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Spray-wall impact handbook of atomization and sprays: Theory
andapplications
Article · January 2011
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Instituto Superior Tecnico, Lisboa
University of Coimbra
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Chapter 21
Spray-Wall Impact
A.L.N. Moreira and M.R. Oliveira Panão
Abstract Spray-wall impact is an important process in numerous applications such
as internal combustion (IC) engines, spray cooling, painting, metallurgy, and many
others. This chapter reviews the main challenges in this dynamic thermofluid event
and attempts to systematize the knowledge developed in the hydrodynamics of
multiple drop impacts and liquid deposition, the statistical analysis of secondary
atomization after impact, and the thermodynamics underlying heat transfer
processes in spray impaction onto heated surfaces.
Keywords Atomization Correlations Deposition Drop Interaction Heat
transfer Impinging spray Liquid deposition Multiple drop impacts Secondary
thermodynamics Spray Spray-wall impact Statistics
Introduction
Spray impingement is a dynamic thermofluid process present in many industrial
applications such as IC engines, spray painting and coating, microelectronic cooling, fire extinguishment, cooling and quenching in metal foundries, ice chiller and
air-conditioning systems, medical inhalators, and dermatological surgery. Besides,
some of these applications consider intermittent spraying, such as fuel injection in
IC engines and cryogen spraying for cooling in dermatological surgery, where the
impingement process is dominated by the transient characteristics of the spray.
Numerous studies are described in the literature, which address the process at a
very fundamental level considering the dynamic phenomena involved at the impact
of each single drop of the spray – a comprehensive and systematic review has been
reported by Yarin [32]. However, the interaction between droplets in the vicinity of
each other alter those phenomena and does not allow to describe the spray as the
A.L.N. Moreira (*)
Instituto Superior Técnico, INþ Centro de Estudos em Inovação Tecnologia e Polı́ticas de
Desenvolvimento, Lisboa, Portugal
e-mail: moreira@dem.ist.utl.pt
N. Ashgriz (ed.), Handbook of Atomization and Sprays,
DOI 10.1007/978-1-4419-7264-4_21, # Springer ScienceþBusiness Media, LLC 2011
441
442
A.L.N. Moreira and M.R.O. Panão
Fig. 21.1 Illustration of spray-wall impact with heat transfer
summation of individual impacts [25, 27], except when it is sufficiently dilute at the
moment of impact. Those phenomena include (1) partial deposition of the impinging mass; (2) emergence of secondary droplets due to hydrodynamic or thermally
induced break-up mechanisms; and (3) heat extraction and vaporization, as illustrated in Fig. 21.1. Together with multiple drop interactions, these are the issues
addressed in the present chapter.
Meeting the challenges of integrating the current available physical information
into a tool, which can accurately describe the resulting complex flow is both
technically interesting and intellectually stimulating. For example, research in
fuel sprays has significantly contributed to the emergence of innovative technological solutions to meet the increasingly stringent regulations imposed by environmental policies and are referred to by most researchers in this field as the main
motivation for studying spray impingement; also important is the research related to
spray impingement for thermal management. Both areas provided the context for
the systematic knowledge reported here.
Multiple Drop Interactions
Despite single drop impact experiments providing understanding on the fundamental mechanisms eventually present on spray impaction, when interaction phenomena occurs between drops with quite dissimilar diameters, impact velocities, and
21
Spray-Wall Impact
443
directions, some hydrodynamic structures are produced, leading to secondary
atomization, but with a morphology that is quite different from the splash and
rebound observed on the impact of single droplets. Roisman et al. [27] identified
those structures as asymmetric corona splash, uprising central jet break-up, splash
from an uprising lamella resultant from multiple drop interactions and film jetting
with subsequent break-up.
The experiments reported in the literature explore the outcome of those interactions studying the influence of length and time scales associated with multiple
drop impacts on the outcome, as described in Fig. 21.2.
