7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
Entropy Transport and Production in TwoFluid Models
E. E. Paladino
Graduate Program in Mechanical Engineering, Federal University of Rio Grande do Norte, Natal/RN 59072970, Brazil
emilio @ct.ufr.br
Keywords: Entropy transport, twofluid model, Irreversibility, thermodynamic optimization
Abstract
This paper presents the equations for the calculation of the local entropy production rates due to heat transfer and flow
friction in multiphase mixtures, including entropy generation through interfacial heat and momentum transfer, by extending
the averaging process to the entropy balance in a differential control volume. Four mechanisms of entropy generation in
multiphase mixtures are identified: Dissipation and heat transfer within phases, dissipation due to interfacial interactions and
interfacial heat transfer. These mechanisms are carefully analyzed and closure relations are proposed for the interfacial
entropy generation. The equations are implemented in a CFD code. As the equation system for mass, momentum and energy
transport is closed, it turns out that entropy generation can be calculated as a prosprocessing procedure. Additionally a
simple onedimensional model is developed in order to have a quick assessment of the entropy generation due to interfacial
interaction. As an application example, the derived equations are applied to a fully developed flow and the contribution of
entropy generation due to phase interactions to the total entropy generated is analyzed.
Introduction
Nomenclature
Second law analysis has been extensively used to
optimize convective heat and mass transfer process
through entropy generation minimization, since pioneering
works from Bejan (Bejan (1978); Bejan (1980)). The
evaluation of the entropy generation, in flow processes,
permits the identification and quantification, of
irreversibility sources, and, in last case, the optimization of
the process by minimizing these irreversibilites. More
recent works have been focused on local or differential
entropy production calculations (Kock & Herwig (21"'4);
Kock & Herwig (2005); Shuja et al. (2001), and other
references) with the aim of implementation in CFD
calculations. This allows the application of these
optimization concepts to problems including complex flow
characteristics and complex geometries for which, flow
details related to entropy generation are unknown.
For a single component fluid, entropy generation is
governed by two distinct well known mechanisms of
thermodynamic irreversibility: flow friction and heat
transfer, which are ultimately related to fluid viscosity (not
zero) and conductivity (not infinite). The evaluation of the
entropy generation, in flow processes, allows the
identification and quantification, at least in comparative
way, of irreversibility sources, and, in last case, the
optimization of the process.
The aim of this paper is to present a theoretical
framework for the analysis of the entropy transport and
generation process in multiphase flows in the context of
the two fluid model which is the most popular approach in
CFD calculations. This will allow the calculation of local
n
GENInt Heat Trans
CGENInt VTscDiss
SVTiscDiss
cGEN HeatTrans
A"
Mk
T
q
J
d,,dD
Re
Nu
Be
Greek letters
a
s
Subsripts
C
D
I
Interface normal vector (unitary)
Entropy generation due to irreversible
interfacial heat transfer
Entropy generation due to viscous
interfacial momentum transfer
Entropy generation within phase k due
to viscous dissipation
Entropy generation within phase k due
to irreversible heat transfer
Interfacial area density
Interfacial momentum transfer per unit
volume
Stress Tensor
Heat Flux
Phase Superficial Velocity
Particle diameter
Reynolds Number
Nusselt Number
Bejan Number
Phase Volume Fraction
Turbulent eddy dissipation
Continuous Phase
Dispersed Phase
Relative to the interface
entropy generation due to irreversibilities in multiphase
process including entropy sources due to interfacial
interactions.
As a first approach, a single component twophase
mixture is considered. Then, entropy can only be generated
due to flow friction and heat transfer. Nevertheless,
conclusions obtained from this analysis related to
"competing irreversibilities" and minimum entropy
generation can be extended to process including interfacial
mass transfer in multicomponenet mixtures. Although
detailed mathematical description including all phenomena,
as entropy generation due to mass diffusion, for instance,
have been presented in literature (Kolev (2005), Ahmadi et
al. (2006), Dobran (1985), Bogere (1996)), entropy
transport equation is usually used, together with second
law of thermodynamics, as a constraint for closure
relations or process direction and, the specific phenomena
of irreversible entropy generation due to interfacial heat
and momentum transfer has not been adequately discussed.
In most of these works, the second law is presented as
inequality, where entropy generation should arise from the
source resulting from entropy balance but actual entropy
generation terms as function of temperature and velocity
gradients are not presented. Although these terms are well
known for single phase flow and have been derived for
turbulent flows as function of averaged and turbulent
variables (as k and e), they have not been well discussed
for the multiphase flows and no closure relations were
encountered for the interfacial entropy generation terms, as
function of averaged variables.
