-
1 Index
1. Introduction
......................................................................................................
22. Model for electrostatic precipitators
................................................................
4
2.1. Gas
properties.................................................................................................................
4
2.2. Estimating the corona current
........................................................................................
5
2.3. Electrical properties of the electrostatic precipitator
...................................................... 5
2.4. Particle charging
.............................................................................................................
6
2.4.1. Field charging
..............................................................................................................
6
2.4.2. Diffusion charging
..................................................................................................
7
2.5. Drift velocity and collection efficiency
.....................................................................
8
3. Centrifugal forces in a flow field
...................................................................
114. Process identification using simultaneous heat and mass
transfer ................ 145. Inertial separation based on
turbulent flows ..................................................
206. The influence of a falling water
film..............................................................
267. Ionic wind Literature review and experiments
........................................... 29
7.1. Literature review
..........................................................................................................
29
7.2. Experimental detection of ionic wind
..........................................................................
31
8. Particle diffusion
............................................................................................
349. Particle
agglomeration....................................................................................
3610. Properties of dust in the paper industry
....................................................... 3911.
Nonspherical particles
..................................................................................
4112. CFD Simulations
..........................................................................................
4313. Experiments
..................................................................................................
52
13.1. Electrostatic collection efficiency
..............................................................................
52
13.2. Determination of a regression model for the new equipment
.................................... 53
13.3. Experiments validating the collection efficiency for the
new geometry .................... 56
14. Model for combined electrical and centrifugal air cleaning
in turbulentflows
...................................................................................................................
57
14.1. MODEL 1 for the air
cleaner......................................................................................
57
14.2. MODEL 2 for the air
cleaner......................................................................................
59
14.3. MODEL 3 for the air
cleaner......................................................................................
59
14.4. Experiments and validation of models
.......................................................................
60
15. Conclusions
..................................................................................................
62References
..........................................................................................................
64Appendix 1
.........................................................................................................
66
-
21. Introduction
Air cleaning in the paper industry has its specific problems
requiring some extraordinary precau-tions and equipment. For
instance, since the dust is inflammable and even explosive, a
normal bagfilter has to be placed outdoors, leading to excessive
piping etc. A wet scrubber on the other handconsumes significant
amounts of water and power.
This is why a concept based on an electrostatic precipitator
with a wetted collection surface, wherethe charging electrode
consists of rings with needles on, was developed (see figure 1). A
project atthe VTT Manufacturing Technology Center was carried out
in order to investigate the electrostaticcleaning properties in
such an air cleaner. From the beginning the concept was like a
normalelectrostatic air cleaner, where the air is lead through the
cleaner as calmly as possible. In the paperindustry, however, the
air can contain large pieces of paper which eventually lead to the
clogging ofthe charging electrode at the centre of the air cleaner.
To avoid this clogging, the inlet was changedto a tangential
position; see figure 1 for a better description.
This project was started in order to establish whether the
conventional model for electrostatic col-lection can be used under
such highly turbulent conditions, and how forces other than
electrostaticones should be considered. A mathematical model to
determine the collection efficiency for differ-ent air cleaner
geometries in varying paper production or conversion industries was
also set as agoal. A Masters thesis was produced within the
project, to determine the properties and distribu-tion functions
for the dust present in different industrial environments
(Dahlbacka 1999). The projectlasted from 1.2.1998 to 31.5.1999.
-
3Figure 1: Full scale electrostatic air cleaner. The reactor
height is 5 m (see figure 6). Thecharging electrode with its rings
can be seen at the center, with the reactor surface actingas the
collection surface. The equipment is situated at the Mets-Tissues
factory in Mntt,Finland.
The experiments described in chapters 4 and 7 are all conducted
on the apparatus in Mntt figure1. Later on during the project a
new, slightly different apparatus was built, to be used for
experi-mental purposes, by Valmet Corp. Air Systems for the
project. The new design was based on mea-surements from the first
apparatus, preliminary calculations from this project and the CFD
calcula-tions described in chapter 12. The new air cleaner (see
figure 36) was built lower, which was onefundamental goal for the
development of the cleaner.
In chapter 13 some different experiments are described. One
important part is the one certifying theelectrostatic collection
efficiency model for a full scale air cleaner.
-
42. Model for electrostatic precipitators
Electrostatic precipitators are well covered by literature and
therefore it should not be necessary todiscuss these things so
thoroughly in this context. A mathematical model based on a
wire-in-tubeelectrostatic precipitator is presented together with
other equations necessary for solving a practicalproblem
numerically. For more detailed information see e.g. (Lehtimki
1998a).
2.1. Gas properties
The mobility of a gas ion, Zi , is defined as the ratio of ionic
drift velocity i to the electric field E,i.e.
E Z= ii (2.1)
The theory dealing with the mobility of gas ions is complicated.
A reasonable approximation for theion mobility can be expressed as
follows:
pi
1 )T k (2 )
m
1 +
m
1)(p 16
e 3( = Z 1/21/2ig
i (2.2)
where p is the gas pressure, mg is the mass of the gas molecule,
m
i is the ionic mass, e is the
elementary charge (e = 1.6021 10-19 As) and is the mean impact
cross section. The mean impactcross section can be determined from
the following polynomial fit curve:
c + m b + m a = i2
i (2.3)
where the coefficients a, b and c have the following values: a =
1.94 10 m/amu, b = 3.62 10-21 m/amu and c = 1.01 10-18 m.
The diffusional motion of gas ions is characterised with the
diffusion coefficient Di which is re-
lated to the ion mobility Zi by
ii Ze
T k = D (2.4)
where k is the Boltzmann constant (k = l.3807 10-23 J/K) and T
is the absolute temperature.
The mean thermal speed of gas ions is calculated from a formula
describing the mean thermal speedin the case of an ideal gas. The
mean thermal speed is given by
-
5)m
T k 8( = c 1/2i
ipi (2.5)
2.2. Estimating the corona current
The electrostatic cleaning efficiency, calculated as described
in the following sections, requires asan input parameter the corona
current. If the calculations are performed for an existing air
cleaner,a measured value for the corona current is normally used,
but in the design phase, at least an ap-proximate value for the
corona current has to be calculated. The corona current depends,
naturally,on several factors e.g. the design of the discharge
electrode and high voltage. The expression pre-sented here can be
used for the electrode configuration as seen in figure 1 (Lehtimki
1998b):
( )2
3x2
2i0 U
rr
LrZKI
= pi (2.6)
In equation (2.6) r2 is the radius of the air cleaner, rx is the
radius of the discharge electrode and Uis the onset voltage. The
factor K is based on experiments . A set of data points for K is
graphicallypresented by Lehtimki, and to these points a curve can
be fitted. Here the following expression forK is suggested:
468.0
x2
r
rr
n/L6278.1K
= (2.7)
In this equation L is the length of the electrostatic collection
section and nr is the number of elec-trode rings. At high voltages,
or with a small distance between the electrode rings, it would
seemthat (2.6) would give somewhat too low values for the corona
current but the expression is anyhowused to estimate the corona
current.
2.3. Electrical properties of the electrostatic precipitator
The electric field between the discharge electrode and the
collection electrode plays an importantrole in the particle
charging and collection processes. The electric field depends on
the voltageapplied between the electrodes. The electric field and
ion concentration are modelled by assuminga simple wire and pipe
geometry, i.e. a thin corona wire on the centre line of the
cylindrical collec-tion tube.
The corona current per unit length of the corona electrode, jL
,is given by
r E(r) n(r) Ze2 = LI
= j iL pi (2.8)
-
6For high corona currents and for r >> r1 , the electric
field can be estimated by
2/1
i0
L0 Z2
jE
pi
(2.9)
An approximate solution for the average ion concentration, n ,
between the electrodes can be calcu-lated:
r
1
Ze2
j 2 n
2i2
L01/2
pi
(2.10)
This solution can be used in the idealised case, i.e. the space
charge due to aerosol particles isassumed to be zero. It has also
been assumed that there is no dust layer on the collection
electrodethat could affect the electrical behavior of the system.
