Etude de levaporation des gouttes dans un spray (nghiên cứu sự bay hơi của hạt trong quá trình phun)

L’ECOLE DOCTORALE MEGA L’ECOLE CENTRALE DE LYON oOo - DIPLOME D’ETUDES APPROFONDIES « THERMIQUE ET ENERGETIQUE » RAPPORT DE STAGE Sujet : ETUDE DE L’EVAPORATION DES GOUTTES DANS UN SPRAY Directeur de Recherche : Mr Jean-Marc VIGNON Etudiant : NGUYEN LE Duy Khai - Juillet 2003 - Study of evaporation of droplets in a spray Mr Jean-Marc VIGNON ACKNOWLEDGMENTS This work was carried out at the Laboratory of Mechanic of the Fluids and Acoustics (LMFA) of the Ecole Centrale de Lyon, from 4/2003 to 6/2003 I would like to thank Professor Denis JEANDEL and Professor Michel LANCE for having agreed to accommodate me in this laboratory I greatly appreciate Professor Jean –Marc VIGNON as my Director of Research I am very happy to work under his directions, whose valuable counsels and idea help me complete well this work I am also grateful to Professor Maurice BRUN for helping me so much in both study and living, particularly after my moving to France Without his help, I would not be able to settle down quickly I would like to thank Professor PHAM Xuan Mai, my Dean of Transportation Faculty, HCM University of Technology He continuously gave me support and encouragement me to advance in my study I would like to deeply thank all the personnel of the laboratory for their sympathy and helps that they bring to me when I work there The financial support that provided by the region Rhone-Alpes is gratefully acknowledgment I would also like to express my sincere gratitude to all my friends who are always beside me in the difficult time of study as well as of life I also make a point of thanking all the people who took time to read this dissertation NGUYEN LE Duy Khai - Dissertation of DEA – 2002-2003 i Study of evaporation of droplets in a spray Mr Jean-Marc VIGNON TABLE OF CONTENTS Acknowledgments i Table of contents ii Nomenclature iv Part – Introduction Part – General study of turbulent free jets 2.1 General considerations 2.2 Equations of motion 2.3 The integral momentum equation 2.4 Estimate the increase in width and the decrease in velocity of the jet 2.5 Dimensional considerations 2.6 Experimental results – The value of constants C1, C2, C3 2.7 Conclusion Part – The one-dimensional integral model of Tamanini 11 3.1 Introduction 11 3.2 Mean flow equations 11 3.3 Turbulence quantities 12 Part – Adaptation model of Peters with Tamanini’s formulations for turbulent jet with the influence of uniform surrounding wind 14 4.1 Introduction 14 4.2 Background equations 14 4.3 Building equations for turbulent jet with the influence of cross-wind 15 4.4 Summary 19 Part – Study the drag force and vaporization of discrete liquid droplet moving through the gaseous medium 21 5.1 Introduction 21 5.2 The aerodynamic force on the object moving through the fluid 21 5.2.1 General consideration of drag force 21 5.2.2 Drag force in the case of spherical drop 24 NGUYEN LE Duy Khai - Dissertation of DEA – 2002-2003 ii Study of evaporation of droplets in a spray Mr Jean-Marc VIGNON 5.3 Vaporization of discrete droplet 25 5.3.1 Quasi-stationary theory of the vaporization of a spherical droplet 25 5.3.2 The d2 law for the vaporizing time 31 5.3.3 The d2 law in case of effect of relative motion 32 Part – Establish a model to calculate the penetration length and the spray characteristics with the distance in steady flow 33 6.1 Introduction 33 6.2 Establish the governing equations 33 6.2.1 Equation of mass balance 33 6.2.2 Equation of dynamic balance 35 6.2.3 Summary 36 6.3 Results and discussion 38 6.3.1 Application for LPG 38 6.3.2 Influence of initial droplet diameter 41 6.3.3 Influence of injection velocity 43 6.3.4 Application for Diesel fuel 45 General conclusion 47 References 48 Annex…… 51 Annex A – Fourth-order Runge-Kutta method 51 Annex B – Programming in Matlab language 53 B.1 Program LPG 53 B.2 Program LPG-D 57 B.3 Program LPG-U 61 B.4 Program Diesel 65 B.5 Program SysKutta4 69 B.6 Program Fsys 71 NGUYEN LE Duy Khai - Dissertation of DEA – 