Faghri & Zhang (2006) presented the averaged entropy
transport equation, neglecting the entropy generation due
to irreversible interfacial momentum transfer and did not
presented closure relations for irreversible entropy
generation due to interfacial transfer. Other references
developed the equation for entropy transport for particulate
systems including surface tension effects (Young (1995),
Bilicki et al. (2002)) and for polydispersed systems, for
cases where important variations of interfacial area
concentration are expected along the domain
(S6roGuillaume & Rimbert (2005)). Nevertheless,
although these works present constitutive equations (not
closure relations, i.e., relate the entropy generation to the
averaged variables) for the entropy generation terms
including surface tension and interfacial area variations
effects (the last related, ultimately to breakup and
coalescence), once again the focus is on the constraints
defined by the second law on equations and not on the
irreversibility minimization, and no actual results for
entropy generation were presented.
In this work, first, the local instantaneous energy and
entropy transport equations for a phase in an Eulerian
context is presented in a conservative way. These equations
are averaged in order to obtain the equations for a
multiphase mixture in the context of the EulerianEulerian
model.
Then, closure relations are proposed for the entropy
generation terms due to phase interactions and an
application for two phase flows in ducts is showed, to
illustrate the implementation of these terms into a general
purpose CFD software. It is important to recall that, for
single component multiphase flow, mass, momentum,
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
energy and state equations for each phase together with
their closure relations for interfacial transfer and, the
"volume conservation" equation, constitute a closed set of
equations. Then there is no need to solve an additional
transport equation for entropy, and irreversible entropy
production can be calculated as a postprocessing phase of
numerical results.
To develop a quick analysis of viscous and heat transfer
entropy generation due to interfacial interactions and
evaluate their significance on the total entropy generation,
a simple one dimensional model for fully developed duct
flow was implemented. This simple model reproduces
adequately the results for single phase flows in ducts and
allows the quick evaluation of the interfacial entropy
generation for parametrical analysis as the impact of
particle diameter or relative velocity, on the interfacial
entropy generation. Through this simple model, it is
demonstrated that there are optimum parameters that
minimize the entropy generation in interfacial heat transfer
processes.
Averaged Equations
This section presents the derivation of the averaged
energy and entropy transport equations. Then, closure
relations are presented for the interfacial entropy
generation in continuousdispersed flows.
The local instantaneous equation for total energy
transport is given by,
k k(e +1V /2)) +V .(pkVkk ( /2))
8t '(1)
_V.qk +v.(r'v Vk )+pkfk Vk
According to the second law of thermodynamics
Clausius inequality, the entropy of a system could vary due
to heat transferred from its boundaries or irreversibilities.
d gen ; ge0
dt T
where the temperature in the first term of the r.h.s.
corresponds to the boundary temperature of the system. In
a differential control volume (Eulerian framework)
contained entirely within a phase k this equation reads,
(PkS ) PkVkSk )+V =k n > 0 (3)
01 Ttk Tk
The energy equation has to be used in order to compute
Sk,. This allows to compute the parcel of the net heat
transferred from/to a differential control volume and work
of surface forces is dissipated into irreversibilities (i.e., lost
of available work). In this work a multiphase single
component flow is considered, and then the entropy
generation is due to heat transfer with finite temperature
gradients and viscous dissipation. Then, the local entropy
generation, rate for a point located within k phase, is given
by,
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
kSk VT+ T"" VVk
gen T2 T 0
k Tk
In order to facilitate the averaging procedure
understand the entropy generation mechanisms
interfaces, the entropy generation term (eq. (4)), is wrii
in a divergent way, then, the local instantaneous entr
transport equation is given by,
a( + d (PVS +
v.rT
(4) (Xkk)= fXkk(r,t)dV= ak Dk (7)
( Vk
and where aGk is the volume fractions of phase k.
at The average of the time and space derivatives are
tten calculated respectively through the general Leibnitz and
opy Gauss theorems, to consider a control volume with a
moving boundary, which represents the phase interface (for
details, see Faghri & Zhang (2006) or Kolev (2005)), as,
( )k = k) A kVznkdA (8)
(5) \ t t AV J
k k k V Tk
+,T
When the energy equation is used to derive the entropy
equation through Gibbs relation, the work of pressure
forces cancel up and the stress appearing in eq. (5)
correspond to the viscous stress. This is obvious as
pressure forces are conservative and their work will not
generate entropy. Then, hereinafter the symbol T represents
the viscous stress tensor, unless in the case of drag
calculation which includes the total local stress at the
interfaces.
The stress Equations (1) and (5) represent the transport
of energy and entropy for single phase flows and are valid
within the region occupied by a phase, since adequate
conditions are specified at interface through the well
known "jump conditions". Nevertheless, in order to make
these equations "solvable" with current computer power,
for complex and quick time and space varying interfaces,
equation have to be averaged. Averaging process is very
well known and has been largely discussed in literature,
mainly, for mass and momentum equations. Although the
averaged energy and entropy transport equations for
multiphase systems have also been presented, as discussed
in previous section, some phenomena evolving entropy
production due to interfacial interactions are commonly
neglected, and others are not adequately discussed. Then,
the averaged equations for energy and entropy transport are
presented and analyzed here.
