Yakıt demetlerinin matematiksel modellenmesi

Abstract

ÖZET Bu çalışmada, bir Diesel motoru enjektöründen durgun ortam içerisine püskürtülen içi dolu konik sıvı yakıt demet lerine ait olayı yöneten denklemler, fazlar arası etkileşimin de dahil edildiği türbülanslı ortam için kartezyen indis. notasyonunda tanımlanmıştır, Eul eriyen yaklaşımla ele alman akışkana ve damlacıklara ait parabolik `kısmi diferansiyel denklemler Patankar-Spalding koordinat sistemine dönüştürü lerek sonlu farklar algoritması ile sayısal olarak çözülmüş tür. Bu çalışmada ayrıca türbTJTanslı akım alanının tanım lanması amacıyla karışım uzunluğu, k-e ve cebirsel gerilme modeli yaklaşımları incelenmiştir. Damlacık konsantrasyonu, damlacıkların ve akışkanın hız bileşenleri ve akım alanına ait türbülans büyüklükleri geliştirilen model kullanılarak hesaplanmış, damlacıkların buharlaşması sonucu damlacık çap değişiminin etkileri ince lenmiş ve elde edilen sonuçlar literatürde mevcut deneysel ve teorik sonuçlarla karşılaştırılarak yeterli uyum sağlan dığı görülmüştür. m

MATHEMATICAL MODELLING OF FUEL SPRAYS SUMMARY The theoretical modelling of the behaviour of a fuel spray is of considerable importance in the design and improve ment of combustion equipment since the influence of certain parameters can be observed in less time and cost. In recent years, the increasing demands for the contrc of the emission of combustion pollutants and the requirements for the increase of combustion efficiency have influenced the necessity for better understanding of the mechanism of fuel injection and combustion. In this study, a theoretical approach has been used for the prediction of liquid fuel spray behaviour injected into a stagnant atmosphere. Events prior to ignition have been modelled for two-phase turbulent free jets of axially symmetrical geometry. A continuum approach has been used in this study for the prediction of the behaviour of gas and liquid phases using separate sets of governing equations with interactive two-way coupling. Her e the fuel spray composed of various number of small droplets is treated as continuum employing equations derived by Marble in Eulerian formulation. The volume fraction of the droplets is assumed to be small, that is the droplets are so widely^spaced, that direct interaction between them can be ignored. But because of the larqe density ratio between fuel and gas, droplets still have significant mass in comparison to the suspending fluid, thus can influence the state of the fluid through the exchange of mass, momentum and energy. The terms showing the drag force on the suspending fluid due to interaction with droplets and the TVimourrt of momentum transfer to the fluid due to the evaporation tf droplets result as source terms in both momentum equations jroviding full coupling. In this study, the governing equations which are ixpressed in terms of primitive variables initially using Cartesian tensor notation are then transformed into modified >atankar-SpaTding co-ordinate system utilising a definition >f new flow function in the presence of evaporating droplets, fhe parabolic partial differential equations are then integrated over a control volume to obtain the finite difference jquati ons which are solved by tri -diagonal matrix algorithm in m implicit manner. The integration process proceeds by iareni ng downstream, having defined the initial and boundary :onditions of the domain of interest. Predictions were initially made for a round clean jet in an effort to test the present mathematical model and to Drovide a base case for comparison with the loaded jets. For this purpouse, several turbulence models haye been employed to predict turbulent viscosity or Reynolds stresses. Various node! s of turbulence have been put forward by different authors, which differ in complexity involving the solution of different number of differential equations. Models of turbulence have been examined here in two main catagories according to whether Boussinesq approximation is used or not. Mixing-length and k-E models of turbulence employed in this study use the Boussinesq approximation by relating turbulent shearing stresses to the mean velocity gradients by the use of turbulent viscosity concept. An -alternative, so-called Reynolds stress model have also been used providing closure by incorporating transportequations for Reynolds stresses. Sufficient results have been obtained by using the algebraic model of Prandtl's mixing-length hypothesisswhere y. is calculated as a local function of the flow field. In order to take the convection and diffusion of turbulence parameters into accounts differential transport equations have been solved for kinetic energy and eddy dissipation of turbulence simultaneously with the main governing equations, employing k-e model of turbulence. Satisfactory results have been obtained with this model for velocity, turbulence kinetic energy profiles and spread rate of the jet with the modifications made according to recommendations of Rodi for the values of the constants given by Launder et al. For complicated flows `where all components of Reynolds stresses are of equal importance, v. becomes direction sensitive, thus differential transport equations for Reynolds stresses which include convection and diffusion terms have to be solved. In this study algebraic stress model proposed by Rodi have been used. Algebraic equations for the Reynolds stresses have been classified according to the form of pressure-strain terms used, as linear and nonlinear models. For both linear and nonlinear approaches the constants of the model have been tunned as a result of the optimizations made accordinq to the experimental data present in the literature and good qualitative agreement is achieved. In the final section of this study, droplet behaviour and the effect of droplets on the flow field have been examined for different droplet diameters and loading factors while solving the governing spray model equations to obtain vidroplet and fluid velocity, concentration and droplet flux profiles at various distancer from the injection nozzle. The predictions of the model are tested against the experimental and theoretical results present in literature and following conclusions have been reached as a result of a series of numerical calculations : (i) The drag force created by the droplets and the amount of momentum transfer due to the droplet evaporation result in an increase of fluid velocity compared to that of the clean jet. Generally fluid velocity increases with increasing loading factor, (ii) The slip velocity between the droplets and the fluid increases with increasing droplet diameter and dec reases with increase in the initial mass loading, (iii) The normalized droplet and fluid velocity profiles are influenced by the changes in droplet size and loading factor, thus similarity concept has not been satisfied contrary to the round clean jet. (iv) The center! ine velocity decay of the fluid is smaller than that of a single-phase jet in the presence of droplets and increase in the loading factor results in a decrease in cehterline velocity decay. Increasing the loading factor also decreases the radial velocity components of both droplets and the suspending fluid, (v) Spread rate of the spray is smaller than that of a single-phase jet. Generally the spray gets narrower with increasing loading factor. Spread rate calculated to bjs 0.086 for the single- phase jet, reduces to 0.084 for a spray of droplets having diameter and loading viifactor values of 50pm and 1 respectively, (vi) For sufficient distances from the nozzle, the velocity of both liquid and gas phases maintain almost balance inbetween them. The addition of liquid droplets to the flow field modifies the turbulence structure resulting in a significant reduction of the turbulence intensity. As the nature of the interaction between the two phases is rather complex and not well understood at present, no attempt has been made in this study to develop a model of turbulence which accounts for this interaction. The applications of the present model show reasonably good agreement with the experimental results present in literature. On the whole, the theoretical model presented here is capable of predicting the behaviour of liquid fuel sprays injected into a stagnant atmosphere with sufficient accuracy. vm