O

N

THE

M

ODELING

OF

P

ARTICLE

T

RANSPORT

IN

E

LECTROSTATIC

P

RECIPITATORS

Hans-Joachim Schmid and Hans Buggisch

Institute of Mechanical Process Engineering and MechanicsUniversity of Karlsruhe

D-76128 KarlsruheGermany

A

BSTRACT

In the literature one can Þnd a lot of different models aiming to describe particle transport and precipi-tation in ESPÕs.

This work focuses on the modeling of particle transport and how it is affected by the electrical conditionsand the turbulent ßow Þeld. The modeling of secondary effects, which may be very important for precipita-tion too, is omitted. For this purpose an attempt to categorize models and to point out the major differencesbetween these approaches is made. There are mainly three different aspects to model: The electric Þeld, theßow Þeld and the turbulent particle transport.

The two main categories to model turbulent particle transport are the Eulerian approach, which is usuallyrepresented by the so-called Ôconvection-diffusion-equationÕ and the Lagrangian or particle-tracking ap-proach which has become popular recently.

A quantitative comparison of results from implementations of both models yields that an Eulerian modelmay be well applicable for practical purposes, if it is implemented elaborately. On the other hand Lagran-gian models are superior, especially for scientiÞc purposes, because the whole history of individual particlesis known, but for the expense of much higher computational effort.

1 I

NTRODUCTION

Since early in this century electrostatic precipitators become widely used to separate particles fromgases, people tried to model the particle transport and precipitation. So Deutsch (1922) proposed his famousmodel of the grade efÞciency, based on crude, but clearly physical assumptions. Immediately this formulawas not only applied to model the size-dependent particle

transport

, but the overall

mass separation efÞ-ciency

. This led to the lumped parameter Ôeffective migration velocityÕ

w

eff

including particle transport

and

the effects of raw gas particle size distribution, reentrainment, rapping losses, sneakage and so on.This paper will focus solely on the modeling of

particle transport

neglecting secondary effects

.

To model particle transport, mainly four different aspects have to be considered (compare table 1 withlist of examples from the literature):

¥ Modeling of the electric Þeld. There are mainly three distinct levels of complexity which can befound in literature: Assuming a constant electric Þeld, using analytical approximate solutions orusing numerically calculated inhomogeneous electric Þelds.

¥ Modeling of particle charging. Many models have to assume saturation charge corresponding to amean electric Þeld strength on all particles entering the precipitator duct. Some models allow theapplication of implementations of some charging kinetics model.

¥ Modeling of the ßow Þeld. Nearly all former models had to assume plug ßow or eventually someanalytical ßow proÞle within the channel. Since recently computational ßuid dynamics becamemore and more readily accessible, there are some efforts to include numerically calculated electri-cally induced secondary ßows in the modeling of particle transport.

¥ Modeling of turbulent particle transport. In this case two different approaches have to be distin-guished, which will be discussed in more detail below: The Eulerian approach, which models theparticles as a second continuum and the Lagrangian approach where individual particle tracks are

calculated. The main advantage of Eulerian methods is a computational effort far less compared toLagrangian calculations.

For this work, different models for the calculation of

turbulent particle transport

and

particle chargingkinetics

are compared, based on numerical calculations of the electric Þeld as well as the electrohydrody-namic ßow Þeld.

2 T

HEORY

2.1 Lagrangian Particle Tracking

This modeling approach is based on the equation of motion for individual particles of mass

m

P

:

(1)

A set of forces

F

i

exerted on spherical particles suspended in an instationary ßow Þeld is given by Maxeyand Riley 1983 and Berlemont

et al.

1990. These forces include drag force, ßuid acceleration force, addedmass force, Basset history force, Saffman lift force, gravity as well as electrical force. A detailed analysisyielded drag force and electrical force to be sufÞcient to model particle transport in electrostatic precipita-tors. Therefore the equation of motion for a particle reads:

(2)

Table 1: Structure of possible approaches to model particle transport in ESPÕs

El. FieldPart.

ChargingFlow Field

Turb. ParticleTransport

Examples

hom

ogen

eous

appr

oxan

alyt

. sol

.

num

eric

al s

ol.

satu

ratio

nch

arge

incl

. kin

etic

s

plug

ßow

anal

yt. ß

owpr

oÞle

CFD

no tu

rb.

tran

sp.

tota

l lat

eral

rem

ixin

gE

uler

ian

0 <

D <

¥

Lag

rang

ian

¥ ¥ ¥ ¥

Deutsch 1922

¥ ¥ ¥ ¥

Schmid

et al.

1996

¥ ¥ ¥ ¥ ¥ ¥

Kihm

et al.

1985

¥ ¥ ¥ ¥

Williams

et al.

1962, Coop-erman 1971, Leonard

et al.

1980, Cooperman 1984,

¥ ¥ (¥) ¥ ¥

Petroll

et al.

1988, Kihm

et al.

1987, Riehle 1995

¥ ¥ ¥ ¥

Lawless, 1996 & 1998

¥ ¥ ¥ ¥

Zamany 1992

(¥) ¥ ¥ ¥ ¥

Riehle 1996

¥ ¥ ¥ ¥

Goo

et al.

