A Turbulence Dissipation Model for ParticleLaden Flow
John D. Schwarzkopf, Clayton T. Crowe, and Prashanta Dutta
School of Mechanical and Materials Engineering, Washington State University, Pullman, WA 99164
DOI 10.1002/aic.11773 Published online May 7, 2009 in Wiley InterScience (www.interscience.wiley.com).
A dissipation transport equation for the carrier phase turbulence in particleladen ﬂow is derived from fundamental principles. The equation is obtained by volume averaging, which inherently includes the effects of the particle surfaces. Three additionalterms appear that reveal the effect of the particles; these terms are evaluated usingStokes drag law. Two of the terms reduce to zero and only one term remains which isidentiﬁed as the production of dissipation due to the particles. The dissipation equationis then applied to cases where particles generate homogeneous turbulence, and experimental data are used to evaluate the empirical coefﬁcients. The ratio of the coefﬁcient of the production of dissipation (due to the presence of particles) to the coefﬁcient of the dissipation of dissipation is found to correlate well with the relative Reynoldsnumber.
V
V
C
2009 American Institute of Chemical Engineers
AIChE J
, 55: 1416–1425, 2009
Keywords: turbulence, particles, dissipation, volume average
Introduction
A recent literature review in the area of modeling particleladen turbulent ﬂows reveals that there is a continuing needto develop a robust model for the effect of the dispersephase on the turbulence of the conveying phase. A shortreview of the modeling status was published by Curtis et al.
1
who pointed out that large industries such as the chemical,pharmaceutical, agricultural and mining industries can beneﬁt from an understanding of particleladen ﬂows. Such abeneﬁt is claimed to affect cost savings and increased productivity. Curtis et al. also identiﬁed several areas of investigation that could beneﬁt industry; these include turbulentgasﬂow interactions, particle clustering, particle shape, frictioneffects, and particle size distribution. Other applicationswhere the effect of particles on ﬂuid phase turbulencebecomes important are ﬂuidized beds, chemical reactors,drug delivery systems, pollution control, and the food processing industries, to mention a few.In particleladen turbulent ﬂow, the mechanisms responsible for turbulence modulation (e.g., the effect of particles oncarrier phase turbulence) are not well understood.
2
As particles are introduced, the statistics of the continuous phaseturbulence are altered. Depending on the particle characteristics such as size, density, mass loading and relative velocitydifference, the level of turbulent kinetic energy (TKE) anddissipation changes relative to the corresponding unladenﬂow. Gore and Crowe
3
showed that there is a relationshipbetween turbulent modulation and the ratio of the particlesize to a characteristic length of the most energetic eddy inthe ﬂow. The primary reason for this modulation is attributedto the twowaycoupled kinetic energy and dissipationbetween the continuous and dispersed phases.The modulation of turbulence due to the presence of particles is attributed to the altered dissipation within the continuous phase caused by the work done at the surfaces of theparticles.
4
The modulation of turbulence in particle ladenﬂows has been demonstrated by extensive experimentationover the last two decades.
5–15
However, a turbulence modelthat adequately predicts these modulations over a wide rangeof data is still lacking.The most robust and widely used model for turbulence insingle phase ﬂows has been the two equation
k

e
model. Theequation for turbulence kinetic energy is:
Dk
t
Dt
¼ ð
P
k
Þ
t
À
e
t
þ
@ @
x
i
m
þð
m
T
Þ
t
r
k
8>:9>;
@
k
t
@
x
i
!
(1)
Correspondence concerning this article should be addressed to P. Dutta at dutta@mail.wsu.edu or dutta@mme.wsu.edu
V
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2009 American Institute of Chemical Engineers
1416 AIChE JournalJune 2009 Vol. 55, No. 6
where
k
t
is the time averaged turbulence kinetic energy
ð
k
t
¼
u
0
i
u
0
i
Þ
=
2
Þ
;
ð
P
k
Þ
t
is the production of turbulent energy dueto mean (i.e., time averaged) velocity gradients,
v
is themolecular kinematic viscosity, (
v
T
)
t
is the turbulent kinematicviscosity (based on time averaging),
e
t
is the time averageddissipation and
r
k
is the effective Schmidt number for turbulent diffusion. The turbulence kinetic energy is affectedby diffusion, the generation due to mean flow velocitygradients and time averaged dissipation. The equation for dissipation is:
D
e
t
Dt
¼
C
e
1
ð
P
e
Þ
t
e
t
k
t
À
C
e
2
e
2
t
k
t
þ
@ @
x
i
v
þð
v
T
Þ
t
r
e
8>:9>;
@
e
t
@
x
i
!