The length scale
corresponds to the
spacing between droplets striking the
impinging surface lspacing ¼ jriþ1 ri j while the normalized time scale between
consecutive impacts (ci) is expressed as
uiþ2 j cos giþ2
j~
tiþ1 ti
liþ1;i
tci ¼
¼
tiþ2 ti liþ1;i þ liþ2;iþ1 j~
uiþ1 j cos giþ1
(21.1)
where ~
u and g refer to the velocity and angle of impact, respectively; i corresponds
to the droplet striking the wall, (i þ 1) and (i þ 2) corresponds to the subsequent
ones. From the reported literature, another parameter can be introduced, expressed
as f ¼ 2ptci , which describes the phase between multiple and consecutive drop
impacts. When multiple impacts are made simultaneously tiþ1 ¼ ti and f ¼ 0
[7, 8], and if droplets impact on the same point and, thus are consecutive, riþ1 ¼ ri,
lspacing ¼ 0, and f ¼ 180 [33].
For impacts with lspacing > 0, the interaction occurs between spreading lamellae,
which give rise to asymmetric uprising sheets. In a complete wetting system, where
each spreading lamella may form a crown, the interaction arises between uprising
crowns, though Barnes et al. [5] observed that, for lspacing < 2D0 there is no time for
crown formation before interaction occurs, while for lspacing < D0 droplets coalesce.
According to Barnes et al. [5], the height of those liquid sheets depends mainly on
z
t
3
z3
t3 =
u3
γ3
2
l32
t2 =
z2
l21
γ2 u2
lspacing 1
z1
r
r2
r1
r3
Fig. 21.2 Scale parameters involved in multiple drop impacts
t1
(l32 + l21)
Cos(γ3)| u3|
l21
Cos(γ2)| u2|
444
A.L.N. Moreira and M.R.O. Panão
drop spacing (lspacing) and the interaction phase (f) (it would be maximum with
lspacing ¼ 2D0 for f ¼ 0 and lspacing ¼ 2.5D0 for f ¼ 180 ). In an attempt to
improve the description of such structures, Roisman et al. [26] report an approach to
estimate the velocity, shape, and thickness of the uprising sheet, the collision line
and the rim motion taking into account the influence of both drop spacing and
interaction phase.
The uprising sheets may further break up into secondary droplets which are, in
general, larger and slower than those generated at single impacts [5, 7]. Despite the
evidence that drop spacing (lspacing) is an important parameter triggering break-up
[7, 8], the establishment of transition criteria has not been an easy task. Only Kalb
et al. [12] address the development of such criteria, though their experiments
consider the interaction of a droplet impacting onto one previously deposited on
the surface. The main difficulties may be attributed to the effects that other impact
parameters have on the length and time scales or even altering the physics of the
interaction. For example, the experiments of Cossali et al. [7] show a strong
influence of the impact velocity on the interaction-driven liquid sheet break-up.
The liquid film thickness influences the size of secondary droplets and slightly
reduces their number, but not the morphological structures formed by interaction
phenomena. Roisman and Tropea [24] further show that crown formation is inhibited in dense sprays and that multiple drop impacts interacting with a dynamic liquid
film generating finger-like jets, later disrupting into secondary droplets with size and
velocity similar to the characteristic scales of the film’s fluctuations. The integration
of interaction phenomena in submodels for spray impingement is still far from being
achieved with accuracy, which means that investigations on this topic are still
at their early stage, and one cannot but realize that multiple drop interaction
phenomena are one of the most prominent subjects open for creative research.
Liquid Deposition upon Spray Impact
In the case of spray impact without heat transfer, the deposited mass fraction and
the mass ratio of secondary to impinging drops are related as
mdep
ma
¼1
mb
mb
(21.2)
where mb is the mass of droplets impacting onto the surface, ma is the mass of
droplets issuing from the surface due to disintegration mechanisms, and mdep is the
mass of deposited liquid. When the surface is wetted, the liquid film may contribute
to the generation of secondary droplets and the ratio ma/mb can be larger than 1,
as observed by Panão and Moreira [20]. Then, the submodel would not apply
because it ceases to depend solely on ma/mb, otherwise mdep < 0. In this case, the
approach of Samenfink et al. [28] can be quite useful because ma/mb depends on
the spray characteristics before impact, though experiments are limited to the range
ma/mb2[0.26–1] and cases where ma/mb > 1 are not considered (see Table 21.1).