This, naturally, is always the case when thewalls are constantly
washed with water.
2.4. Particle charging
Particle charging has been studied for several decades and
therefore relatively good models areavailable. There are two basic
mechanisms responsible for the charging of aerosol particles.
Theseare referred to as field and diffusion charging. Particle
charging in an electric field is assumed to bedue to the ordered
motion of ions under the influence of an electric field. This
approach is appli-cable predominantly for large particles (dp >
0.5 m). Diffusion charging is due to ion attachment toparticles
caused by the random motion of the ions. Particle charging in an
electrostatic precipitatortakes place in a strong electric field
and at high ion concentration. Thus both charging mechanismsmust be
taken into account.
2.4.1. Field charging
Particle charging in a strong electric field can be described by
the classical field-charging theory(Hinds 1982). According to the
field charging theory the mean charge number of a particle is
givenby
+=
s0F tt
ts)t(s (2.11)
where the saturation charge number is given by
-
702
pr
r00 Ed2e
3s
+=
pi(2.12)
and the time constant ts is given by
i
0s Zen
4t
= (2.13)
and t is the particle charging time calculated from the geometry
and the mean flow-velocity. Theconstant r is the dielectric
constant for the particle material.
2.4.2. Diffusion charging
Ion attachment to aerosol particles can also be caused by the
thermal motion of ions (Brownianmotion). This is called diffusion
charging and it is the dominating process for particles smaller
than0.2 0.5 m (depending on source). Due to diffusion charging, the
maximum charging predictedby (2.12) can be slightly exceeded.
The so-called White equation gives the particle charge as a
function of charging time t:
+= tncdd
41ln
dd)t(s i0p
0
pW
pi(2.14)
where the coefficient d0 is given by
Tk2ed
0
20 pi
= (2.15)
When modeling particle charging by corona discharge, both field
and diffusion charging have to betaken into account. The simplest
way of doing this is by combining the two mechanisms:
)t(s)t(ss FW += (2.16)
This equation can be used to estimate the total particle charge
in an electrostatic precipitator.
-
82.5. Drift velocity and collection efficiency
Electrostatic force and aerodynamic forces govern the motion of
a charged aerosol particle in a gas.The electrostatic force FE
caused by the electric field E0 is given by
0E E e s=F (2.17)
where s is the particle charge number and e is the elementary
charge. This force causes a particledrift in the direction of the
electric field. The electrostatic force is balanced by the gas
resistanceforce or drag force, which is given by
p22
pgDD dw3w d 8 C = F pipi = (2.18)
(the middle part of which, is the general form of Newtons
resistance equation and the right handpart is known as Stokes law)
where CD is the drag coefficient, g is the gas density, dp is the
particlediameter and w is the velocity of the particle (Hinds
l982). The drag coefficient can be approxi-mated by
6Re
+ 1 Re24
Cp
2/3
pD (2.19)
where Rep is the particle Reynolds number given by
d w
= Repg
p (2.20)
At mechanical equilibrium the forces FE = F
D and the drifting velocity w for a charged particle in an
electric field can be calculated by
1p
2/3
p
c
6Re
+ 1 d 3CE e s
= w
pi (2.21)
where Cc is the Cunningham slip correction factor (Hinds l982)
the value of which is given by
-
9
d
C -exp B + A d
+ 1 = C pp
c (2.22)
where is the mean free path of the gas molecules (for air = 6.53
10-8 m). The values of thecoefficients A, B and C are A = 2.5l4, B
= 0.8 and C = 0.55 (values for A, B and C vary slightlydepending on
the source). The Cunningham slip correction factor extends the
range of Stokes lawto below 0.01 m (without correction it is valid
only down to 1 m). Thus small particles, using theslip correction,
settle faster by a factor of Cc than predicted by Stokes law. This
correction isneeded when the size of a particle approaches the mean
free path of the gas molecules because theparticle is so small that
it starts slipping between the gas molecules.
In order to predict the collection (or cleaning) efficiency of
the electrostatic precipitator the as-sumption of the complete
mixing of air in the collection system is made, i.e. that no
gradients existin the particle concentration. This should, for a
totally turbulent flow, be rather a good assumption.The collection
efficiency is then given by
VA w
exp - 1 =
&
(2.23)
where A is the collecting area (i.e. the area of the collecting
electrode at the length of the chargingelectrode) and V& is the
air flow. The equation is known as the Deutsch formula (Hinds
1982). Theequation (2.23) is based on the assumption that no
particle re-entrainment takes place. This as-sumption is valid for
liquid particles or if a falling water film wets the collection
surface. As thedrifting velocity w depends on particle size, the
collection efficiency is of course also different fordifferent
particle sizes. As several phenomena are examined in this context,
the collection effi-ciency given by (2.23) is hereafter referred to
as e and the corresponding velocity as we. A typicalcollection
efficiency curve is shown below:
-
10
Collection efficiency (%) as a function of particle size
0 %
20 %
40 %
60 %
80 %
100 %
0.001 0.010 0.100 1.000 10.000 100.000Particle diameter [m]
Effi
cien
cy [%
]
Figure 2: Collection efficiency for an electrostatic
precipitator as a function of particlesize. The calculations have
been made for a system with the following dimensions:
Flow rate V& =3.0 m/s Length of collection section L=2.6 m
Diameter of the collector tube d=1.6 m Corona current I=3.2 mA
-
11
3. Centrifugal forces in a flow field
A rotational air flow will invoke a centrifugal force throwing
airborne particles outwards in theflow field. This property is
commonly utilised in, for example, cyclones. The airflow is
broughtinto rotation by, for instance, bringing the air
tangentially into a larger vessel. The vessel studied inthis
particular case could well be described as a cylindrical reactor
(for cyclones it would then be aconical reactor). A tangential
inlet, as described earlier, is assumed in this study (see figure
1).
In order to determine the centrifugal collection efficiency, the
centrifugal particle velocity wc has tobe determined. Assuming that
the centrifugal force acting on a particle is equal to the drag
forceacting on the same particle (see equation 2.18) the radial
velocity of the particle is calculated by(Ogawa 1984)
( )
18
rdw
2gp
2p
c
= (3.1)
where is the angular velocity, g and p are the gas and particle
densities respectively and r is theradius of the particle path in
the flow. This radius is growing with time since the particle is
slungoutwards by the centrifugal force. Assuming that no other
forces tend to drag the particle towardsthe center of the reactor,
a simplification is done by using an average value for the particle
pathradius, r, calculated from the centerline of the reactor to the
centerline of the tangential inlet. Sincethe measure of the inlet
is rather small compared to the reactor radius, using an average
value for rdoes not drastically affect the result of equation
(3.1).
Equation (3.1) should be completed with the Cunningham slip
correction factor
( )
18
Crdw
c2
gp2
pc
= (3.2)
The meaning of this correction factor was discussed in chapter
2.
Since the inlet and outlet not are necessarily equal in size and
shape, it is therefore justifiable toassume that the angular
velocity varies with the length (or height) of the reactor.
Calculating theangular velocities in and out for the inlet and
outlet from the velocities in the connecting pipes(depending on
geometry it might be necessary to use only the tangential velocity
component) from
r
inin
r
w= and
r
outout
r
w= (3.3)
gives the angular velocity as a function of reactor height:
-
12
hh
)h(r
inoutin
+=
(3.4)
where rr marks the total reactor radius (3.3) and h
r the reactor height (3.4). In a later section these
assumptions can be compared to the results from CFD simulations.
