Commenced in January 2007
Frequency: Monthly
Edition: International
Paper Count: 14392

# Search results for: convection-diffusion controlled model

##### 14392 A Controlled Mathematical Model for Population Dynamics in an Infested Honeybees Colonies

Authors: Chakib Jerry, Mounir Jerry

Abstract:

In this paper, a mathematical model of infested honey bees colonies is formulated in order to investigate Colony Collapse Disorder in a honeybee colony. CCD, as it is known, is a major problem on honeybee farms because of the massive decline in colony numbers. We introduce to the model a control variable which represents forager protection. We study the controlled model to derive conditions under which the bee colony can fight off epidemic. Secondly we study the problem of minimizing prevention cost under model’s dynamics constraints. Downloads 264
##### 14391 A Bi-Objective Model to Optimize the Total Time and Idle Probability for Facility Location Problem Behaving as M/M/1/K Queues

Authors: Amirhossein Chambari

Abstract:

This article proposes a bi-objective model for the facility location problem subject to congestion (overcrowding). Motivated by implementations to locate servers in internet mirror sites, communication networks, one-server-systems, so on. This model consider for situations in which immobile (or fixed) service facilities are congested (or queued) by stochastic demand to behave as M/M/1/K queues. We consider for this problem two simultaneous perspectives; (1) Customers (desire to limit times of accessing and waiting for service) and (2) Service provider (desire to limit average facility idle-time). A bi-objective model is setup for facility location problem with two objective functions; (1) Minimizing sum of expected total traveling and waiting time (customers) and (2) Minimizing the average facility idle-time percentage (service provider). The proposed model belongs to the class of mixed-integer nonlinear programming models and the class of NP-hard problems. In addition, to solve the model, controlled elitist non-dominated sorting genetic algorithms (Controlled NSGA-II) and controlled elitist non-dominated ranking genetic algorithms (NRGA-I) are proposed. Furthermore, the two proposed metaheuristics algorithms are evaluated by establishing standard multiobjective metrics. Finally, the results are analyzed and some conclusions are given. Downloads 463
##### 14390 Quantification of Leachate Potential of the Quezon City Controlled Dumping Facility Using Help Model

Abstract:

The Quezon City Controlled Dumping facility also known as Payatas produces leachate which can contaminate soil and water environment in the area. The goal of this study is to quantify the leachate produced by the QCCDF using the Hydrologic Evaluation of Landfill Performance (HELP) model. Results could be used as input for groundwater contaminant transport studies. The HELP model is based on a simple water budget and is an essential “model requirement” used by the US Environmental Protection Agency (EPA). Annual waste profile of the QCCDF was calculated. Based on topographical maps and estimation of settlement due to overburden pressure and degradation, a total of 10M m^3 of waste is contained in the landfill. The input necessary for the HELP model are weather data, soil properties, and landfill design. Results showed that from 1988 to 2011, an average of 50% of the total precipitation percolates through the bottom layer. Validation of the results is still needed due to the assumptions made in the study. The decrease in porosity of the top soil cover showed the best mitigation for minimizing percolation rate. This study concludes that there is a need for better leachate management system in the QCCDF. Downloads 210
##### 14389 Trajectory Tracking Control for Quadrotor Helicopter by Controlled Lagrangian Method

Authors: Ce Liu, Wei Huo

Abstract:

A nonlinear trajectory tracking controller for quadrotor helicopter based on controlled Lagrangian (CL) method is proposed in this paper. A Lagrangian system with virtual angles as generated coordinates rather than Euler angles is developed. Based on the model, the matching conditions presented by nonlinear partial differential equations are simplified and explicitly solved. Smooth tracking control laws and the range of control parameters are deduced based on the controlled energy of closed-loop system. Besides, a constraint condition for reference accelerations is deduced to identify the trackable reference trajectories by the proposed controller and to ensure the stability of the closed-loop system. The proposed method in this paper does not rely on the division of the quadrotor system, and the design of the control torques does not depend on the thrust as in backstepping or hierarchical control method. Simulations for a quadrotor model demonstrate the feasibility and efficiency of the theoretical results. Downloads 37
##### 14388 First-Principles Modeling of Nanoparticle Magnetization, Chaining, and Motion

Abstract:

