# Analyse:Jem

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The atomization of sprays can be divided into two main processes, primary and secondary breakup. The former takes place in the region close to the nozzle. It is not only determined by the interaction between the liquid and gaseous phases but also by internal nozzle phenomena like turbulence. Atomization that occurs further downstream in the spray due to hydrodynamic interaction processes, and which is largely independent of the nozzle type, is called secondary breakup. The atomization of IC-engine fuel sprays can be divided into two main processes, primary and secondary break-up. The former takes place in the region close to the nozzle at high Weber numbers. It is not only determined by the interaction between the liquid and gaseous phases but also by internal nozzle phenomena like turbulence and cavitation. Atomization that occurs further downstream in the spray due to aerodynamic interaction processes and which is largely independent of the nozzle type is called secondary break-up.
 Anmerkungen The source is not mentioned. Sichter (Hindemith)

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Often an average droplet diameter djk is taken to represent a spray. A general mean diameter is (Lefebvre, 1989)

$d_{jk} = \left[\frac{\int_0^\infty d^j n(d) {\rm d}d}{\int_0^\infty d^k n(d) {\rm d}d}\right]^{1/(j-k)} = \left[ \frac{m_j}{m_k}\right]^{1/(j-k)} \quad$ (1.7)

An example is the Sauter mean diameter (SMD) d32, which is proportional to the ratio of the total liquid volume to the total droplet surface area.

Lefebvre, A.H. (1989). Atomization and Sprays. Hemisphere, New York.

Often an average droplet diameter dmn is taken to represent a spray. In particular,

$d_{mn} = \frac{\int_0^\infty f(d) d^m \, {\rm d}d}{\int_0^\infty f(d) d^n \, {\rm d}d}. \quad$ (1.1)

[...] One example of an average droplet is the Sauter mean diameter d32, which is proportional to the ratio of the total liquid volume in a spray to the total droplet surface area in a spray.

 Anmerkungen No part of this has been marked as a citation. In the original text this piece belonged together with the passage used in Jem/Fragment_005_04. Sichter (Graf Isolan)

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1.6 Numerical Simulation of Sprays

Computational fluid dynamics (CFD) of single- and multi-phase flows has been a rapidly developing research topic over the last years. At this point, advanced CFD codes are a valuable complement to experimental investigations, since they allow a detailed local analysis of the flow. Engineering flow prediction of single-phase flows is standard application of CFD and is widely used nowadays.

[page 520]

1. Introduction

Computational fluid dynamics of single- and two-phase flows in combustors has been a rapidly developing research

[page 521]

topic over the last years. [...] At this point, advanced CFD codes are a valuable complement to experimental investigations, since they allow a detailed local analysis of the flow. Engineering flow prediction of single-phase flows is a standard application of CFD and widely used nowadays.

 Anmerkungen Nothing has been marked as a citation. Sichter (Graf Isolan)

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Currently the most common spray description is based on the Lagrangian discrete droplet method (DDM) (e.g., Rüger, et al., 2000). While the continuous phase is described by the standard Eulerian conservation equations, the transport of the dispersed phase is calculated by tracking the trajectories of a certain number of representative parcels (particles). A parcel consists of a number of droplets and it is assumed that all the droplets within one parcel have the same physical properties and behave equally when they move, breakup, or evaporate. The coupling between the liquid and the gaseous phases is achieved by source term exchange for mass, momentum, energy, and turbulence. Various submodels account for the effects of turbulent dispersion, coalescence, evaporation, and droplet breakup. The Lagrangian method is especially suitable for dilute sprays, but has shortcomings with respect to modeling of dense sprays. Further problems are reported connected with bad statistical convergence and also with dependence of the spray on grid size (Schmidt and Rutland, 2000). Currently the most common spray description is based on the Lagrangian discrete droplet method [8]. While the continuous gaseous phase is described by the standard Eulerian conservation equations, the transport of the dispersed phase is calculated by tracking the trajectories of a certain number of representative parcels (particles). A parcel consists of a number of droplets and it is assumed that all the droplets within one parcel have the same physical properties and behave equally when they move, break up, hit a wall or evaporate. The coupling between the liquid and the gaseous phases is achieved by source term exchange for mass, momentum, energy and turbulence. Various sub-models account for the effects of turbulent dispersion [9], coalescence [10], evaporation [11], wall interaction [12] and droplet break up [13].

[page 3]

This method is especially suitable for dilute sprays, but has shortcomings with respect to modeling of dense sprays where particle interactions are strongly influenced by collisions and parcels have to be rearranged and redistributed very often. Further problems are reported connected with bad statistical convergence [18] and also with dependence of the propagation of the spray on grid size [19].