To obtain the averaged entropy transport equation, time
or volume averages,
yk k (r,t)dt
A t (6)
(6)
k = k (r,t)dV
k V
have to be used. Note that volume integral is performed
along the volume occupied by phase k, although the
average is over the whole control volume. Drew (1983)
and other authors use the concept of "phase indicator
function" Xk which has the value of 1 within phase k and 0
otherwise, to perform these integrals. The phase averaged
of Dk, is then given by,
(V.Vk)=V.(Vk)+ Vk.nkdAk
AV Vd
where A1 is the interfacial area within V and V is a vector
or tensor quantity. The second term of the r.h.s. of equation
(9) is equivalent to (VkVXk) frequently used in
averaging of two fluid models (see Drew (1983), Enwald
et al. (1996)), and corresponds to the average of the scalar
product of the tensor times the local interface normal.
Nevertheless, in our point of view, notation hereby used is
clearer and can be physically interpreted as a balance in a
control volume, performed along the control surface where
part of it corresponds to an interface. Then the second term
of the r.h.s. corresponds to the flux across the interface. In
view of this, entropy transport equation is written in
divergent way, particularly, the entropy production terms,
and the entropy production at the interfaces will arise from
an entropy balance across the interfaces, providing a more
physical understanding to these terms.
To account for the entropy generation due to turbulent
fluctuations, the equations the averaging process must
include the deviation of flow variables due to turbulence
fluctuations.
The deviations of the flow variables in a multiphase
mixture are due to turbulent fluctuations within each phase
and interfacial interactions. Additionally, phase
interactions could induce more fluctuations or dissipate
them depending on interface scales (Crowe (1993),
Kataoka et al. (1992)).
Although these phenomena are coupled, they are usually
treated separately in Eulerian averaged models. Turbulent
fluctuations are modeled through usual turbulence models
for single phase flow or slightly modified to include effects
of interfaces on turbulence production/dissipation (see, for
instance, Serizawa & Kataoka (1990), Kataoka et al.
(1992), Kataoka et al. (1993), Troshko & Hassan (2001),
etc.).
To derive averaged equations flow variables are usually
decomposed into an average and fluctuation parts. Usual
derivation of averaged equation for single phase turbulent
flow uses the well known Reynolds decomposition in
which the averaged part consist of an ensemble or time
average of the variable. To deal with products of density
and velocity fluctuation in compressible flows, a density
weighted is commonly used. Following the same line of
reasoning, Drew (1983) and other authors introduced two
weighted averages for adequate variable decomposition in
multiphase flows. These are the phasicc" and "massic"
averages which uses as weight function the phase indicator
function and the density times the phase indicator function,
respectively. The phasic average is defined through
equation (7) and is equivalent to the Reynolds average in
single phase flows. The massic average is defined by,
(pkXk k) =If Xkk (r, t)dV= kak (,k) (10)
V V
where Pk is the phasic average o density. In this work
the "^" is used to denote phasic average and "" to denote
massic average. The massic average can be considered
equivalent to the density weighted Favre average in single
phase flows. Whit these definitions the flow variables are
decomposed as,
qk = 6k + > phasic average
(11)
c)k = k + cI"k massic average
To obtain the averaged equation in a consistent way, the
transported variables as energy and velocity (momentum /
mass) are averaged weighted by density and other
variables as stress tensor are weighted by phase indicator
function. It is important to note that, in general, the
' = 0 but # 0 Nevertheless, for incompressible
flow, considered in this work, these averages coincide and,
in both cases the average of fluctuation is zero. For
consistency purposes the notation considering Favre
average is used but it has to be kept in mind that the
average of fluctuation is zero.
Introducing these decompositions into the entropy
equation and averaging the averaged entropy transport
equation can be written as,
(aPk )+V. (A^ Vk k) + v.('k,) + v.
at 1k
4k VT q'V77VT' ak'kV /k XkTV : VVk'
qk k k _/
1 1 V (12)
V k k ,
VTk k VTk 4 k
To obtain this equation, the term Tk= T + T' appearing
in the denominator of several terms was expanded in series
to demonstrate that T' appear only in higher order term
and then, was neglected (see, Kock & Herwig (2414)).
Additionally, as the temperature is not a transported
variable (instead is a intrinsic specific variable), its average
corresponds to a simple average over the volume occupied
by phase k, given by eq. (6).
The terms at the second line in eq. (12) (excepting the
last one which corresponds to the entropy transfer due to
phase change) stands for the intraphase entropy
production. For the case of turbulent flow, the entropy
generation mechanisms include: irreversible heat transfer
in a mean and fluctuating temperature field and viscous
dissipation due to gradients of mean velocity fields and
velocity fluctuations. The intraphase entropy production
due to turbulent fluctuations can be calculated using
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
standard turbulence models for single phase flows,.
Therefore, these terms are modeled following the ideas of
Kock & Herwig (2005) but considering the pahse indicator
function in the averaged quantities. Then, these terms are
modeled as,
S _4k VTk Xkq VTk LT k V1
!GENHeafTrh ~ + (k kf1 12k
k k (13)
ykl(W,+W k ):W k qkk
where K and KT are the molecular and turbulent thermal
diffusivities and k is the thermal conductivity.
