EPA-650/2-74-086-b
SEPTEMBER 1974
Environmental Protection Technology Series
-------
EPA-650/2-74-086-b
PROCEDURES FOR MEASUREMENT
IN STRATIFIED GASES
VOLUME II, APPENDICES
by '
A. Zakak, R. Siegel, J. McCoy, S. Arab-Ismali,
J. Porter, L. Harris, L. Forney, and R. Lisk
Walden Research Division of Abcor, Inc.
201 Vassar St. ,
Cambridge, Massachusetts 02139
Contract No. 68-02-1306
Program Element No. 1AB013
ROAPNo. 21ACX-092
EPA Project Officer: William B. Kuykendal
Control Systems Laboratory
National Environmental Research Center
Research Triangle Park, North Carolina 27711
Prepared for
OFFICE OF RESEARCH AND DEVELOPMENT
U.S. ENVIRONMENTAL PROTECTION AGENCY
WASHINGTON, D . C. 20460
September 1974
-------
This report has been reviewed by the Environmental Protection Agency and
approved for publication. Approval does not signify that the contents
necessarily reflect the views and policies of the Agency, nor does mention
of trade names or commercial products constitute endorsement or recommen-
dation for use.
11
-------
TABLE OF CONTENTS
Appendix Title Page
A Automatic Isoklnetlc Sampling Systems 1
B Arrays and Mechanical Traversing Systems... 12
C Diffusion Tube Sampling 19
D Temperature Stabilized Diffusion Tube 31
E Tracer Methods 33
F Thermal Null Probes 35
G Velocity and Concentration Profiles 37
H-l The Effect of Jet Impingement on Mixing
in a Gas Stream 65
H-2 Jet Mixing in a Duct 84
I Combustion and Material Balance Calculations. 89
J Computer Program for Analytical Simulation
of Procedures for Velocity and Emission
Measurements in Stratified Stack Gases
(Theory and User Manual) 102
iii
-------
APPENDIX A
AUTOMATIC ISOKINETIC SAMPLING SYSTEMS
Presented 1n this appendix are a variety of automatic isokinetic
sampling instruments. This set is not intended to encompass all the
particular conceptual or implemented instruments of this type, but only
the devices discovered in our literature survey. However, the instruments
presented herein span the range of the general approaches to automatic
isokinetic sampling and, by extension, the general approaches to automatic
proportional sampling.
-------
1/8" STAINLESS STEEL ROD
I/8"STAINLESS STEEL
TUBING
0 HOLES
ENDS SEALE 0
AND ROUNDED
-1/64 0 H6LES
CUTAWAY VIEW
TAPERED END
TOP VIEW
7"
-*• H jr Tt
1/2' STAINLESS
STEEL TUBING
SfDE VIEW
SUPPORT /
PINS
END VIEW
FIGURE A-l
STATIC PRESSURE NULL PROBE
This 1s an example of a null balance probe reported by Wilson
14*
and Falgout . The distinct advantage of this particular probe
1s that the static taps for the sampling probe are In the plane
of the orifice, not Inside the probe nozzle. When the static ports
are Inside the nozzle, there are entrance losses which require the
probe to be calibrated (see Figure A-2). In this device the static
pressure sensing Is made in the region just upstream of where en-
trance losses begin.
(*) References are Iteted in Vol. I of this report.
-------
STAINLESS
ST EEL TUBE
STAiHL Ec ' SI E t L TUbt
••
•
jTANVVVYS i -ri.r- —
A^vl~ I v .~c?£"
^•77 y frt)>ti i J J ) > n 11 i jJ-rr-r/ > /*> n /
t m i _ T- j i
* ^ ~~ ^ ~~ M ~ . * '•••
WELD'
8 HOLES TOTAL hnjn
SPACED P-ECULARL* ON
THE CIRCUMFERENCE
TC TAL LF u:- TM
'+
.VELD. •
1\ TOTAL LENGTH fc06 mm
" \ TOTAL' LE NC-TH 0^5 V
-JU- \8HCLES TQT.-L Ifr ,'M IN Cl AM ET ER S PACED
• OH THE :i RCUMF-ERENCE
CTIO'I 66
F I j U RE A - 2
STATIC PRESSURE NULL TYPE PROBES
15
MLLL pROtE USES
TAPi 1^3 IDE THE NOZZL6
iEENOTE XVITH FloURE A-1
EXTERNAL STATIC
PRESiUfif LlNr
\ \\\\\>\^ \\V\\\\\\\V.j
N rERNAL STATIC
PRESSURE LINE
-------
Diaphragm of
differential pressure
controller
No.S open-end
piezometer ring
How of gas in flue
•
A/O.6'sampling
piezometer ring
2-2
3-3
4-4
5-5
6-6
7-7
8-8
9-9
10
11
12
Butterfly dampers
Lead from the Piez
to controllers
ometer rings
Piezometer rings
Differential pressure
Differential pressure
Regulating valves
Filter holder
Filter fabric
Suction system to
sampling pipe
recorders
controller
downstream of
FIGURE A-3
AUTOMATIC MULTIPLE SMOKE SAMPLER
16
This Is the only example of a multi-probe 1sokinet1c
sampler found.
-------
Flue Gas
U.
m
\
7 —L
Sampling Tube
At Isokinetic Conditions:
g m
U * U
9 s
Thermocouples
Shielded
Thermocouples
To Control System
FIGURE A-4
THERMAL NULL TYPE SYSTEM9' 10* 17
This is an example of a null type system which makes use of a difference
In stagnation temperature to sense velocity (See Appendix F).
Thermal anemometers in place of thermocouples would provide the same
Information, probably with greater sensitivity at the expense of instru-
mental complication.
-------
Flow
Control
Vacuum
Pump
3-15 psi.
Pressure Differential
Cell
Ratio = 0.80 - 1.20
=1.0 (Average)
o—o
Ratio
Station
3-15 psi
2-5
m
Pressure
Differential
Cell
Orifice
Meter
Filter
FIGURE A-5
CONTINUOUS SAMPLING RATE ADJUSTMENT
BY AUTOMATIC CALCULATION26
-------
Vacuum
Pump
Orifice
Meter Filter
mi
Amplifier
Analog Divider
Analog f~~"fxiO
Multiplier Li
0.10 V
Amplifier
0.20 psia
—
0.05 psial
Captive Analog
Compensator
0.10 V
Thermocouple
0.20 psia
Pressure Transducer
0.3mm Hg |
r I
Analog Multiplier
0.10 V
Wild Analog Compensator
&JO
Figure A-6
CONTINUOUS SAMPLING RATE ADJUSTMENT
BY AUTOMATIC CALCULATION7
This system is an improvement over the system shown in Figure A-7
because temperature and pressure for both the stack and sample gas
is used in the automatic calculation.
-------
S-Type Pi tot Tube.
\\
Transducers
Potenti-
ometer
Temp.'
Sensor
Null Balance
Amplifier
Quadratic
Transconductor
>tor1zed
Valve
Excess Air
Sampling
Pump
Cyclone
Mass Flow
Circuit
Signal Sample
Pump
Programmer
One Revolution
gmof Dial Completes
*VOne Program Cycle
Radial Injection of Air
Sedimentation Tube
Recirculating
FIGURE A-7
CONTINUOUS SAMPLING RATE ADJUSTMENT BY CALCULATION19'20
This system Includes a thermistor to compensate for temperature
fluctuations in the sampled gas.
-------
n
Stack
Gas
Flow
Sensor
Electronics
Temperature
Control
Power
Control
Auto-Isokinetic
and
Auto-Purge
Control
Strip Chart
Recorder
(Ibs/hr or
Gr/SCF)
Analog
iomputation
System
Control
FIGURE A-8
CONTINUOUS SAMPLING RATE ADJUSTMENT
BY AUTOMATIC CALCULATION21
This device has a unique feature tn that the sample temperature is
automatically held at the same value as the stack temperature, thereby
removing the temperature parameter from the gas density calculation.
-------
1 >
h
Samp
1
x Mai
;
-------
A/-
Scavenging
Air Inlet
Inlet
Nozzle
Heated Tube
Particle Charging Section
Particle Collecting Section
Recorder
D.C. Amplifier
High Voltage
Transformer
Heater
FIGURE A-10
PASSIVE, GAS-STREAM DRIVEN AUTOMATIC ISOKINETIC SAMPLER11'12
By design of the inlet nozzle and exit cone, the sample flow to the
particle collection section will be isokinetic, hence proportional
to the gas stream velocity, over a range of operating velocities.
11
-------
APPENDIX B
ARRAYS AND MECHANICAL TRAVERSING SYSTEMS
This appendix presents brief descriptions of several examples of
sampling arrays and two mechanical probe traversing schemes.
12
-------
FIGURE B 1
A RAPID MULTI-POINT OXYGEN ANALYZER
FOR POWER STATION FLUE GASES23
REFRACTORY
FILTERS
^bQILERDUCT
SAM PL IMG PROBES
tv/
f
1
t
>
A
S
1
>
n
f/
i
2
V1
^
\
(
>
•V
Z
I 345
TO WASTE<—-i
v
>TO RECORDER
This system displays a twelve point scan that can carry out a
traverse at the rate of one scan per minute. In this sequential
operation the spatial variation of oxygen concentration 1n the
section of the duct Is obtained.
13
-------
. TO TOTAl PMSSMf tWOUl
SAMPU LINE
N
FIGURE B 2
SAMPLING SYSTEM USED
FOR ON-LINE GAS ANALYSIS
OF JET ENGINE EXHAUST
24
This system displays a sampling probe in the form of
multi-element rakes. Sampling probes are also used for
total pressure measurement.
14
-------
~l
FLOW
SAMPLE OUT
FIGURE B-3
PROBE WITH MULTIPLE SAMPLING HOLES
Representative sample is obtained when the flow is uniform. Design
considerations regarding frictional pressure drop along the probe and
non-equal quantities sampling in stratified concentrations are described
in detail in Ref. 51. Probe length to diameter ratio must be less than
100 and the total sampling hole area less than 1/4 of the probe internal
cross sectional area to obtain a good approximation of the spatial average
°f gas concentration.
15
-------
Equal Annul 1
Static pressure-suction
pressure of the
flow
Integrator
Senses Velocity
FIGURE B-4
ANNUBAR FROM ELLISON INSTRUMENT DIV.2
This system displays an Interpolating tube and sensing ports precisely
located at predetermined positions. It uses the Chebyshef Calculus for
averaging. The result of the measurement is an average velocity.
16
-------
MOTOR AND CRANK
TO SAMPLING TRAIN
SWIVEL POINT
FIGURE B-5
CONSTANTLY-TRAVERSING SAMPLING PROBE25'26
A single sampling probe mechanically traverses in an arc
over the flow channel while removing samples.
17
-------
Soil. 16 OIN 4702 Blott 2
FIGURE B 6. Multi-Points Sampling Techniques In West Germany49
In rectangular ducts averaging 1s done
along the diagonal axis of the cross
section, Including the corner points,
and sampling through a common manifold.
Thtrtncttuntr
-Ovrchmnur «M
Bild 7 o. Abwugekre-Ji b«t cufgAMlxtar S)r&tm»ng*
rung mit !Cr*i»qw«f»«hn;tt
«Wftr mit 1 Bohnjn-
gm. m rffiwn dt OrOHr
Ifl
Bild 7 b. Ab»ouo»lir«ut
rung, d«m
-------
APPENDIX C
DIFFUSION TUBE SAMPLING
C-l. IN STACK DIFFUSION TUBE
A special Instrumental approach, the diffusion tube, has the potential
to average concentration over part of the sample plane and control sample
rate proportionally. A discussion on diffusion tubes is presented below.
Diffusion tubes are devices in which a sample of the species to be
measured is extracted by permeation or diffusion through a surface without
physical or bulk flow of the stream. The simplest configuration of a
diffusion tube is a cylinder with its axis normal to the direction of flow
1n the duct. A carrier gas which mixes with and transports the sampled
species to the monitor flows through the tube.
The rate at which material diffuses depends on the combined resistance
to mass transfer through the stagnant boundary layer outside the tube,
through the tube itself, and through the boundary layer inside. The con-
centration profile in the general case is shown in C-l where C is the
CO
concentration in the bulk (the concentration we want to determine); CQ the
concentration at the outer surface; and Ci the concentration at the inner
surface. We assume the bulk concentration in the carrier stream is zero.
In this case, assuming steady state flow, the rate of transport per unit
area, nfl, is given by:
na ' hl ' C0' ci = concentration (gm/cc)
D = permeability of membrane (cm2/sec)
t - thickness of membrane (cm)
19
-------
Direction of
Steam Flow
Outside
Inside
Direction of Carrier
Gas Flow
0
a. General Case
Diffusion Tube Wall
Direction of Carrier
Gas Flow
Diffusion Tube Wall
b. Permeation Rate Limited
Figure C-l. Concentration Profiles
20
-------
These equations can be combined into a single relationship:
na
and
- + +
u hl h2
where u = overall mass transfer coefficient.
Now if, as usually is the case, the rate of diffusion through the
membrane is much lower than in the films, i.e.:
and
then the overall equation reduces to the simple form:
n - D r
n - tC«
If we now integrate the flow through a finite length (L) of tube of
diameter (d), the total amount of material entering is:
N = £(ird)/CwdL
L
Further, the apparent concentration (S) as determined at the monitor is:
S = =
where
S = measured concentration in gm/cc
q = flow rate of carrier gas, cc/sec
K = Drrd_
tq
21
-------
Thus, if permeability controls, the sensed concentration is proportional to
the integral of concentration along the tube, but not to the integral of
concentration times velocity. It is this latter quantity that is proportion
to emission through the duct. The diffusion tube then is not theoretically
suited to applications in which emission rates are to be determined. On the
other hand, it is an excellent device to obtain a sample proportional to the
average value of concentration.
If this devide is to be used in a circular duct, the diameter of the
tube should increase along the radius proportional to the square of the
radius, in order to obtain proper area weighting.
Materials such as teflon show finite permeation rates for SCL, NCL,
and other small molecular species. The rates of diffusion through films of
sufficient thickness to maintain mechanical integrity are so low that the
rate controlling mechanism will be as assumed above. If the membrane per-
meability could be increased to the same magnitude as the outer boundary
layer coefficient, then the flow along a section becomes:
N = -rrd / u C dL
oo
1.4-*i
and h,D
Thus
= Dud f h! D dL
D/t+h, » QL
rrd I
/
the film coefficient, h,, is a complex function of velocity, depending on
the flow regimes, orientation of tube, etc.
22
-------
C-2. CALCULATION FOR OUT-OF-STACK DIFFUSION TUBE
This appendix shows the detailed calculations for the out-of-stack
diffusion tube. Figure C-l shows a conceptual design for an out-of-
stack diffusion tube and Figure C-2 shows the conditions for this
calculation. Figure C-3 shows a conceptual design for a multi-element
out-of-stack diffusion tubes.
23
-------
Duct Wall
1-0
sample 1n V TTi
Carrier gas Inlet
'Mixing Mnlfold
To analyzer
Constant temperature
Insulated box
Permeable membrane
FIGURE C-l
CONCEPTUAL DESIGN OF AN OUT-OF-STACK DIFFUSION TUBE
(FOR ONE ELEMENT SHOWN)
-------
STACK GAS
SAMPLE
SOg Diffusion Through a
Permeable Membrane
_PTFE Teflon Tube
0.788 cm I.D. x 0.940 cm O.D.
Flux Out
cl
Carrier gas
Permeability of S02 for PTFE
PTFE Teflon Tube
0.788 cm I.D. x 0.94 cm O.D.
T°C
S02 permeability f
cm Hg x cm3 S00(STP)/cm2-sec"1
cm" (pol.)
121°
177°
232°
0.539 x 10
1.81 x 10
-8
4.48 x 10
-8
FIGURE C-2
CONDITIONS
25
-------
Duct Wall
To analyzer
Sample In
Permeable Membrane
Constant Temperature
Insulated Box
Carrier Gas Inlet
FIGURE C-3
MULTI-ELEMENT OUT-OF-STACK DIFFUSION TUBES
-------
The permeability 1s a function of temperature. The equation can be
written as follows:
p = PO exp (-Ep/RT) (i)
To find P for the above tube we can write:
o
f- = exp (-E /RT)
o
2.3 [log P - log PQ] =
At 121°C P = 0.539 x 10"8
Replacing the values in Equation (2)
2.3 jlog 0.539 x 10"8 - log pi = - y-
6990
987 x 394
log 0.539 + log 10-8 + ™ ' *» Po
0.2684 -8x1+ 3.882 = log PQ
- 4.3864 = log P
P0 = 4.1 x 10-5
P • 4 1 x 10'5 e
r H.I x lu e
P - 4.1 x 10"5 e "351Z*86
(2)
(3)
27
-------
It 1s convenient to express the flux "F" in terms of the bulk-phase
partial pressures of the permeating gas ' for steady state and diffusion
of a gas from a hollow cylinder. Also assuming a constant diffusity
and the applicability of Henry's Law:
2TrP(pg-pc)
where: « .
R = cnr (STP) - CTT
sec-au (polymer)-cm Hg
pg - partial pressure of S02 in gas sample, cm(Hg)
pc = partial pressure of S02 in the carrier gas, cm(Hg)
a,b = Inner and outer tube radii, respectively, air
F = diffusive flux, cm3 (STP)/sec - cm length
replacing P, equation (4) becomes
F „ 27r(4.1 xlQ-5 -^p°) (Pg-Pc)
In a/b
By assuming plug flow and neglecting all resistances except the tube
membrane, we can write
dy
• - F' y * mole fraction
By neglecting pc, the partial pressure in carrier gas being very low it may
be neglected, also if pg = Pt x y
.1 x ID'5 -35TL86\ p
F--> , „ T / tg-!^ (5)
In a/b
28
-------
where
Pt 1s the total pressure of the gas stream in cm(Hg)
ySQ 1s the mole fraction of SOg 1n the stream
If the Ideal gas law holds for S02 PF = n'RT at STP
where n1 = number of moles diffusing/sec - cm length
x 1 (6)
0.082 x 273
also for the gas stream PQ = NRT
where N = Total number of moles/unit time in gas stream
3
Q = Flow rate reduced to STP in cm /sec
0.082 x 273
Also jp = n1 but = dy (assuming dNy *#fQ + Ndy)
Ndy = n'
by substitution
In a/bxQ
29
(8)
-------
J
In a/b x Q
(in y? - In yJ (in a/b) x Q
^-T^ - ftL 3*17 fi6\ =
-5^
if T = 100°C = 373°K
Y2 = 50 ppm = 50/106
Y] = 5000 ppm = 5000/106
a = 0.788
b = 0.940
Ptr = 76 cm Hg
>-S
3
Q = 3 liters/min = 50 cm /sec
2. 4.1 xlO-5 x76
1 = 0.2612 x 108 cm
30
dl
an) (10
(in 50 x IP'6 - in 5000 x IP'6) (in gj||) x 50 . 1(cn|)
-------
APPENDIX D
TEMPERATURE STABILIZED DIFFUSION TUBE
A proposed method to stabilize the temperature of a diffusion tube
was brought to our attention in a private communication with J.J. McKinley
(president of KIN-TEK Laboratories). In this scheme, a diffusion tube is
wrapped as a helix around a heat pipe (Figure D-l) which is a temperature
adjustable, isothermal tube. The close contact between the diffusion tube
and the heat pipe maintains the diffusion at a constant temperature.