[8] Dukowicz, J.K., A Particle-Fluid Numerical Model for Liquid Sprays, Journal of Computational Physics, Vol. 35, pp. 229-253, 1980

[9] Gosman A.D. and Ioannides, E., Aspects of Computer Simulation of Liquid-Fueled Combusters, J. Energy, 7, pp. 482-490, 1983

[10] O’Rourke, P.J., Modeling of Drop Interaction in Thick Sprays and a Comparison with Experiments, IMechE - Stratified Charge Automotive Engines Conference, 1980

[11] Dukowicz, J.K., Quasi-steady Droplet Phase Change in the Presence of Convection, Los Alamos Report LA-7997-MS, 1979

[12] Naber, J.D., Reitz, R.D., Modeling Engine Spray / Wall Impingement, SAE 880107, 1988

[13] Liu, A.B. and Reitz, R.D., Modeling the Effects of Drop Drag and Breakup on Fuel Sprays, SAE 930072

[18] Krüger, Ch., Validierung eines 1D-Spraymodells zur Simulation der Gemischbildung in direkteinspritzenden Dieselmotoren, Dissertation RWTH Aachen, März, 2001

[19] Abraham, J., What is Adequate Resolution in the Numerical Computation of Transient Jets?, SAE 970051

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As an alternative, McGraw (1997) developed the so-called quadrature method of moments (QMOM), which is based on the product-difference (PD) algorithm formulated by Gordon (1968). QMOM has been validated for small particles in the study of aerosols in chemical engineering. It provides a precise and efficient numerical method in order to follow the size distribution of particles, without inertia, experiencing some aggregation-breakage phenomena (Marchisio, et al., 2003b). One of the main limitations of QMOM is that since the dispersed phase is represented through the moments of the size distribution, the phase-average velocity of different solid phases must be used to solve the transport equations for the moments. Thus, in order to use this method in the context of sprays for which the inertia determines the dynamical behavior of the droplets, it is necessary to extend QMOM to handle cases where each droplet size is convected by its own velocity. In order to address these issues, the direct quadrature method of moments (DQMOM) has been formulated and validated (Marchisio and Fox, 2005).

Gordon, R.G. (1968). Error bounds in equilibrium statistical mechanics. Journal of Mathematical Physics 9, pp. 655-663.

Marchisio, D.L., R.D. Vigil and R.O. Fox (2003b). Quadrature method of moments for aggregation-breakage processes. Journal of Colloid and Interface Science 258, pp. 322-334.

Marchisio, D.L. and R.O. Fox (2005). Solution of population balance equations using the direct quadrature method of moments. J. Aerosol Sci. 36, pp. 43-73.

McGraw, R. (1997). Description of aerosol dynamics by the quadrature method of moments. Aerosol Science and Technology 27, pp. 255-265.

As an alternative, McGraw [20] developed the so-called quadrature method of moments (QMOM), which is based on the approximation of the unclosed terms by using an ad-hoc quadrature formula. The quadrature approximation (i.e., its abscissas and weights) can be determined from the lower-order moments [21] by resorting to the product-difference (PD) algorithm [22]. QMOM has been extensively validated for several problems with different internal coordinates [23–25]. One of the main limitations of QMOM is that since the solid phase is represented through the moments of the distribution, the phase-average velocity of the different solid phases must be used to solve the transport equations for the moments. Thus, in order to use this method in the context of the multiphase flows, it is necessary to extend QMOM to handle cases where each particle size is convected by its own velocity.

In order to address these issues, the direct quadrature method of moments (DQMOM) has been formulated and validated [26].

[20] R. McGraw, Description of aerosol dynamics by the quadrature method of moments, Aerosp. Sci. Technol. 27 (1997) 255.

[21] H. Dette, W.J. Studden, The Theory of Canonical Moments with Applications in Statistics, Probability and Analysis,Wiley, New York, 1997.

[22] R.G. Gordon, Error bounds in equilibrium statistical mechanics, J. Math. Phys. 9 (1968) 655.

[23] J.C. Barret, N.A. Webb, A comparison of some approximate methods for solving the aerosol general dynamic equation, J. Aerosol Sci. 29 (1998) 31.

[24] D.L. Marchisio, R.D. Vigil, R.O. Fox, Quadrature method of moments for aggregation-breakage processes, J. Colloid Interface Sci. 258 (2003) 322.

[25] D.L. Marchisio, J.T. Pikturna, R.O. Fox, R.D. Vigil, A.A. Barresi, Quadrature method of moments for population-balance equations, AIChE J. 49 (2003) 1266.

[26] D.L. Marchisio and R.O. Fox, Direct quadrature method of moments: derivation, analysis and applications, J. Comput. Phys. submitted for publication.

 Anmerkungen Though in large parts identical, nothing has been marked as a citation. Mark that Jem speaks of "dispersed phase" instead of the original "solid phase". On page 119 this passage is repeated again (see Jem/Fragment_119_12). Sichter (Graf Isolan)

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3.3.2 Outcome of Collision

The binary droplet collision phenomenon is discussed in this section. The outcome of collisions can be described by three non-dimensional parameters: the collisional Weber number, the impact parameter, and the droplet size ratio (Orme, 1997; Post and Abraham, 2002; Ko and Ryou, 2005b).

The collisional Weber number is defined as

$We_{coll}= \frac {\rho_l U_{rel}^2 d_2}{\sigma}$ (3.21)

where Urel is the relative velocity of the interacting droplets and d2 is the diameter of the smaller droplet.

The dimensional impact parameter b is defined as the distance from the center of one droplet to the relative velocity vector, $\vec{u}_{rel},$ placed on the center of the other droplet. This definition is illustrated in Figure 3.6. The non-dimensional impact parameter is calculated as

[$B=\frac{2b}{d_1+d_2}= \sin \theta$(3.22)

where d1 is the diameter of the larger droplet and θ is the angle between the line of centers of the droplets at the moment of impact and the relative velocity vector.]

Ko, G.H. and H.S. Ryou (2005b). Modeling of droplet collision-induced breakup process. Int. J. Multiphase Flow 31, pp. 723-738.

