United States Industrial Environmental Research EPA-600/7-78-189
Environmental Protection Laboratory September 1978
Agency Research Triangle Park NC 27711
Analysis of Cascade
Impactor Data for
Calculating Particle
Penetration
Interagency
Energy/Environment
R&D Program Report
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EPA-600/7-78-189
September 1978
Analysis of Cascade Impactor
Data for Calculating
Particle Penetration
by
Phil A. Lawless
Research Triangle Institute
P.O. Box12194
Research Triangle Park, North Carolina 27709
Contract No. 68-02-2612
Task No. 36
Program Element No. EHE624
EPA Project Officer: Leslie E. Sparks
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
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ABSTRACT
The difficulties of analyzing cascade impactor data to obtain particle
penetrations according to size are discussed. Several methods of analysis are
considered and their merits weighed: interpolation, least-squares fitting,
and spline fitting. The use of transforming functions prior to data fitting
is also discussed. Recommendations for the use of the normal transformation
and spline fitting method are made, and computer programs are provided to
facilitate their use.
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CONTENTS
Abstract ii
Figures iv
Acknowledgment v
1.0 Introduction to the Problem of Impactor Analysis 1
1.1 Impactor Cut-Points and the Cumulative Distribution 1
1.2 The Log-Normal Nature of the Distribution 2
1.3 The Problem of Differentiating Sparse Data 3
2.0 Recommendations 5
3.0 Approaches to Fitting Functions 6
3.1 Interpolating Polynomials 6
3.2 The Method of Least-Squares Fitting 7
3.3 Spline Methods of Analysis 10
4.0 The Normal Transformation 13
5.0 Results of Spline Fits of Data 16
5.1 Method of Generating Test Data 16
5.2 Results of Spline Fits to Untransformed Log-Normal
Distributions 18
5.3 Results of Spline Fits to Transformed Distributions 18
5.4 Spline Fits of Real Data 22
6.0 References 34
7.0 Appendices 35
7.1 Computer Routine for the Normal Transformation 36
7.2 Spline Fit Subroutine 37
ill
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FIGURES
Number Page
1.1 Real impactor data plotted on log-normal probability paper 4
5.1 Spline fit of a single narrow peak on a broad peak 20
5.2 Inlet cumulative distribution for a single run 23
5.3 Inlet frequency distribution from Figure 5.2 24
5.4 Outlet size distribution for a single run 25
5.5 Outlet frequency distribution from Figure 5.4 26
5.6 The penetration curve for a single run 27
5.7 Inlet size distribution (average of six runs) 28
5.8 Inlet frequency distribution from Figure 5.7 29
5.9 Outlet size distribution (average of six runs) 30
5.10 Outlet frequency distribution from Figure 5.9 31
5.11 Penetration curves for the averaged data runs 32
iv
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ACKNOWLEDGMENT
The efforts of the Project Officer, Dr. L. E. Sparks, in the first stages
of using the spline technique are gratefully acknowledged. He appreciated the
power of the technique and stimulated the investigation.
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1.0 Introduction to the Problem of Impactor Analysis
Multistage cascade inertial impactors are particle sizing devices which
rely upon the inertial aerodynamic properties of particles of different
diameters to separate them into distinct groups. By measuring the amount of
particulate collected in each group, the size distribution of the sampled
dust can be inferred. The widespread use of cascade impactors as particle
sizing devices requires that a consistent reliable method for inferring
the particle size distribution be used. This report addresses some of the
problems and solutions encountered in the analysis of cascade impactor data.
1.1 Impactor Cut-points and the Cumulative Distribution
It is assumed for the purposes of this report that whatever type of
cascade impactor is used, it has been adequately calibrated and carefully
used to avoid the problems associated with such measurements. (See for
example, References 1 and 2.) Each stage of an impactor has a characteristic
diameter associated with it, depending on the operating conditions, called the
cut diameter or cut-point which is the diameter particle collected with fifty
percent efficiency. Ideally all particles of larger diameter would be col-
lected with one hundred percent efficiency and none of the particles with
smaller diameter would be collected on that stage. In practice, the collection
efficiency curve is a moderately sharp function near the cut-point with sub-
stantial "wings" on either side of the cut-point. By knowing the precise
nature of the collection efficiency curve, it should be possible to deconvolve
the impactor stage weights to obtain a particle distribution. This approach
would have to be specific to each impactor and could not reasonably be used
unless each stage were calibrated for the flow conditions used. Generally,
attempts to deconvolve real impactor data have not been successful.
Commonly, the impactor cut-points are assumed to represent infinitely
sharp demarcations in the particle spectrum. This assumption will be carried
on through this report.
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Since each impactor stage separates the particle stream into fractions
larger than the cut diameter and smaller than the cut diameter, the amount of
dust on each stage represents the fraction of particles between adjacent cut
diameters. If the total mass collected is determined as well as the mass on
each stage, a cumulative fraction curve can be constructed which represents
the cumulative particle size distribution. It is this cumulative distribution
which the cascade impactor measures directly. The differential particle
distribution, which is the derivative of the cumulative distribution, gives
the abundance of particles for every diameter under consideration. Since
control devices collect different diameter particles with varying degrees of
efficiency, it is necessary to know both the inlet and outlet differential
distributions in order to adequately evaluate the performance of control
devices. A device may have an acceptable total collection efficiency, yet
have so poor an efficiency for some particle diameters that remedial action
is called for. Obtaining the differential distribution from the cumulative
distribution is the heart of the impactor data reduction problem.
1.2 The Log-Normal Nature of the Distribution
If the particle size distribution from a single source, such as a boiler
stack, is measured directly, it will be found to have a rather skewed distri-
bution, with the particle frequency approaching zero at zero diameter and, for
diameters above the mean diameter, approaching zero rather slowly. If the
logarithm of particle diameter is used as the abscissa, rather than the
particle diameter itself, the particle size distribution much more closely
resembles the normal curve:
1 v2/?
4, (x) = -=L^ e'x "• , (1.1)
V2*
where (x) is the normally-distributed function of x. The resemblance is
often so strong that the particles are said to have a log-normal size
distribution.
Associated with the normal curve is the probability integral, defined
as:
Q (x) = fx * (t) dt , (1.2)
where t Is a variable of integration and 4* is given by Equation 1.1. The
probability integral corresponds to the cumulative size distribution
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measured by the cascade impactor. There are graph papers available with
cumulative probability as one variable and logarithms as the other. The
probability integral function, Q(x), where x is a logarithmic variable,
would plot as a straight line on such graph paper.