31
-------
Solenoid Valve
Instr. A1r
Carrier
Gas In
Flow
Control
rs»
Insulation
Install In
Horizontal Plant
Porus Baffle —'
Baffles
temp Control
Sensor
Pressure Sensor
Permeation Sampling Method
FIGURE D-l
SCHEMATIC OF TYPICAL IN-LINE SAMPLER
I
-------
APPENDIX E
TRACER METHODS
In theory, it is implicitly possible to weight, an average value
of concentration obtained by uniform (non-proportional) sampling to reflect
the proper velocity-concentration product average needed to provide a
measure of emissions. This can be done by reference to a tracer compound
introduced into the flowing stream externally, or released or produced during
the same processes forming the material to be determined. If the emission
rate of the tracer is known by reference to the overall stoichiometry of
the process or by metering, then the emission rate of the desired species
can be determined. The basis of this technique is developed in the
following analysis.
Assume we are using a multi-point sampling array, with N identical
probes, each of area a, and with a sampling velocity of vs. Assume also
that each is located within an equal area (A) segment of the duct. (It is
not necessary to make this latter assumption, but it is less complex
analytically.)
The amount of tracer and species sampled per unit time are given by:
N or x,i N or x,i
wnere vg = sample probe inlet velocity
a = sample probe area
Ct i *Cx i = concBntrati°n of tracer and species, respectively,
at ith probe
et'ex ~ rate °^ samPlln9 of tracer and species, respectively
The emission rate of the two materials through the system is given by:
mt = A 2-vi ct,i
mx = A
33
-------
where
A = area associated with each probe
v1 = actual stream velocity at ith probe
mt' mx = emissi°n rate °f tracer and species, respectively
Now take the ratios of sampled rate to emission rate:
Now, if the concentration profile of the tracer is proportional to
the profile of the contaminant species, i.e.,
Ct,1 • k cx,i
then the two ratios are equal.
x i S
x • *t • mt •
where Sx, St are the sensed signals from the monitor in idential units.
Thus, if we know the emission rate of the tracer, the emission rate
of the unknown species can be determined. In combustion systems, one tracer
of particular interest appears to be C02> This specie is produced at the
same time and in the same space as the other gaseous pollutants., SCL, NO ,
etc., so that it should exhibit an identical concentration profile. Further,
the emission rate can be determined accurately by reference to the fuel rate
and fuel ultimate analysis. This method, in theory, is valid for a single
sampling point.
34
-------
APPENDIX F
THERMAL NULL PROBES
Null probes, which use thermocouples to detect an error signal
between the stack gas velocity and the sample gas velocity, potentially
offer instrumental simplicity. The following analysis considers the
magnitude of the error signal produced due to differences in stagnation
temperatures between the stack gas stream and the sample gas stream.
The total (stagnation) temperature (Tt) is equal to the sum of the
static temperature (Ts) and the dynamic temperature of the gas (Ty). For
an ideal probe at rest with respect to the system boundaries, the stag-
nation of an idealized gas. follows the energy equation which mav be
written for the open system, viz.:
6Q + 6W = dh + ~ (1)
-------
TP • Ts + KTv
Where T is the Indicated probe temperature, the value of K is a function
of probe geometry. For the half-shielded type thermocouple, K is about
0.96.
In the thermal null method we are only interested in differences in
temperature between the flue gas and the sampled gases. If the two thermo-
couples are close enough, TS will be the same, but the value of T will
change if the velocity of the flue gases does not equal the velocity of
the gases at the entrance of the sampling nozzle. For example:
If
Vf = 19 ft/sec
Vs = 20 ft/sec
and C = 0.24 B!TU/lb°F
T = - __ — U*) _ , = o n?n
vf 2 x 778 x 32.2 x 0.24 U'UJU
T = -x — -mx -"" = n
vs 2 x 778 x 32.2 x 0.24 u<
If K is taken as 0.96, the difference becomes:
0.96 T
0.003°F
This temperature difference is very small. If a double junction, iron-
constantan thermocouple is used for a differential temperature sensor, an
EMF of about 0.9 uv must be detected. The detection of this magnitude of
voltage is within the state of the art of present electronic technology.
However, some development effort would be required for a system suitable
for industrial conditions.
36
-------
APPENDIX G
VELOCITY AND CONCENTRATION PROFILES
37
-------
Figure 9-a. Case III Normalized Velocity Profile
8.0' Diameter Duct
38
-------
Figure 9-b. Case III Normalized Concentration Profile
8.0' Diameter Duct
39
-------
?**
Figure 10-a.
V4
-------
Figure 10-b. TEMPERATURE PROFILE TO BE TAKEN AS CONCENTRATION PROFILE
CASE IV .
-------
NORTH
CASE V Finure lie (TV')
ISQVELS - INTERVRL = 5 FT/SEC
-------
PLflNE fl-fl WEST SIDE 3-19 -70
2008
1312
CASE v Figure lib (TVA) 46 feet x 24 feet
SQ2 DISTRIBUTION - INT =50 PPM
-------
Figure 12a Case VI
Hypothetical Velocity Profile
20' Diameter Duct
44
-------
90U
Figure 12b. Case VI
Hypothetical Concentration Profile
20' Diameter Duct
45
-------
Figure 13a. Case VII Hypothetical Velocity Profile
20' Diameter Duct
46
-------
90L
Figure 13b. Case VII
Hypothetical Concentration Profile
20' Diameter Duct
47
-------
Figure 14a. Case VIII
Hypothetical Velocity Profile
20' Diameter Duct
48
-------
Figure 14b. Case VIII Hypothetical Concentration Profile
20' Diameter Duct
49
-------
Figure 15a. Case IX Hypothetical Velocity Profile
20' Diameter Duct
50
-------
90'
Figure 15b. Case IX
Hypothetical Concentration Profile
20' Diameter Duct
51
-------
en
ro
Figure 16a. Case X Hypothetical Velocity Profile
15' x 20' Duct
-------
en
Figure 165. Case X Hypothetical Concentration Profile
15' x 20' Duct
-------
en
Figure 17a. Case XI Hypothetical Velocity Profile
15' x 20' Duct
-------
en
en
Figure 17b. Case XI Hypothetical Concentration Profile
15' x 20' Duct
-------
TABLE 6
CASE XII TEST NO. 9-1
VELOCITY AND CONCENTRATION DISTRIBUTION DATA IN A SQUARE SECTION
A Vl
C2
B V
C
C V
. C
D V
C
E V
C
F V
C
G V
C
1
2.1
85
2.6
95
.2.8
105
2.3
120
2,3
100
2.6
80
2.9
60
1-2
2.1
90
2.1
95
2.8
115
2.6
115
2.3
105
3.0
80
3.5
55
2
2.1
85
2.8
106
2.6
125
2.6
125
2.8
110
3,2
85
4.0
55
2-3
3,0
95
3.0
115
3.1
125
3.0
125
3.0
120
3.3
90
4.0
60
3
4.1
105
3.1
120
3.1
130
3.3
130
3.3
115
3.6
90
4.0
60
3-4
3.1
no
3.6
120
3.8
130
3.9
125
3.4
115
3.4
100
4.0
65
4
3.5
no
3.7
120
4.0
130
4.2
120
3.8
100
3.8
105
4.0
70
4-5
3.5
115
3.8
115
3.9
120
4.4
105
3.9
95
3.9
90
3.9
70
5
3.5
120
3.6
no
4.1
115
4.5
105
4.4
85
4,1
70
3.6
60
5-6
3.0
95
3.5
90
4.2
90
4.9
80
4.6
65
3.9
60
3.6
50
6
2.8
100
3.2
85
4.1
90
4.9
70
5.1
50
4.6
45
3.8
45
6-7
2.3
85
3.0
65
4.3
65
4.7
60
5.0
35
4.9
30
3.8
30
7
2.3
95
3.0
75
4.0
65
4.7
55
4.7
35
4.6
25
3.8
25
en
CONDITIONS: Baffle: Inclined 22.5° to horizontal
Ethane Flow Rate: 20.23 x 10"6 kg/s
Temperature: 20°C
Flow Area: 0.18m2 (280m2)
Number of Fans: 2 fans in series
velocity m m/s
concentration must be multiplied by 0.298 to convert to ppa c^ane: Average Velocity • 3.5 m/s (11.6 ft/sec)
Total Flow = 0.64 m3/s (1356 ft3/min) at 20°C Total Emission » 20.62 x 10*6 kg/s
-------
PLRNE fl-fl WEST 4-7-70 RUN 6
NORTH
70.6
en
o
en
ISOVELS - INTERVflL = 5 FT/SEC
Figure .18a. CASE XIII-1
-------
PLflNE fl-fl WEST SIDE 3-IE -70
IBS
en
00
502 DISTRIBUTION - INT.=50 PPM
Figure 18b. CASE XIII-1
-------
PlflNE fl-fl WEST 4-7-70 RUN 6
NORTH
70.6
en
vo
in
•-»
»-«
Q
ISOVELS - INTERVflL = 5 FT/SEC
Figure 19a. CASE XIII-2
-------
PlflNE fl-fl WEST SIDE
502 DISTRIBUTION - I NT.=50 PPM
Figure 19b. CASE XIII-2
-------
PLHNE fl-fl WEST 4-7-70 RUN 5
NORTH
CTl
z
o
ISQVELS - INTERVAL = 5 FT/SEC
Figure 20a. CASE XIII-3
-------
PLflNE fl-fl WEST SIDE
SQ2 DISTRIBUTION - INT.=50 PPM
Figure 20b. CASE XI11-3
-------
PLRNE R-R WEST 4-7-70 RUN 5
NORTH
Ch
CO
tn
«™»
s»
•-«
a
ISQVELS - INTERVRL = 5 FT/SEC
-------
PLRNE fl-fl WEST SIDE
502 DISTRIBUT
INT.=50 PPM
Figure 21b. CASE XIII-4
-------
APPENDIX H. JET MIXING OF FLUE GAS STREAM
APPENDIX H-l
THE EFFECT OF JET IMPINGEMENT ON MIXING
IN A GAS STREAM
A TWO DIMENSIONAL ANALYSIS OF THE
RATE OF SPREADING OF A UNIT IMPULSE IN
CONCENTRATION OF A GASEOUS SPECIES IN
A FREE STREAM DUE TO JET IMPINGEMENT
By
Professor J. H. Porter, M.I.T., Consultant
Professor W. L. Harris, Sr., M.I.T., Consultant
65
-------
I. INTRODUCTION
In the determination of the emissions rate of a contaminant species
in a gas stream emitted from a stack, it is necessary to measure the velocity
of the gas and concentration of the contaminant at several locations within
a stack cross section with probes. The accuracy of the estimate of emissions
rate will depend on the weighting strategy used to associate the individual
probe readings with the appropriate fraction of stack cross sectional area.
However, independent of probe strategy, with any fixed strategy the accuracy
of the estimate will increase, as the number of sampling probes in the stack
are increased. Unfortunately, the cost of the analysis increases with an
increasing number of probes used in the determination. Thus, it is desirable
to consider alternative means of improving the accuracy of emissions rate
determination in stacks.
If a uniform concentration and velocity profile existed in a stack,
one need only take a single measurement of concentration and velocity to
determine the exact emissions rate. However, for a large number of reasons
(e.g., leaks in stack brick work, uneven combustion rates of fuel in
combustors, flow blockage and stagnation in ducts, etc.) rarely, if ever,
will one find uniform velocity and concentration profiles in stacks. In
addition, since the height to diameter ratios in stacks are relatively small
(H/D <50) equilibrium profiles are established before the stack
effluent is admitted to the atmosphere. Thus, if the limiting condition of
using a single velocity and concentration probe to make an accurate deter-
mination of emissions rate is to be approached, external or internal devices
must be placed in the stack to promote mixing normal to the direction of
flow of inequalities in local velocities and concentration.
66
-------
Several devices may be conceived to accomplish this task. The
effectiveness of each device should be measured in terms of the rate
of spreading induced by the device divided by the rate power is consumed
by the device either from an external power supply or internally from
the gas stream. This ratio (radial mass transfer rate/energy input rate)
Is at its maximum value for the most effective mixing device and diminishes
with less effective devices. Thus for each device conceived this ratio
should be determinable and compared to the value of this ratio for all
other devices. In addition, the capital and annual operating cost of
each device should be determinable and compared to the cost of other
devices as well as probe capital and installation costs. Selection
of devices should then be based on the cost of the total system, a
device plus the number of probes required with each device in order to
Produce the same accuracy in emissions determination.
This report is concerned with the determination of the spreading
ratio for a particular device, an induced jet impinging on the stack
9as stream. The analysis because of complexity considers a two dimensional
analysis of a jet stream which is compressed and directed at the stack
9as. The stack gas is assumed to flow with uniform velocity but to contain
a unit impulse in the concentration of a contaminant species at some
distance Afrom the entering jet. Radial diffusion in the free stream
is assumed neglible. Diffusion in the jet is characterized as isotropic
as induced by jet turbulence. The combination of the jet and free stream
induce mixing of the contaminant. Figure 1 is a schematic of the model.
While a two dimensional analysis of the problem is not entirely
accurate, its ease of solution relative to the three dimensional problem,
and the consideration that the appropriate factors which cause spreading
are included in the two dimensional analysis, dictates its choice as a
model.
67
-------
FIGURE H-l
JET IMPINGEMENT ON FREE STREAM
Y
TV
Jet
\No2zle
\
Potential / /
Core Boundary / / '
V- / ' / ' • x
^N^f "•*-. Jet Boundary ' / /
Ax " / / -f
\ ^Jet Centerline / /
\ ^>^(VMax Locus) ^/ /
S>«,
^__ a ^H^^^X -*Jet Boundary ^
Jet Angle \ ^*^ ^
\
cnc°
Contaminant
Free Stream
Concentration
£ _ Profile
C = 0 — 6 *-
Uniform Velocity VFS
m M n tt M M
Free Stream
68
-------
II. BACKGROUND
An essential objective of the proposed model is to determine a
species concentration distribution and the axial rate of change of this
distribution corresponding to an imposed velocity field and specified
boundary conditions. Hence, the reliability of the results obtained
from the proposed model is greatly enhanced by an accurate and logically
self-consistent description of the jet properties and the velocity
distribution along the intersection of the jet and free stream flow.
The particular jet properties of interest are:
(1) The position of the jet center!ine (Locus of points of
maximum axial jet velocity)
(2) The spread (cross section growth) of the jet
(3) The velocity distribution in the jet
(4) The turbulent (eddy) diffusion coefficient distribution in
the jet.
Each of the properties of the jet which are identified above vary with
location, as there is a sequential pattern of development of a jet in a
cross flow. This sequential pattern of development has been discussed
by several investigators [1,2,3,4] and is briefly discussed here.
The jet has three regions with no sharp boundaries between the
regions. The first region is identified as the potential core which
exists before the turbulent shear region, generated along the jet boundary,
extends to the centerline of the jet. Adjacent to the potential core is
the zone of maximum deflection. The high intensity of shear at the jet-
free stream boundary entrains the free stream rapidly into the jet and
causes the flow direction of the jet to bend most rapidly to approach the
"How direction of the free stream. The zone of maximum deflection is
followed by the vortex zone in the far field where the flow is governed
roainly by the motion of a turbulent vortex pair formed downstream of the
bending over region. The degree of entrainment (mixing of the free stream
with the jet fluid) increases monotonically and non-1inearly from the
Potential core, through the zone of maximum deflection, to the vortex zone.
69
-------
The required distribution of the jet properties of interest will be
obtained by least-square fitting these properties from experimental mea-
surements [43, 44, 45, 46]. Contained in references 43, 44, 45 and 46
are the required distributions of the jet properties of interest for a
range of values of the ratio entering jet velocity to free stream velocity
and angle of jet inclination relative to the free stream. The jet Reynolds
number dependence has been shown to be negligible [44].
Consistent with the proposed two-dimensional model, the velocity
distribution along the intersection of the jet flow and free stream flow
will be determined by vectorial addition of the planar velocities involved.
It is noted, however, that since the actual jet contains a pair of vortices
downstream of the potential core, the velocity field is locally three-
dimensional along the intersection.
70
-------
III. MODEL DEVELOPMENT
The model to be developed in this section considers the velocity
and turbulent diffusion coefficient distribution profiles developed by
the interaction between an impinging jet and a free stream, and the
resultant mixing in the free stream created by this interaction.
It is assumed that the system is isothermal and isobaric at
temperature T and pressure P. At a sufficiently far distance upstream of
the jet entry, so as not to be disturbed by the jet, flows a free stream
with a uniform velocity in the X-direction from the jet entry exists a
concentration pulse of some contaminant species of height C° and width <5 in
the free stream. If the width of the free stream flow channel is L (L
is sufficiently large as not to effect the path of the jet) the mean
concentration of the contaminant species is given by,
C * C° 6/L (1)
and the contaminant emission rate is,
Qs • V$C°6 (2)
The entrance velocity of the jet is V emitted uniformly through
a planar nozzle of width DQ at an angle of inclination a with respect
to the negative y direction (a = 0).
Thus assuming an ideal fluid, the mass flow rate of jet fluid
entering the free stream is:
- 1KP\
-w)
71
-------
A. POWER REQUIREMENTS OF JET
Focusing attention first on the jet, the power requirements
are computed by assuming a stagnant fluid at pressure P and T is adiabatically
compressed to a pressure P°, cooled at T, and expanded in a friction!ess
channel to vi
is given by:
channel to velocity V and pressure P. The energy input rate for compression
-1
(4)
Free expansion (isothermal) in a frictionless duct is given
by Bernoulli's equation which when integrated produces:
iff
exp
(5)
Thus combining (3), (4) and (5):
E =
TT n
exP lim
-1
(6)
72
-------
B. MASS TRANSPORT
Considering an area element dx by dy in the plane of the jet and
the free stream the steady state mass conservation equation for any species
is:
E(x'y)H" u(x'y)c * fy E(x'y)" -v(x'y)c* =0
where E(x,y) is the diffusion coefficient and E(x,y) outside the jet
boundaries is zero
U(x,y), V(x,y) are the x and y components of the velocity vector
with the initial conditions:
U(x,-oo) = o U(o,y) = 0
V(x.-») = V (x.--) • 0
C(x,-») = 0 x<4
C(x.~) = 0 x>£+5
= C°
0 (o,y) = 0
The equations describing the velocity and diffusion field (u(x,y),
V(x,y), E(x,y)j will be developed based upon experimental results on jets
injected into cross flow streams. The solution to equation (7) (carried
out numerically) will provide concentration profiles in x and y.
73
-------
C. SPREADING COEFFICIENT
In a completely mixed stream the concentration is uniform and
equal to the average concentration:
C * C°6/L
The deviation of the point concentration from the average, V (x,y) will be
defined as,
-? / ?
\r(x,y) = (C(x.y) - ?) (8)
and the average value at any plane of constant y as
2
[/ (C(x.y) - C) dx
(9)
Initially, \r has a value,
!SL fc+6 L
/ (o-F)2dx +/ (C°- C)2dx +/ (o - F)3dx = C° (f) /I - f)
J0 JSL J£+ 6 a/ ^ L/
The rate of change of ? will be defined as the spreading coefficient:
ay u/ul i^-yj -c rd* do)
= V
and its average value over some axial channel length, H defined by:
H '
a? <
[/ (C(x.y) -IT)2dx
i
(ii)
74
-------
The average speeding coefficient divided by the jet energy input will be
the rating factor to measure the effectiveness of jet mixing
r.f.
(12)
It is noted that
r.f.
«
E. "
(13)
PVoDo
5MV
75
-------
D. VELOCITY AND TURBULENT DIFFUSION PROFILES WITHIN A JET
The development of correlations to predict jet properties is based
primarily upon experimental evidence [43, 44, 45, 46]. These results are
presented herein.
The ratio of the entrance jet velocity to the free stream velocity
is defined as X ,
vs
Referring to Figure 2, if Z is the y location of the jet centerline,
then Z's dependence on x, X and a is given by:
Z = o?qq / X \ * / X cat a
D0X '2293 (06 Asinc,;
If R is the radius of the jet, R is given by:
If yand n are the coordinate axes in the Z and R directions respectively, the
velocities in these directions are given by V(n,^ and U(n,j) respectively.