The ability to effectively design and test magnetic nanoparticles for controlled movement has been an elusive goal in the design of these particles. Magnetic nanoparticles of various characteristics have been created for use towards therapeutic effects, however the challenge of designing for controlled movement remains unmet. A step towards design in this aspect is a first principles model that captures and predicts the behaviors of particles in a magnetic field. The model is governed by four forces acting on the particles, the magnetic gradient, the dipole-dipole forces, the steric forces, and the viscous drag force. The particles are multi-core or single core, and incorporate a preferred magnetization axis. Particles exhibit behaviors, such as chaining, in simulations that are similar to those witnessed through experimentation. Currently, experimental results are being compared to the modeling results for verification of the model, through the analysis of chaining behaviors. This modeling system will be used in designing magnetic nanoparticles for specific chaining and movement behaviors. Downloads 219
##### 14387 Using of Particle Swarm Optimization for Loss Minimization of Vector-Controlled Induction Motors

Authors: V. Rashtchi, H. Bizhani, F. R. Tatari

Abstract:

This paper presents a new online loss minimization for an induction motor drive. Among the many loss minimization algorithms (LMAs) for an induction motor, a particle swarm optimization (PSO) has the advantages of fast response and high accuracy. However, the performance of the PSO and other optimization algorithms depend on the accuracy of the modeling of the motor drive and losses. In the development of the loss model, there is always a trade off between accuracy and complexity. This paper presents a new online optimization to determine an optimum flux level for the efficiency optimization of the vector-controlled induction motor drive. An induction motor (IM) model in d-q coordinates is referenced to the rotor magnetizing current. This transformation results in no leakage inductance on the rotor side, thus the decomposition into d-q components in the steady-state motor model can be utilized in deriving the motor loss model. The suggested algorithm is simple for implementation. Downloads 497
##### 14386 Controlled Release of Glucosamine from Pluronic-Based Hydrogels for the Treatment of Osteoarthritis

Abstract:

Osteoarthritis affects a lot of people worldwide. Local injection of glucosamine is one of the alternative treatment methods to replenish the natural lubrication of cartilage. However, multiple injections can potentially lead to possible bacterial infection. Therefore, a drug delivery system is desired to reduce the frequencies of injections. A hydrogel is one of the delivery systems that can control the release of drugs. Thermo-reversible hydrogels can be beneficial to the drug delivery system especially in the local injection route because this formulation can change from liquid to gel after getting into human body. Once the gel is in the body, it will slowly release the drug in a controlled manner. In this study, various formulations of Pluronic-based hydrogels were synthesized for the controlled release of glucosamine. One of the challenges of the Pluronic controlled release system is its fast dissolution rate. To overcome this problem, alginate and calcium sulfate (CaSO4) were added to the polymer solution. The characteristics of the hydrogels were investigated including the gelation temperature, gelation time, hydrogel dissolution and glucosamine release mechanism. Finally, a mathematical model of glucosamine release from Pluronic-alginate-hyaluronic acid hydrogel was developed. Our results have shown that crosslinking Pluronic gel with alginate did not significantly extend the dissolution rate of the gel. Moreover, the gel dissolution profiles and the glucosamine release mechanisms were best described using the zeroth-order kinetic model, indicating that the release of glucosamine was primarily governed by the gel dissolution. Downloads 155
##### 14385 Mechanical Model of Gypsum Board Anchors Subjected Cyclic Shear Loading

Authors: Yoshinori Kitsutaka, Fumiya Ikedo

Abstract:

##### 14384 Optimization Analysis of Controlled Cooling Process for H-Shape Steam Beams

Authors: Jiin-Yuh Jang, Yu-Feng Gan

Abstract:

In order to improve the comprehensive mechanical properties of the steel, the cooling rate, and the temperature distribution must be controlled in the cooling process. A three-dimensional numerical model for the prediction of the heat transfer coefficient distribution of H-beam in the controlled cooling process was performed in order to obtain the uniform temperature distribution and minimize the maximum stress and the maximum deformation after the controlled cooling. An algorithm developed with a simplified conjugated-gradient method was used as an optimizer to optimize the heat transfer coefficient distribution. The numerical results showed that, for the case of air cooling 5 seconds followed by water cooling 6 seconds with uniform the heat transfer coefficient, the cooling rate is 15.5 (℃/s), the maximum temperature difference is 85℃, the maximum the stress is 125 MPa, and the maximum deformation is 1.280 mm. After optimize the heat transfer coefficient distribution in control cooling process with the same cooling time, the cooling rate is increased to 20.5 (℃/s), the maximum temperature difference is decreased to 52℃, the maximum stress is decreased to 82MPa and the maximum deformation is decreased to 1.167mm. Downloads 267
##### 14383 Numerical Simulation of Urea Water Solution Evaporation Behavior inside the Diesel Selective Catalytic Reduction System