Equations (3.l) and (3.2) hold,only if the Reynolds particle number
is small (equation 2.20), i.e. if the relative velocity betweenthe
particle and the fluid is small enough, whilst only then the drag
force C
D can be given by the
simpler form
pD Re
24C (3.5)
Considering that Rep might be significantly higher than one,
equation (2.19) should be used. Thusequation (3.2) has to be
modified to:
( )
12
h
h p2/3
c2
gp2
p
c hh
dh
6Re
+ 118
Crd
w
2
1
=
(3.6)
A mean value for the velocity wc is obtained from equation
(3.6). In equation (3.6) the equation for
(3.4) is inserted. An expression similar to the Deutsch formula
can be used to calculate thecentrifugal collection efficiency
VA w
exp - 1 = cc
&
(3.7)
Since the centrifugal particle velocity wc depends on the square
of the particle diameter, the result of
equation (3.7) will depend heavily on the particle size. A
typical collection-efficiency curve isshown below (figure 4).
These calculations, as well as the equations presented above,
are made under the assumption thatthe angular velocity changes
linearly through the collector. The angular velocity at the inlet
iscalculated from the mean velocity win in the inlet pipe (
2.11.1ww inmax K ), see figure 3, usingthe total radius of the
collector rr instead of the actual radius of the particle path r
(assuming wmax atthe centre line of the inlet). The same principle
goes for the outlet. The value for win obtained thisway is smaller
than it would be using wmax and the radius r. Thus a compensation
for wall losses isautomatically obtained, giving a flow profile
more according to the somewhat hypothetical oneinserted in Figure
3. The profile can be compared with the ones given by the CFD
calculations(chapter 12).
-
13
rr
r
wmax win
Figure 3: A drawing describing the assumptions made when
calculatingthe collection efficiency for the centrifugal flow.
Collection efficiency (%) as a function of particle size
0 %
20 %
40 %
60 %
80 %
100 %
0.001 0.010 0.100 1.000 10.000 100.000
Particle diameter [m]
Effi
cien
cy [%
]
Figure 4: Collection efficiency in a rotating flow field as a
function of particle size.The calculations have been made using the
following values: Length of collection section L=5 m Diameter of
collection section d=1.6 m w
in=12.3 m/s in a 0.60.4 m duct
wout
=0.5 win
Particle density rp=1000 kg/m
Air properties at 25C
-
14
4. Process identification using simultaneous heat and mass
transfer
This chapter explains how a regression model giving heat- and
mass-transfer coefficients for thestudied air cleaner is obtained.
The coefficients are needed later when calculating the
collectionefficiency for the air cleaner. A more detailed
description can be found in (Berg and Altner 1999).
The studied process is described with the following differential
energy balance for the controlvolume a, b, c and d in figure 5, and
gives:
1wwa0wv0wwac tcm)dii(mtcmdtcmimqd &&&&&&
++=+++ (4.1)
The numerical differentiation of equation (4.1) requires
governing equations for both turbulent heatand mass transfer.
2.
i x i0 x0t
t
5. di/dx i/ x
3. h0 +tcv
BA
4.
6.
dqc.
ix
t
.
ma
i+dix+dxt+dt
dmvt
d
b
c
a
1.
6.t0
.
t1
mv.
mv.
dmv+.
1.
dq.
i1 x1
2.
Figure 5: Heat and mass transfer through the boundary layer in
the cylinder with a falling waterfilm. The air state change during
evaporation is also described by including a principal picture ofa
perspectively transformed humidity chart (Soininen 1994), often
named the i,x-chart. The controlvolume and control surface are
described by the dotted lines connecting a,b,c and d.
Heat transfer
Heat-flow through a boundary layer is usually defined by the
following expression, see figure 5:
dAttqd
=dAtt=qd )( )( &
& (4.2)
We may, in principle, define an average mass flow Amn&
perpendicular to the surface and consider
that the measured heat flow is the difference between the
enthalpies of mass flow away from thesurface, at surface
temperature t, and mass flow of the same magnitude towards the
surface, at themain flow temperature t. This gives:
-
15
)()( ttA
Cttc
Am
tcA
mtc
Am
=
dAqd n
pn
pn
pn
==
&&&&&(4.3)
From a comparison of equations (4.2) and (4.3), we obtain:
AC
cA
m np
n&&
== (4.4)
Mass transfer
If the surface in contact with air is the surface of a drying
material, the humidity of the flow awayfrom the surface is higher
than that of the flow towards the surface. Denoting the temperature
of thesurface with t, the humidity of saturated air at that
temperature with x and the humidity of themain flow with x and
replacing A
mn& combined with a more accurate definition of the specific
heat cpone may write (Berg and Karlsson 1998):
va
nv
cxc
xxxx
Am
dAmd
)()(+
==
&&(4.5)
where the content of moisture in the air, also known as air
humidity x, is defined as:
v
v
a
v
ppp
MM
x
= (4.6)
and the air humidity x (kgH2O/kgda) is the saturation moisture
content defined by the local controlvolumes mass content of water
vapour divided by the mass content of dry air.
The content of moisture in the air is defined using Daltons law
for ideal gas combined with Antoineswater vapour equilibrium curve.
Here the notation is used for the mass-transfer coefficient
in-stead of , that is, the heat-transfer coefficient. The stress
mark indicates that we now refer to themass transfer which occurs
simultaneously to heat transfer. Equation (4.5) is known in the
literatureas the Lewis equation when k
1 in equation (4.7) equals one.
kk 11 = 1 when (4.7)
Let us now examine the experimental process by looking at
figures 5 and 6. Convective heat trans-fer is the product of the
heat capacity mass flow rate of air and the temperature drop of the
air.Whereas mass transfer takes place at the same time, we write
that this convective heat flow is
dAtt )( . Further, when we consider that dxma& represents
evaporation vmd & , equation (4.1) can
-
16
be rewritten as:
)21()()()( dt
dAmd
dAm
c=ttkttdAmd
cttch vwwccvwv0&&&
+++ (4.8)
It is now possible to solve equation (4.8) numerically when
assuming k1=1 and solving an approxi-mate value kc 4 W/m
2C from the free convection theory (Jakob 1949).
Surface temperature t according to the Lewis analogy
During heat and mass transfer, the state of the air is changed
from point 1 to point 6 shown as the di/dx line B in figure 5. The
Lewis analogy surface temperature t is obtained, if the difference
be-tween curve A and curve B in figure 5 is neglected. This is the
case when k1=1 and thus along thisline the direction of the air
state change is indicated by the ratio i/x and can be expressed
with thefollowing statement.
xx
iix
idxdi
x
i
=
=
(4.9)
The Lewis analogy surface temperature t, is therefore, always
different from the measured surfacetemperature when 11 k . The
assumptions that are related to equations (4.8 and 4.9) are
consid-ered, at this stage of the work, as the method basis when
determining . It is often veryinformative to critically analyse the
process by drawing the experimental wet and dry bulb tempera-tures
in an i,x-diagram, and thereby obtain a figurative description
showing if the assumed air statechanges differ from the
experimental ones.
The experiments were carried out on a full-scale apparatus,
shown in figure 6. In order to obtain theheat- and mass-transfer
coefficients ( and ) for the apparatus using the theory presented,
someprocess variables were measured (see table 1 below): dry and
wet bulb temperatures of both theinlet and outlet air flows, as
well as flow velocities. The flow velocities were measured with a
Pitottube and a digital micromanometer. The inlet temperatures were
measured in the vertical duct (seefigure 6) about 2 metres before
the duct entered the cylinder body and outlet temperatures at a
pointabout 2 metres from the cylinder exit. The temperature of the
surrounding hall was measured onseveral occasions, and it could be
considered constant (tc=25 C). The heat-transfer coefficientfrom
the surrounding hall air to the water film was calculated as 4
W/mC. Table 1 also shows theair humidity, calculated from dry and
wet temperatures.