The cumulative fractions of cascade impactor particle measurements also
generally exhibit a linear tread when plotted on such graph paper, and, in
fact, a straight line plot is indicative of a true log-normal particle size
distribution. A plot on log-normal (also called "log-probability") graph
paper of a real impactor data set is shown in Figure 1.1.
Most real impactor data is only approximately log-normal and there are
objections to considering them as such, particularly when the deviations
become large. However, some data reduction schemes rely upon assuming a
log-normal character for the cascade impactor data.
1.3 The Problem of Differentiating Sparse Data
Since it is the derivative of the cumulative distribution which provides
the desired frequency distribution, a method of differentiating the data is
needed. Unfortunately, there are no simple solutions to the problem. Direct
approximation of the derivatives by finite differences has the disadvantages
that the N data points produce only N-l "derivatives" and that the derivatives
so obtained are relatively crude approximations. Moreover, the derivative so
calculated must be assigned to some particle diameter located between the two
cut-points. (The well-known Mean Value Theorem states that the slope of the
chord joining the end points of an interval is equal to the derivative of a
function which passes through those end points at some point within the
interval.) Commonly, this diameter is chosen to be the arithmetic mean of
the logarithms of the cut-points, or equivalently, the geometric mean of the
cut-points themselves. Thus, the points at which the "derivatives" are
evaluated depend on the cut-points and may not be suitably located to compare
the results from different impactors, for example, or for inlet and outlet
samples from a control device. Nor is there any guarantee that the deriv-
atives are correct for these particle diameters.
The general recommendation for differentiation of numerical data, and
especially so when the data are sparse, is to fit a differentiable function
to the data and obtain the derivatives analytically. Methods of doing this
will be addressed in Section 3.
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98
95
90
70
o
DC
S 50
30
10
0.2
0.5
I
I
1.0 2
DIAMETER (ym)
10
20
Figure 1.1. Real impactor data plotted on log-normal probability
paper.
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2.0 Recommendations
Cascade impactor cumulative distribution data can be analyzed in many ways
to obtain a frequency distribution. Some of these ways have been investigated in
this study and the following recommendations can now be made.
1. Prior to any curve fitting technique, the impactor data should
be log-normal transformed: that is, the logarithm of the cut-
point diameters whould be used instead of the diameters themselves
and the normal transform of the cumulative probabilities should
be used instead of the cumulative probabilities themselves. With
this transformation, all of the techniques discussed in this
study will fit a true log-normal distribution exactly, and the
inherent errors in fitting near-log-normal distributions are
substantially reduced. A calculator algorithm and a computer
program for the normal transformation are provided in this report.
2. The natural cubic spline curve fitting method should be used to
fit the transformed data. The properties of such splines are
well-defined mathematically; these splines are the smoothest
curves that can be passed through all the data points, and they
are capable of fitting non-log-normal distributions with small
error. A computer program for calculating a spline passing
through given data points is provided in this report.
3. In cases where the impactor data are known (or suspected) to
contain substantial random errors, a polynomial least-squares
fit of the transformed data is recommended, with the poly-
nomial degree selected judiciously. The variance of the data
from the fitting polynomial should be calculated and used to
establish confidence limits on the resulting frequency distri-
bution. Least-squares fitting is not recommended in cases
where the random error is small, because the spline fit will
make better use of the data in such cases.
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3.0 Approaches to Fitting Functions
3.1 Interpolating Polynomials
The simplest function which can pass through N data points and is
different!able is a polynomial of degree N-l. Such a polynomial exists and
is unique (Reference 3). It is possible that the degree of the interpolating
polynomial is actually less than N-l.
This polynomial can be written in a simple way: consider three functional
values fi, f2, and f~ at the respective points x-,, x«, and Xg. The inter-
polating polynomial I (x) is:
(x-x7) (x-x,) (x-x,) (x-x.J (x-x,) (x-x,,) ,. n
I (x) = f, ? —+f? + f 3 '
(xj-XgHxj-x^ (x2-x.j)(x2-x3) (x3-x.j)(x3-x2)
This reproduces the functional values exactly at their respective points. For
more data points, a correspondingly higher degree polynomial can be generated.
The problem with using the interpolating polynomial for reducing impactor
data is that it is generally a fifth degree or higher polynomial and, except
at the cut-points, the value of the polynomial depends on the cumulative
fractions at all of the cut-points. The first derivative will generally have
3 or more peaks, (the second derivative is a cubic or higher polynomial).
Moreover, all the derivatives will be sensitive to slight changes in the
impactor data, with the result that repeated measurements on the same particle
size distribution may exhibit wide variations in the calculated distributions
due to small perturbations in the stage weights.
It is possible to use piece wise interpolation of a few data points at
a time in order to reduce the degree of the polynomial and the sensitivity
of the derivative to small perturbations in the impactor data. Two such
methods have been described recently (References 4 and 5). The problem with
this approach is that generally the first (and higher) derivatives are dis-
continuous. Since this leads to unphysical particle size distributions,
efforts must be made to smooth out the discontinuities.
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In Reference 4, this smoothing is done by convolving the piecewise inter-
polating polynomials with a smoothing function, essentially performing a running
average whose first derivative is a continuous function. The choice of the
smoothing function is rather arbitrary, though the function used is reason-
able on physical grounds. It appears that this particular smoothing function
would allow discontinuities in the second derivative, equivalent to abrupt
changes in slope of the particle size distribution. A higher order smoothing
function could be chosen to produce a continuous second derivative as well.
In Reference 5, the smoothing procedure produces continuous first
derivatives at the cost of allowing the interpolation to miss the data points.
The interpolation passes close to, but not necessarily through the impactor
cumulative fraction points. The method used also produces discontinuities in
the second derivative of the interpolating function, in a way that cannot be
easily corrected for. However, in this method, the logarithms of the
cumulative fractions are the data points fitted, which reduces the magnitude
of the discontinuities in the second derivatives.
One difficulty that applies to all interpolation schemes is that the
interpolation is valid, or has meaning, only over the range of the data. Any
effort to extrapolate the fitting fractions beyond the first and last data
points is not only unjustified, but generally leads to poor results anyway.
