A REGIONAL AIR
QUALITY SIMULATION MODEL
(A DESCRIPTION OF THE CONNECTICUT
AIR POLLUTION MODEL AS DEVELOPED
BY THE TRAVELERS RESEARCH CENTER)
THE RESEARCH CORPORATION
of NEW ENGLAND
210 Washington Street
Hartford, Connecticut 06106
Phone (203) 527-4101
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,-------
1JW@
THE RESEARCH CORPORATION.
of NEW ENGLAND
210 Washington Street, Hartford, Connecticut 06106
203 527-4101
r
,
A REGIONAL AIR
. .
QUALITY SIMULATION MODEL
(A DESCRIPTION OF THE CONNECTICUT
AIR POLLUTION MODEL AS DEVELOPED
BY THE TRAVELERS RESEARCH CENTER)
L
J
N. E. BOWNE
The Research Corporation of New England
G. D. ROBINSON
The Center for the Environment and Man
September 1971
Prepared under Contract CPA 70-155 for Division of Meteoro1ogyp
National Environmental Research Center, Environmental Protection
Agency, Research Triangle Park, North Carolina
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TABLE OF CONTENTS
Section
Title
1.0
Short-Term Air Quality Models
2.0
The TRC Model.
3.0
The Gaussian Dispersion Formula
4.0
Trajectories and Multiple Sources
5.0
Aspects of the Application of the Gaussian
Formulae in the TRC Model
5.1
Peculiarities of Application
5.2
General Description of the Computation
5.3
Quadrature and Truncation of the Gaussian
Distribution
5.4
Finite Emission Time and the Puff Equation
5.5
Time-Variation of the a(t)t [a (x)] FUnctions
5.6
Space and Time Variations of the Mean Wind
5.7
The Decay of Contaminant
6.0
Operating Procedure
6.1
Derivation of the Mean Wind Field
6.2
Derivation of the Trajectories
6.3
Choice of the Diffusion Parameters
6.4
The Gaussian Weighting Factors
. 6.5
The Treatment of Vertical Diffusion
6.6
Calibration of the Model - Flux across the Boundary
607
Calibration of the Model - Decay of Pollutant
608
Height of Source
6.9
Summary of Operating Procedures - Flow of Com-
putation for One Receptor
7.0
References
Page
1
2
4
8
10
10
11
13.
15
16
16
18
20
20
20
23
24
26
29
30
31
32
34
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A.Regional Air Quality Simulation Model
LO
Short-Term Air Quality Models
The purpose of a short-term (resolution of one hour or so) air
quality model is to allow computation of the concentration of one or
more atmospheric contaminants at one or more locations in a region.
Finite time resolution implies that the computed results are time
averages over the resolution period.
Contaminant concentration is
very variable in space and time. and it is impracticable to specify
all sources in complete detail so that some averaging in space of both
input and output is a feature of all air quality models.
We recognize two major categories of model - those which set up an
equation to describe the transport and diffusion of emitted material
and solve this by forward integration in time. and those which assume
a form of solution of the equation for a single source. insert appropriate
empirical parameters and sum the solutions over all sources.
-1-
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II,
2.0
The TRC Model
The model to be described is of the second kind.
It was devel-
oped between 1966 and 1969 at the Travelers Research Center and it is
convenient to describe it as "The TRC Model".
Major contributors to
its development were G. E. Andersonp F. I. Badgleyp N. E. Bownep K.
Hage and G. R. Hilst.
A brief description has been published by Hilst
(1967).
It has been applied to the State of Connecticut (Hilstp Yocom
and Bownep 1967) and to the environs of Toronto (Bownep Boyer~ Trent and
Cooper p 1971).
Both applications have been subjected to an extensive
verification programp Hilst (1969) and Bowne et al (1971).
Three categories of source are r,?cognized in the model.
a.
Majorp intensep discontinuousp i.e.p point and linep sources.
Examples are the stacks of electrical generating or other major indus-
trial plants and highways~ etc.
b.
Numerous minor sources which are treated as continuous area
sources.
The area of the model is divided by a square grid of side Va?
All locations are referred to by t4e coordinates (i.j.) of this grid.
The minor sources within each grid square are added~ and the total
assumed to be distributed evenly over the square.
Thus~ all. space
variability of the minor sources on scales less than one grid square
is suppressedp and we cannot recover a meaningful pattern of the con-
centration variation caused by the minor sources on any smaller scale.
There is also a limitation on time variability which is discussed later.
c.
Sources beyond the boundary of the model.
-2-
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No model can treat sources beyond its boundary as realistically
as it can treat explicit internal sources~ but ideally it should include
all sources having a major influence on air quality within the region
to which it is to be applied.
The application of the TRC model to the
State of Connecticut was complicated by the proximity of very large
sources, unknown in detail~ in the area of New York City.
Special de-
vices» effectively a sub-model, were needed to deal with this situation.
They are described later.
-3-
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,
I'
3.0
The Gaussian Dispersion Formula
The TRC model employs empirically established formulae - the
Gaussian dispersion formulae - to calculate concentrations of pollu-
tants .
In part the formulae are employed in an unorthodox manner and to
understand the possibilities and limitations of the model it is useful
to examine the conceptual and factual bases of the formulae and relate
them to the way in which the formulae are incorporated into the model 0
The Gaussian formula describing the transport and dispersion of a puff
of material emitted instantaneously from a point source at height H is
X
1
(x~y~z~t) = Q
(21T) 3/2 0' 0'
x Y
[ 2
exp -" (x-~ t) <}
, 2
2 0' '
X
2J
Y
2 ay 2 ,
0'
Z
f [(Z-H)j
exp - 2
2 0'
. Z
+
exp -
r~::m
(1)
where Q is the quantity of material emitted; x~y~z are coordinates with
origin at the earth9s surface perpendicularly below the source~ which is
at (O~ O~ H); u is the mean speed of the wind transporting the material
during the time, t ~ which has elapsed since the ins tantaneous emission;
the x-axis is in the direction of this, mean wind; and the aVs are the
standard deviations,relative to the puff center along axes with origin
(utp 0» 0), of the distribution of emitted material.
Dispersion estimates
represented by the standard deviations depend on atmospheric turbulence
-4-
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and thus are empirical functions of time which are related to atmospheric
I
I
I
I
s tabili ty.
The standard deviations of concentration describe rather than
explain or predict the concentration distribution.
As written. eq. (1) is
quite general; u and the crvs may change with change of the time scale for
which they are defined and there is no restriction to steady meteoro-
logical conditions.
Note that if the wind direction varies with time.
the x and y axes are only locally vcartesianv.
On a larger scale the x
axis is bent to conform to the generally curvilinear wind trajectory.
The y distance is then measured along the local normal to the trajectory.
This system. commonly used in fluid flow analyses. is referred to as a
stream-oriented coordinate system.
If the puff formula is to be useful in prediction or post-facto
explanation of observed concentrations. there must be some generally
applicable specification of the crvs.
It is possible to specify them as
empirical functions of time. (i.e.. time elapsed since emission) and
Vatmospheric stabilityv.
It is also possible by grouping 'atmospheric
stability' into a few (four to six) broadly specified classes. to limit
the number of empirical functions required. the functional relations
being determined by field trials. ',This limitation on cr.t relationships
limits the generality of the formula - it now refers only to circum-
stances when Vatmospheric stability V is steady.
There is no'explicit
restriction to steady mean wind. but wind speed is one of the factors
normally used in the assessment of Vatmospheric stabilityv.
The Gaussian formula describing dispersion of material from a
-5-
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i
continuous point source is
21T a
y
a
z
exp - [ y2~
2 cr .
y
Q
X
2
(x.y.z) =
u
fxp - Dz~:~j + . exp f~:~ J
(2)
where Q is now the rate of emission of matter.
We may regard eq. (2)
either as being empirically established independently of eq. (1) or as
being derived for any (x, y, z) by integration of eqo (1) over time,
regarding the continuous point source as an infinite series of puffs with
non changing a as the puff passes a point,
x
ioeo,
X
2
=
f
-00
X (t) dt
1
Eq. (2) is essentially a steady state formula.
Regarded as an empirical
formula independent of eq. (1) it requires specification of a mean wind
u steady in both magnitude and direction to define the plume axiso
The a's are found to be empirical functions of x, the functional dependence
varying with atmospheric stability.
It can be derived for any x by time
integration from eq. (1) only if ~ is independent of time and if the
a's (which may be functions of x) do not vary explicitly with time.
Application of eq. (2), therefore, presumes steady mean wind and un-
changing atmospheric stability.
The approximation of stream-oriented
coordinates is, however, acceptable if the wind direction changes slowly
wi th time.
In that case, the plume center-line coincides with the
-6-
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I .
trajectory.
The Gaussian formulae eq. (1) and eq. (2) have been confirmed by
observations made over fairly uniform terrain in reasonably steady conditions
of wind and atmospheric stability.
These observations show that the
Gaussian distribution is a reasonable description of the average concen-
tration distribution in an ensemble of puffs, and of the variation of
concentration across a plume for averaging times of order one hour.
There
is more
and more satisfactory evidence for the plume formula than for
the puff formula.
In trials such as these, the aVs are established as
empirical functions of time or distance from source, and of atmospheric
stability.
Deviations of the results of individual trials from the
average of several trials may typically be more than a factor of 2 in
one trial out of five.
Given this degree of imprecision in single ap-
plications of the formulae, it follows that inconsistencies of application
likely to produce random errors of the same order are not serious defects
of a Gaussian formula model.
We must, however, examine all inconsistencies
to ensure that they are not likely to produce systematic error.
-7-
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path of center of mass of puff
~~--~
~
Source
a
~
concentration distribution about b
concentration distribution about a
Fig. 1.
IllustTating the puff formula in two dimensions.
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4.0
Trajectories and Multiple Sources
In explaining how the formulae are incorporated into the TRC model,
it is convenient initially to ignore the dispersion in the vertical
(treatment of which in practice often calls for substantial modification
of the Gaussian formula).
For the present, therefore, we consider only
the horizontal dispersion terms.
In the TRC model, it is assumed that
cr
x
= a
y
When using the model the computational requirements can be reduced
= a.
by considering only those sources making an appreciable contribution to
pollution concentration at chosen receptors.
To do this the traj ectory
of the air which surrounds the receptor point at the required time is
traCked baCkwards across the model grid, and the contributions of only
those sources within a corridor of a given width along the trajectory
are summed to give the total concentration at the receptQr.
Fig. 2
illustrates the method.
Sl and S2 are the strengths of sources (assumed
at ground level for convenience) which emit puffs at time t .
o
At some
later time to + t the concentration at a point RI is
x
=
I
(2'IT)3/2 cr2 cr
z
Gl + 82 exp - (2b:2TI
(3)
i.e.
the emissions at any time from multiple instantaneous sources sur-
rounding a source 51 on the trajectory. can be replaced by the emission
of a virtual source at Sl obtained by summing the intensities of the
sources, and each weighted by a Gaussian factor depending on its distance,
b, from 51 and the appropriate cr.
A similar argument applies to continuous
-8-
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51!..
lit
Fig. 2.
ut
a
Concentration distribution from multiple sources.
b
-------
sources if a is replaced by a , i.e., a distribution of continuously
y
emitting sources on any perpendicular to the trajectory can be replaced
by a single source on the trajectory with an intensity equal to a weighted
sum of all the source intensities.
-9-
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5.0
Aspects of the Application of the Gaussian Formulae in the TRC Model
5.1
Peculiarities of Application
We are now in a position to examine aspects of the application of
the Gaussian formulae in the TRC model.
Some of these are common to all
multiple-source plume or puff models, some are peculiar to the TRC model.
During anyone computation of the concentration at a receptor -
1.
The instantaneous point source ("puff") formula is used for
finite area sources emitting for finite times.
The emission rate for a
single area of side a is assumed to remain constant for the basic defini-
tion time .of the model, usually one or two hours.
2.
The infinite integration limits of the Gaussian formulae are
replaced by finite limits -- many sources are neglected in computing. the
concentration at anyone receptor.
Gaussian formulae require integration
between infinite limits even though 99 percent of the area under a Gaussian
curve lies within the limits of i30.
The integration used in the model,
described in Sections 5.3 and 6.4 assumes 100 percent of the area within
fini te Umi ts .
3.
The model necessarily neglects temporal variation of the
functional dependence of the o's on time and distance.
4.
The model includes spatial variation of the mean wind vector,
but also employs the continuous source (plume) formula for major sources.
The simple Gaussian plume model of eq. (2) assumes straight-line flow at
constant wind speed.
We permit changes in the flow over the region and
assume a plume model as described in Section 5.2.
5.