Orme, M. (1997). Experiments on droplet collisions, bounce, coalescence and disruption. Prog. Energy Combust Sci. 23, pp. 65-79.

Post, S.L. and J. Abraham (2002). Modeling the outcome of drop-drop collisions in Diesel sprays. Int. J. Multiphase Flow 28, pp. 997-1019.

[page 73]

3. Droplet collision

The binary droplet collision phenomenon is discussed in this section. The phenomenon of droplet collision is mainly controlled by the following physical parameters: droplet velocities, droplet diameters, dimensional impact parameter, surface tension of the liquid, and the densities and viscosity coefficients of the liquid and the surrounding gas, but further components may also be important, such as the pressure, the molecular weight and the molecular structure of the gas. From these physical parameters several dimensionless quantities can be formed, namely, the Weber number, the Reynolds number, impact parameter, droplet size ratio, the ratio of densities, and the ratio of viscosity coefficients. Thus, for a fixed liquid-gas system, the outcome of collisions can be described by three non-dimensional parameters: either the Weber number or the Reynolds number, the impact parameter, and the droplet size ratio.

(i) The Weber number is the ratio of the inertial force to the surface force and is defined as follows:

$We = \frac{\rho_d U_r^2 D_S}{\sigma},$(2)

where ρd is the droplet density, Ur is the relative velocity of the interacting droplets, DS is the diameter of the smaller droplet, and σ is the surface tension. [...]

(ii) The dimensional impact parameter b is defined as the distance from the center of one droplet to the relative velocity vector placed on the center of the other droplet. This definition is illustrated in Fig. 2. The non-dimensional impact parameter is calculated as follows:

[Page 74]

$B=\frac{2b}{D_L+D_S}$(3)

where DL is the diameter of the larger droplet.

Orme, M., 1997. Experiments on droplet collisions, bounce, coalescence and disruption. Progress. Energy Combust. Sci. 23, 65–79.

Post, S.L., Abraham, J., 2002. Modeling the outcome of drop-drop collisions in Diesel sprays. Int. J. Multiphase Flow 28, 997–1019.

 Anmerkungen Shortened severely but otherwise left intact. Again, significant details necessary for understanding the formulas have been left out. Nothing has been marked as a citation. No reference to the original author is given. Sichter (Graf Isolan)

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[When the relative velocity of the droplets is higher and the collisional kinetic energy is sufficient to expel] the intervening layer of gas, the droplets will coalesce. If the collisional energy exceeds the value for permanent coalescence, then temporary coalescence occurs. Temporary coalescence may result in either disruption or fragmentation. In disruption, the collision product separates into the same number of droplets which existed prior to the collision. In fragmentation, the coalesced droplet breaks up into numerous satellite droplets (Orme, 1997).

Bounce affects droplet trajectory, but it does not modify droplet size. Coalescence followed by disruption does not have any significant influence on droplet size. Even if some mass transfer occurs, the droplet diameters are not changed in any observable way. However, other types of collision outcomes may influence the DSD, because the sizes of post-collision droplets are different from those of the pre-collision droplets. During fragmentation, a number of small satellite droplets are formed with the accompanying decrease in size. Fragmentation occurs when the relative velocity of colliding droplets is high.

Orme, M. (1997). Experiments on droplet collisions, bounce, coalescence and disruption. Prog. Energy Combust Sci. 23, pp. 65-79.

When the relative velocity is even higher and the collisional kinetic energy is sufficient to expel the intervening layer of gas, the droplets will coalesce after substantial deformation. [...] If the collisional kinetic energy exceeds the value for permanent coalescence, then the temporarily coalesced droplets separate into two or more droplets. [...] Temporary coalescence (regimes IV and V) may result in either disruption or fragmentation. In disruption, the collision product separates into the same number of droplets which existed prior to the collision. In fragmentation, the coalesced droplet breaks up into numerous satellite droplets (Orme, 1997).

It is clear that bounce affects droplet trajectory, but it does not modify the droplet size. Coalescence followed by disruption does not have any significant influence on droplet size. Even if some mass transfer occurs, the droplet diameters are not usually changed in any observable way. Other regions of collision outcomes, however, may influence DSD, because the sizes of post-collision droplets are different from those of the pre-collision droplets. During fragmentation, a number of small satellite droplets is formed with the accompanying decrease in droplet size. Fragmentation occurs when the relative velocity of colliding droplets is high, and since low velocity flows are under examination here, the phenomenon almost never occurs in this investigation.

Orme, M., 1997. Experiments on droplet collisions, bounce, coalescence and disruption. Progress. Energy Combust. Sci. 23, 65–79.

 Anmerkungen Shortened but nonetheless the pieces taken are verbatim. Nothing has been marked as a citation. Sichter (Graf Isolan)

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This technique determines the droplet velocity by standard fringe mode LDA (see section 4.2.1), and establishes the droplet size by measuring the phase shift of light encoded in the spatial variation of the fringes reaching two detectors after traveling paths of different lengths through the droplets (Figure 4.7). The phase shift is measured by two detectors, each looking at a spatially distinct portion of the collection lens. This technique determines the

droplet velocity by standard fringe mode laser anemometry, and establishes the droplet size by measuring the phase shift of light encoded in the spatial variation of the fringes reaching two detectors after traveling paths of different lengths through the droplets (Fig. 1a). The phase shift is measured by two detectors, each looking at a spatially distinct portion of the collection lens.