The experiments were carried out during the cold season, with
very dry air. In this range of dry andwet temperatures, the
influence of daily variations in total air pressure can, in some
cases, be con-sidered negligible when calculating differences in
air humidities, x. The influence of the absolutepressure p
0 on the experimental results should, however, be checked before
continuing with or
without correction.
-
17
windin=0.5 m
woutdout=0.4 m Starting point of
falling water film
See figure 1 forenlargement
Tangential inlet
h4.4 m
dc=1.6 m
Figure 6: Drawing showing the principles of the apparatus used
forexperiments.
Table 1: Experimental data.
Mea
sure
d in
let
Velo
city
[m/s]
.M
easu
red
Dry
Tem
pera
ture
at
inle
t [C]
Mea
sure
d W
et
Tem
pera
ture
at
inle
t [C]
Mea
sure
d Dr
y Te
mpe
ratu
re a
t outle
t [C]
Mea
sure
d W
et
Tem
pera
ture
at
outle
t [C]
Psyc
hrom
etri
c Ai
r Hum
idity
at
inle
t [kg
H 2O
/kgd
a]Ps
ychr
om
etri
c Ai
r H
umid
ity a
t outle
t [kg
H 2O
/kgd
a]15.1 26.2 14.4 21.9 14.2 0.00538 0.006937.5 26.5 14.2 22.2 14
0.00504 0.0065910.6 26.3 13.8 22.5 13.9 0.0047 0.0063614.6 26.1
13.5 22 13.8 0.00447 0.006467.3 26.6 14 22.3 13.8 0.00479
0.006346.4 24.5 12.1 20.1 11.8 0.0037 0.005214.8 23.6 11.1 20.2
12.1 0.0031 0.0054610.8 24.5 12.2 20.1 11.8 0.0038 0.00526.9 24.6
12.5 20.6 12.9 0.00406 0.006114.7 24.5 12.9 20.6 13.1 0.00451
0.00631
Using the numerical data, together with the equations presented,
gives the possibility of calculating. Iterative calculations are
carried out as follows: is given a value, after which the outlet
drytemperature and the outlet air humidity are calculated. The
calculated values are then comparedwith the measured ones, and the
-value adjusted until a satisfactory result is obtained. The
maingoal is to match the dry temperatures, as the measured dry
temperature values can be expected to bethe most reliable ones. As
one can see from the results in table 2, the agreement of the
measure-ments is good, and even the calculated air humidity values
(which were mentioned as being ofsecondary interest at this point)
give good agreement. In addition, the inlet temperature of
thefalling water film had to be adjusted (note: water inlet in the
upper region of the cylinder).
-
18
Table 2: Experimental and calculated values. The transfer
coefficients are adjustedto an accuracy of 0.5 W/mK.
Mea
sure
d Dr
y Te
mpe
ratu
re at i
nlet
[C]
Mea
sure
d Dr
y Te
mpe
ratu
re a
t out
let [
C]
Calcu
late
d Dr
y Te
mpe
ratu
re a
t out
let [
C]
Diff
erence
[%]
Psyc
hrom
etri
c Ai
r H
umid
ity at o
utle
t [kg
H 2O
/kgd
a]
Calcu
late
d Ai
r Hum
idity
at
outle
t [kgH
2O/k
gda]
Diff
erence
[%]
Res
ultin
g Tr
ansf
er
coeffi
cient
[W
/mK
]
26.2 21.9 21.86 -0.19 0.00693 0.00691 -0.34 69.526.5 22.2 22.18
-0.1 0.00659 0.00656 -0.55 32.526.3 22.5 22.5 0 0.00636 0.00635
-0.24 42.526.1 22 21.95 -0.21 0.00646 0.00643 -0.59 67.526.6 22.3
22.31 0.04 0.00634 0.00631 -0.57 30.524.5 20.1 20.13 0.15 0.0052
0.00519 -0.22 27.523.6 20.2 20.19 -0.05 0.00546 0.00544 -0.33
6924.5 20.1 20.12 0.08 0.0052 0.0052 0.01 45.524.6 20.6 20.61 0.05
0.0061 0.00608 -0.43 33.524.5 20.6 20.57 -0.16 0.00631 0.00627
-0.57 70
When looking at the small differences between the measured and
the calculated values for the drytemperatures and air humidity, the
conclusion is that the agreement seems to be acceptable.
Therelation between heat and mass transfer can be expressed as = 1k
where, in this particular case,k11. From the resulting heat- and
mass-transfer coefficients presented in table 2 the Nusselt
andReynolds numbers are calculated from their respective
characteristic equation:
dNu = (4.10)
dwRe m= (4.11)
For separate flow geometries, the characteristic flow velocity
(w) and the characteristic length unit(d) will be defined
differently. Here wm is defined as the mean axial velocity in the
cylinder and d isthe cylinder diameter (dc in figure 6). The
experimentally obtained Nusselt numbers, as a functionof Reynolds
numbers, can be seen in figure 7. A curve is, thereafter, fitted to
the experimentallyobtained Nusselt numbers. As can be seen, a
fairly good agreement is reached using a straight line.This would
give the relation:
Re027.0Nu = (4.12a)
which can be transformed into a more familiar form:
45.01 PrRe032.0Nu = (4.12b)
-
19
Equation (4.12a) is presented graphically in figure 7. The
tangential flow pattern seems to give highNusselt numbers compared
to Reynolds numbers.
Nusselt number as a function of Reynolds number
0
1000
2000
3000
4000
5000
0 50000 100000 150000 200000Re
Nu
Figure 7: The experimentally obtained Nusselt numbers as a
function ofReynolds numbers. The dotted line marks the equation
(4.12a).
The values of in the cylinder might be considered to be higher
than expected. One could, how-ever, compare these values with the
values calculated for the inlet pipe diameter and the flowvelocity
of the air. The expression for Nu in turbulent flow is
(VDI-Wrmeatlas 1953):
cm
bm
3/2PrRe
ld1aNu
+= (4.13)
where a = 0.024, b = 0.786 and c = 0.45 for turbulent flows in
straight circular tubes. In equation(4.13) the physical properties
of air should be calculated using arithmetic mean values, i.e.
tm=(t1+t2)/2, xm=(x1+x2)/2. For the cylinder under observation the
ratio l/d is 4.2 m/1.6 m 2.6. Noting that theconstant in front of
Re in (4.12a and 4.12b) describes the development of the
temperature or veloc-ity profile, where in this case thermal
entrance effects are significant, the same constant togetherwith
the mentioned l/d, should be used in (4.13) when comparing the
levels of heat- and mass-transfer coefficients.
Thus the level of turbulence (i.e. the level of heat and mass
transfer) in the cylinder, depends bothon the inlet geometry and
the kinetic energy of the flow in the inlet pipe. Hence, this is
the mostplausible explanation for the remarkably high exponent b in
Reb. Values of b=0.80.95 can befound in the pertinent literature
for turbulent flows over rough surfaces and impingement drying(e.g.
Nunner 1956 and Mujumdar 1987, pp. 461-474).
Some more experiments and thoughts around regression models for
the studied air cleaner can befound in chapters 7 (same geometry as
here) and in 13 (a new geometry).
-
20
5. Inertial separation based on turbulent flows
The deposition of particles in turbulent flows has been
discussed by many authors (for instanceOgawa 1984 and Friedlander
1977). One physical interpretation of the deposition is that fine
solidparticles with a large inertial force caused by the
fluctuating velocity of the turbulent flow, in thefully turbulent
flow region, penetrate the quiescent region near the wall and reach
the wall surface.In other words, the stopping distance of a
particle is compared with the thickness of the laminarsublayer.