2002-2003 iii Study of evaporation of droplets in a spray Mr Jean-Marc VIGNON NOMENCLATURE Latin character b value of r where U = Um/2 m B SPALDING transfer parameter - ck constant of equation (3-1) - cρ constant of equation (3-7) - cε1, cε2, cε3 constant of equation (3-9) - C1,C2,C3 constant of equation (2-7, 2-8, 2-9) - Cd discharge coefficient - Cp heat capacity J/kg°K CD drag coefficient - d0 initial droplet diameter m dd droplet diameter m dnoz nozzle diameter m D mass diffusivity m²/s Dk total dissipation of k kg/ms3 Dv vapor mass diffusivity m²/s FD drag force N g acceleration of gravity m/s2 Gk production of k due to buoyancy kg/ms3 k turbulent kinetic energy m2/s2 K evaporation coefficient m2/s L characteristic length m Lve effective latent heat J/kg m0 initial droplet mass kg md droplet mass kg & m mass flow kg/s &0 m initial mass flow kg/s &λ m mass flow of liquid traverse a cross-section of jet kg/s &′∞ m mass of ambient air entrained per unit height kg/ms NGUYEN LE Duy Khai - Dissertation of DEA – 2002-2003 iv Study of evaporation of droplets in a spray Mr Jean-Marc VIGNON M momentum flux kgm/s2 M0 initial momentum flux (at the nozzle exit) kgm/s2 Mg molecular weight of ambient air g/mol Mv molecular weight of fuel vapor g/mol & N number of injected droplets per unit time -/s p pressure kg/m² pg ambient pressure kg/m² pinj injection pressure N/m2 Pk production of k due to shear kg/ms3 Q flow rate m3/s r radial coordinate m rd droplet radius m rnoz nozzle radius m R radius of the jet, or distance from the droplet center m s axial coordinate m SMD0 Sauter mean diameter at the nozzle exit m tvap vaporizing time s T temperature °K Tb droplet boiling temperature °K Td surface droplet temperature °K Tg ambient temperature °K u’ fluctuating velocity in the axial directions m/s U mean axial velocity m/s U0 exit velocity m/s Ud axial velocity of droplet m/s Ug radial velocity of vaporizing gas m/s Um mean axial velocity on the symmetric axis m/s Urel relative velocity m/s U∞ co-flowing stream velocity m/s ˆ U area-average axial velocity m/s v’ fluctuating velocity in the radial directions m/s V mean radial velocity m/s W momentum flux kg.m/s² NGUYEN LE Duy Khai - Dissertation of DEA – 2002-2003 v Study of evaporation of droplets in a spray Mr Jean-Marc VIGNON Wd droplets momentum flux kg.m/s² Wg gas momentum flux kg.m/s² YK fuel vapor mass fraction - YK,d fuel vapor mass fraction at the droplet surface - Greek character α half spray angle of the jet deg β spreading coefficient - ε dissipation rate of turbulence m2/s3 θ local angle of the jet axis to the horizontal deg λ thermal conductivity W/m°K μ viscosity kg/ms μλ liquid viscosity kg/ms μv vaporizing gas viscosity kg/ms μτ turbulent viscosity kg/ms νt turbulent kinematic viscosity m²/s ρ density kg/m3 ρλ liquid density kg/m3 ρg ambient gas density kg/m3 ρv vaporizing gas density kg/m3 Non dimensional number Le LEWIS number - Pr PRANDTL number - Re REYNOLD number - Sc SCHMIDT number - Sh SHERWOOD number - NGUYEN LE Duy Khai - Dissertation of DEA – 2002-2003 vi Study of evaporation of droplets in a spray Mr Jean-Marc VIGNON PART INTRODUCTION A spray is a mechanically produced dispersion of droplets with a sufficient momentum to penetrate the surrounding gaseous medium Sprays are involved in many practical applications: in the process industries (spray drying, spray cooling), in treatment applications (humidification, gas scrubbing), in coating applications (spray painting, crop spraying), in medicinal and printing applications, and especially in spray combustions (burners, furnaces, liquid-fuel rockets, aircraft gas turbines, diesel engines and gasoline injection in spark ignition engines), which is the mayor source of energy nowadays In the combustion chamber of the engine, the fuel to be sprayed is injected at a high velocity through an injection nozzle, where the potential pressure energy is transferred to the kinetic energy of the fuel Typically, sprays are composed of a multitude of liquid droplets of different sizes and velocities, whose values