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4.2.3 Operating Conditions

A PDA system from Dantec Dynamics is used in the present study. The light source is a multi-line argon ion air-cooled laser (Spectra-Physics Model 177) operating at 488 nm and 514.5 nm with approximately output of 300 mW. In the transmitter unit, the laser light is color separated (514.5 nm for the axial component and 488 nm for the radial component) and the two beams are split into four and coupled into single mode glass fibers, which conducts them to the transmitting optics unit. One beam of each color is frequency shifted by 40 MHz through a Bragg cell for the purpose of direction recognition of the velocities. The polarization direction is adjusted perpendicular to the scattering plane. The scattering light is collected by a receiving lens where it is coupled into multimode fibers. After color separation by filters, the light reaches the photomultipliers and is transformed into electrical signals. The signals are processed by a covariance-processor (Dantec 58N80) of the PDA system. With this processor, the frequency and the phase shift of the filtered and amplified signals are determined. The signals are processed if the signal-to-noise of the burst of a droplet fulfills a minimum criterion.

The light source was a multi-line argon ion watercooled

laser (Coherent Innova 70) operating at 488 nm and 514.5 nm with approximately output power of 200 mW. In the fibre drive unit, the laser light was colour separated (514.5 nm for the axial component, 488 nm for the radial or tangential component) and the two beams split into four and coupled into monomode glass fibres, which conducted them to the transmitting optics unit. One beam of each colour was frequency shifted by 40MHz through a Bragg cell for the purpose of direction recognition of the velocities. The polarisation direction was adjusted perpendicular to the scattering plane. [...] The scattering light is collected by a receiving lens with a focal length of 400 mm, which is placed at a 30° off-axis forward scatter mode, where the scattering of light by refraction is in the dominant mode and yields a linear relation of particle size and phase between the photodetectors over the detectable size range of the instrument. After colour separation by filters, the light fell onto the detectors regions where it was coupled again into glass fibres to reach the photomultipliers and transformed into electrical signals. The signals were processed by a covariance-processor of the PDA system. With this processor, the frequency and the phase shift of the filtered and amplified signals are determined. The signals are processed if the signal to noise of the every burst of a droplet is -3 dB or better.

 Anmerkungen Nothing has been marked as a citation. Various modification have been made, most prominently the transference from BE to AE. Otherwise the original text has been left mostly intact. Seemingly unnecessary technical details have been cut out. Sichter (Graf Isolan)

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5.4 Eulerian Multi-Fluid Model

In the Eulerian multi-fluid model, the gas and droplet phases are treated as interpenetrating continua in an Eulerian framework. The gas phase is considered as the primary phase, whereas the droplet phases are considered as dispersed or secondary phases. The gas and droplet phases are characterized by volume fractions, and by definition, the volume fractions of all phases must sum to unity:

$\alpha_g + \sum_{q=1}^N \alpha_q =1 \quad$ (5.16)

where αg is the gas volume fraction, αq is the volume fraction of the qth droplet phase, and N is the total number of droplet phases.

2.1. Multi-fluid model for gas–solid flow

The multi-fluid model is an extension of the two fluid model for gas–solid flows [27]. In this model, the gas and solid phases are treated as inter-penetrating continua in an Eulerian framework. The gas phase is considered as the primary phase, whereas the solid phases are considered as secondary or dispersed phases. Each solid phase is characterized by a specific diameter, density and other properties. The primary and dispersed phases are characterized by volume fractions, and by definition, the volume fractions of all phases must sum to unity:

$\epsilon_g + \sum_{\alpha=1}^N \epsilon_{s\alpha} =1 \quad$ (1)

where εg is the gas volume fraction, ε is the volume fraction of αth solid phase, and N is the total number of solid phases.

 Anmerkungen Nothing has been marked as a citation. "Solid phase" has been substituted by "droplet phase". Otherwise the text has been taken verbatim. Sichter (Graf Isolan)

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In the context of CFD modeling, turbulence in two-phase flow is more often modeled using an extension of the standard k-ε model, although other more sophisticated models using higher order moment closures also [exist.] In the context of CFD modelling, turbulence in two-phase

flow is more often modelled using an extension of the standard k-ε model, although other more sophisticated models using higher moment closures also exist.

 Anmerkungen Not marked as a citation. Sichter (Graf Isolan)

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[In the context of CFD modeling, turbulence in two-phase flow is more often modeled using an extension of the standard k-ε model, although other more sophisticated models using higher order moment closures also] exist. This is because the k-ε model offers reasonable accuracy at low computational cost as well as being more stable to execute. It is therefore quite attractive in routine engineering computations. In the context of CFD modelling, turbulence in two-phase

flow is more often modelled using an extension of the standard k-ε model, although other more sophisticated models using higher moment closures also exist. This is because the k-ε model offers reasonable accuracy at low computational cost as well as being more stable to execute. It is therefore quite attractive in routine engineering computations.

 Anmerkungen Not marked as a citation. Continues from the previous page. Sichter (Graf Isolan)

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5.5.3 Turbulence Model for Each Phase

In the dispersed k-ε model, turbulence is associated with the continuous phase, which is assumed to be the dominant phase, i.e. the dispersed phase is present in small quantities. Hence, the dispersed phase can only respond to or modify the continuous phase turbulence. When the phase fraction increases this assumption ceases to be valid as the dispersed phase fluctuations become intertwined with those of the continuous phase. In the limit, when the dispersed phase fraction approaches unity, turbulence becomes associated with the "dispersed" phase. Hence a model catering for the full range of phase fraction values is needed. Such a model, based on the solution of the k and ε transport equations for each phase, is presented here.