Thus, the velocities within the jet in the x and y directions are given by the
relationships,
U(x,y) = V(n,^) cos 0 - U(n,^) sin 3 (16)
V(x,y) = V(n,^ sin B - U(n,^ cos 6 (17)
where 8 is defined by:
Tan 3 = ^ (18) and Z = f(x) is given by equation (14)
76
-------
Jet V.
\
Free Stream
FIGURE 2
POSITION OF JET CENTERLINE
77
-------
Since velocity profiles within the jet are given with respect
to the n.Vaxes, these can be coonverted to velocities parallel to the x
and y axes by the above expressions.
The n,^components of the velocity are given by,
- 0 (19)
and V(n,*) is obtained by correlating the data of Keffer and Baines [1] shown
in Figure 3, and plotted as:
where n* is defined as the location in the jet where the velocity falls to
one half the center line value, Correlations of n* and V(o, ) are shown in
the same figure.
The eddy diffusion coefficient e is given in terms of an entrapment
coefficient E as
rR
where V, » = I 27mV{n,^d7]/TrR2
/ J f /
Since V(n,-^ is given by (20).
The diffusion coefficient, only dependent upon z and uniform in
the n direction is determinate if E is known. Figure 4 is a correlation
for E.
78
-------
l.O
0.8-
Oj6
0.2
**«*.'
0.5
gk
* « *
1.0
1.5
2.0 n/rt* 2.5
Figure 3.
-------
2.0
l.fe
0.8
0.4-
o!
Figure 4.
Z/D,
10
The preceding description thus provides a mechanism
to determine the velocity and diffusion in a jet.
80
-------
IV PROPOSED SIMULATION
A solution of the partial differential equation governing the con-
centration field of the proposed model corresponds to solving the diffusion
equation with variable coefficients, the description of the velocity and
eddy diffusion coefficient fields. The numerical simulation of the partial
differential equation governing the concentration will consist of an implicit
finite difference model of the governing equations and boundary conditions.
Suitable explicit finite difference models may be developed; however, due
to the error propagation characteristics inherent to explicity finite
difference approximations of parabolic equations, it is recommended that
implicit schemes be exploited initially.
81
-------
V. EXPECTED RESULTS
The numerical simulation of the equation describing mass transport will
provide concentration profiles (c = f(x)) for all axial positions y along the
duct. This information is then used to compute the local spreading coefficient
dV(y) =
~3t "
where
H(c(x,y) -
1
2 Vd
L
If (C
-------
VI. NOMENCLATURE
R.f. Jet mixing rating factor - gr-cmv/ERG
spreading coefficient - gr/cc-sec
U(x,y) x velocity component to fluid - cm/sec
V(x,y) y velocity component to fluid - cm/sec
2
E(x,y) Turbulent diffusion coefficient - cm /sec
a Angle of jet inclination - radiants
DQ Diameter of jet nozzle - cm.
A cm distance in the x-direction from the jet entry to the
concentration pulse
L Width of the free stream flow channel, L cm
-------
APPENDIX H-2
JET MIXING IN A DUCT
Assumptions: [47, 48]
1. Jet is directed at right angles to main stream
2. Can neglect the effect of non-uniform velocity in main stream
(assume slug flow)
3. Can neglect the effect of ambient turbulence on mixing near
jet orifice
du ^^^^
4. Jet is turbulent - Re = — >300.
nomenclature
d orifice diameter (b * radius)
D duct diameter
u mean jet velocity at orifice
V mean gas velocity in duct
For a turbulent jet ejected normal to a uniform cross flow, it is
possible to define a momentum length scale
1m = d u_
v
This represents the scale over which the vertical momentum of the entrained
fluid becomes comparable to the initial momentum flux at the orifice, (in
84
-------
a rough sense it represents how far the fluid is projected into the cross
flow within a few orifice diameters downstream.)
Consider now a jet in a duct of diameter D. It is possible to
assure ourselves of geometric similarity by making Im/ fixed. Thus, set
• W
which defines a well mixed jet. (It is possible to show that this relation-
ship will guarantee a local jet width comparable to the duct diameter.) (It
is also possible to show that equation 1 assures mixing within a few duct
diameters from orifice.)
To check on the validity of Equation (1), we can look at the data of
Chilton and Genereaux [47] in which mixing is defined as "good" if it occurs within
2 or 3 duct diameters from the orifice and "poor" if it occurs further down-
stream. This is shown on graph 1 in which u/v is plotted versus D/d for both
good and poor runs. Also shown is the best fit curve to the "good" points.
As it can be seen, Equation (1) holds reasonably well for the "good" points
and that it does not fall near the "poor" runs.
It is possible to define the ratio of the mass flow in the duct to
mass flow out of the orifice
2
MF dm/dt (orifice) _ /d \ /£\
nt ~ dm/dt (duct)\0 ' \V/
and the ratio of kinetic energies per unit time in the duct and jet
85
-------
Graph 1
CO
CT>
Best fit to good points
u/v = D/d
-------
„ _ dE/dt (orifice) . /d \2 /u\3
Ef - HETdT (duct) * Iff) (v)
4 4
Thus since MfEf = ({j-V (y-) = 1 from equation (1) then
Ef = 1/Mf (2)
where Mf <1. This is plotted on graph 2.
87
-------
4. ,
2"
GRAPH 2.
y where M
and
2
2 .. 3
I
1
1..
Eq- (2)
-h— K- 1 ^
.1 .2 .3 .4
^ 1_
.5 .6
.8
-4-
.9
1.0
-------
APPENDIX I
COMBUSTION AND MATERIAL BALANCE CALCULATIONS
Residual Fuel 011 Analysis*
Total Elemental Carbon: 85.36%
Total Elemental Hydrogen 11.45%
Total Elemental Sulfur 2.10%
Mystic Station on 4/24/74
High Sulphur Fuel Oil Reading
Time Tank Gage Volume (m3)
12:40 pm 22-10-1/2 a 120°C 5256.2 (33061.42 bbls)
2:40 pm 22-6-5/8 a 120°C 5181.8 (32593.78 bbls)
fuel usage * 74-4 (467.64 bbls)
.. fuel usage in kg/s assuming a specific gravity of oil = 0.96
3
74.4 x 0.95 x 1 x 10 - g go ka/s
3600 x 2 9'9^ K9/S
Expected Emission
Assuming 3% of the sulfur goes to S03
S02(kg/s) = 9'92 "looVag'97 X M - 405 x 10~3 kg/S
C02 (kg/s) =
Average of duplicate analysis, see laboratory tests reports
89
-------
L U V A K INC
TEi.:«17-MM401
723 MAIN STREET
BOVLSTON.
MASSACHUSETTS 01M6
0-942?
REQUESTED BY Walden Research
Division of Abcor, Inc.
201 Vasser Street
Cambridge, Mass. 02193
ATTENTION M3. A. Zakak
1 sample analyzed for sulfur content in duplicate.
5/2/74
lflvo*c* Number
VF 1525
Cutwiwi Oram Number
CunomWt N«oulMti«« MumMr
A/V./T/L
DMCMIPTION
RfSULTS
SAMPLE IDENTIFICATION
P.O. 15328W
DUPLICATE
SULFUR
2.05*
2.11%
90
-------
IA«O*ATOI»I» AT KUtlkVOBTM, M. J.
CAM*. PACAOINAIHUUBTOMt, V VV
COPPUB CH(tf»T|«, WILMIMBTOM. » "•
CALir.. SCATTLC. POPTLANO.
TAMPICO. MEH-. CHtCAOO.
LICCNSKO
A«O rA»«.
& CO., INC.
IMSMCTIOM Of MTHOLfUM AND OTHEH PRODUCTS
QENCRAU HCAOOUARTERS
4OO BWCNBON OKIVC
KCNItWOMTH. N. J. O7O33
UDVIMO TH€ PITNOKUM INOUtTH V »O« OVID 'O V«AK»
DIMHOMU[ INtMCTIOM M*VK( AT ALL KtNTt Off tMC ATLAHTIC. OULP AMD Metric CO»»T»
CABLC ADOKU* "•AV>DLTOIL"
8UUF COAST HCAOOUAItTCM
P. 0. »0» 4M«
rA*Aouu. n«A» rrsoa
(711) M7.«ITt
WCST COAST HCAOOUAMrKM
P. O. MX 1144
WIIMIMT«H. CAlir. M744
IJIJII
•AST COAST HIAOQUANTIM
400 IWINtOM MlVt
(CINILVOKTIi. H. J, * A PtHlOO OF 4» OAV1UMLCM A LO»«IR PtRlOO IS MCOUf STCO IN WRITINO.
-------
CALCULATIONS A
Moisture Calculation North Duct South Duct
Average Dry bulb temperature (TD), °C
Average Wet bulb temperature (TW)» °C
Relative Humidity (from psy to chart) (% R.H.)
62
49
54
55
47
70
Saturated vapor pressure at dry bulb temp. - -
(V.P.) N/m2 21.5 x 10J 15.7 x 10
Barometric Pressure (B.P.) N/m2 100.9 x 103 100.9 x 103
Static pressure at moisture measuring
device (PS) N/m2 3.7 x 103 3.2 x Ifl3
Absolute static pressure at moisture . 3
measuring device (B-P =P ') N/m2 97.2 x TO"1 97.7 x 10
d •>
% Moisture by volume V.P. x R.H. 21.5 x 50 15.7 x 63
Pj197.2 97.7 ^
%m 11 10
92
-------
CALCULATIONS B
Molecular Weight and Relative Gas Density Calculation
Mfl = * CO. x 44 + 1 Qr x 32 + 10° " <% C°2 + % v
a z z
Ms = Md (1 - .) + 18 x
s luu
M.
Gd '
North Duct South Duct
Average % C02
Average % 02
% M
Md
Ms
Gd
11.6
6.3
11
30.11
28.78
0.99
13.0
5.2
10
30.57
29.40
1.01
93
-------
CALCULATIONS C
Emissions Calculations
Emission from North duct
Duct Area 3.14 x 3.35 = 10.52, m2
No. of Probes 9
% Moisture 11%
ISO, Concentration x velocity 121,000, ppm x m/s
* A
ZCOg Concentration x velocity 1,530 x 10 , ppm x m/s
Average duct temperature 411, °K
2 3
Average static pressure 97.2, N/m x 10
S0« Emission from North duct:
10.52 2 ,000 x 10 f x 64 x 10 (1-0.11) _ -3 . .
9 x 101.33 x 22.414 x 411 „ 1ft-3 " "* x IU Kg/s
97.2 x 273x IU
COp Emission from North duct:
10.52 „ 1,533 x IP4 x IP"6 x 44 x IP"3 (1-0.11) 10 _ . .
—g— x » 1- = ig.g kg/s
101.33 x 22.414 x 411 v ln_3
97.2 x 273x IU
94
-------
Emission from South duct
Duct Area 3.14 x 3.35 = 10.52 m2
No. of probes 9
% Moisture 10%
IS02 Concentration x velocity 125,100, ppm x m/s
£C02 Concentration x velocity 1,638 x 104, ppm x m/s
Average duct temperature 403, °K
Average static pressure 97.6, N/m2 x 103
S02 Emission from South duct:
»* *
97.8 x 273
C02 Emission from South duct
IP.52 ¥ 1,637 x IP4 x IP"6 x 44 x IP"3 (1-0.1)
x
Total S02 Emission from both South and North ducts is equal to
229 x 10~3 + 246 x 10"3 = 475 x 10"3 kg/s
% Error 4754Qg405 x 100 = + 17%
Total C02 Emissions from both South and North ducts is equal to
22.1 kg/s + 19.9 kg/s = 42 kg/s
% Error 42 ;,31 = + 35%
95
-------
In the following calculations (D and E), analysis of the system
showed + 26% error on the measured flow rate, + 4% error on C02 analysis,
and - 9% error on SOg analysis.
Total flow through North and South duct (Dry Basis)
10.52 x 15 x 273 x 97.2 x (1-0.11) 10.52 x 14 x 273 x 97.6 (1-0.1)
411 x 101.33403 x 101.33
89.48 + 86.49 = 175.97
= 176 m3/s at 273°K and 101.33 N/m2 x 103
This is equivalent to 373,000 SCFM (Dry Basis).
* After the end of test and analysis of results the span gas of the
Beckman instrument was checked against the Air Orsat analysis and
was found to be about 4% lower than expected from an analysis made
by the manufacturer, (span gas was believed to be!0%; was found
9.6%).
96
-------
CALCULATIONS D
Theoretical Combustion Calculations
Given Residual Fuel Oil
Carbon 85.36$
Hydrogen 11.45%
Sulfur 2.10%
Water )
Ash j negligible
Assume 18% excess Air
Theoretical combustion air requirement, 1 kg residual fuel oil basis
Standard Conditions
Carbon C + 02 -»• C02
10 -1 9 ?7fi y 10 t
0.8536 x TT= 2.276 kg 02 = 22.414 x 10 ° x 33 = 1.594 mj
or 7.59 m of Air
Hydrogen \\2 + 1/2
0.1145 x 1| = 0.916 kg 02 = 22.414 x 10'3 x 0>9163* 10 = 0.642 m3
or 3.06 m3 of Air
Sulfur s + 02 + S02
3
"3 °'Q2] 10 3
0.021 x || = 0.021 kg 02 522.414 x 10"3 x °'Q2]£ 10 = 0.0147 m
or 0.070 m3 of Air
97
-------
Total Dry Air* Requirement
7.59 + 3.06 + 0.07 = 10.72 m3/kg of oil
10.72 x 1.18 = 12.65 m3/kg of oil
Products of Combustion** at Standard Conditions/kg residual fuel oil
Carbon Dioxide
n A*™ v in3 44 v 22.414 x IP"3 m , RQA 3
U.oOJO X 10 yj- X TT- a 1.594 m
Water from combustion
0.1145 x 103 X 4 x 22A}% * 10'3 - 1.283 m3
Nitrogen
12.65 x 0.79 = 9.99 m3
Sulfur Oixdes as Sulfur Dioxide
0.021 x 103 x ft x 22'4144 10'3 . 0.015 m3
Oxygen
12.65 x (1.18 - 1) x 0.21 - 0.48 m3
Total Dry Volume
1.594 + 9.99 + 0.015 + 0.48 - 12.08 m3/kg fuel oil
* Tambiant = 44°F, R.H. = 70
** Water in fuel or air is not accounted for
98
-------
Estimated sulfur dioxide concentration (assume 3% goes to StL)
0.021 x IP3 x 22.414 x IP"3 x 106 x 0.97
0<5w 10 AO
Jt X It.Uo
ppm
Estimated Carbon dioxide concentration (neglect CO conversion]
0.8536 x 103 x 22.414 x IP"3 x IP6 _ lr> nnn nnm ._ t .
y2—x 12.08 ~ 13Z»000 PPm or 13.2% (Dry Basis)
Expected % moisture is near 10%. Slightly higher results were obtained
(10% for South duct and 11% for North duct) because of leakage of the
feed water at the economizer section during the test period.
99
-------
CALCULATIONS E
Analysis of Results (Based on 18% excess air*)
Excess dilution due to air in leakage at the air preheaters (using 5.8% (L
from Pyrites measurements):
12.08
0.058
x + 0.48 = (12.08 + 4.76x) 0.058
x + 0.48 = 0.7 + 0.276 x
0.724 x = 0.22
x = 0.3
1180 x 12.08 ln,, nnm
Expected S09 concentration after dilution 12.08 + 4.76 x 0.3 UDD pp
* (13.51)
Average S02 reading is equal to 955, i.e., 9% lower
Expected C02 concentration after dilution llioVV^?8* x Q 3 ~ 11.8%
Average C02 reading is equal to 12.3%, i.e., 4% higher.
From Boston Edison personal communication about -18% excess air was used
during the test period
100
-------
All Standard
Conditions
(Dry Basis)
Expected dry flue gas volumetric rate 119.8 m3/s
12.08 x 9.92 = (254,000 SCFM*)
Measured flow rate 176.0 m3/s
(373,000 SCFM**)
If +17% _ }_9%) = +26% error is from flow
measurement then the actual flow rate is equal ,,0 7 3. *
to i?fi v 139.7 m /$*
/0X" X = 0.26 (296,000 SCFM)
Air in leakage is equal to 139.7 - 119.8 = 19.9 m3/s
(42,000 SCFM)
Oxygen in air in leakage 19.9 x 0.21 = 4.2 m3/s
Theoretical oxygen from combustion 0.48 x 9.92 = 4.8 m /s
Total oxygen at the after air preheater -
section 4.2 + 4.8 * 9 nT/s
* oxygen based on dry volume 73977* 10° " ~6.4%
Average Measured % oxygen 5.8%
3> difference 9%
**
Actual Flow rate at - 407°K and 97.4 N/m2 is equal to -455,000 ACFM (DRY)
Measured flow rate ~ 573,000 ACFM (DRY)
101
-------
APPENDIX J.
COMPUTER PROGRAM FOR ANALYTICAL SIMULATION
OF PROCEDURES FOR
VELOCITY AND EMISSION MEASUREMENTS
IN STRATIFIED STACK GASES
(THEORY AND USER MANUAL)
102
-------
I. INTRODUCTION
A. PROGRAM FUNCTION
This program was developed to determine a measure of the error
associated with the number, location, and area proportion assignment
(strategy) for probes measuring effluent concentration and velocity in
circular or rectangular flue gas ducts. It has been used to generate
guidelines for selecting the number and best location of probes in ducts
and the best means of weighting probe measurements (strategy) so as to
determine the total and species effluent rate in a duct.
The program is designed such that the user specifies the duct
geometry, actual concentration and velocity profiles in the duct and the
number of probes and location and area associated with each probe. The
program will compute the actual effluent rates:
CVdA and
VdA
based upon the specified profiles and the rates that would be measured
using the specified number, location, and calculation strategy of the
probe placement. The error in the measured and actual effluent rate is
then calculated as:
Jt CVdA
-L W
i
I
/ VdA -
CVdA
Vi
/VdA
103
-------
By running a series of such cases, graphs may be produced
showing the error in stack gas measurement (i.e., average velocity,
effluent emission rate) as a function of number of probes, probe
location, strategy, and concentration and velocity profile specifica-
tion. These graphs may then be used as guidelines to determine the
best number, location, and strategy for placing probes in a specific
duct if some estimate for measurement (survey) of the duct profiles is
available.
B. REPORT ARRANGEMENT
The main body of this report is made up of six primary
sections as follows:
1. Computational Procedure, which briefly describes the
analytical procedures through which the program operates.
2. Input Data Description, which contains a detailed
description of the input data format for the program.
3. Output Data Description, which contains a brief
description of the normal program output and the error messages
generated by the program.
4. Sample Cases, which demonstrate the execution of
the program.
5. Miscellaneous Operational Information, consisting
of program capacity and modification procedures and of program
and JCL requirements.
6. Exhibits, describing the detailed computational
procedures of each subroutine and a source listing are included
in the last section.
104
-------
II. COMPUTATION PROCEDURE
There are seven general steps in the program's analysis procedure.
In step one, the duct characteristics are read (shape and size) and the
form of the concentration, velocity data are specified (raw data or
polynomials). Step two deals with reading and editing raw concentration
and velocity data and with generating least square polynomial curve
fit coefficients for this data. Step three involves reading the order
and coefficients of concentration and velocity data specified via poly-
nomials. Step four involves printing and punching of polynomial co-
efficients for raw velocity and concentration data calculation and
printing of given versus predicted velocity and concentration data
(for velocity and concentration profiles specified as raw data) and of
the standard deviation in concentration and velocity for these fits.
Step five consists of the analytical integration of the velocity and
concentration polynomials to predict "actual" average emissions,
velocity, and concentration. In step six, the program reads, counts
and edits the specified probe location data. Finally, in step seven.
the program determines and prints the average concentration, velocity
and emissions associated with the probe placement specified in step six,
and determines the error associated with that simulated measurement
procedure.
