EPA-600/3-77-043
May 1977
Ecological Research Series
WALL RE
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EPA-600/3-77-043
May 1977
EFFICIENCY OF GAS-WALL REACTIONS IN A
CYLINDRICAL FLOW REACTOR
by
Henry S. Judeikis and Seymour Siegel
The Aerospace Corporation
El Segundo, California 90245
EPA Grant No. 802687
Project Officer
Jack L. Durham
Atmospheric Chemistry and Physics Division
Environmental Sciences Research Laboratory
Research Triangle Park, North Carolina 27711
ENVIRONMENTAL SCIENCES RESEARCH LABORATORY
OFFICE OF RESEARCH AND DEVELOPMENT
U.S. ENVIRONMENTAL PROTECTION AGENCY
RESEARCH TRIANGLE PARK, NORTH CAROLINA 27711
-------
DISCLAIMER
This report has been reviewed by the Environmental Sciences Research
Laboratory, U.S. Environmental Protection Agency, and approved for publication.
Approval does not signify that the contents, necessarily reflect the views and
policies of the U.S. Environmental Protection Agency, nor does mention of
trade names or commercial products constitute endorsement or recommendation
for use,
ii
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ABSTRACT
Expressions are given for the concentration of a dilute reactive gas
mixed with an inert carrier gas as a function of the radial and longitudinal
distances in a cylindrical reactor and the reaction efficiency. The reaction
efficiency is defined as the fraction of gas-wall collisions that leads to
the disappearance of the reactive gas from the gas phase. The solutions pre-
sented here are applicable for all values of reaction efficiencies and extend
earlier work by other investigators that was applicable only for values of 1
or near zero. In addition to the solution of Pick's second law of diffusion
as applied here (with an additional term for flow in the cylinder), a one-
dimensional random walk analysis is also applied to this problem. The com-
bination of diffusion equation solutions and the random walk analysis leads
to the conclusion that, for a given set of experimental conditions, the reac-
tion efficiency can be uniquely determined only if it lies within a certain
range of values. Small values of the reaction efficiency will produce insuf-
ficient reaction and large values will yield diffusion-limited results. The
sensitive range for the reaction efficiency can be changed by appropriate
adjustment of various experimental parameters. However, adjustment of the
total pressure gives the greatest effect and is generally one of the easiest
parameters to vary in an experiment.
This report was submitted in fulfillment of Grant No. 801340 by The
Aerospace Corporation under the sponsorship of the U.S. Environmental
Protection Agency. This report covers the period June 1971, to January 1973,
and work was completed as of January 1973.
iii
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SECTION 1
INTRODUCTION
In a recent study (1), we attempted to establish the efficiency of NO -
wall reactions in a cylindrical flow reactor. In particular, we tried to
measure the reaction efficiency , defined as the fraction of NO -wall col-
lisions that leads to the removal of NO from the gas phase (via adsorption
or reaction). The general case of gas-wall reactions has been treated pre-
viously by Paneth and Herzfeld (2) and Wise and Ablow (3) for « 1 and <}> =1.
(Related solutions have also been given by Jost (4) and Carslaw and Jaeger
(5)). The intermediate case ( § comparable to unity but not necessarily equal
to unity), which was of interest in our work, was not treated by these au-
thors.
In this report, we derive a general solution applicable for all values
of cf>. This solution becomes identical to the published results for the lim-
iting cases of « 1 and =1. In addition, we perform a one-dimensional
random walk analysis of gas-wall reactions, which, in conjunction with the
diffusion equation solution, gives a clear physical picture of the collision-
reaction process. These solutions are applied to our specific case of inter-
est (N09~wall reactions) as an example.
The utility of these solutions, however, is wide spread. They can be
used to obtain wall reaction rate constants for many reactions since the in-
formation content of fitting experimental results to diffusion-limited kinet-
ic expressions is limited. In many cases, such as heterogeneous catalysis
and surface-initiated reactions, the specific information desired is lost in
attempts to analyze experimental results in terms of limiting cases of wall
collision efficiencies.