This is particularly troublesome when inlet and outlet impactor samples are
obtained from a control device to assess its graded penetration, since the
valid range will only include those data points common to both samples. For
this reason, it is worthwhile to try to match the impactor cut-points at
inlet and outlet.
3.2 The Method of Least-Squares Fitting
In contrast to interpolation methods, which effectively consider the
impactor data as completely reliable, least-squares fitting methods assume that
there are inherent errors in the data which partially conceal the true nature
of the functional dependence of the variables. The method of least-squares
assumes a functional dependence, either from theoretical considerations,
educated guesses, or reasons of simplicity, and then finds the "best" function
which fits the data. "Best" is defined in terms of the deviations of the data
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from the functional values, with the quantity
s2 = E ty-fCxj))2 (3.2)
where y. is the ordinate corresponding to abscissa x^, being minimized for
the "best" function f. It f is a linear function of the coefficients of powers
of x, then a system of linear equations results which has a unique solution
(Reference 3). This means that any functional dependence which can be
expressed as a polynomial in x can be easily used with the method of least-
squares.
Often the function used is in terms of variables transformed from the
actual data. For instance, the logarithm of the diameter may be used as the
independent variable, rather than the diameter itself. Such transformations
may greatly simplify the required functional form and the method for solving
the resulting equations. In particular, we will discuss the normal trans-
formation in some detail later, by which log-normal functions are transformed
to straight line functions.
If impactor data fall on a nearly straight line when plotted on log-
probability graph paper, it may well be said that they should obey a linear
relationship in those coordinates, and a linear least-squares fit might be
used. If the data show an obvious, but gentle curvature in those coordinates,
a quadratic least-squares fit, using powers of x up to the second, might be
used. Higher powers could also be attempted. If there are N data points, then
a least-squares fit with an N-l degree polynomial will give a perfect fit and
will in fact produce the interpolating polynomial for that data.
The choice of functional dependence is a matter for judgment, since there
is no strong theoretical reason that the particle size distribution should indeed
be log-normal. Since the method of least-squares is a method of accounting
for random measurement errors, it is reasonable to use it only if the errors
associated with the measurements are at least of the order of magnitude of
the deviations of the data from the fitted function. The method cannot legiti-
mately be used to remove "errors" that are in fact really present in the
measured quantities.
If a functional dependence is assumed for the data, then there is some
justification for extrapolating that dependence into the regions beyond the data
8
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at either end of the size distribution. This extrapolation must be done with
due regard to the nature of the function; that is, a linear function extra-
polates well, while higher order polynomial functions may quickly reach
unphysical regions. Extrapolation is one of the useful properties of least-
squares methods, particularly so with impactor data, where inlet and outlet
samples may not have good overlap of cut-points.
The second major advantage of the least-squares method is that it provides
an intrinsic measure of the total random error associated with the fit. If
the deviation of the fitted function f(x^) from the measured datum yi is defined
as 6.., then the variance of the fit is given by
-l), (3.3)
where N is the number of data points and M is the degree of the fitting poly-
nomial. It can be seen that if the number of data points equals M-l, the
variance is undefined because all the deviations are zero. Thus, no measure
of the error can be obtained for this case, which is equivalent to the inter-
polating polynomial.
The variance is a statistical measure of the random error in the fit and
provides information for determining the overall confidence level that can be
assigned to the data. A strategy for using the variance as a fitting parameter
would be to fit successively higher degree polynomials to the data until the
variance was minimized. A minimum usually exists because the denominator in
Equation 3.3 approaches zero as the degree of the polynomial approaches N-l.
In summary, the least-squares method provides a statistical measure of
the error in the data and the ability to extrapolate the fitted function, but
at the cost of assuming a functional dependence for the data. As an aside, the
method of Tschebyschev fitting is similar to the method of least-squares in
that a functional dependence must be assumed. The procedure is different in
that the maximum (absolute value) deviation of the data from the fitting function
is minimized. The Tschebyschev method would be more applicable in situations
where the randown error in the data is known to be small, but for which a certain
functional form is desired (Reference 3).
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3.3 Spline Methods of Analysis
A draftsman's spline is a thin flexible strip that can be made to conform
to plotted data points by means of weighted pivots. The curve so constructed
passes through all the data points and has continuous first and second derivatives.
Mathematical splines are a class of polynomial functions with properties that
are generalizations of those of the draftsman's spline.
The particular spline of interest in the analysis of impactor data is
a third order natural spline. It consists of a series of cubic polynomials,
joined at the data points with continuous first and second derivatives and
with the second derivatives at the first and last data points equal to zero.
Outside the data range, the cubic polynomial is reduced to a linear polynomial,
so that the natural spline has some utility in situations calling for extra-
polation.
Splines have well-defined mathematical properties, and odd-order splines
have several special properties that are of interest in the impactor analysis
problem. These are:
1. If F(x) is a continuous function, with continuous first and
second derivatives, then the natural spline of third order,
S(x), which attains the values F(xn-) at points a=x^, < x
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4. If the number of interpolating points in the inteval [a,b] is
increased so that the maximum separation of any two adajcent
points tends toward zero, then the spline and its first
derivative, S(x) and S'(x), converge uniformly to F(x)
and F'(x). (Reference 6)
The first and third properties describe the best approximation properties
of the spline. The second property describes the "smoothness" of the spline
curve itself, and the fourth property describes the behavior of the spline
function and its first derivative at points away from the junction points.
These properties can be extended to higher order splines and higher
order derivatives (Reference 7), but that is not of further interest at this
time.
A fifth property that the third order natural spline has that is less
mathematically defined is its insensitivity to variations at a single point.
Changes at a single point in the value of the function being fitted changes
the value of the spline function only at that point, changes the first
spline derivatives mainly at the point and its two nearest neighbors, and
changes the second derivatives of the spline mainly at the point and the two
adjacent points on either side. Furthermore, the changes are largest at
the central point, diminishing with distance from the point. Thus, an error
in the original data does not propagate throughout the entire fitted region,
as can happen with the interpolating polynomial.
The third order natural spline is calculated from the following set of
equations.
If there are N data points with abscissas at x^< ...xk-..< XN and values
F(x.), and if lk = xk - xk-1 (k = 2,N), then the spline second derivatives
Mk = S"(xk) are given by
Vl .. F(xkt1)-F(xk) F(xk)-F(xk,1)
\t 1 ^ k fi k+1 1
K- I 0 Iv O KT I I. .
with M] = MN = 0.
