THEORETICAL INVESTIGATION ON VARIABLE DENSITY SPRAYS

Atomization and Srays, vol. 1,. 39 68, 00 THEORETICAL INVESTIGATION ON VARIABLE DENSITY SPRAYS A. de Risi, T. Donateo, and D. Laforgia Diartimento di Ingegneria dell Innovazione, Università di Lecce, Lecce, Italy The aim of the resent investigation is the analysis of the influence of liquid-fuel comressibility on the simulation of srays roduced by high-ressure injection systems. Two different equations have been introduced in the KIVA3V code to calculate liquid-hase density. The first one determines fuel density by using a second-order function of dro temerature and ressure, while the second one also takes into account the quantity of air dissolved in the fuel. Breaku, vaorization, and collision models as well as the energy, momentum, and air-sray mass exchange equations were modified so that each drolet would have a different density, according to its osition and evolution. A comarison between exerimental and numerical data for srays injected in a constant-volume vessel at ambient temerature and ressure has been carried out to test the ractical caability of the modified KIVA3V subroutines. The redicted and measured results of enetration versus time and dro size distribution showed good agreement. An in-deth study of the influence of gas temerature on the drolet vaorization rate has been erformed for a single drolet and for srays injected in a high-temerature, medium-ressure, constant-volume chamber. The effect of fuel density variability on vaorizing noncombusting srays has been investigated for both models. The air dissolved in the fuel was found to affect liquid-hase density only at low ambient ressure. Finally, the exerimental data measured on a small-bore diesel engine have been used to verify the rovisional caabilities of constant- and variable-density models. NO and soot redictions have shown to be deendent on the model used for liquid-hase density. INTRODUCTION An accurate simulation of the vaorization rocess is a very imortant issue for engine investigations, because the gas-hase temerature and the amount of fuel vaor available for combustion are affected by the vaorization rate. Many effects should be considered in the case of dense srays, such as drolet drolet and drolet gas interactions, which influence dro size distribution and vaorization rate. In the resent research, the KIVA3V code has been used to test the effect of fuel density variability on drolet collisions, sray-gas mass and energy exchanges, vaorization rocess, liquid internal energy, and drolet size distribution. Fuel air mixing rate and burning rate are known to be deendent on the ratio of ambient density to injected fuel density [13]. Thus, the effect of fuel density is exected to be more imortant for evaorating srays injected into relatively low ambient density. In the KIVA3V code [17, 19], srays are reresented with a set of arcels, with each arcel consisting of a certain number of dros. The dros contained in one arcel have the same thermodynamic and hysical roerties, such as temerature, radius, local osition, velocity, etc. They differ from arcel to arcel. Only fuel liquid density is considered con- The authors would like to thank Prof. R. D. Reitz for the comments and suggestions which originated the resent study. Coyright 00 by Begell House, Inc. 39

40 A. DE RISI ET AL. NOMENCLATURE a arent dro radius b imact arameter B mass transfer number b cr critical imact arameter C d drag coefficient c l liquid secific heat D m mass diffusivity of fuel in air D article drag function f drolet distribution function F dro velocity rate of change g acceleration of gravity H enthaly er unit mass H lat latent heat of vaorization I internal energy er unit mass i4 comutational cell i4m momentum cell M mass in the momentum cell mf inj total injected fuel mass N c number of collisions N number of dros in the arcel Oh Ohnesorge number = (We ) 0.5 /Re ressure P va vaor ressure Q heat transferred Q d rate of heat conduction to the dro r dro radius R drolet radius rate of change R * universal constant of gases Re Reynolds number r noz nozzle radius Sc Schmidt number Sh Sherwood number t time T temerature tn arc total number of injected arcels u gas velocity vector u gas turbulent velocity vector v dro velocity vector V cell volume W We x y Y t Κ turbulent energy source Weber number dro osition vector drolet distortion from shericity drolet distortion rate of change time ste wave number Λ wavelength µ dynamic viscosity ρ density σ liquid surface tension τ breaku time Ω wave growth rate χ mass fraction Subscrits 1 larger dro smaller dro a air d drolet f fuel mixture l liquid hase arcel of dros q new dros v vaor hase Suerscrits A values at the end of hase A B values at the end of hase B n value at the revious time ste s sray udated quantities after breaku and collision calculations stant, and its value is assigned in the fuel library subroutine. In articular, the density of C 14 H 30 is ρ f = 76.55 kg/m 3, and the density of gas oil is ρ = 84 kg/m 3. Therefore, arcels which have different temerature and different ressure have, in the original code, the same density. Many researchers have found [1 6] that fuel density influences the structure and the evolution of srays by affecting ti enetration, sray angle, and vaorization rate. Moreover, fuel density is known to be very imortant in the heat-u eriod of dros because it increases the dro radius during that eriod and modifies the surface mass ratio [7]. Priem et al. [8] develoed a sray simulation model that takes into account fuel density variability versus temerature. Gonzales et al. [7] comared the KIVA3V evaoration

