Environmental Protection Technology Series
iIGN OF MINIMUM-WEIGHT DIFFUSION
BATTERIES
Industrial Environmental Research Laboratory
Office of Research and Development
U.S. Environmental Protection Agency
Research Triangle Park, North Carolina ffl\\
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U.S. Environmental Protection Age-ncy, have been grouped into
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This report has been assigned to the ENVIRONMENTAL PROTECTION
TECHNOLOGY series. This series describes research performed
to develop and demonstrate instrumentation, equipment and
methodology to repair or prevent environmental degradation from
point and non-point sources of pollution. This work provides the
new or improved technology required for the control and treatment
of pollution sources to meet environmental quality standards.
I
EPA REVIEW NOTICE
This report has been Ire viewed by the U.S. Environmental Protection
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This document is available to the public through the National
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EPA-600/2-77-001
January 1977
DESIGN
OF MINIMUM-WEIGHT
DIFFUSION BATTERIES
by
A. L. Marcum, L.E. Dresher. A. Wojtowicz, and W. H. Hedley
Monsanto Research Corporation
Dayton Laboratory
Dayton, Ohio 45407
Contract No. 68-02-1320, Task 15
ROAP No. 21ADL-034
Program Element No. 1AB012
EPA Task Officer: Geddes H. Ramsey
Industrial Environmental Research Laboratory
Office of Energy, Minerals, and Industry
Research Triangle Park, NC 27711
Prepared for
U.S. ENVIRONMENTAL PROTECTION AGENCY
Office of Research and Development
Washington, DC 20460
-------
SUMMARY
The relationship between functional requirements and the dimen-
sions of rectangular channel diffusion batteries was analyzed to
determine the optimum configuration with respect to weight and
volume.
For a given penetration and minimum gas flow, the minimum weight
and volume of a diffusion battery are determined by the thick-
ness and spacing of the plates which form the walls of the
channels. The number of batteries required to cover a given
range of particle sizes is determined by the spread between the
minimum and maximum available gas flow rates.
The extent to which battery weight can be reduced is ultimately
limited by the stiffness of the plate material and by manufac-
turing tolerances.
ii
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TABLE OF CONTENTS
Section
Summary
Figures
1
3
4
Introduction
Technical Discussion
2.1 Theory of Operations
2.2 Core Volume and Weight
2.3 Total Battery Weight
2.4 Effect of Reynolds Number and
Pressure Drop Limitations
2.5 Residence Time as a Function of
Battery Geometry
2.6 Number of Units Required
Conclusions
References
Appendix - Diffusion Battery Design
Page
11
iv
1
3
3
9
11
13
15
16
17
19
21
iii
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FIGURES
Number Pap.e
1 Typical Plot of Penetration vs y for constant
Mean Radius and Varying Standard Deviation of
Particle Radius. '5
2 Particle Radius vs Diffusion Coefficient 7
3 Frontal Area of Core 9
4 Area of Spacers 11
5 External Dimensions of Core and Internal
Dimensions of Shell 11
6 Relationship Between Core Weight and
Parameter VAin 23
7 Core Assembly 25
8 y vs V 27
CL
9 Penetration vs Flow for Unit 1 28
10 Penetration vs Flow for Unit 2 29
11 Penetration vs Flow for Unit 3 30
12 Penetration vs Flow for Unit 4 31
iv
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1. INTRODUCTION
A widely used technique for determining the size of particles
in aerosols makes use of the mechanism of molecular diffusion
to change the particle concentration in the aerosol by an amount
that is proportional to particle size. The separation is carried
out in an array of small bore tubes or channels, collectively
called a diffusion battery, through which the aerosol is drawn
at velocities in the viscous flow range.
As the particles are carried through the battery, their random
E^rownian motions displace some of them to the channel walls
where they are held by Van der Waals and other adhesive forces.
Since the magnitude of the Brownian motion is a function of
particle size, the rate at which particles strike the wall and
are removed from the stream is also a function of their size.