The term corresponding to the entropy generation due to
viscous dissipation, can be easily obtained using the
definition of the turbulent energy dissipation, e. The
derivation of the equation for entropy production due heat
transfer, which includes the average and fluctuating
temperatures involves several considerations and details
are beyond the scope of this work and can be encountered
in {Kock, 2004 1969 /id} and {Kock, 2005 797 /id}.
Our main concern is on the modeling of entropy
production mechanism at the interfaces. The first term at
third line of eq. (12) represents the interfacial entropy flux.
As phase averaged temperatures at both side of interfaces
are different (see Figure 1), the balance of entropy flux
through interfaces will result in an entropy source which
corresponds to the entropy generation due to irreversible
interfacial heat transfer. The other terms on third and
fourth lines represent the entropy generation due to
interfacial viscous momentum transfer.
Next section focuses on closure relations for the
calculation of entropy production due to interphase
interactions.
Interfacial Entropy Transport and Generation
This objective of this section is to analyze the interfacial
terms in the entropy transport equation and develop closure
relations for these terms in particulate systems.
The irreversible entropy generation does not appear
explicitly in equation (12) but is embedded in entropy flux
balances across interfaces, involving local and averaged
variables. In order to explicit the total entropy generation
due to interface interactions, these terms are summed for
all phases and resulting imbalances will represent the
irreversible entropy generation in the mixture, due to
interface interactions.
An important concept behind the derivation of interfacial
entropy generation in the context of the twofluid model is
that, assuming standard hypothesis of no slip and thermal
equilibrium at interfaces, i.e., Va = V,6 and Td = T,6, no
entropy will be generated at interfaces in single component
mixtures1. Actually, entropy will be generated due to
temperature and velocity gradients near the interfaces and,
1 For multicomponent mixtures, equilibrium coefficient at interfaces are
not necessarily one, and "jumps" can be expected in component
concentration fields, generating a local entropy source at the interface.
if the instantaneous local equations are solved, entropy
generation will already be considered, through intraphase
entropy generation terms. Nevertheless, in the context of
twofluid model, information of detailed fields near
interfaces is lost in the averaging process and entropy
generated within these regions due to temperature and
velocity gradients has to be taken into account through
constitutive models. This concept is depicted in Figure 1.
The entropy calculated through temperature and velocity
"jumps" will represent, in the averaged equations, the
entropy generated due to local gradients at interfaces.
Control Volume
at Interface
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
1 1*;.v,A; T';. n,dA=
V T>' VJ k ndA V" V ndA =
VTk VTk
VkA, 'k A,
V" V"
V" S.n kdA+ N Jk nkdA= 0
SI kA k
VTk A, Vk k A,
Assuming continuity of viscous stresses at the interface
T, nl = 2 ni (valid when surface tension effects are
neglected) and using the definition of interfacial
momentum transfer (see, Enwald et al. (1996), Faghri &
Zhang (2006))
MIk Tk nkdA
SA,
U
4
Figure 1 Schematic local instantaneous and averaged
distributions, near an interface. The entropy production due
to interfacial interaction in two fluid model, arises from the
entropy generation within the control volume
For the sake of simplicity, only two phases are
considered in deriving closure relations. Summing up
entropy generation terms, bearing in mind that
n, = n2 = n and taken in account the continuity of heat
fluxes at interfaces, the entropy generation per unit volume
due to interfacial heat transfer is given by,
5 NIntHeat 1 41 ^
SGENJtHe 1Jq1.ndA qi J ndA=
1V 2 A 4 (14)
T2 T HI TT.
it "1 fj4,n*tdA= 21Q1
T1 VA, 1 T1
where Q, is the total interfacial heat transfer per unit
volume, which can be calculated using well known
correlations for interfacial heat transfer.
For the entropy generation due to the work of viscous
forces, first we consider the terms at the last line of
equation (12) and eq. (11), to take r'=Tri. Knowing
that the averaged values can be taken out from the average
term (w=T w) and that the average of a fluctuation is
zero, these terms can be written as (see Troshko & Hassan
(2001)),
Kk *VXk)
Custom Notation
in literature
As stated, for the calculation of total drag force the stress
r represents the total stress at the interface. Nevertheless, it
is made clear that the contribution to viscous entropy
generation is due to interfacial momentum transfer
originated by viscous interactions, as drag force. Although
equation (15) includes the nondrag forces, these forces
wil not contribute to entropy generation.
Then the entropy generation due to interfacial viscous
dissipation is given by,
V V
SGENIft VTscDiss M= '1 1 +
2 VT1
2 l (162
fV,1.T, nd4A Vr ndA
VT, VT2 ,
The terms on the second line correspond to the
dissipation due to the work done by viscous stresses
exactly at the interface. For general flow patterns these
terms have to be modeled adequately to each case.