The model includes temporal variation of the mean wind vector.
The wind fields are permitted to change at the basic definition time of the
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model, usually one or two hours.
This permits the wind field to be
changed. to account for time changes at an interval that reasonab ly re-
presents observed information.
Of these five points the third refers to an aspect of the real physical
situation which is not accounted for by the model because the Gaussian
formulae do not permit it. Points four and five refer to aspects of the
real physical situation which the model attempts to account for in spite
of the fact that formally the Gaussian formulae do not permit it.
Points
one and two concern approximations which are essential in order to restrict
computation to acceptable limits.
We will consider them in turn and discuss
the errors, random and systematic, which they are likely to introduce, but
before doing so it is necessary to outline the computational procedure.
5.2
General Description of the Computation
The first step is to compute from the windfield the trajectory of a
materia~ particle which occupies the receptor point at the time at which
pollutant concentration is required.
Details are given in Section 5.6 and
the result is illustrated in Fig. 3.
Here ~(I,J) is the receptor point
and the trajectory is identified (stored in the computer) by the points
at whiCh the trajectory crosses the grid -- (n~~) and (~,v) are repre~entative
points in Fig. 3.
At these points one coordinate (i or j) is integral, the
other normally non-integral.
The windfield and source inventory are held
constant for two-hour periods.
They may be changed at the beginning of
each period.
In Fig. 3, points a and y on the trajectory illustrate the
effects of such a change.
The trajectory itself is defined by the straight
line segments between the crossing points.
Distance along the trajectory
-11-
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I (~J)~
:/ .....t
~ /
/
/
V
,I
(QP) pm) /
c I '}B C ~ ~ B
.
A..... ~ ,/
'/~//
B B (17 r.)
I(U\J =
- ~/ %
Ioo""'"y ij// jI
C B C
Fi q. 3.
Sources, trajectory and receptor.
-------
and elapsed time between the receptor and all crossing points are known, as
is the ~ime spent within each grid square.
The contribution. of real and
virtual sources within each on-trajectory grid square to the concentration at
~ is then computed from the puff formula as expressed in eq. (3), the puff
intensity being taken as the product of the time the wind takes to cross
the grid square and the total rate of emission of sources within the square
at the time at which the trajectory crossed the square.
(Section 5.4).
The elapsed time (or source -- receptor distance) required for selection of
the appropriate cr values is taken as the time or distance on trajectory from
the receptor to first contact with the source square (e.g., for Square A in
Fig. 3 time or distance between (I,J) and (n,~), plus a time 'correction'
a/2~ or a distance 'correction' a/2 where ~ is the mean wind at ~(I,J) speed
and a is the length of the grid side.
(Section 5.4).
Major point sources are treated according to the plume formula eq. (2).
In the Connecticut application they are considered to be located at the
nearest grid intersection, e.g., in Fig. 3 a major source actually at
(O,P) is considered to be located at (M,N).
The source-receptor distance
(x) used in the comput~tion is the distance from receptor to the trajectory-
grid crossing-point nearest to th~ conventional location of the source
(distance (I,J) - ()J,\!) in Fig. 3). The off-axis distance (y) is taken as
the shorter of the distances from the conventional source position to the
trajectory measured along the grid axes, (distance (M,N) - ()J,\!) in Fig. 3).
These approximations are required because the trajectory is not more closely
defined in the computer store than by the crossing-points.
They are not
essential features of the TRC model but are used in the Connecticut applica-
tion.
In general neither x nor y need necessarily be an integral multiple
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of the grid side.
In the Toronto application, locations of point sources
are not.restricted to integral grid points.
5.3
Quadrature and Truncation of the Gaussian Distribution
We are now in a position to consider the peculiarities of the model
set out in Section 5.1.
The first concerns the use of the instantaneous
point source formula with a finite area source.
Although we have described
the receptor as a 'point' the suppression of source detail on scale less
than the grid square implies that it is meaningless to compute a concentra-
tion pattern on less than this scale.
The 'receptor' becomes an area equal
to the grid square, centered on the point chosen as the receptor (e.g., the
area surrounding tt in Fig. 3).
It then also. becomes meaningless to employ
the continuous Gaussian distributions which would imply concentration varia-
tions across the receptor area.
The Gaussian functions are therefore re-
placed, in the area source computation, by a rectangular stepped distribution
of the type shown in Fig. 4; each continuous two-dimensional Gaussian dis-
tribution being replaced by (2n + 1)2 factors where n is the number of
square grid rings surrounding the central source square.
This procedure
also facilitates truncation of the.Gaussian distribution with corresponding
limitation of the number of area sources considered in the computation of
the concentration at anyone receptor.
In the Connecticut application
truncation was at n = 1, as illustrated in Fig. (4).
At Toronto n = 2
has been used.
The method of calculation of the factors is explained in
Section 6.4.
Referring again to Fig. 3, we see that the contribution to
the concentration at (I,J) which can be attributed to the time the center
of mass of the air surrounding (I,J) spent in square At between (nts)and
-13-
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B .
A
Fig. 4.
Quadrature and truncation of the Gaussian
distribution.
-------
1-
(~,v) is computed by summing, for square A and the eight contiguous squares,
the product of the time spent between (n,~) and (~,v), the source intensity,
and the appropriate Gaussian factor A, B, or C.
The total contribution of
area sources to the concentration at A is the sum of such sums computed for
each square transected by the trajectory.
(Note that we are still not con-
sidering vertical diffusion.)
In the Connecticut application of the model the continuous Gaussian
distribution was used in computing the contribution of major continuous
point sources, a procedure not consistent with the treatment of area sources,
but not an essential feature of the model.
A more consistent procedure
would be to attribute to each receptor the average of the continuous distribu-
tion of concentration over a segment of y of .1ength a (the grid-1engt~), centered
on the receptor point.
There is no obvious reason why these procedures, spatial averaging and
truncation, should produce errors so long as computed grid-square averages
of concentration are compared with actual grid-square averages.
In fact,
spatial averages of contaminant concentrations are not measured and it is
obvious that there can be both random and systematic differences between the
actual point measurements commonly made and true area averages.
The procedures
should not produce systematic errors in grid-square averages of conce~tration
if the whole area of the model is considered, because the truncated Gaussian
distributions are normalized to the integral between infinite limits.
How-
ever, it seems likely (because there are prevailing winds) that locally
systematic errors could occur close to concentrations of strong sources.
A fairly even spread of sources over the model area would be conducive to
small random or systematic error.
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I -------
5.4
Finite Emission Time and the Puff Equation
TWo major difficulties arise in using the puff equation'with an ex-
tended source and a finite emission time.
The first is to establish the
source strength Q.
This has been taken as the product of the time the
wind takes to cross the grid square and the total rate of emission of
sources within the grid square at the time at which the trajectory crossed
the square.
This is an assumption:
its validity has not been quantitatively
analyzed.
Intuitively it appears reasonably sound when the segment of
the trajectory falling within the grid square is substantial, or when
there is no large change in source intensity between contiguous grid
squares.
It might be responsible for considerable errors, systemati~
only because there are prevailing winds, at receptor points close to regions
in which there are steep spatial gradients of source intensity.
The second major difficulty in using the puff equation with a source
of finite emission time is in deciding the appropriate value of 0, which
is, a non-linear function of time elapsed since emission.
The problem is
obviously most serious for those sources near the receptor where the finite
emission time is comparable with the elapsed time.
The elapsed time actually
used to calculate the Gaussian factor for emissions from any square ~s the
trajectory time to the contact point nearest to the receptor' (Le., (11 ,Z;)
for square A in Fig. 3) plus a/2 ~ where ~ is the mean wind speed at the
receptor.
This is obviously a reasonable device not likely to lead to
major random or systematic error, but capable of making minor contributions
to both.
-15-
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5.5
Time-Variation of the aCt), [a(x)] Functions
The general structure of the model allows change in the intensity
of the sources and the meteorological parameters - wind speed, direction
and stability - on a discontinuous basis.
In the Connecticut application
the change was allowed at two-hourly intervals.
Changes in wind field
and source inventory are accommodated in the manner described in Section
5.2.
In Section 3 we have seen that change of stability presents one
difficulty which is inherent in the Gaussian formulae - the functional
relation between a and elapsed time or distance cannot change along a
trajectory.
The convention adopted in the TRC model is to use the a's
appropriate to the stability existing at the time for which the concentra-
tions are required.
It is not obvious whether this convention, acting
through the horizontal diffusion parameter a , will produce systematic
y .
error.
Sensitivity analyses, Hilst (1970), shows that the overall effect
of changing a is small, at least for the Connecticut source distribution.
y
However, systematic error might be introduced through a , because concen-
z
tration computations made at the end of a period of increasing vertical
stability (decreasing a ) will overestimate concentration, and computations
z
at the end of a period of decreasing stability (increasing a ) will under-
z
estimate concentration.
In general, underestimation in the hours after
dawn and overestimation in the hours after dusk would be expected.
The
existence of elevated sources complicates this argument, but they are not
a major feature of the Connecticut application of the TRC model.
5.6
Space and Time Variations of the Mean Wind
The method of computing the wind field allows variation in speed
-16-
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and direction over the grid.
A new wind field is computed for each two-
hour period.
In Section 5.2 we have seen how these variations are in-
corporated into the trajectory computation.
No difficulty arises from this
variation of u in the instantaneous point source formula, and in the
extension to grid square sources u enters only in the computation of the
time for which a particular square is contributing to the required con-
centration.
Section 3 shows that u within the source square should be,
and is, used in determining this time.
Difficulty arises with the continuous source formula used for major
point sources, which requires a constant u to define the plume axis as
well as the source strength.
It is clear that so far as source strength
is concerned, the u to use in the formula is that appropriate to the
source.
In the TRC model the u of the Gaussian formula is replaced by
distance along trajectory, i.e., the mean wind used implicitly to compute
source-receptor distance and to select 0 is not necessarily the same as
that used where it appears explicitly in the concentration formula.
Fig.
5 illustrates the plume model with a curved centerline.
The lower illus-
tration in Figure 5 shows the conventional Gaussian plume model applied
at distance x downwind from the source and at distance y perpendicular
to the plume centerline.
The upper illustration shows how the IRC model
assumes the plume centerline to be parallel to the trajectory and y re-
presents the distance from the trajectory point to the source point, but
not necessarily a perpendicular distance.
This use of non-perpendicular
distances could result in systematic underestimation of the effect of a
point source on the receptor.
The major causes of systematic spatial wind variations over an area
-17-
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~x
x
Fi g. 5
Plume with curved trajectory.
-------
the size of Connecticut or less are judged to be topography and differen-
tial thermal effects.
(Both are treated in the Toronto application of the
TRC model; topography is allowed for in the Connecticut version.)
The
model is considered to account satisfactorily for the grid scale features
of topographic effects upon the windfield.
In the Connecticut application
no attempt is made to vary the dispersion parameters (o's) which are
known to be affected also by topography and thermal stability.
In the
Toronto application the effect of the city on diffusion is recognized by
the use of o's appropriate to urban conditions.
The model might be subject
to systematic error in localities affected, for example, by persistent
lee eddies on the scale of the grid side or less.
If this error exists,
it is not a product of any procedure adopted in the mode~, but rather
of an omission of detail.
5.7
The Decay of Contaminant
Gaussian formulae such as eqs. 1 and 2 were developed to account
for the distribution of conserved contaminants -- the normalizing con-
stants ensure that the total source material is accounted for if the plume
or puff is integrated over space.
'Material emitted into the atmosphere
may be lost or transformed in various ways --it may react chemically,with
another contaminant or a permanent atmospheric constituent, it may fall
out or be washed out, or it may decay spontaneously, in the manner of a
radioactive substance.
This las t type of decay can be handled by the
formulae, since the continuity equation for mass is satisfied if we re-
place the source term Q by Q exp (-AT) where A is a decay coefficient
lx - - Ax
at -
-18-
-------
and T is the elapsed time since emission.
The device adopted in the TRC model to account for loss of con-
taminant is to characterize this loss, in fact due to processes unknown
in detail, by a decay coefficient A or a half-life T, and regard the
source as virtually time-dependent, amending eqs. (1) and (2).
then becomes
E - 2
( . t) - Q exp (-0.693 t/T) - (x-ut)
X x,y,z, - 3/2 exp 2
1 (2~) cr a a 2 a
x y z x
txp
- fiZ-H~~ + exp!<~+H ]
Ua J L2a
z z
and eq. (2) becomes
X
2
= Q exp (-0.693 X/UT)
(x,y,z) exp
a
z
2iru a
y
- G ::~
[exp -
2
(z-H) +
2 a 2
z
(z+H) 2 J
exp - .. 2
2 a
. z
-19-
Eq. (1)
.~
... -L-
2 a 2
y
(4)
(5)
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6.0
Operating Procedure
6.1
Derivation of the Mean Wind Field
When the TRC model was under development it was assumed that a
basic wind field over the model region would be obtained by interpolation
between observed "surface" winds.