-------
SECTION 2
GENERAL SOLUTION FOR FLOW, DIFFUSION AND WALL
REACTIONS IN A CYLINDRICAL REACTOR
We consider a system in which the cylindrical walls (but not the ends)
of a tubular flow reactor are coated with catalyst. We assume that C, the
reacting gas, disappears via a heterogeneous first-order process (adsorption
or reaction) that occurs on the walls with an efficiency <}>. If the reacting
gas is introduced into the system as a dilute component in a large concentra-
tion of inert gas, we can assume that the total pressure P and therefore D,
the diffusion coefficient of C in the mixture, are approximately constant
during the course of reaction. Fick's second law of diffusion with an addi-
tional term for flow in the direction of the cylinder axis (under steady-state
conditions) is (2-4)
1 *\f^ *\2r« I *\ri
= 0 (1)
where r = radial cylindrical coordinate
x = longitudinal cylindrical coordinate
v = the linear flow rate in the x direction (equal to F/A, the
volume flow rate divided by the cross-sectional area of the
cylinder)
The following boundary conditions are applicable to the solution of
Eq. 1 for the case of interest here
C (r, x=0) = C
o
|| (r=0, x) = 0 (2)
D|£ (r=R, x) = 4> k C-
O(j L
-------
where C = the initial concentration of C
o
R = the cylinder radius
k = the average velocity of the reacting molecule in the
radial direction
r *
where k = the Boltzmann constant
T = the absolute temperature
m = the mass of the reacting gass molecule (6)]
C' = the concentration of C one mean free path away from the wall
The last boundary condition in Eq. 2 arises from our assumption that a frac-
tion of reactant molecules <|> is removed from the gas phase on collision
with the walls via adsorption or reaction. Alternatively, we might con-
sider a reaction scheme involving simple first-order adsorption, desorp-
tion, and reaction. Then the last boundary condition in Eq. 2 would be
replaced by
D i£ - k C' - k.C = Y (1 ~ f) k C' - k.C
•JQ a da r da
where k = the rate constant for adsorption
3.
k = the rate constand for desorption
C = the concentration of adsorbed C
a
Y = the sticking coefficient
f = the fractional surface coverage
If we assume a steady state for C , we may write
3,
dCa = Y(i - f)K C' - kJC - k C =0
j. r d a s a
dt
where k C = the rate of surface reaction of adsorbed C.
S 3,
Solving the latter equation for C and substituting the result into the
3
former equation we find
D _8C = Y(! - f)
-------
Comparison of this expression with the last boundary condition in Eq. 2
indicates that for the case discussed here,
yd - ^' k
Paneth and Herzfeld (2) and later Wise and Ablow (3) assumed C(R) ~ C'
and obtained the solution of Eq. 1 on the basis of the boundary conditions
in Eqs. 2. This assumption, however, is valid only for <|> « 1. In general,
C(R) = (1 - )C'. If we apply this assumption and the boundary conditions
in Eqs. 2, the solution of Eq. 1, following the methods described by other
workers (2-5), is
oo f
£ _ V"^ o PiR [exp(a x) ] (3)
where
.
C -t P,(l + 6* p
J and J, are Bessel functions of the first kind, p. is the ith root of the
o 1 i
equation
and
2 AD ~ 2 AD T R . (6)
The application of the limiting cases for $ « I or = 1 to Eq. 3 gives the
results previously reported (2-4). (Alternatively, the application of these
conditions to the boundary conditions in Eqs. 2 gives the same boundary
conditions specified by the previous authors for « 1 or = 1.)
P and D are related by (7)
3000 kT KTV/2 (7)
8TrdzP 2y
-------
where y is the reduced mass of the reacting and inert gas molecules, and d =
(d + d )/2. where d and d are the molecular diameters of the reacting
m n m n
and inert gas molecules treated as hard spheres.