Authors: Kumaresh Selvakumar, Man Young Kim

Abstract:

Selective catalytic reduction (SCR) converts the nitrogen oxides with the aid of a catalyst by adding aqueous urea into the exhaust stream. In this work, the urea water droplets are sprayed over the exhaust gases by treating with Lagrangian particle tracking. The evaporation of ammonia from a single droplet of urea water solution is investigated computationally by convection-diffusion controlled model. The conversion to ammonia due to thermolysis of urea water droplets is measured downstream at different sections using finite rate/eddy dissipation model. In this paper, the mixer installed at the upstream enhances the distribution of ammonia over the entire domain which is calculated for different time steps. Calculations are made within the respective duration such that the complete decomposition of urea is possible at a much shorter residence time. Downloads 326
##### 14382 Pressure-Controlled Dynamic Equations of the PFC Model: A Mathematical Formulation

Authors: Jatupon Em-Udom, Nirand Pisutha-Arnond

Abstract:

The phase-field-crystal, PFC, approach is a density-functional-type material model with an atomic resolution on a diffusive timescale. Spatially, the model incorporates periodic nature of crystal lattices and can naturally exhibit elasticity, plasticity and crystal defects such as grain boundaries and dislocations. Temporally, the model operates on a diffusive timescale which bypasses the need to resolve prohibitively small atomic-vibration time steps. The PFC model has been used to study many material phenomena such as grain growth, elastic and plastic deformations and solid-solid phase transformations. In this study, the pressure-controlled dynamic equation for the PFC model was developed to simulate a single-component system under externally applied pressure; these coupled equations are important for studies of deformable systems such as those under constant pressure. The formulation is based on the non-equilibrium thermodynamics and the thermodynamics of crystalline solids. To obtain the equations, the entropy variation around the equilibrium point was derived. Then the resulting driving forces and flux around the equilibrium were obtained and rewritten as conventional thermodynamic quantities. These dynamics equations are different from the recently-proposed equations; the equations in this study should provide more rigorous descriptions of the system dynamics under externally applied pressure. Downloads 235
##### 14381 Mathematical Modeling of Switching Processes in Magnetically Controlled MEMS Switches

Abstract:

The operating principle of magnetically controlled microelectromechanical system (MEMS) switches is based on controlling the beam movement under the influence of a magnetic field. Currently, there is a MEMS switch design with a flexible ferromagnetic electrode in the form of a fixed-terminal beam, with an electrode fastened on a straight or cranked anchor. The basic performance characteristics of magnetically controlled MEMS switches (service life, sensitivity, contact resistance, fast response) are largely determined by the flexible electrode design. To ensure the stable and controlled motion of the flexible electrode, it is necessary to provide the optimal design of a flexible electrode. Downloads 72
##### 14380 Improving Ride Comfort of a Bus Using Fuzzy Logic Controlled Suspension

Authors: Mujde Turkkan, Nurkan Yagiz

Abstract:

In this study an active controller is presented for vibration suppression of a full-bus model. The bus is modelled having seven degrees of freedom. Using the achieved model via Lagrange Equations the system equations of motion are derived. The suspensions of the bus model include air springs with two auxiliary chambers are used. Fuzzy logic controller is used to improve the ride comfort. The numerical results, verifies that the presented fuzzy logic controller improves the ride comfort.