Considering a pipe flow, as Friedlander did, gives the stopping
distance for a particle, based on theradial fluctuating velocity of
turbulent flow, w
r:
2f
w9.0wr = (5.1)
The equation is the result of experiments (Laufer 1953). Here w
is the mean gas-velocity in the pipeand f is known as the fanning
friction factor, which is given by the Blasius equation (Bennett
andMeyers 1982)
25.0Re0791.0f = (5.2)
Using the fluctuating velocity wr given by equation (5.1) gives
the stopping distance for a particle
according to the Stokes equation
18wd
S r2
pp= (5.3)
Friedlander gives two separate equations for the transport (or
migration) velocity of particles in theturbulent flow. When the
stopping distance is much smaller than the thickness of the viscous
sub-layer, the following expression gives the deposition
velocity
2/5
45
52p
2p
2
2f
101.6
Udk
=
(5.4)
where k is the transport velocity, f is the fanning friction
factor and U is the average gas velocity.The upper limit of this
equation is given as
-
21
52fUS 2/1
(5.7)
In figure 8 the group of lines denoted as A are calculated with
equation (5.4), while those denotedas B represent equation (5.6).
The lines B are not drawn to the upper limit S/d < 0.5.
Already,before reaching the upper limit the empirical equation
(5.6) tends to collapse (i.e. the transportvelocity falls
dramatically). The theory, presented above, gives the equations for
two extreme situ-ations. The discontinuity of the lines (see figure
8) also causes problems when modelling the clean-ing efficiency for
turbulent flows.
-
22
In order to obtain a continuous function and to be able to
secure equations for varying geometries,a slightly different method
though based on the same principles as described above is used.
Thestopping distance is compared with the thickness of the laminar
sublayer (Berg 1998):
1g
21
1w
ky
= (5.8)
where k1=2.5 and w1 gives the flow velocity at the distance y1
from the wall:
g
210
1k
w
= (5.9)
which gives a critical diameter, dp,crit
1pgr
21
2crit,p
ww
k18d
= (5.10)
Particle fractions larger than dp,crit are assumed to deposit
totally, and consequently no particlessmaller than dp,crit would
deposit. Since this is experimentally verified as being false, the
function ismade continuous by proportioning the stopping distance
of an observed particle to the stoppingdistance of the critical
particle size. A constant kp is introduced:
critp S
Sk = (5.11)
where Scrit
is the stopping distance for the particle having the diameter
dp,crit
. The maximum value ofk
p is 1, since the turbulent eddies cannot transport particles
faster than the transporting velocity (it
was earlier stated that particles larger than dp,crit
are assumed to deposit totally). See also later sec-tions.
The velocity of the particles towards the collection surface is
determined by the flows which arenormal to the pipe surface. The
heat- and mass-transfer coefficient for a system is, in fact,
theaverage heat-capacity rate of flows occurring in the direction
normal to the surface (Soininen 1994).For turbulent flows in pipes
is given by (VDI-Wrmeatlas 1953):
dPrRe024.0 45.0786.0
= (5.12)
-
23
where the physical properties of air (in Re and Pr) should be
calculated using arithmetic meanvalues, i.e. tm=(t1+t2)/2,
xm=(x1+x2)/2. Thus the velocity for the turbulent flow (i.e.
eddies) in thedirection normal to the surface is given by (Soininen
1994):
gpn
cw
= (5.13)
The velocity wn can be said to transport particles towards the
collection surface in the air cleaner,whilst the stopping distance
is determined by wr (Friedlander 1977). The product np wk is
equiva-lent to the transport velocity of the particles. This way,
the upper limit of kp given above, alsobecomes natural.
Transport velocity as a function of Reynolds number
1
10
100
1000
1000 10000 100000 1000000
Re
[cm/m
in] 2 m
5 m
8 m
Figure 9: Particle transport velocity in a turbulent pipe flow
as a function ofReynolds number. Pipe diameter d=25 mm and particle
density r=7800 kg/m.The critical particle size is obtained by
comparing equations (5.3) and (5.8).
Since the radial fluctuating velocity wr given by equation (5.1)
actually gives the velocity of the gasjust outside the buffer
layer, the stopping distance should be compared with the thickness
of thelaminar and the buffer layer, not just the laminar sublayer
as above. The thickness of the laminarand buffer layer, y2, is
given by (Berg 1998)
1g
212
w
kky
= (5.14)
(k2=22.5) and therefore the expression giving the critical
particle size becomes
-
24
1pgr
212
crit,pww
kk18d
= (5.15)
Figures 9 and 10 can be said to describe two extreme situations,
the first with S=y1 and the second
with S=y2. The actual case will probably be somewhere between
these two values, which also can
be seen from figure 11 showing some experimental results (Ogawa
1984). Unfortunately theseexperimental values are covered by
equation (5.4) and never go into the region covered by
equation(5.6).
Transport velocity as a function of Reynolds number
1
10
100
1000
1000 10000 100000 1000000Re
[cm/m
in] 5 m
8 m2 m
Figure 10: Particle transport velocity in a turbulent pipe flow
as a function ofReynolds number. Pipe diameter d=25 mm and particle
density r =7800 kg/m. The critical particle size is obtained by
comparing equations (5.3) and (5.14).
Figure 11: The deposition of fine solidparticles. Pipe diameter
d = 25 mm.Experiments presented in (Ogawa 1984).
In agreement with the assumption of no re-entrainment, made in
the Deutsch formula, equation(2.23), the collection efficiency of
the turbulent deposition is given by
-
25
=
VAwk
exp1 npT & (5.16)
Collection efficiency (%) as a function of particle size
0 %
20 %
40 %
60 %
80 %
100 %
0.001 0.01 0.1 1 10 100
Particle diameter [m]
Effi
cien
cy [%
]
ABC
Figure 12: Collection efficiency in a turbulent flow field as a
function of particlesize. Using equation (5.10) for the critical
particle size gives (B), using equation(5.15) gives (A and C). The
difference between A and C is explained below.The calculations have
been made with the following values: Pipe diameter d=0.5 m Flow
velocity w=15.1 m/s Particle density r
p=1000 kg/m
Collection area A=25.1 m
Since the velocity of the fluctuating velocity, wr, is given
just outside the buffer layer, the transport-ing velocity, wn also
should be calculated at the same location. Equation (5.13) gives wn
as a func-tion of the overall heat- and mass-transfer coefficient.
It can easily be shown (Berg 1998) that theheat- and mass-transfer
coefficient for the turbulent core can be expressed as:
tot3 32 K= (5.17)
In figure 12, C is calculated using tot5.2 in equation
(5.13).
The maximum level for collection efficiency is consequently
determined by the transporting veloc-ity w
n, whereas the point at which the maximum level is reached, is
determined by the thickness of
the laminar (y1) or laminar and buffer layer (y
2). Using
3 for (w
n) and y
2 is, at this point, considered
the most logical way since wr is used to determine the stopping
distance and by definition w
r is the
fluctuating velocity at the location where 3 describes the heat
and mass transfer rate.
The values used to calculate the lines in figure 12 correspond
to the inlet pipe in the studied aircleaner (Figure 1). The
collection area is the area of the air cleaner. This calculation is
made just toget an overall picture of the turbulent collection
efficiency. The selection of geometrical parameterswill be
discussed more in detail in a later section.
-
26
6. The influence of a falling water film
The collection surface of the air cleaner is constantly washed
with water falling down as a thin film.The water is applied to the
uppermost region of the cleaner (see figure 13) along the whole
width ofthe wall. To estimate the properties of the falling water
film, some equations from (Bird et al l960)are used.