Early in the verification program for
the Connecticut application of the model it was found that this method
was unsatisfactory.
Too few observations were available, and those that were
were clearly affected by local influences of sUb-grid-scale size.
I twas
decided to develop a specification of the wind field which accounted for
the influence of topography on scales greater than the grid scale.
The
method in principle assumes that there is a height in the atmosphere
above which the effect of topography can be neglected; in particular- there
is no vertical motion through this surface.
At the ground there is ver-
tical motion -- the wind blows parallel to the surface, and thus has a
vertical component.
The average horizontal wind 1:etween the ground and the
lid is computed from the mass-conservation equation.
Details of the topographic wind analysis were described by Anderson
(1971) .
6.2
Derivation of the Trajectories
The model computes concentration at chosen receptor points at chosen
times.
The first stage in the computation is determination of the tra-
jectory of a material particle travelling with the mean wind which, at the
chosen time, is situated at the receptor point; i.e., the air surrounding
the receptor point is followed backwards in time.
The requirements are to
-20-
-------
identify the grid squares through which the imaginary material particle
has passedt and to determine the time spent in traversing ea~ of these
squares.
The wind field is stored as a stream function ~ on each grid square
so that local velocity components are derivatives of the stream function.
The derivatives and finite difference forms are:
u =- l.!f = u(itj) - [1/J(itj+l) - ~(i,j-l)] /2a (6)
ay
v = ~ = v(i,j) = [~(i+l, j) - 1/J(i-lt j)] /2a (7)
ax
where a is the grid-length.
The wind field is assumed to be constant for some basic period, typically
, '
two (2) hours, and then to change discontinuously.
The routine is entered
at grid point ~(I,J), (Fig. 3) representing the location of the desired
air quality estimate.
A countert kt is set to one and will change by
one increment for each time step.
The local u and v wind components are
obtained from eqs. (6) and (7) and an initial time step ~t (typically 15
minutes) is chosen to determine the local trajectory line segment.
Then:
~TI(l) = - u(l) ~t/a
~K(l) = - vel) ~t/a
Vel) = (u2(1) + v2(1»1/2
(8)
where ~K and ~TI are the j and i components of displacement expressed in
-21-
-------
terms of the grid length a and (1) denotes that the displacements cor-
respond to the first time step.
Tests are applied to determine if 0.3<
(lm or boK) < 1.
If not, then bot is doubled or halved and the process is
repeated.
The upper limit ensures that at each step i and J do not change
by more than 1, while the lower limit is used to limit the number of
computations required.
bot may be different for different portions of the
traj ectory .
Further points on the trajectory are computed from
1T(i+l)
=
1T(i) + bo1T
K (i+l)
=
K(i) + boK
(9)
These points are tested to see if the edge of the model grid array has
been reached, if not the process is repeated.
A secondary check limits
the total number of trajectory points to a maximum of 200.
At this point a trajectory has been calculated baCkwards from the
point of interest to the edge of the regional grid with a series of short
line segments and has been defined by the stored (1T,K), coordinates of the
end points of the line segments.
The corresponding wind speeds have been
calculated and stored in a companion array.
An example of such a trajectory pattern is shown in Fig. 6.
The
number above the point represents the counting index k associated with it.
The next step is to express the trajectory as the series of all the points
at which it (the trajectory) intersects the lines of the basic grid,
i.e., points for which either of 1T, K, but not necessarily both, is an
integral value of j or i.
This step results in a set of trajectory points
-22-
-------
j
j
20
? 1
~ -
.
JI~
Il'
JI
- -'jJ
-
~
-ZC 71
",
15
10
5
4
3
2
1
1 2 3 4 5 6 7 8 9 1 0 12 14 1 6 18 20
;
Fig. 6. Trajectory and stored points.
20
~ -
~
L
~
V
~ .......
-
./
./'
""
15
10
5
4
3
2
1
1 2 3 4 5 6 7 8 9 10 12
;
18 20
14 16
Fig. 7. Trajectory and grid crossing points.
-------
as illustrated in Fig. 7, where horizontal and vertical lines represent
the crossing points.
It is now possible to determine from the crossing
points and the mean wind speed how long the air parcel in question is being
affected by the sources in a given grid square.
The logical and arithmeti-
cal procedures used to convert the description of the trajectory from
the initially computed series .of points (7T ,k) to a description in terms of
intersections with the grid are most conveniently set out in the form of
a computer program, and will be found in the flow diagrams and program
which accompany this report.
The trajectory is stored as the series of crossing points and the
wind speed between successive pairs of crossing points.
6.3
Choice of the Diffusion Parameters
As we have seen, the standard deviations a , a , and a which appear
x y z
in eqs. (1) and (2) are, strictly speaking, descriptive parameters which
vary from occasion to occasion, but the formulae have, of course, no pre-
dictive value unless some numerical values are attached to the a's.
The
concept of the cr's as functions of travel distance, as proposed by Meade
(1960) and Pasquill (1961, 1962) is .illustrated in Turner (1968) and Slade
(1968) .
The magnitudes chosen derive in the main from Pasquill's inter-
polation of the experimental data.
In the model six stability categories
are considered and the a's are defined in terms of the following empirical
equations, which closely reproduce Turner's curves but are not exactly
equivalent to any previous formulations.
Equations of form
In a = a + B
2
(In x) + 'f (In x)
(10)
-23-
-------
where x is downwind distance in meters were assumed.
The coefficients
were det~rmined statistically from data read off the published curves.
The coefficients are:
Stability Vertical Lateral
a 8 y a 8 y
A 1.23090 -.43984 .15842 -1.02935 .97355 -.00702
B -0.82953 .52251 .03857 -1.55774 1.01215 -.00801
C -1. 76510 .74932 .01334 -1. 89130 .98147 -.00495
D -2.61432 .91491 - .00557 -2.44760 1.01255 -.00670
E -3.70065 1. 25914 -.04205 -2.66891 .99002 -.00515
F -4.19033 1. 29181 - .04535 -3.06583 .98858 -.00504
The effective distance is computed as set out in Section 5.2 and
5.4, incorporating the time or distance correction for finite grid size.
As explained in Section 5.5, the stability category used is that appropri-
ate to the time at the receptor, (Le., the time at which concentrations
are being calculated).
6.4
The Gaussian Weighting Factors
In Section 5.3, we discussed the rationale of the replacement of the
continuous Gaussian distribution by sets of (2n + 1)2 weighting factors.
Fig. 4 illustrates a 9-square array, where the factors A, Band Care
computed by integration of the Gaussian distribution according to the
following scheme.
A(s) =
1
(1/2 (1/2
) du J dE;, exp-
-1/2 -1/2
E;,2 + u2
2 s2
2
2 1T S
-24-
-------
J 1/2 f -1/2 ~2 + u 2
B(s) = 1 du d~ exp
2 2
2 1T s -1/2 -co 2 s
(11)
-1/j. -1/2 ~2 + u 2
1 J du d~ exp
C(s) = 2 2
2 1T S 2 s
-co -co
A + 4 B + 4 C = .1
All lengths are expressed in terms of the grid square side a.
Then
(x - ut) = a ~» Y = a u» a = as
and it is assumed a
x
= a
y
= a.
In Fig. 2 the concentration at R1 at ground
level due to source emission rate Q(81) at position 81 is given by
x =
d 1
Q (8 1) . A .
u (21T)1/2a
z
(12)
where diu represents the distance across the grid square from (n» s) to
(~» v) in Fig. 3 divided by the mean wind speed in that square.
In the application of the model to the Toronto area» a 25-grid
square approximation to the Gaussian distribution is proposed.
The weigh t-
ing factors attributed to the squares of this grid» are derived from
tabulated values of the bivariate Gaussian distribution.
Pr =
1
JdU Jd<
,"2 2
.., + u
exp -
2 s2
2
2 1T S
-25-
-------
w(5) = Pr(~>2/s, u>2/s)
w(4) = Pr(~>l/s, u>2/s) - w(5)
w(3) = Pr(~>l/s, u>l/s)- [2 w(4) + w(5)]
w(2) = Pr(~>O, u>2/s) -[2 (w(4) + w(5»]
w(l) = Pr(~>O, u>l/s) -[ (w(2) + 2 w(3) + 4 w(4) + 2 w(5»]
w(O) = 1 -[4 (w(l) + w(2) + w(3) + 2 w(4) + w(5»]
The weights corresponding to six values of s are shown in Table 1.
Table 1
Weights applied to area sources to account for horizontal diffusion
s
0 0.5 0.75 1.0 1.25 1.5 1. 75
w 1.0 .911072 .650280 .466060 .332108 .244572 .186876
o
wI 0 .021686 .075540 .092782 .090510 .079758 .067622
w2 0 .000028 .002520 .015532 .031580 .045228 .055086
w3 0 .000516 .008490 .018471 .024666 .026011 .024469
w4 0 .000001 .000429 .003091 .008607 .014718 .019933
w5 0 0 .000022 .000518 .003003 .008364 .016238
6.5 The Treatment of Vertical Diffusion
Eq. (4) contains two implied assumptions concerning vertical diffusion.
The first is that contaminant is not absorbed at the surface.
The general
. validity of this assumption is doubtful, but it is standard practice to
make it, and in the absence of specific information concerning the actual
-26-
-------
L______--- -
pollutants considered in the model the assumption is retained.
Some ad-
justment. for absorption at the surface may be included in the empirical
determination of the half-life in the exponential decay term, during 'cal-
ibration' of the model (Section 6.7).
The second assumption. of eq. (4)
is that diffusion in the vertical can be characterized by the single
parameter 0 (x), i.e., that the stability category as measured at ground
z .
level is a sufficient description of vertical diffusion.
This is rarely
so and a most important set of cases can be characterized by imposing
a fixed 'lid' to diffusion at a 'mixing height' H.
Thus, a common physical
situation is that turbulence scales are determined by ground level stabil-
ity until they grow large enough to be damped by an elevated inversion.
Further growth is then limited if the inversion is strong enough.
Various
methods have been used to accommodate this fact.
Fortak (1970) for example,
replaces the Gaussian term in 0 by a different analytical expression in-
z
volving 0 and H which yields computed concentrations asymptotic to zero
z
at height H. Turner (1968) advocates the use of the Gaussian function
until 0 (x) = 0.47H, uniform concentration in the vertical at twi ce this
z
distance, and linear interpolation in between. Turner's method is adopted
in the TRC model.
Quoting Turner -."The dispersion computation can be
modified for this situation by considering the height of the base of the
stable layer, L.
At a height 2.15 0 above the plume centerline the con-
z
centration is one-tenth the plume centerline concentration at the same
distance.
When one-tenth the plume centerline concentration extends to the
stable layer, at height L, it is reasonable to assume that the distribution
starts being affected by the 'lid'. II The following method is suggested to
take care of this situation.
Allow 0
z
to increase with distance to a value
-27-
-------
of L/2.15 or 0.47 L.
At this distance ~, the plume is assumed to have
a Gaussian distribution in the vertical.
Assume that by the time the plume
travels twice this far, 2XL' the plume has become uniformly distributed
between the earth's surface and the height L, i.e., concentration does
not vary with height.
For the distances greater than 2XL' the concentra-
tion for any height between the ground and L can be calculated from:
x = Q
/i;o
y
exp - C ::2 )
(13)
L u
for any z from 0 to L, for x >2 xL; XL is where 0z = 0.47L.
this equation assumes normal or Gaussian distribution of the plume only
Note that
in the horizontal plane.
The same result can be obtained from the fo1-
lowing equation where 0zL is an effective dispersion parameter because
;Z; L = 2.5066L and 0.8 ~ L = 2.5lL.
x =
Q
exp - (zl:/ )
(14)
~ a a L u
y z
for any z from 0 to L, for x >2xL; XL is where 0z
=
0.47L.
The value of
0zL = 0.8 L.
Application of Turner's method to the major point sources is straight-
forward; the formula applies directly.
For the area sources the distance
XL in Turner's formulation must be measured backwards from the receptor
point; for grid squares more distant than 2XL from the receptor the constant
. factor a = 0.8L applies.
z
-28-
-------
6.6
Calibration of the Model - Flux across the Boundary
Th~re are two aspects in which the model is not self-contained and
requires empirical calibration.
These concern sources outside the grid
boundaries and the half-life characterizing the apparent decay or loss of
material.