Keywords: ride comfort, air spring, bus, fuzzy logic controller

##### 14379 Application of Fractional Model Predictive Control to Thermal System

Authors: Aymen Rhouma, Khaled Hcheichi, Sami Hafsi

Abstract:

The article presents an application of Fractional Model Predictive Control (FMPC) to a fractional order thermal system using Controlled Auto Regressive Integrated Moving Average (CARIMA) model obtained by discretization of a continuous fractional differential equation. Moreover, the output deviation approach is exploited to design the K -step ahead output predictor, and the corresponding control law is obtained by solving a quadratic cost function. Experiment results onto a thermal system are presented to emphasize the performances and the effectiveness of the proposed predictive controller. Downloads 270
##### 14378 Modeling and Control Design of a Centralized Adaptive Cruise Control System

Authors: Markus Mazzola, Gunther Schaaf

Abstract:

A vehicle driving with an Adaptive Cruise Control System (ACC) is usually controlled decentrally, based on the information of radar systems and in some publications based on C2X-Communication (CACC) to guarantee stable platoons. In this paper, we present a Model Predictive Control (MPC) design of a centralized, server-based ACC-System, whereby the vehicular platoon is modeled and controlled as a whole. It is then proven that the proposed MPC design guarantees asymptotic stability and hence string stability of the platoon. The Networked MPC design is chosen to be able to integrate system constraints optimally as well as to reduce the effects of communication delay and packet loss. The performance of the proposed controller is then simulated and analyzed in an LTE communication scenario using the LTE/EPC Network Simulator LENA, which is based on the ns-3 network simulator. Downloads 406
##### 14377 A Data-Mining Model for Protection of FACTS-Based Transmission Line

Authors: Ashok Kalagura

Abstract:

This paper presents a data-mining model for fault-zone identification of ﬂexible AC transmission systems (FACTS)-based transmission line including a thyristor-controlled series compensator (TCSC) and unified power-ﬂow controller (UPFC), using ensemble decision trees. Given the randomness in the ensemble of decision trees stacked inside the random forests model, it provides an effective decision on the fault-zone identification. Half-cycle post-fault current and voltage samples from the fault inception are used as an input vector against target output ‘1’ for the fault after TCSC/UPFC and ‘1’ for the fault before TCSC/UPFC for fault-zone identification. The algorithm is tested on simulated fault data with wide variations in operating parameters of the power system network, including noisy environment providing a reliability measure of 99% with faster response time (3/4th cycle from fault inception). The results of the presented approach using the RF model indicate the reliable identification of the fault zone in FACTS-based transmission lines. Downloads 340
##### 14376 Development of a Human Vibration Model Considering Muscles and Stiffness of Intervertebral Discs

Authors: Young Nam Jo, Moon Jeong Kang, Hong Hee Yoo

Abstract:

Most human vibration models have been modeled as a multibody system consisting of some rigid bodies and spring-dampers. These models are developed for certain posture and conditions. So, the models cannot be used in vibration analysis in various posture and conditions. The purpose of this study is to develop a human vibration model that represent human vibration characteristics under various conditions by employing a musculoskeletal model. To do this, the human vibration model is developed based on biomechanical models. In addition, muscle models are employed instead of spring-dampers. Activations of muscles are controlled by PD controller to maintain body posture under vertical vibration is applied. Each gain value of the controller is obtained to minimize the difference of apparent mass and acceleration transmissibility between experim ent and analysis by using an optimization method. Downloads 314
##### 14375 Conditions for Model Matching of Switched Asynchronous Sequential Machines with Output Feedback

Authors: Jung–Min Yang

Abstract:

Solvability of the model matching problem for input/output switched asynchronous sequential machines is discussed in this paper. The control objective is to determine the existence condition and design algorithm for a corrective controller that can match the stable-state behavior of the closed-loop system to that of a reference model. Switching operations and correction procedures are incorporated using output feedback so that the controlled switched machine can show the desired input/output behavior. A matrix expression is presented to address reachability of switched asynchronous sequential machines with output equivalence with respect to a model. The presented reachability condition for the controller design is validated in a simple example. Downloads 246
##### 14374 An Approach on Robust Multi Inversion of a Nonlinear Model for an Omni-Directional Mobile

Abstract:

In this paper, a nonlinear controller design for an omnidirectional mobile is presented. The robot controller consists of an inner-loop controller and an outer-loop controller, the ﬁrst is designed using state feedback (robust allocation) and the second controller is designed based on Robust Multi Inversion (RMI) approach. The objective of RMI controller is rendering the robust inversion of the dynamic, when the model is affected by uncertainties. A model nonlinear MIMO of an omni-directional robot (small-league of Robocup) is used to simulate the RMI approach. The parameters of linear and nonlinear model are varied to cause modelling uncertainties among the model and the real model (real system) generating an error in inner-loop controller signal that must be compensated by RMI controller. The simulation test results show that the RMI is capable of compensating the uncertainties and keep the system stable and controlled under uncertainties. Downloads 441
##### 14373 Preparation and Evaluation of Multiple Unit Tablets of Aceclofenac