41 VARIABLE-DENSITY SPRAYS model with Priem s theory and found that KIVA3V redicts a longer drolet lifetime under all conditions of dro size, injection velocity, gas temerature, and ressure. Other vaorization models have been develoed to consider multicomonent drolets [9] or heterogeneous temerature distribution inside the drolet [10]; many of them assumed that fuel density variability was negligible. Moreover, the lowering of average fuel density due to the cavitation henomenon, should be considered, as shown by Verhoeven et al. [11]. Hublin et al. [1] found that, excet for NO x, the lowest emissions are obtained with lowdensity fuels in the case of light-duty diesel engines. The effect of fuel density variability on the rediction of soot and NO emissions of a high-seed, small-bore, last-generation diesel engine has been tested for different oerating conditions. NUMERICAL MODELS As reorted by Ficarella [14], all combustible oils have the same thermodynamics behavior (i.e., the same density ressure temerature deendence), which can be reresented by the following exression: ρ ( ) ρ l T, = ρ 0( T, 0) 1 + a( T) + b( T) (1) where a(t) and b(t)are second-order functions of temerature given by a(t) = a 0 + a 1 T + a T, b(t) = b 0 + b 1 T + b T ; ρ 0 (T, 0 ) is the fuel density at temerature T and atmosheric ressure 0, which can be calculated as ρ ( T, ) = ρ ex( ρβt ) ex( βt) 0 0 ref ref () The coefficients in Eqs. (1) and () are listed in Table 1. Equation (1) was also used by Arcoumanis et al. [15] to consider density change with ressure in the simulation of fuel injection systems for direct-injection (DI) diesel engines. Fuel recirculation is usual ractice in automotive injection systems and roduces a certain amount of air to be traed in the liquid as microbubbles; this is a noteworthy issue because the resence of dissolved air has been observed to affect fuel density for ressure lower than.0 MPa. The effect of air can be considered by using the following equation: Table 1 Coefficients for Fuel Density Correlation Eq. Coeff. i = 0 i = 1 i = (1) a i 3.7175 10 10 m /N 5.39464 10 10 m /N K 6.88769 10 10 m /N K (1) b i 6.1866 10 18 m /N.56408 10 10 m /N K 3.11703 10 3 m /N K () ρ ref = 850 kg/m 3 T ref = 93.1 K β = 7 10 5 K 1

4 A. DE RISI ET AL. ρ ( T,, χ ) = l a 1 1 1 1 χa ρ ( T, ) ρ ( T, ) ρ ( T, ) l l a (3) where ρ l (T, ) is calculated from Eq. (1) and χ a is the mass fraction of air. The resence of air also affects the calculation of fuel internal energy, which in this case has to be evaluated according to the following equation: Il( T,, χ a) = Il( T, ) χa Il( T, ) Ia( T, ) (4) It should be noticed that Eqs. (3) and (4) are written under the hyothesis of negligible molecular interactions between oil and dissolved air, and of negligible surface tension of the air bubbles. The air mass fraction has been calculated by measuring the sound velocity through the fuel, and a value of 7.4 10 3 % (5% in volume) was found by Ishihara et al. [16] for ambient temerature and ressure. This value was found to affect only wave roagation, while mass flow rate and vaorization rate under steady-state conditions were not influenced. The KIVA3V code calculates liquid-hase internal energy from the values of the vaorhase internal energy, which is assumed to be a function of temerature only. Thus, the actual value of liquid-hase internal energy is calculated at a given drolet temerature T by H ( T) = H ( T) H ( T) l v lat (5) When fuel density is not constant [see Eq. (1)], liquid internal energy does not deend on temerature only, but is also affected by the local value of ressure. Using the definition of enthaly, its exression is given by * va( T) Il( T, ) = Iv( T) + R T Hlat( T) ρ ( T, ) l (6) If the resence of dissolved air is considered, fuel internal energy can be calculated with the following equation: I f ( T,, χ a) = Il( T, ) χa Il( T, ) Ia( T, ) (7) where ρ l (T, ) in Eq. (6) is relaced by ρ l (T,, χ a ). As liquid-hase internal energy deends on the actual values of ressure and temerature, it has to be calculated for each arcel searately. This was erformed by using dro temerature and ressure (i.e., the value of the ressure in the comutational cell that contains the arcel under consideration).

43 VARIABLE-DENSITY SPRAYS In the case of two-hase flows, the conservation equations include terms due to the exchange of mass, momentum, and energy between the liquid and gas hases. In the case of srays, these terms are obtained by summing the rates of changes of mass density [Eq. (8)], momentum [Eq. (9)], internal energy [Eq. (10)], and turbulent kinetic energy [Eq. (11)] of all drolets at a given osition x and time t. The distribution of dro sizes, velocities, and temeratures is accounted for by using a drolet distribution function f. df dt s s dρ = ρ π dt f 4 d r R dv dr dtd dy dy 4 d( F g) = ρ π + 4π v 3 dt v f 3 d r r R d dr dtd dy dy (8) (9) dq dt s f r R I ( T ) ( ) = ρ π + v u 1 4 d l d 4 dt + π ( ) ( ) 3 + F g v u u dt 3 d r cl dv dr dtd dy dy dw dt s 4 = ρ π ( F g) u 3 f 3 d r dv dr dtd dy dy (10) (11) The revious equations are discretized both in sace and time as detailed in the Aendix. Collision Model In the KIVA3V code, two colliding dros can either coalesce or searate; dros coalesce when the imact arameter b is less than the critical value b cr given by [18] b r r f cr = ( 1 + ) min 1.0,.4 ( γ)/we (1) where f ( γ ) = γ3.4γ +.7γ γ= r r 1 We ρ = v v 1 1 σ( T ) r T Tr ρ + Tr ρ = 3 3 11 1 r3 3 1ρ 1+ rρ Tyically, two colliding arcels have different temeratures and consequently different densities. Thus, when considering the coalescence between arcel 1 and arcel (r 1 > r ), the radius of the resulting drolets is calculated by a mass-weighted average value:

44 A. DE RISI ET AL. ρ r = r + N r q 3 3 1 3 1 ρ c q ρ ρ q (13) where N c is the number of coalescences for arcel 1 and is calculated according to O Rourke [18]. Moreover, the new drolet velocity and temerature are to be evaluated with a massweighted average rocedure that takes into account the different arcel densities. Their values are given by v q q v r ρ + v r ρ N = 3 3 11 1 r3 3 1ρ 1+ rρnc Tr ρ + Tr ρ Nc T = 3 3 11 1 r3 3 1ρ 1+ rρnc c (14) (15) In the case of grazing collision, dros kee the same radius and their velocities after collisions are calculated by alying the conservation of linear and angular momentum as well as energy conservation [17]. In the frame of reference used by the KIVA3V code [19], the ugraded drolet velocities can be calculated, under the hyothesis of sherical dros and variable fuel density, with the following equations: v v v r ρ + v r ρ + ( v v ) r ρ Z 3 3 3 1 1 1 1 1q = r3 3 1ρ 1 + rρ v r ρ + v r ρ + ( v v ) r ρ Z 3 3 3 1 1 1 1 1 1 q = r3 3 1ρ 1+ rρ (16) (17) where b bcrit Z = r + r b 1 crit Vaorization Model Due to the heat exchange with the surrounding gas, drolets warm u and vaorize. Drolet temerature and mass exchanges are calculated [0] by solving the energy and mass conservation equations in Lagrangian form, which follows for the reader s convenience. Mass conservation gives 3 ( r ) 1 d ρadmb Sh ρ = 3r dt r (18)

45 VARIABLE-DENSITY SPRAYS where 1/ ln(1 + B) Sh = 1/3 ( + 0.6 Red Sc ) B is the Sherwood number. Energy conservation gives 4 dt d 4 ρ π 3 dt dt πρ = π 3 3 3 rc l r Hlat( T) 4 rq d (19) Equations (18) and (19) can be rearranged as 4 dt ρ π π ρ = π 3 dt 3 Sh rcl admb r Hlat( T) 4 rqd (0) Equation (0) was used to calculate dro temerature rate of change due to heat and mass exchanges. To integrate Eq. (18), fuel density and mass transfer number were assumed to be linearly deendent on time during the comutational time ste t. Thus: ρ () t =ρ ( t0) + mt Bt () Bt ( ) B t = 0 + m [ 0 0 ] for t t, t + t (1) where m and B m are defined as m = B m ρ ( t + t) ρ ( t ) 0 0 t Bt ( 0 + t) Bt ( 0) = t () (3) Moreover, by assuming that ρadm Sh k = = const during the time ste t, Eq. (18) becomes

46 A. DE RISI ET AL. dr 3 ρ() t r() t + 3 m r() t + kb() t = 0 dt (4) This equation reresents the drolet radius rate of variation and allows the calculation of the new dro radius. By integrating Eq. (4) with Eq. (1), the value of drolet radius is given by 1 /3 { ( ρ 310 ( t0) + mt m ) r () t = 9 k ρ ( t0) + mt /3 /3 5 Bt ( 0) m Bm m t 3 Bm ( t0) 3 ( t0) + ρ + ρ 15 Bt ( 0) m k 9 Bm k ρ ( t0) + 5 m r( t0) } (5) where r (t 0 ) = r 0 is the initial drolet radius (i.e., the value of radius calculated at the revious time ste). In the original KIVA3V code [17], drolet radius was calculated according to the following exression: r () t r ( t ) Bt ( ) + Bt ( ) k t 0 = 0 ρ f (6) It is interesting to note that Eq. (6) is obtained by Eq. (5) in the limit of m 0 (i.e., constant fuel density). By using Eq. (5), the dro radius increase due to the heating of the liquid hase, which affects the vaorization rate by changing the volume-to-surface ratio, can be also redicted. Injection Model A stochastic aroach similar to that of Dukowicz [1] was used to erform liquid injection. Injected fuel is simulated as arcels of big dros, or blobs, having a radius equal to the effective nozzle radius. All injected arcels have the same velocity, the same thermodynamic roerties, the same initial mass, and therefore the same number of blobs. The initial mass is calculated by using Eq. (7), and the number of dros or blobs in each arcel is given by Eq. (8): m inj mf = tn inj arc (7)

47 VARIABLE-DENSITY SPRAYS n 0 m = 3 πρl 4 inj ( r ) noz 3 (8) The number of arcels injected in the unit time is determined from the measured mass flow rate. Breaku Model After injection, blobs undergo a breaku rocess that removes mass from the blobs and forms new child dros, whose radius r can be redicted by using the stability analysis of cylindrical liquid surfaces erturbed by an axisymmetric dislacement [, 3]: r = B 0 Λ if B0Λ a 0.33 0.33 3πau 3 aλ r = min, ( Ω) 4 if B Λ> a (9) 0 where B 0 is a constant equal to 0.61 and a indicates the blob radius whose rate of change due to breaku is assumed to be equal to da dt = ( a r) τ (30) where τ = 3.76B 1 a/ Ω and B 1 is an adjustable constant that deends on the nozzle geometry. The wavelength and the maximum growth rate of the fastest-growing waves can be calculated with analytical correlations as a function of the sray arameters, i.e.: ( 1+ 0.45 Oh0.5 0.7 )( 1+ 0.4S ) 1.67 0.6 ( 1 0.87 We a ) Λ = 9.0 a + ( 0.34 0.38 Wea ) ( 1 Oh 0.6 )( 1 1.4S ) 3 0.5 1.5 a ρ + Ω = σ + + (31) (3) where S = Oh (We) 0.5. However, different rates of enetration are observed within and beyond the breaku length because of the resence of an intact core close to the nozzle [4]. To account for this behavior, a model including both Kelvin-Helmholtz (K-H) and Rayleigh-Taylor (R- T) instability criteria has been roosed by Reitz [5]. According to this model, breaku is

