Brennen Ch.E. Fundamentals of Multiphase Flow

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Figure 11.4. Non-dimensional attenuation, Im{−κc

/ω}, as a function

of reduced frequency for a droplet-laden gas ﬂow with ξ =0.01, γ =1.4,

=1 and c

= 1. The dashed line is the result without phase

change; the solid line is an example of the alteration caused by phase

change. Adapted from Marble and Wooten (1970).

from the other relaxation times when the loading is small. At higher loadings

the eﬀect merges with the velocity and temperature relaxation processes.

11.5 OTHER LINEAR PERTURBATION

ANALYSES

In the preceding section we examined the behavior of small perturbations

about a constant and uniform state of the mixture. The perturbation was

a plane acoustic wave but the reader will recognize that an essentially sim-

ilar methodology can be used (and has been) to study other types of ﬂow

involving small linear perturbations. An example is steady ﬂow in which

the deviation from a uniform stream is small. The equations governing the

small deviations in a steady planar ﬂow in, say, the (x, y) plane are then

quite analogous to the equations in (x, t) derived in the preceding section.

11.5.1 Stability of laminar ﬂow

An important example of this type of solution is the eﬀect that dust might

have on the stability of a laminar ﬂow (for instance a boundary layer ﬂow)

and, therefore, on the transition to turbulence. Saﬀman (1962) explored

the eﬀect of a small volume fraction of dust on the stability of a parallel

ﬂow. As expected and as described in section 1.3.2, when the response times

279

of the particles are short compared with the typical times associated with

the ﬂuid motion, the particles simply alter the eﬀective properties of the

ﬂuid, its eﬀective density, viscosity and compressibility. It follows that un-

der these circumstances the stability is governed by the eﬀective Reynolds

number and eﬀective Mach number. Saﬀman considered dusty gases at low

volume concentrations, α, and low Mach numbers; under those conditions

the net eﬀect of the dust is to change the density by (1 + αρ

/ρ

)andthe

viscosity by (1 + 2.5α). The eﬀective Reynolds number therefore varies like

(1 + αρ

/ρ

)/(1 + 2.5α). Since ρ

 ρ

the eﬀective Reynolds number is

increased and therefore, in the small relaxation time range, the dust is desta-

bilizing. Conversely for large relaxation times, the dust stabilizes the ﬂow.

11.5.2 Flow over a wavy wall

A second example of this type of solution that was investigated by Zung

(1967) is steady particle-laden ﬂow over a wavy wall of small amplitude

(ﬁgure 11.5) so that only the terms that are linear in the amplitude need be

retained. The solution takes the form

exp(iκ

x − iκ

y) (11.48)

where 2π/κ

is the wavelength of the wall whose mean direction corresponds

with the x axis and κ

is a complex number whose real part determines

the inclination of the characteristics or Mach waves and whose imaginary

part determines the attenuation with distance from the wall. The value of

is obtained in the solution from a dispersion relation that has many

similarities to equation 11.45. Typical computations of κ

are presented in

ﬁgure 11.6. The asymptotic values for large t

that occur on the right in

this ﬁgure correspond to cases in which the particle motion is constant and

Figure 11.5. Schematic for ﬂow over a wavy wall.

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Figure 11.6. Typical results from the wavy wall solution of Zung (1969).

Real and imaginary parts of κ

/κ

are plotted against t

U/κ

for various

mean Mach numbers, M = U/c

, for the case of t

=1,c

=1,

γ =1.4 and a particle loading, ξ =1.

unaﬀected by the waves. Consequently, in subsonic ﬂows (M = U/c

< 1)

in which there are no characteristics, the value of Re{κ

/κ

} asymptotes to

zero and the waves decay with distance from the wall such that Im{κ

/κ

}

tends to (1 − M

)

. On the other hand in supersonic ﬂows (M = U/c

> 1)

Re{κ

/κ

} asymptotes to the tangent of the Mach wave angle in the gas

alone, namely (M

− 1)

, and the decay along these characteristics is zero.