This is a tridiagonal set of equations which can be solved easily by
several methods. A program using successive overrelaxation is shown in the
11
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Appendices. A program for programmable calculators is given in Reference 11.
The value of the spline and its derivatives are determined by integration of
these second derivatives from the junction points to the points of interest.
Because of its excellent approximating properties and relatively simple
method of calculation, the spline fit is the best method for analyzing
impactor data under any conditions where the random error is known to be small
12
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4.0 The Normal Transformation
All of the fitting methods described so far can be applied to any set
of data, whether from impactors or not. There is one type of data for which
these methods all exhibit intrinsic errors (and in which we are interested):
log-normal distributions. We have stated that a log-normal distribution is
one in which the logarithm of the independent variable is normally distributed.
We now write it explicitly.
If y = log (d) and the cumulative function Q (y) is given by
/•y 0-t /2
Q(y) = / ^-=^ dt, (4.1)
= f
then Q is log-normally distributed in d. The variable t may be related to
the mean value of log(d), m, and its standard deviation, o, by the equations:
(4.2)
u
and
„(„, . / £!^! . (4.3)
/
Now, the point is that this function, Q, is linear near x = 0, but exhibits
strong curvature (with infinitely many derivatives) for x away from zero. Thus,
any fitting scheme based on polynomial approximations is bound to contain
intrinsic errors in attempting to fit this function. Some analysis schemes,
that of Reference 5 for example, reduce the errors by fitting log Q instead of
Q itself; but the errors are still present. The spline can deliver an excellent
fit to Q if the points of junction are close enough, but begins to deteriorate
rapidly if they are too far apart.
Because the log-normal distribution is often very close to impactor-
measured size disbributions, it would be worthwhile to require any fitting scheme
13
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to be able to reproduce the log-normal distribution as closely as possible
under all conditions. Fortunately, all of the methods discussed before
are capable of fitting the distribution exactly by use of the normal trans-
formation.
The normal transformation is defined as the inverse function to the
probability integral Q. In other words,
N(Q(y)) =y . (4.4)
Whereas Q(y) ranges from 0 to 1, N(Q(y)) ranges from -« to +~ in a one-to-one
correspondence with y, i.e. it is a linear function. The normal transformation
is equivalent, then, to plotting on probability paper, and any scheme that
can fit a straight line can fit the transformed variable. The inverse function
N is not analytic, but can be approximated by a polynomial expression:
2
cn + c,t + c9t
-N(Q) = -y = t * 3 2 (4 5)
where t = \l log I—-~] 0 < Q < .5
and cQ = 2.515517
c1 = 0.802853
C2 = 0.010328
d1 = 1.432788
d2 = 0.189269
d3 = 0.001308
with an absolute error less than 4.5 x 10 (Reference 8). Since this is valid
for Q over only half its range, the relation is needed that:
Q(-x) = 1 - Q(x) . (4.6)
14
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This function is easily evaluated on a pocket calculator, and is even
available as a stored function on one model (Hewlett-Packard HP 32E). It
is shown as a computer subroutine in the Appendices. A calculator program
for the function is given in Reference 11.
With the use of this transformation, linear or higher order least-squares
methods can be used to fit nearly log-normal data, and interpolating poly-
nomials or splines can give good results with even widely spaced data points.
A discussion of some of these results will be deferred to a later section.
Since it is the particle frequency distribution that is desired from
the cumulative data, the derivative of the fit to the normal transfored data
must itself be transformed back to the linear space of the impactor data.
That is, if the transformed function F is given by
F(x) = N(f(n)) , (4.7)
then the derivative of the function f is given by
2
f'(x) = e"F(x) /2 - F'(x) . (4.8)
Since all of the methods discussed give F(x) as well as F'(x), the inverse
transformation is easy to evaluate. If the cumulative function f is also
desired, it must be evaluated from the probability integral (Equation 4.1)
with y = F(x). Polynomial approximations to the probability integral are
available in Reference 8. Note that if F'(x) is constant, then f(x) is a
normally distributed function. In particular, for the natural spline fit,
the extrapolation regions outside the range of the data for which the spline
is of degree one represent normally distributed tails which join smoothly with
the spline interpolation. This does not justify the extrapolation, but at
least it guarantees a well-behaved extrapolation.
15
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5.0 Results of Spline Fits of Test Data
The analysis of impactor data by splines was tested both with and without
the normal transformation. Tests were performed on the ability of the analysis
to reproduce the test frequency distributions for a single impactor and to
reproduce the test penetrations for inlet and outlet impactors. The errors
produced in fitting inlet and outlet impactor data were not always independent
with the results that the propagated error in penetration could be smaller
than the errors in the individual fits.
5.1 Method of Generating Test Data
Log-normal frequency distributions were specified by mass median diameters
(HMD) and geometric standard deviations (o ). The logarithm of a is the
standard deviation that appears in the normal distribution (Equations 4.2 and
4.3). At a series of abscissas corresponding to the logarithms of particle
diameters in the range of 0.2 to 20 ym, the frequency distribution appropriate
to the MMD and o was calculated by polynomial approximations (Reference 8).
This formed the "true" data to which could be compared to the results of the
impactor analysis. For testing the spline method without the normal trans-
formation, an inlet distribution and an outlet distribution were used. The
ratio of outlet to inlet at each diameter gave an effective penetration for
that diameter.
These distributions were "sampled" by ideal impactors in the following
way. Typical impactor cut points were chosen (independently) for the inlet
and outlet samples. Then the cumulative fractions of the appropriate log-normal
distribution were calculated for these cut points, one set for the inlet and
one set for the outlet. Generally, six or seven cut-points were chosen.
Then the spline method was used to obtain a fit to each cumulative dis-
tribution, and the spline derivatives were evaluated at each diameter for
which the "true" inlet or outlet was known. These derivatives were compared
to the corresponding distribution values and the ratio of outlet to inlet at
each diameter was compared to the "true" penetration.
16
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When the normal transformation was used, a different set of data was
required, since log-normal data would be transformed to a linear function,
which the spline could fit exactly. For this case, two log-normal distributions
were added together at the inlet. The two distributions could have their MMD's
and o 's set separately and the ratio of their amplitudes was adjustable. The
outlet distribution was allowed to remain a single log-normal distribution.