The decrease in particle concentration in the aerosol as it
flows through the battery provides a basis for calculating the
diffusion coefficient of the particles, and their size.
Until recently, the measurement of particle sizes in aerosols
was largely a laboratory exercise. Currently, however, par-
ticulates in the atmosphere and in the industrial exhaust gases
are being monitored extensively in the field.
While the weight and volume of laboratory apparatus is seldom
of concern, field work is often seriously hampered by equipment
that is heavy or bulky.
-------
The diffusion batteries currently in use weigh in excess of
fifty pounds and are often the heaviest piece of equipment in
a test setup. For this reason, it was felt that the possibility
of optimizing the dimensions of the battery for minimum weight
should be investigated.
The objectives of this study were: first, to analyze the
relationship between the physical dimensions of the battery and
operational parameters to determine if an optimum configuration
exists, and second, to design a series of optimum weight batteries
based upon the results of the study.
The technical discussion which follows is limited to the first
of these objectives. The theoretical foundation of the dif-
fusion method has been extensively covered in the literature and
is not considered here in detail. The design calculations for
a specific series of batteries are included in the appendix.
-------
2. TECHNICAL DISCUSSION
2.1 Theory of Operations
The function of a diffusion battery is to decrease the particle
concentration In a sample stream by an amount which is propor-
tional to the diffusion coefficient of the particles. The
accuracy with which particle sizes can be determined by the
diffusion method depends to a considerable degree upon the
level to which modes of transport other than diffusion, primar-
ily turbulence and gravitational settling, can be reduced.
Thus, the operating range of the diffusion battery is limited
to the viscous flow region and to particles smaller than
lu diameter.
The number of particles per unit volume of gas leaving a diffusion
battery is related to the number entering by an equation of
the general form.
...(1)
n
o
in which n is the particle concentration leaving the battery,
n is the initial concentration, A and B are experimentally
O XX
determined constants, D is the diffusion coefficient, and y is
a function of battery dimensions and gas flow rate. The ratio
— Is called penetration.
o
If the values of the constants In this equation are known, the
diffusion coefficient, D, can be calculated from the known
flow rate, and the measured values of n and n . This applied
however, only to monodisperse aerosols. For polydisperse
-------
aerosols, the apparent diffusion coefficient varies as a function
of diopersity as the flow rate is changed. The relationship
between dispersity, flow rate, and penetration, provides a means
of determining the particle size distribution in real aerosols,
which are usually more or less polydisperse. The calculations
required in this method, however, are laborious, and if a large
number of tests are involved it is practical only if a computer
is available.
l'\johs et al (1) have proposed a semigraphical method of determining
size distribution directly from test data, which appears suita-
ble for data reduction in the field. Their method involves
matching curves of penetration (—) vs log y to pre-plotted
curves for various distributions and values of the mean particle
radius and standard deviation. For a given mean particle radius,
the standard deviations (a) give a family of curves which intersect
at — ^ .4 as shown in Figure 1.
no
To determine the distribution by this method, at least three
n
points on the — vs log y curve are required, with the middle
near — = 0.4 and the upper and lower points near 0.8 and 0.2
respectively.
The preplotted curves are calculated from a modification of an
equation derived by DeMarcus (2) for batteries having rectangular
channels:
IL. = .9149 e'1-885 Dy + .0592e~22-33 Dy + .026e~isi ^ ...(2)
-------
o
c
o
•H
-P
-P
OJ
0)
•r,
Log 1.885 y
Figure 1. Typical Plot of Penetration vs y for Constant
Mean Radius and Varying Standard Deviation
of Particle Radius.
-------
where D = kT
in which
k = Boltzmann Constant 1.38047 x 1CT16 erg/K
T = Absolute temperature °K
Z = Particle radius (cm)
y = 18.3 x 1CT5 gm/cm-sec is the viscosity of air at 23°C
and 760 mm hg
1 = .653 x 10~5 (cm) is the mean free path
A = 1.246
B = .042
b = .87
L_ ... (3)
and y =
L = Length of battery channel
b = h width of a channel
v = average gas velocity
A plot of D vs particle radius is given in Figure 2.