(2)where (
P
e
)
t
is the production of dissipation due to mean (i.e.,time averaged) velocity gradients,
C
e
1
,
C
e
2
are empiricalconstants and
r
e
is the turbulent Schmidt number for diffusionof dissipation. As with the turbulence kinetic energy, thedissipation is affected by diffusion of dissipation, productionof dissipation and dissipation of dissipation.The development of the equation for dissipation is not asstraight forward as for the turbulence kinetic energy equation. Bernard and Wallace
16
identify the time averaged dissipation as:
e
t
v
@
u
0
i
@
x
j
@
u
0
i
@
x
j
8>>>>:9>>>>;
(3)They show the development of an equation for dissipation bytaking the gradient of the momentum equation, multiplying itby twice the kinematic viscosity and by the gradient of thefluctuating velocity and then time averaging the entireequation, which is mathematically represented by:
@ @
x
j
½
NS
i
8>>:9>>;
Ã
2
v
@
u
0
i
@
x
j
8>>:9>>;
The instantaneous velocity, pressure, and shear stress aredecomposed into the sum of a time averaged and a fluctuatingproperty to obtain:
D
e
t
Dt
¼ À
2
v
@
u
k
@
x
j
@
u
0
i
@
x
j
@
u
0
i
@
x
k
!
À
e
ik
ð Þ
t
@
u
i
@
x
k
À
2
v u
0
k
@
u
0
i
@
x
j
!
@
2
u
i
@
x
j
@
x
k
À
2
v
@
u
0
i
@
x
j
@
d
u
0
k
@
x
j
@
u
0
i
@
x
k
!
À
v
@ @
x
k
u
0
k
@
u
0
i
@
x
j
@
u
0
i
@
x
j
!
À
2
v
q
@ @
x
i
@
u
0
i
@
x
j
@
P
0
@
x
j
!
þ
v
@
2
e
t
@
x
2
k
À
2
v
2
@
2
u
0
i
@
x
j
@
x
k
8>>:9>>;
2
ð
4
Þ
where (
e
ik
)
t
is the dissipation tensor. A discussion of each of the terms is presented in Bernard and Wallace.
16
They makearguments for grouping terms together and modeling of other terms to yield the equation for dissipation, namely Eq. 2.A typical approach to obtain twoequation models for turbulence energy and dissipation in dispersed phase ﬂows is tobegin by adding a source term to the singlephase momentum equation to account for the surface effects, namely,
@ @
t
q
u
i
ð Þ þ
@ @
x
j
q
u
j
u
i
ÀÁ
¼ À
@
P
@
x
i
þ
l
@ @
x
j
@
u
i
@
x
j
8>>:9>>;
þ
a
d
q
d
s
p
f v
i
À
u
i
ð Þ ð
5
Þ
where
u
i
is the instantaneous carrier phase velocity,
v
i
is theinstantaneous dispersed phase velocity,
a
d
and
q
d
are thevolume fraction and material density of the dispersed phase,respectively,
s
p
is the particle response time, and
f
is the dragfactor. The derivations then proceed using the same Reynoldsaveraging procedures employed for single phase incompressible flows. This additional term is the drag force per unitvolume on the continuous phase. The concept of adding apoint force to represent the effect of a cloud of particles isopen to argument.A test to assess the viability of a turbulence model for disperse phase ﬂows is to apply the model to the simplest possible ﬂow conﬁguration. This idealized case would be a uniform, steady ﬂow through a cloud of particles ﬁxed in position over a large region of space with no walls (shown inFigure 1). The volume averaged properties of the ﬂow wouldbe homogeneous and steady. In this case the production of turbulence by the particles would be equal to the dissipationbecause there would be no diffusion or production due tomean velocity gradients. Applying the turbulence modelsderived from Eq. 5 to this conﬁguration typically yields azero or negative value for the dissipation.
17
The momentum equation shown in Eq. 5 cannot bederived from fundamental principles; even for very diluteﬂows, the momentum equations cannot be reduced to Eq. 5.The two valid approaches for developing the momentumequations for dispersed phase ﬂow are volume averaging
18
and ensemble averaging.