21
Spray-Wall Impact
445
Table 21.1 Submodels to predict the mass ratio between secondary (a – after) and primary
droplets (b – before)
Subparameters and validation domain
Reference
ma/mb
(
21 9:2133
K
¼ We0.5Re0.25
Mundo et al. [18]
ðsmoothÞ
3:9869 10 K
Bai et al. [4]
Kuhnke [13]
Samenfink et al.
[28]
Bai et al. [4]
Stanton and
Rutland [31]
Han et al. [10]
8:0350 1011 K 4:1718 ðroughÞ
0:2 þ
0:6 rndð0::1Þ
T 0:8
ð 1 BÞ þ B
min 1;
1:1 0:8
0:0866ðscd 1Þ0:3188
ð90 ab Þ0:1223 df0:9585
0:2 þ 0:9 rndð0::1Þ
27:2 þ 3:15u 0:1164u2
þ 1:4 103 u3
0:75 1 exp 107 ðH Hcr Þ1:5
B ¼ 0:2 þ 0:6 rndð0:1Þ
T ¼ Tw =Tb
scd ¼ 241 Reb Lab0:4189 2 ½1; 5;
La 2 ½5; 000; 20; 000;
ab 2 ½0 ; 85 ;
df ¼ hf =db 2 ½0:3; 3
u ¼ Ub ðr=sÞ1=4 n1=8 f 3=8
f ¼ Ub =db
H ¼ We Re0:5
Hcr ¼ KH 1 þ 0:1Re0:5 min df ;0:5
KH ¼ 1;500 þ 650ðRa =DiD Þ0:42
Senda and
For We 300
Fujimoto [29]
0:423 0:096df þ 1:61d2f
Kuhnke [13]
Kalantari and
Tropea [11]
1:47d3f þ 0:367d4f
For We>300; 0:8
mlf T 0:8
ð1 BÞ þ B
min 1þ ;
mb 1:1 0:8
For lWeb <0:1
B ¼ 0:2 þ 0:9 rndð0::1Þ
T ¼ Tw =Tb
lWeb ¼ Wetb =Wenb
6:74 103 Wenb 0:204
For lWeb
0:1
35Wenb 1:63
Condition: Dry/wetted
The various submodels proposed in the literature are summarized in Table 21.1,
where Ra is the average surface roughness; r, s, and n are the liquid density, surface
tension, and kinematic viscosity, respectively; Wenb and Wetb are the Weber
numbers (¼rUb2D/s) based on the normal (Unb) and tangential (Utb) velocity
components of impinging droplets, respectively.
Statistical Description of Secondary Atomization
Secondary atomization upon spray impact can be generated by diverse mechanisms. Models have been devised from experiments with single droplet impacts,
which give the number, velocity, and direction of secondary droplets. Incorporation
446
A.L.N. Moreira and M.R.O. Panão
of these models on CFD codes gives innumerable secondary droplets, such that
tracking them all becomes prohibitive. To resolve this issue a stochastic approach is
usually followed by calculating a statistical sample of the full population, where
each computational secondary droplet actually represents a parcel of real droplets.
Some older models originally assigned number mean values to these parcels within
an appropriate range, which were randomly selected. However, experiments indicate that secondary droplets are better described by characteristic probability
density functions (PDF).
Basically, the distributions reported by several authors for secondary atomization processes depend on a curve fit to data collected for a wide range of operating
conditions and atomizer designs, which is designated as the empirical approach.