In all those models, turbulence is associated with

the continuous phase, which is assumed to be the dominant phase, i.e. the dispersed phase is present in small quantities. Hence, the dispersed phase can only respond to or modify the continuous phase turbulence. When the phase fraction increases this assumption ceases to be valid as the dispersed phase fluctuations become intertwined with those of the continuous phase. Indeed, in the limit when the dispersed phase fraction approaches unity, or when phase inversion occurs, turbulence becomes associated with the “dispersed” phase. Hence a model catering for the full range of phase fraction values is needed. Such a model is proposed here and is based on the derivation of the k and ε transport equations for the mixture of the two phases.

 Anmerkungen Nothing has been marked as a citation. Sichter (Graf Isolan)

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6.1 Introduction

Spray models have a common basis at what can be called "the kinetic level" under the form of a probability density function (PDF or distribution function) satisfying a Boltzmann type equation, the so-called Williams equation (Williams, 1985). The variables characterizing one droplet are the size, the velocity, and the temperature. Such a transport equation describes the evolution of the distribution function of the spray due to evaporation, to the drag force of the gaseous phase, to the heating of the droplets by the gas, to breakup phenomena, and finally to the droplet-droplet interaction such as collision. The spray transport equation is then coupled to the gas phase equations. The two-way coupling of the phases occurs first in the spray transport equations through the rate of evaporation, drag force, and the heating rate, which are functions of the gas phase variables, and second through exchange terms in the gas phase equations.

There are several strategies in order to solve the liquid phase.

Williams, F.A. (1985). Combustion Theory. Addison-Wesley, Redwood, CA.

1. Introduction

[...]

Spray models (where a spray is understood as a dispersed phase of liquid droplets, i.e. where the liquid volume fraction is much smaller than one) have a common basis at what can be called ‘‘the kinetic level’’ under the form of a probability density function (p.d.f. or distribution function) satisfying a Boltzmann type equation, the so-called Williams equation [6–8]. The variables characterizing one droplet are the size, the velocity and the temperature, so that the total phase space dimension involved is usually of twice the space dimension plus two. Such a transport equation describes the evolution of the distribution function of the spray due to evaporation, to the drag force of the gaseous phase, to the heating of the droplets by the gas and finally to the droplet-droplet interactions (such as coalescence and fragmentation phenomena) [2,8–13]. The spray transport equation is then coupled to the gas phase equations. The two-way coupling of the phases occurs first in the spray transport equations through the rate of evaporation, drag force and heating rate, which are functions of the gas phase variables and second through exchange terms in the gas phase equations.

There are several strategies in order to solve the liquid phase.

[6] F.A. Williams, Spray combustion and atomization, Phys. Fluids 1 (1958) 541–545.

[7] F.A. Williams, Combustion theory, in: F.A. Williams (Ed.), Combustion Science and Engineering Series, Addison-Wesley, Reading, MA, 1985.

[8] F. Laurent, M. Massot, Multi-fluid modeling of laminar poly-diperse spray flames: origin, assumptions and comparison of sectional and sampling methods, Combust. Theory Modelling 5 (4) (2001) 537–572.

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Seite(n): 1, Zeilen: left col. 22-27
Currently the most common spray description is based on the Lagrangian discrete droplet method (DDM) (e.g., Rüger, et al., 2000). While the continuous phase is described by the standard Eulerian conservation equations, the transport of the dispersed phase is calculated by tracking the trajectories [of a certain number of representative parcels (particles).]

Rüger, M., S. Hohmann, M. Sommerfeld and G. Kohnen (2000). Euler/Lagrange calculations of turbulent sprays: The effect of droplet collisions and coalescence. Atomization and Sprays 10, pp. 47-81.

Currently the most common spray description is based on the Lagrangian discrete droplet method [8]. While the continuous gaseous phase is described by the standard Eulerian conservation equations, the transport of the dispersed phase is calculated by tracking the trajectories of a certain number of representative parcels (particles).

[8] Dukowicz, J.K., A Particle-Fluid Numerical Model for Liquid Sprays, Journal of Computational Physics, Vol. 35, pp. 229-253, 1980

 Anmerkungen Not marked as a citation. Text has already been used on page 9 (see Jem/Fragment_009_14). Takeover continues into the next page (see Jem/Fragment_118_01). Sichter (Graf Isolan)

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A parcel consists of a number of droplets and it is assumed that all the droplets within one parcel have the same physical properties and behave equally when they move, break up, or evaporate. The coupling between the liquid and the gaseous phases is achieved by source term exchange for mass, momentum, energy, and turbulence. Various submodels account for the effects of turbulent dispersion, coalescence, evaporation, and droplet breakup. The Lagrangian method is especially suitable for dilute sprays, but has shortcomings with respect to modeling of dense sprays. Further problems are reported connected with bad statistical convergence and also with dependence of the spray on grid size (Schmidt and Rutland, 2000).

Schmidt, D.P. and C.J. Rutland (2000). A new droplet collision algorithm. Journal of Computational Physics 164, pp. 62-80.