change according to the operating conditions Atomization is the process in which the injected liquid is broken up into droplets The combustion properties of the engine are significantly influenced by the atomization processes into a combustion chamber Therefore, it is important to understand atomization mechanism and the spray properties of the liquid jet injected through a nozzle There are a lot of physical models describing liquid fuel atomization At low exit velocity and low injection pressure, the fuel remains in the liquid phase up to distance S bu (break-up length) from the injection nozzle Under the aerodynamic instabilities, it splits up into numerous droplets at different sizes But in the “common rail injection system”, while the injection pressure is up to 1700 bar, the fuel leaves the nozzle hole in the form of a two-phase flow rather than as pure liquid.1 In most of advanced computer codes such as KIVA-II, the Lagrangian modeling is used The liquid fuel is injected as discrete parcels of drops or “blops” Then they undergo the break-up process2 Another simpler approach assumes that the spray is already atomized at the nozzle exit (the Assumption of a Fully Atomized Jet – AFAJ)3 This method is easier and older than NGUYEN LE Duy Khai - Dissertation of DEA – 2002-2003 Study of evaporation of droplets in a spray Mr Jean-Marc VIGNON break-up modeling, but it does not represent the real physical processes and can not be universal Droplets in the spray are influenced by vaporization and drag force while they move through the ambient gaseous medium Many investigations have been carried out to set up the influence of operating conditions such as fuel characteristics, injection pressure, ambient gas properties, geometric shapes of the nozzle 4-8 etc on the spray characteristics: droplet size (Sauter mean diameter – SMD), droplet size distribution, droplet velocities, spray angle, spray tip penetration, air entrainment All results give an overview of the spray characteristics versus lifetime of the droplets, but no research presents usual relations between spray behaviors and axial distance s from the nozzle exit of the droplet The present dissertation proposes a model for the calculation some specific spray parameters versus axial distance s: droplet velocity, droplet diameter, droplet mass, droplet momentum flux as well as gas velocity, gas momentum flux, gas mass flux Then the spray penetration and the spray angle are deduced The model accounts for the droplet vaporization and gas aerodynamic force The remainder of this dissertation is divided into sections The first section discusses the general turbulent free jet The second section presents a one-dimensional integral model of Tamanini, which uses the approach to the modeling of turbulent processes introduced by the k-ε-g technique The third section discusses the adaptation model of Peters with Tamanini’s formulations for turbulent jet with the influence of uniform surrounding wind The next section studies the drag force and vaporization of discrete liquid droplet moving through the gaseous medium The last presents a model to calculate the penetration length of the spray and the other spray characteristics with the axial coordinate This last part tries to put together the approaches of the previous chapters, with the aim to have comprehensive description and mechanism The result is a one-dimensional approach taking into account the simplified model of air entrainement NGUYEN LE Duy Khai - Dissertation of DEA – 2002-2003 Study of evaporation of droplets in a spray Mr Jean-Marc VIGNON PART GENERAL STUDY OF TURBULENT FREE JETS A jet is a source of fluid mass and momentum in a fluid reservoir A plume is a buoyant flow arising from a source of heated, and therefore less dense, fluid in a reservoir Fluid dynamic jets and plumes constitute omnipresent phenomena in nature The jets of technology dominate our lives, from propelling the aircraft which moves us across continents to providing the simple air hoses of the machine shop, from the stacks which spew the