Nevertheless, considering that the velocity exactly at the
interfaces in continuous dispersed flows can be considered
equal to the dispersed phase velocity (for dilute mixtures,
Ishii (1975))
S1CENIt fVacCDISS J T2V2 nndA
1 A, (17)
_V, M M )I
VTD IV112
1 fT T
"GENInt VLscD.iss 7 V2 i
iv 2T12
V2v V V2 2VR MI
VT1 VT T,
T
* o
where subindex "1" corresponds to the continuous phase.
For particulate flows, this result is consistent with the
assumption of no local velocity gradients within particles
or bubbles or droplets (This does not means the averaged
velocity fields of dispersed phases, cannot vary in space).
Then local velocity gradients are present only within the
continuous phase. This means that entropy generation due
to velocity deviations resulting from interfacial interactions
is only present in continuous phase and it is reasonable
consider the continuous phase average temperature to
compute the entropy generation due to interfacial drag. On
the other side, the entropy generation due to heat transfer
arises from entropy flux (q" T) imbalances and difference
of the inverse of the phase temperatures have to be used.
For the Sint,,s term, Bogere (1996), obtained,
^V,. ( M V .,
This term represents only the variation of kinetic energy
due to interfacial stresses, assuming that the total work
done by interface stresses in equation (16) is zero. But
work done drag force is not totally irreversible.
A very simple example can explain this question: if we
drop some rocks (particles) in a liquid at rest, hitting liquid
already with terminal velocity, it is true that some time
after the rocks reached the container bottom all work done
by gravity will be dissipated by viscosity. Nevertheless, in
the proximity of the interfaces there will a local increase of
the liquid velocity, or kinetic energy that could be,
eventually, transformed in work.
The entropy generation terms have to be consistent with
heat transfer correlations
These terms take into account the entropy generation due
to temperature and velocity deviations from averages,
resulting from local gradients near interfaces.
Applications and Discussion
This section is divided into two main parts; first an
analysis of the entropy generation due to interfacial
momentum and heat transfer in particulate systems is
presented for fully developed duct flow, with the aid of a
simple one dimensional model. For the case of particulate
systems, by "fully developed", it is understood that
velocity profiles are developed and the relative velocity
between phases is constant and equal to the terminal
velocity.
Additionally the onedimensional model was applied for
the study of free shear flows with particles, which can
represent for instance, pneumatic transport in wide isolated
sections, heat transfer in refrigeration columns or bubble
columns, i.e., when entropy generation due to shear and
heat transfer within the continuous phase is negligible, and
entropy generation will be, mainly, due to interfacial
interactions. Nevertheless, beyond the specific application,
this study allowed to isolate and analyze the phenomenon
of entropy generation due to interfacial interactions, which
lead to some interesting conclusions discussed hereafter.
In the second part of this section, the developed
equations for the interfacial entropy generation are
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
implemented in a CFD software and some results for duct
flow are presented to show the radial distribution of
entropy generation terms.
Onedimensional Model
To allow a quick evaluation of the entropy generation
due to interfacial interactions and understand the impact of
interfacial irreversible heat and momentum transfer on
total entropy generation, a one dimensional model for
hydrodynamic fully developed duct flow was implemented.
Assuming constant mean phase velocities and
incompressibility of both phases, only the energy the phase
equations have to be solved.
Additionally it is assumed that there are no interaction
among particles and no heat or momentum transfer with
particles and walls and temperature and velocity within
particles is constant. Then particles interact only with
continuous phase and entropy generation takes place only
in the continuous phase
These hypothesis (also considered in twodimensional
axissymmetric model, which results are presented in next
section) are consistent with particle models for dilute flows
and allowed a better understanding of the entropy
generation due to interfacial interactions. Basically, this
means that continuous phase consist in a stream
exchanging heat and momentum (viscous, then
irreversible) with solid boundaries and with dispersed
phase. Nevertheless, equations developed in previous
section are general and can be used for any application
with EulerianEulerian model.
Under these hypotheses, the one dimensional energy
equation for continuous and dispersed phases are,
m d( T q, P+hA (TD Tc) + PVc +MV19)
dz
d (CDTD) = hA"(T TD)
dz
Here continuous and dispersed phases can be constituted
by any fluid as, gas and droplets or solid particles, liquid
and bubbles, liquid and droplets, etc.
For constant diameter, the interfacial area density is
given by,
,6a
A"' ba
dD
For the cases of polydispersed systems, dD can be
replaced by the Sauter mean diameter. The conclusions
regarding optimum particle diameter are affected by this
fact and can be taken in a mean sense.
ShillerNewmann and RanzMarshal models were used
for interfacial momentum and heat transfer. For high
interfacial Re number the drag coefficient becomes
approximately constant with value about 0.44, then,
24
CD =max I(1+0.15 Re0687),0.44
Rez (21)
Nu, = 2+0.6Re105 PrC03
To account for the entropy generation due to continuous
phase heat transfer and viscous dissipation, wall friction
factor and Nusselt number are calculated, using classic
correlations for duct flows, as,
f, = 0.316Re 025
(22)
Nu,, = 0.023Rec08Prc04
The Nusselt at the duct wall is necessary to compute the
wall temperature, to calculate the entropy generation due to
external heat transfer. These correlations do not account for
the influence of particulate phase, and are valid only for
low dispersed volume fractions, but can be corrected using
some classic correlation as those based on
LockhartMartinelli parameter and, they do not affect the
conclusions in this work, as are based on qualitative
analysis.