1997, Meroth 1998

m

P

u

P

dtd

---------

F

i

i

å

=

u

P

dtd

---------

u

F

u

P

Ð

t

P

------------------

c

w

Re

( )

c

w Stokes

,

Re

( )---------------------------------

×

Q

P

E

m

P

-----------+=

with the particle relaxation time (3)

For the calculation of the Cunningham-Slip correction

Cu

and the ßuid viscosity

h

F

the interested readermay refer to Hutchins

et al.

1995.Integrating eq. 2 leads to the particle trajectory. To do this integration, at each point the ßuid velocity

has to be simulated. This is done by a superposition of the mean ßow velocity and a ßuctuating componentmodeled by a random number generator. Some different approaches to model this ßuctuating turbulent ve-locity have to be distinguished:

The eddy-lifetime model uses a gaussian probability distribution for this ßuctuating velocity with amean value of zero and with the rms-ßuid velocity as standard deviation. A new value of the ßuctuating ve-locity

independent

from the one in the predecessing step is drawn after a characteristic time, called the ÔeddylifetimeÕ. During this time the ßuctuating velocity is supposed to be constant (see e.g. Gosman and Ioan-nides 1981, Shuen

et al.

1983, Meroth 1998). Kallio and Reeks (1989) improved this model by using anexponential probability distribution for the eddy lifetime instead of a constant value. With the average eddylifetime as the Lagrangian integral time scale of the ßuid ßow this model proved to predict long time turbu-lent particle dispersion in homogeneous turbulence quite well. The main drawback of the eddy lifetimemodel are cases where there is a relative velocity of particles to the ßuid (e.g. due to gravitational settlingor electrical drift) or inhomogeneous turbulence. Hence several corrections are necessary to apply eddy life-time models to predict particle transport in electrostatic precipitators (Kallio 1997).

A different approach is called Ôcontinuous random walk methodÕ (described in more detail e.g. by Lu

etal.

1992 or Wang and Stock 1992). The method adapted in this work is only sketched in brief. For furtherdetails see Lu

et al.

1992.For

each

time step necessary to integrate the equation of motion a new ßuctuating velocity componentu

i

Õ(t) is generated

correlated

with the value in the previous time step u

i

Õ(t-

D

t):

(4)

a

i

denotes the Lagrangian velocity correlation along the particle path between two subsequent timesteps. It can be calculated if the Lagrangian velocity autocorrelation

f

L

(

D

t

) and the Eulerian velocity corre-lation between the new particle location and the ßuid location

f

E

(

D

s

) (which is particular important if thereis a particle drift) is known:

(5)

g

i

denotes the new random number with a mean value of zero and a standard deviation

s

g,

i

of:

(6)

The modeling of the correlation functions is somewhat critical and only few experimental data is avail-able on this topic. Frenkiel (1948) compared several statements and proposed the following functions:

(7)

Hence the Lagrangian integral time scale

of the ßuid ßow

t

F,L

and the eulerian integral length

of the ßuidßow

L

F,E

have to be determined. From dimensional analysis the following relations emerge:

(8)

These constants are most times Þtted to measurements and differ therefore signiÞcantly in the literature.For the value of

C

1

one can Þnd values of 0.2 .. 2.0 (Shirolkar

et al.

1996, Tennekes and Lumley 1989, Hinze1975). Most authors use values for C

1

slightly above 0.2.The proposed value of

C

2

varies from 1.05 to 2.94 (see e.g. Shirolkar

et al.

1996, Tennekes and Lumley1989, Hinze 1975). According to Hinze (1975) the Eulerian length scale is different

in

the direction of the

t

P

r

P

x

2

18

h

F

-------------

Cu

=

u

F

u

'

i

x t

,( )

u

i rms

,

x

( ) a

i

u

'

i

t

D

t

Ð

( )

u

i rms

,

x

t

D

t

Ð

( )

----------------------------------

×

1

a

i

Ð

( ) g

i

×

+=

a

i

f

L

D

t

( )

f

E

D

s

( )

=

s

g

i

,

1

f

L

2

f

E

2

Ð=

f

L

D

t

( ) D

t

t

L

-----Ð

î þí ýì ü

exp= and

f

E

D

s

( ) D

s

2

L

E

----------Ð

î þí ýì ü D

s

2

L

E

----------

î þí ýì ü

cosexp=

t

L

C

1

u

rms

2

e

-----------

×

= and

L

E

C

2

t

L

u

rms

× ×

=

line between the two points under consideration (superscript II) and the perpendicular direction (superscriptI). Hence the following relation is used:

(9)

This set of equations and constants proved to be well suited to describe a number of experiments satis-factorily (e.g. Lu

et al.

1993).

2.2 Convection-Diffusion-Modeling (Eulerian)

This modeling approach treats the dispersed phase as a second continuum phase characterized by itsconcentration c which may be based on the number or the mass of particles per volume respectively. Thetransport equations can be deduced from a balance of particle ßuxes

j

P

for an inÞnite small volume readingin the stationary case:

(10)

Using a Reynolds decomposition for all instantaneous quantities into a mean value (denoted by a bar)and a ßuctuating component (denoted by a Õ) leads to:

(11)

The second term in eq. 11 accounts for the particle transport due to turbulent ßuid velocity ßuctuations.In Eulerian models this transport is modeled in analogy to Brownian diffusion with a gradient statement:

(12)

The proportionality factor

D

P

should not be confounded with a Diffusion constant and is therefore calledfrom now on

Dispersion

coefÞcient. It is a function of particle properties and mostly of local turbulent ßowÞeld properties.