19
It has been shown
18
that the additional source term needed in the momentum equations toaccount for the surface effects of particles arises from volume averaging the momentum equations (i.e., a volumelarger than the continuum limit is necessary). The averagevelocities in the volume averaged or ensemble averagedequations do not represent the local (point wise) instantaneous velocity of a given ﬂow and thereby are not amenable tothe Reynolds averaging procedures used in single phaseﬂows. In other words, the temporal ﬂuctuations of the averaged velocities do not reﬂect the ﬂow turbulence.Crowe and Gillandt
20
developed an equation for carrier phase turbulent kinetic energy by volume averaging the mechanical energy equation. When applied to the basic testcase, the equation reduces to the expected result; the rate of
Figure 1. An idealized case used to test twoway coupled turbulence models in particleladenﬂows.
For such a case, it is assumed that there are no wall effectsand the particles are stationary; therefore the generationmust balance with the dissipation.
AIChE Journal June 2009 Vol. 55, No. 6 Published on behalf of the AIChE DOI 10.1002/aic 1417
turbulence generation by the particles is equal to the rate of dissipation. The turbulence kinetic energy equation developed by Crowe and Gillandt
20
has been used by Lain et al.
21
in bubble ﬂow. The comparison showed that the numericalresults using Crowe and Gillandt’s
20
turbulence kineticenergy equation slightly underpredicted the average velocitybut was an improvement over conventional models. Zhangand Reese
22,23
also performed an extensive study on Croweand Gillandt
20
turbulent kinetic energy equation and foundgood comparison with the data of Tsuji et al.
8
The purpose of this article is to present an approach toobtain a volume averaged dissipation equation for particleladen turbulent ﬂows. By deriving the dissipation equationfrom ﬁrst principles, it is possible to establish the effects of the dispersed phase on the turbulence dissipation of the carrier phase.
Review of Volume Averaging Concepts
In single phase ﬂows, the turbulence equations are developed by Reynolds averaging the Navier Stokes equationswhich describe the instantaneous properties at a point. Turbulence is described as the temporal velocity ﬂuctuationfrom its time averaged value at a point in the ﬂow. In dispersed phase ﬂows it is not possible to describe the ﬂowproperties at a point without the inclusion of the effect of the neighboring particles (illustrated in Figure 2).Volume averaging and ensemble averaging provide ascheme to include the effects of the dispersed phase withoutthe necessity of including the details of the surface interaction. Crowe et al.
18
and Slattery
24
provide a detailed description of the volume average concept. The local volume average of a property B is deﬁned as:
B
¼
1
V
Z
V
c
BdV
(6)where
V
is the volume of the mixture and
V
c
is the volume of the continuous phase in the mixture volume. The phasevolume average of the property B is defined as:
h
B
i ¼
1
V
c
Z
V
c
BdV
(7)The phase average property is related to the local volumeaveraged property by:
B
¼
a
c
h
B
i
(8)where
a
c
is the volume fraction of the continuous phase, oftenreferred to as the ‘‘void fraction.’’ The averaging volume mustbe large enough to maintain a stationary average yet smallcompared to system dimensions to enable the use of differential operators.Aside from temporal averaging, another way of deﬁningturbulence is by the velocity deviation from the volumeaveraged velocity at a point in time, such as:
u
i
¼
u
i
h iþ
d
u
i
(9)where
u
i
is the instantaneous velocity,
h
u
i
i
is the phase volumeaveraged velocity as illustrated in Figure 3. The turbulenceenergy is then defined as
k
¼
1
V
c
Z
V
c
d
u
i
d
u
i
2
dV
¼
d
u
i
d
u
i
2
()
(10)The advantage of using volume averaging is that the effects of the surfaces are easily distinguished from the effects of thefluid. One of the identities introduced is the volume average of the gradient of a property, which brings out the effects of eachparticle surface within the domain. This identity is expressedas:
@ @
x
j
B
i
ð Þ ¼
@
B
i
@
x
j
À
1
V
Z
S
d
B
i
n
j
dS
(11)where the integration is performed over the particle surfaces,
S
d
, inside the control volume.A very important advantage in applying the volume average concept to dispersed phase ﬂows is that jump boundaryconditions and moving meshes are not needed as would bein a point wise ﬂow analysis (such as the RANS equations).In the presence of millions of particles, this approach wouldbecome computationally expensive. Although DNS practicesuse point wise forces to represent particles, the same
Figure 2. A qualitative comparison of the volume averaging approach and temporal averagingapproach to modeling effects of particles.