As an example, let us apply Eq.3 to the case of our experiments with
N09-wall reactions in a cylindrical flow reactor. Nitrogen was used as the
inert carrier gas with conditions such that P /PNQ > 50Q^ Thlg fulfilled
the requirement that P and therefore D be approximately constant. The hard
sphere diameters of N0? and N at 298 K, which is the usual temperature
for our experiments, are 0.444 and 0.371 nm, respectively (6). With these
values and typical values for R, F, and P, we calculated the concentration
distribution of C from Eq.3 for selected values of $. Some calculated
concentration profiles are shown in Figure 1. We also plotted C/C as a
function of x at a given value of r. This is shown in Figure 2 for r = 0
and the same parameters as in Figure 1 as well as additional values of .
In our N09 experiments, we determined the average value of the N0«
concentration at r ~ 0 by measuring the decrease in intensity (due to
optical absorption by NO at 475 nm) of a narrow beam of light passing
through the center of the cylinder (8). Calculated values corresponding
to the average N0? concentration at r = 0 may be obtained by integrating
the curves in Figure 2 over the length of the cylindrical reactor, or by
setting r = 0 in Eq.2 and integrating. In the latter case,
/-C-\ -^0 "fe" r=OdX 2 ^o^ ^^i^ " l
\ o / r=0 A a 1=1 a.H p . (1 + 5 2p .2 ) J (p .)
dx
(8)
where £ is the length of the cylinder (~ 40 cm in our experiments).
In Table 1 ^C/C? is calculated for selected values of T, R, F,
and H versus P and . Note that for large values of , \3/C/ _„ approaches
a constant value because the reaction becomes diffusion-limited under such
conditions. (Note also that the limiting value of C/C __ becomes
independent of pressure below ^10 torr -*- because at a very low pressure,
convert from torr to newton/meter 2, multiply by 1.333 22 E+02.
-------
D » F, and the limiting value of \C/CJ _0 depends only on R/l, or R only
for cylinders sufficiently long that C(l)-H).) Conversely, for low values of
, C/C _ approaches unity. Significant variations in\3/C ^ __ occur
only for intermediate values of ; the exace range is dependent on the total
pressure. Of course these variations would also be observed for \C/C } ,n
» o'rfO
and for C/C (averaged over r) as a function of x.
-------
0
1 1
III ill
111 1 1
1 1
0
10
cm
Figure 1. Concentration Contours Calculated from eq. 3 for
Selected Experimental Parameters and = 1. P = 700 torr
(D = 0.15 cm2/sec), T = 298°K (kr = 9.3 x 103 cm/sec), R = 2 cm,
o L
and F = 10 cmj/sec.
-------
Figure 2. C/CO at r = 0 Calculated from Eq. 3 for Selected Values
of P, T, R, and F and Various Values of . P = torr (D = 0.15
cm2/sec), T = 298°K (k = 9.3 x 103 cm/sec), R = 2 cm, and
- « "5 i J-"
F = 10 cm-Vsec.
-------
P,
torr
700
100
10
1
0. 1
0. 01
0. 001
1
0. 138
0.038
0.028
0.027
0. 027
0.027
0. 027
TABLE 1.
lo-1
0. 138
0.038
0.028
0.028
0. 038
0.085
0.249
C/C
o
o
1C'2
0. 138
0.038
0.029
0.038
0.089
0.260
0. 584
n vs P
r=0
for
t> -
ID'4
0. 159
0.064
0. 101
0.271
0. 589
0. 834
0. 943
io-5
0.341
0.259
0. 366
0.620
0.839
0. 944
0. 982
ID'6
0.828
0.806
0.821
0.883
0. 949
0.982
0. 994
Calculated from Eq. 8 assuming the values given in Figure 1 for P, T, R,
and F, and £ = 40 cm.