If particle inertia is neglected the instantaneous as well as the mean particle velocity

u

P

can be writtenas a superposition of the respective gas velocity

u

F

and particle drift velocity

u

P,D

:

(13)

In electrostatic precipitators where gravity usually can be neglected the drift velocity

u

P,D

of a particlewith diameter

x

is proportional to the local electric Þeld strength

E

with a mobility

µ

P

:

(14)

At least for particles with diameters less than 10 µm the drag force may be calculated according toStokes and therefore the second factor in the formula for the particle mobility approaches unity.

Finally one can derive from the Poissonian equation for electric potential a relation for div(

E

) as a func-tion of the space charge

r

el

and the permittivity

e

0

:

(15)

This Þnally leads to the modeled convective diffusion equation:

(16)

If the electric Þeld is assumed to be homogeneous or free of space charge the second addend is zero.This assumption is made in most models published in the literature. On the other hand if the electric Þelddistribution is calculated numerically including the space charge this term can easily be included althoughit turned out in our calculations, that it is small compared to the other terms. The last addend in eq. 16 ac-counts for an inhomogeneity of the dispersion coefÞcient. Since most models use a constant value of the

C

2

I

1.25= and

C

2

II

2.5=

div

j

P

div

c u

P

×è øæ ö

0= =

div

c u

P

×( )

div

c

'

u

P

'

×( )

+ 0=

c

'

u

P

'

×

D

P

c

Ñ×

Ð=

u

P

u

F

u

P D

,

+=

u

P D

,

m

P

E

with:

m

P

×

Q

P

Cu

3

ph

F

x

-----------------

c

w Stokes

,

Re

( )

c

w

Re

( )

---------------------------------

×

= =

div

E

( )r

el

e

0

-------=

u

F

m

P

E

+

( )

grad

c

( )

c

m

P

r

el

e

0

-------

D

P

c

D×

grad

c

( ) grad

D

P

( )

×

+

( )

Ð+ 0=

dispersion coefÞcient this term was neglected most times. Section 2.3 will deal in more detail about disper-sion coefÞcients in inhomogeneous turbulence.

A set of boundary conditions completes the problem. At the inlet usually a homogeneous distribution ofparticle concentration is assumed, but arbitrary concentration proÞles may easily be adapted. At the centerof the duct a symmetry boundary condition may be applied assuming that there is no net ßux across thisboundary. At the outlet usually a zero gradient boundary condition is used. At the wall two different bound-ary conditions may be applied: Either assuming zero concentration or zero gradient of particle concentrationat the wall. Whereas the zero concentration assumption applies well for pure diffusion precipitation in thecase of electrostatic precipitation particle drift vastly dominates all other precipitation mechanisms andhence the zero gradient assumptions seems to be most appropriate in this case implying that particle trans-port in the viscous sublayer at the wall is solely due to electrical particle drift.

Hence eq. 16 together with the chosen boundary conditions must be solved. This may done either byanalytical solutions if some further assumptions are made (e.g. constant electric Þeld strength, particle havealready saturation charge at inlet, homogeneous particle dispersion coefÞcient) or by numerical calculationswith Finite Difference methods.

An alternative method of numerically solving the convection diffusion problem is to use a Finite Volumemethod, which is based on a direct balance for a Þnite volume, analogous to eq. 10 with the particle ßuxesat the boundary faces of the control volume

j

P,BF

:

(17)

The balances for each control volume lead to a mean value of particle concentration which are assignedto the center of the cell. To get the values at the control volume surfaces a method of interpolation must bechosen (see section 3.1). The boundary conditions are similar to the Finite Difference method except at thecollecting electrode. Now it seems to be appropriate to use the assumption of

D

P

= 0 or any other distinctvalue at the control volume face adjacent to the wall, implying no or any extent of turbulent deposition re-spectively. The Finite Volume method has two main advantages: First it is intrinsically conservative with theconservation of mass always fulÞlled in contrast to the Finite Difference method, where numerical errorsmay lead to a Ônumerical lossÕ of mass. Second the boundary condition at the wall has a physically clearmeaning. Hence all Eulerian calculations presented subsequently are done with a Finite Volume method.

2.3 On the modeling of the Dispersion coefÞcient

This is the most crucial point in Eulerian models but as will be shown in this section this problem isequivalent to the problem of getting the right correlation functions and constants in the case of Lagrangianmodels.

Many authors using the convective diffusion equation for modeling the particle transport in ESPÕs justgave the solutions in terms of the Peclet-number

Pe

thus eliminating the problem of giving speciÞc valuesof

D

P

(e.g. Kihm

et al.

1987, Riehle 1995):

(18)

Most other authors use values ranging from 1 to 50 cm

2

/s. But Cooperman (1971) used a physically un-realistic high value of the dispersion coefÞcient in

axial

direction of 100 to 1000 cm

2

/s making the obtainedresults somehow doubtful.