The details of these seven steps are outlined in greater detail,
below.
105
-------
Step 1. Determine duct characteristics and form of concentration velocity
data by reading the first control card:
Geometry Code (NGC): o circular cross sectioned duct
1 rectangular cross sectioned duct
>1 end of Program (call EXIT)
Dimensions of the Duct: Radius for circular (A).
(A,B) Length and width for rectangular
Profile Data Code (NIC): 1 concentration and velocity profile data
are given as the coefficients of the
polynomial
..
. ,\ . .
YORTJ'' XORR1"'
/
N
Z=
YORT: IjeTJy °r Theta,
XORR: (JL or (£))
\ "/ t M /
GO TO STEP 3
: O Concentration and velocity raw data are given
GO TO STEP 2
106
-------
-SteP 2- Read, edit and fit raw concentration and velocity data
a. READ, COUNT, and EDIT concentration and velocity raw data.
(A blank card with 9999 on column 77-80 after the last concentration or
velocity data will provide a method to determine NCR, number of con-
centration data points, and NVP, number of velocity data points).
EDITING CRITERIA: concentration >0
for |X'| < j
rectangular £
duct |Y'| £ |
for 0
-------
Y'
i >
X'
R'
Y'
108
-------
c. Read the number of maximum and minimum points in concentration
and velocity input data, with respect to (X,Y) or R, 0.
(MCXOR, MCYOT, MVXOR, MVYOT) (See note 5, input data).
d. Determine the order of the polynomials [{V(X,Y), C(X,Y)},
{V(R,0), C(R,0)}] to be generated.
Order of C(X,Y) or C(R,0) in X or R =
Order of C(X,Y) or C(R,G) in Y or 0 =
Order of V(X,Y) or V(R,0) in X or R «
Order of V(X,Y) or V(R,0) in Y or 0 =
NC = 2*MCXOR + 3
MC = 2*MCYOT + 3
NV = 2*MVXOR + 3
MV = 2*MVYOT + 3
If NC * MONCP; NC = /RCF, MC
NCR
UT
If NV * MV>NVP; NV
MV =
NVP
e. Call Subroutine COEFG and CROUT (See Exhibits D and E) to find
the coefficients of the least square polynomials for C and V. (Note that
output arrays of these routines are printed as additional information).
GO TO STEP 4.
Step 3. Read the order and coefficients of concentration and velocity
polynomials.
GO TO STEP 4
Step 4. Print and punch curve fit data.
a. PRINT the coefficients of the polynomials. If NIC = 1, GO TO
STEP 5, otherwise, continue as follows
b. PUNCH the coefficients of the polynomials, and PRINT the actual
(given) and predicted (computed) velocity and concentration for each input
data point.
109
-------
c. Calculate and PRINT the standard deviation in C and V
en < ^ E(computed C - actual C)
S.D. 1n c
V2
^(computed V - actual V)
NVP
Step 5. Evaluate average C, V, and CV. (based on polynomials)
f*
CI »
/. CdA
VI r JA
/dA
L
A'
VdA
AdA
CVI = JA
f CVdA
JA
dA
Step 6. Read, count and edit the probe location data by reading the probe
assignment control card.
If NST - 1: READ, COUNT, and EDIT, probe assignment data.
a. READ:
NASEG: (Number of equal area segments).
XORR(I), YORT(I), IARSEG(I) (Area segment number).
110
-------
b. COUNT: The blank card with 9999 on column 77-80 after the
last probe data card indicates number of probes (NPR)
c. EDIT: for rectangular ducts: IXORR'^^, |YORT'|<|
for circular ducts: o1 CALL EXIT (also reserved for future modification of program)
Step 7. Determine average concentration, emission, and velocity for the
given strategy.
a. PRINT the probe locations and probe area segment number.
b. CALL Subroutine PRBANS (see Exhibit G) to find average concen-
tration (CBAR), velocity (VBAR), and emission (CVBAR) based on the
EAUAL AREA STRATEGY.
Ill
-------
c. Find the errors: El « |1. -
CBAR*VBARi
CV1I
VBAR,
d. PRINT actual (VI, CVI) and predicted (VBAR, CVBAR, CBAR) average
velocities, emissions, and concentrations.
e. PRINT, El, E2, and E3
GO TO STEP 6 to read the next probe assignment control card.
112
-------
HI. INPUT DESCRIPTION
A. GENERAL DESCRIPTION
Input data to the program are specified by use of thirteen basic
card formats specified in Figure 1. Card types one - two are used for
specification of duct geometry and concentration-velocity data format.
Card types three - nine are used for specification of the concentration
and velocity profiles. Specifically, card types three - six are employed
if the profiles are specified as raw data; types seven - nine are for
use with data specified via polynomials. Card types ten - thirteen are
used for specification of sampling strategy, and probe location.
Card type one is used for inputting duct shape (rectangular or
circular), duct dimensions (X, Y or R), and input velocity-concentration
data format (raw data or polynomial coefficients). Card type two is a
single comment card employed by the user for Identification of the duct.
Card type three is used to specify concentration and geometric
coordinate data (x, Y or R,0) for raw concentration data. Card type
four is an indicator card used to signify the end of the concentration
Input data. Card type five is for use with actual velocity data, whereon
the user specifies velocity and geometric coordinate data (x, Y or R,e).
Card type four is then reused as an indicator to signify the end of the
input velocity data. Card type six follows the raw concentration and
velocity data, specifying the number of maxima and minima in planes of
constant Y(or 0) and X (or R) 1n the concentration and velocity profiles.
If the user specified concentration and velocity profiles via
polynomials, card types three - six are omitted and card types seven -
nine employed. On card type seven, the user specifies the order + 1 of
the concentration and velocity polynomials as functions of X(or R) and
Y(or ©)• In card types eight and nine the concentration and velocity
polynomial coefficient are respectively specified.
113
-------
figure T. JKP«r DAT*
Card
Type
1
Number
1 Only
'1GC. GEOMETRY CODE: (rectangular, l)t(Circular.O)
d. (flote 1)
B, {Note Z)
NIC. (Note 3)
Card
Column
I
T
It
Y
f
,(
*
<1
>
n
f
it.
3
•
JY
it.
ij
t1
:*'
^
«•>
)
;'
9
it
11
•*•
1C
ii
S-
*i
21
9
j*
J*
11
"
';
0
"
LI
"*'
JU.
If
**
•ff
e
I'-
ll
jr
f
*'
^r
<".c
"
*
•J
V
•
•
it
*v
•*
?4
fi.
*
u
Ii
•"
»/
S'f
w
y
31
i
***
M
if
""
3*
Tf
a?
*J
ss
M
21
ii
23
i/
»*
to
t.
J"
af *> v .u.
V5 y* V? it
*' ******
" V "' V
NOTES:
1} A ii radius of circular duct or rectangular dimension in X direction.
2) B 1s 0 for circular duct and dimension in Y direction of Sectanqular duct.
3] KIC * 0 Input data on profiles as raw ilata points, use card forrs 3. 5. and 6 but not 7. 8. and 9.
,
PUT after the last card 3, last card 5, last card 13.
-------
Figure I. 1HPUT MTU FOPH. Coat.
tn
Card
Type
6
Number
Hone or 1
MCXOB. (Note 5) <4
HCYOT. (Note 5) <4
MVXOR. (Note 5) <4
HYYCT. (Note 5) <4
Card
Column
1
1
.5
f
3-
t
tf
s
7
Hone or 1
NC.Order + 1 of polynomial C as * function of x or R
tC.Order + 1 of polynomial C *s a function of T or 8
NV .Order + 1 of polynomial V as a function of x or R
HY, Order * 1 of polynomial V as a function of T or 6
1
i
f
r
im
1
t
8
8
9
t!C*MC
None or '"• "*•
"tone or •
C(l,l) (See note 6)
C(Z.l)
'•
C(MC.l)
C(HC.HC)
V(1.1) (Note 6)
V(2.l)
:
V(NC,1)
V(MC.NC)
"
4J
VJ
ts
"
t-
'
61
it
iv
1*
5~T
*
s
If
IJ-
11
t
"'.
•\
".
r
•".
'••
".
'
¥f
"
•».
"
*'
''
~ 1
T*
•••
M
7T-.
*. C
.-
• -•
'*
-'
- *
5) I" order for the prosraa to find the least squar concentration and velocity polynomials f{V(x Y) Cfx Y)!
the order of these polynomials by specifying MCXOR. HCYOT. HVXOR. and HVYOT iherel uvtx'Y'' clx.Y».
1 • order of C(X.Y) or C(R.e) in X or R . NC • 2«MCXOR *3 1 * order of V(X,Y) or V(R.O) in X or R
o «i nn in ...
R.e). C(R.c-))]. the user ™st provioe
HV
°f
2*MVXOR + 3
3
"C«OR: Nu«nber of ma«ima and minima in a plane of constant Y or e In o-xentntion profile
«*CI: ',«x:r>tr of nu.-iia »?>d minima in a plane of constant x or R in ti •.. entration profile
»¥«OR: Muroer of ««»:,« and minima in a plane of constant Y or 6 in velocity profile
*Vrt)T: tfanter of Wd/ima and minima in a plane of constant x or R in velocity profile
6) Coefficients of concentration and velocity polynomials must be In the order of (1,1). (2,1).
C(J.I) (-^) (^) , Velocity -V* V"
1=1 j-l ft jTT
(MC.t). . (KC.NC).
Contention •
V(J,1) (
-------
Figure 1. INPUT DATA FORM. Cont
Card
Type
10
11
12
Number
1
1
1
I.'ST. Strategy Code, (Note 7)
Strategy Consent Card
MSEC, Number of equal area segments
Care
CoU
/
t
/?
1 ;
£r>
/
2.
j
f4
u
v«-
(f
sr
13
Less than 300
X or R probe location -2Lxiy. 0<.Ri
-•'
.1
•f
s
6
it
f
•
'•)
•
t
TT~
•J
Tr-
io
j--»
//
rr-
/2
jy
HOTES:
7) NST « 0 For new duct
" 1 ^'J Are* Strategy; followed by card types 11, 12, U. 4
• 2, 3, 4, and 5 is reserved for other strategies; (Call EXIT).
IMPORTANT NOTE:
f *• "• *» *• "• concentration, and velocity rust be consistent, e.g..
Concentration 1n Ib or 1b mole/ft3
Velocity 1n ft/sec
6 In radian
r
ft3
-------
Card types ten - thirteen are used for specification of sample
probe location. Card type ten Is used as an indicator of the strategy
type and to signify the end of the examination of a given duct and profile
set. Card type el evert is a comment card for the user to specify the type
of sampling strategy under evaluation. Card type twelve 1s used with equal
area strategies and indicates the number of equal area segments involved in
the sampling strategy. Card type thirteen is employed to specify each
probe geometric location (X,Y, or R,0) and area segment assignment.
B. DETAILED DESCRIPTION OF INPUT DATA
Figure 1 clearly details the format of the input data. The user
is advised to be assured of the following special considerations:
1. If raw data is specified for the concentration and velocity
profiles, no more than 300 discrete concentration and 300 discrete velocity
points may be entered.
2. The order of the velocity and concentration polynomials
is specified as the order + 1. A maximum order of 10 (order +1 =11)
is allowed, i.e., NC, NV, MC, MV < 11.
3. For each probe assignment case, the maximum number of probes
and equal area segments is 300.
4. Any set of self-consistent units are acceptable in the
analysis procedure.
Figure 2 Illustrates the input data structure for the program.
The user may examine one duct and one strategy (Case 1), one duct and
n strategies (Case 2) or m ducts and n strategies (Case 3). There are no
internal limits on m (number of ducts) and n (number of probe assignment
strategies In the program). The program is terminated by reading card
type one with geometry code of 2, as shown on Figure 2.
117
-------
INPUT DATA STRUCTURE
00
(Card 1
, I NRC=?
r
Cards
10
Card 10
NST=0
: (11,
lat
Urf*^P •
if-
w
End of Job
Card 2
fCard 1 NGC«0
orl, NCI=0 or 1
Probe Data
Profile Data
Subroutines and Main Program
(Including System Control Cards)
Figure 2. Case 1. One Duct and One Strategy Only
-------
Card 10
f Card 1
1 MP
fCard 10
:ards: (11,
13.
0
12,
4)
C»2
NST=1
'End Job
.ast Probe Ass.
1
(Cards: (11, 12,
13. A)
Card 10
NSl
fCards: (11, 12,
I 13, 4)
1
Card 10
JIST«1
Cards:(3,4,5,4,
6) or (7,8,9)
fCardl
I N
fCard 2
I Comment
NGC=0 or 1
Main Sub-
routine and Sys.
"xitrol Cards
2nd Probe Ass,
Mst Probe Ass.
Profile Data
Figure 2. Case 2. One Duct Only With More Than One Strategy
-------
IN3
O
f
Card 1
f Cird 10
NST-0
fCards: (II. 12. 13,
1 41
fCardlO
MST-1
•2
Cud
Card 1
NGC-0 or 1
f Card TO
KST-0
fCirdT: Qj. 1Z. 13.
yfi»1n. Subroutines
I System Control Cirds
'Ust Prob« An.
Utt Duct
f Cirdir (3.4,5.4.6)
I or (7.8.9)
List Probe Ass.
of 1st Duct
f Cards: (3,4.5.4 6)
or (7.8.9)
Figure 2. C«t« 3. Multiple Ducts tnd Multiple Strctegles In Each Duct
-------
IV. OUTPUT DESCRIPTION
The output of the program consists of both printed and punched .data.
The data falls Into two main categories: normal output and error messages.
The normal output consists of the printed and punched results usually
obtained with each run of the program. If any difficulties are encountered
in the solution of a case, one or more error messages are printed. These
messages are diagnostic statements which describe the nature of the
difficulty. They are described in greater detail in sub-section IV-B.
A. NORMAL OUTPUT
Case 1: When the velocity and concentration profiles are provided
as raw data, printed output consists of:
1. Number of input concentration and velocity data cards.
2. Input and output arrays from subroutine CROUT for concentration
and velocity.
3. Stack geometry.
4. Concentration and velocity polynomials coefficients.
5. Results of profile correlation (actual versus predicted
velocity and concentration values at input locations).
6. Probe location data for strategy.
7. Actual and predicted average velocity, average rate of
emission, and average concentration.
8. Error of predicted average velocity and emission for
the strategy.
Printed outputs 6, 7, and 8 are repeated for each specified strategy.
Concentration and velocity polynomial coefficients (Item 4 above) are
also punched for subsequent studies of the same duct - profiles.
121
-------
Case 2: When the velocity and concentration profiles are specified
through polynomial coefficients, there is no punched output and the printed
output is the same as Case 1, except the above Items 1, 2, and 5 are omit-
ted.
NOTE: For details, see output of sample cases.
B. ERROR MESSAGES
Five diagnostic error messages are printed by the program, each
specific to an error condition which might occur in a given duct - strategy
case. These conditions, the error message, and the program response are
detailed below:
ERROR
MESSAGE
ERROR
CONDITION
PROGRAM
RESPONSE
1. Error in entering
concentration
profile data
(Card No. I)
CON(I)<0
for rectangular duct:
|XORR(I)l
for circular duct:
The program edits the
remaining data, but th"
duct is aborted. The
next duct is evaluated)
if available.
XORRJ[I)<0, XORRH)>A
YORT(I)<0, YORT(I)>2ir
2. Error in entering
velocity profile
data (I)
for rectangular duct
|XRVtl)|>|, |YTVtD|>|
for circular duct
XRV'(I)<0 XRV'(I)>A
YTV(I)<0 YTV(I)>2ir
Same as response
immediately above.
122
-------
ERROR
MESSAGE
ERROR
CONDITION
PROGRAM
RESPONSE
3. Unable to solve
by this method
problem aborted.
CROUT routine unable to
find the coefficients
(division by zero or small
numbers unavoidable).
Same as above response*
4. Probe (I) located
outside duct
boundaries.
Problem aborted.
for rectangular duct
|XORR'(0|:
B
|YORT'(I)|>f
for circular duct
XORR'(IJA
YORT'(I)<0, YORT'(I)>2ir
The program edits the re-
maining data of this probe
assignment strategy, but the
analysis is aborted. The
program executes the next
case (if it exists).
5. Probe (I) assigned
to a non existing
area segment.
Problem aborted.
IARSEG(I)<0
IARSEG(I)>NASEG
Same as response immediatly
above.
123
-------
V. SAMPLE CASES
This section consists of the following;
A. PROBLEM 1
(a) Problem Statement
(b) Data Preparation
B. PROBLEM 2
(a) Problem Statement
(b) Data Preparation
C. PROBLEM 3
(a) Problem Statement
(b) Data Preparation
D. SAMPLE CASES INPUT DATA
E. SAMPLE CASES OUTPUT DATA
F. RESULTS AND CONCLUSIONS
(a) Problem 1
(b) Problem 2
(c) Problem 3
124
-------
A. PROBLEM 1
(a) Problem Statement
The concentration and velocity raw data for a duct of rectangular
cross section 1s given as follows:
-1.0
-1.0
-1.0
-1.0
-1.0
-0.5
-0.5
-0.5
-0.5
-0.5
o.o
0,0
0.0
0.0
0.0
0.5
0.5
0.5
0.5
0.5
1.0
1.0
1.0
1.0
1.0
-1.0
-0.5
0.0
+0.5
+1.0
-1.0
-0.5
0.0
0.5
1.0
-1.0
-0.5 '
0.0
0.5
1.0
-1.0
-0.5
0.0
0.5
1.0
-1.0
-0.5
0.0
0.5
1.0
0,0
0.0
0.0
0.0
0.0
0.0
0.5625
0.7500
0.5625
0.0
0.0
0.7500
1.0
0.7500
0,0
0.0
0.5625
0.7500
0.5625
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.5626
0.7500
0.5625
0.0
0.0
0.7500
1.0
0.7500
0.0
0.0
0.5625
0.7500
0.5625
0.0
0.0
0.0
0.0
0.0
0.0
i
I
B
I
A * B - 2 ft,
125
-------
a.l. Find the concentration and velocity polynomials, and
average C, V, and CV" (emission).
a.2. What is the predicted average c, v, and cv, if 4 probes
are located as shown.
probes
a.3. Repeat a.2. for one probe in center.
(b) DATA PREPARATION
Card type
Data
(1)
(2)
(3)
(4)
(5)
NGC * 1 (rectangular)
A * 2
B * 2
NIC * 0 (raw data)
Comment
Sample problem one
Concentration raw data
C X Y
• * *
* *
* *
Velocity raw data
C X Y
9999
(4)
.9999
126
-------
(6) Based on the raw data, we assumed that the concentration and
velocity polynomials were second order in x and y .'. MCXOR=MCYOT=MVXOR=MVYOT=0
(10) Strategy Code
NST = 1
(11) Strategy Comment
Problem one, a.l (4 probes)
(12) NASEG = 4
(13) Probe Locations
X.
-.5
-.5
.5
.5
JT
+.5
-.5
.5
-.5
Area Sefl. No,
1
2
3
4
(4) 9999
(10) Strategy Code
NST = 1
(11) Strategy Comment
Probelm one, a.2 (1 probe in center)
(12) NASEG = 1
(13) Probe Locations
J( _Y Area Seg. No.
0. 0. • 1
(4) 9999
(10) Strategy Code
NST = 0 (End of problem 1)
127
-------
B. PROBLEM 2
(a) Problem Statement
The concentration and velocity polynomials for above rectangular
duct is given as:
2
A = B * 2
a.l) What is the predicted average C, V, and (CV) if 4 probes
are located as shown:
I
(b) Data Preparation
Card Type Data
(1)
(2)
NGC = 1 (rectangular)
A * 2
B * 2
NIC » 1 (polynomial coefficients given)
Sample Problem 2
128
-------
Card Type
(7)
Data
NC = 3
MC = 3
NV = 3
MV = 3
(8)
C =
Concentration polynomial is,
(0) () -
(0)
(1)
en] (£L\ _ (£L\
\V I \ D ) \ D /
(o) (V) -
r> ^
(0)
• *' ]• I
(0) + (0)
21 ov 2
2Y
(0)
(0)
Therefore the concentration coefficients are as follows:
21
31
b!2
C22
c32 = o.