Flux of material from the greater New York City area was a major
problem in the application of the model to Connecticut.
The late Professor
Ben Davidson advised that 106 tons/year of S02 were emitted in New York
City and for distant receptors could be considered to be released along a
line 35 km long centered at the Battery orientated perpendicular to the
mean wind direction.
This source strength was further broken down into 50%
basic and 50% distributed by degree days.
The standard continuous line
source Gaussian equation was used to calculate the concentration of 502
at the border of the Connecticut grid for the appropriate wind directions.
One of 25 verification periods of observations (Section 7) was used to
confirm these initial estimates of boundary concentration, and at the
same time establish a decay constant for S02 of the form discussed in
Section 5.7 (see also Section 6.7).
The validity of the new estimates
was supported by subsequent verification data.
Sources from other regions were not known.
It was assumed that land
use in New York (outside New York City), Massachusetts and Rhode Island
is similar to that in Connecticut and that, apart from that originating
in the New York City area, there is no net import or export of pollutant
from the model area.
Using the model, a calculation was made of the con-
centration at the eastern border of the grid from Connecticut sources for
a west wind.
The simulated pattern is shown in Fig. 8.
The mean con-
-29-
-------
cent ration from this calculation was used as the concentration at all points
on the b.oundary with an inward-flowing wind component, other than any
affected by the New York City source.
Ground level mobile monitors and vertical soundings with airborne
S02 sampling equipment confirmed the general validity of these assumptions,
including the "raggedness" of the concentration profile.
In an application
of the model, direct measurements of contributions of sources beyond the
grid boundary could be incorporated if available.
6.7
Calibration of the Model - Decay of Pollutant
When the flux from New York City, predicted by the line source model
with no allowance for loss of pollutant, was compared with that measured
near the model boundary, it was apparent that either diffusion was much
more rapid than anticipated or that a removal process was active.
Measured
vertical profiles of S02 were consistent with the assumed diffusion con-
ditions so the discrepancy was considered to be the result of a removal
pro cess.
The same feature was noted in the calculation of the flux of
material out of Connecticut.
The only way in which a removal process can
be incorporated into a Gaussian model is set out in Section 5.7.
Turner
(1964) found that in St. Louis, Mo., the removal process for S02 could be
approximated by a decay characterized by a "half-life" of a few hours.
A
three-hour half-life was employed on the "calibration" test data and
yielded satisfactory results.
This half-life was adopted for all subse-
quent tests.
It is known that S02 does oxidize to S03 in the atmosphere
and time scales for the reaction have been measured or inferred from
several sets of observations and from experiments.
Although a wide range
-30-
-------
of results has been found, a three-hour half-life is well in the middle of
this range.
6.8
Heigh t of Source
Eqs. (1) and (2) contain the parameter H, the effective source
height.
There is considerable literature on the evaluation of this para-
meter for industrial stacks where the emitted gases possess buoyancy
an~ upward momentum, but there is no theoretically satisfactory and gen-
era11y accepted procedure for determining an "effective height" for
insertion into the Gaussian formulae.
There are several empirical pro-
cedures:
that adopted for major point sources in the Connecti.cut appli-
cation of the TRC model was to multiply the actual stack height by the
factor 1.5.
The minor (area averaged) sources were assumed to be at a
height of 10 m.
In the Toronto application the empirical formula by
Stumke (1963)
H = h + 6H = h + 1.5 ~ D + 65
u
D3/2
6T1/4
T
s
(15)
u
was used for major sources where wis exit velocity of the stack gas, D
is stack diameter, 6T is the excess of stack gas temperature over ambient
and T is the stack gas temperature.
s
Temperatures are in degrees Kelvin,
-1 d d' ,
velocities in m sec , an 1stances 1n meters.
Note that the formula
is not dimensionally self-consistent.
6.9
Summary of Operating Procedures - Flow of Computation for One Receptor
We assume all pertinent information has been read and the program is
-31-
-------
ready to operate on the input data at this point.
The first receptor
point is. selected and its i, j coordinates noted.
The following sequence
of events occurs (with reference to subroutine names in the flow diagrams
in the Appendix).
1.
Enter Trajectory (TRAJEC)
a)
Load the wind field stream values for the current
prediction period.
b)
Calculate the trajectory to the border of the grid,
incorporating new wind field stream values if re-
quired.
c)
Calculate the trajectory grid crossing points to
the border and the associated wind speed for each
segment.
2.
Enter Concentration (CONC)
a)
Calculate the distances, from the lengths of the
line segments, from the receptor to source grid
points.
b)
Enter vertical distribution routine (SIGZZ) to
obtain the appropriate cr value.
. z
c)
Enter horizontal distribution routine (SIGYY) to
d)
obtain the appropriate cr value.
y
Enter QWT routine to weight the central source square,
being used together with its eight neighbors.
Each
of the nine sources is weighed according to the
e)
value of cr .
y
Calculate the concentration from these sources and
-32-
-------
3.
add to those for previous line segments.
f)
Go back to 2 a) and repeat until all line segments
have been accounted for and summed.
g)
Enter the Boundary Correction Routine (BOUND) to
calculate the concentration to be added to account
for material emitted outside of the specific grid
area.
h)
Calculate the contribution from point sources
(QPTSRC) that may affect this receptor point.
This requires use of subroutines FILET (to iden-
tify points within 3 grid squares of the trajectory
line) and SIGYY and SIGZZ (for the appropriate
diffusion coefficients).
i)
Add the concentrations from area sources, point
sources and boundary sources to obtain the total
pollutant concentration.
Select the next receptor point and repeat from Step 1
until list of receptor points is exhausted.
-33-
-------
7.0
References
Anderson, Gerald E. 1971:
Mesoscale influences on wind fields, J. Appl.
Meteorol., 10, 377-386.
Bowne, N. E., A. E. Boyer, K. E. Trent, D. G. Cooper, 1971: An air quality
model for metropolitan Toronto, Paper 71-94 64th Annual Meeting,
Air Poll. Control Assoc.
Atlantic City, New Jersey.
Fortak, H. G. 1970:
Numerical simulation of the temporal and spatial
distributions of urban air pollution concentrations, Proceedings
of the Symposium on Multiple Source Urban Diffusion Models, Chapel
Hill, North Carolina. U. S. Environmental Protection Agency AP-86.
Hilst, Glenn R. 1967:
An air pollution model of Connecticut, IBM Scien-
tific Computing Symposium on Water and. Air Resource. Management '.
Yorktown Heights, N.Y.
Hilst, Glenn R., J. E. Yocom and N. E. Bowne, 1967:
A simulation model
for air pollution over Connecticut, Final Report to Connecticut
Research Commission under grant RSA-67-4, Hartford, Conn.
Hilst, G. R., 1969: Verification by observation of the application of
our air pollution model to the State of Connecticut, Final Report
to the Connecticut Research Commission, The Travelers Research
Corporation.
Hilst, G. R., 1970:
The sensitivities of air quality predictions to
input errors and uncertainties.
Proceedings of the Symposium on
Multiple Source Urban Diffusion. Models, Chapel Hill, North Carolina.
U. S. Environmental Protection Agency AP-86.
Meade, P. J.
1960:
The estimation of ground level concentration from
an elevated source, Int. J. Air Poll., 1" 303-313.
-34-
-------
Pasquill, F.
1961 :
The estimation of dispersion of windborne material,
Meteorol. Ma~., 90, 33-49.
1962:
Atmospheric Diffusion, D. Van Nostrand Company, Ltd.,
London.
Slade, David H., editor, 1968:
Meteorology and atomic energy 1968,
u. S. Atomic Energy Commission.
Stumke, H. 1963: Vorsch1ag einer empirischen Forme1 f~r die Schorstein-
uberhohung im Anschu1ub und eine Uberprufung bekannter Forme1u mit
zusatz1ichem Beobacktungs-materia1, Dem Ausschub II der VDI-Kommission
ItReinha1tung der Luftlt vorge1egt a1s wissenshaft1iches gutachten,
Institu fus Gasstromungen der Technischen Hockschu1e, Stuttgart,
West Germany.
Turner, D. B., 1964: A diffusion model for an urban area, J. Appl.
Meteoro1., 3, 83-91.
Turner, D. B., 1968:
Workbook of atmospheric dispersion estimates,
National Air Pollution Control Admin., U. S. Department of Health,
Education and Welfare.
-35-
-------
A P PEN D I X
Flow Diagrams
-------
Interrelational Structure of the Subroutines
in the
TRC Regional Air Quality Simulation Model
-rnERO
Main WINDS ~RAJEC
SUBM ---f.:ONC
SIGZZ
SIGYY
QWT
BOUND
QPTSRC .;. FILET
-------
"/1"/71
F3RTR4~ ~~DUlE
C 4Q('\ ~(l
,
,
1
,
9
10
II
12
13
1<
IS
I'
11
10
19
20
21
22
ZJ
24
25
76
U
20
29
30
31
'2
,~
34
35
36
31
"
"
<0
Ild::'1J IIIJCII,JI,J=l,n,J=l,~SPI
IJC - I.J R410S OF conRnl~HES OF R31NTS OF INTF>EST
READII.1D41 HlSIII
NHlS-I 4T THIS TIHE
~il$= S:1iIQ::~ HEfG~ AVE~Ar;E F~~ 1\ LAY'fR ''II MFTERS
DEAf) 11,101) ~AXP
~UR - '0. OF Rnl~T SOURCE CO~RO"4TES
DJ 1 1=1,~A.XP
J=2-ef
R-J-I
1 ~=Ar) Il.1')C;t l"'IJ(KI,I~IJ(J),'1PTIIJ,f)PTC"
I,J PAP 3F ::D')~nl'4ATES FJ~ PllNT SJJ~::ES
HPT = HEIGHT nF pnlNT $(HJR:E ~AOVF GOI:ULJ") IN ~ETE{S
OPT = S::WI):E STP.EN:;TH DF D]I'" $f]U;):f= AT I,J n"E PE~ CARD
In""1
00 20 lR-I.IZ
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wRITI;C4' rr"..
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no 21 lP-l,3
WRITE(,,' 1041 Q
21 CONTI'UE
REwt~n 10
REAI)(l,lOS) IBIJG,IPRI~T
CONI-zn
cnN2=T~
C~N3=T
CON4-DT
I)J '200 lP~6.I"'=l,"1\4""
READCl.1:)C)I (04
CALL $tl8M(R~EM,~SP)
/J-CO'I
TO=CON2
T-CO'3
OT-CJN4
11'1 5) llL=1,NSP
50 Q"!E\42Cl1l,lP""AINI=~~F."'lllLI
200 CJNT! 'UE
WRITFI12.10CI TITL
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15 OJ 71 N\lL=I,~SP
77 WRITEft2,1:)GJ 'lNL,IR"'E"'21~~Nl,'I,I=1,"'~\jI.
GO TJ 51
16 0) 19 NNL=1,N$P
79 WQITEf1Z,11C'1 ~NL,(Rt.tE"'2("JNL.II,I=1,"'MM)
81 STOR
E~D
....
64~O
6470
649n
6500
61 ~O
~ tt~O
6200
621':1
6220
6230
624~
t-')I:)r.1
6260
6Vif'
t,1h(";
63H
63AC
61QC.
6400
6410
6.20
-'14 ~('
6lt4r.
-------
0~/15/71
CHART TITLE - PROCEOURES
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03.15--->1 +------------------>1
I I I 11
- 10 I *-------------------*
. - 1 I MVC 0(4,51,PI I
o - I I I
. CLI * EQUAL I 1 MVC 0(4,6I,OLJ I
. ROUT+3,X'OZ' 0-----+ I I
. - I A 5 . -F ' 4' I
.. I I
. - I A 6,-F'4' I
. *-------------------.
IUNEQL ,"
I 1
I ' I
I (00CZ821 .
I - -
I . *
1 234.0(011 11 * CLC . lJNEOL
.-------------------* . ((1.,6It'lllF.O. *-+
, S1M 1,8.GRAB I . 0
*-------------------* ...
I * .
1
I
1 234.0CII-
*
18
12
I EOUAL. 1.
I . 14 .
I
1
1
07.C4--->1
ZOUNDS' "19
.-------------------*
L 6 , NUM 8
-
-
UGH
*
UNEOL. CLI *
+-----. IND~IT,X'OI' *
I + 0
I + .
I .
I *
I I EQUAL
I I
1 I
I I
I I
I I
I 1 234. 1C3I' 13
I +-------------------.