For the normal-transformed data, the error in the penetration would then be
in proportion to the error in the inlet distribution.
Two measures of error were used. The first was the root mean square
relative deviation, defined as:
- f , \2\ 1/2
(5.1)
The function f could present either a distribution function or a penetration
calculation, for which true values were known. The number of points averaged
was variable, usually ten to fifteen. The second measure of error was the
relative maximum error, defined as:
max (|ftrue-fcalcl)
o - —
This measure took the maximum deviation and related it to the maximum value
of the function (which was always positive).
The RMS error e gives a measure of the overall discrepancy between the true
values and the fitted curve while the relative maximum error, 6, gives an
error that is not exceeded in the range of the test. Both types of measure are
useful, and in general, 6 was of the order of twice e.
The geometric standard deviation of a log-normal distribution is defined
as the ratio of particle sizes that are one standard deviation apart; for
instance, the ratio of the size corresponding to a cumulative fraction of
0.841 to the size corresponding to a cumulative fraction of 0.500. Since over
68 percent of the total mass of the particles are included in the interval
from one standard deviation below the MMD (MMD/o ) to one standard deviation
above the mean (MMD x a ), the geometric standard deviation provides a quick
measure of the sharpness of the distribution. A a of 4, for example, means
17
-------
the distribution is spread over a substantial range from one-fourth the MMD to
four times the MMD.
5.2 Results of Spline Fits to Untransformed Log-Normal Distributions
The spline was capable of fitting the untransformed log-normal distributions
quite well, provided that the cut-points were suitably placed. The magnitude
of errors in the penetration was about five percent if both the inlet and
outlet impactor cut-points were well-distributed over the central part of the
curve (MMD/o to MMD x o ). As fewer impactor cut-points were placed in this
region, so the error rose, reaching values of sixty percent or greater. If a
distribution was broad, the impactor cut-points could be far apart without
serious effect. If the distribution were sharp, the cut-points had to be
closely spaced to keep the error low.
For these cases, a special advantage was found in matching the inlet and
outlet impactor cut-points. Apparently, the errors in each fit compensated
each other and the error (e) in penetration could be as low as one percent.
Changing the location of a single cut-point by as little as twenty-five
percent raised the RMS error to the level of five percent.
It is rarely possible to match inlet and outlet impactors cut-points
when testing a control device, and it is also difficult to make sure the
cut-points are well-distributed over each distribution and still maintain
sufficient overlap to calculate the penetration over a wide range. It is
also not reasonable to try to use the spline extrapolation in the untrans-
formed case, because the extrapolations are constant values for the distribution,
a very unphysical result. These problems lead us to look further, specifically
to the transformed case.
5.3 Results of Spline Fits to Double Mode Transformed Distributions
The first tests of the normal-transformed spline fits used relatively
broad distributions (MMD.| = 20, o = 4.5; MMD2 = 3.0, og = 2-3) in various
ratios. The errors (e) in these cases were all less than one percent, often
less than one-tenth percent. In those cases where the impactor cut-points
were well-distributed over the distribution, the untransformed spline also
performed creditably (e < 0.05). However, the transformed spline showed no
sensitivity to choice of impactor cut-points in these tests. There were no
significant differences between matched or unmatched cut-points, or between
18
-------
well -distributed and poorly distributed cut-points. No comparisons of the
extrapolated fits were attempted for these tests. Even mixtures of positive
and negative distributions were fitted well, as long as the total frequency
distribution remained positive at all points.
Then a group of tests were performed combining a broad distribution
(MMD = 10, o = 3.5) with a very narrow distribution (MMD = 3, o = 1.05),
using normal cut-points. Very high errors resulted, with e's in the range of
forty to two hundred percent, in proportion to the ratio of the sharper
distribution to the broader distribution. In effect, the spline was completely
missing the sharp peak in the distribution, because it fell almost totally
between the impactor cut-points. The effect of this on the frequency distribu-
tion is shown in Figure 5.1. In this example, the total mass in the sharp
peak is one-tenth the total mass in the main peak and produced a thirty-five
percent error. If more mass is placed in the sharp peak, the error is greater,
and if less mass is placed in the sharp peak, the error is less. The spacing
of the cut-points does not have a significant effect in this case, as long as
the sharp peak affects no more than one cut-point in the cumulative distribution.
This test provides a measure of the sensitivity of the spline technique
to errors in the data points. If the distribution being fitted is truly log-
normal, except that one data point is in error, then the spline will produce
a distribution that is in error from the log-normal distribution. Based on
the results of this series of tests, if x. is the mass error at a single data
point in the cumulative curve divided by the total mass of the distribution,
then the RMS relative error, e^ , is given approximately by
ei = 6.7 x1 (5.4)
This error holds mainly in the neighborhood of the erroneous point, and
decreases to near zero beyond two cut-points away from the erroneous point.
The effect of errors in all the data points could be approximated by summing
the squares of the errors in the neighborhood of each point:
e = i e (5.5)
1 j = i-2 J
19
-------
1.0
0.7
0.5
0.3
o 0.2
Z3
i—i
£ 0.15
O.I
0.07
0.05
0.03
1.5
3
DIAMETER
10
Figure 5.1.
Spline fit of a single narrow peak on a broad peak. Dashed
line is the true distribution (broad peak: MMD = 10, og = 3;
sharp peak: MMD = 3, ag = 1.05). Solid line is the spline
curve. Arrows indicate location of impactor cut-points.
20
-------
The usefulness of this error estimate is somewhat limited because of
its empirical nature, but it does provide a method of assigning error bars
to points in the frequency distribution if some estimate of the errors in
the data can be made. For example, if the weighing error for each stage
were 0.1 mg and the total mass collected were 20 mg, then the x.'s for each
cut-point would be 0.005 and the overall error (Equation 5.5) at each cut-
point would be 0.075 for interior cut-points (those located two or more
points from the end points) and would decrease to approximately 0.035
(Equation 5.4) at the ends. These are very rough estimates based on the
examination of limited data.
More precise error limits can be established from the mathematical
properties of the spline, provided that a bound on the magnitude of the
fourth derivative of the cumulative function can be estimated (Reference 10)
There is no physical basis for doing this, and some arbitrary "smoothness"
criterion would have to be used.