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SECTION 3
RANDOM WALK ANALYSIS OF GAS-WALL REACTIONS
We consider a one-dimensional problem in which a molecule can move toward
or away from a wall in equal steps and with equal probability (except at the
wall, where the probability of reflection is unity). We assume that the mole-
cule starts at a distance of R -r cm, or n steps away from the wall (n > o),
o o o —
and takes n steps. Gnedendo (9) has calculated the probability that the mole-
cule will end up at a distance R - r,. cm, or n steps from the wall, after
taking n steps. This probability is equal to the sum of two probabilities,
the probability of the molecule arriving at R - rf after n steps, plus the
probability of the molecule arriving at R + rf if there were no reflecting
barrier at R, Here, however, we desire to calculate the probability (pB"> that
a molecule starting at R - r cm, or n steps from the wall, will collide with
o o
the wall 6 times during the n steps.
We obtain the following solution by induction
" (n-Q) 1 (L + M )!1 (L. - M.)l
i 7 i i i i i
The forms of Q, L., and M. depend on the values of n , R, and n + n . The
11 o o
various solutions are tabulated in Table 2. Two points should be made regard-
ing the use of Eq. 9. For given values of n and n , there is a maximum number
of collisions (6 ) that can occur with the wall. The value of g can be
max max
calculated from
3 =1 (n - n ) + 1 3+ (-1) (n~V + A (10)
max "2 ° ~9
where A * 0 for n > 0, and A = -2 for n =0. In addition, the use of ^q. 9
o o
is subject to the condition the pg = 0 if M^ > L^. Otherwise the equation is
10
-------
TABLE 2. SOLUTIONS TO RANDOM WALK PROBLEM
n + n n
o o
odd o
odd
o
L>O
even o
even
o
>0
P3 Q L: M:
>1 P + 1 n-p-1 P
and 1'
or
and si
P n-p n+p-1
>1 P n - p p
and 1
or
and 2 1
P-l n-p + 1 n+p-1
Except for a few cases (n = o or 1), the probability that a molecule will not
Pmax
collide with the wall, can only be calculated from p = 1 - / p_, where
P = l
P is the maximum number of collisions possible for a given n and n .
Kmax v 6 o
12
-------
We now apply this analysis to the example in Figure 2. We assume that
the time required for the molecule to take one step is T =X/k , where X is the
S
mean free path, and k is the average velocity in space. By using the expres-
S
sions for k (= 4 k ) and the mean free path of one gas in a two-component mix-
S r -10
ture, we obtain T= 1.0 x 10 sec for the example given in Figure 2. We also
assume that the length of the step (in the radial direction) is equal to X x k
divided by the average velocity in space, or 1/4X. The latter value is equal
to 1.3 x 10~ cm for the example given in Figure 2. In addition, n = 2/(1.3
x 10 ) = 1.54 x 10 for a molecule starting at the center of the cylinder.
In Figure 2, the linear velocity v down the cylinder is 0.80 cm/sec. Thus,
in the 1.25 sec required for the molecule to traverse 1 cm in the x direction,
11 take 1.25/T = 1.25 x 1010 randoi
cm length in the radial direction.
it will take 1.25/T = 1.25 x 10 random steps (except at the wall) of 1.3 x
Results of calculations for t = 1.25 sec (n = 1.25 X 1010) versus
R - r or n and 8 are given in Table 3. Only 11% of the molecules will
collide with the wall during this interval , and those that do collide with
the wall do so many times (10^ < 3 < 10 ). Results of calculations for the
same example integrated over r for various values of t or x are given in
Table 4. Here again, those molecules that collide with the wall do so many
times. Of course, the total fraction of molecules that collide with the wall
decreases as t and x decrease. In addition, the peak in the distribution of
molecules that collide with the wall shifts to lower values of 3 as t and x
decrease.