Figure 13: Flow of liquid film under the influence of gravity
with no rippling. Sliceof thickness Dx over which the momentum
balance is made. The y-axis is pointingoutward from the plane of
the paper.
The maximum velocity of the falling film (Figure 13) is the
velocity furthest away from the wall,and can be calculated by
2cosg
w2
max = (6.1)
where is the fluid (water) density, is the wall angle and is the
film thickness. The integratedaverage velocity is given by
3cosg
w2
z = (6.2)
The film thickness may be calculated from the average velocity,
the volume rate of flow or fromthe mass rate of flow per unit width
of the wall. Here, the equation for volume rate of flow only,
isgiven, for the others please see (Bird et al 1960).
-
27
3cosgWQ3
= (6.3)
where Q is the volume rate of flow and W is the wall width
(perpendicular to the falling direction).
Film velocity as a function of film temperature
0.2
0.3
0.4
0.5
0.6
0.7
0 20 40 60 80Film temperature [C]
Film
v
eloc
ity [m
/s
Q=20 l/min Q=30 l/min
Figure 14: Velocity of the falling water film for two flow
rates. Cylinderdiameter d =1.6 m.
For instance, having a wall width W = 5.0 m (i.e. d = 1.6 m) and
Q = 20 litres/min would give amaximum velocity wmax = 0.34 m/s
(fluid properties at 12 C), with a film thickness = 0.30 mm.
Ahigher temperature results in a higher film velocity and a thinner
film. Thus it should be consideredwhich property to keep constant
in the air cleaner with seasonal temperature changes the flowfield
or the film thickness.
When considering the flow profile of the upward flowing air in
the air cleaner, the following ques-tion arises: how does a water
film with an opposite direction influence the profile? Closest to
thewater surface, the air will have the same velocity and direction
as the water (the velocity of whichwill slightly diminish from the
value given by equation (6.1) under the influence of the air. This
is,however, ignored in this context).
Since particles drift towards the collecting surface as they
travel through the cleaner, they will, atsome point, enter the
region assumed to have a velocity directed downwards. This should,
slightly,influence the cleaning efficiency in a positive manner.
Particles entering this region can be assumedto be separated since
their relaxation time reaches infinity. The thickness of the
laminar and bufferlayer (y2) can be diminished to represent only
the part of y2 being directed upwards. The followingmodification is
suggested to the theory presented in chapter 5: The velocity of the
falling waterfilm, equation (6.1) is added to the axial velocity
component of the main flow. The critical particlesize is calculated
by comparing equation (5.3) with the thickness of the reduced
laminar and bufferlayer given by
-
28
=+
max2
maxmax222
w
wwyy (6.4)
(which is slightly less than y2) where wmax is given by (6.1)
and w2max is calculated from the originalw2 increased with
wmax.
Changing the thickness of the laminar and buffer layer (from y2
to y2+) will slightly change w2 (and
consequently w2max). The radial velocity component is, however,
dominant compared to the axialcomponent considered here. The result
is that minor inaccuracies in the value of w2 will not
signifi-cantly affect the second factor at the right hand side of
equation (6.4). This somewhat simplifiedprocedure results in a
minor error of less than 1% in equation (6.4).
-
29
7. Ionic wind Literature review and experiments
In an electrostatic precipitator a continuous stream of ions is
established, due to the potential differ-ence between the charging
and collecting electrodes. Because of this stream of ions,
surroundinggas molecules are also brought into motion in the same
direction. This movement of gas (air mostly)is often called ionic
wind (or corona wind, electric wind). In this text a short summary
of somepapers on the subject is presented first. Later, some
experiments to evaluate ionic wind are re-ported. A more extensive
analysis of the experiments in chapter 7.2 is found in (Berg and
Altner1999).
7.1. Literature review
As a summation of the papers studied, it can be said that they
all imply that ionic wind gives ahigher rate of turbulence in the
precipitator. Whether this is a positive or a negative factor
whenconsidering collection efficiency, varies from one paper to
another.
Zhibin & Guoquan (1994) for instance, reach the conclusion
that when increasing the corona cur-rent in relation to the flow
velocity, the collection efficiency increases, but when increasing
thecurrent over a certain point diminishes the collection
efficiency. Furthermore, they state that areduction of turbulence
results in a higher collection efficiency. An explanation for this
maxi-mum in collection efficiency, when varying the corona current,
could be that the ionic wind in-creases the level of turbulence.
Increasing the turbulence only a little improves the collection
effi-ciency, but when the corona current is further increased,
re-entrainment occurs, resulting in a low-ered collection
efficiency. Strictly speaking, the statement made by Zhibin &
Guoquan, that a re-duction of turbulence gives a higher collection
efficiency, would mean that ionic wind is a negativefactor for the
collection efficiency.
Leonard et al (1982) found experimentally that the intensity of
the corona-induced turbulence wascomparable to or less than the
background turbulence present, without an electrical field for
gasvelocities exceeding 1.5 m/s, but increased rapidly when
compared to background turbulence fordecreasing gas velocities.
Generally, it is assumed that there will be a progressive
deterioration inthe effective migration velocity at increasing flow
velocities due to re-entrainment. According to(Leonard et al 1982),
to achieve improved precipitator performance with a negative corona
throughturbulence control, the gas velocity should be approximately
equal to or larger than 1.5 m/s, i.e. theratio of corona-induced
turbulence to background turbulence should become lower. This
meansthat the ionic wind could be seen as a negative factor with
regard to collection efficiency. Both(Zhibin & Guoquan) and
(Leonard et al) seems to have made their experiments with dry
collectionsurfaces. The fact that re-entrainment occurs at higher
flow velocities is valid only for dry surfaces with a wet surface
no re-entrainment occurs.
Kercher (1969) made some interesting observations in his
experiments. First of all, in quiescent air,no ionic wind could be
observed for a wire discharge electrode. Some needles or points had
to bepresent at the electrode before any ionic wind could be
measured. For a system with a 43 kV voltage(and needle electrode),
increasing the distance, between the charging and collecting
electrode, tonot more than 120 mm, decreased the ionic wind to a
minor level. For air in motion, a system with50 kV was used. When
the flow velocity was 0.5 m/s the ionic wind was still clearly
measurable,but when the flow velocity was increased to 1.2 m/s no
ionic wind could be detected. Overall theconclusion made by Kercher
is that ionic wind has a positive effect on collection efficiency
in an
-
30
electrostatic precipitator because of the corona-induced
turbulence pointing towards the collectingsurface. It is especially
beneficial when considering small particles that easily form space
charge.Due to the turbulence, the space charge is displaced towards
the collecting surface leading to onlya minimal reduction in the
corona current. The experiments presented by Laufer were carried
outwith clean air only.
Liang and Lin (1994) gave a ratio between the flow velocity and
the characteristic velocity of ionicwind, described as:
0ew
eV
U = (7.1)
where is the gas density, ew
is the charge density and V0 is the applied voltage. They
concluded
that when the ratio was less than 0.2, turbulent kinetic energy
increased significantly due to ionicwind, thus diminishing the
collecting efficiency. When the ratio was over 0.5 the electric
forceacting upon the flow field was not prominent, and the kinetic
energy of the flow was near that of afully developed flow without
an electric field. Furthermore, the authors stated that under
normaloperating conditions, the influence of ionic wind on the flow
field was insignificant. Decreasing theflow velocity (i.e.
decreasing the ratio mentioned to below 0.2) the ionic wind became
more obvi-ous and had a negative effect on the collection
efficiency. The results were theoretical only, and thereduction in
collection efficiency seemed to be due to re-entrainment caused by
turbulence. Againit can be said that this does not apply to wet
surfaces.