waste products of industry to the diffuser arrays which disperse effluent into our streams and rivers A flow is called laminar if in this flow, all fluid elements move deterministically along distinct and traceable streamlines A laminar flow is characterized by laminar of fluid which smoothly slides by each other At sufficiency high Reynolds numbers, laminar flow must become unstable and evolves into turbulent flow Turbulence is a phenomenon in which the chaotic mixing of viscous fluid flows induced by large-scale eddies which spawn ever smaller-scale until the turbulent energy is dissipated by viscosity by the smallest eddies Turbulent flow will be called free turbulent flow if it is not confined by solid walls H SCHLICHTING9 distinguished into three kinds of turbulent free flow (Fig 2.1): Fig.2.1 03 kinds of free turbulent flow : a) Jet boundary b) Free jet c) Wake ‰ Jet boundary (Fig 2.1.a): Occurs between two streams which move at different speeds in the same general direction Such a surface of discontinuity in the velocity NGUYEN LE Duy Khai - Dissertation of DEA – 2002-2003 Study of evaporation of droplets in a spray Mr Jean-Marc VIGNON B.2 PROGRAM LPG-D %********** Program LPG-D ************** % Solve the system of differential equations to calculate % the penetration length of droplet and the changing of % droplet diameter with the distance in steady flow % when the initial droplet diameter is changing clear all tic global K rov muv B Dv rog mug beta N % The initial conditions: % - Liquid (fuel) : LPG (Liquid Petroleum Gas: 50%C3H8 + 50%C4H10), td = 25°C % - Ambient gas : Air, pg = bar, tg = 65°C % - Nozzle : dnoz = 0.6mm, pinj = 45 bar rol = 570; Td = 273 + 25 ; Tg = 273 + 65 ; pg = 1*10^5 ; dnoz = 0.6*10^-3 ; R = 8.314 ; Mg = 29; % liquid density (kg/m3) % droplet temperature (°K) % gas temperature (°K) % gas pressure (N/m2) % nozzle diameter (m) % universal gas constant (J/mol°K) % gas molecular mass (g/mol) % Calculation the initial necessary parameters: mug = (3.6e-8*Tg^3 - 6.95e-5*Tg^2 + 8.05e-2*Tg - 0.30)*10^-6; muv = (-5e-6*Tg^2 + 2.95e-2*Tg - 0.1)*10^-6; rog = 10^-3*(pg*Mg)/(R*Tg); rov = 1.73*rog; % gas viscosity at Tg (kg/ms) % vaporizing viscosity (kg/ms) % gas density (kg/m3) % vaporizing density (kg/m3) Cp = 2.449; Lv = 365.2; B = Cp*(Tg - Td)/Lv; % specific heat of LPG (kJ/kg°K) % latent heat of vaporization (kJ/kg) % Spalding transfer parameter B Dv = muv/rov; K = 8*Dv*rov*log(1+B)/rol; % vapor mass diffusivity (m2/s) % evaporation coefficient (m2/s) % Calculation the initial conditions: vald = [10e-6 15e-6 20e-6 25e-6 30e-6]; valsf = [0.05 0.09 0.13 0.17 0.21]; c = ['r' 'b' 'g' 'm' 'k']; NGUYEN LE Duy Khai - Dissertation of DEA – 2002-2003 57 Study of evaporation of droplets in a spray Mr Jean-Marc VIGNON for i = 1:length(vald) t0 = 0; d0 = vald(i); m0 = pi*d0^3*rol/6; U0 = 19; Ug0 = 0.1; R0 = 0.001; mf0 = pi*R0^2*rog*Ug0; md0 = pi*dnoz^2*rol*U0/4; Wd0 = md0*U0; Wg0 = mf0*Ug0; % initial time (s) % initial droplet diameter (m) % initial droplet mass (kg) % initial droplet velocity (m/s) % initial gas velocity (m/s) % initial radius (m) % initial gas mass flux (kg/s) % initial droplet mass flux (kg/s) % initial droplet momentum flux (kg.m/s2) % initial gas momentum flux (kg.m/s2) N = md0/m0; alpha = 19; beta = tan(alpha); tvap = d0^2/K; % number of droplets / second % half spray angle (deg) % transfer parameter % evaporation time (s) % Solve the system of differential equations by 4-order Runge-Kutta method s0 = 0; % initial axial coordinate hdebut = 0.00000001; % step of s in the first loop sfdebut = 35000*hdebut; % final value of s in the first loop i0 = 1; [s,d,m,U,Wd,Wg,mf,Ug,t] = syskutta4('fsys',i0,s0,d0,m0,U0,Wd0,Wg0,mf0,Ug0,t0,sfdebut,hdebut); s1st = s(i0:floor((sfdebut-s0)/hdebut)+2); d1st = d(i0:floor((sfdebut-s0)/hdebut)+2); m1st = m(i0:floor((sfdebut-s0)/hdebut)+2); U1st = U(i0:floor((sfdebut-s0)/hdebut)+2); Wd1st = Wd(i0:floor((sfdebut-s0)/hdebut)+2); Wg1st = Wg(i0:floor((sfdebut-s0)/hdebut)+2); mf1st = mf(i0:floor((sfdebut-s0)/hdebut)+2); Ug1st = Ug(i0:floor((sfdebut-s0)/hdebut)+2); t1st = t(i0:floor((sfdebut-s0)/hdebut)+2); ndebut = 2+floor((sfdebut-s0)/hdebut); sf = valsf(i); % final axial coordinate h = 0.0001; % step of s in the second loop [s,d,m,U,Wd,Wg,mf,Ug,t] = syskutta4('fsys',ndebut,sfdebut,d(ndebut),m(ndebut), U(ndebut),Wd(ndebut),Wg(ndebut),mf(ndebut),Ug(ndebut),t(ndebut),sf,h); n = length(find(imag(d)==0 & d>=0)); s = [s1st s(ndebut+1:n)]; d = [d1st d(ndebut+1:n)]; m = [m1st m(ndebut+1:n)]; U = [U1st U(ndebut+1:n)]; Wd = [Wd1st Wd(ndebut+1:n)]; NGUYEN LE Duy Khai - Dissertation of DEA – 2002-2003 58 Study of evaporation of droplets in a spray Mr Jean-Marc VIGNON Wg = [Wg1st Wg(ndebut+1:n)]; mf = [mf1st mf(ndebut+1:n)]; Ug = [Ug1st Ug(ndebut+1:n)]; t = [t1st t(ndebut+1:n)];; R = mf./sqrt(pi*rog*Wg); % spray radius (m) md = N*m; % droplet mass flux (kg/s) ma = mf - md; % air entrainement per second (kg/s) % Draw the results figure(1) plot(s,d,c(i)), grid xlabel('Axial distance s (m)') ylabel('d (m)') title('Change of droplet diameter d (m) with s (m)') hold on figure(2) plot(s,m,c(i)),grid xlabel('Axial distance s (m)') ylabel('m (kg)') title('Change of droplet mass m (kg) ') hold on figure(3) plot(s,U,c(i)),grid xlabel('Axial distance s (m)') ylabel('U (m/s)') title('Change of droplet velocity U (m/s) ') hold on figure(4) plot(s,Wd,c(i)),grid xlabel('Axial distance s (m)') ylabel('Wd (kgm/s2)') title('Change of droplet momentum flux Wd (kgm/s2)') hold on figure(5) plot(s,Wg,c(i)),grid xlabel('Axial distance s (m)') ylabel('Wg (kgm/s2)') title('Change of gas momentum flux Wg (kgm/s2)') hold on figure(6) plot(s,mf,c(i),s,md,c(i)),grid xlabel('Axial distance s (m)') ylabel('mf (kg/s)') NGUYEN LE Duy Khai - Dissertation of DEA – 2002-2003 59 Study of evaporation of droplets in a spray Mr Jean-Marc VIGNON title('Change of gas mass flux mf (kg/s)') hold on figure(7) plot(s,Ug,c(i)),grid xlabel('Axial distance s (m)') ylabel('Ug (m/s)') title('Change of gas velocity Ug (m/s)') hold on figure(8) plot(s,R,c(i)),grid xlabel('Axial distance s (m)') ylabel('R (m)') title('Change of spray radius R (m)') hold on figure(9) plot(t,s,c(i)), grid ylabel('Axial distance s (m)') xlabel('t (s)') title('Change of distance s(t)') hold on, end toc hold off NGUYEN LE Duy Khai - Dissertation of DEA – 2002-2003 60 Study of evaporation of droplets in a spray Mr Jean-Marc VIGNON B.3 PROGRAM LPG-U %********** Program LPG-U ************** % Solve the system of differential equations to calculate % the penetration length of droplet and the changing of % droplet diameter with the distance in steady flow % when the injection velocity is changing clear all tic global K rov muv B Dv rog mug beta N % The initial conditions: % - Liquid (fuel) : LPG (Liquid Petroleum Gas: 50%C3H8 + 50%C4H10), td = 25°C % - Ambient gas : Air, pg = bar, tg = 65°C % - Nozzle : dnoz = 0.6mm, pinj = 45 bar rol = 570; Td = 273 + 25 ; Tg = 273 + 65 ; pg = 1*10^5 ; dnoz = 0.6*10^-3 ; R = 8.314 ; Mg = 29; % liquid density (kg/m3) % droplet temperature (°K) % gas temperature (°K) % gas pressure (N/m2) % nozzle diameter (m) % universal gas constant (J/mol°K) % gas molecular mass (g/mol) % Calculation the initial necessary parameters: mug = (3.6e-8*Tg^3 - 6.95e-5*Tg^2 % gas viscosity at Tg (kg/ms) + 8.05e-2*Tg - 0.30)*10^-6; muv = (-5e-6*Tg^2 + 2.95e-2*Tg - 0.1)*10^-6; % vaporizing viscosity (kg/ms) rog = 10^-3*(pg*Mg)/(R*Tg); % gas density (kg/m3) rov = 1.73*rog; % vaporizing density (kg/m3) Cp = 2.449; Lv = 365.2; B = Cp*(Tg - Td)/Lv; % specific heat of LPG (kJ/kg°K) % latent heat of vaporization (kJ/kg) % Spalding transfer parameter B Dv = muv/rov; K = 8*Dv*rov*log(1+B)/rol; % vapor mass diffusivity (m2/s) % evaporation coefficient (m2/s) % Calculation the initial conditions: valU = [10 20 30 40 50]; valsf = [0.15 0.22 0.27 0.32 0.35]; c = ['r' 'b' 'g' 'm' 'k']; NGUYEN LE Duy Khai - Dissertation of DEA – 2002-2003 61 Study of evaporation of droplets in a spray Mr Jean-Marc VIGNON for i = 1:length(valU) t0 = 0; % initial time (s) d0 = 30*10^-6; % initial droplet diameter (m) m0 = pi*d0^3*rol/6; % initial droplet mass (kg) U0 = valU(i); % initial droplet velocity (m/s) Ug0 = 0.1; % initial gas velocity (m/s) R0 = 0.001; % initial radius (m) mf0 = pi*R0^2*rog*Ug0; % initial gas mass flux (kg/s) md0 = pi*dnoz^2*rol*U0/4; % initial droplet mass flux (kg/s) Wd0 = md0*U0; % initial droplet momentum flux (kg.m/s2) Wg0 = mf0*Ug0; % initial gas momentum flux (kg.m/s2) N = md0/m0; alpha = 19; beta = tan(alpha); tvap = d0^2/K; % number of droplets / second % half spray angle (deg) % transfer parameter % evaporation time (s) % Solve the system of differential equations by 4-order Runge-Kutta method s0 = 0; % initial axial coordinate hdebut = 0.00000001; % step of s in the first loop sfdebut = 35000*hdebut; % final value of s in the first loop i0 = 1; [s,d,m,U,Wd,Wg,mf,Ug,t] = syskutta4('fsys',i0,s0,d0,m0,U0,Wd0,Wg0,mf0,Ug0,t0,sfdebut,hdebut); s1st = s(i0:floor((sfdebut-s0)/hdebut)+2); d1st = d(i0:floor((sfdebut-s0)/hdebut)+2); m1st = m(i0:floor((sfdebut-s0)/hdebut)+2); U1st = U(i0:floor((sfdebut-s0)/hdebut)+2); Wd1st = Wd(i0:floor((sfdebut-s0)/hdebut)+2); Wg1st = Wg(i0:floor((sfdebut-s0)/hdebut)+2); mf1st = mf(i0:floor((sfdebut-s0)/hdebut)+2); Ug1st = Ug(i0:floor((sfdebut-s0)/hdebut)+2); t1st = t(i0:floor((sfdebut-s0)/hdebut)+2); ndebut = 2+floor((sfdebut-s0)/hdebut); sf = valsf(i); % final axial coordinate h = 0.0001; % step of s in the second loop [s,d,m,U,Wd,Wg,mf,Ug,t] = syskutta4('fsys',ndebut,sfdebut,d(ndebut),m(ndebut), U(ndebut),Wd(ndebut),Wg(ndebut),mf(ndebut),Ug(ndebut),t(ndebut),sf,h); n = length(find(imag(d)==0 & d>=0)); s = [s1st s(ndebut+1:n)]; d = [d1st d(ndebut+1:n)]; m = [m1st m(ndebut+1:n)]; U = [U1st U(ndebut+1:n)]; Wd = [Wd1st Wd(ndebut+1:n)]; NGUYEN LE Duy Khai - Dissertation of DEA – 2002-2003 62 Study of evaporation of droplets in a spray Mr Jean-Marc VIGNON Wg = [Wg1st Wg(ndebut+1:n)]; mf = [mf1st mf(ndebut+1:n)]; Ug = [Ug1st Ug(ndebut+1:n)]; t = [t1st t(ndebut+1:n)];; R = mf./sqrt(pi*rog*Wg); % spray radius (m) md = N*m; % droplet mass flux (kg/s) ma = mf - md; % air entrainement per second (kg/s) % Draw the results figure(1) plot(s,d,c(i)), grid xlabel('Axial distance s (m)') ylabel('d (m)') title('Change of droplet diameter d (m) with s (m)') hold on figure(2) plot(s,m,c(i)),grid xlabel('Axial distance s (m)') ylabel('m (kg)') title('Change of droplet mass m (kg) ') hold on figure(3) plot(s,U,c(i)),grid xlabel('Axial distance s (m)') ylabel('U (m/s)') title('Change of droplet velocity U (m/s) ') hold on figure(4) plot(s,Wd,c(i)),grid xlabel('Axial distance s (m)') ylabel('Wd (kgm/s2)') title('Change of droplet momentum flux Wd (kgm/s2)') hold on figure(5) plot(s,Wg,c(i)),grid xlabel('Axial distance s (m)') ylabel('Wg (kgm/s2)') title('Change of gas momentum flux Wg (kgm/s2)') hold on figure(6) plot(s,mf,c(i),s,md,c(i)),grid xlabel('Axial distance s (m)') ylabel('mf (kg/s)') NGUYEN LE Duy Khai - Dissertation of DEA – 2002-2003 63 Study of evaporation of droplets in a spray Mr Jean-Marc VIGNON title('Change of gas mass flux mf (kg/s)') hold on figure(7) plot(s,Ug,c(i)),grid xlabel('Axial distance s (m)') ylabel('Ug (m/s)') title('Change of gas velocity Ug (m/s)') hold on figure(8) plot(s,R,c(i)),grid xlabel('Axial distance s (m)') ylabel('R (m)') title('Change of spray radius R (m)') hold on figure(9) plot(t,s,c(i)), grid ylabel('Axial distance s (m)') xlabel('t (s)') title('Change of distance s(t)') hold on, end toc hold off NGUYEN LE Duy Khai - Dissertation of DEA – 2002-2003 64 Study of evaporation of droplets in a spray Mr Jean-Marc VIGNON B.4 PROGRAM DIESEL %********** Program Diesel ************** % Solve the system of differential equations to calculate % the penetration length of droplet and the changing of % droplet diameter with the distance in steady flow clear all tic global K rov muv B Dv rog mug beta N % The initial conditions: % Liquid (fuel) : Diesel fuel C16H34 % Ambient gas : Nitrogen N2, at MPa and 170°C % Nozzle : d = 0.2mm, p = 17 MPa x = 16; y = 34; rol = 850; Td = 273 + 80 ; Tg = 273 + 170 ; pg = 2*10^6 ; Mg = 28; dnoz = 0.2*10^-3 ; Cd = 0.75 ; pinj = 17*10^6 ; Yinf = 0.0005 ; R = 8.314 ; % Fuel