[page 1]

A parcel consists of a number of droplets and it is assumed that all the droplets within one parcel have the same physical properties and behave equally when they move, break up, hit a wall or evaporate. The coupling between the liquid and the gaseous phases is achieved by source term exchange for mass, momentum, energy and turbulence. Various sub-models account for the effects of turbulent dispersion [9], coalescence [10], evaporation [11], wall interaction [12] and droplet break up [13].

[page 3]

This method is especially suitable for dilute sprays, but has shortcomings with respect to modeling of dense sprays where particle interactions are strongly influenced by collisions and parcels have to be rearranged and redistributed very often. Further problems are reported connected with bad statistical convergence [18] and also with dependence of the propagation of the spray on grid size [19].

[8] Dukowicz, J.K., A Particle-Fluid Numerical Model for Liquid Sprays, Journal of Computational Physics, Vol. 35, pp. 229-253, 1980

[9] Gosman A.D. and Ioannides, E., Aspects of Computer Simulation of Liquid-Fueled Combusters, J. Energy, 7, pp. 482-490, 1983

[10] O’Rourke, P.J., Modeling of Drop Interaction in Thick Sprays and a Comparison with Experiments, IMechE - Stratified Charge Automotive Engines Conference, 1980

[11] Dukowicz, J.K., Quasi-steady Droplet Phase Change in the Presence of Convection, Los Alamos Report LA-7997-MS, 1979

[12] Naber, J.D., Reitz, R.D., Modeling Engine Spray / Wall Impingement, SAE 880107, 1988

[13] Liu, A.B. and Reitz, R.D., Modeling the Effects of Drop Drag and Breakup on Fuel Sprays, SAE 930072

[18] Krüger, Ch., Validierung eines 1D-Spraymodells zur Simulation der Gemischbildung in direkteinspritzenden Dieselmotoren, Dissertation RWTH Aachen, März, 2001

[19] Abraham, J., What is Adequate Resolution in the Numerical Computation of Transient Jets?, SAE 970051

 Anmerkungen Not marked as a citation. Text has already been used on page 9 (see Jem/Fragment_009_14). Takeover continues from the previous page (see Jem/Fragment_117_19). Sichter (Graf Isolan)

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Seite(n): 8-9, Zeilen: 8:8-18; 9:3-4.6-14
[This linear system can be written in matrix form as

$\mathbf A \mathbf x = \mathbf b\quad\quad$ (6.16)

where the 2N x 2N coefficient matrix A = [A1A2] is defined by

$\mathbf A_1 = \left[ \begin{matrix} 1 & \cdots & 1\\ 0 & \cdots & 0\\ -d_1^2 & \cdots & -d_N^2\\ \vdots & \ddots & \vdots\\ 2(1-N)d_1^{2N-1} & \cdots & 2(1-N)d_N^{2N-1} \end{matrix} \right]\quad\quad$ (6.17)

and]

$\mathbf A_2 = \left[ \begin{matrix} 0 & \cdots & 0\\ 1 & \cdots & 1\\ 2d_1 & \cdots & 2d_N\\ \vdots & \ddots & \vdots\\ 2(N-1)d_1^{2N-2} & \cdots & 2(N-1)d_N^{2N-2} \end{matrix} \right]\quad\quad$ (6.18)

The 2N vector of unknowns x is defined by

${\mathbf x} = \left[S_{\omega_1} \cdots S_{\omega_N} \; S_{\delta_1} \cdots S_{\delta_N}\right]^T \quad\quad$ (6.19)

and the known right-hand side is

${\mathbf b} = \left[ \overline{S}_{m_0} \cdots \overline{S}_{m_{2N-1}} \right]^T \quad\quad$ (6.20)

As shown below, with the DQMOM approximation, the right-hand side of Eq. (6.10) is closed in terms of N weights and abscissas. As N increases, the quadrature approximation will approach the exact value, although at a higher computational cost.

If the abscissas dq are unique, then A will be full rank. For this case, the source terms for the transport equations of the weights ωq and weighted diameters δq can be found by inverting A in Eq. (6.16):

There are cases for which the matrix A is not full rank (the matrix is singular). These cases can occur when one or more of the abscisses dq are non-distinct.

[page 8]

This linear system can be written in matrix form as:

$\mathbf A \mathbf \alpha = \mathbf d,\quad\quad$ (13)

where the 2N x 2N coefficient matrix A = [A1A2] is defined by

$\mathbf A_1 = \left[ \begin{matrix} 1 & \cdots & 1\\ 0 & \cdots & 0\\ -L_1^2 & \cdots & -L_N^2\\ \vdots & \ddots & \vdots\\ 2(1-N)L_1^{2N-1} & \cdots & 2(1-N)L_N^{2N-1} \end{matrix} \right]\quad\quad$ (14)

and

$\mathbf A_2 = \left[ \begin{matrix} 0 & \cdots & 0\\ 1 & \cdots & 1\\ 2L_1 & \cdots & 2L_N\\ \vdots & \ddots & \vdots\\ 2(N-1)L_1^{2N-2} & \cdots & 2(N-1)L_N^{2N-2} \end{matrix} \right].\quad\quad$ (15)

The 2N vector of unknowns α is defined by

${\mathbf \alpha} = \left[\alpha_1 \cdots \alpha_N \; b_1 \cdots b_N\right]^T = \left[ \begin{matrix} {\mathbf a}\\ {\mathbf b} \end{matrix} \right],\quad\quad$ (16)

and the known right-hand side is

${\mathbf d} = \left[ \overline{S}^{(N)}_0 \cdots \overline{S}^{(N)}_{2N-1} \right]^T. \quad\quad$ (17)

[page 9]

As shown below, with the DQMOM approximation the right-hand side of Eq. 18 is closed in terms of the N weights and abscissas. [...] As N increases, the quadrature approximation will approach the exact value, albeit at a higher computational cost.