Shu & Lai (1995) in their paper, studied improved heat
transfer due to an electric field. The electricfield results in a
secondary air-flow (i.e. ionic wind) directed towards a grounded
surface, enhanc-ing the heat-transfer in the system. The net effect
of this secondary flow is an additional mixing ofthe flow and
destabilisation of the thermal boundary layer leading to raised
heat-transfer coeffi-cients. The enhancement is most significant
regarding small air-flows, giving a heat-transfer sev-eral times
higher than one without electric field. The results of Shu &
Lai are theoretical. Theyfurther suggest that the enhancement in
heat-transfer is possible due to the oscillatory
flow-field.Whenever two flows, being of the same order of magnitude
are opposed or perpendicular to eachother, some degree of
oscillation is to be expected.
Of the papers cited here discussing ionic wind and particle
collection, the only one implying animprovement in particle
collection efficiency due to ionic wind is Kercher (1969), and that
regard-less of whether it is a wet or dry collecting surface. A wet
surface might influence the conclusionsof the other studies.
The paper of Shu & Lai gives an opportunity to consider the
results in chapter 4, and also to try tomeasure the ionic wind in
the air cleaner studied in this project. The heat- and
mass-transfer coef-ficients are measured with and without an
electric field, and any difference in levels would then beexplained
by the ionic wind. In the following sections some experimental
results are presented.
-
31
7.2. Experimental detection of ionic wind
The experiments were made using the equipment described in
chapters 1 and 4. A more detaileddiscussion about regression models
and also about ionic wind can be found in (Berg and Altner1999).
The regression model presented in this chapter is an improvement to
that presented in chap-ter 4. See also chapter 13.
The experimental results were entered into the overall energy
balance for the system, giving as aresult, the heat- and
mass-transfer coefficient . Thereafter, a regression model of the
same type asbefore is generated. The energy and mass balances give
the following equations from which t2 and are calculated. Solving
the equations, assuming a steady temperature t1 = 6 C and
calculatingthe Reynolds and Nusselt numbers in accordance with
equations (4.10) and (4.11), we obtained aset of points which are
shown in figure 15.
( )ww
totc
w1wtot1
ctotcc1a2ael2
cm2A
ctmA2tAtxmxmq
t&
&&&&
++=
(7.2)
and
( )
tot2121
elv21
aa21
A2
tt
2tt
qc2
xxcmtt
+
+
+
++
=
&&
(7.3)
where t represents the temperature in the vicinity of the
evaporating surface, the lower indexes 1and 2 denote the properties
at the inlet and outlet, and elq& is a small amount of
electrical lossesadded as heat to the system, measured in form of a
corona current. In this case, the corona currentcomes from the fact
that the air is lead through a passage between two high voltage
electrodes(Ogawa 1984). The overall effect of the corona current
was in this case measured as elq& 400 W.
Here elq& is added completely to the air since the
resistance of air, compared to other resistances isof a different
order of magnitude. Anyhow, in this case the influence of elq&
400 W is so small thatit could just as well be omitted. Furthermore
using elq& in equation (7.3) assures a maximum influ-ence of
the input electrical effect, in the form of ionic wind, on the
heat- and mass-transfer coeffi-cient, and yet no measurable
differences can be detected.
-
32
Nusselt number as a function of Reynolds number
0
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
0 50000 100000 150000 200000 250000Re
Nu
Experiments (without electricity) Experiments (with electricity)
Regression modell
Figure 15: The experimentally obtained Nusselt numbers as a
function of theReynolds numbers and a regression model (equation
7.4a, shown as a drawn line)fitted to the Nusselt numbers.
From the data points in figure 15 no significant difference can
be seen between those cases with orwithout an electric field, i.e.
Nusselt numbers and therefore can be held as equal in both
cases.Shu & Lai (1995) amongst others, wrote about an
additional mixing of the gas due to the appliedhigh voltage (i.e.
ionic wind), resulting in an enhancement in the heat-transfer, but
they also statedthat the influence would be greatest at small flow
velocities (i.e. low Reynolds numbers). In thecase under study (the
system was described earlier in chapters 1 and 4) the level of
turbulence isextremely high and even a voltage as high as 130 kV
(i.e. elq& 400 W), should not cause measur-able differences in
the heat- and mass-transfer coefficients.
From these data points the regression model becomes:
95.0Re032.0Nu = (7.4a)
or, to use a more familiar form
45.095.0 PrRe037.0Nu = (7.4b)
Considering the ratio d/l and using the more general equation
(4.13) gives for (7.4b)
45.095.03/2
PrReld1024.0Nu
+= (7.5)
-
33
When comparing these equations with the equations (4.12a) and
(4.12b), one can see that the expo-nent of Re does not equal one,
as suggested in the previous section, but is a little bit lower,
whichwas already predicted as the most probable case. Some further
experiments giving regression mod-els of the same kind as equations
(7.4a) and (7.4b) are described in chapter 13.
-
34
8. Particle diffusion
In order to evaluate the need to consider particle diffusion
when modeling an air cleaner of the kindstudied during this
project, some equations are presented. The diffusion coefficient
for a particle isgiven by (Hinds 1982)
p
c
d3CTk
Dpi
= (8.1)
where k is the Boltzmanns constant, T is the absolute
temperature and Cc is the Cunninghamscorrection factor (see chapter
2). The equation is called the Stokes-Einstein equation for
aerosolparticle diffusion coefficients.
For deposition on a collecting surface by diffusion from a
turbulent flow, the situation is compli-cated, and no explicit
solution exists. The customary assumption made, is that the
turbulent flowprovides a constant concentration, n0, which is
uniform beyond a thin diffusive boundary layer(the same assumption
concerning the concentration was made earlier for the Deutsch
formula). Inthis thin layer, the concentration decreases from the
concentration of the main flow to zero at thesurface. The
deposition velocity is given by
DVdep = (8.2)
where is the thickness of the diffusion boundary layer. The
value of depends on flow mecha-nisms, the nature of the velocity
boundary layer and the size of the particles. One suggestion for
found in (Hinds 1982, page 148) is
4/1
g
8/7
4/1t
Re
Dd5.28
=
(8.3)
where dt is the tube diameter. Larger particles get thrown
partway into the diffusion boundary layerand have a shorter
distance to diffuse, whereas smaller particles, having smaller
inertia, have todiffuse the whole distance. To express the
diffusive deposition in terms of collection efficiency,
thefollowing is given
=
VAV
exp1 depD & (8.4)
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35
In the equation (8.4) A is the collection surface area. For
decreasing particle sizes the diffusionaleffects become more
pronounced, whilst for larger particles, inertial effects
increasing with particlesize, govern collection efficiency. In
between, a gap appears where neither of the processes workproperly.
This gap in the efficiency curve can be seen for particles in the
0.1 to 1.0 m range.
When calculating the collection efficiency for diffusional
deposition in a turbulent flow for the aircleaner studied in this
project, a circular tube with a flow, giving an equal level of
turbulence (i.e.-value) is chosen in order to achieve collection
efficiencies of the right magnitude. Therefore apipe of d=0.5 m
with the flow velocity w
g=15 m/s is used. To get approximately the same residence
time as in the air cleaner, a tube length L52 m is used,
resulting in a collection surface area A82m.
Collection efficiency (%) as a funtion of particle size
0
20
40
60
80
100
0.001 0.01 0.1 1 10 100
Particle diameter [m]
Effi
cien
cy [%
]
Figure 16: Collection efficiency for deposition by diffusion in
a turbulent pipe flow.
Analogous to diffusive and field charging, chapter 2, where
equations (2.9) and (2.12) were added,equations (5.16) and (8.4)
could be combined to one - the error being very small because of
theabove-mentioned gap in the collection efficiency. The resulting
equation would become
( )
+
=+ VAVwk
exp1 depnpDT & (8.5)
The influence of the diffusional deposition, however, is
hereafter neglected since, as seen in figure16, it is significant
in a particle size range with very little practical interest to the
paper industry.