2 24 -1
The total number of gas-wall collisions for this case is a.2.2 X 10 sec
compared with 2.4 X 10^ sec~l calculated from simple kinetic theory(5)
13
-------
TABLE 3. MOLECULE - WALL COLLISIONS vs r
R -r
o'
1.95 -
1.90 -
1.84 -
1.79 -
1.67 -
<1.67
cm
2.00
t.SS
1.90
1.84
1.79
Fraction of
Molecules
in Interval
0.05
0.05
0.05
0.05
0. 10
0.70
Fraction ol Molecules In Interval With
n
o
1.94X 104
5.83 X 104
9.95X 104
1.41 X 105
2.06X 105
>2.06 X 105
P = 0
0,139
0,401
0,625
0,792
0.929
>0,980
1105
0.291
0.159
0.077
0.016
0.007
<0.001
0-2
I. 00
0.890
0.000
0.000
0.001
0.011
0.070
'Calculated for the example in Figure 2 with t = 1.25 sec, the time required for the average molecule to
traverse 1 cm in the x direction.
0.028
14
-------
a
TABLE 4. MOLECULE - WALL COLLISIONS vs t or x
Fraction of Molecules With
tb, x, •
sec cm B = 0 Kp<10 10<(3<102 102
-------
SECTION 4
DISCUSSION
The results obtained from the solution of the differential equation for
diffusion and flow in a cylindrical reactor indicate that only intermediate
values of $ ('vlO -10 ) could be determined for the example cited. For
smaller values of <(>, the reaction would be too slow. For higher values, the
reaction becomes diffusion-limited and independent of <{>.
Examination of the behavior of individual molecules via the random walk
analysis shows that, for the example cited, those molecules that collide with
the wall do so many times within a short interval of time, or x. The peak in
the distribution of molecules that collide with the wall (for all but very
4
small values of t and x) lies at 3 * 3 X 10 . This is about what would be
expected in an experiment where the limiting sensitivity for is - 10
We have investigated these results further to determine the conditions
under which higher or lower values of <}> could be resolved experimentally. It
was found that by increasing the cylinder length or decreasing the flow rate
or cylinder radius, alone or in combination, the sensitivity for lower values
of .
Decreasing the pressure would also give a greater distinction of high values
of <}>. However, at very low pressures, few gas-wall collisions would occur,
and the flow rate would have to be decreased or the reactor length increased
in conjunction with the pressure decrease.
We have also examined the possibility of obtaining greater definition by
running the analyzing light beam near the wall, rather than down the center
of the cylinder, or perpendicular to the cylinder axis. Although greater
definition would be obtained in this manner, experimental difficulties would
be greater and, depending on the particular experimental conditions,
extremely accurate measurements of distances would be required.
16
-------
Although our discussion of the experimental variation of the sensitive
range of $ is based on a particular example, the conclusions are generally
applicable. Thus, we conclude that making appropriate changes in flow rates
or pressure is the easiest and most accurate method for establishing the
experimental sensitivity range for . Experimental sensitivities for detect-
ing changes in reactant concentration will probably be limiting, as will very
high flow rates or pressures. In the latter cases, drag at the walls may
become significant, and the condition implicit in Eq. 1 that v is independent
o
of r (and x) may no longer be valid. Changes in the dimensions of the reactor
will affect the sensitivity range. However, practicle limits for variation
of these parameters are generally more severe than those for flow and partic-
ularly pressure. The examination of changes in reactant concentration near
r - R provides an alternative method for varying the sensitivity range for .
The general limits in this case concern the accuracy and the resolution at
which distances from the wall can be measure.
17
-------
REFERENCES
1. Judeikis, H.S., and S. Siegel. Particle-Catalyzed Oxidation of Atmos-
pheric Pollutants. Atmos. Environ., 7(6): 619-631, 1973.
2. Paneth, F., and K. Herzfeld. Free Methyl and Free Ethyl. Z. Elektrochem,
37: 577-582, 1931.
3. Wise, H., and C.M. Ablow. Diffusion and Heterogeneous Reaction. I. The
Dynamics of Radical Reactions. J. Chem. Phys., 29: 634-639, 1958.
4. Jost, W. Diffusion in Solids, Liquids, Gases. Academic Press Inc., New
York, New York, 1960. pp. 51-54.