48 A. DE RISI ET AL. due to K-H instability until the sray reaches the breaku length and then breaku occurs due to both K-H and R-T mechanisms. The value of the breaku length, L b, is given by L b = ρ l C r ρg (33) where C is an adjustable arameter that deends on the nozzle characteristics. Senecal [6] suggests that the value of the constant C, in the limit of high Weber numbers, is equal to B 1 /. This is because in the hyothesis of negligible liquid viscosity and gas hase, Weber number aroaches infinity, and the breaku time in the K-H model (τ = 3.76B 1 a/λω) can be reduced to τ= B r v u ρ 1 l ρ a (34) Thus, the breaku length, which is assumed to be L b = τu, can be calculated as L b = v u B r 1 ρl ρ a (35) By comaring Eq. (35) and Eq. (33), it is ossible to obtain for the constant C the value of B 1 /. Neglecting liquid viscosity and considering surface tension only, the analytical fastest-growing frequency, Ω t, and the corresonding wave number, Κ, redicted by the R-T instability criteria are given by [5] Ω = t gt( l g) ρ ρ 3 3σ ρ +ρ l g 3/ (36) Κ= g t l g ( ρ ρ ) 3σ (37) where g t is the acceleration in the direction of travel. Other Models The modified RNG κ ε turbulence model roosed by Han and Reitz [7] has been adoted in the resent investigation. This model differs from the standard RNG κ ε

49 VARIABLE-DENSITY SPRAYS model through the resence of an extra term in the dissiation equation that accounts for the comressibility of the flow. The modified RNG κ ε model [7] reroduces the largescale flame structures in a more realistic way, imroving the rediction of the high-temerature domains and of NO and soot emissions. The multi-ste Shell ignition model [9] was used together with the laminar and turbulent characteristic-time combustion model [9] to describe the entire combustion rocess. A temerature threshold of 1100 K for switching from ignition chemistry (T < 1100 K) to combustion chemistry (T > 1100 K) was chosen. The soot emission models adoted in this study are the Hiroyasu formation model [30] and the Nagle and Strickland-Constable oxidation model [31]. COMPARISON BETWEEN EXPERIMENTAL AND NUMERICAL DATA Exerimental investigation are often erformed by using a constant-volume vessel filled with inert gases. The reasons for this choice are manifold. The air motion in the chamber of a diesel engine affects sray gas interactions and increases uncertainties on sray calculations. Moreover, the values of temerature and ressure in an engine cylinder are not uniform, and show significant cycle-to-cycle variation. Finally, inert gases revent combustion in the chamber and allow the study to be erformed in high-density environments without combustion. Thus, a constant-volume chamber was first considered to test the effect of variable fuel density on the KIVA3V results. For the simulations at constant volume, a cylindrical sector has been used with a number of divisions along the x, y, and z axes equal to 30, 36, and 15, resectively. The sray was simulated to be injected 1.5 mm from the axis of a cylinder of 10 mm diameter. The cylinder diameter was sufficient to avoid imingement on the wall. Initially the effect of grid resolution was investigated by increasing the number of divisions along the x, y, and z axes u to 60, 36, and 30, resectively. No significant differences between the two sets of results were observed. The rovisional caability of both standard and modified models has been checked by using exerimental data of enetration and Sauter mean radius (SMR) distribution acquired in a revious investigation [3]. These measurements refer to srays injected by a high-ressure common-rail injection system into a vessel at ambient ressure and temerature. Since vaorization is negligible under these oerating conditions, this case will be henceforth referred to as nonevaorating sray. All the investigated oerating conditions are reorted in Table. Table Oerating Conditions for Nonevaorating Srays Units Case A Case B Case C Energizing time µs 3000 65000 101 Needle oening time µs 59000 95600 136 Injected quantity mm 3 16.4 8.6 00050. Total fuel mass mg 13.6 3.7 00041.6 Ambient ressure MPa 00.1 00.1 00000.1 Ambient temerature K 8800 8800 88 Injection ressure MPa 900 900 090 Injection temerature K 31300 31300 313