formula CxHy % liquid density (kg/m3) % droplet temperature (°K) % gas temperature (°K) % gas pressure (N/m2) % gas molecular mass (g/mol) % nozzle diameter (m) % discharge coefficient % injection pressure (N/m2) % vapor mass fraction at infinitive % universal gas constant (J/mol°K) % Calculation the initial necessary parameters: Mv = 12*x + y; % liquid molecular mass (g/mol) mul =(-1.2527*10^-8*Td^4 + 1.3867*10^-5*Td^3 % liquid viscosity (kg/ms) -5.1605*10^-3*Td^2 + 6.4807*10^-1*Td)*10^-3; mug = (2.5*10^-8*Tg^3 - 5.4*10^-5*Tg^2 % gas viscosity (kg/ms) + 7.2*10^-2*Tg + 0.44)*10^-6; muv = mug; % vaporizing viscosity (kg/ms) rog = 10^-3*(pg*Mg)/(R*Tg); % gas density (kg/m3) rov = rog; % vaporizing density (kg/m3) pv = 6894.757*exp(12.12767 - 3743.84/(Td - 93));% partial vaporizing pressure (N/m2) Yd = (pv*Mv)/(pv*Mv + (pg-pv)*Mg*Td/Tg); % fuel vapor mass fraction B = (Yd - Yinf)/(1 - Yd); % Spalding transfer parameter B a = 1.864 - 2.992*10^-3*Tg + 3.357*10^-6*Tg^2 - 1.394*10^-9*Tg^3; % coefficient for calculation D Dv = 1.218*10^-4*Tg^1.5/(pg*a); % diffusivity (m2/s) K = 8*Dv*rov*log(1+B)/rol; % evaporation coefficient (m2/s) NGUYEN LE Duy Khai - Dissertation of DEA – 2002-2003 65 Study of evaporation of droplets in a spray Mr Jean-Marc VIGNON % Calculation the initial conditions: t0 = 0; d0 = 15*10^-6; m0 = pi*d0^3*rol/6; U0 = Cd*(2*(pinj - pg)/rol)^0.5; Ug0 = 0.1; R0 = 0.001; mf0 = pi*R0^2*rog*Ug0; md0 = pi*dnoz^2*rol*U0/4; Wd0 = md0*U0; Wg0 = mf0*Ug0; % initial time (s) % initial droplet diameter (m) % initial droplet mass (kg) % initial droplet velocity (m/s) % initial gas velocity (m/s) % initial radius (m) % initial gas mass flux (kg/s) % initial droplet mass flux (kg/s) % initial droplet momentum flux (kg.m/s2) % initial gas momentum flux (kg.m/s2) N = md0/m0; tvap = d0^2/K; beta = 0.7*(rog/rol)^0.5; alpha = atan(beta)*180/pi; % number of droplets / second % evaporation time (s) % spreading coefficient % half spray angle (deg) % Solve the system of differential equations by 4-order Runge-Kutta method s0 = 0; % initial axial coordinate hdebut = 0.0000001; % step of s in the first loop sfdebut = 35000*hdebut; % final value of s in the first loop i0 = 1; [s,d,m,U,Wd,Wg,mf,Ug,t] = syskutta4('fsys',i0,s0,d0,m0,U0,Wd0,Wg0,mf0,Ug0,t0,sfdebut,hdebut); s1st = s(i0:floor((sfdebut-s0)/hdebut)+2); d1st = d(i0:floor((sfdebut-s0)/hdebut)+2); m1st = m(i0:floor((sfdebut-s0)/hdebut)+2); U1st = U(i0:floor((sfdebut-s0)/hdebut)+2); Wd1st = Wd(i0:floor((sfdebut-s0)/hdebut)+2); Wg1st = Wg(i0:floor((sfdebut-s0)/hdebut)+2); mf1st = mf(i0:floor((sfdebut-s0)/hdebut)+2); Ug1st = Ug(i0:floor((sfdebut-s0)/hdebut)+2); t1st = t(i0:floor((sfdebut-s0)/hdebut)+2); ndebut = 2+floor((sfdebut-s0)/hdebut); sf = 0.5; % final axial coordinate h = 0.0001; % step of s in the second loop [s,d,m,U,Wd,Wg,mf,Ug,t] = syskutta4('fsys',ndebut,sfdebut,d(ndebut),m(ndebut), U(ndebut),Wd(ndebut),Wg(ndebut),mf(ndebut),Ug(ndebut),t(ndebut),sf,h); n = length(find(imag(d)==0 & d>=0)); s = [s1st s(ndebut+1:n)]; d = [d1st d(ndebut+1:n)]; m = [m1st m(ndebut+1:n)]; U = [U1st U(ndebut+1:n)]; NGUYEN LE Duy Khai - Dissertation of DEA – 2002-2003 66 Study of evaporation of droplets in a spray Mr Jean-Marc VIGNON Wd = [Wd1st Wd(ndebut+1:n)]; Wg = [Wg1st Wg(ndebut+1:n)]; mf = [mf1st mf(ndebut+1:n)]; Ug = [Ug1st Ug(ndebut+1:n)]; t = [t1st t(ndebut+1:n)];; R = mf./sqrt(pi*rog*Wg); % spray radius (m) md = N*m; % droplet mass flux (kg/s) ma = mf - md; % air entrainement per second (kg/s) % Draw the results figure(1) plot(s,d), grid xlabel('Axial distance s (m)') ylabel('d (m)') title('Change of droplet diameter d (m) with s (m)') figure(2) plot(s,m),grid xlabel('Axial distance s (m)') ylabel('m (kg)') title('Change of droplet mass m (kg) ') figure(3) plot(s,U, s,Ug,'.