On the other hand there is a lot of work in the literature to derive the dispersion coefÞcients from ßuidßow data. Due to limited space we can only sketch a few principle facts here. For more details the interestedreader may refer to the cited articles. Assuming homogeneous, isotropic turbulence and a gaussian concen-tration proÞle of particles injected at a distinct point and subsequent dispersion by the turbulent ßow ÞeldTaylor (1920) derived an expression for the (time dependent) dispersion coefÞcient depending on theLagrangian autocorrelation of the particle velocity

R

P,L

:

j

P BF

,

å

0 with:

j

P BF

,

u

F

m

P

E

×

+

( )

c D

P

grad

c

( )

×

Ð= =

Peu

P D

,

s

×

D

P

------------------=

(19)

For ÔlongÕ times (which means long related to the Lagrangian integral time scale) the integral over theautocorrelation tends to the Lagrangian integral time scale, thus leading to:

(20)

The difÞculty is to determine the particle ßuctuating velocity and the particle lagrangian integral timescale with respect to particle properties and ßuid ßow.

Tchen (1947) made the assumption that the particle is always surrounded by the same ßuid element.With this assumption the equation of motion is linearized and by performing a Fourier transformation onthe energy spectrum of the ßuctuating ßuid velocity, calculating the particle response to ßuid oscillations,back transformation of the resulting energy spectrum of particle velocity ßuctuations he got the correspond-ing values of

D

P

. But this assumption is not justiÞable Ð even in homogeneous turbulence Ð if particle inertiaplays some role. Especially in the case of a particle drift (e.g. due to the electric Þeld in ESPÕs) this methodis not applicable, although some attempts were made to use this model even in the case of particle drift (Pi-cart

et al.

1986).Pismen and Nir (1978) as well as Reeks (1977) did some calculations without the simplifying assump-

tion of Tchen. They could show that particle inertia does not changes the dispersion coefÞcient very much.This may be explained by two contra effects: On the one hand, an increasing particle inertia will lead to a

decreasing

amplitude of the particle ßuctuating velocity

u

P,rms

for given ßuid velocity ßuctuations. On theother hand the particle Lagrangian integral time scale will

increase

for increasing particle inertia. Hence

D

P

increases only about 25% from inertialess particles to Ôvery heavyÕ particles.For small particles of less than 10 µm in diameter, which are most interesting in ESP modeling, the par-

ticle relaxation time

t

P

is much smaller compared to the Lagrangian integral time scale

t

F,L

of the ßuid ßow.For typical situation in ESPÕs we got a ratio of always less than 10

-3

. Hence particle inertia may be totallyneglected, leading to a much simpler situation, where in homogeneous turbulence without particle drift

D

P

equalizes the dispersion coefÞcient of the ßuid:

(21)

If there is a particle

drift

the dispersion coefÞcient is reduced, because the particle changes its ßuid-ÔneighborhoodÕ more rapidly and hence the particle Lagrangian integral time scale drops. Csanady (1963)proposed a semi-empirical model to account for this effect, often called ÔCrossing Trajectory Effect (CTE)Õ.Under the neglecting of particle inertia

D

P

for particles with a drift velocity of

u

P,D

may be calculated ac-cording to:

(22)

Eq. 22 allows to calculate

D

P

from the properties of the ßow Þeld and the particle drift velocity and theproblem is reduced to the determination of

t

F,L

and

L

F,E

(eq. 8) as discussed at the end of section 2.1. Henceif the model of Csanady is sufÞciently correct it should yield results comparable to Lagrangian calculationsif the same constants

C

1

and

C

2

are used in eq. 8!To compare the Eulerian results with the Lagrangian calculations particle concentration proÞles for

given boundary conditions can be compared. Additionally a more sensitive measure of accordance is thecalculation of

D

P

from Lagrangian calculations: According to Batchelor (1949), in homogeneous, isotropicturbulence with concentration proÞles of gaussian distribution

D

P

can be calculated from the change in timeof the variance of particle locations X

i

:

(23)

D

P

t

( )

u

P rms

,

2

R

P L

,

t( ) t

d

0

t

ò

= with

R

P L

,

t( )

u

P

¢

t

( )

u

P

¢

t

t

+( )

×

u

P

¢

t

( )( )

2

--------------------------------------=

D

P

¥,

u

P rms

,

2

t

P L

,

×

=

D

P

¥,

D

F

¥,

u

F rms

,

2

t

F L

,

×

= =

D

P

¥,

u

F rms

,

2

t

F L

,

×

u

P D

,

2

t

F L

,

L

F E

,

------------

è øæ ö

2

1+

-----------------------------------------------=

D

P

t

( )12---

tdd X t

( )

X t

( )Ð

( )

2

×

=

Hence at different time steps the locations of an ensemble of simulated particle trajectories is stored andsubsequently the dispersion coefÞcient is calculated according to eq. 23.

3 C

OMPARISON

BETWEEN

E

ULERIAN

AND

L

AGRANGIAN

C

ALCULATIONS

3.1 Inßuence of Numerics on Eulerian Calculations

Eulerian methods are always based on numerically discretizing the computational domain. If a FiniteVolume method is chosen (comp. eq. 17) for each cell (Þnite volume) a mean concentration is calculated.To evaluate eq. 17 an appropriate interpolation procedure for the concentration c and the other quantities atthe boundary faces have to be applied. A method of second order accuracy in space discretization is the cen-tral scheme, where the mean value of the two adjacent cells is used as the value at the boundary surface. Theso-called upwind method uses the value of the cell upstream. In that case at each boundary the sign of thelocal particle velocity decides which value is taken as value at the boundary surface.