50 A. DE RISI ET AL. In the resent article, unless otherwise secified, comarisons between exerimental and numerical data are described only for case A, since the comarison for cases B and C leads to very similar results. The aim of the exerimental investigation [3] was to show the different behavior of the five injector holes. Ficarella et al. [3] carried out a hotograhic characterization of ti enetration and sray angle for each nozzle hole as well as a comlete characterization of the drolet size distribution, by means of Malvern technique. The acquisition time for Malvern investigation was set to 100 µs before the end of the injection, and the laser beam diameter was 9 mm. Measurements were erformed at two axial distances (4 and 48 mm) from the injector hole and at -mm-ste radial ositions from the sray axis. More information about the exerimental setu and results can be found in [3]. In the resent investigation, the corresondence between exerimental and numerical measurement ositions was obtained by using as control volumes hexahedrons 9 mm on a side at stes of 4 mm along the radial direction. The Sauter mean radius in each osition was calculated as SMR N r 3 = N r (38) where the summation is over all arcels in the control volumes, at 100 µs before the end of the injection. Figure 1 shows a comarison between the comuted and exerimental images of the sray as well as the axial distance at which the SMR was measured. Figure a comares the numerical ti enetration with the exerimental data. Since the environment ressure is constant and the drolet temerature changes are very small, differences between the constant-density and variable-density models are small. The value of density versus time redicted by both the tested models was roughly constant and equal to ρ (T, ) = 819 kg/m 3 and ρ (T,, χ a ) = 777 kg/m 3, resectively. In the case of quiescent air and constant dro radius, dro deceleration is given by dv 9µ = dt ρ r l ( u v) (39) Thus, ti enetration increases as fuel density increases. This is confirmed by the results in Fig. a. Dro size distribution has been studied by many researchers, who demonstrated the influence of atomization, collision, and drag resistance on the SMR characteristics of dense srays [4]. The granulometric characteristics of dense srays are the results of cometition between breaku, which tends to reduce drolets size, and collision, which acts in the oosite direction. Thus, drolet size distribution in the axial and radial directions of srays deends on the secific oerating conditions under which the measurements are carried out, and they can be quite different. Payry et al. [34] and Coghe et al. [35], while investigating diesel srays injected in a high-density environment, found that the SMR is

51 VARIABLE-DENSITY SPRAYS Drolet density a) exerimental image b) ρ const = c) ρ = f ( T, ) d) ρ = f ( T,, χ a ) Fig. 1 Comarison between exerimental and numerical sray images at 100 µs before end of injection. aroximately constant along the radial direction. On the other hand, in the case of turbulent disersion of fuel drolets in an unsteady sray, Kosaka et al. [36] found a dro size distribution with larger diameters in the sray erihery than in the center region of the sray. They justified such behavior by assuming that large drolets, due to their higher mass, are more easily centrifuged to the erihery by the motion of large head vortices in the sray ti. According to this hyothesis, centrifugal forces lay an imortant role in the determination of axial and radial SMR distributions. Figures b and c show a comarison between numerical and exerimental dro size distributions. Exerimental data are ensemble-averaged over 0 injections. Due to the asymmetric behavior of Valve Covered Orifice (VCO) nozzles, each nozzle hole generates a differently shaed sray; thus, an uncertainty bar is lotted in Figs. b and c to reresent the maximum difference between the lowest and the highest values of SMR for the five holes. Although dro average size is the same for both numerical and exerimental data, KIVA3V redicts larger dro diameters in the inner region of the sray at 100 µs before the end of injection. The tendency of KIVA3V is to roduce a radial distribution with higher values of SMR along the axis in the case of dense sray due to the high collision robability near the core of the sray. At a 4-mm axial osition, drolets with highly unequal radii exist because of the resence of both an intact liquid core and dros generated by the breaku

5 A. DE RISI ET AL. a) Ti enetration versus time. b) SMR radial distribution at a 4 mm axial osition c) SMR radial distribution at a 48 mm axial osition. Fig. Comarison between exerimental and numerical data for case A. rocess. The robability of coalescence deends on the ratio of the drolet radii, and Eq. (1) shows that dros of unequal size are more likely to coalesce than dros of nearly equal size. Thus, higher values of dro diameter are redicted in this region. Figure 3 shows, for the three tested fuel density models, the axial SMR distribution versus time at 48 mm downstream of the injector nozzle. During most of the injection event and for all the investigated cases, the SMR is greater in the inner region of the sray than in the erihery. As the injection aroaches the end (time = 0.0), the SMR calculated at the center of the sray diminishes as the drolet size at the erihery increases. This is caused by the lower energy of the incoming sray at the end of the injection event. By com-

53 VARIABLE-DENSITY SPRAYS a) ρ = const b) ρ = f ( T, ) c) ρ = f T,, χ ) ( a Fig. 3 SMR radial distribution versus time (t = 0 at the end of injection). aring the grahs in Fig. 3, it is ossible to observe the effect of fuel density on sray SMR. Note that in the case of higher average densities, ρ (T, ) = 819 kg/m 3, the redicted SMR is smaller than in the other cases. Exerimental results for nonevaorating sray confirm that the breaku mechanism and the drolets air interaction are deendent on the gas-toliquid density ratio [3]. In articular, an increase in this ratio leads to larger drolet sizes. Comutational fluid dynamics (CFD) calculations carried out by the authors have also shown that by increasing the gas-to-liquid density ratio, a reduction of the average drolet velocity, and therefore a decrease of the collisional Weber number, were observed. This