-r'),grid xlabel('Axial distance s (m)') ylabel('U (m/s)') title('Change of droplet velocity U (m/s) ') figure(4) plot(s,Wd),grid xlabel('Axial distance s (m)') ylabel('Wd (kgm/s2)') title('Change of droplet momentum flux Wd (kgm/s2)') figure(5) plot(s,Wg),grid xlabel('Axial distance s (m)') ylabel('Wg (kgm/s2)') title('Change of gas momentum flux Wg (kgm/s2)') figure(6) plot(s,mf,s,md,':r'),grid xlabel('Axial distance s (m)') ylabel('mf (kg/s)') title('Change of gas mass flux mf (kg/s)') figure(7) plot(s,Ug),grid NGUYEN LE Duy Khai - Dissertation of DEA – 2002-2003 67 Study of evaporation of droplets in a spray Mr Jean-Marc VIGNON xlabel('Axial distance s (m)') ylabel('Ug (m/s)') title('Change of gas velocity Ug (m/s)') figure(8) plot(s,R),grid xlabel('Axial distance s (m)') ylabel('R (m)') title('Change of spray radius R (m)') figure(9) plot(t,s), grid ylabel('Axial distance s (m)') xlabel('t (s)') title('Change of distance s(t)') toc hold off NGUYEN LE Duy Khai - Dissertation of DEA – 2002-2003 68 Study of evaporation of droplets in a spray Mr Jean-Marc VIGNON B.5 PROGRAM SYSKUTTA4 function [s,d,m,U,Wd,Wg,mf,Ug,t] = syskutta4(fsys,i0,s0,d0,m0,U0,Wd0,Wg0,mf0,Ug0,t0,sf,h) % Solve the system of differential equations by 4-order Runge-Kutta method % [s,d,m,U,Wd,Wg,mf,Ug,t] : table of the values of the functions d(s), m(s), % i0,s0,d0,m0,U0,Ug0,Wd0,Wg0,mf0,Ug0,t0 : initial conditions % sf : final value of s %h : step % fsys : M-file containing the differential equations ts(i0) = s0; td(i0) = d0; tm(i0) = m0; tU(i0) = U0; tWd(i0) = Wd0; tWg(i0) = Wg0; tmf(i0) = mf0; tUg(i0) = Ug0; tt(i0) = t0; j = i0+floor((sf-s0)/h); for k = i0:j s = ts(k); d = td(k); m = tm(k); U = tU(k); Wd = tWd(k); Wg = tWg(k); mf = tmf(k); Ug = tUg(k); t = tt(k); [dd1,dm1,dU1,dWd1,dWg1,dmf1,Ug,dt1] = feval(fsys,s,d,m,U,Wd,Wg,mf,Ug,t); s = ts(k) + h/2; d = td(k) + dd1*h/2; m = tm(k) + dm1*h/2; U = tU(k) + dU1*h/2; Wd = tWd(k) + dWd1*h/2; Wg = tWg(k) + dWg1*h/2; mf = tmf(k) + dmf1*h/2; Ug = Wg/mf; t = tt(k) + dt1*h/2; [dd2,dm2,dU2,dWd2,dWg2,dmf2,Ug,dt2] = feval(fsys,s,d,m,U,Wd,Wg,mf,Ug,t); NGUYEN LE Duy Khai - Dissertation of DEA – 2002-2003 69 Study of evaporation of droplets in a spray Mr Jean-Marc VIGNON s = ts(k) + h/2; d = td(k) + dd2*h/2; m = tm(k) + dm2*h/2; U = tU(k) + dU2*h/2; Wd = tWd(k) + dWd2*h/2; Wg = tWg(k) + dWg2*h/2; mf = tmf(k) + dmf2*h/2; Ug = Wg/mf; t = tt(k) + dt2*h/2; [dd3,dm3,dU3,dWd3,dWg3,dmf3,Ug,dt3] = feval(fsys,s,d,m,U,Wd,Wg,mf,Ug,t); s = ts(k) + h; d = td(k) + dd3*h; m = tm(k) + dm3*h; U = tU(k) + dU3*h; Wd = tWd(k) + dWd3*h; Wg = tWg(k) + dWg3*h; mf = tmf(k) + dmf3*h; Ug = Wg/mf; t = tt(k) + dt3*h; [dd4,dm4,dU4,dWd4,dWg4,dmf4,Ug,dt4] = feval(fsys,s,d,m,U,Wd,Wg,mf,Ug,t); ts(k+1) = ts(k) + h; td(k+1) = td(k) + h*(dd1 + 2*dd2 + 2*dd3 + dd4)/6; tm(k+1) = tm(k) + h*(dm1 + 2*dm2 + 2*dm3 + dm4)/6; tU(k+1) = tU(k) + h*(dU1 + 2*dU2 + 2*dU3 + dU4)/6; tWd(k+1) = tWd(k) + h*(dWd1 + 2*dWd2 + 2*dWd3 + dWd4)/6; tWg(k+1) = tWg(k) + h*(dWg1 + 2*dWg2 + 2*dWg3 + dWg4)/6; tmf(k+1) = tmf(k) + h*(dmf1 + 2*dmf2 + 2*dmf3 + dmf4)/6; tUg(k+1) = Ug; tt(k+1) = tt(k) + h*(dt1 + 2*dt2 + 2*dt3 + dt4)/6; end s = ts; d = td; m = tm; U = tU; Wd = tWd; Wg = tWg; mf = tmf; Ug = tUg; t = tt; NGUYEN LE Duy Khai - Dissertation of DEA – 2002-2003 70 Study of evaporation of droplets in a spray Mr Jean-Marc VIGNON B.6 PROGRAM FSYS function [dd,dm,dU,dWd,dWg,dmf,Ug,dt] = fsys(s,d,m,U,Wd,Wg,mf,Ug,t) % M-file contains the differential equations global K rov muv B Dv rog mug beta N dd = -K*(1+0.3*(rog*(U-Ug)*d/mug)^0.5)/(2*U*d); dm = -2*pi*rov*Dv*d*log(1+B)/U; dU = -3*pi*mug*d*(U-Ug)*(1+0.0923*(rog*(U-Ug)*d/mug)^0.803)/(m*U); dWd = N*(U*dm + m*dU); dWg = -dWd; dmf = beta*sqrt(pi*rog*Wg) - N*dm; dt = 1/U; NGUYEN LE Duy Khai - Dissertation of DEA – 2002-2003 71 ... flux Then the spray penetration and the spray angle are deduced The model accounts for the droplet vaporization and gas aerodynamic force The remainder of this dissertation is divided into sections... Therefore, it is important to understand atomization mechanism and the spray properties of the liquid jet injected through a nozzle There are a lot of physical models describing liquid fuel atomization... neglected z Quasi-steady gas-phase around the droplet z Uniform pressure distribution around the droplet z Uniform liquid temperature z Constant liquid density z Ideal latent heat z The heat capacity