Unless the particle surfaces are treated as boundary conditions, a temporal averaging approach does not include theeffect of neighboring particles at a point in the ﬂow.
Figure 3. An illustration of the volume deviation velocity used to deﬁne turbulence in a volumeaveraged setting.
The instantaneous velocity, at any point in the ﬂow, is thesum of the volume average and the volume deviation velocities at a point in time.
1418 DOI 10.1002/aic Published on behalf of the AIChE June 2009 Vol. 55, No. 6 AIChE Journal
problem is imposed. In addition, the effect of the particlevolume is ignored. An interim solution to this complex problem is volume averaging.The difﬁculty with the application of volume averaging tomultiphase turbulence equations is in the comparison toexperiments. Typical experiments are set up for point wisemeasurements, however with newly developed instrumentation such as Particle Image Velocimetry (PIV) there ispotential to perform volume averaged measurements.
Derivation of the Dissipation Equation
To close the volume averaged turbulence equation set, anequation for the time rate of change of dissipation is needed.The following derivation is analogous to the derivation of the time average dissipation equation, provided by Bernardand Wallace.
16
The deﬁnition of volume average dissipationintroduced by Crowe and Gillant
20
is:
e
¼
v
@
d
u
i
@
x
j
@
d
u
i
@
x
j
()
(12)To begin, a spatial gradient of the NavierStokes equation istaken. This is multiplied by the volume deviation velocitygradient (
q
d
u
i
/
q
x
j
) and twice the kinematic viscosity. Finally,the result is volume averaged, which is represented mathematically by:
@ @
x
j
½
NS
i
8>>:9>>;
Ã
2
v
@
d
u
i
@
x
j
8>>:9>>;
which can also be expressed as:2
v
@
d
u
i
@
x
j
@ @
x
j
q
@
u
i
@
t
8>:9>;
þ
2
v
@
d
u
i
@
x
j
@ @
x
j
q
u
k
@
u
i
@
x
k
8>:9>;
¼ À
2
v
@
d
u
i
@
x
j
@ @
x
j
@
P
@
x
i
þ
2
v
@
d
u
i
@
x
j
@
2
s
ik
@
x
j
@
x
k
ð
13
Þ
Decomposing the instantaneous terms in Eq. 13 into volumeaverage and deviation terms and assuming the flow isincompressible, the above equation can be rewritten as:2
v
@
d
u
i
@
x
j
@ @
x
j
@
u
i
h i
@
t
8>:9>;
þ
2
v
@
d
u
i
@
x
j
@ @
x
j
@
d
u
i
@
t
8>:9>;
þ
2
v
@
d
u
i
@
x
j
@
h
u
k
i
@
x
j
@
h
u
i
i
@
x
k
þ
2
v
h
u
k
i
@
d
u
i
@
x
j
@
2
h
u
i
i
@
x
j
@
x
k
þ
2
v
@
d
u
i
@
x
j
@
h
u
k
i
@
x
j
@
d
u
i
@
x
k
þ
2
v
h
u
k
i
@
d
u
i
@
x
j
@
2
d
u
i
@
x
j
@
x
k
þ
2
v
@
d
u
i
@
x
j
@
d
u
k
@
x
j
@
h
u
i
i
@
x
k
þ
2
v
d
u
k
@
d
u
i
@
x
j
@
2
h
u
i
i
@
x
j
@
x
k
þ
2
v
@
d
u
i
@
x
j
@
d
u
k
@
x
j
@
d
u
i
@
x
k
þ
2
v
d
u
k
@
d
u
i
@
x
j
@
2
d
u
i
@
x
j
@
x
k
¼ À
2
v
q
@
d
u
i
@
x
j
@ @
x
j
@
h
P
i þ
d
P
@
x
i
þ
2
v
q
@
d
u
i
@
x
j
@
2
@
x
j
@
x
k
ðh
s
ik
i þ
ds
ik
Þ ð
14
Þ
Applying the volume averaging concept to each term in Eq. 14shows (details are shown in Appendix):
v
@ @
t
a
c
@
d
u
i
@
x
j
@
d
u
i
@
x
j
()!
