A contribution to the understanding of isothermal diesel spray dynamics

J.M. Desantes, R. Payri

*

, J.M. Garcia, F.J. Salvador

CMT-Motores Te´ rmicos, Universidad Polite´ cnica de Valencia, Camino de Vera s/n, E-46022 Valencia, Spain

Received 11 May 2006; received in revised form 11 October 2006; accepted 12 October 2006Available online 10 November 2006

Abstract

A research on diesel spray dynamic injected into stagnant ambient air in a chamber is reported in this paper. As a result of a theo-retical reasoning based on momentum ﬂux conservation in the axial direction of the diesel spray, a mathematical model which relates themomentum ﬂux with proﬁles of velocity and concentration is obtained. The main contribution of the present investigation, which makesit diﬀerent from previous work in the literature, is the consideration of local density variations and the deduction of the model for ageneric Schmidt number. A PDPA system and additional measurements of spray momentum, mass ﬂow rate and spray cone angle wereused in order to validate the model in high density environment and real injection pressure conditions.

2006 Elsevier Ltd. All rights reserved.

Keywords:

Diesel spray; Schmidt number; Spray momentum

1. Introduction

Although sprays are commonly used in many industrialapplications their study has always been diﬃcult due to thecomplex phenomena involved: atomisation, mixing, coales-cence, transfer of mass and momentum and evaporation[1].This complexity is accentuated in the particular case of sprays in direct injection diesel engines because of the highfrequency transient operation and the small characteristictime and length (

1 ms and 25 mm). In such adverse con-ditions from the point of view of the experimentation, thespray characteristics that can be measured are very limited.The most typical are spray tip penetration and spray coneangle [2 – 12] which are macroscopic characteristics, and
droplet velocity and droplet diameter, which are micro-scopic features [13 – 19]. In general, macroscopic measure-
ments are more reliable than microscopic ones and itwould be interesting to discover how the formers relateto the latter’s. One of the key parameter which relatesmicroscopic and macroscopic characteristics of the sprayis momentum ﬂux. It is considered by several authors asone of the most important parameter governing the spraydynamics [5 – 7,20 – 22]. The momentum ﬂux brings together
the eﬀective ﬂux velocity at the oriﬁce outlet, the fuel den-sity, and the eﬀective diameter of the nozzles oriﬁces. Ricouand Spalding [21] found that diﬀerent gas jets behave in asimilar way if both momentum ﬂux and exit velocity arethe same. Another important aspect concerning spraydynamics is that it depends on air density rather than onair pressure. This issue was pointed out by Lefe`bvre, [1]and more recently conﬁrmed by Naber and Siebers [4]and Desantes et al. [11], who performed a test of sprayinjected into two diﬀerent gases with the same gas densitybut at diﬀerent pressures. In the case of Naber and Siebersthe spray behavior was exactly the same in both cases. Inthe case of Desantes et al. very small diﬀerences werefound.In this paper, as a result of a theoretical approach basedon momentum ﬂux conservation along the spray axis, amathematical model is presented which relates the momen-tum ﬂux with axial proﬁles of velocity and concentration.The main point of the model, which makes it diﬀerentfrom previous work in the literature [3,4,10,12], is the

0016-2361/$ - see front matter

2006 Elsevier Ltd. All rights reserved.doi:10.1016/j.fuel.2006.10.011

*

Corresponding author.

E-mail address:

rpayri@mot.upv.es (R. Payri).

www.fuelﬁrst.com

Fuel 86 (2007) 1093–1101

consideration of local density variations and the deductionof the model for a generic Schmidt number.As far as the structure of the paper is concerned, thearticle is divided in four parts. In Section 2, the develop-ment of the spray model is shown and the assumptionstaken into account are justiﬁed. At the end of this sectiontwo simpliﬁcations of the model are presented: the ﬁrstone concerns the case of Schmidt number equal to one,and the second simpliﬁcation consists of assuming constantlocal density equal to the air density in the chamber.In Section 3, after two studies about the inﬂuence of theSchmidt number on the axis velocity and about the accu-racy of the model, the validation of the model is performed.In order to carry out this validation, measurements of droplet velocities with a phase doppler particle anemome-try system (PDPA) device have been performed. Finally,in Section 4, the most important conclusions of the workare drawn.