At the other extreme, the asymptotic values as t

approaches zero corre-

spond to the case of the eﬀective gas whose properties are given in section

11.2.2. Then the appropriate Mach number, M

, is that based on the speed

of sound in the eﬀective gas (equation 11.16). In the case of ﬁgure 11.6,

=2.4M

. Consequently, in subsonic ﬂows (M

< 1), the real and imag-

inary parts of κ

/κ

tend to zero and (1 − M

)

respectively as t

tends

to zero. In supersonic ﬂows (M

> 1), they tend to (M

− 1)

and zero

respectively.

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11.6 SMALL SLIP PERTURBATION

The analyses described in the preceding two sections, 11.4 and 11.5, used a

linearization about a uniform and constant mean state and assumed that the

perturbations in the variables were small compared with their mean values.

Another, diﬀerent linearization known as the small slip approximation can be

advantageous in other contexts in which the mean state is more complicated.

It proceeds as follows. First recall that the solutions always asymptote to

those for a single eﬀective gas when t

and t

tend to zero. Therefore, when

these quantities are small and the slip between the particles and the gas is

correspondingly small, we can consider constructing solutions in which the

ﬂow variables are represented by power series expansions in one of these

small quantities, say t

, and it is assumed that the other (t

) is of similar

order. Then, generically,

Q(x

,t)=Q

(0)

,t)+t

(1)

,t)+t

(2)

,t)+.... (11.49)

where Q represents any of the ﬂow quantities, u

, u

, T

, p, ρ

, α

, etc. In addition, it is assumed for the reasons given above that the slip

velocity and slip temperature, (u

− u

)and(T

− T

), are of order t

so that

(0)

= u

(0)

= u

(0)

; T

(0)

= T

(0)

= T

(0)

(11.50)

Substituting these expansions into the basic equations 11.6, 11.7 and 11.8

and gathering together the terms of like order in t

we obtain the following

zeroth order continuity, momentum and energy relations (omitting gravity):

∂

∂x



(1 + ξ)ρ

(0)



= 0 (11.51)

(1 + ξ)ρ

(0)

∂u

(0)

∂x

= −

∂p

(0)

∂x

∂σ

D(0)

Cik

∂x

(11.52)

(0)

+ ξc

)

∂T

(0)

∂x

= u

(0)

∂p

(0)

∂x

+ σ

D(0)

Cik

∂u

(0)

∂x

(11.53)

Note that Marble (1970) also includes thermal conduction in the energy

equation. Clearly the above are just the equations for single phase ﬂow of

the eﬀective gas deﬁned in section 11.2.2. Conventional single phase gas

dynamic methods can therefore be deployed to obtain their solution.

Next, the relaxation equations 11.22 and 11.25 that are ﬁrst order in t

282

Figure 11.7. The dimensionless choked mass ﬂow rate as a function of

loading, ξ,forγ

=1.4 and various speciﬁc heat ratios, c

as shown.

yield:

(0)

∂u

(0)

∂x

= u

(1)

− u

(1)

(11.54)

(0)

∂T

(0)

∂x







(1)

− T

(1)

) (11.55)

From these the slip velocity and slip temperature can be calculated once the

zeroth order solution is known.

The third step is to evaluate the modiﬁcation to the eﬀective gas solution

caused by the slip velocity and temperature; in other words, to evaluate

the ﬁrst order terms, u

(1)

, T

(1)

, etc. The relations for these are derived

by extracting the O(t

) terms from the continuity, momentum and energy

equations. For example, the continuity equation yields

∂

∂x



ξρ

(0)

(1)

− u

(1)

)+u

(0)

(ρ

(1)

− ξρ

(1)

)



= 0 (11.56)

This and the corresponding ﬁrst order momentum and energy equations can

then be solved to ﬁnd the O(t

) slip perturbations to the gas and particle

ﬂow variables. For further details the reader is referred to Marble (1970).

