Baumgarten

154 4 Modeling Spray and Mixture Formation

where av = 15.903 cm3/mol and bv = 1.21 cm3/mol2. The combination of Eq. 4.198) and Eq. 4.199 results in [77]

Tcrit ac av avbcT acbvT bcbv\ .

(4.200)

Typical results obtained with the multi-component evaporation model by Pagel [100] are shown in the following figures. First, the temperature rise inside a droplet (diesel fuel) during evaporation is shown in Fig. 4.43. The corresponding change of the distribution function is given in Fig. 4.44. In contrast to singlecomponent fuels that first heat up and keep a constant temperature until the end of evaporation, Fig. 4.43 shows the typical strong temperature rise at the end of evaporation due to the multi-component fuel modeling.

Fig. 4.43. Change of liquid temperature during evaporation of a multi-component diesel

droplet (initial values: D = 100 µm, u = 0 m/s, Tdrop = 300 K, Tgas = 973 K, pgas = 0.1 MPa) [100]

Fig. 4.44. Change of liquid composition during evaporation of a multi-component diesel

droplet (initial values: D = 100 µm, u = 0 m/s, Tdrop = 300 K, Tgas = 973 K, pgas = 0.1 MPa) [100]

4.5 Evaporation	155

Fig. 4.45. Vapor phase fuel mass fractions for evaporating n-C14H30 (left) and continuous

multi-component fuel (right) sprays in nitrogen after 2 ms, mfuel = 40 mg, Tfuel = 300 K, pgas = 5.7 MPa, Tgas = 800 K, pinj = 110 MPa [100]

The application of Raoult’s law for every component of the distribution guarantees that the diffusive mass transport of the components with low molecular weight and high volatility is larger than that of the heavier components. Hence, most of the evaporated fuel mass consists at first of the lightweight components, and only a few heavier components leave the liquid drop. Thus, the distribution function is shifted to larger molecular weights, and its variance decreases, Fig. 4.44. The value f(I) of the distribution function increases because the curves are normalized and the area below is always 1. The droplet temperature is controlled by the balance of energy that is transferred from the hot gas to the drop and of energy that leaves the drop due to the evaporation of liquid mass. As long as the mean molecular weight of the evaporated mass is low, the liquid temperature can be kept down. As evaporation proceeds, the mean molecular weight of the evaporated mass increases. Due to the lower volatility of the heavier components (Raoult’s law), the droplet temperature increases until the energy loss due to an enhanced evaporation at higher temperature compensates the energy input from the hot gas again. If the droplet composition does not change again, a steady-state as described in the case of a single-component fuel, Fig. 4.38, could be reached.

At the very end of evaporation, only a few components with increasing molecular weight and strongly decreasing volatility are left. Now the heat flux to the droplet results in a strong increase of droplet temperature.

In Fig. 4.45, the evaporation of a complete spray under engine-like conditions is shown in the case of multi-component diesel fuel and tetradecane (n-c14H30), which is usually used in standard single-component evaporation models in order to represent the behavior of diesel fuel. The figure shows the overall fuel mass fraction in the gas phase two milliseconds after the start of injection. Although the mean properties of the multi-component fuel and tetradecane are very similar,

156 4 Modeling Spray and Mixture Formation

Fig. 4.46. Representation of different molecule classes by different distribution functions

significant differences concerning the overall mixture formation process are visible. Due to the stronger initial evaporation of the more volatile components near the nozzle, evaporation is enhanced, droplet sizes reduce faster, and the specific aerodynamic forces are increased resulting in a smaller penetration and a broader fuel vapor distribution. The simulations of Pagel [100] have also revealed that due to the strong evaporation at the edges of the spray the droplets at these positions have a higher molecular weight than those in the center of the spray. Although initial evaporation is increased in the case of multi-component diesel, ignition processes start later compared to a tetradecane spray. At first, the lightweight components with high volatility but low ignitability evaporate, and the molecules with longer carbon chains that produce radicals much more easily are the last ones to enter the vapor phase. In the case of tetradecane, its long molecules can earlier start to initiate ignition.

It should be mentioned that it is not always possible to represent the relevant fuel properties of multi-component fuels by using only one distribution function. Although the only use of the distribution of alkanes shows accurate results in the case of diesel, this simple approach is no more sufficient if for example tailored HCCI-fuels consisting of two or more groups of completely different molecule classes are considered. In this case, at least one more distribution function must be implemented describing the behavior of the second molecule group, Fig. 4.46. Most recently, a combination of two distribution functions has been investigated by Fischer et al. [37]. This way of modeling multi-component fuels promises a much more accurate representation of the evaporation behavior and is especially needed in the case of HCCI engine simulations, where ignition delays are long and the low-temperature ignition processes strongly depend on the temporal composition of the fuel vapor phase, see also Sect. 6.4.