2. Theoretical analysis

2.1. Background and global considerations

Fig. 1 shows an image of a typical transient sprayinjected with constant nozzle exit velocity. Two diﬀerentregions in the spray can be observed: a conical-shaperegion from the nozzle to approximately 70% of the totalpenetration, which is denoted as the steady region, andan elliptical-shape region, which has been denoted as thetransient region or unsteady region [2,22].Regarding the transient region, located at the spray tip,it has a very complicated vortical structure that makes itdiﬃcult to study. There is a lot of information about thissubject available in the literature, but there is no clearagreement among the diﬀerent authors. For this reason,several authors [2 – 4,12,22] have tried to avoid this region
and focus their study on the steady part of the spray.Following the gas jet analogy, which will be later ana-lyzed, the internal structure of the steady region of thespray can be divided into two other regions: the initialregion, located near the oriﬁce of the nozzle, where the fuelon the spray axis has not been perturbed by the entrainedair (and therefore fuel concentration can be considered as

Fig. 1. Example of a spray image taken 825

l

s after the start of theinjection. Injection pressure is 80 MPa, nozzle diameter is 0.14

l

m and gasdensity is 20 kg/m

3

.

Nomenclature

A

,

B

,

G

variables used in the integration of the spraymomentum equation

a

,

b

,

c

,

d

variables used in the deﬁnition of the hypergeo-metric function in Appendix A

C

(

x

,

r

) mass concentration in the coordinate (

x

,

r

) of thespray

C

axis

(

x

) mass concentration in the coordinate

x

of thespray’s axis

i

counter of Taylor’s series

M

o

momentum ﬂux at the nozzle outlet oriﬁce

_

m

f

fuel mass ﬂux

m

f

fuel mass

m

a

air mass

P

back

backpressure

P

in

injection pressure

D

mass diﬀusivity

r

radial coordinate

R

radius of the spray

S

spray tip penetration

Sc

Schmidt number

t

time

U

axis

(

x

) velocity in the coordinate

x

of the spray’s axis

U

o

oriﬁce outlet velocity

U

(

x

,

r

) axial velocity in the coordinates (

x

,

r

) of thespray

x

axial coordinate

Greek symbols

a

coeﬃcient of the Gaussian radial proﬁle for theaxial velocity

/

o

outlet diameter of the nozzle’s oriﬁce

n

non-dimensional radial coordinate (

n

=

r

/

R

)

q

a

ambient density

q

f

fuel density

q

(

x

,

r

) density in the coordinates (

x

,

r

) of the spray

m

viscosity

p

Pi number

f

variable used in the integration of the spraymomentum equation

h

u

spray cone angle

1094

J.M. Desantes et al. / Fuel 86 (2007) 1093–1101

unity, and the local velocity is still the same as the exitvelocity), and the main region or fully developed region,where the fuel in the whole section of the spray has beenperturbed by the entrained air [2]. A schematic view of these two regions is presented in Fig. 2. The present theo-retical development will focus on the main steady regionof the spray.Many advances have been made in the ﬂuid mechanicsof jets in the past, and the quantitative and qualitative basisestablished for the jet theory can be conveniently utilizedfor the spray phenomenon as well. Adler and Lyn [23] wereone of the ﬁrst authors to propose a study of sprays using acontinuous model of a gas jet, stating that this was justiﬁeddue to the similarity between gas jets and sprays from thepoint of view of basic mechanism. Since then, many otherresearchers have followed this path, as for example Rifeand Heywood [24] who developed a model to predict spraybehavior based on gas jet equation, or Prasad and Kar [25],who performed an investigation in order to analyse theprocesses of diﬀusion of mass and velocity obtaining quan-titative data for treating the diesel spray as a turbulent jet.These and many other investigations imply that manyresults from the literature concerning gas jets will bedirectly applicable to sprays. The main diﬀerence betweena turbulent gas jet and a spray is that, for a given nozzlegeometry, the jet has a constant cone angle [26] and it doesnot depend neither on injection pressure nor on ambientdensity, whilst the diesel spray has a cone angle thatdepends on the operating conditions [4 – 14,17]. In this
statement, the cone angle is assumed to be that correspond-ing to the main region of the jet or spray (see Fig. 1).An additional important feature concerning the radialevolution of axial velocity and fuel concentration is self-similarity. Rajaratnam [27] among others [1,25 – 31], found
that, for any section in the fully developed region of thespray, if the velocity at any radial position is divided bythe centerline velocity and plotted versus the normalizedradius (

r

/

R

), where

r

is the radial coordinate and

R

thespray radius, it has a unique evolution.This result can be expressed as

U

ð

x

;

r

Þ¼

U

ð

x

;