A particular useful application of the slip perturbation method is to the

one-dimensional steady ﬂow in a convergent/divergent nozzle. The zeroth

order, eﬀective gas solution leads to pressure, velocity, temperature and

density proﬁles that are straightforward functions of the Mach number which

is, in turn, derived from the cross-sectional area. This area is used as a

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surrogate axial coordinate. Here we focus on just one part of this solution

namely the choked mass ﬂow rate, ˙m, that, according to the single phase,

eﬀective gas analysis will be given by

˙m

∗

)

=(1+ξ)



1+γ



(γ+1)/2(γ−1)

(11.57)

where p

and ρ

refer to the pressure and gas density in the upstream

reservoir, A

∗

is the throat cross-sectional area and γ is the eﬀective speciﬁc

heat ratio as given in equation 11.14. The dimensionless choked mass ﬂow

rate on the left of equation 11.57 is a function only of ξ, γ

and the speciﬁc

heat ratio, c

. As shown in ﬁgure 11.7, this is primarily a function of

the loading ξ and is only weakly dependent on the speciﬁc heat ratio.

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SPRAYS

12.1 INTRODUCTION

Sprays are an important constituent of many natural and technological pro-

cesses and range in scale from the very large dimensions of the global air-sea

interaction and the dynamics of spillways and plunge pools to the smaller

dimensions of fuel injection and ink jet systems. In this chapter we ﬁrst ex-

amine the processes by which sprays are formed and some of the resulting

features of those sprays. Then since, the the combustion of liquid fuels in

droplet form constitute such an important component of our industrialized

society, we focus on the evaporation and combustion of single droplets and

follow that with an examination of the features involved in the combustion

of sprays.

12.2 TYPES OF SPRAY FORMATION

In general, sprays are formed when the interface between a liquid and a gas

becomes deformed and droplets of liquid are generated. These then migrate

out into the body of the gas. Sometimes the gas plays a negligible role in the

kinematics and dynamics of the droplet formation process; this simpliﬁes the

analyses of the phenomena. In other circumstances the gasdynamic forces

generated can play an important role. This tends to occur when the relative

velocity between the gas and the liquid becomes large as is the case, for

example, with hurricane-generated ocean spray.

Several prototypical ﬂow geometries are characteristic of the natural and

technological circumstances in which spray formation is important. The ﬁrst

prototypical geometry is the ﬂow of a gas over a liquid surface. When the

relative velocity is suﬃciently large, the interfacial shear stress produces

waves on the interface and the breakup of the waves generates a spray that

285

is transported further into the gas phase by the turbulent motions. Ocean

spray generated in high wind conditions falls into this category as does

annular, vertical two-phase ﬂow. In some fuel injectors a coﬂowing gas jet is

often added to enhance spray formation. Section 12.4.2 provides an overview

of this class of spray formation processes.

A second, related conﬁguration is a liquid pool or ocean into which gas is

injected so that the bubbles rise up to break through the free surface of the

liquid. In the more quiescent version of this conﬁguration, the spray is formed

by process of break-through (see section 12.4.1). However, as the superﬁcial

gas ﬂux is increased, the induced liquid motions become more violent and

spray is formed within the gas bubbles. This spray is then released when the

bubbles reach the surface. An example of this is the spray contained within

the gas phase of churn-turbulent ﬂow in a vertical pipe.

A third conﬁguration is the formation of a spray due to condensation in

a vapor ﬂow. This process is governed by a very diﬀerent set of physical

principles. The nucleation mechanisms involved are beyond the scope of this

book.

The fourth conﬁguration is the break up of a liquid jet propelled through

a nozzle into a gaseous atmosphere. The unsteady, turbulent motions in

the liquid (or the gas) generate ligaments of liquid that project into the

gas and the breakup of these ligaments creates the spray. The jet may be

laminar or turbulent when it leaves the nozzle and the details of ligament

formation, jet breakup and spray formation are somewhat diﬀerent in the

two cases. Sections 12.4.3 and 12.4.4 will summarize the processes of this

ﬂow conﬁguration.

One area in which sprays play a very important role is in the combustion

of liquid fuels. We conclude this chapter with brief reviews of the impor-

tant phenomena associated with the combustion of sprays, beginning with

the evaporation of droplets and concluding with droplet and droplet cloud

combustion.