54 A. DE RISI ET AL. Table 3 Oerating Conditions for Single-Parcel Investigation Pressure 1 Bar Dro temerature 313 K Total injected mass 3.5 µg Dro velocity 14 m/s Dro radius 10 µm means that the aerodynamic force decreases comared to the drolet surface tension, and the robability of breaku decreases, which in turn leads to larger drolets. EVAPORATING SPRAYS Fuel density variability was exected to have a larger influence on evaorating srays because of the larger temerature gradients. The modified vaorization model [Eq. (5)] allows the increase of dro radius, due to a reduction in fuel density, to be redicted during the heatingu eriod. Therefore, a correct volume/surface ratio for each arcel can be determined. To understand the effect of the variable-density models, the injection of a single arcel was simulated in a high-temerature environment. In this way, the effects of breaku and collision on the vaorization rate were eliminated. Three temerature values (400, 600, and 800 K) for the chamber were analyzed, and the conditions listed in Table 3 were chosen. Figure 4 shows arcel density and vaorized mass for the three temerature values using the two density models and the constant value of 80 kg/m 3 (corresonding to gasoil density at the injection conditions: 313 K, 50 bar). If fuel density is allowed to change, a density decrease versus time in both cases, ρ = f (T, ) and ρ = f (T,, χ a ), can be noticed, and this is due mainly to the heating of the dros. Moreover, the resence of air in the fuel roduces an additional decrease in density that results in an increased vaorization rate. Two effects must be considered to understand the correlation between dro density and vaorization rate: the increase in heat exchange due to the dro surface increasing, and the influence of drolets mass on the heating rate. The key factor in the study of a single-arcel case is the injection model used by KIVA3V. From Eqs. (8) and (7), it is ossible to notice that for a given injected mass, the arcel created by the blobs injection model contains a number of drolets inversely roortional to the initial liquid fuel density and radius equal to the nozzle radius. Under the same conditions, drolets with lower density have lower mass; therefore they are rone to a faster vaorization. Figure 5 shows the effect of the heating-u eriod on dro size when fuel density variability is considered, and the results are comared with the constant-density case. In the case of low ambient temerature (400 K), the drolet exansion is reduced comared with the others; thus, the effect of the increased heat exchange caused by the dro surface increasing can be neglected and the vaorization rate deends mainly on the drolet mass, i.e., on the initial value of their density. In this case the arcel is formed by a nearly equal number of drolets for both cases at ρ = f (T, ) and ρ = const since the initial density is nearly the same,as shown in Fig.4a.Thus,the drolets generated in the two

55 VARIABLE-DENSITY SPRAYS Parcel density Vaorized mass 400 K a) b) 600 K c) d) 800 K Fig. 4 e) f) ρ 3 = 0.8 g / cm f ( T, ) Drolet density and vaorized mass for the single-arcel case. ρ = ρ = f ( T,, χa ) cases have about the same mass. When dro density is calculated as ρ = f (T,, χ a ), the initial value of density is much less than 80 kg/m 3, so a higher number of dros with lower mass than in the other two cases is obtained, and the resulting heating rate is higher. As ambient temerature increases, the two effects (dro surface increasing with temerature and heating rates deending on drolet mass) coexist, and the vaorization rate seems to be roortional to the time-averaged arcel density (Figs. 4c f ).

56 A. DE RISI ET AL. a) 400 K b) 600 K c) 800 K Fig. 5 Drolet radius versus time for the single-arcel case. Further considerations are needed to exlain the dro radius trends versus time reorted in Fig. 5. At the beginning, an increasing dro diameter can be redicted by Eq. (5), due to the heat exchange with the surrounding air. This effect is not resent in the constant-density case since the temerature increase has no influence on fuel density. On the other hand, drolet size decreases due to vaorization; when this effect revails, the dro radius diminishes and the rate of decrease deends on the vaorization rate. By examining Fig. 5c for 800 K ambient temerature, it is ossible to notice that a longer drolet lifetime is redicted under the hyothesis of constant density than in the cases of variable density. Similar results were also found by Gonzales et al. [7].

57 VARIABLE-DENSITY SPRAYS Although the increase of drolet radius versus time could be considered negligible, it must be taken into account when fuel density is allowed to change. This statement can be roved with the use of a reductio ab absurdum, by considering the algorithm used by KIVA3V to calculate the arcel vaorization rate. In the case of nonconstant density, the mass vaorized from a single arcel is calculated in the vaorization subroutine with the following difference: A A ( ) ( ) 4 3 3 m = N π r ρ r ρ 3 (40) The following hyotheses will be considered for the roof: 1. The drolets of the arcel undergo a rocess of heating u without vaorizing during the time ste.. Parcel density deends on drolet temerature and ressure. 3. The dro exansion is not taken into account (i.e., drolet size is indeendent of density). 4. The number of drolets in the arcel does not change. Due to the heat exchange between the arcel and the surrounding air, drolet temerature increases. Thus, the arcel density ρ decreases between hase A and hase of the calculation (ρ A < ρ ṗ ) On the other hand, the drolet radius kees the same value (r A = r ṗ ). As a result of these effects, Eq. (40) redicts a quantity m of vaorized arcel mass, which contrasts with the hyothesis of nonevaorating drolets. Thus, the imortance of calculating the effects of heat exchange on dro radius rate by Eq. (5) instead of Eq. (6) is roven. The temerature law of variation during the heating-u and vaorization eriods is the same [Eq. (0)] in both cases of constant and variable density. However, the rate of temerature increase is underestimated when dro radius variation with density is not taken into account, as the heat exchange between the liquid hase and the surrounding air is roortional to dro surface. After this reliminary study on a single arcel, the same oerating conditions of the nonevaorating case (see Table ) were considered to investigate three cases of evaorating srays with environment temeratures of 400, 600, and 800 K and a ressure of 1 bar, resectively. Figure 6 shows that the effect of the air dissolved in the fuel on the average dro density is quite small in the case of 1 bar. Simulations carried out for an ambient ressure higher than 0 bar have shown virtually no differences due to the resence of air dissolved in the fuel. By comaring the results of Fig. 6 for the three values of temerature, it can be noticed that the trends of vaorized mass versus time are different from those observed in the single-arcel case when the air dissolved in the fuel is taken into account. In articular, even though the average fuel density calculated as ρ = f(t,, χ a ) is lower than the other cases, the vaorization rate at 400 and 600 K follows a contradictory trend. This henomenon was not observed in the case of the single arcel (see Fig. 4), and it could be exlained as follows. When srays are considered, many mechanisms, such as collision and breaku, affect drolet vaorization. Moreover, very small dros heat u and vaorize ear-