Abstract:

The present research is aimed at fabrication of multiple-unit controlled-release tablet formulation of aceclofenac by employing acrylic polymers as the release controlling excipients for drug multi-particulates to achieve the desired objectives of maintaining the same controlled release characteristics as that prior to their compression into tablet. Various manufacturers are successfully manufacturing and marketing aceclofenac controlled release tablet by applying directly coating materials on the tablet. The basic idea behind development of such formulations was to employ aqueous acrylics polymers dispersion as an alternative to the existing approaches, wherein the forces of compression may cause twist of drug pellets, but do not have adverse effects on the drug release properties. Thus, the study was undertaken to illustrate manufacturing of controlled release aceclofenac multiple-unit tablet formulation. Downloads 317
##### 14372 Development of Nanostructrued Hydrogel for Spatial and Temporal Controlled Release of Active Compounds

Authors: Shaker Alsharif, Xavier Banquy

Abstract:

Controlled drug delivery technology represents one of the most rapidly advancing areas of science in which chemists and chemical engineers are contributing to human health care. Such delivery systems provide numerous advantages compared to conventional dosage forms including improved efficacy, and improved patient compliance and convenience. Such systems often use synthetic polymers as carriers for the drugs. As a result, treatments that would not otherwise be possible are now in conventional use. The role of bilayered vesicles as efficient carriers for drugs, vaccines, diagnostic agents and other bioactive agents have led to a rapid advancement in the liposomal drug delivery system. Moreover, the site avoidance and site-specific drug targeting therapy could be achieved by formulating a liposomal product, so as to reduce the cytotoxicity of many potent therapeutic agents. Our project focuses on developing and building hydrogel with nanoinclusion of liposomes loaded with active compounds such as proteins and growth factors able to release them in a controlled fashion. In order to achieve that, we synthesize several liposomes of two different phospholipids concentrations encapsulating model drug. Then, formulating hydrogel with specific mechanical properties embedding the liposomes to manage the release of active compound. Downloads 345
##### 14371 Estimation of Respiratory Parameters in Pressure Controlled Ventilation System with Double Lungs on Secretion Clearance

Authors: Qian Zhang, Dongkai Shen, Yan Shi

Abstract:

A new mechanical ventilator with automatic secretion clearance function can improve the secretion clearance safely and efficiently. However, in recent modeling studies on various mechanical ventilators, it was considered that human had one lung, and the coupling effect of double lungs was never illustrated. In this paper, to expound the coupling effect of double lungs, a mathematical model of a ventilation system of a bi-level positive airway pressure (BiPAP) controlled ventilator with secretion clearance was set up. Moreover, an experimental study about the mechanical ventilation system of double lungs on BiPAP ventilator was conducted to verify the mathematical model. Finally, the coupling effect of double lungs of the mathematical ventilation was studied by simulation and orthogonal experimental design. This paper adds to previous studies and can be referred to optimization methods in medical researches. Downloads 434
##### 14370 A New Nonlinear State-Space Model and Its Application

Authors: Abdullah Eqal Al Mazrooei

Abstract:

In this work, a new nonlinear model will be introduced. The model is in the state-space form. The nonlinearity of this model is in the state equation where the state vector is multiplied by its self. This technique makes our model generalizes many famous models as Lotka-Volterra model and Lorenz model which have many applications in the real life. We will apply our new model to estimate the wind speed by using a new nonlinear estimator which suitable to work with our model. Downloads 527
##### 14369 Experimental Study of Upsetting and Die Forging with Controlled Impact

Authors: T. Penchev, D. Karastoyanov

Abstract:

The results from experimental research of deformation by upsetting and die forging of lead specimens wit controlled impact are presented. Laboratory setup for conducting the investigations, which uses cold rocket engine operated with compressed air, is described. The results show that when using controlled impact is achieving greater plastic deformation and consumes less impact energy than at ordinary impact deformation process. Downloads 278
##### 14368 Implicit Force Control of a Position Controlled Robot - A Comparison with Explicit Algorithms

Authors: Alexander Winkler, Jozef Suchý

Abstract:

This paper investigates simple implicit force control algorithms realizable with industrial robots. A lot of approaches already published are difficult to implement in commercial robot controllers, because the access to the robot joint torques is necessary or the complete dynamic model of the manipulator is used. In the past we already deal with explicit force control of a position controlled robot. Well known schemes of implicit force control are stiffness control, damping control and impedance control. Using such algorithms the contact force cannot be set directly. It is further the result of controller impedance, environment impedance and the commanded robot motion/position. The relationships of these properties are worked out in this paper in detail for the chosen implicit approaches. They have been adapted to be implementable on a position controlled robot. The behaviors of stiffness control and damping control are verified by practical experiments. For this purpose a suitable test bed was configured. Using the full mechanical impedance within the controller structure will not be practical in the case when the robot is in physical contact with the environment. This fact will be verified by simulation. Downloads 390
##### 14367 Elasto-Plastic Behavior of Rock during Temperature Drop

Authors: N. Reppas, Y. L. Gui, B. Wetenhall, C. T. Davie, J. Ma

Abstract:

A theoretical constitutive model describing the stress-strain behavior of rock subjected to different confining pressures is presented. A bounding surface plastic model with hardening effects is proposed which includes the effect of temperature drop. The bounding surface is based on a mapping rule and the temperature effect on rock is controlled by Poisson’s ratio. Validation of the results against available experimental data is also presented. The relation of deviatoric stress and axial strain is illustrated at different temperatures to analyze the effect of temperature decrease in terms of stiffness of the material. Downloads 43
##### 14366 Realization of a Temperature Based Automatic Controlled Domestic Electric Boiling System

Authors: Shengqi Yu, Jinwei Zhao

Abstract:

This paper presents a kind of analog circuit based temperature control system, which is mainly composed by threshold control signal circuit, synchronization signal circuit and trigger pulse circuit. Firstly, the temperature feedback signal function is realized by temperature sensor TS503F3950E. Secondly, the main control circuit forms the cycle controlled pulse signal to control the thyristor switching model. Finally two reverse paralleled thyristors regulate the output power by their switching state. In the consequence, this is a modernized and energy-saving domestic electric heating system. Downloads 381
##### 14365 Model Free Terminal Sliding Mode with Gravity Compensation: Application to an Exoskeleton-Upper Limb System

Abstract:

This paper deals with a robust model free terminal sliding mode with gravity compensation approach used to control an exoskeleton-upper limb system. The considered system is a 2-DoF robot in interaction with an upper limb used for rehabilitation. The aim of this paper is to control the flexion/extension movement of the shoulder and the elbow joints in presence of matched disturbances. In the first part, we present the exoskeleton-upper limb system modeling. Then, we controlled the considered system by the model free terminal sliding mode with gravity compensation. A stability study is realized. To prove the controller performance, a robustness analysis was needed. Simulation results are provided to confirm the robustness of the gravity compensation combined with to the Model free terminal sliding mode in presence of uncertainties. Downloads 36
##### 14364 Hypergraph for System of Systems modeling

Authors: Haffaf Hafid

Abstract:

Hypergraphs, after being used to model the structural organization of System of Sytems (SoS) at macroscopic level, has recent trends towards generalizing this powerful representation at different stages of complex system modelling. In this paper, we first describe different applications of hypergraph theory, and step by step, introduce multilevel modeling of SoS by means of integrating Constraint Programming Langages (CSP) dealing with engineering system reconfiguration strategy. As an application, we give an A.C.T Terminal controlled by a set of Intelligent Automated Vehicle. Downloads 398
##### 14363 Numerical Investigation of the Electromagnetic Common Rail Injector Characteristics

Abstract:

The paper describes the modeling of a fuel injector for common rail systems. A one-dimensional model of a solenoid-valve-controlled injector with Valve Closes Orifice (VCO) spray was modelled in the AVL Hydsim. This model shows the dynamic phenomena that occur in the injector. The accuracy of the calibration, based on a regulation of the parameters of the control valve and the nozzle needle lift, was verified by comparing the numerical results of injector flow rate. Our model is capable of a precise simulation of injector operating parameters in relation to injection time and fuel pressure in a fuel rail. As a result, there were made characteristics of the injector flow rate and backflow.

Keywords: common rail, diesel engine, fuel injector, modeling