If the abscissas Lα are unique, then A will be full rank. For this case, the source terms for the transport equations of the weights ωα and weighted lengths ${\mathcal L}_\alpha$ can be found simply by inverting A in Eq. 13:

If at any point in the computational domain two abscissas are equal, then the matrix A is not full rank (or the matrix is singular), and therefore it is impossible to invert it.

 Anmerkungen Nothing has been marked as a citation. Sichter (Graf Isolan)

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DQMOM is implemented in the CFD code Fluent 6.2 by representing each node of the quadrature approximation as a distinct droplet phase. In the Eulerian multi-fluid model each droplet phase has its own momentum balance, giving the DQMOM-multi-fluid model the ability to treat polydispersed droplets, which have their own inertia and size-conditioned dynamics. Equation (6.25) is solved in the multi-fluid model as a set of user-defined scalars. [page 1]

DQMOM is implemented in the code by representing each node of the quadrature approximation as a distinct solid phase. Since in the multi-fluid model each solid phase has its own momentum balance, the nodes of the DQMOM approximation are convected with their own velocities.

[page 7]

It is thus clear that each node of the quadrature approximation is calculated in order to guarantee that the moments of the PSD are tracked with high accuracy but, at the same time, each node is treated as a distinct solid phase giving the DQMOM-multi-fluid model the ability to treat polydisperse solids.

[page 10]

The second equation in Eq. 22 is solved in the multi-fluid model as a set of user-defined scalars.

 Anmerkungen Patchwork. Nothing is marked as a citation. Sichter (Graf Isolan)

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In this case, convection in physical space will solve the stability problem. However, it is important to highlight that the frequency of this event is very low in the simulations. In this case, convection in physical space will "solve" the singularity. If the second approach is used, the source vector α is estimated from the average of the source vectors from neighboring cells. However it is important to highlight that the frequency of this event is very low in the simulations.
 Anmerkungen Not marked as a citation. Sichter (Graf Isolan)

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10.2.4 Initial Droplet Size Distribution

Several empirical relationships have been proposed to characterize the distribution of droplet sizes in a spray, e.g. Rosin-Rammler, Nukiyama-Tanasawa, log-normal, root-normal, and log-hyperbolic, etc. (Lefebvre, 1989; Babinsky and Sojka, 2002).

Babinsky, E. and P.E. Sojka (2002). Modeling drop size distributions. Prog. Energy Combust Sci. 28 , pp. 303-329.

Lefebvre, A.H. (1989). Atomization and Sprays. Hemisphere, New York.

Several empirical relationships have been proposed to characterize the distribution of droplet sizes in a spray, e.g., Rosin-Rammler, Nukiyama-Tanasawa, log-normal, root-normal, and log-hyperbolic, etc. (Lefebvre, 1989; Babinsky and Sojka, 2002).

Babinsky, E.B., Sojka, P.E., 2002. Modeling drop size distributions. Progress in Energy Combustion Science 28, 303–329.

Lefebvre, A.H., 1989. Atomization and Sprays. Hemisphere Publishing Corporation, New York

 Anmerkungen Though identical not marked as a citation. Sichter (Graf Isolan)

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Quelle: Ko and Ryou 2005
Seite(n): 1304, Zeilen: 5-14, 16-17
However, there are at least three limitations in the Fluent model. The first is that the Fluent model assumes that a given parcel may collide with another parcel only if these two parcels lie in the same computational cell. As indicated by Schmidt and Rutland (2000) and Nordin (2001), this assumption may be inappropriate. Under this assumption, the collision between two spatially very close parcels is a priori ignored if they reside in different computational cells. Contrary to this, the collision may occur for a pair of possibly far distant parcels in the same computational cell. As a result, the collision model strongly depends on the computational cell sizes. The second limitation is linked with non-uniformity of the spatial distribution of parcels in the domain. Aneja and Abraham (1998) indicated that the Fluent approach is not suitable for sprays, where the variation in number density is large even inside one cell. Finally, the Fluent model considers only two collision regimes such as separation and [permanent coalescence.]

Aneja, R. and J. Abraham (1998). How far does the liquid penetrate in a Diesel engine: computed results vs. measurements? Combust. Sci. and Tech. 138, pp. 233-255.

Nordin, P.A.N. (2001). Complex Chemistry Modeling of Diesel Spray Combustion. Ph.D. Thesis, Chalmers University of Technology, Sweden.

Schmidt, D.P. and C.J. Rutland (2000). A new droplet collision algorithm. Journal of Computational Physics 164, pp. 62-80.

However, there are at least three limitations in the O’Rourke model. The first is that the O’Rourke model assumes that a given parcel may collide with another parcel only if these two parcels lie in the same computational cell. As indicated by Gavaises (1997) recently, this assumption may be inappropriate. Under this assumption, the collision between two spatially very close parcels is a priori ignored if they reside in different computational cell. Contrary to this, the collision may occur for a pair of possibly far distant parcels in the same computational cell. As a result, the collision model strongly depends on the computational cell sizes. The second limitation is linked with the ignorance of the preferred directional effects of droplets. Nordin (2000) has indicated that the O’Rourke’s approach is not suitable for sprays, where the variation in void fraction is large even inside one cell and the parcels have the so-called ‘preferred directions’. [...] Finally, the O’Rourke model considers only three collision regimes such as separation, permanent coalescence, and grazing bounce.