The parameter y, eq. (3) is the experimental variable which is
varied during a test by varying the average velocity v. Since
the range of particle sizes of interest is known, the range of
y values required can be calculated from equation 2, and from
this, the dimensions of the battery and flow rates required can be
-------
10
-7 r
o
-------
determined from equation 3. Generally the available flow rates,
both maximum and minimum, are also known so that the basic
dimensions of the battery core are fixed.
To convert (3) into a more useable form,
Let A = total flow area, cm2
a
b = s where w is the width of 1 channel (cm)
/cm3\
Q = total volume flow rate (^—}
\seci
then
N = -jr cm/sec
a
Substituting into 3,
i /sec_\
w2Q Icm
But LA = flow volume V
a
so
y = a /sec\ ...(5)
w2Q
lcm2J
From (5), for a fixed value of y, the value of Va varies
inversely with Q and w2. For a given channel width, w, and
minimum flow rate Q .
Va . ymaxSnin w2 (cm3) ...(6)
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2.2 Core Volume and Weight
The total volume of the battery core consists of the volume of
the channels and the volume of the plates which form the walls
of the channels. The weight of the core is simply the volume
of the plates times the density of the plate material. To
simplify the equations relating plate volume V to other parameters,
the number of plates is assumed to be equal to the number of
channels. Since the number of channels will be large for the
larger batteries, this approximation results in less than 1%
error for the size range in which weight is of concern.
Channel
1
I
/
I/
T
M
/
\
/
1
/
/
/
7
1
1
Plate /
/
/
/
/
/
k-tp
< W >
H
Figure 3. Frontal Area of Core
rom eq. (6)
a .
mm
r LA
a
Y Q .
max mm
^
y Q 4
max min
w
-------
From Fig. 3
fl YV1M1 • • • \ | /
Where N = number of channels
c
and
Va = WNCHL ...(8)
Vp = tpScHL ...(9)
Total volume, V , therefore is
(W + t )N HL ...(10)
^J \-s
And the core weight is
Wm = V p
Tc pMp
= t N HLp
pep
Solving (7) for N
\
Nc = wl ...(12)
A
But A = /ymaxSnin \ 2
a ( - rr— - ) wz
and N
c
/y Q . \
[''max min\ w
\ 5" J HL
10
-------
Substituting this value of N into (11)
WTc = tpw
max min\
—*—K
2. 3 Total Battery Weight
The total weight of the battery includes both the weight of the
core and the weight of the shell and end plates. To the basic
weight of the core must be added the weight of spacers which
separate the plates.
h
W
t
H
Lh
Figure i|. Area of Spacers
Figure 5. External Dimensions of Core and
Internal Dimensions of Shell
11
-------
Prom Figure >\, the combined area of plates and spacers in the area
of the spacers, is 2hw and the volume is 2 hwl. Since WL = V
Tc
H
V
s
H
and
From Eq. (10)
V • <» + V
Substituting into (15)
= [2h(w+tp)NGL]pp ...(16)
From (14)
,Q
min\ W
/ H
giving
WTs = [2h(w + tp)(ymaffiln) g]Pp ...(17)
Equation 17 indicates that the added weight due to the spacers
decreases as the height of the core increases. In practice, the
height of the core will be limited by the flatness of the plate
material and by the problem of uniformly distributing the
incoming flow over a long narrow face.
The weight of the external shell can be estimated using the
dimensions shown in Figure 4, as internal dimensions and
assuming flat end plates.
12
-------
W = 2 {(H'W + (H' + W)L} t p
ex s s
where H1 = (2h + H). The minimum weight in this case occurs
when H1 = W+L. These proportions, however, would not be practical
in a large battery for the reasons given above.