In Þg. 1 a comparison between concentrationproÞles for two distinct cases are presented withconstant particle drift velocity and homogeneous,isotropic turbulence. A

low Peclet-number

(seeeq. 18) corresponds to a low drift velocity com-pared to the Dispersion coefÞcient leading to arelatively ßat concentration proÞle. In this caseboth Eulerian methods yield quite similar results.Further these results are in good agreement withthe results of the particle tracking method. But itshould be mentioned that concentration proÞlescalculated with the Lagrangian method showsome scatter although these results are based on10,000 particle trajectories. For a smoother curveeven more (e.g. 100,000 particle tracks were nec-essary). For

high

Pe-numbers the situation is quitedifferent. The high drift velocity combined withthe low turbulent dispersion leads to a steep gra-dient in the concentration proÞle as predicted bythe Lagrangian method. The central scheme isable to reproduce this steep gradient approxi-mately but for the expense of physically unrealistic oscillations where the proÞle has a high curvature. Thisis a well known effect of numerical instability of central schemes applied to convection-diffusion problems.On the other hand the upwind scheme is absolutely stable (which is true in

all

cases) but for the expense ofhigh artiÞcial numerical diffusion, what is caused by the much higher truncation error (Leonard 1979). Thisleads to a proÞle which is much too ßat compared with the ÔcorrectÕ proÞle.

To overcome this problem a higher order stable method must be applied, such as the ÔQUICKÕ scheme(Leonard 1979). Normally the gradients in ESPÕs are not as steep as in the presented case of ÔhighÕ Pe-num-ber and hence an upwind scheme should be most appropriate for a Þrst approach providing a stable solutionin all cases.

3.2 Turbulent Dispersion

Fig. 2 presents particle dispersion coefÞcients recalculated from Lagrangian particle trajectories in ho-mogeneous isotropic turbulence without particle drift. All particles were released at one point and then theparticle tracks are modeled like sketched in section 2.1.

Fig. 1: Concentration proÞles for Lagrangian as well as different numerical implementations of Eu-lerian FV method.

0

0.2

0.4

0.6

0.8

1

0 0.02 0.04 0.06 0.08 0.1

Lagrangian

UpwindCentral

c /

c o

Lateral Position / m

low Pe-number

high Pe-number

For both cartesian coordinates the variance ofparticle locations can be calculated yielding twodistinct curves. This Þgure shows a typical coursewith respect to time. The dispersion coefÞcientstarts at zero and reaches his Þnal value after atime t approximately equal to four times theLagrangian integral time scale of the ßuid. Thisbehavior can readily be explained if eq. 19 is in-spected: for very short times the velocity correla-tion is approximately one and hence there is alinear increase of

D

P

with respect to time. Forlonger times the integral approaches theLagrangian integral time scale and eq. 19 passesover to eq. 20 with

D

P

reaching its Þnal value.Although the value of

D

P

shows signiÞcantscatter for t/

t

L,F

> 4, the

mean

values of 10.7 (inmean ßow direction) and 10.4 cm

2

/s (in lateraldirection) taken over the remaining time of 4..20agrees very well with the dispersion coefÞcient ofthe ßuid of 10.6 cm

2

/s as obtained from eq. 21 in-dicating that it is appropriate to neglect inertia inEulerian turbulent dispersion calculations, at leastfor particles with less than 10 µm in diameter asused in these calculations. From these results it can also be concluded that long-time dispersion is only de-pending on the integral time scale and not on the spectrum. Hence the chosen correlation function for cal-culating the particle trajectories is not critical.

All models described in section 2.3 to calcu-late particle dispersion coefÞcients from ßowÞeld properties are originally derived for homoge-neous turbulence whereas in ESPÕs there will bean

inhomogeneous

turbulence Þeld. Hence somecalculations of particle trajectories in inhomoge-neous turbulence were conducted. This was doneby adequately increasing the

u

F,rms

and

e

(comp.eqs. 8 & 21) to get a constant incline of the

D

F

anda constant Lagrangian integral time scale of0.25 s. The results presented in Þg. 3 show thatthe particle dispersion coefÞcient after a time ofjust about 1 s (the time needed to approximatelyreach the Þnal value) closely follows the ramp ofthe ßuid dispersion coefÞcient. This is very im-portant because this integral time scale is realisticfor ßow Þelds in ESPÕs and hence it can be con-cluded that the particle dispersion very quicklyfollows the changes in the ßuid dispersion. There-fore at each point in an Eulerian approach

D

F,

¥

could be used.Another Ôtest caseÕ is shown in Þg. 4. This time

u

F,rms

and

e

have been varied to get a jump of the dis-persion coefÞcient at a distinct axial position and a constant time scale in the whole domain. Now the par-ticle dispersion coefÞcient derived from particle trajectories can not follow this jump any more. Due to the

Fig. 2: Particle dispersion coefÞcient calculated from Lagrangian calculations according to eq. 23. Homogeneous, isotropic turbulence (

D

F,

¥

= 10.6 cm

2

/s).

0

2

4

6

8

10

12

0 5 10 15 20

Variance x

Variance y

DP

/ c

m2 /

sDimensionless time t / tL,F

Fig. 3: Dispersion coefÞcient for inhomogeneous turbulence from Lagrangian calculations ac-cording to eq. 23.