I I LM 1,8.IJOINOX I
I I I
I I MVC 01",11,0151 I
, I I
I 1 MVC 014.81,0161 I
I 1 I
I I LA 1.4111 I
I '-------------------0
I I
I I
1 1 234./j( 111 14
1 0-------------------+
1 I LA 8,4181 I
I , I
I I STM 1.8.IJOINOX I
I I I
I I MVI ,
I I INOH!T,X'OO' 1
I I I
I I L 7,SUBNUMB I
I *-------------------0
I I
I I
I 1 234.0111 I 15
I +-------------------*
1 I A 7,=F'I' I
1 1 I
liST 1.SUBNUMB 1
I --------------------*
I 1
+------------>1
SVOBA8F 1 16
*-------------------.
I LA 5,8151 I
I I
1 LA b,8161 1
I I
1 LM 7,8,GRAR I
*-------------------.
MVC
Clio ,61 ,SUA "IlJ MB
.-------------------*
I
01.15--->1
EXITO '20
---------
ExIT
*
.
---------
RETURN 114.121
. 2.03.
... START
-------
03/15171
FOJnRAN MODUl E
CARO "10
1
2
3
t.
5
6
7
~
~
In
11
12
1'3
14
15
16
17
1fi
19
20
21
27
23
24
25
26
27
2'\
29
30
31
3~
3~
34
35
36
17
3'3
:n
4~
41
42
43
44
45
46
47
4F1
49
INPUT Ll STPJG
AUTnFlD~ CHA~T SET - CJM8.E~G
(LlSTl,NA'1S0)
"'*.*
cn~TENTS
SUBRJUTINE CONC(TITl)
COM "'ON Al S (247) , I N I J I 20 J ) , IT R ".jON ( ') 0) , J TO ( 501 , I I I 2" 0 ) ,
x J J ( 200) , H I I 20) , I S TC 20) ,... LS ( 5) , x ( 2 J 0) , Y ( 2 o() ) , V H A R ( 200) , P ( 2 () n ) ,
XQl(200) ,CHI,IT01,TO,N,T,DT,GS,N~lS,XMI"I,XMAX,Y'1r~,Y'1AX,IBUG,DG,OS,
XSIG'-O,SIGl,SIGDT,NG,l,TQ,I,J,U,V,DX,Dy,~l,IR,KL,'1,ll,KK,M'1,S,Tl,F,
XA,B,TT,D,SIGy,Dl,SI~llN,HT,TI,OT,ITa,PI,K,lNO,C,BGHI,lO,JST(2J),
X PC HI, I P R r NT , MA X P, OP T (50) , H P T ( 50) , N GR [) S , MJ
COM'10N/COM01/ I04,ID5
COMMON/COM02/ 0(105,75)
DIMENSION TITl(20)
REAI)(4'ID4) 0
TT=T
0=-1;S/4.
TQ= O.
NG=1
M=l
SIGlO=5.
SIGY=.25
Kl=l
l=l
10 KL=Kl+-l
OL = ( I P ( K l ) - P ( K L -1 ) ) . * 2 ) + ( ( QU K L ) -;}L ( K L -1 ) I H 2 )
Dl=SQRTlOL) * GS
f)=D+[)l
TQ=TQ+(Dl/VBAR(Kl»
IF (TQ-TT) 13,12,12
12 TT=TT+T
M=M+l
13 Cflll SIGll
CALL SIGVY
Cflll OWT IQ,TITl)
E=((-.5*HT*HT)/(SIGl*SIGl»
E=FXP(E)*EXPITQ*lO)
IF( IRUG)2990,2990,29()';
2995 TI=EXP( TI I
Cflll PDU~P (SIGlO,SIGl,5,S,QT,5)
299r1 CHI=:H!+( (( 2.*QT*Dl) , (P I*VBAR «L) -s IG1».I:)
IFIKl-K)10,l1,l1
11 CAll ROLIND
IF(IPPINT) 23,23,22
22 WRITE (3,5111 RCHI
511 FORMAT (40X,' BOUNDA~'( C')NHIRUTIOf\! = '.lPE13.51
23 CAll OPTSRC
IF (IPRINT) 25,25,24
24 WRITE (3,502) PCHI
502 FORMAT (40X,' POINT S~URCE CONTRI8UTIJNS = ',1~E13.5)
25 CHI=:HI+8:HI+PCrlI
RETURN
END
*....
6020
6050
6rJ60
00003430
00003431
00003440
00003450
00003460
00003410
00003480
0000349(1
00003500
00003510
0(1003520
00003521
3522
3523
3524
3530
3540
3550
0('1)03560
00003580
3581
35QC
00003600
00003610
3614
3615
3616
3617
3618
3610
3620
362l
. 3622
0'0 or 1 63 r
-------
01/1<;/71
CHART TITLE - SU~ROUTI~E
C~~C( TlTl 1
-----------
CONC
-----------
01
----------------
I REAO OEV 4 I
I P C[) 1 D4 I
I IN I
IINTERNAL FOR~ATI
I INTO THE LIST I
----------------
I
I
1 NOTE OZ
. . . . . . . .
LlST.O *
. . . . . . . .
,
I
I 03
.-------------------.
I TT . T ,
, I
I 0 . - GS/4. ,
I I
1 TO . O. I
I ,
, NG . 1 I
I I
I ~ - 1 I
I I
1 S I ~ZO . 5. I
* * *
o
* . 0
.-------------------.
I
I
I 04
.-------------------.
I SIGY . .Z5 I
I I
I U . 1 I
I ,
1 L - 1 I
*-------------------8
I
01 o1R---> I
10 I 05
--------------------.
I KL = KL + I
I
, ~L = ((PIKLI -
I PIKL - III"ZI +
, IIOU1
13 I 09
.-------------------.
I I H
I I S I Gll H
, I H
I I H
I I H
.
.-------------------.
PAGE
01
AUTaFLO~ CHA~T SET - COMA.ENG
+------------------)*
I I
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I I I ISIGZO,SIGZ,5, ~
1 I 1 S,OT,51 ~
I I , ~
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, I
+------------> ,
Z990 I 17
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I II I Z .OOT.Oll I
1 I(P(OV8AR(KLI I
I oSIGZI'OEI ,
.-------------------«
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+-----.
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+------------------)*
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1 I I O~T H
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I 1 I ~
1 1 I H
I .----- -------------*
I I
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1 .-------------------0
I IE' I
I I II -. S*IfT*IfTl I
I I IISIGZ.SIGZII I
I I I
I I E ~ I
I I EXP(EI.EXP(Tg*ZJI I
I .-------------------0
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18
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1 I
I 1
I ZZ I ZZ
I ----------------
I 1 WRITE TO DEV 1
I I 3 I
I 1 VIA FORHAT I
I I 511 I
I I FROM THE LIST I
, ----------------
I I
I I
, I NOTE Z 3
1** . * * * * * * 0 0
I * LIST. 8tHI *
I .. * . * . . . * . *
I I
+------------>1
Z3 I Z4
.-------------------.
1 I H
I I QPTSRt If
I I H
, I If
I I H
*-------------------*
I
I
f
o
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.
.
25
.
(-/0) .
+-----.
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1
I NOTE Z7
I .............
I . LIS T . PCHI .
I ...........
I J
+-----------> I
Z5 I Z8
.-------------------*
I CHI ~ CHI + I
I RC~I + PCHI I
.-------------------.
I
I
I ZQ
IPRINT
o
.
o
Z4
.
I 1+1
I
I
I
f
,
I
Z6
----------------
I WRITE TO OEV
I 3
I VIA FORMAT
I 50Z
I FROM THE LIST
I
I
I
I
I
----------------
---------
EXIT
.
---------
-------
Il
03/15111
FORTIUN MODULE
CARD r-jO
1
2
~
4
5
6
7
8
9
11)
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
21
2R
29
30
31
']2
~~
34
35
36
31
VI
3Q
40
41
42
43
44
45
46
47
4R
4<:1
50
51
52
5'\
54
55
56
51
58
<;9
60
61
INPUT LISTING
AUTOFLOW CHA~T SET - COMB.ENG
(LlSTl,NAMSO)
*.**
CONTENTS
....
SUBRJUTINE Qwr (Q,TITL)
CDM~JN ALS(241J,INIJ(200J,ITRNON(50J,JTO(501,II(za01,
XJJI2001,HI(201,ISTI201,HLS(51,XI200J,V(2001,V8AR(2001,Pf200J,
XQL(200),CHI,IT01,TO,N,T,OT,GS,NHLS,XMIN,XMAX,VMI~,VMAX,IBUG,OG,DS,
XSIGl),SIGl,SIGDT,NG,L,T~,I,J,U,V,OX,OV,NL,IR,KL,M,LL,KK,MM,S,TL,F,
XA,B,rT,D,SIGV,DL,SIGZLN,HT,TI,Or,ITO,PI,K,LNO,C,8CHI,ZO,JST(20I,
XPCHI,IPRINT,MAXP,QPT(50J,HPT(5~),NGRIDS,MJ
COM~ON/COMOI/ I04,ID5
DIMENSION TITLf20J,QII05,751
IFIL-MJ 1,22,1
1 L=L+l
REAO(It'ID4. Q
IF(NG-NGRIDSI 2,2,98
2 NG=NG+l
22 IJN=1
XOM=.315
3 IfISIGY-XOM'll,ll,S
5 XOM:zXOM+.250
IFfSIGY-XOMI12,IZ,6
6 XOM=XOM+.250
IF(SIGV-XOMJI3,13,7
7 XOM=XOM+.250
IFISIGY-XoMJI4,14,8
8 XOM=XOM+.250
IFISIGY-XoMI15,15,16
11 S=CIPIKl)+P(KL-I)I/2.)
TL=ICQlIKLI+QLIKL-II)/2.)
1=(<;+.5)
J=ITl+.5)
QT=QII,JJ
RETURN
12 A=.000518
R=.911072
C=.021714
GO TO 17
13 A=.009310
8=.650280
C=.078060
GO TO 17
14 A=.029<'02
R=.482184
C=.100252
GO TO 17
15 1\=.Olttt883
8=.332108
C=.122090
GO TO 17
16 A=.066359
8=.233004
C=.125390
17 S=CIPIKlJ+PCKl-IJ)/2.1
TL=I(OlCKLI+OLIKL-IIJ/2.)
I=S+.5
J=TL+.5
OT = ( At< I 0 (1-1 , J -11 +Q C 1+1, J-l ) +Q C 1-1 , J + 11 +0 I 1+ 1, J + 1 J ) J + ( B. ( Q ( I , J) II +
XIC.IQII,J-IJ+Q(I-I,JI+Q(I+I,Jt+QII,J+111)
RETURN
98 WRITED,501
50 FClR"IATI' INSUFFICIENT SOURCE FIelDS' 1
RETU~N . .
END
6020
6050
6060
-------
,
I
,
I
,
,
1
,
I
1
,
,
I
,
I
,
,
,
I
I
,
,
,
1
I
1
,
I
I
I
I
I
I
1
1
I
I
1
I
I
1
I
C +1 1
*-+ I
,
I
,
,
I
,
,
90 1
I
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I
,
1
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1
,
I
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I
11 ,
I
,
I
I
I
,
,
1
+- -- -------- --...
11/15171
CH~QT T !TLf~ - SURPUlITl'JE
-----------
l.HH
-----------
«('I) .
+-----*
L - ~
I
I
1
,
,
I
I
I
,
I
,
I
1
I
I
,
,
I
1
1
1
1
I
I
,
,
,
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I
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1
,
1
1
,
I
I
, 11-10). 1.
I I. ? H .
I ,
, ,
, I
, ,
, '00
I .-------------------*
I ~r, ~ ~r, ... 1
I *-------------------*
, ,
"----------"'->1
2 ~ ,
II -1+1
1
,
I
,
,
1
*-------------------$
L = L + 1
*-------------------*
----...-----------
1 PF~O OFV 4 1
1 OC~ 104 1
1 I~ 1
/I\JTr:P\Jht FD~~,'\TI
1 I~T~ THF L!ST 1
----------------
".JOTF "4
. .. * ~ .. . , .. . ~ .
L!ST=Q .
*....
"G - %p 1 OS
*-------------------*
IJ~ = 1
)(0'1 = .375
*-------------------*
str;y ... X'llol
, (+)
,
,
,
I
,
I
4-------------------*
I XU~ = X('~ .. .~~0
*--------------------
01
°2
J3
'"
07
CQ
.-+
1)9
Q"TIQ,TITU
+------------------)*
S!;Y - Xl~
.
6
*-------------------*
I XOM = XO~ + .250
.-------------------*
.
S!~Y - XJ~
.
*-------------------.
I xn~ = XO~ + .250
.-------------------.
SIGY - Xll~
.
.-------------------*
, XJ~ = Xl" . .250
*--------------------
SI~Y - xn'1
.
. 1.2~.