In the case of conventional diesel engines, the Shell-model (see Sect. 4.9.1) is normally used in 3D CFD codes today to describe the ignition process. Numerical investigations have shown that the cetane number of the fuel has an important effect on the ignition behavior. The larger the cetane number, the larger the ignitability. Thus, the effect of a variable cetane number due to the change of fuel vapor

4.5 Evaporation	157

composition in the gas during the evaporation of a multi-component fuel has to be taken into account. Heywood [51] has given a correlation describing the activation energy for auto-ignition as function of the cetane number (CN):

EA	618840	.	(4.201)
	CN 25

This approach was implemented by Ayoub [9] in the standard Shell-model. The expression for the activation energy of the chain propagation reaction

	f4 kp		(4.202)
R		o R Q ,

(Sect. 4.9.1), which has been shown to be a crucial path for the imtermediate ignition species to transform into the branching species [68], was modified:

	E* f 4	E f 4	65		.	(4.203)
			CN	25

In the case of diesel fuel with a cetane number of 40, the activation energy remains unchanged. In order to apply this approach to multi-component fuels, further equations describing the relation between cetane number and mean molecular weight of the distribution function are necessary. Rose and Cooper [120] have measured the cetane number of pure hydrocarbons, Fig. 4.47. The relation between cetane number and mean molecular weight of the fuel vapor can approximated by the function

CN paraffins 4.2438 10 6T3 1.7080 10 3T2 0.14675T 29.295 . (4.204)

Fig. 4.57. Cetane number as function of molecular weight of paraffins (alkanes), data from [120]

158 4 Modeling Spray and Mixture Formation

However, diesel fuel does not only consist of n-alkanes but also of other components like iso-alkanes and aromats. Thus, the cetane number of Eq. 4.204 must be modified by another correlation given by Glavincevski et al. [42]:

CN 30 0.221182 CN paraffins 47.18, 48 d CN paraffin d110 .

(4.205)

This approach has been successfully used by Lippert [77] and Pagel [100] in order to simulate the ignition behavior of a multi-component fuel using the Shellmodel.

4.5.3 Flash-Boiling

When a liquid, initially in a subcooled state, is rapidly depressurized to a pressure sufficiently below the saturated vapor pressure, it can no longer exist in the liquid state, and a rapid boiling process called flash-boiling is initiated. A portion of the fuel then evaporates instantaneously and cools the rest of the liquid down. This sudden evaporation results in a significant increase of spray volume and a faster spray break-up. In the case of high-pressure diesel injection, the phenomenon of flash-boiling can only be achieved if the fuel is sufficiently preheated before injection. In the case of gasoline injection, flash-boiling is much easier to obtain due to the lower boiling curve. Especially if gasoline is injected in the intake manifold, where the static pressure can fall below the saturated vapor pressure of some hydrocarbon fuel components. Such a condition will result in unintended flashboiling. This causes significant changes in the fuel spray distribution and the fuelair mixing.

Fig. 4.48 shows the conventional and the flash-boiling fuel injection in a pres- sure-enthalpy diagram. Subcooled liquid exists to the left of the liquid saturation line, and superheated vapor exists to the right of the vapor saturation line.

Fig. 4.48. Comparison of conventional injection and flash-boiling injection [99]

4.5 Evaporation	159

Superheated liquid can exist for a significant period of time without phase transition in a metastable condition between the liquid saturation line and the liquid spinodal, while to the right of the liquid spinodal there is no metastable state and liquid and vapor must coexist. During injection, the highly pressurized fuel leaves the nozzle through the injection hole, in which the liquid is strongly accelerated and the pressure decreases. In the case of conventional injection (line 1’- 2’), the fuel temperature, and thus the enthalpy, is too low to cross the liquid saturation line during pressure decrease. In the case of flash-boiling injection, the increased fuel temperature results in a higher fuel enthalpy, and the fuel undergoes a pressure reduction from point 1 to point 4 while passing through the nozzle hole. Between point 2 and point 3, vapor bubbles are formed and begin to grow. If there were be enough time, an equilibrium vapor fraction would be achieved. As point 3 is approached, the nucleation rates become large, and beyond point 3 the transition from vapor to liquid becomes explosively rapid.

The flash-boiling process consists of three stages: nucleation, vapor bubble growth, and atomization [99]. A nucleus is a vapor bubble in a metastable equilibrium with the surrounding liquid. The size r of the nucleus depends on a force balance between the surface tension force tending to reduce the bubble radius, and an opposing force due to the difference of saturated vapor pressure psat(T) inside the bubble and the static pressure p in the surrounding liquid,

	r0	2ς		,	(4.206)
		psat T	p

where ς is the surface tension of the liquid in contact with its vapor. The larger the degree of superheat at a constant pressure of the liquid, the larger the saturated vapor pressure, and the smaller the bubble radius in equilibrium. A decrease of surrounding pressure p will result in bubble growth. Nuclei can form in crevices and other imperfections at the nozzle hole wall (e.g. roughness), on solid particles in the flow, etc. Small air bubbles entrained in the flow can also provide nucleation sites. At high degrees of superheating, nucleation even occurs at random locations throughout the liquid in the absence of particles or air bubbles.