Babinsky and Sojka [3] have reviewed also two other methods or approaches for
predicting size distributions: (1) the maximum entropy (ME); and (2) the discrete
probability function (DPF). Although these methods rely on a physical interpretation of the droplet generation process as nondeterministic (ME method), or as an
ensemble of deterministic and nondeterministic portions (DPF method), one may
question whether the resulting distributions fully capture the true nature of the
atomization process. Moreover, even within the common empirical approach, what
is the reason for stating that a Weibull, w2, Rosin–Rammler, Nukiama–Tanasawa,
or a Log-normal distribution function is more suitable for describing, or predicting,
secondary atomization?
Liquid atomization is a process for converting a bulk liquid volume of fluid into
a myriad of single particle elements of multiple sizes (drops), which can be statistically described. Therefore, it is worth synthesizing the underlying statistical
principles associated with a certain distribution function and the atomization
process itself.
When we do this, the distributions usually considered for describing secondary
atomization could be encompassed in two groups. One purely empirical, concerned about the shape and scale of the secondary drop size distribution (Weibull,
w2, Rosin–Rammler, Nukyiama–Tanasawa), and the second, a semiempirical
group associated with the multiplicative meaning of the Log-normal distribution
function.
The distribution function for the first group could be generally expressed as
p q q d
d
f ðd Þ ¼ exp d d
d
(21.3)
where p is a shape parameter, d and q are the scale parameters. While d clearly
shifts the size distribution within its range because it is intrinsically linked with
a characteristic drop size, q appears to affect, mainly, the frequency range and,
consequently the size range for “conservation” reasons since the integral of f (d)
must equal 1 (see Fig. 21.3), and p, as seen on the top left plot in Fig. 21.3, affects
the distribution shape. In the case of Weibull or Rosin–Rammler, the shape and
scale parameters are the same as in (21.3), with p ¼ q 1; and the w2 case is similar,
but with q ¼ 1.
21
Spray-Wall Impact
0.05
0.05
q=1
d = 30
p=2
0.04
frequency (-)
0.04
frequency (-)
447
0.03
0.02
0.01
d = 30
0.03
0.02
0.01
0
0.00
20 40 60 80 100 120 140 160 180 200
d (μm)
0.14
20 40 60 80 100 120 140 160 180 200
d (μm)
q=3
p=2
0.12
frequency (-)
0
0.10
0.08
0.06
0.04
0.02
0.00
0
20
40
60
d (μm)
80
100
Fig. 21.3 Effect of shape and scale parameters on drop size distributions
The known Nukiama–Tanasawa empirical function, expressed as:
f ð DÞ ¼ a Dp exp½b Dq (21.4)
can be considered a particular case of the general form written in (21.3), with
a ¼ q=dq ; p ¼ q 1; b ¼ dq , but is unable to independently control the shape and
scale parameters of the distribution, given that a and b are nonlinearly dependent on
The distribution functions for secondary drop size reported in the
both q and d.
literature are summarized in Table 21.2.
In Stanton and Rutland [31], the shape and scale parameters depend on the
Weber number, using the axial component of the impinging drop velocity in the
calculations, but the authors are unclear about the characteristic mean size, whether
it is arithmetic (AMD), volumetric (VMD, or D30) or Sauter (SMD). The approach
in Bai et al. [4] depends on the mass ratio between secondary and primary drops,
which is randomly determined, thus introducing some nondeterminism into the
formulation. In his approach, Lemini [14] fitted three polynomial functions f1, f2, f3
to the data reported in Mundo et al. [18], with the objective of establishing different
scale parameters q1 and q2, aiming at an independent control of the frequency and
size range of the distribution, respectively.
3/2
1
qi
da
da
da
Lemini [14]
Web
1
di
q–1
2
0
Web
0.463d32
d30
0.217.69 10
5
f3 ðK Þ ¼ 1:05522 103 K 2 þ 0:34599K 26:130703
f2 ðK Þ ¼ 5:3313 105 K 2 þ 1:53062 102 K 0:936192
f1 ðK Þ ¼ 1:96602 104 K 2 9:1349 102 K þ 10:419257
d1 ¼ f1 ðK Þdb ; q1 ¼ Gð2=f3 ðK ÞÞ=½f3 ðK Þ f1 ðKÞ=f2 ðKÞ
d2 ¼ f2 ðK Þdb ; q2 ¼ f3 ðK Þ
d32 is the Sauter mean diameter
1=3
6 ma =mb
db
d30 ¼ 0:3218
Web =Webc 1
ma/mb is on Table 21.1 and the critical Weber Webc on
Bai et al. [4]
p q2 q1 d
d
f ðda Þ ¼ exp d2
d1 d1
Web ¼ rUb2db/s
Other parameters
Roisman et al.