1
I
I
I
,
I
,
I
I
1
I
I
I
I
I
,
,
,
I
I
. (-/3) 1
.-----1-----+
1 I
I I -----------
I' 12
1 , -----------
" I
,-----1------------>1
I 1 20
I .-------------------*
, A = .000510
I
I
,
I
I
,
,
I
I
I
I
I
,
,
I
I
,
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I -----------
I 1 ,
I -----------
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+------------> I
I
... 16
, I"
I
,
,
1
I
,
.
I (+)
I
,
I
,
I
I
1 (+1
I
I
I
1
I
I
, (+)
I
I
,
,
I
1
1~
. (-13 I
*-----+
12
14
III (-/:)
*-----+
10
. (-/J)
.-+
1
I
I
I
1
I
I
,
I
I
,
,
,
,
,
I
I
I
I
I
I
I C = .125)90
, ._------------------.
V I
+----------01.20'-->'
I 17' 25
I *-------------------*
I S = l(rCKU .
I PI"' - \1112,)
I
,
1
I
+------------...+
A
,
,
I
I
I
,
I
I
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1
I
,
I
,
I
,
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,
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,
,
,
,
~UTnFLnw CHA~T SFT - COM~.r~G
-----------
11
-----------
,
01.0°--->1
I
*------------...------*
'S=«(PIKU+ ,
I P,"l-IIII2.) I
, ,
1 TL = ((OLCKLI + ,
1 OLIn - \I)/~.I 1
I I
1 1=(S+.5I I
I I
'J=(Tl+.,) I
*------------------....
*-------------------.
11
~T=QC1,J)
*-------------------*
I
1
I 19
---------
EX! T
---...-...---
\3
" = .Q11072
( = .021114
«-------------------*
15
I
I
I
I
I
I
,
I
I
,
I
,
,
I
I
I
I
,
I
I
I
I
1
I
I
,
,
I
,
,
1
I -----------
, 14
I -----------
, ,
+------------------>1
,
*----------------...--*
~ = .OOQ170
4 = .65C?An
:: = .07R060
*-------------------*
+-------------+
1 .
. 23 .
15
*-------------------*
!I. = .O?t')zn,
A = .4~?1~4
(". = . I"'::) 2 ')'
*------------------...*
+-------------+
17
I.
21
?2
~ Ar, F
01
-----------
15
-----------
,
01.16---> I
, 23
*-------------------*
~ = . r44~"3
~ = .~32l"~
C = .r 270Q3
*-------------------*
+-----------_:.+
-----------
16
-----------
,
01.1'>--->1
I
24
0-------------------*
A : .C6f.359
" = .7.3'004
TL = IIULlKLJ +
QL(KL - 1111201
I = ~ + .,
.J = TL + .5
*-------------------*
,
,
I 26
*--------------...---....
, OT = (A' I () 11 -
11.J-!)+Qll.
, I.J - II + Q(l -
11.J+ll+OII+
Il,J+l)))+
I I"'(QIJ.JIII +
I IC'IQII.J - 11 +
I (JI 1 - 1, J) +
I Q( I + 1,.J) +
I ~11.J+III)
*-------------------*
I
I
in
---------
rx 1 T
---------
-----------
QB
-----------
1
01.""---)1
I
2°
----------------
1
I ,O:ITt- TI lr::V
1 >
I VI" FrjP~hT
1 ~r
1
1
1
1
I
----------------
I
I
I ,'9
---------
r XI T
---------
-------
0311')171
FORTRA~ "OCUlE
CARC NO
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
1 I:
17
18
19
20
21
22
23
24
25
26
27
2R
29
3C
31
32
33
34
35
36
37
38
'9
41)
H
42
l,3
l,4
4')
46
47
48
l,<;
50
~ 1
~2
53
~4
~5
~6
~ 7
58
5S
1:0
tl
1:2
t3
64
1:5
66
67
fB
69
70
71
72
73
[t,plT liSTING
AUTOFlOW CHART SET - cn~R.ENG
IL ISTl ,~A"St'
._**
CCNTENTS
SUBROUTINE QpTSRC
CC""O~ ~lSI2471,!NIJC200',ITRNONI50',JTO(50',II(2001,
XJ J C 200 I ,H I ( 201 , [ S TI 20 I ,HL S ( 51 ,X 1200 1 ,Y 12001 ,V RA R 1200 I, I' (200 I ,
XCLI2001,C~[,IT01,TO,N,T,DT,GS,NHLS,x~IN,X~AX,YMIN,YMAX,IBUG,DG,OS,
XSIGZC,SIGZ,S[GDT,NG,l,TQ,[,J,U,V,DX,Dy,Nl,[R,KL,~,lL,KK,MM,S,Tl,F,
XA,B,TT,C,SIGY,DL,SIGZLN,HT,TI,QT,ITC,pl,K,LNO,C,BCHI,ZO,JSTI201,
XPCHI,IpRINT,MAXP,QpTI501,HpTI501,NGRIDS,MJ
lOCO FCR"AT 141X,'POINT SOURCE IN GRID UNIT 'I
CO 2 [=1,241
2 AlSI[I=O
pCH[=O.
CALL F[lETI2,AlS,NUMI
CALL FIlETll,ALS,NUMI
IF IIBUGI 10,10,5
5 WRITE 0,5011 NU",cITRNONI[I,JTOI[I,I=I,NUMI
~Ol FCR"AT 120[51
10 IF I~U~I 20,20,40
20 RETLR~
40 no 150 K[=l,NUM
TQ=-TO
!=ITR~OIK[1
J=JlCIK[1
N"AX=2-,.,AXP
r.0 50 Kl=1,NMAX,2
Ip=!I';IJIKlI
. J 1'= [N ! J I K L +11
IF. ([-II'I 50.60.50
60 [F (J-Jpl 50,7C,50
5:) CC~TI~UE
10 "K=IIL/2 +1
N=l
&A A = S Q R T ( I ( 1'111 -! 1** Z I + I ( QL 111- J 1** 2 , I
~X=AAA
CO 80 l=2.K
'I''I'=SCRT II (I'I L I-! 1**21 +IIOllll-J I- *2))
!FIXX-YYI81,81,90
90 XX=YY
N='..l
80 CONTINUE
81 D =0.
F=O.
SU,.,=O.
!F(t,-21100,lCO,110
100 WRITEI3.1COCI
IIJ CO 120 l=2.N
D l = S Q H II ( PILI - I' IL -11 .**2 I + II Oll L 1- CL Il -1 I ,.. Z I I
SU~=SUM.VBAR I LI
C=O'Dl*C;S
F=F'1
FT=(Dl-GSI/VRARILI
TC=ltHT
12') CC~TINUE
V=SU/l/F
N=l
/I=! I TO+TOIITI +1
UllSHlZ
CAll SIGVY
SIGY=S[GY*GS
YY=YY*GS
H =HpT I" ~ I
IF II BUG 1 140,14C,130
130 WR[TE 13,,)001 SIGY,SIGZ.YY,H,XX,V,DT,TO,QpTI"KI
~CC FCR"Al I1pI0E12.31
140 El=-.S-CIH*.ZISIGZ.*21 + IY'I'**2/SIGY*-211
E2=-.S*II~.*2/S[GZ.-21 + IIYY-.50.GS...2/S[GY**ZI'
E3=-.r;*(lHU2/SIGZ."21 + IIYY-.25*GSI*.2ISlGY.*ZII
E4=-.S-IIH..2/SIGZ*-ZI + IIYY+.2S*GSI.*ZISIGY.*ZII
E5=-.5*IIH.-ZISIGZ**21 . I(YY+.50*GSI*.2/SIGY.*211
I:l=EXPIEll + EXPIE2) + ExptE31 + EXPlc41 + EXI'1E51
E=I El/r;..-EXPITQ*ZOI
1 ~ C PCH I =PCH I + II I tP T I "K '*2. II I P I**Z *V*5 [GY. S I GZ II*E ,
RETURN .
E,,"O
.*.-
6020
6050
6060
-------
2111<;171
CHART TITLE - 5lE"rLTINF
-----------
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-----------
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I
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--------------------*
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OR
.. 1-/0 I
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I "'". ~ ,... .'. ~ .. )I;. II. .. ..
I I
'-----1------------>1
I I ILQ I II
I I 0-------------------"
I I I EI = -
I I I .~.((Hn2/SIr.l~'1I'
I I I ?I .
1 I I IYY'"Z/SIGY"?1 I
I I I
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I I I. ':-*' (H~.t2/ ;IGZ-t-
I I I 21. I 1 YY -
I I 1 .5: 'GS I
I I I "Z/SIGY"ZII
I 1 .-------------------.
I I
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PAGE-
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+ -- ---- - --- -- )"
130
(9
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------- ---- -----
17
~-------------------(
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I .~.((H..?ISIr.l"'.
I 21.((YV-
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I "2/SIGY"711
I
I F4 = -
I .r;-(H*.2/)IG1..
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~:-------------------.
13
*-------------------~
I Eo = - I
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FORI.AN .COULE
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11
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17
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19
20
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42
43
44
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71
72
73
74
75
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77
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111
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AUTOFLOW [HA.T SET - CO~R.ENG
ClI5T1,~I\IIISCI
..... 4
CONTENTS
SUP~CUTI~E TRAJECITITLI
CC..C> AL5C2411,INIJI2001.lTRNCNI501.JTOI50I,1112001.
XJJ(2I1CI ,I-H2C.,ISTt ZO),HLS(';) ,XC 7001 ,YC lOOI ,VBAR(2001 ,PfZO,)!,
XOLllC')1 ,(HI, I TOI ,rO,N, T ,OT, G<;, ,.""HlS XMIN, X~AX, yM IN,YMAX, IRlJG,OG, OS,
XS IGlG, SIG1, S IGOT ,NG ,L, TO, I, J,U,V ,DX ,DY,NL, (P, I(L, "',ll ,KK ,MM, S, TL ,F,
)(A, 1:\, TT, C, S IGY ,Ol ,5 IG7LN,HT, Tt ,QT, ITO,PI ,K,LNO,C ,eCHt,ZO ,JSTI20 I,
XPCHI,IPR'~T,~AXP,CPT(50I,HPT(51.,NGRIOS,~J
cc,..,.,(t\ICC~OII 10',105
r("'~(~/C(w!21 Q(105,151
c I ~ E r-. s 1C N PSI (t J 5,15 I , T IT L( 2t~ )
fOUIVAlENC!:(O(l,lt,PSlf 1,111
QF:~Cfl)' )CSI PSt
10;C.
IT;T
>111;1 ll>lI
YIII;JJ'>lI
[;1
2 t=x4L1
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u= ( f PS I ( I , J + II-PS 1 ( 1, J-l ) ) -OS I
v= -(PSlll+l,.JI-PSIII-l,.JlJ*OSt
~ CX=fU*O(;)
r.Y;I~.C(1
OXI=AASICXI
OYl=~BSI CY t
IFCCXl-l.t4,4,5
4 IFfOYl-1.lb,f:,5
., CG=HC:/Z.I
cr=' r.1 12. I
GO TO 3
6 IF 101-.31 9q5.~q5.qqo
GC;I) IF ID'f}-.3) c;C;1,GC;1,QQb
QCj7 CG=2..C(
DT=2.IIIoCT
GO TC ~
CG6 lJV=«U..2t+(V..ZJ)
VPA.IlI=SORTIUVI
l =l + 1
IF(L-19<;) 112,113,113
113 X(l)=:cll-ll+0X
Yll)=Y(L-lt+OY
X(L+ll=)lCLI
Y(l+II=Y(LI
roc TC 1)0
112 XILI'XlL-lI+OX
v-ILI=IYIL-II.OYI
TC.IC.OI
IF I )I(ll-XMINI130,130,1
7 IFtX(Lt-XMI\X)A,130,110
q IF(YCL)-Y,",INI130,I~O,q
g lFCYfll-YMI\XllO,130,130
IJ IF IIC-111 2.11,11
11 TT=T T + T
IFIIC~-12ql 4C1,4C1,C;12
4(7 REJI[IS'IC51 D$(
GC 10 2
C12 WRITE 13,Qq31
>III;C.
YILI=C.
993 F(R~H (40X,'RF6Ct-IED FND OF I)ATA RF.FQRF REACH[NG GRID BOUNDARY')
I:CP(U=JlllJ
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Ve~"lll'~PA.Il-11
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(ALL POU~P IXlll,VRARC2JO),S.