During bubble growth, fuel evaporates at the bubble wall, and the vapor is added to the bubble volume. The theory of vapor bubble growth is based on the same basic equation as the theory of bubble collapse in the case of cavitation. Usually the Rayleigh-Plesset equation (e.g. [108, 48]) or some advanced forms of this equation are used in order to describe the bubble dynamics. In the case of vapor bubble growth due to flash-boiling, the latent heat of vaporization, which is transferred from the liquid to the bubble surface, must be included (e.g. [99]).

Atomization is the final stage of the flash-boiling injection process. Three possible mechanisms of atomization are discussed in the literature: bubble coalescence, inertial shattering, and-micro explosions of droplets. In the case of bubble coalescence, it is assumed that bubbles grow until they touch each other, resulting in a transition from bubbles in a liquid matrix to liquid drops in a vapor matrix. The release of surface tension energy may then result in further atomization.

160 4 Modeling Spray and Mixture Formation

Fig. 4.49. Comparison of flashing and non-flashing full-cone sprays [98]

In the case of inertial shattering, the rapid bubble growth outside the nozzle causes a momentum in radial direction that finally results in disintegration. In addition to these effects, bubble growth inside primary droplets that are still superheated can produce micro-explosions and result in further atomization.

There are three potential benefits of flash-boiling injection: enhanced atomization, increased initial spray cone angle for faster fuel-air mixing, and reduced spray penetration. These advantages can be attractive in direct injection diesel and stratified-charge gasoline engines as well as in HCCI combustion, in which fuelair mixing rates and spray penetration must be carefully matched to the combustion chamber geometry and to the gas temperature and pressure in order to avoid wall wetting. Further on, a better atomization and fuel-air mixing due to flashboiling might also enhance cold-starting processes.

Fig. 4.49 shows a comparison between a non-flashing gasoline spray from a multi-hole injector (full-cone spray plumes) and the flashing condition, which is achieved by an increase of initial fuel temperature from 80°C to 100°C (vapor pressure exceeds ambient pressure). Both sprays are injected into atmospheric air. As can be seen, the spray pattern changes completely: the instantaneous evaporation results in an increase of air-fuel mixing (the single spray plumes are no more visible) and in reduced penetration.

In contrast to Fig. 4.49, Fig. 4.50 shows a comparison between a non-flashing hollow-cone iso-octane spray and the corresponding flashing spray. The fuel has

been injected into a pressure chamber (Tchamber = 323 K, pchamber = 50 kPa, prail = 8 MPa). Again, the spray structure changes significantly if flash-boiling occurs: the

hollow-cone spray collapses into a full-cone structure, but this causes a decrease of cone angle and an increase of spray penetration due to the compact spray structure. The droplets at the tip of the spray induce air motion and reduce the relative velocity between air and the following drops, and since the droplets are surrounded by saturated fuel vapor, evaporation is inhibited and drop sizes reduce slower [126].

Numerous experimental studies of flash-boiling injection have been performed to examine the phenomenon in detail. Gerrish and Ayer [41] observed an increase of spray cone angle when diesel fuel was preheated prior to injection. Kim et al. [62] performed flash-boiling studies with alcohol, in both an engine cylinder and a test chamber. The measurements of drop sizes, spray cone angle and penetration confirmed that flash-boiling provides the benefits listed above. Similar results

4.5 Evaporation	161

Fig. 4.50. Comparison of flashing and non-flashing hollow-cone sprays [126]

were obtained by more fundamental experiments in atmospheric pressure chambers (e.g. [146, 99]).

The experimental investigations have revealed that there are two main categories of flash-boiling sprays, dependent on the degree of superheat: external flashboiling and internal flash-boiling (e.g. [99, 103]). In the case of external flashing, evaporation and rapid bubble growth occur outside the nozzle in the spray. The rapid bubble growth shatters the liquid jet and results in an increased spray cone angle and a reduced penetration. In the case of internal flashing, rapid bubble growth occurs already inside the nozzle hole, resulting in an under-expanded compressible two-phase flow that expands immediately upon leaving the nozzle.