The volume fraction is calculated as,
( D S+1
UDa
DD i
J,=Uc(1 a) ; JD
and
S (1aD) R
77
S J=
(1 a D)
UR = U UD (25)
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
to a minimization of entropy generation. The seek for these
optimal values is, precisely, the objective of the method of
optimization through entropy minimization.
To analyze this behavior Paoletti et al. (1989) introduced
a useful parameter which relates the entropy generation
due heat transfer to total entropy generation, called "Bejan
Number" 2, defined as the entropy generation due to
irreversible heat transfer and the total entropy generation.
As the focus of this work is on the entropy generation
due to interfacial interactions, it is introduced here the
"interface Bejan Number", defined as,.
e ~SInt Heat
Be, = S_. He,
IntVlsc Int Heat
For Bez values above 1/2 the viscous dissipation governs
the irreversible entropy generation (or exergy destruction),
and for values below, the heat transfer dominates the
process.
For particulate systems (continuousdispersed
morphologies), the local entropy generation due to
interfacial heat transfer and viscous momentum transfer,
per unit volume, can be computed as,
6aD Nuk (T D C)2
GENnt Heat Trans Nuk (7
dD dD TCTD (28)
(28)
6cx aD VR2 IV
SGENInt VicDiss CD D C
dD 2 Tc
Then, the particle Bejan number can be expressed as,
Be, =1
STDdDCD PcV
2Nu kc (TD TC)
where S is slip ratio and UR is the relative velocity.
Considering the relative velocity equal to the terminal
velocity, the volume fraction of dispersed phase can be
calculated iteratively (for details, see Paladino (2005)).
Integrating equations (14) and (18) across the channel
section, the total entropy generation per unit length is
given by,
=gen qw PTw Nu k A'(TD CT
Tw Tc dD TD T
SHeat
+ rPV MDVR
T, T,
vsc "Intrisc
SInt Heat
Depending on flow conditions entropy generation can be
governed by heat transfer or viscous dissipation. This
depend on several flow parameters, but as stated in Bejan
(1995) or Bejan (1997) and other references, there are
specific geometrical parameters which optimal values lead
It is clear that the increase of interfacial area density
(decrease of particle diameter, for the case of particulate
systems) lead to a increase of momentum and heat transfer.
As can be seen in Figure 2 entropy generation due to
viscous dissipation and heat transfer in fully developed
flow, are competing phenomena when varying the particle
diameter and there is a particle diameter that minimizes the
entropy generation. This is not explicit in equations (28)
because the relative velocity is a variable but, depending
on flow type, it is closely related to the particle diameter.
Particularly, in order to evidence the existence of optimal
particle diameter, the one dimensional model implemented
is used to compute the interfacial entropy generation for
different particle diameters. Additionally, the model
accounts for the variation with dD of CD Nu1 depends on
Rez calculated through terminal velocity which, on its
turns, depends on CD making all these calculations
iterative and not allowing for an explicit equation for the
interfacial entropy generation as function of particle
diameter.
Figure 2 shows the local interfacial entropy generation
for airwater flow for different particle diameters for a
2 honoring Professor A. Bejan at Duke University
fixed temperature difference of 10C. Two limiting cases are
presented, airdroplet and waterbubble flows. In the first
case, the continuous phase has low thermal conductivity
and low viscosity and, in the second case the continuous
phase has high values of these properties. As stated, it is
considered that temperature and velocity is constant within
particles (droplets or bubbles) and then, entropy generation
takes place only within continuous phase.
Nevertheless this very simple analysis, lead to the
interesting conclusion that there is an optimum particle
diameter or optimum particle Reynolds number, for
heat transfer between phases. Although, the "minor
possible" diameter, which gives higher interfacial area
concentration, certainly enhances heat transfer between
phases, this is not the one which minimizes entropy
generation in processes involving interfacial heat transfer.
IU
10
101
100
101
102 
102
103
1(
103
 SGEN Interfacial
AA SGEN Int. Heat Transfer
BE] SGEN Int. Visc. Dissipation.
104 103
dp [m]
(a)
102 101
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
This simple example is presented to illustrate the
concept of minimum entropy generation in interfacial heat
transfer.
A hot air stream has to be refrigerated by injecting a
quantity of liquid water in cocurrent flow. Actually, in
terms of interfacial entropy generation, the fact of being
cocurrent flow is only relevant as it affects inlet
temperature of phases, as momentum and heat transfer
depends only on relative velocity, which is assumed to
reach instantaneously the terminal velocity for a given
droplet diameter3. This assumption could not be exact for
larger droplet diameters as it could take some distance
(equal to the continuous phase velocity times the relaxation
time) to the relative velocity reach the terminal velocity,
but does not invalidate the conclusions from this analysis,
taken on a qualitative basis. To have a more general picture
of entropy generation due to interfacial entropy generation,
the whole equation system, including mass and momentum
transport has to be solved. This is accomplished by
implementing the entropy generation terms in a CFD
software, as described in next section.