0

5

10

15

20

Variance xVariance yDF,¥

0 0.5 1 1.5 2 2.5 3

DP

/ c

m2 /

s

Time t / s

axial dispersion some particles are subjected to the higher turbulence earlier then a ßuid particle travelingwith mean ßow velocity. Hence there is an increase in the dispersion coefÞcient

before

the mean ßowreaches the jump. Then the particles need again a time approximately equal to four times of the integral timescale to reach the Þnal value of the dispersion coefÞcient.

From these calculations one can conclude that jumps (or very steep gradients in turbulence intensity anddispersion coefÞcients) are not adequately modeled by a local value of

D

F,

¥

. Nevertheless

long-time disper-sion

is modeled quite good again even if local values of

D

F,

¥

are used in Eulerian calculations. This mayalso be seen from Þg. 4b: The proÞles calculated with Lagrangian and Eulerian approaches agree very well,substantiating that long time dispersion (with Prof. 2 ÔsufÞcientlyÕ downstream of the location of jump) ismodeled adequately with Eulerian models and

D

P

according to eq. 21.It is also obvious that the assumption of gaussian concentration proÞles if all particles are injected at a

distinct location made to derive eqs. 19 and 23 is justiÞed.

Thus it can be concluded that for particles less than 10 µm suspended in a turbulent ßow Þeld with nodrift long time dispersion is well modeled by an Eulerian calculation using local values of

D

F,

¥

. The result-ing concentration proÞles are identical with Lagrangian calculations within the straggling of the data.

3.3 Inßuence of Particle Drift

As already mentioned in section 2.3 in the presence of a particle drift the particle is transported in regionsof lower correlation and hence the dispersion coefÞcient will drop. This effect is often called the ÔCrossingTrajectory Effect (CTE)Õ.

For Lagrangian calculations no further changes have to be made because the CTE is accounted for in theLagrangian and Eulerian velocity correlation functions (eqs. 5 to 8). For the Eulerian integral length scaleone have to distinguish between the correlation

in

the direction of particle displacement relative to ßuid dis-placement and the

perpendicular

one. These correlations are calculated for each time step according to theactual displacements. The resulting dispersion coefÞcients are subsequently evaluated for the mean ßow di-rection and the perpendicular drift direction. Due to the higher length scale in the direction of the displace-ment the dispersion coefÞcient in drift direction is higher than in ßow direction, as may be seen in Þg. 5.

Fig. 4: Jump in turbulence intensity: Dispersion coefÞcient from Lagrangian calculations according to eq. 23 (left) and comparison of concentration proÞles from Lagrangian and Eulerian FV-calculations (right).

0

0.2

0.4

0.6

0.8

1

1.2

c /

c ma

x,1

Position

LagrangianEulerian

A B

A B

Prof 1

Prof 2

jump

0

5

10

15

20

25

Lagr. Variance xDF,¥

0 0.5 1 1.5 2 2.5 3

DP

/ c

m2 /

s

Time t / s

But nevertheless, the dispersion coefÞcients calculated according to Csanady (eq. 22) with the correspond-ing Eulerian integral length scales (eq. 8) yields higher values of

D

P

in drift direction and lower values inmean ßow direction. Hence the differences in these two directions are partly equalized by the particle tra-jectory calculations. Nevertheless the agreement between calculated concentration proÞles at different po-sitions seems to be very good. Only for the third proÞle one can see a slightly smaller particle dispersion forthe Lagrangian calculations.

Thus it can be concluded that the CTE have to be considered. The semi-empirical formula of Csanadyseems to be sufÞciently exact to describe turbulent particle transport with particle drift in the framework ofEulerian modeling if the same modeling constants to calculate the integral time and length scales are used.

3.4 Application to Modeling of Particle Transport in ESPÕs

Until now only

model test cases

with given distributions of turbulence and steady constant particle drifthave been considered. In this section exemplary calculations for a model ESP are discussed. Due to an in-homogeneous electric Þeld within the precipitator particle charging kinetics now plays also a signiÞcantrole. The main object of this section is to discuss the inßuence of the inhomogeneity in ßuid ßow and electricÞeld distribution on the obtained results.

3.4.1 Modeling Charging Kinetics.

There are some models to predict particle charging with differentcomplexity. The simplest models are Þeld charging (valid for large particles and high Þelds) and diffusioncharging theory (deemed valid for small particles and low electric Þelds) respectively (White 1963). But itis well known that for particles about 2 µm and less even a linear superposition of both charging mecha-nisms does not lead to satisfactory results. Cochet (1961) published a semi-empirical formula to calculateparticle charging kinetics which in fact uses the kinetics of Þeld charging but modiÞed for the saturationcharge depending on the Knudsen-number (

Kn = 2á

l

/x

). It is supposed to model particle charging fairlygood although there are some doubts about the

kinetic

of charging. The formula of Cochet is widely used inmodeling particle charge in ESPÕs because it gives particle charging kinetics with an analytical formula andhence very low computational effort.

Fig. 5: Particle dispersion in the presence of a particle drift perpendicular to the mean ßow: dispersion co-efÞcients from Lagrangian calculations (left) and concentration proÞles in lateral (drift) direction from Eulerian as well as Lagrangian calculations (right).

4

6

8

10

12

0 5 10 15 20

DP

/ c

m2 /

s

Particle drift velocity uP,D / cm / s

Drift dir.

Drift dir.Mean dir.

Mean dir.

Csanady

Lagr.