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36
9. Particle agglomeration
The phenomena studied so far in this text are all collecting
particles from the gas flow regardless ofparticle concentration.
For particle agglomeration (or coagulation) this is different as
agglomera-tion occures due to collisions between particles. No mass
is actually separated from the gas flowdue to agglomeration, but
the number of particles is reduced.
The theories for particle agglomeration assume that when two
particles collide with one anotherdue to a relative motion between
them, they attach to each other and form larger particles. The
sizeof the particle growing with the collision, is not affected
that much, but for the other particle thechange is much more
dramatic it vanishes completely. The result is a decrease in number
concen-tration, and as the agglomeration is more significant for
small particles, the number size distribu-tion is also displaced
towards larger particles.
The mechanisms for particle collisions are various, and some
idealistic models can be found in thepertinant literature. When the
relative motion between particles is due to Brownian motion
theprocess is called thermal agglomeration (or coagulation),
whereas when external forces are in-volved, the process is called
kinetic agglomeration. Such forces are gravity, electrical forces
orturbulent eddies.
The simplest case is thermal agglomeration of monodisperse
spherical aerosols. The theory is basedon how particles diffuse to
the surface of an observed particle. The rate of collisions, which
is equalto the rate of change in number concentration, can be given
by (Hinds 1982):
2p NDd4dt
dNpi= (9.1)
where D is the diffusion coefficient for particles (equation
8.1). If the change in number concentra-tion with time due to
agglomeration is to be described, the following equation is
obtained by inte-grating equation (9.1):
tDd4N1N)t(N
p0
0pi+
= (9.2)
where N0 is the number concentration at t=0. The diffusion
coefficient D decreases rapidly withincreasing particle size.
Therefore N(t) will differ from N0 only for small particles. The
same can besaid when the time period is short. As an example, the
net result of agglomeration can be neglectedover a 10-minute period
if the number concentration is less than 106/cm. Since the
residence timein the studied air cleaner is given in terms of
seconds rather than minutes, and since the numberconcentration
reaches 106/cm only for the smallest particles, the influence of
(thermal) agglomera-tion (of monodisperse spherical aerosols) can
be neglected in the paper industry. The curve Num-ber distr out in
figure 17 represents equation (9.2) with t=5 seconds.
For polydisperse aerosols the rate of agglomeration is
significantly higher. Small particles having a
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37
large diffusion coefficient D diffuse well to larger particles,
having a higher surface area. Thusagglomeration between particles
of different size will affect the number size distribution faster.
Thebigger difference in size, the faster the agglomeration will
proceed. For the mass size distributionshown in figure 17, which
should represent a typical distribution curve for paper converting
dust,having a total mass concentration of 21 mg/m, the increased
agglomeration for polydisperse aero-sols will not significantly
change the number size distribution curve coming out of the
system.
1.E-10
1.E-08
1.E-06
1.E-04
1.E-02
1.E+00
1.E+02
1.E+04
1.E+06
1.E+08
0.001 0.01 0.1 1 10 100 1000Particle diameter [m]
Nu
mbe
r di
stri
butio
n [1
/cm
]
0.0
0.5
1.0
1.5
2.0
2.5
Mass
di
stri
butio
n [m
g/m
]
Number distr in Number distr out Mass distr
Figure 17: Mass and number distributions for an aerosol (rp=1000
kg/m, spherical
particles). The total mass concentration in this example is 21
mg/m.
When observing an airflow, there will always be velocity
gradients which result in different veloci-ties for particles.
Equations can be found for both laminar and turbulent flows, only
the latter beingof practical interest. As the turbulent eddies
result in significant velocity gradients, the followingequation is
given:
=
t
3g
2p
dUf2
D64db
ionagglomeratThermalionagglomeratTurbulent
pi (9.3)
where b is a constant of order 10, U is the average velocity in
the duct with the diameter dt and f isthe fanning friction factor
(equation 5.2). Equation (9.3) gives the ratio of turbulent
agglomerationto thermal agglomeration. With U =15 m/s and dt = 0.5
m the ratio becomes one (i.e. turbulentagglomeration exceeds
thermal agglomeration) for approximately 0.7 m, and increases
rapidlywith increasing particle size. However, as the number
concentration decreases rapidly with increas-ing particle size, the
conclusion drawn from figure 17 will not change.
For the agglomeration of charged particles, particles of
opposite charges would attract each otherand form agglomerates
(clusters) more easily. The net result is a minor change compared
to theagglomeration of uncharged particles, as the effect is
balanced by collisions between particles of
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38
same sign repelling each other.
In an electrostatic air cleaner the particles all have a
negative (or positive) charge (possibly formingdipoles) and
therefore the net result should be, not just equal to agglomeration
of uncharged par-ticles, but even lower.
The conclusion is, for the air cleaner studied in this project
with paper dust, that the number ofparticles is too low for
significant agglometation, no matter which theory for agglomeration
isconsidered. The conclusion is the same even when the particles
are charged.
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39
10. Properties of dust in the paper industry
The different mechanisms for particle collection described in
earlier chapters, all to some extent,include properties of the
particles, and the material of the particles. The particle size,
density, di-electric constant and particle shape for individual
particles must be known and since the collectionefficiency depends
heavily on particle size, the particle size distribution function
for the processeddust must be known in order to properly estimate
the final (mass) collection efficiency of the plannedair cleaning
system.
Some of the properties are described here in brief, but the
shape factor and some aspects concerningthat are presented in a
later chapter. The properties presented are based on the Masters
thesisundertaken within the project (Dahlbacka 1999).
Particle size distribution can be split up in three separate
fractions; the particles with the smallestdiameter are referred to
as the nucleous fraction, the next is called the accumulation
fraction and thelargest particles are defined as the coarse
fraction. Whether the sizes of particles match any defini-tion for
the names used is irrelevant. The names are just used to indicate
which fraction is beingdiscussed.
Since the physical properties of the particles vary for the
different fractions, the collection effi-ciency as a function of
particle size will also be different for the different fractions.
The threedifferent fractions have to be treated separately and if a
final curve giving the collection efficiencyas a function of
particle size for the whole particle distribution is desired it
can, for example, becalculated from the difference between input
and output dust. This might lead to some unlinearitiesin the
collection efficiency curve.
The nucleous and accumulation fractions are treated as ideal
spheres, the coarse fraction as fibersdescribed as cylinders
(Dahlbacka 1999). A study was made in order to determine the
particle sizedistribution function for dust from different paper
qualities in different paper production or conver-sion processes.
The qualities were coated paper, newsprint and tissue paper. The
relative size distri-butions are shown in figure 18.
For the different distributions the following values are used
(table 3), based on the study mentionedand literature:
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40
Table 3 : Particle properties.
Nucle
ous
Accu
mula
tion
Coar
se
r 4 6 2.5 -Tissue paper
p 640 950 800 kg/mk 0.0085 0.261 0.731 - 0.4289 10.9 174.8 m 1.7
2.51 76.9 m
Newsprint p 640 850 800 kg/mk 0.0278 0.255 0.717 - 0.4456 6.55
174.8 m 1.79 2.34 76.9 m
Coated paper p 640 790 800 kg/mk 0.0818 0.567 0.351 - 0.485 6.27
174.8 m 1.81 2.21 76.9 m
Relative particle size distribution for different paper
qualities
0.0 %
2.0 %
4.0 %
6.0 %
8.0 %
10.0 %
12.0 %
14.0 %
16.0 %
18.0 %
0.1 1 10 100 1000Particle diameter [m]
Fra
ctio
n [%
]
Tissue paper Coated paper Newsprint
Figure 18: Relative particle size distribution for different
paper qualities.