0.
'23
'33
•1.
0.
1.
(9) Similarly, the velocity coefficients are:
v21 = o.
'31
-1.
23
'13
'23
'33
-1,
0,
1.
r2Y
129
-------
Card Type Data
00) NST » 1
(11) Problem 2, a.l (4 probes)
(12) NASEG * 2
(13) Probe Locations
X.
1.
0.
0.
-1.
Y.
1.
.5
- .5
-1.
Area Seg. No.
1
1
2
2
9999
(10) NST = 0 (End of Problem 2)
130
-------
C. PROBLEM 3
(a) Problem Statement
The concentration and velocity raw data for a duct of circular
cross section is given as follows:
e
R (Radians) C
0.0
0.0
0.25
0.25
0.5
0.5
0.75
0.75
1,0
1.0
R
0.0
0.0
0.0
0.0
0.0
0.25
0.25
0.25
0,25
0.25
0.5
0.5
0.5
0.5
0.5
0.75
0.75
0.75
0.75
0.75
1.0
1.0
1.0
1.0
1.0
0.0
3.14159
0,0
3.H159
0.0
3.14159
0.0
3.14159
0.0
3.14159
9
0.0
1.57080
3.14159
4.71239
6.28318
0.0
1.57080
3.14159
4.71239
6.28318
0.0
1.57080
3.14159
4.71239
6.28318
0.0
1.57080
3.14159
4.71239
6.28318
0.0
1.57080
3.14159
4.71239
6.28318
2.0
2.0
2.0625
2.0265
2.25
2.25
2.5625
2.5625
3.0
3.0
V
2250
2250
2250
2250
2250
1687.5
1652.34375
1640.625
1652.34375
1687.5
1125
1078.125
1062.5
1078.125
1125
562.5
527.34375
515.625
527.34375
562.5
0.0
0.0
0.0
0.0
0.0
A = 1
131
-------
a.l) Find the predicted average c, v, and (cv) 1f 4 probes
located as shown:
a.2} Repeat a.l for 4 probes located as below:
a.3) Repeat a.l for one probe in center.
132
-------
(b) Data Preparation
Card Type Data
! NGC = 0 (circular)
A = 1 (radius)
B = 0 (dummy)
NIC = 0 (raw data)
(2) Sample Problem 3
(3) Concentration Raw Data
C R 6
(4) 9999
(5) Velocity Raw Data,
V. R. ©
• • •
• • •
(4) .'........ 9999
(6) it is clear that concentration profile is independent of e
(zero order) and is second order in R: Therefore:
MCYOT = -1
MCXOR = 0
The velocity profile appears to be second order in both R
and 6, therefore:
MVXOR * 0
MVYOT = o
(10) NST = 1
(11) PROBLEM 3 - a.l (4 probes)
(12) NASEG * 1
133
-------
Card Type Data
(13) Probe Locations
H i Area Seg. No.
.5 0.0 1
.5 1.57080 1
.5 3.14159 1
.5 4.71239 1
(4) 9999
(10) NST = 1
(11) Problem 3 - a.2 <4 probes)
(12) NASEG = 2
(13) Probe Locations
i 1 Area Seg. No.
.333 0.0 1
.333 3.14159 2
.6666 0.0 1
.6666 3.14159 2
(4) 9999
(10) NST - 1
(11) Problem 3 - a.3 (one probe)
(12) NASEG * 1
(13) Probe Locations
i i Area Seg. No.
0 0 1
(4) 9999
(10) NST - 0 (End of Problem 3)
(1) NGC = 2 (End of Job)
134
-------
D. SAMPLE CASE INPUT DATA
Completed input data sheets are shown in Figure 3 for the three
sample case problems.
E. SAMPLE CASE OUTPUT DATA
Figure 4 is the printed output generated by the program for the three
problems specified as the sample cases.
135
-------
IBM
Figure 3.
FORTRAN CWi«f Fun
Pruned .A U.S.A.
MOCUtt
KCMMMI |p,,,
C
(
1
0
0
0
0
C
.-..-- ., !..",»
FO1TCAN STATtMEMT
2 .
.
.
.
.
•
.562
.7 5
.562
.0
.0
.7 5
. 0
.7 5
.0
.0
562
.7 5
.562
.0
. 0
. 0
5
5
- 1 .
- 1 .
- 1 .
- 1 .
- 1 .
-0.5
-0.5
-0.5
-0.5
-0.5
0 .
0 .
0 .
0 .
0 .
0 . 5
2 .
O.s!
0 .51
0 . 5
0 . 5
1 .
1 .
0
AMP LE
- 1
- 0
0
0
1
. - 1
- 0
0
0
1
-1
-0
0
0
1
-1
-0
0
0
1
-1
-0
P R OB
,
. 5
. 5
.
. 5
. 5
.
.
. 5
5
5
5
5
L EM 0
N E
---
<*•*•>• .
* •, •
"
. _ ..
\
i. —
p
—
U S* 1>
1
--
11 a a « u
t
.«•!/ -.7
<'«"«•->•••""••
14 *T M 4V
M ., ,?
>JtN:M.C'TiOM
J ;« 71 f» 77 71 *V Kl
CO
-------
IBM
FORTRAN Coda* Farm
6X38-7327-6 U/MOSO"
Primed in U.S Jk.
"x— i «.-*,«
»?«.»..
OA.I (
G..MC
h-NCH
rut 2 Of 7
CJKE^CnOMJuKI-
-------
vmur OX28-7317-61I/M050"
lOm FORTRAN CMO* F«n PrinMd In U.3A
f JG---W
,«.-K-«ww*
J*fl
*M>i>«a
4-«*T«- C'lTM
C* ***•»*.
r. MCII
^a 3 o. 7
C«OlU,Tt0 .***«.
*
(;•
1
L
i
•.T4tf*«:M
-•*•
! •
1
0 0
4
-
•
i
i_ i
•
0
1
0 3 03 0
5 KMIIAN STATEMENT
. 0
. 0
0 . 0
0 . 0
D . 0
) 0
PROS
. 5
. S
. 5
. 5
PROS
.
i
2 .
9 3
LE H 0
L EM 0
1 .
1 .
1 .
1 ..
1 ,
N E
-0.5
0. 5
-0. 5
0.5
NE
0 .
S A
A.. 1
A . 2
2 .
HP L E
' _ '
- 0
0
0
1
(4 PR
1
2
3
4
ONE
1
1
R OB L
. 5
. 5
.
3 B E S)
ROBE
N 2
I N C
E NT ER
H v
-------
IBM
j •
1
j.
*
i
0
0
-
1
*WW*4
. J . »
2
-
fOaikAN SlATEMiNI
: j . j :i .! i- — . i. i. •• •< .- • — . •- -. ^ -. . • — : — — — — - — : ~
.
.
.
'ROB
.
.
.
1 .
SAW
.
.
. 06
. 06
. 2 5
. 2 5
.562
.562
•
.EM 2
PL E P
5
5
5
5
1 .
0 . 5
-0.5
- 1 .
ROB L
0 .
0 .
0 . 2
. 2
. 5
. 5
. 7
. 7
1 .
0.
1 .
0 .
1 .
A .1
M 3
5
5
5
5
(4 PR
1
1
2
2
0
0
\
0
;
0
\
0
.
0
OB E S)
f
. 1 41 5
.
.1415
m
.1415
.
. 1 41 5
•
-1 .
0 .
- 1 .
0 .
9
9
9 '
9
1
•
0 .
1 .
0 .
1 .
•
0 .
0 .
'
*«..— .,0«I1J. .,_-:^. V «•—,—.„., i. „„.,!,_ .—— - ,
IJtNdfCAtlUN
-Ji+l-fl
99 99
OJ
-------
IBM
FORTRAN C*4u| Fira
CfinicdmUSA
**£.**••*•!*
.*' !*.:»« 1
* 2Jj
1 2 2 5|
2 2 si
225
22 5
168"
i
1 6 5^
1 64<
1 6 Si
1 6 83
1 1 2
1 0 7
106
1071
1 1 2
s 6 ;
5 2 '
SI <
S 2 '
5 62
<
(
i i i • > <
|
3 .
0 .
D .
3 .
) .
) .
'•»
Si. 343
) . 6 25
? .343
r .5
3. 1 25
2 .5
J .12 5
; .
I . 5
r . 343
5 .6 25
1 . 343
! . 5
) .
) .
7 5
7 5
75
7 5
1 .
i ° •
, 0 .
0 .
0 .
0 .
. 2
. 2
. 2
. 2
. 2
. 5
.5
.5
. 5
. 5
. 7
. 7
. 7
. 7
. 7
1 .
1 .
T
1
!s
5
5
5
5
5
S
5
5
5
G
1
3
4
6
0
1
3
4
6
0
1
3
4
6
0
1
3
4
6
0
1
1.1415
.
.5 7 0 £
.141!
.7123
.2831
•
.5708
.1415
.7123
.2831
»
.5708
.1415
.7123
.2831
.
.5708
.1415
.7123
.2831
.5708
rot IRAN
|9
i
I
ib ""
i9
J9
£
0
9
9
8
0
9
9
8
0
9
9
8
0
STA^tMf.Nt
1
.
1
|
1 1-
T —
't
t
1
1
|
_
~
1
+ -
i
r •
,„, 5
jt*....^ .
"
,.. 7
4JtNfttlw4lKI*<
"••*»•»«>
9999
• •Ml* »M •*<•*•
-------
IBM
TORTRAK
OX»-7M7-«U/MOW
MOGKMMf * | Mn
I
^
1
0
t *Ii k I NT
?».v*i» i
0
0
0
0-1 0
01
1
1
0
1
0
02 '
01
r-ff<*tr.^
w«»oe»jNs
GWttC
fjtfn
rust 6 w 7
UCUUMftiuEr
FOITBAN SlAttMfNI
•
•
0
PROS
. 5
. 5
. 5
. 5
. _ .
PROS
.333
.333
. 6 66
. 6 66
P ROB
L EM 3
\ .
\ .
1 .
0 .0
A . 1 (
1 . 5 |7 0 JB 0
3 ,1
4 .7
L
L EH 3J
3
3
6
6
L EN 3
0 .
3 .1
0 .
4159
239
3
4
6
4 PRO
1
1
1
.1415
.7 1 23
. 2 83 1
B ES )
1
^
A . 2 j( 4 PRtOB E S)
4 1 59
3.14159
A . 3
1
2
1
2
( ON E
PRO BC
)
-
" '• " » "
'
1
UNTtnCATlON
UCUNCI
• ' . 1
9999
. • .
9999
9 999
-------
IBM FORTRAN CMuf Fprm
Mov«»
0
2
i
r~
-
i
•
* • • • •
>.-..
0 KJHTIAN StAKMtm
D . 0
1
1
0. 0
I
j
1
1
r
f-
r —
]
._
-
1
1
i —
i
- - -
—
,
GXO-TO?* UfU 050' •
^ 7 » 7
MOIUiULttMtr
1M)SSo0"
9999
-------
r f'..' LAU.
;.L rf C"j. :
IfJi 1,1 ANE/
A CAKCS ;•, :;(... CF v?L. DATA
AORY- c.r LRCJ7 KMJTJN FUR CC'iti
c.it.a.lift,C,Ct 02 0.0
O.O O.C
C..^ o.O
"."• (..C
C.i; V.CCf l;- f:.t.
O.i/ o.e2
01 U.O
0.125K'OE 02 0.0
Oi O.O O.C
02 C.C
c.o
c.c
t.o
O.O
f..C
C.O
01
p.° 0.12iOOCt: P? 0.0 0.6?VCOOt 01
SrLooe 01 ii:H001 01 £:r°°ot °l ii;h>c' " £:Hct tc
o.c?i,coofc 01 o.o r,.o o.o c. /, no
r..f?t(iC3i 01 t-.f. ... o.iBUiot o: o.o _ _ j.o p j uo
"o*C ^'r «'« O.IU.2501 02 0.0 * c".53l"iJOc 01 i>Iv37iOOt 00
- ^os», S:S s:S 2:; tS.u»«.,-t?1—-"Bl.u«i«!:!«««
t-4.-J.iXOC L2 C.C
i».j 0.0
02
o.o
^
C. i^
0.ol'..Ctt.t M C.f-
.. -
0* O.C 0.0
O.tZtiOOCE 01 0.0
f.;-l»7!.0f Cl O.C
«•" o.o o.o o.o
01 o-o o.o o.o o.o
0.125GOOr 02 0.0 C.SOOtOOt CO O.C C.C
°.0 0.625000E 01 0.0 o.O 0 0
0.62SOOOE 01 0.0 O.21E75OE Ol O.O O.O
°/° °'° - - 0.0 0.4375COE-CO.-0.0....
°-° °'° 0.0 0.0 0.2187506 01 0.0
0.0. . .._ ... _ 0.2&OOOOE_OO 0.1OOOOOE_OJ_
O.iOOOOOf 00 0.0 o.O
O.C 0.5000OOL 00-0.1000006 Ol
0.0
o.o
0.0
ANb OUlr-UT ARKY OF tkOUT RUUTIN FOK V?L
C*>
0.2tSOCO6 02 0.0
o.c c.i2srooe
o.u o.o
0.0 o.O
O.C 0.0
C.12&OCOE 02 0.0
0.0
c.c>2St;ooe
0 12SCOO, 02 00
a— »
0.t25000E 01 0.0
0.c2iCOCl 01 (i.O o.O
fc.o o.ssi^sce 01 o.o
O.JTiOOOOfc 02 C.O O.SOOOOOE 00 0.0
0.0 0.125000E 0? 0.0 0.0
0.1/50006 02 0.0 O.O7500C 01 0.0
o"«
'
0.6250001 01 0.0
'
0.21E7!.0£ 01 0.0
C.O
0.0
02 a
C.218750E 01 0.0
0.0
o.o
0.0
^U ikLC
0.0
0.765625E 00 O.lOOOOOt 01
01
0.0
0.0
*
0.0
00
°'C
0.0
'
01
0.125COOt 02 0.0 0.625COOt 01 O.e25000£ 01 t
l » :E - -
0°:S ---- S-S - lo0 ----- SS --------
?:?Ot25CE 02 olS _ _ Si!,™* 01 S&TMHK 0.
0.0 0.531250t 01 0.0 0.0
0.531250E 01 O.C C.*5i&63t Ci C.140625E 00
fc.C t.SlOCCCE 00 C'.O
0.t25000E 01 0.0 0.0
f.*1 O.2It750t OJ C'.ti
C.O C.O
o.o o.c
O.C O.O
.sOOOOCt 00 0.0
0.0
o.?
0.0
01
0.0 0.0
O.U 0.0
- 0.0 o.O ...
o.o o.soooooe oo-o.io&ooor ai
0.0 O.?187iOi. 01 0.0 0.0 ',.
0.?1C75GE 01 O.C O.V6^62^E 00 0.IOOOOOE 01
Figure 4. Sample Case Output
-------
-SAKPLt-PROBLEM ONt ...
STACK GEOKfcTRV
CONCENTRATION COEFFICIENTS
C 1, 1« 0.1OOOOOOOOE 01
C 2« 1- O.b
C 3, l=-0.1GCKHlOOOOr 01
C I, ?* 0.0 _
C 2, 2« 0.0
C ?• i* 0.0
c i» 3=-o.iccu;ccoot ci
C i» 3* O.I'
C 3» 3- O.lOOOOOOOOt bl
ii: FFKUHTS
v i, i> o.icooooocot 01
V 2( 1= 0.0
v 3t IK-O.ICOOOOOOOE 01
. V 1, 2« 0.0 ________
v ;» ** o.o
V 3i i«. 0.0
V 1, 3«-O.lCOCOCCOOE Cl
V if 3* C.C
V i» 3>* 0.10OOOOCOOF 01
-------
2Y/B
CONCINTRATION
-1.IK'(»'O
-l.U/C'OO
-l.ococu
-l.COOOO
-i.iooro
-o.sootu
-o.soooo
-0.50000
-C.EOOCO
-0.500OO
i c.t>
Vi 0.0
o.o
o.o
o.c
O.!>0000
(i.SCOCG
L.^OCClt
O.tCfK-0
c.^cuoo
l.COOOO
1.00000
l.CCOOO
1.00000
1.00000
-l.CH'OM.
-O.WKlC'O
O.i)
0.i/OO(-(>
i.roft-o
-l.OOObb
— O.Si>0()O
0.0
0.500 CO
l.CCOOO
-I.CCCOO
-o.tc-ooo
6.0
o.toooo
I.OOOOO
-1.00000
-c.iooco
0.0
O.il-OOO
l.OCOCO
-i.OiOCO
-O.S.COOO
0.0
o.^^ooo
l.oooco
o.o
0.0
0.3
o.c
0 .0
0.0
O.btZSO
0.75.000
0.5J.250
0.0
c.c
0.7iOOO
i.cooco
O. 76000
0.0
0.0
c.9t,no
0.71000
o.stzto
0.0
o.o
0.0
c.6
0.0
0.0
RLS.ULTI, Cf PkCFItt CCKRELMUN
INPUT
VtLOCITY
PRIPICTLD
CONCENTRATION VELOCITY
0.0 O.C .. ... . ...
O-.0 0.0>
0.0 0.0 . ...
0.0 0.0
O.o
O.7&OOO
0.5t2i.O
C.O
0.0
0.750CC
1.00000
0.7MKHJ
0.0
0.0
O.562JO
0.75SOO
C.S62SO
0.0
0.0
O.O
C.O
c.c
o.o
0.0
<•. 75000
O.t.6210 . . .. .
0.0
P.O
O.TiOOO
i.ooooo -
0.75000
C.O
C.O
&.7SCOC
olo *
C.O ...-•.
o.o
C.O
C.O . .
-------
Ol
2X/A
2Y/1
ktSUtTS Of PKCMtE CCKACLATItN
INPUT
PRCCICTtO
- .OOIMHI
- .OOOOO
- .cctco
- .toooo
- .GOO C.C 0.0
1.COC6C 0.0 0-0
-1 .OOOOO
-0.50000
0.0
O.FOOOC
1. OOOOO
-!.«..( Ot-0
-O.i.'OOO
0.(*
o,ir«o&
t.f^eon
-I. OOOOO
-0.5(400
O.G
C.!'C-CUf
I .000 00
-1. OOOOO
-o.»oooo
0.0
l.OOCGO 0.5V000
1. 00000 1 .00000
STANDAhO DLVIAT10K IN C- O.O
0.0
0.&f>250
0.75000
0.562SO
O.G
0.0
0./5000
1.00000
0.7*000
*.«
0.0
O.^6250_..
0.75000
0.56750
o.o
0.0
0.0
0.0
0.0
0.0
IN V* 0.0
B.O
0.56250
0.71.000
0.56250
C.O
0.0
0.75000
1 .00000
0.75000
«<•
0.0
. O.StiSO.
0,75f;OO
0,56250
0.0
0.0
o.o
n.o
o.o
o.o
WflOC'TY
0.0 . . . ....
o.o
0.0
C.O
ft.n
0.0
O.S«?50
0.75000
0.56250 . _ .
0.0
°-n , , , , - - -
0.75000
i.ocnoo
S^500^ .......
C.O
0.!>B?«0
0.75000
0.56250 . ... .... . _^
o.o
0*0 ..._._.. ... . . _.....
o.o ~" ~ " -•---.
0.0
o.o
0.0 . .
