in Nile tilapia. These markers could be used in marker-assisted selection to select genotypes that give a higher percentage of males for commercial production.

Keywords: Heat treatment, Microsatellite, Nile tilapia, sex-determination.

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Authors: P. Srinivas, P. V. N. Prasad

Abstract:

Since torque ripple is the main cause of noise and vibrations, the performance of Switched Reluctance Motor (SRM) can be improved by minimizing its torque ripple using a novel control technique called Direct Torque Control (DTC). In DTC technique, torque is controlled directly through control of magnitude of the flux and change in speed of the stator flux vector. The flux and torque are maintained within set hysteresis bands.

The DTC of SRM is analyzed by two methods. In one method, the actual torque is computed by conducting Finite Element Analysis (FEA) on the design specifications of the motor. In the other method, the torque is computed by Simplified Torque Equation. The variation of peak current, average current, torque ripple and speed settling time with Simplified Torque Equation model is compared with FEA based model.

Keywords: Direct Toque Control, Simplified Torque Equation, Finite Element Analysis, Torque Ripple.

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Authors: David Lávicka

Abstract:

This paper looks into detailed investigation of thermal-hydraulic characteristics of the flow field in a fuel rod model, especially near the spacer. The area investigate represents a source of information on the velocity flow field, vortex, and on the amount of heat transfer into the coolant all of which are critical for the design and improvement of the fuel rod in nuclear power plants. The flow field investigation uses three-dimensional Computational Fluid Dynamics (CFD) with the Reynolds stresses turbulence model (RSM). The fuel rod model incorporates a vertical annular channel where three different shapes of spacers are used; each spacer shape is addressed individually. These spacers are mutually compared in consideration of heat transfer capabilities between the coolant and the fuel rod model. The results are complemented with the calculated heat transfer coefficient in the location of the spacer and along the stainless-steel pipe.

Keywords: CFD, fuel rod model, heat transfer, spacer

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Authors: J. J. Peña, J. Morales, J. García-Ravelo, L. Arcos-Díaz

Abstract:

The Point canonical transformation method is applied for solving the Schrödinger equation with position-dependent mass. This class of problem has been solved for continuous mass distributions. In this work, a staggered mass distribution for the case of a free particle in an infinite square well potential has been proposed. The continuity conditions as well as normalization for the wave function are also considered. The proposal can be used for dealing with other kind of staggered mass distributions in the Schrödinger equation with different quantum potentials.

Keywords: Free particle, point canonical transformation method, position-dependent mass, staggered mass distribution.

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Authors: Yehia Manawi, Ahmad Kayvani Fard

Abstract:

Membrane distillation is an emerging technology which has been used to produce freshwater and purify different types of aqueous mixtures. Qatar is an arid country where almost 100% of its freshwater demand is supplied through the energy-intensive thermal desalination process. The country’s need for water has reached an all-time high which stipulates finding an alternative way to augment freshwater without adding any drastic affect to the environment. The objective of this paper was to investigate the potential of using the industrial low grade waste heat to produce freshwater using membrane distillation. The main part of this work was conducting a heat audit on selected Qatari chemical industries to estimate the amounts of freshwater produced if such industrial waste heat were to be recovered. By the end of this work, the main objective was met and the heat audit conducted on the Qatari chemical industries enabled us to estimate both the amounts of waste heat which can be potentially recovered in addition to the amounts of freshwater which can be produced if such waste heat were to be recovered.

By the end, the heat audit showed that around 605 Mega Watts of waste heat can be recovered from the studied Qatari chemical industries which resulted in a total daily production of 5078.7 cubic meter of freshwater.

This water can be used in a wide variety of applications such as human consumption or industry. The amount of produced freshwater may look small when compared to that produced through thermal desalination plants; however, one must bear in mind that this water comes from waste and can be used to supply water for small cities or remote areas which are not connected to the water grid. The idea of producing freshwater from the two widely-available wastes (thermal rejected brine and waste heat) seems promising as less environmental and economic impacts will be associated with freshwater production which may in the near future augment the conventional way of producing freshwater currently being thermal desalination. This work has shown that low grade waste heat in the chemical industries in Qatar and perhaps the rest of the world can contribute to additional production of freshwater using membrane distillation without significantly adding to the environmental impact.

Keywords: Membrane distillation, desalination, heat recovery, environment.

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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: Sarun Phibanchon, Michael A. Allen

Abstract:

A variational method is used to obtain the growth rate of a transverse long-wavelength perturbation applied to the soliton solution of a nonlinear Schr¨odinger equation with a three-half order potential. We demonstrate numerically that this unstable perturbed soliton will eventually transform into a cylindrical soliton.

Keywords: Soliton, instability, variational method, spectral method.