12C5 ~CF"'A1(2QX,IIO,4FI?21
12IF()llll-XfL-lIH3,15,14
13 LL.2
GO Tr It
14 II = ~
r;o rf: 16
15 ll=1
16 IFIY(l I-YfL-IH11,l9,l8
17 I( 1(=3
GO 10 2C
18 I('t<=t
roc 10 20
11 KK=C
2) "''''=KK +ll
GO TC( 14,1,II)O,160,110,lAC ,1 Q"\,l 1:10,111:- ,112l'! ),MM
11~a IFIl.-LI112,q3,o~
C;l P(KU=XCl!
Ulll(ll='dll
VE~R(l
-------
,
!
I
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,.
13/1':/11
(/R[: NU
114
1I~
116
117
11 ~
119
1 ]t;
121
122
12 ,
124
125
126
127
12"
12 9
l3e
1;1
1~2
1'3
1 ~"
135
1~6
I? 1
138
1 ~Q
14~
141
142
143
144
145
146
147
148
149
ISO
1 ~ 1
I S2
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1 ~4
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156
IS7
IS8
1~C;
160
1t1
162
113
I"
If 5
166
In
1 t R
169
170
11\
172
173
174
115
176
177
178
170
100
lEI
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10
104
l' 5
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107
IPP
IP9
190
101
192
1<:3
1°4
195
196
197
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212
en
214
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216
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n3
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22;
226
727
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R= C I ~. C ~-)( (L-ll . . +V (l-I) .
IF('/!-SI45,45,46
45 PIKlI=S
PCKl+ll=A
CLCKL+U=Tl
~eIlRCl'=X(l-11+.5
IY=YIlI+.5
lYf"=Y(L-lt+.5
IF (U(-l)(~1 4H,t.,q,48
"'=~+1
JF fl'(-LYMI 50,~1,50
""="'+2
IF (MI ~2,:2,53
L=l+l
Gr Tr 11~O
I: 1 XL =l x'"
Vl"lY
~=)(l+.5
TL=Yl+.:
51) F = f IV (ll-Y C l-l I ,,( x (ll-I( (L -11 I I
r.c T[':(IH,1)7,58),~
!:6 P(Kll=S
OLlKLI= II H I S-X I L-IIIIHI l-III
VEARIKlI=V~ARILI
r:n TO 61
!:7 (If I1,=«(1=*1 ~-X(L-U f)+YtL-l II
4F
49
5C
51
~2
AUTOFLOW CHART SET - COMS.ENG
CONTENTS
-.*.
r.OOt')09Qi)
00001000
00001010
00001020
000010,0
"0~OI040
000010S0
CC.00I060
00001070
00001080
00001090
00C'1100
00001110
00001120
00001130
00001140
<10001150
(0001160
00001170
OOOOIIFO
00001190
00001200
00001210
00001210
GC001230
COOOl240
00001250
00001260
ooao 127 0
000<112RO
00001200
00001300
00001310
00001320
00001330
00001340
000'11'0
00001360
00001370
00001380
00001300
')0001400
00(01410
00001420
00001430
OOOQ1440
00001450
0P.0014tO
00001470
f"J00014BO
00001490
00001500
00001510
COO~1540
aOf')1)15c;C
0000156L
0000157C
000015QO
00001500
0.0001600
OOrJOlf,IO
OGO~1620
00001630
00001640
00001650
00001660
')00')1670
00001680
000016QO
OJrOl700
ooonl 71 0
00001770
00001730
OCOC114i)
OQ001750
00001760
00001770
000')17P.J
0000179n
0"'.001 ROC
OOOOIRIO
00001"70
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)C )01850
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'~c. '101 Q7G
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00002020
00002030
00002C4C
00002050
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lJ = - .E2t;C;~
.-------------------*
. I. C9.
... 1 R
------._---
1 ~
-----------
I
01.02--- > I
I 05
*-------------------.
1 B = . 749 ~ 2 I
I 1
I C = . C 13 3< 1
I I
I . = - 1.7t~1 1
*-------------------.
. I. C 9.
... Ie
----------.
I'
-.---------
I
01.02--->1
I
*-------------------.
I '=-3.Z<1tQ I
I I
I p. = 1.1113E 1
I I
I (=-.CI~12 1
*-------------------*
. I.C9.
... IP
SIGZl
C6
-----------
15
-----------
1
01.02--->1
I 07
*-------------------*
1 ~ = - 3.70065 I
1 1
1 R = 1.25914 I
I 1
I C = - .04205 I
$-------------------*
I
I
I
. 1.09.
... Iq
1
'----------01.100-->1
I I 1'5
I *---------------~---.
1 I SIGl = SIGlO I
I .------------7------0
I 1
I 01.14--->1 .
, 22 I 16
I .-------------------0
I 1 HT : O. 1
I *-------------------*
I I
I 1 (------------.
I 23 1 17 I
I - -------- 1
I . E KI T 0 I
I --------- I
I 1
I 1
I I
I I
I I
I ----------- I
1 24 I
I ----------- 1
1 I 1
I '------------>1 1
o '-/01 I I I I R I
. SIGZ - SIGlO 0-----+ I 0-------------------0 I
o A I I HT=HLSfLNOI I I
I I I I I
I I 1 SIG10 = SIGl I I
I I 0-------------------0 I
'I 1 I
" , I
I 1 .-------------.
1 1
I I
I I
, I
I I
I I
I I
I I
I 1
, 1
I 1
1 I
I 1
.. (-/01 I 1
.-----1 -----.
I
I
,
1
1
1
I
I
I
I
I
1
I
1
(-I 1
*-----+
AUTOFLOW CHART SET - crMB.ENG
-----------
-----------
I
01.02--->1
, OB
*-------------------*
A = - 4. 190 3 ~
B = 1.29161
c = - .04535
*-------------------*
I
(>1.C4.-->1
16 I 09
*-------------------*
1 SIG1LN = 'A' I
I IBOALOGIOI + I
/ (CO,ALOGI0,.o21/, I
I I I
I I
I SIGl = I
, EXPISIGlLNI I
.-------------------*
I
1
1
o
o
o
25
*-------------------*
PI! = ...OHIIMI I
*-------------------*
I
I
1
0 12
o
.
0 SIGl - PII
o
..
I f + I
I
I
I
I
I
,
19 13
o
HI (M I -
OH[fM-II°
o 0
o
1 (0 I"
J
I
I
I
I
20 I 14
*-------------------*
I
I
S!G10 = SIGl I
.-------------------*
SIGl = HI(~I
. 1.16.
... 22
PAGE
Cl
----...------
16
21
-----------
10
o
o
o
, 1'1
,
1
I
,
I
1
11
-------
01/15/71
FOR-IRAt\ ~OCUlF
CtRf: NC
1
2
3
4
'5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
;;0
, 1
22
23
24
15
Hj:LT LISTING
AUTUFLOW CHA" ) , 1ST ( 20 ) , HL Sf 5 I , X ( 20;) ) ,Y ( 21"':)) ,V j M< ( 7. r; 0) , P ( 2 OJ ) ,
XQl(20n),C~I,ITOl,TO,~,T,OT,GS,NHLS,X~I~,X~AX,YMTN,YMAX,IBUG,OG,nS,
XSIGlO,SIGl,SIGOT,NG,L,TQ,I,J,U,V,OX,OY,Nl,IP,KL,M,LL,KK,MM,S,1L,F,
XA,B,TT,C,SIGY,DL,SIGlLN,HT,TI,QT,ITO,PI,K,LN1,C,RCrll,lO,JST(ZO) ,
XPCHJ,TPRIt\T,MAXP,QPT(50),HPT('5J),NGRIOS,MJ
TI=TQ+TC
TI=ALCr,(TIt
GC T(l,2I,N
1 ~J=IST(M)
2 GC TC(11,12,13,14,15),MJ
11 S I G YL t\ = ( 1. 2 <; 5 21 9+ ( . 947684 * T I ) - ( .02? Z 892 * (T I * ~;<"'21))
GO TO 11:
13 SIGYLt\=(-.57577+(1.193406*TT)-(.037105P7*(Tl**Ztt)
GO TO 1 I:
14 S IGYLN= (-1.1f.482E+( 1. 310C8*TI)- (.04C815 5* (T 1**2) t)
GO 10 It
1 5 S [ G Y L N == ( - 3. 0 94+ ( 1 . 3 5 q '5 1 8'* T I ) - ( . J ~ 7 ') 3 2 * C T I * ~ 2 ) ) )
16 SIGY=F.XP(SIGYLNt
STG'r=SIGY/GS
RETURN
END
hOZO.
6050
606C
-------
03/1S/71
CH~~T TITLE - S~EPOUTINE
ALJTOFlOW CHART SET - COMB.ENG
PAGE
01
S ICVY
-----------
-----------
SIGVY
13
I
-----------
-----------
01
*-------------------*
1
01.04--->1
I 07
.-------------------*
I SIGYLN = I
I (-.57577 . I
1 11.193400*TII - I
1 1.0371G5R7*fTI** I
I 2111 I
$--------------------
TI= TQ + TO
TI = HOGITII
*-------------------*
1
I
I 02
$-------------------*
I COMPUTED GO TO I
I FOR N ,
*-------------------*
I 1 1.03 I
I 2 1.04 t
*-------------------*
+-------------+
I
1
I
I
I
I
I
I
I
I
I -----------
I 14 /
I -----------
I I
I 01.04--->1
1 1 08
I *-------------------*
I I S I GYl N = I
I I (-1.764R2R. I
, 1 (1.31C08*TII - 1
I I (.04)8155*ITI**21 I
1 I II I
I *-------------------*
I I
I I
1(------------+
1
I
I
I
I
1
I
1
I
I
I -----------
I 15
I -----------
I I
I 01.04--->1
I I 09
1 *-------------------*
1 I SIGYlN= I
+-------------+ I I 1-3.094 . I
I I I 11.359518*T II - I
I I I (.017532* fT1U2l) I
I I I I I
I I *-------------------*
I V I
+----------01.05*-->1
A 16 I 10
I *-------------------*
I SIGY = I
I F.XPf SIGYlNI I
I I
I SIGY = SIGY/GS I
I *-------------------*
I I
I I
I I 11
I ---------
I .. EX IT
I ---------
I
I
I
I
I
I
.-------------+
I
I
I
IF OUTSIDE THE
,
01.02---> I
1 I
RANGE
03
*-------------------.
I ~J = ISTI~I I
.-------------------$
1
C 1.02--->1
2 I
04
*-------------------*
I COMPUTED GO TO I
I FOR MJ I
*-------------------.
1 11 1.05 I
I 12 1.06 I
1 13 1.07 I
I 14 1.08 I
I 15 1.0C; I
*-------------------*
I
I
I
IF OUTSIDE THE RANGE
I
Cl.04--->1
11 I 05
*-------------------*
I SIGYLN = I
I (1.295219 + 1
I (.947684*TI/ - I
I (.0252R92"(TI;*21 I
I II I
.-------------------~
-----------
12
-----------
I
(1.14---> I
I
06
*-------------------*
I SIGYLN =
I 1.2200925 +
I (I.C941>6R*TI J -
I 1.0326761E*(TI~*
I ? II I
.-------------------*
-------
')3/15/71
FJRTRAN MnOUlE
CAQn \JO
1
?
~
4
c;
6
7
1
9
10
11
12
13
1/+
15
IS
1 7
1~
19
21
?l
:?2
21
24
2')
26
~7
?~
? ,)
30
31
3~
33
~ t..
~,
3')
INPUT lISTI\!:;
L\IJTiJFLLH., CHA~T s,:r - Cn"1n.r.r~r;
(L1STl,NAMSQ)
****
Cn":TE'-JTS
\,(~;X'&;
SUH~JUTtNE QWT9 (Q,TITL)
C,)~"'O"J ALS(?47t, INIJ(Z'))), ITqr>,JCJ~('5Q) ,JTO(50), 11(21:),
X J J ( ?:) 0) ,'"i t ( 20) ,IS T( 20) ,-i L S ( 5 ) , X ( ~ 1 (I) ,Y ( 2 CO) , V R ~ P ( 2 ~ ') ) , p ( 2') C ) ,
X OL ( 2:) 0 I ,C HI, I T 01 , TO, \I , T , 0 T , GS , \I H L S , X '-1 IN, X M A X , Y'1 I \J , Y ~ A X , I RUG, f) G , n S ,
X S I G l J , S I G l , S 1 G D T , NG , L , T J ,I ,J,;J, V ,OX, DY , N L , I R , i( l , "1 , L L , K K , M £-1, S, T l , F,
XL\,R,TT,D,SIGV,~L,SISlLN,HT,TI,~T,ITiJ,PI,K,LNQ,C,3CHI,lO,JST(20),
XPCHI,IPR[NT,~AXP,QPT(50),HPT(50),NGRIDS,MJ
DIMENSION 1(105,75),TTTL(20)
IF(SIGV-.375)10,10,1
1 IF(SIGV-.625)9,9,2
2 IF(SIGV-.875)8,8,3
3 TF(SIGV-l.125)7,7,4
4 IF(SIGY-l.375)o,o,5
5 Fl=.25810e
F2=.483784
G'J T J ( 11 ,12 ) , I R
6 Fl=.211856
F2=.516288
GCJ TJ(1l,12),IR
7 Fl=.158656
F?=.'d~268P.