2.4 Effect of Reynolds Number and Pressure Drop Limitations
Reynolds Number
The upper limit of the Reynolds number for viscous flow between
parallel plates is approximately 2000. Below this limit, turbu-
lence is eventually damped out and laminar flow established.
Since turbulence is damped out more quickly at lower Reynolds
numbers, an upper limit of Re = 200 is generally used in diffusion
battery design. This also reduces the effect of entrance dis-
turbances which reduces the effective length of the battery by
creating a region of turbulence for a distance down stream from
the entrance.
The Reynolds number for viscous flow between parallel plates is
given by:
Re = wv (^J . . .(18)
where p is the density of the gas, and y is the viscosity.
Substituting ^— for v,
a
Re = jQ fB.\ . m .(19)
a
13
-------
Solving for Q,
A /Rep\
Q '
and
max w
A /Re y
a / max , (?Q)
From
YmaxQmin\ w2
a L
Substituting into (20)
,-. W / max min\ / emaxu\
Qmax = L\ 5 ) \—p ) .-.(21)
which is the maximum flow for a battery designed for the given
values of Ymax and Qmln, as limited by the allowable Reynolds
number.
Pressure Drop
For viscous flow between parallel plates the pressure drop, AP is
AP =
b2
Making the appropriate substitutions,
AP - ...(22)
-------
12yLQ
or AP =
w2Aa
Solving for
Q = WAa A?max
~ -
Substituting for A^ from (13)
3,
YmaxQmin w_^ 12y ...(25)
ymax 5 T 5 AP
L 2 max
In general, Reynolds number and pressure drop limits are of
concern only in the smallest batteries, that is, for low values
of y, and maximum flow rates.
2.5 Residence Time as a Function of Battery Geometry
Residence time is of interest since it determines the minimum
length of time required for a test run at some value of flow rate,
It can be shown that residence time, for a given value of y, is a
function only of channel width.
From equation (5)
Y =
W2Q
but V = T, is the time required for a particle to pass through
Q~
the battery, or residence time, therefore
w'2
...(26)
15
-------
2.6 Number of Units Required
Let YI = Ymax> Y2 = Ymin
Qi = Qmln, Q2 = Qmax
Then, from Equation (5)
4 LA
w 2
= Y2Q2= Constant
Log Y! + Log Qx = Log Y2 + Log Q2
Log Y!- Log Y2 = Log Q2 - Log Q],
ALOE Y!_2 = Log(|f)
If Qi and Q2 are constant for the full range of Y, then ALog Yt 2
is the same for all units and
Nu = L°S Ymax ~ Log Ymin
ALog Yi-2
Y
T , max
Log
Y
min/ --.(27)
, max\
Log ( )
7
16
-------
3. CONCLUSION?
The foregoing analysis indicates that when the limits of the
parameter y, and the minimum flow rate have been established,
the minimum weight and volume of the battery are determined
primarily by the thickness and density of the plate material and
the width of the flow channels. The thinner the plates., and the
narrower the channels, the lighter the battery. This is true,
in particular, of the largest, batteries in which the weight of
the plates is large relative to the weight of the external shell.
In these cases, so long as the required flow volume is maintained,
the shape of the volume has relatively little effect upon the
total weight of the battery.
Depending upon the /spread between maximum and minimum flow rates,
battery weight decreases rapidly as the required y value
decreases. In practical terms, only the largest battery in a
series is of concern from a weight or volume standpoint.
The maximum weight of a battery, therefore, depends primarily
upon the properties of the plate material and the tolerances
which can be maintained upon channel width during manufacture.
The practical minimum to which plate thickness can be carried is set
by the degree of flatness which can be maintained in the plates.
as installed. This is essentially a function of the height of
the plates and their initial flatness, if it is assumed that the
plates are thick enough to support themselves without buckling.
17
-------
The minimum channel width for a particular application depends
upon the inherent waviness of the plate material and the antici-
pated assembly tolerances. In general, waviness is more diffi-
cult to account for and, in most cases, will be the limiting
factor. Additional leveling operations will probably be required
on sheet material to be used for plates at channel widths below
one millimeter.