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Authors: Chih-Chieh Chen, Chih-Hao Chen, Guan-Wei Wu, Sih-Li Chen

Abstract:

A high thermal dissipation performance polyethylene terephthalate heat pipe has been fabricated and tested in this research. Polyethylene terephthalate (PET) is used as the container material instead of copper. Copper mesh and methanol are sealed in the middle of two PET films as the wick structure and working fluid. Although the thermal conductivity of PET (0.15-0.24 W/m·K) is much smaller than copper (401 W/m·K), the experiment results reveal that the PET heat pipe can reach a minimum thermal resistance of 0.146 (^oC/W) and maximum effective thermal conductivity of 18,310 (W/m·K) with 36.9 vol% at 26 W input power. However, when the input power is larger than 30 W, the laminated PET will debond due to the high vapor pressure of methanol.

Keywords: PET, heat pipe, thermal resistance, effective thermal conductivity.

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Authors: A. Chatterjee, A. Sarkar, N. Gayathri, P. Mukherjee, P. Barat

Abstract:

An experimental study is presented on the effect of microstructural change on the Portevin-Le Chatelier effect behaviour of Al-2.5%Mg alloy. Tensile tests are performed on the as received and heat treated (at 400 ºC for 16 hours) samples for a wide range of strain rates. The serrations observed in the stress-time curve are investigated from statistical analysis point of view. Microstructures of the samples are characterized by optical metallography and X-ray diffraction. It is found that the excess vacancy generated due to heat treatment leads to decrease in the strain rate sensitivity and the increase in the number of stress drop occurrences per unit time during the PLC effect. The microstructural parameters like domain size, dislocation density have no appreciable effect on the PLC effect as far as the statistical behavior of the serrations is considered.

Keywords: Dynamic strain ageing, Heat treatment, Portevin-LeChatelier effect

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Authors: V. Nikkhah Rashidabad, M. Manteghian, M. Masoumi, S. Mousavian

Abstract:

In this research, the changes in bubbles diameter and  number that may occur due to the change in heat flux of pure water  during pool boiling process. For this purpose, test equipment was  designed and developed to collect test data. The bubbles were graded  using Caliper Screen software. To calculate the growth and  nucleation rates of bubbles under different fluxes, population balance  model was employed. The results show that the increase in heat flux  from q=20 kw/m2 to q= 102 kw/m2 raised the growth and nucleation  rates of bubbles.

 

Keywords: Heat flux, bubble growth, bubble nucleation, population balance model.

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Authors: Fábio A. S. Mota, Mauro A. S. S. Ravagnani, E. P. Carvalho

Abstract:

In the present paper the design of plate heat exchangers is formulated as an optimization problem considering two mathematical modelling. The number of plates is the objective function to be minimized, considering implicitly some parameters configuration. Screening is the optimization method used to solve the problem. Thermal and hydraulic constraints are verified, not viable solutions are discarded and the method searches for the convergence to the optimum, case it exists. A case study is presented to test the applicability of the developed algorithm. Results show coherency with the literature.

Keywords: Plate heat exchanger, optimization, modeling, simulation.

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Authors: O. S. Pradeep, S. Vigneshwaran, K. Praveen Kumar, K. Jeyendran, V. R. Sanal Kumar

Abstract:

The influence of injector attitude on wall heat flux plays an important role in predicting the start-up transient and also determining the combustion chamber wall durability of liquid rockets. In this paper comprehensive numerical studies have been carried out on an idealized liquid rocket combustion chamber to examine the transient wall heat flux during its start-up transient at different injector attitude. Numerical simulations have been carried out with the help of a validated 2d axisymmetric, double precision, pressure-based, transient, species transport, SST k-omega model with laminar finite rate model for governing turbulent-chemistry interaction for four cases with different jet intersection angles, viz., 0^o, 30^o, 45^o, and 60^o. We concluded that the jets intersection angle is having a bearing on the time and location of the maximum wall-heat flux zone of the liquid rocket combustion chamber during the start-up transient. We also concluded that the wall heat flux mapping in liquid rocket combustion chamber during the start-up transient is a meaningful objective for the chamber wall material selection and the lucrative design optimization of the combustion chamber for improving the payload capability of the rocket.  

Keywords: Combustion chamber, injector, liquid rocket, rocket engine wall heat flux.

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Authors: Kourosh Parand, Jamal Amani Rad

Abstract:

In this paper, Exp-function method is used for some exact solitary solutions of the generalized Pochhammer-Chree equation. It has been shown that the Exp-function method, with the help of symbolic computation, provides a very effective and powerful mathematical tool for solving nonlinear partial differential equations. As a result, some exact solitary solutions are obtained. It is shown that the Exp-function method is direct, effective, succinct and can be used for many other nonlinear partial differential equations.

Keywords: Exp-function method, generalized Pochhammer- Chree equation, solitary wave solution, ODE's.

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