[27]
da/Da30
1.94
0.813
q–1
and K ¼ Web0.5Reb0.25 with K 2 ½133; 187
1=2
24Db30 Reb
for Reb >500
Da30 ¼
b 252
for Reb >500
Db30 0:65 þ 0:017 exp Re73:5
Reb ¼ rUbdb/m and Db30 is the Volumetric Mean Diameter
Acronyms: a ¼ secondary drops (after impact); b ¼ primary drops (before impact). In Lemini [14] q1 is equal to 1, but the expression presented is the result of
the PDF normalization
2.719.25 10
da/db
Stanton and
Rutland [31]
Han et al. [10]
Bai et al. [4]
4
Table 21.2 Distribution functions for secondary drop size reported in the literature
Reference
d
q
P
d
448
A.L.N. Moreira and M.R.O. Panão
21
Spray-Wall Impact
449
The second group of distributions, which describes secondary atomization is
based on the Log-normal distribution function, expressed as:
2 !
lnðda Þ ln dg
1
pffiffiffiffiffiffi exp f ð da Þ ¼
2g2g
da gg 2p
(21.5)
where dg and gg are the geometric mean diameter and standard deviation, respectively. As earlier mentioned, the Log-normal is related to the multiplicative nature
of the statistical process involved. In fact, it is not hard to imagine such multiplicative nature within the disintegration process itself, for example, one large droplet
could split into two smaller ones, these two into four and so on. However, since the
Log-normal distribution is being applied to describe secondary drop sizes, the fact
that some authors found a good fitting to their measurement data using it, means
that secondary atomization triggered by hydrodynamic impact mechanisms (e.g.,
rebound, splash), has also a multiplicative physical nature. In fact, mechanisms
such as splash will produce a certain number of droplets on the impact of a single
one. Even with some degree of interaction on multiple impacts, this multiplicative
factor remains. Nevertheless, as reviewed in the “Introduction” section, the knowledge on multiple drop impacts is still in its early stages, justifying that integrating
the statistical description of secondary atomization with its physical interpretation
remains open for further research.
Heat Transfer on Spray Impact
Heat transfer can be analyzed based on the rate form of the conservation of energy
equation for the open thermodynamic system depicted in Fig. 21.4, where liquid
vaporization occurs with the extraction of a heat flux q_ 00w from the surface. The mass
flux of impinging droplets (m_ 00in ) may deposit and accumulate on the surface in
the form of a liquid film (Dm00 /Dt with temperature TLF); move away from the
surface in the form of secondary droplets (m_ 00s ); or, vaporize and mix with the
surrounding environment (m_ 00vap ) with an average temperature between the wall
and boiling values.
Tvap »
Tw + Tb
2
.
.²
m²s
mvap
Fig. 21.4 Open system
considered in the analysis of
heat transfer at spray impact
.²
qw
Tf
.
m²in
TLF
Tw
Dm²
Dt
450
A.L.N. Moreira and M.R.O. Panão
One of the key parameters in the analysis is the initial surface temperature as,
according to the classical boiling theory, it establishes the regime by which a liquid
removes heat from a heated surface. In the film evaporation and boiling regimes, a
liquid film forms onto the surface, and these are, therefore, jointly considered as a
wetting regime. In vaporization/boiling, bubbles emerge from the surface due to
nucleation, which may result in thermo-induced secondary atomization mechanisms
[9, 16, 17]. In the transition regime, the liquid is in contact with the surface only
intermittently, due to separations from the surface caused by vapor expelled from the
liquid. Above the temperature of the local minimum in the boiling curve, occurs the
Leidenfrost phenomenon, characterized by the appearance of a thin vapor layer
between the liquid and the surface, being thus referred to as a nonwetting regime.