58 A. DE RISI ET AL. Average dro density Vaorized mass 400 K a) b) 600 K c) d) 800 K e) f) ρ = 3 0.8 g cm f / ( T, ) ρ = ρ = f ( T,, χa ) Fig. 6 Average dro density and vaorized mass for the sray case. earlier, thus the average size of dros increases due to the dynamics of the vaorization rocess. According to the atomization model used for the simulations [5], the radius of the dros roduced by the breaku rocess deends on both liquid and gas density. The robability of collision deends on the characteristics of the colliding dros, esecially on their radius and density [Eq. (1)]. The combined effects of breaku and collision henomena have been already analyzed in the case of a nonevaorating sray. In the same

59 VARIABLE-DENSITY SPRAYS % of arcels a) ρ = f ( T, ) % of arcels b) ρ = f ( T,, χ a ) Fig. 7 Drolet density distribution versus time at 400 K environment temerature (t = 0 at the end of injection). way it is ossible to state that the interaction among the rocesses involved in sray evolution roduces manifold results which are hard to quantify. In Fig. 7 the drolet density distribution versus time is shown for the two variabledensity models. Note that a different dro density distribution is redicted when the resence of air dissolved in the fuel is taken into account, even if the average dro density is roughly the same for the two variable-density models at 1 bar. This can be exlained by noting that the density variability effect is mitigated by the drolet drolet interactions. Thus, the overall drolet characteristics are only weakly influenced, as shown in Fig. 8. Finally, it is ossible to state that dro radius distribution is not strongly affected by the density model because of the combined influence of the several henomena, as shown in Fig. 9. COMPARISON OF ENGINE EXPERIMENTAL AND NUMERICAL DATA CFD codes are often used to exlore injection strategies for engine ollutant emission control. Thus, the assessment of the rovisional caability of these codes is a very imortant

60 A. DE RISI ET AL. SMR Average dro temerature 400 K a) b) 600 K c) d) 800 K e) f) ρ = 3 0.8 g / cm ρ f ( T, ) = ρ = f ( T,, χ a ) Fig. 8 SMR and average dro temerature for the sray case. issue in the case of engine simulations. Once again, the standard KIVA3V and the modified version resented in the resent investigation were used to comare the calculated and measured emission levels of a direct-injection diesel engine equied with a common-rail injection system. The exerimental data were acquired by using a commercial small-bore, high-seed engine, rovided with adequate instrumentation for the control of the oerating conditions and for the measurement of emissions and erformance arameters. The engine secifications are summarized in Table 4.

61 VARIABLE-DENSITY SPRAYS Fig. 9 Drolets radius distribution vrsus time for a 400 K environment temerature. The injection was erformed by using a six-hole VCO nozzle centrally located in an axisymmetric geometry. Thus, to reduce the comutational time, a 60 sector mesh has been used. The mesh was characterized by 0 cells in the radial direction, 0 cells in the longitudinal direction, and 37 cells in the axial direction, with a minimum of 17 cells at to dead center (TDC). The cylinder dome was of conic shae, while the bowl was of reentrant tye. The resulting comutational grid at TDC is shown in Fig. 10. The combustion and emissions models used for the simulations have been reviously described in the models section. For the breaku model, the values of 5 and 1.5 were chosen for B 1 and C, resectively [Eqs. (33) (35)]. The engine was run at 800 rev/min and at 50% load, and data were acquired at four different values of the start of injection. The investigated oerating conditions are summarized in Table 5.

6 A. DE RISI ET AL. Table 4 Engine Secifications Number of cylinders 4 Bore 88.0 mm Stroke 88.4 mm Connecting rod length 149 mm Comression ratio 19 Intake valves closing (IVC) 173 BTDC Number of nozzles 6 Nozzle diameter 169 µm Injection system Common Rail A comarison between exerimental and numerical ressure traces for cases 1 4 is shown in Fig. 11. It is interesting to notice the resence of fluctuations in the exerimental ressure traces. These fluctuations are believed to be caused by the resence of noise due to ressure wave fluctuation at the osition of the sensor in the combustion chamber. In fact, the frequency of the ressure fluctuation in the exerimental data (16.8 khz) corresonds to a length of about 48 mm, which is about twice the distance between the ressure sensor and the facing wall. Although the comuted comression and exansion ressure traces match the exerimental ones, there is a difference of about 11% between exerimental and numerical ressure eaks. This ga could be reduced by adjusting the arameters of the combustion models that affect the laminar and turbulent characteristic times. However, the standard values of laminar and turbulent times were chosen because redicted NO and soot levels have been shown to be strongly deendent on the values of these arameters [33]. Comarisons of soot versus NO trade-offs for the three density models are shown in Fig. 1 as the injection timing was varied over a range of 15 to 3 ATDC. For comarison, the exerimental values are also reorted, and for each datum, the start of the injection is dislayed. In site of the significant differences between exerimental and numerical ressure eaks, the code reasonably redicts the ignition delay and the soot NO trade-off as the injection timing is varied. As far as the effect of fuel density variability is concerned, lots of Fig. 11 show there are no noteworthy differences among the results obtained by using the three density models. These results are consistent with the constant-chamber outcomes for evaorating srays, which revealed that only the local air drolet and drolet drolet interactions Table 5 Oerating Conditions for Engine Simulations Total injected mass 7.8 mg Injection duration 17.4 Pressure at IVC 0.17 MPa Temerature at IVC 35 K Engine seed 800 rm Start of injection Case 1 = 15 BTDC Case = 11 BTDC Case 3 = 7 BTDC Case 4 = 3 BTDC