The final group of test data was aimed at establishing how narrow the
secondary mode could be and still be resolved. It was quickly determined
that if the distribution mean was no more than one standard deviation away
from the second nearest cut-point, the RMS error was acceptably small. In
other words, a secondary mode should be centered between cut-points and
span them comfortably. Under such conditions, a narrow secondary peak in
the distribution could have a peak amplitude twice that of the main broad
peak with an RMS of the order of five percent. Smaller peak amplitudes in
the secondary peak reduce the RMS error considerably, to less than one
percent at an amplitude ratio of one to one.
Again, these observations are based on limited tests, but in general,
a secondary peak that spans two or three cut-points will probably be
accurately resolved from the data, assuming the data itself is free from
error.
It should be noted that, in this section, we have referred to the RMS
error in the particle size distribution, rather than in the penetration.
The reason for this is that a constant log-normal distribution was assumed
for the outlet distribution, which was fit exactly under the normal trans-
formation. Therefore, no compensating errors could occur, and the error in
the distribution is as good an indication of the fit as is the error in the
penetration.
21
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5.4 Spline Fits of Real Data
As a final demonstration of the curve fitting results, several sets of
data for inlet and outlet impactor samples of an operating electrostatic
precipitator (the EPA-IERL Pilot Scale Precipitator) are presented. Figures 5.2
through 5.6 are concerned with a single day's results while Figures 5.7 through
5.11 cover an average of six inlet and six outlet samples. (The impactor flow
conditions were stable enough to allow direct averaging of the stage weights.)
In each group a comparison is shown between the results obtained with a
least-squares fit of the log-normal-transformed data and the results with a
natural cubic spline fit of the transformed data. In some cases, a linear
least-squares fit gave a correlation coefficient just as good as the quadratic
least-squares fit, and so the linear fit was used for simplicity. However, in
most cases, the quadratic fit gives a substantially better correlation coef-
ficient and a visually better fit of the data.
Figure 5.2 shows the inlet cumulative distribution, represented by the
open circles, the fitted spline function passing through the points, and the
quadratic least-squares line passing near the points. It can be seen that
both fits represent the data well, except that the quadratic fit does poorly
at the smallest particle sizes. If this were an isolated set of data, one
might be inclined to use the quadratic fit and attribute the deviations to
error in the data. However, the behavior of the data at the small sizes is
consistent from run to tun, or in averages of many runs, indicating that
random error cannot account for all the deviation.
The derivatives of the cumulative function fits are shown in Figure 5.3,
with arrows indicating the cut-point diameters. The spline fit indicates a
secondary peak in the distribution which the least-squares curve completely
ignores. By the criteria of the preceding section, this secondary peak
should be accurately represented because the data points are well-distributed
over it. Whether the peak is actually in the particle distribution or
represents an anomalous response of the impactor is a question that cannot
be answered here.
Figures 5.4 and 5.5 show the cumulative and frequency distribution for
the outlet impactor sample. Again the secondary peak is present and well-
resolved in the spline fit.
22
-------
98
95
90
80
70
I 60
L«J r- «
a. 50
UJ
g 40
1 30
20
10
0.3
0.5 0.7 1.0
2 3
DIAMETER (ym)
10
20
Figure 5.2.
Inlet cumulative distribution for a single run. The solid curve
is the spline fit passing through the data points (open circles)
and the dashed line is the quadratic least-squares fitting curve.
23
-------
10
O.I
0.01
O.I
Figure 5.3.
1.0
10
too
DIAMETER (ym)
Inlet frequency distribution from Figure 5.2. The solid line is the
derivative of the spline fit curve and the dashed line is the
derivative of the least-squares curve. The arrows indicate the
positions of the original data points.
24
-------
O
98
95
90
80
70
60
50
40
30
20
10
T r
I I
0.2 0.3 0.5 0.7 1.0
2 3
DIAMETER (urn)
5 7 10
20
Figure 5.4. Outlet size distribution for a single run. The solid curve is
the spline fit and the dashed line is a linear least-squares
fit of the data.
25
-------
10
O.I
0.01
O.I
Figure 5.5.
1.0
10
100
DIAMETER (ym)
Outlet frequency distribution from Figure 5.4. The solid line
is the spline curve derivative and the dashed line is the
least-squares curve derivative. The arrows indicate the
positions of the data points.
26
-------
1.0
O.I
o
i—i
i
0.01
0.001
O.I
Figure 5.6.
1.0
10
100
DIAMETER (urn)
The penetration curve for a single run. The solid curve is the
ratio of outlet to inlet spline derivatives multiplied by the
ratio of dust concentrations at the outlet to the inlet. The
dashed curve is the corresponding representation for the least-
squares fits. The horizontal arrow shows the range of data
common to both inlet and outlet samples.
27
-------
99.99
99.9
99
o
UJ
CL.
90
70
50
30
10
O.I
0.01
0.2
0.5
1.0 2
DIAMETER (ym)
10
20
Figure 5.7. Inlet size distribution (average of six runs). Spline fit is the
solid curve and quadratic least-squares fit is the dashed curve.
28
-------
10
O.I
0.01
O.I
Figure 5.8.
1.0
10
100
DIAMETER (ym)
Inlet frequency distribution from Figure 5.7. Solid line is
spline derivative and the dashed line is the least-squares
derivative. Arrows indicate the positions of the data points.
the
29
-------
99.99
99.9 -
0.01
1.0 2
DIAMETER (ym)
10
20
Figure 5.9. Outlet size distribution (average of six runs). Solid curve is
the spline fit and the dashed curve is the least-squares fit.
30
-------
10
O.I
0.01
O.I
Figure 5.10.
1.0
10
DIAMETER (ym)
100
Outlet frequency distribution from Figure 5.9. Solid curve is
from the spline fit and the dashed curve is from the least-
squares fit. Arrows indicate the location of the original
data points.
31
-------
1.0
O.I
LU
LU
Q.
O.OI
0.001
O.I
Figure 5.11.
1.0
10
100
DIAMETER (ym)
Penetration curves for the averaged data runs. Solid line Is
from the spline fits and the dashed line Is from the least-
squares fits. The horizontal arrow indicates the range of
data common to both inlet and outlet samples.