þ
v
@ @
x
k
a
c
h
u
k
i
@
d
u
i
@
x
j
@
d
u
i
@
x
j
()!
¼ À
2
v
a
c
@
h
u
k
i
@
x
j
@
d
u
i
@
x
j
@
d
u
i
@
x
k
()
À
2
v
a
c
@
h
u
i
i
@
x
k
@
d
u
i
@
x
j
@
d
u
k
@
x
j
()
À
2
v
a
c
@
2
h
u
i
i
@
x
j
@
x
k
d
u
k
@
d
u
i
@
x
j
()
À
2
v
a
c
@
d
u
i
@
x
j
@
d
u
k
@
x
j
@
d
u
i
@
x
k
()
À
v
@ @
x
k
a
c
d
u
k
@
d
u
i
@
x
j
@
d
u
i
@
x
j
()!
À
2
v
q
@ @
x
i
a
c
@
d
u
i
@
x
j
@
d
P
@
x
j
()!
þ
v
2
@
2
@
x
2
k
a
c
@
d
u
i
@
x
j
@
d
u
i
@
x
j
()!
À
2
v
2
a
c
@
2
d
u
i
@
x
j
@
x
k
8>>:9>>;
2
*+
À
v
2
V
@ @
x
k
Z
S
d
@
d
u
i
@
x
j
@
d
u
i
@
x
j
n
k
dS
þ
2
v
q
V
Z
S
d
@
d
u
i
@
x
j
@
d
P
@
x
j
n
i
dS
À
2
v
2
V
Z
S
d
@
d
u
i
@
x
j
@ @
x
j
@
d
u
i
@
x
k
n
k
dS
ð
15
Þ
By identifying the volume averaged dissipation tensor as
e
ik
¼
2
v
@
d
u
i
@
x
j
@
d
u
k
@
x
j
DE
and noting that the volume averagedcontinuity equation for incompressible flow is
@
a
c
@
t
þ
@
ð
a
c
h
u
k
iÞ
@
x
k
¼
0, Eq. 15 can be rewritten in the form:
a
c
D
e
Dt
¼ À
2
v
a
c
@
h
u
k
i
@
x
j
@
d
u
i
@
x
j
@
d
u
i
@
x
k
()
À
a
c
e
ik
@
h
u
i
i
@
x
k
À
2
v
a
c
d
u
k
@
d
u
i
@
x
j
()
@
2
h
u
i
i
@
x
j
@
x
k
À
2
v
a
c
@
d
u
i
@
x
j
@
d
u
k
@
x
j
@
d
u
i
@
x
k
()
À
v
@ @
x
k
a
c
d
u
k
@
d
u
i
@
x
j
@
d
u
i
@
x
j
()!
À
2
v
q
@ @
x
i
a
c
@
d
u
i
@
x
j
@
d
P
@
x
j
()!
þ
v
@
2
@
x
2
k
a
c
e
½ À
2
v
2
a
c
@
2
d
u
i
@
x
j
@
x
k
8>>:9>>;
2
*+
À
v
2
V
@ @
x
k
Z
S
d
@
d
u
i
@
x
j
@
d
u
i
@
x
j
n
k
dS
þ
2
v
q
V
Z
S
d
@
d
u
i
@
x
j
@
d
P
@
x
j
n
i
dS
À
v
2
V
Z
S
d
@ @
x
k
@
d
u
i
@
x
j
@
d
u
i
@
x
j
!
n
k
dS
ð
16
Þ
The above equation is the general dissipation equation for twoway coupled particle laden flows. The assumptionsassociated with Eq. 16 are incompressible flow and no masstransfer between the dispersed and continuous phase. If thevoid fraction is unity and no dispersed phase surfaces arepresent, Eq. 16 reduces to the singlephase flow dissipationequation:
AIChE Journal June 2009 Vol. 55, No. 6 Published on behalf of the AIChE DOI 10.1002/aic 1419
Production of Dissipation
D
e
Dt
¼ À
2
m
@
h
u
i
i
@
x
k
@
d
u
j
@
x
k
@
d
u
j
@
x
i
()
À
e
ik
@
h
u
i
i
@
x
k
À
2
m d
u
k
@
d
u
i
@
x
j
()
@
2
h
u
i
i
@
x
j
@
x
k
À
2
m
@
d
u
i
@
x
j
@
d
u
k
@
x
j
@
d
u
i
@
x
k
() zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ}ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{
À
m
@ @
x
k
d
u
k
@
d
u
i
@
x
j
@
d
u
i
@
x
j
()! ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ}
À
2
mq
@ @
x
i
@
d
u
i
@
x
j
@
d
P
@
x
j
()! ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ}
þ
m
@
2
@
x
2
k
½
e
À
ﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄ}
2
m
2
@
2
d
u
i
@
x
j
@
x
k
8>>:9>>;
2
*+ ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ}
Transport Pressure Viscous Dissipationof Diffusion of Diffusion of of Dissipation Dissipation Dissipation Dissipation(17)which is the volume average equivalent to time averagedissipation equation presented by Bernard and Wallace.