0

Þ

f

ð

r

=

R

Þ ð

1

Þ

where

f

is a radial proﬁle for

U

. The same result is obtainedif fuel concentration is considered:

C

ð

x

;

r

Þ¼

C

ð

x

;

0

Þ½

f

ð

r

=

R

Þ

Sc

ð

2

Þ

where

Sc

is the eﬀective Schmidt number.The Schmidt number is the ratio of eﬀective momen-tum diﬀusivity to eﬀective mass diﬀusivity, and representsthe relative rate of momentum and mass transfer, includ-ing both molecular and turbulent contributions. It isdeﬁned as

Sc

¼

m

D

ð

3

Þ

with

m

the viscosity, and

D

, the mass diﬀusivity.An immediate consequence of self-similarity is that asigniﬁcant simpliﬁcation can be made when presentingresults: only centerline axial velocity and fuel concentrationare required, as values for any other point can be deducedfrom centerline values. According to the studies presentedby the previous authors, this result is only valid in thesteady region of free sprays (without wall impingement orinteraction). That is the reason why this assumption willbe used throughout the present work.

2.2. Model hypotheses

The hypotheses assumed to carry out the theoreticalderivation of the model are based on the evolution of anisothermal spray under steady boundary conditions:

•

The environment is quiescent and so no axis deﬂectionexists.

•

Air density in the injection chamber is constant duringthe whole injection process.

•

Constant temperature is assumed for both the injectedfuel and air.

•

Momentum, and thus, injection velocity and mass ﬂowrate, are constant during the whole injection process.

•

A Gaussian radial proﬁle is assumed for the axial veloc-ity. At this point it is necessary to point out that radialdistributions of axial velocity are not well known insprays. As commented before, some authors use gas jet distributions as a ﬁrst approximation. Experimentalsimilarities between them have been always remarkedby other researchers [4,12,22,23,25,27,29]. Diﬀerent
expressions for radial proﬁles can be found in the liter-ature [12,22,29 – 32]. Correas [12] made a comparative
study of all of them, and proposed a modiﬁcation of the expressions by Hinze [30], which has been usuallyconsidered the proﬁle that better ﬁts the available exper-imental data in the literature. This proﬁle [12] was alsoassumed by Pastor et al. [33] and Desantes et al.[22,34] in more recent studies. Even though it is includedhere as an assumption, results obtained with a PDPA(phase doppler particle anemometry) system will be pre-sented in the following sections, which will show that theGaussian proﬁle is a reasonable approach for the type of sprays within the scope of the present work.Besides all these hypotheses, the study will be focused onthe steady regions (and so on the conical part) and in thefully developed region of the liquid spray. This conﬁrms

Fig. 2. Initial and main region in a jet.

J.M. Desantes et al. / Fuel 86 (2007) 1093–1101

1095

the validity of the previous hypotheses, according to theprevious background section.

2.3. Theoretical derivation

In order to rigorously impose momentum ﬂux conserva-tion in a free gas jet or a diesel spray, it is necessary to takeinto account the radial evolution of both axial velocity andfuel concentration. For any section perpendicular to thespray axis in the steady region of the gas jet or diesel spray,momentum ﬂux is conservative, and thus equal to thatexisting at the nozzle exit [6,8]. Consequently, the following
equation can be written:

M

o

¼

M

ð

x

Þ ð

4

Þ

where

M

(

x

) and

M

o

are the momentum ﬂux through aspray cross-section at a distance

x

and at the oriﬁce outlet,respectively. It can be assumed that the radial proﬁle of thevelocity at the nozzle exit is ﬂat, and thus

M

o

=

m

f

U

o

,where

m

f

is the mass ﬂux, and

U

o

the oriﬁce outlet velocity.Inﬂuences of a non-ﬂat proﬁle have been studied by Postet al. [35]. They show that, if the mass and axial momentumﬂuxes are the same, the inﬂuence of the proﬁle shape is con-ﬁned to the initial region. In the main jet region, which isthe focus of the present paper, the distribution of axialvelocity is identical.In Fig. 3, a coordinate system (

x

,

r

) is considered locatedat the srcin of the spray, in such a way that the

x

-coordi-nate coincides with the spray axis, and the

r

-coordinate isthe radial position (perpendicular to the spray axis). Inorder to develop expression (4), momentum must be inte-grated over the whole section:

M

o

¼

M

ð

x

Þ¼

Z

R

0

2

pq

ð

x

;

r

Þ

rU

2

ð

x

;

r

Þ

d

r

ð

5

Þ

where

q

(

x

,

r

) is the local density in the gas jet or dieselspray, and

U

(

x

,

r

) is the axial velocity.The density at an internal point of the spray, taking intoaccount the local concentration, can be written as

q

ð

x

;

r

Þ¼

q

f

1

C

ð

x

;

r

Þ

1

q

f

q

a

þ

q

f

q

a

ð

6

Þ

with

q

f

, the fuel density,

q

a

the air density and

C

(

x

,

r

) thelocal fuel mass concentration deﬁned as

C

¼

m

f

m

a

þ

m

f

ð

7

Þ

with

m

f

the local mass of fuel, and

m

a

the local mass of air.For the developed region in the spray, fuel concentra-tion and axial velocity can be considered to follow a Gauss-ian radial proﬁle:

U

ð

x

;

r

Þ¼

U

axis

ð

x

Þ

exp

a

r R

2

ð

8

Þ

C

ð

x

;

r

Þ¼

C

axis

ð

x

Þ

exp

a

Sc r R

2

ð

9

Þ

with

Sc

the Schmidt number, and

a

the shape factor of theGaussian distribution. As stated before, these are the func-tions that better ﬁt the available experimental data in theliterature [20,30,33].
Substituting Eqs. (6), (8) and (9) in Eq. (5), the momen-
tum in any section of the spray can be expressed by Eq.(10):

M

o

¼

2

p

U

2axis

q

f

Z

R

0

r

exp

2

a

r R

2

C

axis

1

q

f

q

a

exp

a

Sc

r R

2

þ

q

f

q

a

d

r

ð

10

Þ

Integrating Eq. (10) and taking into account that the radiusof the spray

R

, can be expressed with respect the spray coneangle:

R

¼

x

tan

h

u

2

ð

11

Þ

the following expression is obtained (Appendix A):

M

o

¼

p

2

aq

a

tan

2

h

u

2

x

2

U

2axis

X

þ1

i

¼

02

Sc

2

Sc

þ

i

C

m axis

q

f

q

a

q

f

i

ð

12

Þ

It can be demonstrated [34] that

C

axis

¼

1

þ

Sc

2

U

axis

U

o

ð

13

Þ

Therefore, Eq. (12) transforms into Eq. (14):

M

o

¼

p

2

aq

a

tan

2

h

u

2

x

2

U

2axis

X

þ1

i

¼

02

Sc

2

Sc

þ

i

U

axis

U

o

1

þ

Sc

2

q

f

q

a

q

f

i

ð

14

Þ

The previous equation is very interesting because it relatesmomentumﬂuxwithvelocityontheaxisforagivenposition,

Fig. 3. Generic spray.1096

J.M. Desantes et al. / Fuel 86 (2007) 1093–1101

density in the chamber, spray cone angle and

Sc

number.This result is valid, as long as the underlying hypothesesare realistic. Such hypotheses have been already discussedin Section 2.2, and the subsequent sections will conﬁrm thatthey can be held for the main region of an isothermal spray,which is the focus of this paper.

2.4. Possible simpliﬁcations to the model 2.4.1. Case of Sc = 1

Although it is diﬃcult to determine the values for the

Sc

number in diesel sprays, values found in the literature arebetween 0.6 and 0.8 [25,34]. Nevertheless the simpliﬁcation
of considering

Sc

= 1 is normally used in order to do someapproximate analysis. In this particular case, Eq. (14)transforms into Eq. (15):

M

o

¼

p

2

aq

a

tan

2

h

u

2

x

2

U

2axis

X

þ1

i

¼

0

2

ð

2

þ

i

Þ

U

axis

U

o

q

f

q

a

q

f

i

ð

15

Þ

from which the following expression is derived:

M

o

¼

p

2

aq

a

tan

2

h

u

2

x

2

U

2axis

þ

paq

a

x

2

tan

2

h

u

2

X

þ1

n

¼

3

1

iU

n

axis

U

n

2o

q

f

q

a

q

f

n

2

ð

16

Þ

Taking into account the logarithm Taylor’s series theexpression (16) for

Sc

= 1 can be written as

M

o

¼

pa

x

2

q

a

q

f

q

a

tan

2

h

u

2

q

f

U

o

U

axis

q

f

q

f

q

a

U

o

Ln

1

U

axis

U

o

q

f

q

a

q

f

ð

17

Þ

Both Eq. (12) for generic

Sc

, and Eq. (17) for

Sc

= 1, couldbe simpliﬁed in the cases where the density of the fuel ismuch higher than density of quiescent air. Normal valuesfor diesel fuel are between 820 and 850 kg/m

3

dependingon pressure and temperature [36].For such conditions, the term

q

f

q

a

q

f

is approximatelyunity and so Eqs. (14) and (17) transform into Eqs. (18)
and (19), respectively:

M

o

¼

p

2

aq

a

tan

2

h

u

2

x

2

U

2axis

X

þ1

i

¼

02

Sc

2

Sc

þ

i

U

axis

U

o

1

þ

Sc

2

i

ð

18

Þ

M

o

¼

pa

x

2

q

a

tan

2

h

u

2

U

o

U

axis

U

o

Ln

1

U

axis

U

o

ð

19

Þ

2.4.2. Case with constant local density (

q

=

q

a

)

The consideration of a constant density in the chamber(and thus inside the spray) equal to the air density in theinjection chamber could be expected to be a good approx-imation because at a short distance of the spray nozzle, theair mass ﬂow is much higher than the srcinal injected fuelmass ﬂow.This approximation has been extensively used in the bib-liography [2 – 5,7,10,22,26]. The error involving the use of
this approach instead of the general expression (Eq. (14))will be quantiﬁed next.In fact, if

q

(

x

,

r

) =

q

a

, the integration of Eq. (5) simpli-ﬁes and it leads to Eq. (20) (the complete theoretical deduc-tion can be found in Desantes et al. [22])

M

o

¼

p

2

aq

a

tan

2

h

u

2

x

2

U

2axis

ð

20

Þ

A comparison of Eq. (20) with the general case (Eq. (14))
shows that the ﬁrst equation coincides with the secondone if only the term for

i

= 0 in the series is considered.It means that the numerical sequence from

i

= 1 to inﬁniterepresents the error made when a constant local density isconsidered. In the next paragraph this error will be quanti-ﬁed for a practical case.

3. Experimental support and validation

In this section a validation of the model is performed.For the validation, the velocity of fuel droplets in diﬀerentsections and radial positions of the spray were character-ized with a PDPA system (phase doppler particle anemom-etry). Furthermore, additional measurements of spraymomentum, mass ﬂow rate and spray cone angle were alsoneeded in order to validate the model. For the validation, amono-oriﬁce nozzle with a 146

l

m diameter mounted on acommon rail injector was used. For this particular nozzlethe droplet velocity, instantaneous momentum ﬂux andspray cone angle at diﬀerent injection pressures and back-pressures were measured. Details of the PDPA (phasedoppler particle anemometry) technique and system conﬁg-uration are given in [37] and the complete set of experimen-tal results is part of a work to be published by the authors.As far as the momentum ﬂux measurement is concerned, apressurized test rig with nitrogen is used. The measuringprinciple of this technique is explained in [6,8], and consists
of measuring the impact force of the spray on a surfacewith a piezo-electric sensor. As long as the whole cross-sec-tion of the spray is impacting on the sensor, this force isequal to the momentum ﬂux at that cross-section. If themeasurement position is close to the nozzle exit, the timeevolution of the impact force is equal to the nozzle momen-tum ﬂux

M

o

, which is the main input to the spray model.Typical injection strategies result in a constant momentumﬂux during the time the needle is open. Accordingly, anaverage value of the momentum ﬂux will be used for mod-eling purposes.The combination of the momentum ﬂux measurementwith the mass ﬂow rate measurement for the same pressurecondition made it possible for the fuel velocity at the nozzleexit to be determined [6,8]. For the mass ﬂow rate measure-
ments an injection rate test rig based on the Bosch methodwas used [36].

J.M. Desantes et al. / Fuel 86 (2007) 1093–1101

1097