-------
PROfcUM ONt
£R OF
2X/A
PPOS-t LOCATION DATA
2Y/t
— C.6C{.OO
-O.5OOOO
0.5CGCO
O.SOOOO
-o.soooo
0.50000
-0.50OOO
O.5(<00u
AKtA SEGMENT
I
?
3
t.
ACTUAL
AVCRAUF V AVEf.AGf C*V
AVERAGE v
CTHPUTtt
AVLRAC-t C»V
AVtRAOt C
CKkCR IN CVbAk=
tRRUR IN CbAR*V£AR=
tkKClk IN V6AK=
0.11236
0.11236
C. it it 2
flNt
PftOf.fc IN CEUltR)
2X/A'
0.0
LCCATICW DATA
OF_PRli.tS= 1. .
ARFA SF&KFNT
1
0.0
ACTUAL
V AVtKACc C*V
«* ..... _0.2c4
tPROR tt. QBAK*ytAK'
rrshCK. IN VLAk-
AViSAf.f V
.... l.CCO
Sf C*V
l.OCQ.
vi C
1.000-
i. 51562
-------
PL£ PROfcLEH.
STACK OEQHttlvY
. ... . ...... - . RECTANGULAR-AC FT O=__JtJMQQO __ BtH.»l»____2*OOOOO_
PRfUIU DATA
CONLCNTRATIG>V COEFFICIENTS . .
C t* I* 0.1COCOOOOOE Ol
C 7§ I- o.n
C 5. l*-O.HCC.CCCOCfc CJ
C J» 2* O.d . .... __________________ ___ ___ _____
C. 2, 2<- 0.0
_j C 3i i- O.O
» C it ?*-0.100CCCOOUc 01
00 C i't 3* O.C
C 3t 3* O.lCCCtCCOCt Cl
V It I- U. l&CtOOb^CiE 01
V 2, IB O.C
V :*t 1*-O.H'OCOOOOOE 01
V it 2- C.O _____________
V 2t i= O.O
V i, ?«- 0.0
V 1. 3**&.lCOCaK!C-GE Cl
v it 3» o.a
V it 3«- O.lOOCCOObCt 01
-------
PR.OLUM 2 .. ._ A.XU rRC'&lS)
(,(-
i.ccccc
O.O
0.0
-1.00000
LC CAT ION DAI*
2Y/E
i.rcccf.
O.50COO
-0.5COCO
-l.OOOCO
tC.tt SICMtNT
i
1
2
.2
ACTUAL
V AVERAGE C*V
LkKOR I
CVtAR=
Ci.AK*VtAR=
_ ..0.375 .
COMPUTED
AVERAbF C*V
. 0.2SI
AVERAGE C
£.371
0.011i3
O.iOJti
-------
no i-f cnn. DATA CAKCS ic nc.. OF ¥£•„. DATA CARCS 25
INPUT Afto OUTPUT ARRY OF CRCUT ROUTIN FOR CO.MCrNTRATTPN
C.K/OOOOc 0? O.SGbOOOE 01 0.37SC3OE Ol 0.2?75OOE O2 0.5OOOCCE 01 0.3750OOf 01 O.3I2500E 01 0.151250E 02 0.3750COE 0' G.3125UOt 01
Cl C.102656h 02 ........'.
C.l&CCuCfc C C.5COCCCE 00 O.375GOOE OO 0.2COOOOE 01 0.50000OE 01 0.12SOOOE 01 0.10000OE Ol 0.0 0.375000L 01 0.12SOOOE 01
I'.lOvi'/ii 00 O.100000E 01__
INPtl AUi> OUTPUT ARKY OF CROUT KDUTIN FCR VEL.
..!•»;•:.< i!(t w i.ir-'.j-li Ci C.c24r 01 0.12SrCO£ f>2 0.6249«>9E 01 0.46fc749t Ci G.9375001..01-0.468749*.-01 O.351562E-OL
Cl O.AtimVE 01 C.3VC6J3E 01 0.46A749E 01 0.361S6?fe 01 0.292966E 01 0.137190E 05
.V^itl-Vt'lii C.7i..ouvr Cl l».c>tl622?' 01 0.345701E 01 C.3515626 01 0.2WS.fcSi 01 O.25V276E Cl 0.1O3702E: 05
.lii(.Ct.c fi O.£,?'.-J?VE 01 0.4£fV49f 01 O.S;7S/--fI3t Cl w.«^749t 01 0.351!>62E 01 O.T:929u<-:£ 01 O.i'OtiiE Ol C.?*."?9t-*i£' Ol 0.2441'.C6 01 O.34l79fcE O4
.4M-7'.rt Ol C.5TOi2^i 01 C.3-V5.7CIJ. Cl 0.3il"62t Cl C..?^iV<£[ Ci fc.rt5'£. 01 0»3<:0624E 01 P.2S'i«'6?E 01 0.t9140£E_01 0.3457C3£-01 0.239277L OJ. .0.240331£ .04-
. «ii.tV4Vr I'.. O.3* lioir. Oi O.:vi%odl Cl 0.ivCpl*£ f! O.i'VISoEE 01 C.T-4414CE Cl C.3^57C3t 01 O.f>9i'77t Cl 0.2160C-4E 01 0.170l6t.E 04
.i>:itl^-v*il <-.:^'.ciE Cl 0...5VS7tE i-1 c.i9i:tttt 01 0.2»*14&E £•! 0.21tOi3E 01 0.259277E. 01 0.21oOi4E 01 C.191216E Ol 0.126&O7E 04
/litit Oft C.37.is;ot CO O.SCiOCOE 00 0.25tOriCt CO 0.1t>74V^t 00 0.375000E OO O.U7iCOE 00 O.14O625E OO O.224999E 04
li"?4ii i.--. : 0.1COC13t Cl 0.4?fci'l«E 00 0.37S4S8t CO-O.i24S90E 04
tl'-'-.'^rt ;!•! «:;ii,/< V«.F Oi-t..S5-.-. 6!;-4j: 01 0.3"731lf 00 O.i32«74b-01-C'.700457t-Ol-*.?15O2iE-Ol 0.9939O5f 00. 0.10012tE 04
".'STvOJl 01 C.i46V3i.i-Ol-0.15Vl4?b-<»l 0.3!i'379£ 01 O.4U)543C-02-().23313SC-02 0.273iitt 'if> 0,49fc5?7c 00 0.373725E 00-0.9497O76-OI
Cl O.lltll3r 01 O.luCl^BE CC C.1UWSF 01 (».397SlDr CO 0.2iOS4«£-Ol 0.10207^£ OO e.:*(.CS(.k*-Gl O.2932>9t-02-C.l001l2E O4
-------
SAMPLE
STACK GEOMETRY
CIRCULAR-RADIUS*
PROFILE DATA
COEFFICIENTS
0.2(/bOtbbGGE 01
o.o
. l.OOOOOFT. ____ _. ____
C 1* 1=
c i, 2=
C 1, 3' O.lOOOOOOOOt 01
VELCC1TY, COEFFICIENTS
V 1. 1= 0.2i^99i,70.ni-Hin
10o2.«.f:A13
562.50854 ...
515.61279
-O.CC8t:t . .
0.00774
-------
R/A
O.75CCO
0.7>((C
1 .C.OOCO
J.OOvOO
l.&OCOO
i.POCOO
THtTA/?Pl
0.3
C.O
0.0
0.0
0,0
u.isooa
C.2SC-CO
C.iUCCt
0.25 OOO
0.251)00
O.bOW.o
d.Jwi'.OO
o.toooo
o.ttcco
i'.&ooro
0.7i(-0d
P.O
0. lt.*15
O.50000
fr. 75000
! .ooor<>
o.o
0.25f-OC
t. ccoo
0.75000
1 • <--iC 00
O.I.
C.2TCOC
o.snccx^
&.75&CO
] . DC'Cf <>
O.O
0.750CO
l.OtOCO
0.0
U.50OOO
0.75000
i.roooo
r-n TN c =
RttULTi OF P*OFIU
INPUT
COKCLNTRATION
CKtDlCUO
e.o
VELOCITY CONCENTRATION
?250. or ooo «.ooooa
2250.00000 3.00000
2250. OC CCO 2.0OOOO
225G.OCCOO 2.OOOCO
I<.fi7.50000
1652^34375
Itt7.50000
1062150000
lO-7i.l25-00
1 !.?£ » 00000
5t>2 * 5 OC'OO
577.313/5
515.62500
527.34375
olo
0.0
olo
O.O
IN V= C.l«,3C3t-Cl
J.&6250
2.06iiO
2.0625O
2.0&250
?.0e?50
2.25COO
2.25000
2.5*250
2.5625C
2.56250
3 I 00000
3.00000
3.00000 __
3.00000
3.00000
VELOCITY
2249.9S706 ._
2250.013<>2
225C.OZ515
2250.00317
?249. 95337
!6e7.5(.MO
U52.3<351
U40. 62158
U52. 3*058 ... „ ._. _..
1078. 1 1546
1062.4B413 _ ....
107E.119K
1125.02^02 .
562.5085*
515.61279
5i7.33v36 _ . .
-olocecfc .
D.C0491
0.0077*.
0.0009B
-0.015S7
-------
_ ' PROtLEH 3 .. A.II* PROBES!
LOCATION LATA
UF ^£,ttS«-_V
THETA/2PI
AREA SFGMtNT NUMBER
O.SOOOO 0.0 1
O.iOOOU O.iiOGf; I
O.tOuCO O.tOOOO 1
Q.YtOCO 1
tfl
OJ
ACTUAL COMPUTED
AVERAGE V AVERAGE C»V AVERAGE V AVERAGE C*V AVERAGE C
722.218 It5e.326 10>5r«33 2443^34$ a.?M
ERROR IN CVBAR* 0.^7338
ERROR IN CBAR*VbAR» 0.47338
ERROR IN VBAR- 0.50361
-------
PRCbLtH 3 A.Z.U P"*OtlSI
PROFt LOCATION DATA
NU«BER_QF.P«CfctS«_ 4
A Tr.iTA/iPI
C.333JO
O.CC.66C
0.6(660
0.0
o.ioooo
0.0
0.50000-
ARtA SEGHtKT NUMfcER
1
2
1
ACTUAL
AVERAGE V AVfRAGE C»V
722.218 16^8.326
ERROK IN CVBAR* C.46952
tkROk IN CfcAk*VtAP= 0.5071V
CRhOH, IN Vb*R« 0.^1939
AVFRAGF V
C*V
2436.938
AVERACE C
2.279
PROEf LOCATION DATA
NUH5ER. OF PRCtfeS*.. J.
R/A THFTA/?P1
0.0
0.0
ARfcA SEOMENT NUMBER
1
ACTUAL
AVE.RA&E V AVt^AGl C*V
7<<.?lb 16*6.32*
FRfcfW IN CVbAK« 1.713M
tkkOft IN CbAR*VftAR« 1.71356
tkkOR II VbAR*
AV'RAGt V
C*V
AVERAGfc C
.. . 2.0CO.
-------
F. RESULTS AND CONCLUSIONS
a) Problem 1
The concentration polynomial was determined to be
c - 1 + (0) <) - (i) -(* + [o + (o) (jjj) + (0)
where A * B = 2
2 2 2 = 2 2
C = i . y . (+1 . Y) X or C = (1-X) (1-Y)
Similarly, the velocity polynomial was found to be:
V = (1-X2) (1-Y2)
Standard deviation in C « 0 (perfect fitting)
Standard deviation 1n V = 0 (perfect fitting)
Actual average velocity - .444
Actual average C*V (emission rate) • .284
(1) For the four probes case:
Predicted average velocity • .563 — error of 26.62
Predicted emission rate = .316 — error of 11.231
Predicted average concentration * .563
(2) For the case of one probe In center
Predicted average velocity = 1, error * 1253!
Predicted emission rate s 1, error = 251%
155
-------
b) Problem 2
Actual average velocity = .444
Actual emission rate * .284
(1) For the 4 probes case:
Predicted average velocity = .375 , error of 15.63S
Predicted emission rate - .281 .error l.U
Note that if the emission rate is to be found by multiplying the average
concentration and average velocity, the predicted error is 50.52
c) Problem 3
Concentration polynomial C * 2 + (1) (£)2
A - 1
therefore: C * 2 + R2
Velocity polynomial
V = 2249.98 + (.1875) (Jj-) - .2209
- 2249.90 - 1001 (^-) + 1001
-0.0949 + 1001 (^) - 1001 (
where A = 1
Considering the order of magnitudes of the coefficients/the
polynomial can be approximated as:
[ft a ol F « o2l 3
2250 + 1000 {^j-HOOO (^T R* 1000 (-J-) - 1000 (^-) R or_
llf [090!)
V • 1000 1 - R 2.25 + R (^-)Z - (JL)
L J v L J /
where at R » 1 V « 0 and at e = 0 and e - 2ir
V = 2250 [1-R]
156
-------
Standard deviation in C = 0 (perfect fitting)
Standard deviation in V « 0.014
Actual average velocity « 722.218
Actual emission rate • 1658.326
(1) For the case of 4 probes shown below:
Predicted average velocity = 1085.933
error = 50%
Predicted emission rate = 2443.349
error = 47%
Predicted average concentration =2.25
(2) For the case of 4 probes as shown below
Predicted average velocity = 1097.333
error - 51.9%
Predicted emission rate = 2436.938
error = 46.9%
Note that if the emission is based on the product
of average velocity and on average concentration
error is equal to 50.7%
(3) For the case of one probe in center:
Predicted average velocity = 224.987
error =211%
Predicted emission rate = 4499.973
error =171%
157
-------
VI. MISCELLANEOUS OPERATIONAL INFORMATION
A. PROGRAM CAPACITY
The program requires approximately 200K core locations during
loading and execution on the IBM 360/65 computer. Actual program length
Including buffers 1s approximately 114K.
The execution time for the program depends primarily on the number
of maxima and minima in planes of constant Y (or e) and X (or R) 1n the
concentration and velocity data. A typical operating case such as the
sample problems in Section VI requires approximately 11.63 minutes of
execution time (CPU time is 19.50 seconds for compilation and 4.31 seconds
for execution) on the IBM 360/65.
No tapes are required by the program.
B. PROGRAM MODIFICATIONS
1. The maximum number of Input concentration or velocity data
or number of probes can be Increased or decreased by modifying the dimension
statements 1n the main program nad COEFG subroutine. (Variables CON, VEL,
XORR, YORT, XRV, YTV 1n Main and variables Z, X, Y 1n COEFG subroutine.)
2. The maximum order In X (or R) or Y (or e) of the velocity and
concentration polynomials can be changed by modifying the DIMENSION statements
in the main program and all subroutines. (Variables; C(M,N), V(M,N)»
AR(M*N, M*N+1), and CORV(M.N) where M and N are the max. order 1n Y (or 6)
and X (or R) respectively.)
3. The present program computes the emission, velocity, and
concentration on the basis of EQUAL AREA STRATEGY (NST « 1, PRBANS Routine).
The values 2, 3, 4, and 5 of NST are reserved for future expansion for using
different strategies (1t is evident that for this purpose, additional
routines must be added). Note that the user 1s able to use the present pro-
gram for any kind of strategy, provided he prepares the Input data based on
the equal area strategy.
158
-------
4. The maximum number of probes and equal area segments can
be changed by modifying the dimension statements 1n the main program and
PRBANS subroutine. (Variables; XORR. YORT, IARSEG, and IUSEG).
C. JCL REQUIREMENTS
The following JCL Statements applied when the program was compiled
and executed for the test sample case. Since JCL Is strongly dependent on
both machine and facilities, all necessary changes should be exercised where
appropriate by consultation with the systems people at any new facility
The following JCL Statements are used:
// Job Card
// EXEC F0RTGCG,PARM.F0RT=N0LIST,TIME.60=20
//F0RT.SYSIN DO *
(MAIN PROGRAM
I SUBROUTINES
/*
//SYSDUMP OD DUMMY
//GO.SYSIN DO *
(INPUT
(DATA
/*
159
-------
EXHIBITS
EXHIBIT A Over-All Program Structure
EXHIBIT B Description of Principal Parameters
EXHIBIT C Main Routine
EXHIBIT D Subroutine COEFG
EXHIBIT E Subroutine CROUT
EXHIBIT F Subroutine CVINT
EXHIBIT G Subroutine PRBANS
EXHIBIT H Function POLGEN
EXHIBIT I The CROUT Method for Solving Sets of Linear
Simultaneous Equations
EXHIBIT J Source Listings
160
-------
EXHIBIT A
OVERALL PROGRAM STRUCTURE
The program consists of a main routine, four subroutines (COEFG,
CROUT, CVINT, and PRBANS), and a function subprogram POLGEN.
The program is written In FORTRAN IV. The modular tree diagram of
the program Is shown In Figure 5.
MAIN
COEFG CROUT POLGEN CVINT
PRBANS
POLGEN
Figure 5. Modular Tree Diagram
161
-------
EXHIBIT B
DESCRIPTION OF PRINCIPAL PARAMETERS
A)
INPUT CODES:
Code
Value
Meaning
1. NGC
0
1
2
2. NIC 0
1
3. MCXOR <4
4. MCYOT <4
5. MVXOR <4
6. MVYOT <4
NOTE: Items 3
7. NST 0
1
2
3
4
5
8. NASEQ £300
Circular Cross sectioned duct
Rectangular Cross sectioned duct
End of Job.
Raw Data Points Submitted for actual duct profiles.
Polynomial coefficients submitted for duct profiles
Number of maximum and minimum points 1n concentration
profile In x 1f rectangular duct or R If circular
duct, (see Note 5 - Input data form)
Number of maximum and minimum points 1n concentration
profile 1n y 1f rectangular duct or 6 If circular
duct, (see Note 5 - Input data form)
Number of maximum and minimum points 1n velocity
profile In x 1f rectangular duct or R 1f circular
duct, (see Note 5 - Input data form)
Number of maximum and minimum points 1n velocity
profile 1n y 1f rectangular duct or 0 1f circular
duct, (see Note 5 - Input data form)
6 are specified only 1f NIC » 0
Another duct (new duct)
Equal area strategy code
CALL EXIT (FUTURE EXPANSION)
CALL EXIT (FUTURE EXPANSION)
CALL EXIT (FUTURE EXPANSION)
CALL EXIT (FUTURE EXPANSION)
Number of equal area segments.
162
-------
B) OTHER INPUT DATA
If NGC - 0 A - Radius of circle
=1 A, B - Rectangle Dimensions in x and y directions respectively
If NCI =0 (CON(I), XORR(I). YORT(I))
(VEL(I), XRV(I), YTV(I))
Concentration and velocity at x, y or R, 6 (Theta)
If NCI * 1 ((C(J,I). J-1.NC), I-l.NC)
((V(J.I), J-1.MV), I-1.NV)
C(J.I), and Y(J.I) are the coefficients of
or
i\M /J_)°~ in the concentration and velocity polynomials respectively,
' V2ir
MC, MV is the order +1 of polynomial in y or 0
NC, NV is the order +1 of polynomial in x or R
YORT(I) IARSEG(D) P»*ot>e location coordinates and area segment
, iui\ \ it number.
C) OTHER PARAMETERS (NOT INPUT)
NCP: No. of concentration data points (<300)
NVP: No. of velocity data points (<300)
NPR: No. of probes (<300)
NARM: Error indicator for the duct profile
NARM = 0 no error
NARM * 1 error
NALARM: Error Indicator for the probe assignment data
NALARM • 0 no error
NALARM * 1 error
163
-------
EXHIBIT C
MAIN ROUTINE
The main routine reads and prints the data and supervises the sub-
sequent calling of the four subroutines and one function subprogram.
For detailed Information see Computational Procedures* Section II.
Figure 6 1s the main routine flow chart.
164
-------
6. Program Flo* Chart
SN • I./A
SJ • 1./2*
Read
NC.MC.NV.MV
((C(j.l).J-I.HC).i-I.HC)
I
Read
Duct
Comment
Card
Profile Error End.