The air is considered saturated to avoid the need o mass
transfer calculation which is beyond the scope of this work.
It is emphasized that this example has the objective of
illustrate the existence of minimum entropy generation
points in interfacial heat transfer process and not to
develop an accurate quantitative analysis. Then any pair of
"inert" fluids can be chosen.
The superficial velocities of both phases are equal to
Im/s. This gives a volumetric fraction of 0.5 for zero slip,
i.e., very small droplet diameters. Nevertheless, as droplet
diameter increases so do the terminal velocity and the
droplet volume fraction, which is calculated iteratively
through equations (23) to (25). This parameter together
with terminal velocity are shown in Figure 3
0.10
 SGEN Interfacial
A A SEN Int. Heat Transfer
S EE] S,, Int. Visc. Dissipation.
S10 01    ' 
S10 0
c 101
102 
103
104 103 102
dp [m]
(b)
Figure 2 Local interfacial entropy gene
volume for air water flow for a tempe
(continuous) water droplets. (b) Water (con
** Droplet Vol. Frac.
A A Terminal Velocity
CD
2
5
1
0x100 4x104 8x1 04 1x103
d,[m]
I I Figure 3 Terminal velocity and void fraction as
0 function of droplet diameter
The global entropy generation for different droplet
ration per unit diameters is shown in Figure 5. The viscous dissipation
nature. (a) air and heat transfer components are also presented. These
tinuous) plots were obtained for values of gas and liquid inlet
Example: Gas cooling by mixing with cool liquid
3 This is the hypothesis underling the "DriftFlux" model (Hibiki & Ishii
(2003), and other references)

7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
temperatures respectively of 310 K and 300 K. Figure 6
presents the plots of total entropy generation rates for
different droplet diameters, for various gas inlet
temperatures. It can be seen that the droplet diameter
which minimizes the entropy generation rate varies with
gas temperature.
The results from this example point out again that there
is a diameter which minimizes the entropy generation.
This behavior can be explained in a simple way from an
energy point of view: The work of buoyancy or weight
force (depending on which is the heavier phase), which is
proportional to dp3, and is conservative is transformed in
kinetic energy which is dissipated by drag (oc dp2) and their
balance results in a terminal velocity which augments with
droplet diameter. As in any heat transfer system, lager
velocities lead to lower heat resistances, reducing the
entropy generation due to irreversible heat transfer. On the
other side the increase of the drag force, which is
accounted in entropy generation terms, as "entropy
generation due to interfacial viscous dissipation", will
decrease the terminal velocity and so increase the entropy
generation due to irreversible heat transfer.
For readers used with the topic of entropy generation
minimization in heat exchangers, the concept can be made
clearer trough a simple analogy, depicted in Figure 4.
Suppose that particles are fixed in space exchanging heat
with a fluid moving at a velocity equal to the relative
velocity and the exchange area is given by the interfacial
area. The pressure drop along the exchanger will be a
proportional to the sum of drag on particles, and is related
to the entropy generation due to viscous. In this scenario, it
is known that, the higher the relative velocity, the higher
entropy generation due to viscous dissipation and the lower
entropy generation due heat transfer, and viceversa. It has
to be pointed out that, for the case of multiphase heat
transfer, both, the interfacial area (or "exchange" area) and
relative velocity varies with particle diameter.
103
10
Zi
105
106
SS, SENInterfacial
AAA GENInt. Heat Transfer
FR SGEN Int. Vise. Dissipation.
dp[m]
Figure 5 Total, heat transfer and Viscous dissipation
interfacial entropy generation rates for gasdroplet flow for
gas and liquid temperatures of 310 K and 300 K
01 
Q
Q
lttltltt
Particles fixed
in space
0 0001
107 106 105 104
dp [m]
V = Terminal
Velocity
Figure 4 Analogy between interfacial heat transfer and
a heat exchanger
103 102
Figure 6 Total interfacial entropy generation rates for
gasdroplet flow for different gas inlet temperatures and
liquid inlet temperature of 300 K
The one dimensional model had the objective of analyze
the entropy generation terms, providing a quick analysis of
them and indicate some trend to be expected as varying
some characteristic parameters of particulate flows as
particle diameter, relative velocity, etc. Nevertheless, in
order to apply the concepts developed in this paper, with
the main objective of entropy generation minimization in
process involving multiphase flow, to industrial
applications, the closure relations developed have to be
implemented in a general purpose CFD software. Nesxt
section illustrates this implementation, by a simple
application to the same example studied in this section.