}}

0

0.2

0.4

0.6

0.8

1

1.2

c /

c ma

x,1

Lateral Position

LagrangianEulerian

A B

Prof 1

Prof 3Prof 2

BA

B

A 1 2 3

On the other hand there are some models which apply a complex numerical treatment of the Þeld-diffu-sion problem (e.g. Liu and Kapadia 1978). But the computational effort is much too high to use these modelsin Lagrangian particle tracking codes, where the charging along each individual particle trajectory must becalculated. Even for Eulerian codes these approaches seem to be too costly. Recently Lawless (1996) pub-lished an approximation called ÔField-ModiÞed Diffusion (FMD)Õ for the charging kinetics which seems togive best agreement with complex models and experimental values in the continuum regime (particle radiusgreater than ionic mean free path). His model consists of a simple differential equation for the particlecharge

Q

P

in time

t

:

(24)

with a surface factor

a

, a dimensionless Þeld strength , the saturation charge according to Þeld charg-ing

Q

S

and a characteristic charging time

t

Q

deÞned as follows:

(25)

In Lagrangian calculations eq. 24 was solved foreach individual particle along the trajectory as addi-tional differential equation using local values of Þeldstrength

E

and ionic space charge density

r

I

.For Eulerian calculations the situation is much

more complex because the particles are treated as con-tinuum and no individual particle tracks are known.Several possible approaches are sketched in Þg. 6: Thesimplest model is to use saturation charge (e.g. accord-ing to Cochet and a mean electric Þeld strength) for thewhole precipitator domain. This approach does notmake much sense for numerically solved convection-diffusion equations. But to enable an analytical solu-tion this is the common assumption. Another possibleapproach to calculate the particle charges

before

solv-ing for the concentration Þeld

c

is shown in Þg. 6b:The charging kinetics is calculated from the respectivecell upstream with any charging model, e.g. the FMD.The time step required for the particles to travel fromone cell to another is guessed from the mean velocityin streamwise direction. The modeling of particlecharging kinetics sketched in Þg. 6c needs some morecomputational effort: A

mean

particle charge for each cell is calculated based on the charges carried by theparticle ßuxes into the cell. Therefore mean particle charge and particle concentration for each cell is cal-culated iteratively. This approach should be able to account to some extent for particles which get highly

Q

P

d

td

----------

14---

t

Q

Q

S

1

Q

P

Q

S

-------Ð

è øæ ö

2

a E

÷

( )

2

pr

I

m

I

kTx

e

----------------------------+ 0

Q

P

Q

S

£ £

a E

÷

( )t

Q

Q

P

Q

S

Ð

Q

P

Q

S

Ð

( )

e

2

pe

0

kTx

----------------------------

î þí ýì ü

exp 1Ð

------------------------------------------------------

Q

P

Q

S

³

îïïïïíïïïïì

=

E

÷

a E

÷

( )

1

E

÷ 0.457+

( )

0.575

---------------------------------------

E

÷ 0.525

³

1

E

÷ 0.525

<îïíïì

= and

E

÷

xe

2

kT

----------

E

=

Q

S

3

pe

0

e

r

e

r

2+--------------

x

2

E

= and

t

Q

e

0

r

I

m

I

-----------=

Fig. 6: Different approaches to model charging ki-netics for Eulerian calculations: a) Satura-tion charge b) Simple charging kinetics c) charge balance.

conv.?

calc. c

recalc. QP

end

guess QP

no

yes

end

calc. c

calc. QP

QP = QS

QCQU

Dx

QC = QU + DQDt

DxUF

Dt =

jQ Q cjP

Q = S jQ,inS jP,in

a

b

c

jQ,in = jP,iná Q+ DQDt( )

U

}

charged in the vicinity of the discharge electrodes and hence possess high drift velocities leading to Ôcross-ing trajectoriesÕ with particles of lower charge which entered the duct in regions with lower Þeld strengths.

3.4.2 Exemplary Calculations for a model ESP

. All of these results are based on calculations of theelectric Þeld and ionic space charge distribution according to Meroth (1997) and subsequently conductedCFD calculations of the electro-hydrodynamic ßow Þeld with a standard k-

e

-model. The modeled wire-plate-type precipitator consists of a duct with ßat plates and Þve smooth wires. The duct width is 200 mmand the length of the collecting electrodes amounts to 500 mm. The applied voltage in the presented calcu-lations was 50 kV and the mean ßow velocity was 1 m/s. Comparative calculations with the Lagrangian ap-proach and Eulerian approaches (as described above with FV method and upwind scheme) with differentimplementations of the charging kinetics (see previous section) were done all based on the same inhomo-geneous ßow and electric Þeld.

Some calculated concentration proÞles are plotted in Þg. 7. Even though the Lagrangian calculations arebased on 10,000 trajectories there is some obvious scatter. Nevertheless the proÞle shows a signiÞcant peakresulting from particles with high charges acquired near the DEÕs. An evaluation of the local charge

distri-butions

from the particle tracking calculations yields a broad distribution extended to higher particle charges(see also experimental investigations of Schmid and Umhauer 1998). An Eulerian calculation will not beable to account for this charge

distribution

, Ð except separate balances for different

charge classes

are made.But this would lead to an enormous increase in computational effort and hence would compensate the mainadvantage of Eulerian methods.