G'J TJ(ll,I2),IR
13 Fl=.OQ6PO
F2=.8G640
GO TJ(11,12),IR
9 Fl=.:)22750
F2=.95450
Gl Tl( 11 ,12),IR
10 Fl=,').
F2=L
GJ TJ(ll,1Z),IQ
11 QT=Fl*«()(f,J-1 )+()(I,J+1> )+(F'-*(.)( I,J»
Q ~ T' J~ '\J
1 2 r.' T = F II!< ( 'J ( I - 1 , J ) + Q ( I + 1 , J I ) + ( F ? '* Q ( I , J I )
PrTURr\J
ff\JG
6n20
6:)C;O
6060
-------
01/1 J)/71
:~APT TITlF - SU~RnIJTI~E
-----------
-----------
SIGY - .375
o
. SIGY - 1.125
. SIr.V - 1.1:7"
.
~-------------------*
FI ~ .25610"
F2 = .4!!:'C7A4
~-------------------*
~-------------------*
C1J"'DUT FI") r.1 f'J I
FnQ I Q I
*-------------------*
t! 1.1 q I
12 1.7(' I
--------------------*
I 1. OR
,'hlT~fQ,TITL)
f)o I
'. I ~~
.-------------------.
F1 ~ .21l65b
1=2 = . 5767.q~
*-------------------.
O~
*--------- -...---...-...--.
~J~PUTEn GO TJ I
FIlP IP. I
*------------...------.
11 101 6 1
12 1.2~ I
*-----...-------------t
I
I
I
IF JUTSIOE THt R.~GF
I
01. J4---> I
7 I D
*-...----...------------t
FI ~ .15665'
F2 = . ~A2bJ:l!t
...._-----------------.
11
*-------------------t
:O.PUTr-O GO TJ
Ft)1'l I R
*-------------------t
II
12
101' I
1.20 I
*-----+
I
I
I
I
,
I
I
I
1
I
I
I
I
I
o «-1011
Sl~V - .b2~ .-----1-----+
I ,
I I
I I
I I
I I
I I
I I
I I
I I
, 1
I I
I I
I I
I I
. (-/~ I I I
SIGV - .87!i *-+ I I
I 1
I I
I I
I I
1 . I ,
. 12 . I 1
I I
I I
I I
I I
I I
I I
I I
I I
. (-/01 I I
I I
I I
I I
I I 11
I I .-------------------..
1 . I I C:1MPIJT(1J r.n T')
. 10 . I I F OR I.
I I h------------------.
I I II 1.1 B I
1 I 12 1.2J I
I I *-------------------.
1 I I
I I I
I I I
1 I IF OUTSlnE THE :;J:A'4GE
. (-/C I I I I
I .------------> I
I 9 I I'
I *----...--------------*
I FI = .022750
I
1 . I
. 0' . I
I
I
I 15
I --------------------*
I C,l~P'JTFO GO T'1
, F'1P "
I .-------------------.
I ! 1 1.1. I
I 12 1.20 ,
I .-------------------"
I I
I I
I I
I IF ,1UTSIIJE THE P.A'.;GF
I I
+------------------)1
lJ I l'
.. - -- -- - - -- ------ - - - -.
* --- -- ---- -...------ --.
I
I
1
IF OlJTSIOF THE
I
n1.03---) I
. 1
R. ~ Gf
12
*-------------------..
Fl = .0Q6AO
F2 = .AC640
*-------------.-----.
F? = .Qr;4':)!1
..-------------------"
F 1 .: (.
F 2 = I.
---------------------
+ - -- - - - - - - - - - --+
+------------- -----)*
I
17
*-------------------*
CO~PlJTEO GO TO
FO~ IQ
.-------------------*
I 11 1.19 I
I 12 1.2Q 1
.-------------------.
I
I
I
IF OUTSIDE THE
I
01.07*-->1
11 I
R.~GE
16
*-------------------.
I Qf = FIO(Q( loJ - I
I i) + f)( I, J + I
I III + IF2*Q(I,JII I
*-------------------.
I
I
I I q
---------
EX IT
---------
----------...
12
-----------
I
ol.n.-->,
I
2C
*--- -----------------
OT ~ FI'(Q(I - I
1. J 1 + Q( 1 + I
1,JH + I
IF?*Q(!,JII 'I
*--- -- -- ---- - ----- -- '"
I
I
I 21
---------
EXIT
---------
PM;[
01
-------
'J~IlSf7l
J:"lQT>: ~'I ","n:JI J:
[" " ;; n 'I~J
1
.,
.)
.:.
')
6
7
T'.:PlJT LT STT'JS
A!J r F L'11' r I',~ "< T S:: T -
( '""' \1 1'< . E 'J :~
( L T S Tl , 'Jt '1 S '.J)'
~~1't't
~l\iTC\JTS
S I JP '0 lIT J 'J ElF :<"1 ( A P 1\ Y , !\J )
)IM::"JSIi1"! 6PAY(11
[1:) 10 T=1,':
A'1I\'( J )=".;1
10 CJNTI"JUE
RFTIJR'IJ
E'J[)
~ ;r.;'t't
- -- -- ------
f
l~ <:;10
I
-- -- -- -----
HJTE ')1
*'****
:#
* . .tc ~ JCIC 't-
...
""
!\«::GI"J '"""In Lf""I'lP
1~T=1,~j
...
:r.:. -/: ir 'II ~ II:'"
I
+------------>1
I I ') 2
I ~-------------------~
I !\H.Y( II = :).~.
I ~-------------------~
I
I
I
I
I
I
I N.J"
+-----*
~ . Q. t~ r
1')
...
03
'"'
~
*'
,.
..
ec.
E"Jn '1= ["In
LW1P?
*
*
~
'"
*'
."
IYF5
I
I
I
I
I
I Cl.
---------
Ie
E XI T
~~
---------
-------
t'~/b/ll
H-;~ lRA~ ~CCUL r
r~qn 'lJ1I
1
2
::
'+
~
6
7
p
q
1-;
11
12
13
14
I"
H
17
I g
1 '~
70
;; 1
2?
n
24
7'5
76
27
tel
2<)
')
? 1
,??
?3
3"-
,~
~":
H
;~
,q
4<::
II\PLl LI STING
AtiTUfLOW CHART SET - COMA.ENG
ILlST1,~A"C;CI
.-. .
CCNTENTe;
C;UERQUTINE wiNDS
CC""(II/C(Mrlll 104,100;
CGMMCII/C(Mu21 Q(lC5,751
CI~EIISI[N GII05,75I,PNS( l'16,77I,PE"'1 ICI'>,771
E Q LlIJ flU II C E 1 Q (l ,11 ,G 11 , 11 1
100 FCRMATI:F10.21
101 F(R~~T(20F4.01
nc :20 J=l,77
I{FH{ldOl) (PNS(I,JI,I=l,lOAI
, 20 cr 1\ T I ~I JE
00 ,30 J=l, 77
REH(I.lC11 IPE..II,JI,J=l,lO/',1
'3J CCI\lII\UE
C:.~f:22/;t;J.O
1(5=1
nc 30 LPP=l,lS
C~LL lFFO(G,7P7~1
REtC(l,1001 UQ,VO
C HF SIGNS nF II AND R OEP~II;D CN THE INITlfIl WINO fJlF
-------
" 'II ;/71
Ct-'~T T ITLt - SUfPCUTINE
WINOS
--_.-------
WINO~
-----------
1
I
1 NOTE 01
.. . . . . . . . * . .
. REr.IN OC LCOP .
32f) J = 1, 77
...........
I
+------------>1
I 1 02
I ----------------
I I OI'D FPCO OEV I
I I 1 I
I I VI' FCPMU I
I I 101 I
I I INTO Tt-E LIST I
1 ----------------
I I
I I
I I 'CTE 03
I .".*0......
I LIST. .
, (PNStI ,JI ,I =
I I.JC< I
I ........
1
1
I
1
I
1
I
? 20
O'
NC
+-----*
E'C CF CC
lCCP?
I YES
1
1
I
I
1
I NOTE 05
. . . . . . . . . .. "
. AEGIN DC LCCP .
. 330 J = \, 17 .
......111*...
1
+------------ > I
I I 0'
I _u_------------
I I PE'O FOCO OEV I
I I I I
I / viA FeRMAT /
I I 101 I
I I INTO TH LIST I
I ----------------
1 I
1 1
I I NerE 07
I ...........
I LIST =
1 IPE~I I .JI ,I =
I 1, lOt I
I ............
I
I
I
I
I
I
1
?10
(P
'C
+-----*
E'O CF cr
LrCP?
IYFS
I
I
I
I
1
1
Oq
;$1----- --------------.
( . . ~t22UOO.O
105 = 1
.-------------------.
.....
I
1
, NCTE 10
......
BEG IN O( LOOP
30lPP=I,15
..,...".....
AUTOFLOW CHART SET - cooe.ENG
+---------- --------)*
1 I 20
I *-------------------*
I I ONSI2 = PNSII +
I I I.J + II -
I I PNS(I,J + 11
I 1
I I SUMl' (OEWI +
I I OEWI21".5 + SUMl
1 1
I I SUM' = IONSI +
I I 0~SI21".5 + SUM4
I *-------------------*
1
I
I
I
I
1
1
1
I
I
I
I
I
I
I NO
I +-----*
I I
I I
I 1
I I
I 1
I I
I I
1 I
I 1
I I
I I
I I
I I
I I
I I
I I
1 I
I I
1 I
I I
I I
I I
I I
I I
I I
I I
I I
I I
NOTE 17 I I
........... I I
AEG!' 00 Lnop I I
21J=1,74 I I
.. " . .. , , .. .. .. ".. I I
I I I
1<------------1------
I I
GP'OIENT OF PHI EAST I
\oIF. co TIN I DIP EC T I ON I
1 I
I I
I If I
*-------------------* I
DEW! = P'"II' I 1
1,JI-PEWCI,JI I I
.-------------------* I
I 1
I I
I I
~Q~nIENT OF PHI N1NTH I
SCUTH IN I OI"FCTlflN I
I I
I I
I lq I
*-------------------" I
ONSI = P~S(I - I
I,JI - PNS(I,JI I
I
I
I
I
I
I
I
+-------------+
-- -------- ------
I RE~O FPOM
I 1
I VIA FORMU
I 100
I INTO THE LIST
------- ---- -----
....".
NOTE 13
.. . . "" .
+----------01.10*-->.
I . I 11
I .-------------------*
, I I H
I 1 1 ZERO H
I I I I G, 7875 I H
I I I H
1 I I H
I *-------------------*
I
1
I
I
,
I
1
I
1
I
I
I
I
I
1
I
I
1
I
I
I
I
1
1
1
I
I
I
I
I
1
1
I
I
I
I
I
I
I
1
I
I
I
I
1
I
1
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
1
I
I
,
I
I
I
I
I
I
I
,
I
I
I
1
I
I
I
,
I
I
+------------- +
LIS T .. UC, VI')
* . * .. . .. . . . - ..
1
I
I
THF. SIGNS nF A ANn B
DEPEND ON THE INITIAL
wIND OIPECTIDN
ESTIMATE.
14
*------ -------------*
I . = VO I
I I
1 B . uo I
*-- ------------- ----*
NOTE 15
......
" BEGIN 00 LOOP
* 2) I = 1, 1(14
,,1I...t-......
I
01.23--->1
I 16
. -------------------.
CjUt-'3 = o.
~UM4 = D.
*-- ---- ---- ---------*
DEWI2=PEW(I+
1,J+1)-
PEW ( I ,J + 11
.------ -------------.
12
DEV I
I
I
I
I
21
*-------------------*
1 r.tl,Jt = IA*I + I
f K"J + C*UO.S1J-"l + ,
I C.VO-SUM4)-1524. I
*-------------------.
21
22
E~n OF nn
LOOP?
IYES
I
I
I
1
1
I
20
23
END OF DO
LnDP1
NO
"-+
.
IYE~
I
I
I
1
I
I
1
16 .
24
----------------
I WPITE OEV 5 I
I R en 105 I
I I N I
/INTt;RNI\I ""nl)"IAT/
I FPpv THF. LIST /
----------------
NIJTE
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