12.3 OCEAN SPRAY

Before proceeding with the details of the formation of spray at a liquid/gas

interface, a few comments are in order regarding the most widely studied

example, namely spray generation on the ocean surface. It is widely accepted

that the mixing of the two components, namely air and water, at the ocean

surface has important consequences for the global environment (see, for ex-

ample, Liss and Slinn, 1983, or Kraus and Businger, 1994). The heat and

286

mass exchange processes that occur as a result of the formation of bubbles in

the ocean and of droplets in the atmosphere are critical to many important

global balances, including the global balances of many gases and chemicals.

For example, the bubbles formed by white caps play an important role in

the oceanic absorption of carbon dioxide; on the other side of the interface

the spray droplets form salt particles that can be carried high into the atmo-

sphere. They, in turn, are an important contributor to condensation nuclei.

Small wonder, then, that ocean surface mixing, the formation of bubbles and

droplets, have been extensively studied (see for example Monahan and Van

Patten, 1989). But the mechanics of these processes are quite complicated,

involving as they do, not only the complexity of wave formation and break-

ing but the dynamics of turbulence in the presence of free surfaces. This, in

turn, may be aﬀected by free surface contamination or dissolved salts be-

cause these eﬀect the surface tension and other free surface properties. Thus,

for example, the bubble and droplet size distributions formed in salt water

are noticeably diﬀerent from those formed in fresh water (Monahan and

Zietlow, 1969). Here, we shall not attempt a comprehensive review of this

extensive literature but conﬁne ourselves to some of the basic mechanical

processes that are believed to inﬂuence these oceanic phenomena.

There appears to be some general concensus regarding the process of spray

formation in the ocean (Blanchard, 1983, Monahan, 1989). This holds that,

at relatively low wind speeds, the dominant droplet spray is generated by

bubbles rising to breach the surface. The details of the droplet formation

process are described in greater detail in the next section. The most proliﬁc

source of bubbles are the white caps that can cover up to 10% of the ocean

surface (Blanchard, 1963). Consequently, an understanding of the droplet

formation requires an understanding of bubble formation in breaking waves;

this, in itself, is a complex process as illustrated by Wood (1991). What is

less clear is the role played by wind shear in ocean spray formation (see

section 12.4.2).

Monahan (1989) provides a valuable survey and rough quantiﬁcation of

ocean spray formation, beginning with the white cap coverage and proceed-

ing through the bubble size distributions to some estimate of the spray

size distribution. Of course, the average droplet size decays with elevation

above the surface as the larger droplets settle faster; thus, for example, de

Leeuw (1987) found the average droplet diameter at a wind speed of 5.5m/s

dropped from 18µm at an elevation of 2m to 15µm at 10m elevation. The

size also increases with increasing wind speed due to the greater turbulent

velocities in the air.

287

Figure 12.1. Stages of a bubble breaking through a free surface.

Figure 12.2. Photographs by Blanchard (1963) of a bubble breaking

through a free surface. Reproduced with permission of the author.

It is also important to observe that there are substantial diﬀerences be-

tween spray formation in the ocean and in fresh water. The typical bubbles

formed by wave breaking are much smaller in the ocean though the total

bubble volume is similar (Wang and Monahan, 1995). Since the bubble size

determines the droplet size created when the bubble bursts through the

surface, it follows that the spray produced in the ocean has many more,

smaller droplets. Moreover, the ocean droplets have a much longer lifetime.

Whereas fresh water droplets evaporate completely in an atmosphere with

less than 100% relative humidity, salt water droplets increase their salin-

ity with evaporation until they reach equilibrium with their surroundings.

Parenthetically, it is interesting to note that somewhat similar diﬀerences

have been observed between cavitation bubbles in salt water and fresh wa-

ter (Ceccio et al. 1997); the bubbles in slat water are smaller and more

numerous.

12.4 SPRAY FORMATION

12.4.1 Spray formation by bubbling

When gas bubbles rise through a pool of liquid and approach the free surface,

the various violent motions associated with the break through to the cover

gas generate droplets that may persist in the cover gas to constitute a spray.

288