18
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4. REFERENCES
1. Fuchs, N. A., I. B. Stechkina, and V. I. Starosselkii, 1962
Brit. J. Appl. Phys. Vol. 13.
2. DeMarcus, W. C. 1952
U.S. Atomic Energy Comm. ORNL 1413.
19
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APPENDIX'
DIFFUSION BATTERY DESIGN.
Input Variables
A series of batteries covering the following range of variables
is required.
1. Particle size .3v dia. to .Oly dia.
2. Penetration n _ 2 to — = .8
no
3. Flow range 1 liter/min. (16.6? cmVsec) to
4.5 liter/min. (75 cmVsec)
4. Maximum Pressure Drop 2500 dyne/cm2
5. Maximum weight 50 pounds
Calculations
The range of y required to give the desired penetration can be
calculated from equation (2). Neglecting the second and third
terms, which are small relative to the first.
S- = .9149 e-
o
Solving for y,
y =
-1.885D
The maximum value of y occurs when — = .2 and the particle
diameter is . 3u. From Figure 2, for°r = .15, D = 1.25xlO-6
21
-------
Substituting in the above equation,
I"/ -2 \
y = V.9199/
(-1.885) (1.25x10-6)
And for the minimum value of y, — = .8, D = 5.3X10-1*
o
(-1.885X5.3x10-"
V--L . t
= 134 sec/cm2
From equation (6)
V n
amin
.43 x 10"5 x 16.67\ 2
11 I W
= (26.88 x 105)w2 cm3
And, from (14)
WTc = (tpw) (Zsax^l) ^ gm
WTc = (26.88 x 10~5) (tpw Pp) gm
Figure 6 showSgthe relationship between core weight and the
oarameter max mln
K I* for various values of (tpw) and aluminum
plates.
Assuming a maximum weight of 40 pounds for the core, as defined
by equation (1^) and aluminum plates,
(tpw)
(26.88xlO~5) (2.7D
= 2.487 x 10"3 cm2
22
-------
100,000
10,000 -
11,000 -
o
EH
1,100 -
10 102 103 101* 105 106 107
max min
(cm)
A
Figure 6. Relationship between Core Weight and Parameter max mln
23
-------
If tp = W,
W = (2.48? x 10~3K = tp
.0499cm (.0196 in)
It is assumed that the spacers will be bonded to the plates.
Bond line thickness will be approximately .001 in (.0025 cm)
Spacers of .016 (.0406 cm) aluminum with two bond lines give,
channels .0457 cm wide,
The active volume then becomes
V = (26.88 x 10 ) (0.457) cm3
CL
= 5614 cm3
Assuming a length of 50 cm,
A, . (5611) - 112 o,
To maintain channel width within acceptable limits, using-
commercially available aluminum sheet, the height of the
channels will be limited to 2.5 inches (6.35 cm) giving
an area of 0.290 cm2 per channel, and a total of
N = "MHln = 386 channels
c
With the spacer arrangement as shown, the weight of the
core is 40.4 pounds . (See Figure 7.)
24
-------
t
125
375
typ
X
\
7
6.57m >
plates
Figure 7. Core Assembly
-------
If the external shell is made of .125 aluminum plate, the
weight of the shell will be approximately 8 pounds giving a
total weight of 48.4 pounds exclusive of accessories such
as tube fillings and handle.
Number of Units Required
Using equation (28)
. 6.45xl05
l_iX 1 -»~o~ J t
N = = 6 units
Ln 16?67
However, it is not necessary that the range of y be completely
covered. In this case 4 units appear to provide sufficient
coverage. The relationship of the individual units to y and V is
a
shown in Figure 8.
Figures 9 through 12 show the relationship between flow rate
and penetration for various particle diameters for each
unit, and can be used to determine the flow rate for a desired
penetration.
26
-------
103
102
CO
E
o
10
10
102
/
7
>^ -/ Unit #1
Q=16.6?