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
300.0
Twodimensional axissymmetric flow in ducts
The objective of this section is to illustrate the
implementation of the closure relations developed into a
CFD package. All capacities of a general purpose CFD
software can be used as different models for drag
coefficient and interface Nusselt number, which includes
particle deformation. Additionally a multidimensional
approach can be employed. As stated, as the equation
system is closed for the calculation velocity, pressure and
temperature through mass, momentum and energy
transport equations, there is no need of additional
calculation in the solution procedure, and entropy
generation, including that due to interfacial interactions,
was implemented in the postprocessing stage of the
simulations.
The example described in previous section is
implemented and analyzed in a twodimensional model.
Radial distributions of entropy generation, due to
interfacial sources are obtained. A free slip and adiabatic
wall is considered, in order that the only entropy source be
the generation due to interfacial interactions. To illustrate
an application with heat and viscous dissipation within the
continuous phase, an additional case is simulated, with the
same conditions of example, but imposing a no slip
condition at the wall and a heat flux of 1000 W / m2. The
duct length is taken to be enough for the hydrodynamic
development of the flow.
Figure 7 presents the entropy generation rates along
radius for the cases of isolated wall and with a heat wall
flux of 1000 W/m2.
3.5 
3.0
2.5
c(1
2.0
1.5
on
1.0
0.5
S200.0
100.0
0.0
A A SGEN Interfacial Heat Trans.
AA SGEN Interfacial Visc. Diss.
0 SGEN Heat Transfer
* SGEN Viscous Diss.
0 0.002 0.004 0.006 0.008 0.01
r [m]
AA A SGEN Interfacial Visc. Diss.
0.4 SEN Viscous Diss.
0 0.002 0.004 0.006
r [m]
(c)
0.008 0.01
Figure 7 Radial distributions of entropy generation
rates for airwater (droplets) flow in a duct. (a) Isolated
free slip wall. (b) No slip wall with heat flux of 100 W/m2
(c) detail showing only the entropy generation due to
viscous dissipation
S Interfafcial Heans
GEN ..... ..
A AA SGEN Interfacial Visc. Diss. For the case of isolated free slip wall no "intraphase"
S  SGEN Heat Transfer entropy generation is expected as, a priori, the only
SE SN Viscous Diss. sources of entropy are the interfacial irreversible heat
 .. .  _transfer and viscous dissipation. Nevertheless small radial
temperature variations (less than a Celsius degree) take
S........ place and an interesting coupled effect is observed in
0 0.002 0.004 0.006 0.008 0.01 results for this case: continuous phase temperature fields
r [m] are affected by interfacial heat transfer and then there will
(a) be an entropy generation within continuous phase, even
when domain is isolated. This phenomenon is not captured
by one dimensional model, unless the gradient of variances,
which account for non constant radial distributions be
included in entropy generation rates.
For the case of noslip wall with heat flux, it can be seen
that, although for these flow condition heat transfer
entropy production dominates, the production due to
interfacial interactions is of the same order of intraphase
production for both viscous dissipation and heat transfer
production. The peak of entropy production due to
interfacial heat transfer near wall is related to the
difference of phase diffusivities. Although it was imposed
that only the continuous phase exchanges het with duct

A
wall, the continuous phase temperature increases near wall
and the interfacial heat flux is intensified as droplet
temperature remains almost constant du the high liquid
thermal inertia.
Figure 8 presents the axial variations of entropy
generation rates along the duct for the same cases. The
values for each axial position are calculated as the area
averaged of the entropy generation rates.
104 
AA SEN Heat Transfer
AA S Vise Diss
0 S_ Int Heat Transfer
S4 S,,Int Vise Diss
0 0.2 0.4
Z [nm]
1.0x103
1.0x104
1.OxO' 
S1.OxlO
1.0x10 
AAA Sc
SS
00 S
 Sc
0.6 0.8 1
SHeat Transfer
SVise Diss
SInt Heat Transfer
SInt Vise Diss
Th~
0 0.2 0.4 0.6
Z [m]
0.8 1
(b)
Figure 8 Area averaged axial distributions of entropy
generation rates for airwater (droplets) flow in a duct. (a)
Isolated free slip wall. (b) No slip wall with heat flux of
100 W/m2
Summary
A systematic derivation of averaged entropy transport
equation in the context of the EulerianEulerian model has
been presented.
A one dimensional model for channel flow was
implemented for quick analysis of the terms behavior.
The model was implemented as a portprocessing
procedure in a CFD software and ...
The main conclusion we arrived in this work is that,
depending on the process the idea of "maximum interfacial
area" in multiphase heat transfer, does not necessarily lead
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
to the maximum process efficiency in terms of irreversible
entropy generation. As multicomponent interfacial mass
transfer mechanisms are similar to the transfer, specifically
in terms of interfacial area, this conclusion can be extended
for process involving these phenomena, as refrigeration
towers, spray drying, etc., although formal equations and
closure relations for multicomponent systems must be
derived to account for the interfacial entropy generation
due to irreversible mass transfer.
Acknowledgements
Author is grateful to CAPES and FAPERN for the
financial support of this research.
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