NAAM • 0
2./A
2./B
Read Con. Data
Last Carrt
NN t 0
NCP - I - 1
165
-------
Print:
Error Mcstigt.S*t
HUM
DlMMlMlMl
/PH
Error Htt
HARM
•».. 1 ..
|1N*T|<|
For ClrciiUr:
0w 0
-------
(NCP.CON.TMT.
KOM.M.MC.
HC)
Print
fM(j.D,J'i
MC **C * 1).
HC *NC)
C«H
CROUT
(MARKI.Mt.C.
HC.NC)
Print
((*R(J,O.J'l
HC* K * J). <
for Concentration
For Concentration
If: NARH1 / 0
HARM • 1
Print Error
Htistge
C«11 COEF6
(NVP.VEt.YTV,
XRV,AR.HV.
MV)
\
Next
Pige
167
For Velocity
-------
M.W * W + 1).
1-1.m • my
CiM
Crout
(NMN2.M.V,
IW.NV)
For Vtloelty
Print
J-1.W* NV + J).
m|
Print Actual and Prtdicttd DaU for Concentration and velocity
. XOHR.YOrr,
^ *«« / Con. (9lv«n).
"> / Conpuud Con.
and VtJ.
MV.m.
Vtl. (gi»tfl).
Conputed Con.
«nd Vil.
Punch
168
-------
169
-------
V
(Muter ilonlesi
XMR • XORR • SN
YORT • VORT * SZ
Print Prob« (tot*
COMMENT. NPR,
(XORR(I).YOHT(I),
!ARSCC(I ,1*1,
NPR)
Ctll PRBAMS
(IARSC6.XORR,
VORT.NPR.KASEG,
•C «fw tW* t * t"* t
NV.CBM.VBAR,
CVBAR)
,
E, -
12 ' II.
CVBAR
-
Print
VI.CVI
VBM.CV8M.CBAR.
O
170
-------
EXHIBIT D
SUBROUTINE: COEFG
A. PURPOSE:
The least square curve fitting of the polynomial,
N M j i j i
ZaZ F ^ x
i«i j-i
results In a set of (N*M) linear equations with (N*M) unknowns. The purpose
of this routine 1s to generate and store the coefficients of that set of
equations in an array.
B. USAGE: CALL COEFG(NP, Z, Y, X, AR, M, N)
C. DESCRIPTION OF PARAMETERS:
1. INPUT TO THE ROUTINE:
NP: Number of data points
Z: Dependent variable data, size: Z (NP)
y, x: Independent variable data, size: x(NP), y(NP)
M-l: Order of polynomial In y
N-l: Order of polynomial In x
2. OUTPUT FROM THE ROUTINE
AR: Matrix of coefficients of equations, Size: AR(M*N, M+N +1)
171
-------
D. REMARKS:
0<(N,M)<11
M*N
-------
since:
N M
zk * F T
£l £l
NP ,
Z
4i4i NP / N M
J-'y "-I s T-> IT"* ^
xk LIZ z;
k=l \s=l r=l
= V* Z v J"^Y
L. Vk xk
k=l
or
Thus, providing NP>.N*M, a set of (N*M) linear equations with N*M unknowns,
which are linearly independent are found. The function of COEFG routine
is to find the coefficients of these equations and store them in matri
AR (M*N. M*N+1). The unknowns (ars, coefficients of the polynomial) are
found by CROUT-JORDAN REDUCTION METHOD in CROUT ROUTINE.
173
-------
EXHIBIT E
SUBROUTINE: CROUT
A. PURPOSE: To solve M*N linear equations with M*N unknowns, which
are linearly independent.
B. USAGE:
Call CROUT (NALARM9 AR, CORV, M, N)
C. DESCRIPTION OF PARAMETERS:
1. INPUT TO THE ROUTINE:
AR: Matrix of coefficients (output of COEFG). This array is
rearranged during execution with the last column becoming the solution of
equations.
Size: AR (M*N, M*N + 1)
M-l: Order of polynomial in y ore
N-l: Order of polynominal in x or R
2. OUTPUT FROM THE ROUTINE:
CORV: Matrix of coefficients of the polynomial,
N M t -I j i
V V CORV(j,i)YJ'V '
1=1 j=1
Size: CORV(M,N)
NALARM: 0 if the routine is able to solve the problem,
1 if the routine is unable to solve the problem.
(very rare, unavoidable division by zero or small numbers).
174
-------
D. REMARKS: 0<(M,N)<11
E. SUBROUTINES AND FUNCTION SUBPROGRAMS REQUIRED: None
F. DESCRIPTION AND METHOD:
The Input coefficients whose absolute values -are less than TOL = lo"9 are
zerorlzed, then the CROUT-JORDAN REDUCTION METHOD (see Appendix I) is used to
solve the N*M equations with N*M unknowns. In addition raw transformation Is
performed, wherever necessary, to avoid division by zero or small numbers
(less than TOL). The solution of the equations (the last column of AR matrix)
Is stored 1n polynomial coefficients matrix, CORV(M.N).
175
-------
EXHIBIT F
SUBROUTINE: CVINT
A. PURPOSE:
Obtain average velocity and average emission from the given velocity
and concentration polynomials.
B. USAGE:
CALL CVINT(NGC, C, NC, MC, V, NV, MV, CVI, VI)
C. DESCRIPTION OF PARAMETERS:
1. INPUT TO THE ROUTINE:
NGC: Geometry code; 0 circular duct.
1 rectangular duct*
C: Matrix whose elements are coefficients of the concentration
polynomial;
NC M
.* I 4 i
* x ^for rectan9u1««r)
Concentration
or
NC MC
Concentration = £ £ C^'V"1 (for circular)
where (y,x) or 6,R are dlmensionless variables;
X V R B
y s * R * "1* and 6 9 - S1ze: C(MC» NC)
176
-------
NC: ORDER + 1 of concentration polynomial In x or R
MC: ORDER + 1 of concentration polynomial In y or theta
V: Matrix of velocity polynomial coefficients similar to C
Size: V(MV, NV)
NV: ORDER + 1 of velocity polynomial 1n x or R
MV: ORDER + 1 of velocity polynomial in y or theta
2. OUTPUT FROM THE ROUTINE:
CVI: average emission per unit time and per unit area
VI: average velocity
D. REMARKS: 0<(NC, MC, NV, MV)<11
E. SUBROUTINE AND FUNCTION SUBPROGRAMS REQUIRED: None
F. DESCRIPTION AND METHOD:
Given velocity and concentration polynomials:
For rectangular duct:
NV MV , , 4,
V(x,y) . VyJ V
1-1 j-1
NC MC im} 4.1
C(x,y) - CyJ 1xt
1-1 j-1
Xj y .
where x and y are dimension! ess (^ » 577)
177
-------
For circular duct:
NV MV
V(R,0) -
c(R,e)
, , . ,
V
NC MC
and;
VI
I VdA
* JA
CVI =
Thus, for Rectangular Duct:
1 1
VI
•I I V(x,
4} -LI
y)dxdy/2*2
n
r E
-i -. 1sl jal
VCdA
VI
dxdy/4
NV MV V.< . ,.
E Z' ijf ^xl
-1
-1
where A = area
178
-------
VI s VJ1
s
1-1
4J1
Note: if 1 and j both be odd[l-(-l)JJ [l-(-l)1'] = 4
otherwise = o
CVI
ff
'I I C(x,y)V(x,
J-W
y)dxdy/4
dxdy
NC MC NV MV
cvi • z z z z
1=1 j»l k«l Jl-1
Note: if
and(1+k-l) both are odd;
-4-
otherwise
179
-------
For Circular Duct:
VI =J j V(R,6)RdRde'/Tr(l)2 where 61 - 2*6
Jo Jo
2V(R,6)RdRde
fl J NV MV , , . ,
f 2y; r v.^J-1 ^!
I *-L~i £—1 J1
v Jb 1-1 j-i
d de
NV MV 2V
•E Z 3T
1=1 j-1
and
CVI « I 2C(R,6)V(R,9)RdRde
j*l r1 NC MC NV MV
NC MC NV MV 2C...V
CVIaE E E "
1=1 j=l k-1
180
-------
EXHIBIT G
SUBROUTINE: PRBANS
A. PURPOSE
Given velocity and concentration polynomials and probes loca-
tions and equal area segment numbers, find average concentration, velocity
and emission based on equal area strategy.
B. USAGE
CALL PRBANS (IARSEG, XORR, YORT, NPR, NASEG, C, MC, NC,
V, MV, NV, CBAR, VBAR, CVBAR)
C. DESCRIPTION OF PARAMETERS
1. |nput to the Routine
IARSEG(I): No, of area segments where probe I Is
located. Size: lARSEG(NPR)
XORR(I):
YORT(I):
NPR:
NASEG:
C:
X or R of probe I
Size: XORR(NPR)
Y or THETA of probe I
Size: YORT(NPR)
No. of probes
Total no. of area segments
Matrix of coefficients of concentration
polynomial. Size: C (MC, NC)
Order + 1 of concentration polynomial in
Y or THETA
181
-------
NC: Order + 1 of concentration polynomial In
X or R.
V: Matrix of coefficients of velocity poly-
nominal. Size: V {MV, NV)
MV: Order + 1 of velocity polynomial in Y or
THETA.
NV: Order + 1 of velocity polynomial In X or R.
2. Output from the Routine
CBAR: Average concentration.
VBAR: Average velocity.
CVBAR: Average emission.
0. REMARKS
0 <(NPR, NASEG) £300
0 <(NC, MC, NV, MV) £11
E. SUBROUTINES AND FUNCTION SUBPROGRAMS REQUIRED:
POLGEN (Function).
F. DESCRIPTION AND METHOD
NPR
c
SEG (lARS
IUSEG (lARSEG(IJ)
c = No. of area segments with probe (ICZ)
NPR v
£ IUSEG (IAI
IUSEG (lARSEG(I))
132
-------
CV
IUSEG
:SEG(T7T
ICZ
where IUSEG(N) 1s the number of probes in area segment N. Example:
Probe Probe Number
Area Segment
Number
•3
•4
NPR - 5
NASEG = 4
IARSEG (1) = 1
IARSEG (2) * 1
IARSEG (3) = 2
IARSEG (4) = 4
IARSEG (5) = 4 '
IUSEG (1) = 2
IUSEG (2) = 1
IUSEG (3) = 0
IUSEG (4) - 2
ICZ - 3
It means that probes 1, 2, 3, 4 and 5 are
respectively located In area segment
1, 1, 2, 4 and 4.
It means that there are 2 probes 1n area
segment 1, 1 probe In area segment 2,
0 probes In area segment 3, and 2 probes
1n area segment 4.
There are 3 area segments with probe.
and similarly for V and CV.
183
-------
EXHIBIT H
FUNCTION POLGEN
A. PURPOSE
41 41
Y X
Evaluates the polynomial,
N -M
£ £(CORV
1=1 j=l
at a given x and Y.
B. USAGE
A = POLGEN (CORV, M. N, X, Y)
C. DESCRIPTION OF PARAMETERS
CORV: Matrlon of polynominal coefficients.
Size: CORV (N, N)
M: Order + 1 of polynominal In Y
N: Order + 1 of polynomlnal 1n X
X,Y: Location where the polynomlnal 1s desired to be
evaluated.
D. REMARKS
0 < (M, N) £11
E. DESCRIPTION AND METHOD
N M
POLGEN = E E (CORV)
1=1 j=l
184
-------
EXHIBIT I
The Grout Method for Solving Sets
of Linear Algebraic Equation*
The method of Crout, for solving a set of n linear algebraic equations in
n unknowns, is basically equivalent to the method of Gauss (see footnote
to page 4). However, the calculations are systematized in such a way
that they are conveniently carried out on a desk calculator, with a minimum
number of separate machine operations. Furthermore, a very considerable
saving of time and labor results from the fact that the recording of auxiliary
data is minimized and compactly arranged.
Only a description and illustration of the method is given here; an
.lytic justification is included in the original paper.*
The calculation proceeds from the augmented matrix of the system,
M-
Oi,:
'[•jc],
(1)
.o,j 0,1 • • • a,.: c,.
which may be considered as partitioned into the coefficient matrix a and
the column vector c, to an auxiliary matrix
i
M'-
Oji ' ' * flu:
"M ' ' ' "l«i !
'(•'ic'J,
(2)
.0»J 0,1 ' ' '
of the same dimensions, and thence to the required tolution vector
Xi
xt
(3)
It is convenient to define the diagonal element of any element to the
right of the principal diagonal of a matrix as that element of the principal
• See Reference 7 to Chapter 1.
185
-------
diagonal which lies in the same row as the given element. The diagonal
clement of any element below the principal diagonal is denned as that
element of the principal diagonal which lies in the same column as the given
element.
With this definition, the procedure for obtaining the elements of M'
from thoae of the given matrix M may be described by the four rules which
follow:
1. The elements of M' are determined in the following order: elements
of the first column, then elements of the first row to the right of the first
column; elements of the second column below the first row, then elements
of the second row to the right of the second column; and so on, until all
elements are determined.
2. The first column of M' is identical with the first column of M. Each
element of the first row of M' except the first is obtained by dividing the
corresponding element of M by the leading element an.
3. Each element Oiy on or below the principal diagonal of M' is obtained
by subtracting from the corresponding clement a
-------
3. Each element *< of x is obtained by subtracting from the correspond-
ing element e< of fc' the sum'of the products of elementa in the t'th row of a'
b/correspbndi«g elements of -the column x, &U uncalcuUvted elements of z
being imagiried'to be zeros. Thus there follows
n ' •
&*>£- 5)
The solution may, of course, be 'checked completely by substitution
irilo t?i'6'njrbasid']i'near eqiiatibnis. tfowever, if desiied, a check column may
be carried aibtig in 'th'e calculation to provide & continuous check on the
W6r^';j^iich:'eiem^ntlttrthe-initia^ check column, corresponding to the
dugmfrhtedttatri.'tM;-i9:the-6um of th« elements of the corresponding row
of M. If this column is recorded to th6 right of the 'augmented matrix,
dfa?':&*irc"tiUd''inl tite same' manner as Ike column cy corresponding check
columns are thus obtained for the auxiliary matrix M', and for the solution
vector x. Con.tiiiupus checks on the calculation are then afforded by the
two rules whfcli follow:'
1. In thpi'av,\Hiary .-matrix, any element, of the check column should
exceed by unity the sum of, the other elements in its row which lie to the
riyht of tjhg princi'pa,f
-------
M achieved by recording auxiliary data in the spaces of A which would other-
wise be occupied by ones and teros.
The t'th diagonal element of a' happens to be the coefficient by which
the t'th equation is divided, before that equation is used to eliminate the
t'th unknown from succeeding equations in the Gauss reduction. Since all
other steps in the reduction do not affect the determinant of the coefficient
matrix, it follows that the determinant of a is equal to the product of the
diagonal elements of a',
I»I - o'uaj, • • • <£,. (8)
Thus the Grout procedure is useful also in evaluating determinants, the
columns c and c', as well as the final vector z, then being omitted.
In the reference cited, it is shown that the method can be extended to
the convenient treatment of equations with complex coefficients, and to
the case of m equations in n unknowns.
In order to illustrate the procedure numerically, we apply it to the system
554.11 *, - 281.91 xt - 34.240*. - 273.02,
-281.91 *,+ 226.81 *,+ 38.100*, - -63.965,
-34.240*,+ 38.100*,+ 80.221*,- 34.717,
with the augmented matrix (and associated check column)
Check
M
S54.ll -281.91 -34.240 273.02
-281.91 226.81 38.100 -63.965
-34.240 38.100 80.221 34.717
510.98
-80.965
118.80
The auxiliary matrix (and associated check column) are obtained in the
form
Cheek
554.11 -0.60876 -0.061793 0.49272] 0.92216
M'"
-281.91 88.385 0.24801 0.89870 • 2.14668
-34.240 20.680 72.976 0.45224J 1.45228
and the solution vector (and final check column) are found to be
Check
10.92083] 1.92080
0.78654*7 1.78650^
0.45224) 1.45228
if all calculated values are rounded off to five significant figures throughout
the calculation. Thus there follows
*, - 0.92083, x» = 0.78654, *, « 0.45224,
188
-------
where reference to the final check column indicates the possible presence
of round-off errors of the order of four units in the last place retained.
Such erron would be decreased by retaining additional significant figures
in the formation of M' and x.
The elements of the first column of M' are identical with the correspond-
ing elements of M; the elements of the first row of M' following the first
element are obtained by dividing the corresponding elements of M by
554.11. The remaining elements of M' are determined as follows:
a,', - 226.81 - (-281. 91) (-0.50876) - 83.385.
«;, - 38.100 - (-0.50876X-34.240) - 20.680.
/ 38.100 - (-0.061793X-281.91) 20.680
-- 83.385 " O
-63.965 - (0.49272X-281.91)
a'tt - 80.221 - (-0.061793X-34.240) - (0.24801) (20.680) - 72.976.
, 34.717 - fl).49272)(-34.240) - (0.89870) (20.680)
ct -. --- ~ 72.976 "
72.976
The last element of x is identical with c{. The remaining elements of z
are determined as follows:
2, - 0.89870 - (0.24801)(0.45224) - 0.78654.
*i - 0.49272 - (-0.50S7G){0.78054) - (-0.061 793) (0.45224) - 0.92083.
It may be noticed that, because of the symmetry of the coefficient
matrix, it is not actually necessary to calculate a'ts and a'lt independently.
If a',, is calculated, the numerator (20.680) may be recorded as ai, before
the final division is effected.
The advantages of the procedure (and the additional simplifications
introduced by symmetry of the matrix a) increase with the number of
equations involved. , , . ,
It is particularly important to notice that the calculation of each
element of either M' or x involves only a single continuous machine operation
(a sum of products, with.or without a final division), without the necessity
of intermediate tabulation or transfer of auxiliary data.
If the determinant of the coefficient matrix were required, it would be
obtained aa the product of the diagonal elements of •':
|«| - (554.11)(83.385)(72.976) - 3.3718 X 10'.
189
-------
JXHIBIT J
SOURCE LISTING
190
-------
-.r,.V IV i.
.'-'.Alt:
ll-Ti. - TA-Cut
Fif-t OOCi
'-'(-.; I- A. i
ct.'"i»i:rrs int ?Kf.r,t AiSrcmtu WITH crwiPUTU-o ifTAL ___
MACK. wAi, IH-U-tM ANC SPECIFIC SPECIES EFFLl'INT FuR SKcCIFIC
ALLOOTIIi'4 ff VtLCCITY Af;C. CtVJCrMR A7 ICN f-KT)[.f£ IN ClfCLlAf C-K
CUtlS KHLKfc THE AReA ALLOCATIOM STRATEGY IS SPlCIHECi
AMI; Tr'.t ACTUit Vttr'ClTY A^O CC'JCtNTVATION PkCiFILtS IN Iht LUCT Af.t
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
THIS ?•••.:' c
MACK. -A;,
ALLOOTIl!
k i_LT ANC.17L
AMI; Tr'.t A
E?LCIf It£
Cl-LE VAl
1 .NuC (
/•.NIC I
i..VCXCR < =
^ . V.CYOT <=
&.HVXOR <=
6.KVYOT <=
7.NST
6.NASEG
IF NGC-
•
IF NCI*
•
IF NCI*
IxriKRU
OTKf
MCP
MVP
NOTt
0
1
2
3
OF IWT CCC^S
CIRCULAR CSOSS SECIIONEfi CUCT
RECTANGULAR CROSS SECTIONED DUCT
^AW DATA J»OIMS iUSHITTEt' fO« ACTUAL O'JCT
PfiLYNC'HTtL trCFFICIENTS JUlMITTtti FCf- DUCT
:.UHt£R OF MAX I ::# AND ^ir4l-.UM PniNTi IN C(=KC . PROPILEL
iri X IF KFC1ANUJLA.-N DUtl OR R If CIKCULAH CJCT
MJ«btR CF MAXIMA AMD MIMIHUM POINTS IN CONC. PROFILE
IV Y IF MXTAKiGULAR DUCT CR TKLTA IF CIRCULAR DUCT
WUMEER CF MAXIMA AND MINIMA IN PROFILE UF VELOCITY .