J.
_L
103 104
y (sec/cm2)
cmJ
sec
106
Figure 8 . y vs V
27
-------
c
c
o
•H
-P
ctf
1.0
.9
.8
.7
.6
.5
.3
.2
.1
0
_L
2 3
Flow (liters/minute)
r = .ly
r = ,05y
Figure 9. Penetration vs Flow for Unit 1
28
-------
o
c
1.0
.9
.8
.7
.6
fl
o
•H
-P
Ctf
-P
0)
0
.3
.2
.1
0
r =
r = .05u
1234
Plow (liters/minute)
Figure 10. Pentration vs Plow for Unit 2
29
-------
1.0
.9
.8
c
£
f\
O
•P
«J
0)
P-!
.7
.6
.2
.1
0
r = .ly
r = .05w
r = .Oly
r = .005u
01 2 3 4
Plow (liters/minute)
Figure 11. Penetration vs Plow for Unit 3
30
-------
1.0 -
.9 .
.8 -
.7
.6
C c
o • ->
•H
4^
O)
C
0)
PL,
.3
.2
.1
0
r = .05
= .005n
1.2 3 4
Plow (liters/minute)
Figure 12. Penetration vs Plow for Unit 4
31
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TECHNICAL REPORT DATA
' Inttntctioni on the rtvene btfort completing)
(Please read /ntlruclloni on
1. REPORT NO.
EPA-600/2-77-001
2.
3. RECIPIENT'S ACCESSION-NO.
4. TITLE AND SUBTITLE
Design of Minimum-Weight Diffusion Batteries
6. REPORT DATE
January 1977
6. PERFORMING ORGANIZATION CODE
7. AUTHOR(S)
A.L.Marcum, L.E. Dresner, A. Wojtowicz, and
W.H. Hedley
8. PERFORMING ORGANIZATION REPORT NO.
MRC-DA-452
9. PERFORMING ORGANIZATION NAME AND ADDRESS
Monsanto Research Corporation
Dayton Laboratory
Dayton, Ohio 45407
10. PROGRAM ELEMENT NO.
1AB012; ROAP 21ADL-034
11. CONTRACT/GRANT NO.
68-02-1320, Task 15
12. SPONSORING AGENCY NAME ANO ADDRESS
EPA, Office of Research and Development
Industrial Environmental Research Laboratory
Research Triangle Park, NC 27711
13. TYPE OF REPORT AND PERIOD COVERED
Task Final; 9/73-9/74
14. SPONSORING AGENCY CODE
EPA-ORD
is.SUPPLEMENTARY NOTES J.ERL-RTP task officer for this report is G.H. Ramsey, 919/549-
8411 Ext 2298, Mail Drop 61.
16. ABSTRACT.
The report gives results of an analysis of the relationship between functional
requirements and the dimensions of rectangular-channel diffusion batteries, to deter-
mine the optimum configuration with respect to weight and volume. For a given
penetration and minimum gas flow, the minimum weight and volume of a diffusion
battery are determined by the thickness and spacing of the plates which form the
walls of the channels. The number of batteries required to cover a given range of
particle sizes is determined by the spread between the minimum and maxiumum
available gas flow rates. The extent to which battery weight can be reduced is
ultimately limited by both plate material stiffness and manufacturing tolerances.
17.
KEY WORDS AND DOCUMENT ANALYSIS
DESCRIPTORS
b.lOENTIFIERS/OPEN ENDED TERMS
c. COS AT I Field/Group
Air Pollution
Size Determination
Measurement
Diffusion
Design
Aerosols
Flue Gases
Air Pollution Control
Stationary Sources
Diffusion Batteries
Particulates
13 B
14B
07D
2 IB
13. DISTRIBUTION STATEMENT
Unlimited
19. SECURITY CLASS (This Report)
Unclassified
21. NO. OF PAGES
34
20. SECURITY CLASS (This page)
Unclassified
22. PRICE
EPA Form 2220-1 (9-73)
32
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