Though these heat transfer regimes can be applied to the impact of a spray, the
temperature values defining the transition between regimes depend on the impact
conditions. Moreover, if the spray is intermittent, transition criteria are dynamic and
multiple regimes can be simultaneously present within the entire impact area [15, 20].
Also, heat transfer must be analyzed either locally or globally within the entire
area where the liquid is in contact with the surface because, depending on the flow
conditions, initial surface temperature and its geometry, or roughness and heat
sources, the heat fluxes can either be enhanced or inhibited, at a local or overall
perspective.
From a local heat-transfer perspective, the outcome of this characterization is
mainly in the form of empirical correlations for the Nusselt dimensionless number
(Nu ¼ hDd/k), summarized in Table 21.3. Most correlations were derived from
experiments within the context of IC engines, except for the dynamic correlation
reported in Panão and Moreira [21], which considers the transient characteristics of
droplets of an intermittent spray along an injection cycle, and a newly introduced
dimensionless parameter l corresponding to the average number of droplets
impinging in the vicinity of each other [24].
An important issue concerns the evaluation of the cooling performance. Usually,
the first-law of thermodynamics is used to compare the amount of heat extracted by
the impinging spray with the total amount of heat that would be removed if all the
mass impinging on the surface vaporized (sensible and latent heat components):
e¼
q_ 00 w
m_ 00f cp DTwb þ hfg
(21.6)
where m_ 00f is the mass flux rate of impinging droplets, DTwb is the superheating
degree, and cp and hfg are the specific heat and latent heat of vaporization,
respectively. It has been recently argued in Panão and Moreira [22] that a performance analysis should also include the point of view of the second-law of thermodynamics. Basically, the best performance is achieved with the lowest production
of irreversibility in the process, or else, the main sources should be identified in
order to act accordingly. Assuming that a liquid film, regardless of its thickness, is
present and that w represents the mass faction of fluid vaporized, according to the
analysis in Panão and Moreira [22], the rate of entropy generation is given by
For the definition of dimensionless parameters, see Chap. 3, section “Three dimensional sheet instability”
Acronyms: LFS, leading front of the spray; EOI, end-of-injection
Table 21.3 Heat transfer correlations for spray impingement
Reference/liquid
Correlation
Arcoumanis and Chang [1]/diesel
We0:94
Nu ¼ 0:34 0:53 0:33
Re Pr
1:08 0:057
Arcoumanis et al. [2]/diesel
Pr 3:94 We1:59
U
db32
Nu ¼ 0:0012
U þ Vc
Re1:89
db32 þ Zimp
1:51
Moreira et al. [16, 17]/gasoline
5 Re
Nu ¼ 3:4 10
Ja0:254
8
37 1:028
Panão and Moreira [21]/HFE-7100
>
l
La9:287 Ca0:984 ; timpact <t timpact þ DtLFS
4:283
10
>
<
and acetone
Nu ¼ 5:8 105 l0:581 Ja0:137 La1:11 Ca0:745 ; timpact þ DtLFS <t tEOI
>
>
: 4:191 104 l0:272 Ja0:365 La1:085 Ca0:901 ; t>t
EOI
Local dynamic correlation
Local correlation
Local correlation in crossflow conditions
Obs.
Local correlation
21
Spray-Wall Impact
451
452
A.L.N. Moreira and M.R.O. Panão
1
1
00
00
00
_
S gen ðwÞ ¼ ½’L ðwÞsL þ ’V ðwÞsV Dm_ þ q_ w
þ
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
TLF Tw
|fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl}
S_00
00
gen;Dm_
8
’L ðwÞ ¼ w 3w w2 3
>
>
<
’ V ð wÞ ¼ w w2 2ð w 1Þ
>
>
:
Dm_ 00 ¼ m_ 00 in m_ 00 s
S_00 gen;q_ 00 w
(21.7)
where the first parcel of the right-hand side corresponds to the irreversibility
associated with evaporation (S_00gen;Dm_ 00 ) and the second parcel corresponds to the
irreversibility associated with the heat extracted from the surface (S_00gen;q_ 00 w ). The
ratio between the former and the latter gives the irreversibility distribution ratio fsc.