External flashing is difficult to obtain because a smooth nozzle entry geometry and/or low injection velocities must be used in order to avoid regions of locally low pressure inside the injection holes and to suppress internal flashing. Further on, careful matching of injection pressure and fuel temperature with combustion chamber pressure is necessary during the time of injection, because the degree of superheat and thus the change from the external to the internal flashing regime is very sensible to the chamber pressure. Hence, the relevant flashing regime is the internal flashing mode. However, internal flashing results in a reduction of effective cross-sectional area inside the injection hole and thus reduces the mass flow through the nozzle. At high degrees of superheat, the nozzle hole can become va- por-locked, and the mass flow reduces drastically [115]. In engine applications this has to be avoided at all costs.

Hence, the optimum degree of superheat is difficult to control, and this is one of the reasons why flash-boiling is not used today in series production engines. However, flash-boiling might become an important effect for future DISI and HCCI engines because of the low gas pressures inside the combustion chamber during early injection.

The development of flash-boiling models is challenging for several reasons. In the case of internal flashing, the nozzle hole flow must be closely linked to the primary spray formation process. Thus, some kind of nozzle hole flow modeling must be included. Non-equilibrium effects have to be included by sub-models describing the bubble dynamics, the nucleation rate in a metastable zone as well as the inception locations. Further on, the multi-component nature of the fuel (no dis-

162 4 Modeling Spray and Mixture Formation

tinct boiling temperature and vapor pressure, but rather continuous curves) has to be accounted for.

In spite of all these difficulties, flash-boiling models have already been developed and implemented in CFD-codes. Some authors have extended conventional spray models and included the effect of flash-boiling by changing the starting and boundary conditions. Others included already detailed nucleation and bubble growth models.

An atomization model for superheated fuel sprays from pressure-swirl atomizers including the effect of flash-boiling has been recently developed by Zuo et al. [150]. The model is based on the linearized instability sheet atomization model (LISA) of Senecal et al. [129] and Schmidt et al. [125], Sect. 4.1.6. It is assumed that under superheat conditions a hollow-cone spray sheet is still formed from the pressure-swirl atomizer, and the sheet flash-boiling is controlled by the rate of heat that can be conducted inside the sheet with an effective thermal conductivity. Hydrodynamic instability, cavitation and bubble growth finally break the sheet up to form drops. Models for the subsequent drop vaporization account for heat transfer under flash-boiling and sub-boiling conditions.

Further models considering the effect of flash-boiling on spray atomization, including detailed nucleation and bubble growth models, are published, for example, by Fujimoto et al. [38], Kawano [61] and Zeng and Lee [148].

4.5.4 Wall Film Evaporation

In Sect. 4.8, it is shown that spray wall impingement and the formation of liquid films may have an important effect on spray atomization and mixture formation. In the case of film formation on a hot wall, which can happen, for example, in a small-bore direct injection diesel engine if the spray impinges on the hot piston surface, its evaporation strongly influences the mixture formation process in the near wall region and must be included in CFD models.

The description of wall film evaporation is based on the wall film energy equation

						DE	¦Q ,	(4.207)
						Dt
where	the left hand side represents the material							derivate of the energy E
( E
	Ulcv,lTl		dxdydz , Tl		: mean film temperature, cv,l: liquid specific heat), and the

right hand side is the sum of all external energy fluxes changing the energy inside a film cell like energy fluxes due to conduction, impinging droplets or splashing.

An early model for the simulation of wall film evolution and heat transfer from the wall to the film or to the impinging drops was developed by Eckhause and Reitz [31] and is described in Sect. 4.8.3.

A more detailed model of film evaporation has been developed by O’Rourke and Amsden [94]. The temperature profile in the film normal to the surface is approximated to be piecewise linear, Fig. 4.51, varying from the wall temperature

4.5 Evaporation	163

Twall to Tl in the lower half of the film, and from Tl to a gas surface temperature Ts in the upper half of the film. The film energy equation is

ªwT

&

º

Ul cv,l dxdydz «

l

v film s Tl »

wt

¬

¼

(4.208)

Twall

ªTs Tl

Tl

º

O dxdy

«

»

Q

Q

,

l

G / 2

G / 2

imp

splash

¬

¼

where the coordinate system is shown in Fig. 4.52. The liquid specific heat cv,l and

the liquid heat conductivity Οl are temperature-dependent. Qimp = m imp·el(Tdrop) is the energy flux due to impingement (el(Tdrop): specific internal energy of the impinging droplet mass), and Q splash = msplash el(Tl ) is the energy flux due to splashing. The left hand side of Eq. 4.208 is the material derivate and consists of the

time derivate and the convective term due to film movement. The first term on the

right hand side expresses the effect of heat conduction (Q cond = -Οl dx dy (7/ z)) is the sum of the heat transfer between gas and film (upper half of the film) and

between film and wall (lower half of the film), Fig. 4.51.

Dividing Eq. 4.208 by the wall surface area Awall = dx dy and remembering that

		yields
dz = Γ and Q / A q

Fig. 4.51. Piecewise linear film temperature profile

Fig. 4.52. Liquid film element and coordinate system