As can be clearly seen in Þg. 7, a charge balance for each cell (see Þg. 6c) leads by far to the best agree-ment with particle trajectory calculations. Only after the 5th DE the concentration proÞle is slightly shiftedtowards the CE due to remotely higher particle charges there. This is caused by the method of calculating

mean

particle charges in each cell. The other methods of calculating particle charges lead to signiÞcantlydifferent results in concentration proÞles as well as precipitation.

Therefore Eulerian methods seem to be applicable if the charge is calculated iteratively for each cellbased on charge balances. For practical purposes this seems to be sufÞciently correct with the advantage ofcomputational effort which is at least 20 times less than the corresponding particle trajectory calculations.But for basic research a Lagrangian approach or alternatively an Eulerian approach with separate chargeclasses per particle size class are superior because they are able to account for the particle charge distributionin each cell.

Fig. 7: Comparison between Lagrangian and Eulerian (FV with upwind scheme) calculations for different implementations of charging kinetics.

0

0.2

0.4

0.6

0.8

1

1.2

0 0.02 0.04 0.06 0.08 0.1

c /

c o

Lateral Position / m CEDE's

Position 2(after 5th DE)

1 2

0

0.2

0.4

0.6

0.8

1

1.2

Lagr. Particle TrackingFV / Charge balance Ñ cFV / Simple charg. kin. Ñ bFV / QS (Cochet) Ñ a

0 0.02 0.04 0.06 0.08 0.1

c /

c o

Lateral Position / m CEDE's

Position 1(at 3rd DE)

4 S

UMMARY

AND

C

ONCLUSIONS

Different approaches to model particle transport in Lagrangian and Eulerian frame respectively werediscussed.

In an Eulerian modeling approach, the long-time turbulent dispersion may be calculated from ßuid ßowdata if particle inertia is negligible (particle relaxation time signiÞcantly smaller than turbulent integral timescale). Both, the dispersion coefÞcient needed in Eulerian calculations as well as Lagrangian particle trajec-tory calculations are based on the turbulence intensity and the Lagrangian integral time scale of the ßuid.Therefore in homogeneous turbulence Lagrangian and Eulerian calculations give identical results if themodeling constants for calculating the time scale from turbulence quantities are identical. For inhomoge-neous turbulence a continuous change will be well described by an Eulerian modeling approach if local val-ues of

D

F,

¥

are used. For very rapid changes or jumps in the ßuid dispersion coefÞcient only long-timedispersion (dispersion times longer than 4á

t

F,L

) is modeled correct with an Eulerian approach using localvalues of

D

F,

¥

.If there is a particle drift, the semi-empirical formula for the dispersion coefÞcient of Csanady leads to

satisfactory results if the Eulerian integral length scale is chosen identical as in the Lagrangian calculations.Only the dependence on the drift direction is more pronounced in the Eulerian case, but for practical pur-poses an anisotropic Dispersion coefÞcient will lead to a good agreement with particle tracking calculations.

In modeling the particle transport in ESPÕs accounting for the inhomogeneities of electric Þeld and ßowÞeld as well as the modeling of charging kinetics is very important. Hence assuming a constant electric Þeldstrength and saturation charge in the whole duct may give misleading results. Model calculations based onnumerically calculated electric and ßow Þeld yielded that Eulerian calculations may lead to comparable re-sults as an Lagrangian approach, if the Eulerian calculations are based on a charge balance for each compu-tational cell. A FV / upwind scheme seems to give reasonable and stable results but a higher order, stablescheme would reduce numerical diffusion and hence further improve correspondence.

Although results from Eulerian calculations show some small differences to Lagrangian calculations dueto their lack of modeling the charge distribution, for

practical purposes

it appears to be reasonable to useEulerian approaches based on local values of velocity and electric Þeld because particle tracking calcula-tions need a computational effort which is at least 20 times higher or even more.

5 A

CKNOWLEDGMENTS

The authors would like to express their gratitude towards the German Science Foundation (DFG) for Þ-nancially supporting this work (Project-No. Schm 810/11-2).

6 R

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Int. J. Multiphase Flow

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Turbulence

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5th ICESP

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ESPVI 4.0A

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Numerical electrohydrodynamics in electrostatic precipitators

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PhD thesis, Technical University of Denmark, Copenhagen.

7 N

OMENCLATURE

c

concentration

a, b

correlation factor

C

1

,C

2

modeling constants

e

0

dielectric constant

CE

collecting electrode

e

r

relative dielectricity

Cu

Cunningham slip correction

g

random number

c

w

drag coefÞcient

h

F

dynamic viscosity

D

dispersion coefÞcient

L

E

Eulerian integral length scale

DE

discharge electrode

m

mobility

E

electrical Þeld strength

r

density

e

elementary charge

s

standard deviation

f

correlation function

t

L

Lagrangian integral time scale

F

i

external forces

t

P

particle relaxation time

FMD

Field modiÞed diffusion

FV

Finite volume method Subscripts:

j

ßux

BF

at boundary face

k

Boltzmann constant

E

Eulerian

m

P

particle mass

el

electrical charges

Q

charge

F

ßuid

R

velocity correlation function

I

ionic

Re

Reynolds-number

L

Lagrangian

s

half gap width

P

particle

t

time

rms

root mean square

T

Temperature

S

saturation

u

velocity

Stokes

according to Stokes

x

particle diameter

¥

long-time

particle location

X

i

particle location Superscripts:Õ ßuctuating component

_

mean value

x