16
It isnow apparent that the effects of the surfaces of the dispersedphase are associated with the following terms:
À
v
2
V
@ @
x
k
Z
S
d
@
d
u
i
@
x
j
@
d
u
i
@
x
j
n
k
dS
þ
2
mq
V
Z
S
d
@
d
u
i
@
x
j
@
d
P
@
x
j
n
i
dS
À
m
2
V
Z
S
d
@ @
x
k
@
d
u
i
@
x
j
@
d
u
i
@
x
j
!
n
k
dS
(18)The integrals in Eq. 18 represent the dissipation effects causedby surfaces of the particles. Within these integrals are spatialgradients of volume deviation properties. To evaluate theintegrals, the coupled gradients must be captured. A firstattempt to evaluate these terms is based on the assumption thatthe particles are much smaller than the equivalent Kolmogorovlength scale and that the drag force is given by Stokes draglaw; in addition, the effects of particle rotation are neglected.The particlefluid relative velocity (
U
i
) can be expressed as theinstantaneous velocity between the particle and the surrounding fluid as:
U
i
¼
u
i
À
m
i
(19)which is the same as taking a particle at rest in a flow fieldwith velocity
U
i
. Substituting Eq. 9 into Eq. 19 shows arelationship between the local velocity deviation and therelative velocity in the form:
d
u
i
¼
U
i
þ
m
i
À h
u
i
i
(20)The spatial velocity gradients shown in Eq. 18 are evaluated atthe surfaces of the particles. Taking a spatial gradient of theabove equation shows:
@
d
u
i
@
x
j
S
d
¼
@
U
i
@
x
j
S
d
þ
@
m
i
@
x
j
S
d
À
@
h
u
i
i
@
x
j
S
d
(21)where
S
d
represents the surface of the dispersed phaseparticles. If the particle is rigid and rotational effects areneglected, then the velocity of the particle is constant acrossthe particle. Therefore, the last two terms on the right handside of the above equation are zero when evaluated along theparticle surface, reducing Eq. 21 to:
@
d
u
i
@
x
j
S
d
¼
@
U
i
@
x
j
S
d
(22)The assumption that the particles are much smaller than theequivalent Kolmogorov length scale (i.e., Stokes flow) allowsthe relative velocity gradients to be solved for analytically.The reason for this assumption is to obtain a simplified form of each term in Eq. 18 and then through empiricism extend theseforms to flows containing particles that are larger than theequivalent Kolmogorov length scale. By transforming thegradients and unit normal vectors between Cartesian andspherical polar coordinate systems, the terms in Eq. 18 can bedirectly solved for. Evaluation of the diffusion term shows thatthe integral over the surface of each particle is zero:
À
m
2
V
@ @
x
k
Z
S
d
@
d
u
i
@
x
j
@
d
u
i
@
x
j
n
k
dS
¼
0 (23)which is primarily because of symmetry. However, this maynot be the case if the particle is larger than the equivalentKolmogorov length scale. Evaluation of the second term inEq. 18 shows:2
mq
V
Z
S
d
@
d
u
i
@
x
j
@
d
P
@
x
j
n
i
dS
¼
2
mq
V
Z
S
d
@
u
r
@
r
@
d
P
@
r dS
þ
2
mq
V
Z
S
d
1
r
@
u
r
@
h
À
u
h
r
!
1
r
@
d
P
@
h
dS
ð
24
Þ
1420 DOI 10.1002/aic Published on behalf of the AIChE June 2009 Vol. 55, No. 6 AIChE Journal