Gavaises, M. (1997). Modeling of diesel fuel injection processes. Ph.D. thesis, Imperial College of Science and Technology and Medicine, Department of Mechanical Engineering, University of London.

Nordin, N. (2000). Complex chemistry modeling of diesel spray combustion. MS. thesis, Chalmers University of Technology, Sweden.

 Anmerkungen Not marked as a citation. Interestingly enough, although the model used is different, the text is mostly taken verbatim. At the end the text modifications needed to become more massive. Sichter (Graf Isolan)

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In the separation regime the colliding droplets loose only their momentum, but retain their sizes after impact. As referred by Georjon and Reitz (1999), this could lead to an overprediction of the coalescence phenomenon because the collision-induced breakup process is ignored.

Georjon, T.L. and R.D. Reitz (1999). Drop-shattering collision model for multidimensional spray computations. Atomization and Sprays 9, pp. 231-254.

Moreover, even in the separation regime, the colliding droplets lose only their momentum, but retain their sizes after impact. As referred by Bai (1996), the O’Rourke model could lead to an over-prediction of the coalescence phenomenon because the collision-induced breakup processes are ignored.

Bai, C. (1996). Modeling of spray impingement processes. Ph.D. thesis, Imperial College of Science and Technology and Medicine, Department of Mechanical Engineering, University of London.

 Anmerkungen Nothing has been marked as a citation. Continues from previous page (see Jem/Fragment_209_19). Again some adaptation to the model at hand has taken place Sichter (Graf Isolan)

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### Fragmente (Verwaist)

1 Fragment

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[The non-dimensional impact parameter is calculated as]

$B=\frac{2b}{d_1+d_2}= \sin \theta$(3.22)

where d1 is the diameter of the larger droplet and θ is the angle between the line of centers of the droplets at the moment of impact and the relative velocity vector.

[Figure 3.6: Illustration of the definition of geometric parameters of the droplet collision.]

The droplet size ratio is given by

$\gamma = \frac{d_1}{d_2}\quad$ (3.23)

The possible outcomes of collisions are illustrated in Figure 3.7. Droplet bounce will occur if there is not enough time for the gas trapped between the droplets to escape and the surfaces of the droplets do not make contact due to the intervening gas film. When the relative velocity of the droplets is higher and the collisional kinetic energy is sufficient to expel [the intervening layer of gas, the droplets will coalesce.]

[page 73]

The non-dimensional impact parameter is calculated as follows:

[Fig. 2. Illustration of the definition of impact parameter b.]

[Page 74]

$B=\frac{2b}{D_L+D_S} \quad$(3)

where DL is the diameter of the larger droplet.

(iii) The droplet size ratio is given by

$\Delta = \frac{D_S}{D_L}. \quad$ (4)

It should be clear that $\Delta \le 1,$ although some authors prefer to use the reciprocal $\gamma = \frac{1}{\Delta}.$

[...]

When two droplets interact during flight, five distinct regimes of outcomes may occur, as listed in Section 1, and depicted in Fig. 3 in the B–We plane for four different values of Δ. [...] If the relative velocity of the droplets is higher, there is not enough time for the gas to escape and the surfaces of the droplets do not make contact due to the intervening gas film, so the droplets become deformed and bounce apart. The corresponding domain in Fig. 3 is regime II. When the relative velocity is even higher and the collisional kinetic energy is sufficient to expel the intervening layer of gas, the droplets will coalesce after substantial deformation.

 Anmerkungen Continued from the previous page. The formulas and the figure are slightly adapted. The comments are taken verbatim without any reference given. Sichter (Graf Isolan)

### Quellen

Quelle Autor Titel Verlag Jahr Lit.-V. FN
Jem/Behzadi et al 2004 A. Behzadi, R.I. Issa, H. Rusche Modelling ofdispersed bubble and droplet at high phase fractions 2004 no no
Jem/Hadef and Lenze 2005 R. Hadef, B. Lenze Measurements of droplets characteristics in a swirl-stabilized spray flame 2005 no no
Jem/Ko and Ryou 2005 Gwon Hyun Ko, HongSun Ryou Droplet collision processes in an inter-spray impingement system 2006 yes yes
Jem/Kollar et al 2005 László E. Kollár, Masoud Farzaneh, Anatolij R. Karev Modeling droplet collision and coalescence in an icing wind tunnel and the influence of these processes on droplet size distribution 2005 no no
Jem/Laurent et al 2003 Frédérique Laurent, Marc Massot, Philippe Villedieu Eulerian multi-fluid modeling for the numerical simulation of coalescence in polydisperse dense liquid sprays 2004 no no
Jem/Schmehl et al 1999 R. Schmehl, H. Rosskamp, M. Willmann, S. Wittig CFD analysis of spray propagation and evaporation including wall film formation and spray/film interactions 1999 no no
Jem/Tatschl et al 2002 Reinhard Tatschl, Christopher v. Künsberg Sarre, Eberhard v. Berg IC-engine Spray modelling – status and Outlook 2002 no no

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