32
-------
The penetration curves in Figure 5.6 were computed by dividing the outlet
frequency distribution at the points of interest by the inlet frequency dis-
tribution at the same points and multiplying by the ratio of the outlet dust
concentration to the inlet dust concentration. The limits of the data range
common to both samples are indicated by the arrows, but the extrapolated regions
are well-behaved for reasonable distances outside the data range. The spline-
derived penetration curve shows considerably more variation than the least-
squares curve, due to the more detailed frequency distributions of the spline
fits.
The second group of data, averaging six impactor runs in the inlet and
outlet cumulative curves, is similar. The cumulative curves are somewhat
smoother, and that is reflected in the frequency distribution curves for the
spline fit. This smoothness may be due to the reduction of random error in
the data by the averaging process. Note that the secondary peaks in the spline
fits are less prominent, but still present. Again, the distribution of cut-
points promises good representation of the secondary peaks.
The penetration curve, Figure 5.11, is also similar to Figure 5.6.
Since similar conditions prevailed in these two data groups, temperature
being the major difference, such a result is expected.
These data were not particularly difficult to fit, since a linear least-
squares method also gave very reasonable (> 0.99) correlation coefficients in
all cases. The additional resolution, obtained in the spline fits seems real,
based on the consistency of the location of the secondary peaks in the
frequency distribution, and the spline fits would be preferred in these cases.
Spline fits of less consistent data would normally give very complex penetration
curves, and a least-squares method might be preferable.
33
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6.0 References
1. K. M. Gushing, G. E. Lacey, J. D. McCain, and W. B. Smith, "Particulate
Sizing Techniques for Control Device Evaluation: Cascade Impactor
Calibrations," U.S. Environmental Protection Agency, Office of Research
and Development, Washington, D.C. 20460. EPA-600/2-76-280.
2. L. G. Felix, G. I. Clinard, G. E. Lacey, and J. D. McCain, "Inertial
Cascade Impactor Substrate Media for Flue Gas Sampling," U.S. Environmental
Protection Agency, Office of Research and Development, Washington, D.C.
20460. EPA-600/7-77-060.
3. E. L. Stiefel, "An Introduction to Numerical Mathematics," (Academic Press,
New York, 1963).
4. G. R. Markowski and D. S. Ensor, "A Proceedure for Computing Particle Size
Dependent Efficiency for Control Devices from Cascade Impactor Data,"
presented at the annual meeting of the Air Pollution Control Association,
Toronto, Canada (1977).
5. J. D. McCain, G. Clinard, L. Felix and J. Johnson, "A Data Reduction System
for Cascade Impactors," presented at the Symposium on Advances in Particle
Sampling and Measurement, Asheville, N.C. (1978). Sponsored by Environmental
Protection Agency, Industrial Environmental Research Laboratory, Research
Triangle Park, N.C. 27713.
6. J. L. Walsh, J. H. Alberg, and E. N. Wilson, J. of Mathematics and Mechanics
H, 225 (1962).
7. C. DeBoor, J. Mathematics and Mechanics 12., 747 (1963).
8. M. Abramowitz and J. Stegen, "Handbook of Mathematical Functions" (Dover,
New York, 1964).
9. T. N. E. Greville, "Mathematical Methods for Digital Computers, Vol. II,"
A. Ralston and H. S. Wilf, eds. (John Wiley and Sons, New York, 1967).
10. P. M. Prenter, "Splines and Variational Methods" (John Wiley and Sons,
New York, 1975).
11. L. E. Sparks, "Cascade Impactor Data Reduction with SR-52 and TI-59
Programmable Calculators," EPA report in Press (1979).
34
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7.0 Appendices
Two computer programs are included for use in implementing the log-normal
transformation and the natural third order spline fit. Appendix 7.1 is a
Fortran realization of the algorithm given in Section 4.0. The algorithm
itself is taken from Reference 8. (As presented in Reference 8, the
algorithm has a sign error; the version shown here operates correctly.)
Appendix 7.2 is a Fortran program taken almost exactly from Reference 9; the
structure is the same except for the addition of the extrapolation calculations
and the omission of an integral calculation.
35
-------
Appendix 7.1
Computer Routine for the Normal Transformation
i*
2* C THIS FUNCTION CALCULATES THE INVERSE OF THE PROBABILITY INTEGRAL. P RUST
?« C BE IN THE RANEE OF C TO 1.
4*
5* FUNCTION APPOB(P)
6* FLfe = C
7« PS = P
8* IF(P.LE.O.C) 60 TO 1
9* IMP.6E.1.0) 60 TO 2
10* IF(P.GT.G.5> 60 TO ?
11* * T = S«RT(ALC6C1.0/P**2»
12* TS = T*T
13* TC = TS *T
U* APROB = -/(TC*0.001308 «
15* 1 TS«0.159?e<; * T*1.432788 * 1.0) * T
16* IFCFL6.EQ.G) APROB =-APROP
17* P = PS
IP* RETURN
19* 1 APROB = -1.CE37
20* RETURN
21* 2 APROB - 1.CF37
22* RETURN
23* 3 FL6 = 1
2** P = 1.C - P
2S* 60 TO t
26* END
36
-------
Appendix 7.2
Spline Fit Subroutine
1*
2*
3* C THIS SU9ROUMNE FITS A NATURAL CUBIC SPLINE THROUGH N DATA POINTS* PASSFD BY
4* C WEANS OF ThEIR COORDINATES IN VECTORS X AND Y. THE ORDINATES IN * MUST BE
5* C ARRANGED IN ASCENDING ALGEBRAIC ORDER. THF * POINTS AT WHICH CALCULATIONS
6* C WILL BE PERFORMED ARE PASSED IN THE AIRAT T, IN ANY ORDER.
7*
8* C THE VECTOR SS OUTPUTS THE VALUE OF THE SPLINE FIT AT EACH OF THF POINTS IN Tf
9* C AND THE VECTOR SS1 OUTPUTS THF DERIVATIVE OF THE SPLINE FIT AT THE POINTS IN T
10* C . THE SECOND DERIVATIVE IS ALSO AVAILABLE INTERNALLY, AND IT CAN BE OUTPUTTED
11* C . FOR POINTS OUTSIDE THE RANGE OF THE INPUT DATA, LINEAR EXTRAPOLATIONS FRO"
12* C THE END POINTS CF THE FIT ARE PFRFCRHFD, USING THE VALUE OF THE FITTED SLOPE
13* C AT EACH END POINT.