5. Carslaw, H.S., and J.C. Jaeger. Conduction of Heat in Solids. 2nd ed.
Clarendon Press, Oxford, 1959. pp. 188-213.
6. Moelwyn-Hughes, E.A. Physical Chemistry. 2nd ed. Pergamon Press, New
York, New York, 1961. pp. 36-62, 609-610.
7. Present, R.D. Kinetic Theory of Gases. McGraw-Hill Book Co., Inc., New
York, New York, 1958. pp. 52-55.
8. Hedgpeth, H., S. Siegel, T. Stewart, and H.S. Judeikis. Cylindrical Flow
Reactor for the Study of Heterogeneous Reactions of Possible Importance
in Polluted Atmospheres. Rev. Sci. Instrum., 45(3): 344-347, 1974.
9. Gnedanko, B.V. The Theory of Probability. 2nd ed. Chelsea Publishing
Co., New York, New York, 1962. pp. 32, 118-121.
18
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TECHNICAL REPORT DATA
(Please read Instructions on the reverse before completing)
1. REPORT NO.
EPA-600/3-77-043
2.
3. RECIPIENT'S ACCESSION NO.
4. TITLE AND SUBTITLE
EFFICIENCY OF GAS-WALL REACTIONS IN A CYLINDRICAL
FLOW REACTOR
5. REPORT DATE
May 1977
6. PERFORMING ORGANIZATION CODE
7. AUTHOR(S)
H.S. Judeikis and Seymour Siegel
8. PERFORMING ORGANIZATION REPORT NO.
ATR-73 (7256)-2
9. PERFORMING ORGANIZATION NAME AND ADDRESS
The Aerospace Corporation
El Segundo, California 90245
10. PROGRAM ELEMENT NO.
1AA603 AH-04 (FY-77)
11. CONTRACT/GRANT NO.
802687
12. SPONSORING AGENCY NAME AND ADDRESS
Environmental Sciences Research Laboratory - RTF, NC
Office of Research and Development
U.S. Environmental Protection Agency
Research Triangle Park, NC 27711
13. TYPE OF REPORT AND PERIOD COVERED
Interim 6/71-1/73
14. SPONSORING AGENCY CODE
EPA/600/09
15. SUPPLEMENTARY NOTES
This report complements EPA-650/3-74-007, Grant No. 801340.
16. ABSTRACT
Expressions are given for the concentration of a dilute reactive gas mixed with an
inert carrier gas as a function of the radial and longitudinal distances in a cylin-
drical reactor and the reaction efficiency. The reaction efficiency is defined as
the fraction of gas-wall collisions that leads to the disappearance of the reactive
gas from the gas phase. The solutions presented here are applicable for all values
of reaction efficiencies and extend earlier work by other investigators that was
applicable only for values of 1 or near zero. In addition to the solution of Pick's
second law of diffusion as applied here (with an additional term for flow in the
cylinder) , a one-dimensional random walk analysis is also applied to this problem. The
combination of diffusion equation solutions and the random walk analysis leads to the
conclusion that, for a given set of experimental conditions, the reaction efficiency
can be uniquely determined only if it lies within a certain range of values. Small
values of the reaction efficiency will produce insufficient reaction and large values
will yield diffusion-limited results. The sensitive range for the reaction efficiency
can be changed by appropriate adjustment of various experimental parameters. However,
adjustment of the total pressure gives the greatest effect and is generally one of the
easiest parameters to vary in an experiment.
17.
KEY WORDS AND DOCUMENT ANALYSIS
DESCRIPTORS
b.IDENTIFIERS/OPEN ENDED TERMS
c. COSATI Field/Group
*Air pollution
*Chemical reations
*Gases
*Walls
^Chemical reactors
-'Diffusion
^Pressure
*Random walk
13B
07D
13M
07A
12B
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RELEASE TO PUBLIC
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UNCLASSIFIED
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23
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