63 VARIABLE-DENSITY SPRAYS Fig. 10 Engine grid at TDC. were affected by liquid-hase density distributions, while the combined effects of breaku, collision, and vaorization rocesses resulted in macroscoic sray characteristics which were weakly influenced by the fuel density variability. Moreover, the average values of ressure and temerature during the injection event are very high for the analyzed oerating conditions. Thus, the vaorization rate is very high and almost indeendent of the liquid fuel density. a) CASE 1 b) CASE c) CASE 3 d) CASE 4 Fig. 11 Measured and redicted engine ressure traces.

64 A. DE RISI ET AL. Fig. 1 Measured and redicted NO soot trade-off for the engine case. Since ollutant emissions are controlled by details of the turbulent fuel air mixing and combustion rocesses, differences in liquid-hase density distribution affect soot NO trade-off. Note that the influence of variable fuel density is lower when the injection occurs close to TDC, i.e., in the case of very high values of ressure and temerature in the combustion chamber, because of the faster vaorization and combustion. On the other hand, the effect of dro density variability is more significant for early injections because, in these conditions, the vaorization rate tends to be more effective. Since the functions ρ = f (T, P) and ρ = f (T,, χ a ) give about the same value of density at ressures higher than 0 bar, the different behavior of the code in the two cases could be exlained by considering that the liquid internal energy is influenced by the resence of air dissolved in fuel [Eq. (4)], so that drolet thermodynamic roerties are different. SUMMARY AND CONCLUSIONS The objective of this study was the imlementation of a model for liquid-hase density calculation in the KIVA3V code. The model allows the arcel density to be calculated with a function of drolet temerature and ressure, and the air dissolved in the fuel can also be taken into account. The rovisional caability of the modified subroutines was tested by using exerimental data for non evaorating srays injected into a vessel at ambient temerature and ressure. Good agreement was found between exerimental and numerical results for both the original and modified codes. The model rediction of the drolet vaorization rate was studied for different values of temerature for both single-arcel and evaorating srays. A direct correlation between vaorization rate and arcel density can be found in the case of a single arcel. In the case of evaorating srays, because of the combined effect of breaku and collision, the

65 VARIABLE-DENSITY SPRAYS vaorization rocess becomes a not univocal function of liquid density. However, a difference of about 10% in the vaorization rate was observed by using different density models. Dro size and density distribution were found to be influenced by the model chosen for the fuel density calculation, while the jet macroscoic characteristics are not affected by changes of the liquid density. Engine investigation confirmed that the overall sray evolution is not affected by fuel density variability. The redicted levels of NO and soot emissions are influenced by fuel density variability deending on the local vaorization, air fuel mixing, and combustion henomena. APPENDIX: NUMERICAL DIFFERENCING OF THE CONSERVATION EQUATIONS Because of fuel density variability, new terms including density contributions arise in all the gas liquid exchange equations. The conservation equations can be written as follows: ( ) ( ) dρ 1 4 dt V t s 3 3 i4 = N A A π r ρ n r ρ i4 3 i4 dw 1 4 dt V t 3 ( r ) ( ) s i4 = N A A t πρ n i4 v v u 3 i4 (A1) (A) where the sum is over all arcels in the cell i4 and v t is a artially udated article velocity obtained as described by Amsden et al. [17] by solving the article acceleration equation: v t D t v + u D t = + 1+ D t 1+ D t t un i 4 m (A3) Internal energy exchange rate is calculated according to Amsden et al. [17] 3 3 {( ) ( ) ( ) ( ) dq 1 4 dt V t s i4 = N A A A π r ρ Il T n r ρil T i4 3 i4 3 ( r A A t t ) ( ) ( n v v v u i 4 m u ) 1 ( r A) ( ) } 3 A 3 t n r i4m + ρ + ρ ρ v u (A4) Under the hyothesis of variable fuel density, air sray momentum exchange can be calculated with the following equation:

66 A. DE RISI ET AL. B ( B n n 4 ) 4 4 ( ) 4 4 E B B B i i 4 ( ) ( ) 3 3 M ui M ui = i N π r vρ r v ρ 3 i4 (A5) where E i4 includes all contributions to the gas momentum, excet for the effect of sray gas interaction. After some maniulations, Eq. (A5) can be written as follows: ( B ) ( ) B n n i4 i4 i4 i4 M u M u ( ) ( ) 4 = E πρ 3 3 3 B B B i4 N r v r v i4 i4 ( ) 3 4 + B N π r v ρd ρ 3 (A6) The udated velocity v B is obtained by solving the drolet acceleration equation: v v = + t ( u 4 u v ) B D B B i (A7) Thus, the udated dro velocity is given by: v B D t v + u D t = + 1+ D t 1+ D t ub i 4 (A8) where the drag function D is D u + u v = ρ n B 3 ρn i4 i4 CD 8 A f r ( Re ) (A9) By using Eq. (A8), the gas momentum equation can be arranged as [( M ) B i4 ]u B i4 ( M ) n u n i4 = E i4 i4 R i4 T i4 (A10) where 4 ( ) 3 t D B B Si 4 = N πρ r 3 1+ t D i4 (A11)

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