U*
15*
16* SUBROUTINE SPL1NE(P*N ,X,Y,T,SS*SS1>
17*
18* DIMENSION X(10),Y(1C),7(12),SS(12),SS1(12),SS?(12>,H(10),OELY(10),
19* « H2(10>*B(1C)vDELSQV(1C>vS2<1C)*CCir:),S3C10)
20* DATA OtEGA/1.C7179(8/
21*
22* C THE FINITE DIFFERENCE APPRCXIPAT IONS TO THE DERIVATIVES ARE HADE FOR EACH OF
23* C THE DATA POINTS* WITH THE SECOND DERIVATIVES AT THE END POINTS FIXED AT ZFRC.
24*
25* N1 = N - 1
26* DO 51 1 - 1*M
27* Ml) = X(I-»1) - X(l)
28* 51 DELY(I) = (Y(I«1> - VU»/H(I>
29* DO 52 I - 2,N1
30* H2 = HC1-1) * H(I)
31* FJU) = C.5«H(I-1)/M2
32* DELS6YC1) = (DELY(I) - DELY (1-1 ))/H?
35* S2<1) = O.C
36* S2(N) - O.C
37*
3J»* C A RELAXATION TfcCHNIOUE IS ISED TO SOLVE FOR THE INTERIOR SECOND DERIVATIVES,
39* C RFFEATED UNTIL ThF RESIDUAL, FTA, IS SPALL FNOUGH.
*0«
41* 5 ETA - C.C
42* DO 10 1 - 2*N1
43* b (CO - bCl)*£2(I-1> - «F6A
37
-------
44* IFCABSCh).faT.ETA) ETA = ABS(W)
45* 10 S2CI) = S?CI) 4 U
46* 1FCETA.6E.1.CE-06) 60 TO 5
47*
4F* C THE FINITE APPROXIMATIONS TO TMF THIRD DERIVATIVES ARE CALCULATED.
49*
50* DO 53 I = 1.M
51* 5? S3CI) - (S2C141) - S2Cl))/hCI)
52*
53*
54* C THE VALUES OF THE SPLINE AND ITS DERIVATIVES ARE CALCULATED AT THE POINTS OF T
55*
56* DO 61 J = 1,H
57* 1=1
5E* IFCT(J) - XCD) 5*, 17,55
59* 55 IFCTU) - XCN)) 57,59,62
60* 56 IF(TCJ) - XCD) 60,17,57
61* 57 I = I 4 1
62* 60 TO 56
61*
64* C EXTRAPOLATION BELOb THE FIRST DATA POINT
65*
66* 5P SS10NE - DELVC1) - HCD/CHC1) 4 H (2) )*2 .C* C DEL V C2) -.DELVC1))
67* SSCJ) = VCD 4 CTCJ) - XC1))*SS10«E
6»* SS1CJ) - SS10NE
69* 60 TO 61
7P*
71* C EXTRAPOLATION APOVE THE LAST DATA POINT
72*
73* 62 SS1N = DELVCN1) 4 HCNl)*S2CND
74* SSCJ) = TCN) 4 CTCJ) - XCN»*SS1N
75* SS1CJ) = SS1N
76* 60 TO 61
77*
7P« C CALCULATION IN THE RANGE OF THE DATA POINTS
79*
fcO* 59 I = N
t1* 60 1-1-1
62* 17 HT1 = TCJ) - XCI)
fc?* HT2 = TCJ) - XCI41)
t4* PROD = HT1*HT2
e5* SS2CJ) = S2CI) 4 HT1*S?CI>
fc6* DELSQS= (S^C1) 4 S2CI41) * SS2Cj))/6.
87* SSCJ) = VCD 4 HT1*DELVCI) 4 PROD«DELSOS
fcP* SS1CJ) - DELVCI) 4 (HT1 4 HT2)*DELSOS 4 PRQD*S3CI
89* 61 CONTINUE
VO*
V1* RETURN
92* END
38
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TECHNICAL REPORT DATA
(Please read Jiiunictions on the reverse before completing)
. REPORT NO.
EPA-600/7-78-189
2.
3. RECIPIENT'S ACCESSION-NO.
4. TITLE AND SUBTITLE
Analysis of Cascade Impactor Data for Calculating
Particle Penetration
5. REPORT DATE
September 1978
i. PERFORMING ORGANIZATION CODE
7. AUTHOR(S)
Phil A. Lawless
8. PERFORMING ORGANIZATION REPORT NO.
9. PERFORMING ORGANIZATION NAME AND ADDRESS
Research Triangle Institute
P.O. Box 12194
Research Triangle Park, North Carolina 27709
10. PROGRAM ELEMENT NO.
EHE624
11. CONTRACT/GRANT NO.
68-02-2612, Task 36
12. SPONSORING AGENCY NAME AND ADDRESS
EPA, Office of Research and Development
Industrial Environmental Research Laboratory
Research Triangle Park, NC 27711
13. TYPE OF REPORT AND PEI
Task Final: 6-8/78
RIOD COVERED
14. SPONSORING AGENCY CODE
EPA/600/13
is. SUPPLEMENTARY NOTES lERL-RTP project officer is Leslie E. Sparks, Mail Drop 61, 919/
541-2925.
16. ABSTRACT
The report discusses the difficulties of analyzing cascade impactor data to
obtain particle penetrations according to size. It considers several methods of analy-
sis (interpolation, least-squares fitting, and spline fitting) and weighs their merits.
It also discusses the use of transforming functions prior to data fitting. It recom-
mends the use of the normal transformation and spline fitting method, and provides
computer programs to facilitate its use.
17.
KEY WORDS AND DOCUMENT ANALYSIS
DESCRIPTORS
b.IDENTIFIERS/OPEN ENDED TERMS
c. COSATI Held/Group
Pollution
Dust
Penetration
Size Determination
Measurement
Analyzing
Impactors
Interpolation
Least-squares
Method
Computer Programs
Data Processing
Pollution Control
Particulate
Cascade Impactors
Normal Transformation
Spline Fitting
13B
11G
14B
131
12A
09B
18. DISTRIBUTION STATEMENT
Unlimited
19. SECURITY CLASS (TinsReport)
Unclassified
21. NO. OF PAGES
44
20. SECURITY CLASS (Thispage)
Unclassified
22. PRICE
EPA Form 2220-1 (9-73)
39
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