Application of the entropy generation minimization (EGM) method to (21.7)
allows identifying the optimal parameters to be considered in the optimization [6].
Concerning the entropy generated by the evaporated mass flux, that parameter is the
fraction w of evaporated mass and its optimal value is found by equating to zero the
partial derivative of (21.7) with respect to w. The optimal evaporated mass fraction
is thus found to be
wopt
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
3sL 2sV 3sL sV 2s2V
¼
3ð s L s V Þ
(21.8)
which requires a positive term inside the square root, such that sL (2/3)sV. The
equality in the former expression sets the lower limit optimal value, wopt ¼ 0. The
upper-limit of the inequality is sL ! sV, which could only occur near the critical
temperature of the cooling liquid and implies that only half of the impinging mass
is vaporized (wopt ¼ 0.5). This analysis indicates that energy losses should be
minimum if the cooling liquid vaporization is minimized, favoring the presence of
a liquid film on the surface, as in some continuous spray cooling concepts (see
[23, 30]).
Application of the EGM method to the entropy generated by heat transfer at the
surface (which should be maximum in a spray-cooling application), identifies the
temperature as the parameter to be accounted for in the optimization process. Here,
the derivative of (21.7) produces a single solution for the optimization of the wall
temperature as Tw,opt ¼ Tb þ 2Tf, which can be interpreted with respect to the
superheating degree (DTwb,opt ¼ Tw,opt – Tb) as
DTwb;opt ¼ 2Tf
(21.9)
This simple criterion based on the least irreversibility is technically interesting
for choosing the most adequate fluid in the optimal design of a certain spray-cooling
application. It should be stressed that the optimal wall temperature corresponds to
the working temperature of the heat-dissipating surface, which the spray-cooling
system is required to maintain at a constant value.
21
Spray-Wall Impact
453
When designing a spray-cooling system, the optimization analysis should consider the relative importance of each term in (21.7) to the irreversibility distribution
ratio fsc.
Concluding Remarks
The application of spray-wall impact to engineering systems relies on the accurate
knowledge of four research subjects: (1) multiple drop impacts; (2) liquid deposition; (3) secondary atomization; and (4) heat transfer on spray impaction.
Research on multiple drop impacts evidences that hydrodynamic structures
depend on interdrop length and time scales, having a different morphology from
that commonly found in single drop impacts, such as splash with crown development. Secondary droplets produced by uprising liquid sheets resulting from multiple drop interaction on the impact site are larger and slower than those usually
generated by single impacts.
The deposition of liquid by spray-wall impact can be positive in thermal
management systems, or spray-painting applications, but it can be negative in IC
engines, since it is directly related to the emission of pollutants. Several methods,
deterministic and nondeterministic, have been reviewed.
A systematic analysis has been made for the statistical approach to describe
secondary drop size distributions. Two groups were identified. An empirical one
based on the Weibull distribution where the scale and shape parameters can change
according to the degree of control desired over the size and frequency range. The
second group is semiempirical and is associated with a log-normal distribution
function. The statistical meaning of the log-normal expresses the multiplicative
nature of the secondary atomization process.
The heat transfer process on spray-wall impact can be viewed at from a local, or
an overall perspective. From a local perspective, a synthesis has been made of the
correlations found between energy exchanges and the characteristics of the impinging spray. From an overall perspective, the analysis argues that the performance can
be improved if, besides taking into account the efficiency based on the first-law of
thermodynamics, it also takes into account the second-law. The insights given by
the EGM method can be useful not only to physically interpret heat transfer on
spray impact, but also to provide criteria for optimizing the performance of the
system from the viewpoint of cooling.
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