IN X IF RECTANGULAR OR H IF CIRCULAR DUCT
MUMtiR OF MAXIMA AND MINIMA _ ........
IN Y IF RFC! ANGULAR OR THETA IF CIRCULAR DUCT
ITEMS 2.-6 A«t SPECIFIED ONLY IF NIC ~ 0
ANOTHER DUCT I NEW DUCT)
EOUAL AREA STRATEGY CGOE
NUMiLR OF ECUAL AREA SEGMENTS
OTHER INPUT OATA
0 A - RADIOS CF CIRCLE
I A,b- SHORT AND LONG DIMENSIONS RFCTANGLE . _
IF NCI*0 «CON(I).XURR(I),YnRT(l),l=l,NCP) CONCENTRATION AND VELOC.
(VELdl. XRV(I). YTV(I),I«=1,NVP) AT X,Y OR K.THETA
IF NCI=1«{CU,U,J=1,MC),I»1,NC>
<(V(J,I).J=lfMV)tI=l,NV»
IXnkKlI),YCkT(I),IARSCG(I),I=l,NPR)
WHERE C(Jtl) A.MD V
-------
IV & UVhL 21
KAIN
DATE =
13/06/37
PAGE 0002
«O
C'OCl
0002
C-OC3
I.Ut'b
( cte
I'-' i-c
I'Vi J
f. ?•?<
NPR KUvtbR OF ERODES.
HARM Ef.k^fi INDICATHR Ft* D'JCI PROFILES. _ __________ ..... _
NALARK t«ROP. IfcSICATOR FOR PRGfit ASSIGNMENT DATA.
DTHENS1CN AH|1?1 ,1221 rCDNC 300) »VEL( 3001 tXORR{300)tYOHT(300),YTVC30
1U), C(lI*111iV(lltllI» IARSE&(300) »XRV(300)
COHLNTC20) . -. _ _
C
C
C
2 FORKATU3J
3
A
&
6
7
6 FORMAT*10X.'STACK GEOHETRY'I .
9 FLRHATdSX,'CIRCULAR-* ASIUS = »iF12.5,'FT. •>
10
13
15
FORMAtt^IZ)
16.SJ
16 F(.JrKAT(30X, 'RESULTS OF PilOFItE COKRFLATIOMVA2X. • INPUT ' ,21X, »PKEOI
1CTIC.')
17 eir
li, f ;,K'..'
IV- ff ;,f--,.
IUC:TV«I
201
21
Afef> DLVIATICfl IN
.EA ffGKtKT NUV.lfcRM
c;?,; ii.b,i>.,Fii*i,7x,ij>
« t £12*5, 5X, »1N
/5>t 'ERROR IN
Ifl VtA;v-'t«X,f 12.5)
2& FLKKAICIXt 'E«ROR IN ENURING CONCENTRATION PROFILE DATA.* *5Xf 131
2fc H'f.K*TtlXt'LkROh IH tMTCfllNG Vf LCCITY PROFKE DATA. • ,5X,13)
27 FlkMAT|«,llJ
(9. Ff.\MATIlX,*P*CHl SI3f • LOCATfeO PUTSlOE DUCT 60UNCARIES.PRC8LEM .Afi_
*(.ul £.:..' >
L« fr- .-1A7UX,«^KU!.£',!3,' AS£ICNf& TO A Nt'H-EXISTEHT AREA SEGMENT. PR
LOCA110N. OATA'/lOX, "HUMbtR OF PRCbES=',I3J
31 FtSWAKlHO)
^2 rii!.XAHl?X,»ACTtlAL't^lX,'Cnf1PUltC»/*X, 'AVERAGE V t6X,« AVERACE_C*V1_
ltl2X,»AVfKAiC V«,7X,'AV£:RAtE C*V«,7X, 'AVERAGE C«l
33 K^
rcr.7
-------
IV G LEVtL 21
MAIN
DATt * 74065
13/06/37
PAGE 0003
IO
CJ
OC3V
0040
OC41
OiK2
0043
C044
0045
OOAt
outl
C«»!.2
Oi.tb
(*tSV
OOtO
00e.fr
00 t 7
lH)f f
0073
fc^7*
5075.
«IQ7«
J&77
6t£0
00 U.
&6b2
OC83
tf64
20GO FORMAT 1 IX. IOS13.M
2001 FORMAT! « 1N?UT AND OUTPUT ARRY OF_ CROUT ROUT1N fOH.CONCEHIRAT ItML.
*•/!
2002 FfcRMATtlHl,* ROH DATA.'/1 NO OF CON. DATA CARDS *,I3,5X,
* "NO. t-fr VEL. DATA CARDS »,I31
2003 FQRXATC//* INPUT AMD OUTPUT ARRY OF CROUT RQUTIN FOR VEL.V/J
6000 FfRMATI' UNABLt TO SOLVt BY THIS HETHOO PROBLEM ABORTED*)
99 R^AOIStil NGC,A,B,NIC ________________________ ________________
IFtNGC-1) 100,101,152
100 f-0.
SM*H./A *
SZ=l./t. 2831653 ,
GO TO 102 , „
101 sfi=2,/A ., ,, _______________ ..... _____ !:„'_:.; __ ......... ., ... - . ..... ...
LI=2./fc
102 KC*CI5»14011 COHENT
103*103t125
,j) Cp»dtI)»XCRRII),YORTCII,NN
1FINM) 1122, 11?3, 1122
1123 If (CDHtin 104,105,105
10* «RITLC6,2i.) I
105 If 1I.CCT 10i,10t,110
106 IF IXCKKtII '- A| 107,107,104
107 IF (XORKI1U 104,108,108
108 If (VoKTUI'-et?831B53» 109,109*104
109 IF (YbRTtD 1104,112, 112
110 IF rAbi(>DF.KlIH-.6*A| 111,111,104
111 IF eASMYpRim»-.5*BI 112,112,104
*'
CC ,TO. 112
1122 NCP*I-1 t
1201 1«I*1
Kr>D(5,3) VgtCl>,X-ITE(f,?AI I ..._ ..... . _'.
MAKM'l
CO TO 12O1
115 IF IXRvmi 114^116.116
lit IF |YTV(I)-t.*631653| 117,117,114
-------
KikiKAN w G LEvti 21 MAIM DATE = 74Cb* 13/0^/37 PAGE 0004
(.Of.* 117 It- (YTVCD) 114,120,120
G0d& _ 116 lf (A6StXRVm)-.5*A) 119.119,114 .
0087 119 IF CAl.S(YTVmi-.5*B> 120,120,114
OOEB 120 XRVI I) = S**XKV
0096 f
0097 t
OOVB . _ .. . r
Ou99 IF (M-NVP) 122,122,121
01 CO 121 RFL-WP
0101 RtL=SCRT(RtLI
0102 I
0103 I
o;RCL
0109 HC-NCP/JIC
0110
0111 MSP1-MS1ZF+1
0112 »'(>. lit (£,2002) MCP.NVP
0112 CALL CutFG(fJCP,CON,YCRT,XCfiR, AR,HC,NC)
WK1TEU.2001)
WRIT!(6,2000) I(AR(I,J).J*l,MSPll,I*l,MSI?E>
HRlTtlfc.31) . .. ._... ....__.
1.117 CALL CRCiOKNARNl ,Ak,C,MC,NC)
Ollf WRITE (6,2000) ((AR(I,J),J=1,MSPH,I«=1,MSIZE)
I.11S II-(NAKMl) 1240,1241,1240
01^0 1240 KSITf (6,oOCf»)
_174l
0124 CALt COEFG(NVP,VtL,YTV,XRV, AR.NV.NV)
WKITM6,2000» ( ( ARI I, J), J'l.HSPl) ,I«1,HSIZEI
Ml* 7
wli't. CALL t»CliTC«ARM2 ,AR,V,HV,NV) .'.
O.^S kK11tU.,?OCO) ((AR(I,J),J«1,MSP1),I«1,MS12EI
C13C IF(NAkM2) 1242,1743,1742
0131 1242 UtI1f(6,6000)
0132 NAKM=1
-------
IV C UVIL ?l
DATE
U/Ot/37
0005
C133
cn
013o
i>137
C136
O14G
0143
C144
O146
01*7
0146
0149
0150
0141
0152
0153
0154
0155
0156
C157
015C
015V
0160
0161
0162
0163
G164
OltS
C166
01 67
0166
C169
O170
O171
017?
0173
0174
0175
01 7ft
0177
017*
017V
0160
1261,126t13M
Rt*D(5t27) HC.KC.KV.MV ..
(tCU.I),J-l,MC),l=l.NC)
WRITtl6,14B3) ChHENT
WklTt(6,8)
IF I»;GCI 127,127, U6
127 KRlTl!t.V> A
CO Tf !?«»
126 WXlTt(6«10) A.B
129 '
IF I fJlC » 130,130,136
130 WKlTtO.M «CU,I>.J«l.HC),I-l,'lC}
(&> HVIJ,l»,J«ltMV»tI»l,MV>
WKITt16*16)
If- INCCI 131*131*132
131 HKITfc <*•,!$)
On Tft 133 _ _ _
132 liltmUitTt
133 KRITHfc.m
DO 134 1=1,NCP
CVAL-PLLGtNtC,KC,NC,XGRRm,YORTmi
^CVAt-CONUl l»CCVAL-COf.MU )
134 WR1TK*»20) XCIRRIIItYOHTf Il.CONtI)
.CVAL.WAL
It- tfcGCt 1341,1341,1342
1341 WRITE (6, 16 1
GO TO 13*3 . _________
1342 vnmie*m
1343 Wf lit (6, 19)
DO 13S 1*1 i NVP
CVAL^PdLGlN
-------
f-C'MK«N iV G LfcVEL 21
MAIN
DATE « 74085
13/06/37
PAGE 0006
UJ i 1
., : -2
01 £6
fl'f,
C'i-.v
U -0
01S1
01 /2
01V3
01*4
01V7
0199
OiOO
0201
«/; 02
W05
0204
0207
0209
0210
C/ill
6212
0215
0216
022 <•
0227
WX.lTfe.lf , 1J.J
13t CALL CVifiTIKC-C,
1361 Kt*DC>,6> NST
IF(MSTI 95,9*.
1362 GO TO (136,l37,137tl37t137»,NST
.„ 137 CALL EXIT .......... ......
138 HtAb(5>«l'f01) CQ«INT
,V,NV,HVtCVl,VI J
NALAHM=0
1=0
1392 1=1*1
RtADIl.,71 XORRm.YORT-LIJ*IARS£Gm_
1HNNI 1390,1391,1390
1391 IF (NGCI
139 IF IXOHftm-Al 141.1A1.140
140 WRITFI6»28I I
GO TO 1392 •
ll
143
144
14S
146
147
146
1481
1390
1464
1485
IF IXCPR(I»> 1*0,142,1*2
IF »yr*Tm-6.2e3i&53)
IF IVOKTIIM 140,1*6,14*
1^ (ALSJXORRIin-.t*A> 145,145.140
IF I4fc,l46,140
IF. (IARSlC(II-NA$EGl_L4ajl49tlVr
WRITEU»29I I
NALAkH=l
GO TO 13*2
IFU/RStGU)) 147, 147, 1461
X(!RRII)>SN*XOKR(I)
YtRTII»«S2*YC«T{U
(a TO 1392
'NPM>I-l
IFINALAMHI 1361,1484,1361
IF(NARM) 1361, 1465, 13M
«KITE(6,14B3) COMENT
IF (fJGCI 140,149,150
149 WKI1t(f,16l
CO 10 151
150 MklTFlfc.m
151 HKITE(6,?2I
W«<1TE«.,23) (XCRR|II,YORT«Il,IARStGIII.l*ltM?R) ____ ....... ___
CALL PK&AN^{IARStt,XORR,YORTtNPR, NASEG.C.KC ,NC, V,HV,W,
E 1 -Ab$l 1 ,-CVbAR/CVI I
C2*AtS U .-Cf AR*Vt.*R/CVI )
-------
FLHlkAN IV G LtVIL 21 MAIN DATE * 7*085 13/06/37 PAGE 0007
Oilb r2=A6itl.-VfcAR/VlY
022V KRITH6,151 .. '..'•'- .__ i : —
0230 WRlTt(6t32) < " .:<;.-',:' ^ " ';,-
Oiil Mknt«6,33> VI,CVltVBAR,CVBAR,CbAR • ... .
,0?32 WRITHe,2«»> fcl»E2,E3
'0233 WUTtlfc,151 •,.,...
023* tO TO 13tl
0235 152 CALL EXIT; _£l:\ ii : .'.-.•-. ; :
0236 tNO
-------
vAh IV U LFVCL
MAIN
DATI & 746*5
13/06/37
PAGE 0006
vo
00
SUBPROGRAMS CALLED
ILCf-il*
LUT
SYMfcPL
HOC
Si
r v?
r;v
ML
i-.Ah.M2
KNCP
NAitG
CVtAR
SYM6OL
AR
YTV
CONENT
SYHbOL
1
6
11
16
201
ii
30
1463
COCO
LOCATION
346
35C
LOCATION
4A8
4bC
4UO
4FB
iOC
520
534
546
LOCATION '
558
FECO
11098
LOCATION
110E6
11116
11167
11200
112t>3
1136A
11462
11*17
115t&
SYMBOL
CCtFC
PR DANS
SYHbOL
A
NARN
KCXOR
MV
HSIZE
EKC
RNVP
LOCATION
34C
360
SCALAR NAP
LOCATION
4AC
4CO
404
AFC
510
524
MALARH 538
El
SYMBOL
CON
C
SYHbOL
2
7
12
17
21
26
31
2000
54C
ARRAY HAP
LOCATION
ECOO
10370
FORMAT STATEMENT
LOCATION
110FA
line
jijsa „ „
1123B
112C7
113A1
11493
11525
SYMBOL
CftnUT
SORT
SVMtOL
8
I
HCYOT
N'C
MSP I.
ERV
CVI
NPR
E2
SYHBOL
VEL
V
HAP
SYMBOL
9
8
j3
18
22
27
32
2001,
LOCATION
350
364
LOCATION
460
4C4
4De
4CC
500 r.
514
528
53C
550
LOCATION
FOBO
10554
LOCATION
IIOFE
11128
nice.
11240
112F4
11303
11498
1152E
SVK6OL
PQLGC.N
SYMBOL
NIC
NN
KVXOR
MC
J
CVAL
VI
CBAR
E3
SYNBOL
XORR
IARSEC
SYNBOL
4
9
14
19
23
28
33
2002
LOCATION
354
LOCATION
^t^
4C8
. . .. 40C . ...
4FO
CfVfc
518
52C
540
554
.^LOCATION
F560
- 10738
LOCATION
11109.
1113F
*M r 7
11263
1130D
113D9
114F6. _ .
1156C
SYHBOL
CVI NT
SYMBOL
SN
NCP
NVYOT
N
MAfrttl
WAL
- NST
VBAR
, SYNBOL
VORT
XRV
SYNBOL
5
10
is
20
2,4.
29
1401
2003
LOCATION
— - —-358
LOCATION
4LB
4CC
4EO
4F4
•hfiA
5IC
-530 ... •-
544
LOCATION
FA10
10BE8
LOCATION
1110F
11150
1 ] IFh
1129F
. 1131D .
1141A
11511
115H6
•Cf'TIONt IN EFFECT* ID,EBCDIC.SOURCEtNOLTST.NODECK,LOAD.MAP
*Oi-TI(!.M IH FfPFCT* NAhF * MAIN ( LINECNT & 50
*S1AI1STICS* SOURCE STATEKtNTS > 236,PROGRAM SIZE *
*STA7I1TICS« NO DIAGNOSTICS GtNfRATEO . . :
767-94
-------
KK1RAN IV G CFVtL
COtFG
DATE = 740U&
13/04/37
PAGE 0001
5 00 6 L=1*N2
CO CONTINUE
200 CONTINUE
30O CONTINUE
ICQL«ICOL*1
AR(1ROW,ICOL)«0,
00 400 K«itNP
IFINll 10,10,11
11 00 12
12 >CWEh
10 If(N2) 13,13,14
1* IO I* L«1,N2
13
400 COMT1NUE
SOO Ct'hTINUt
tOO CL»N1 1NUC
kfT.UAN
END
-------
Ki.TkiN IV (, LLVEL /I
U:CFG
CATt
13/06/37
PAGE 0002
i
I
1R
K
SYHLfiL
LOCATION
FS
IOC
120
LOCATION
SCALAR MAP...
SYMBOL LOCATION
N FC
1COL 110
HP
ARRAY MAP
SYMBOL LOCATION
Y 138
f'S IN fcFPECT* IC-»E£COIC,SCUR
*C-HK:Ni IN tFKCT* NAKf = CC£FC , LINtCNT = 50
•STATISTICS* SOURCE STATEMENTS = «3,PROGRAM SIZt
"STATISTICS* NO DIAGNOSTICS GENERATED
N SYMBOL
J
IROW
POWtRl
LOCATION
mo
114
128
N SYNhOL LOCATION
X 13C
NOLIST.KODECK, LOAD, MAP -
SYMBOL
K
Nl
POWER2
&YHCOL
AR
LOCATION
104
na
. 12C
LOCATION
140
SYMtOL
IS --
N2
L
SYMBOL
LOCATION
IDS
11C
130
- LOCATION -
1364
-------
IV C LCVfL 21
OO01
0002
00031
0004
CUOS
OO06
0007
000ft
000%
COLO
COll
i.012
0015
OU14.
CGii.
CO 16
CG17
0018
OO19
OOIO
OOii
0022
CG24
0025
0026
0027
0028
OO2<»
0030
0031
0032
0033
003*
0036
OO*7
cose
0039
OO40
00* 1
00*2
UO43
t C*«.
&047
0048
CRPUT
DATE » 7*064
13/06/37
PACE 0001
NALARK=0
tAR*CORVtHtNI
CQhV(lI,n),A«tl?l,122l
CO 100 1=1. 1A
PO 100
101 AKU.JV*U
1OO CCNTINUF
,
J-TCH 101.101,100
300,301,300
301 J=l
CO TO 2C9
SCO DC 1 J~2,IAP1
DO 6 KO=2.1A
MJ 200 K>1,IHCIX
200 &y~AX-M(J,K)*AK(K.J)
!K(AeS(AX)-TULI 201,201,202
,2Ol IFCJ-IAI 200,205,205
209 KO>J«-1 '" ~ .....
AMAX>0.
DO 203 I=NO,IA
SAVf«ARII,J|
IFUfaSCSAVE ». rAftSlAMAXl) 203,203. ?OA
2O4 IHAX>1
203 CANTtNUt
IFIIHAXI 205,205,206
206 DO 207 I-1.IAP1
207 CONtlNUE
IPfJ-l I 300,300,202
205 NALAkM«l
202 tO * I^MC.IA ......
CO 2C K»l, INDEX
AR(I.J)*ARII.J)-AR(I,K)*AR(K,J1
2O CONTINUE
-------
FCfRTkAN IV C UVtL ?X
CRCUT
OATf
13/06/37
PACE 0002
PO
CC4*
'0050
3 CONTINUE
l«*D
DU 5 J-IS.1AP1
10 40 KtJ,INDEX
CC1.S-
O0i6
0058
00 S9
C-CtC.
GO 61
OCti
GGc.3
0064
ObtS
00(6
OCt.7
Ot-t-V
*.i070
0071
0072
0073
O074
007S
40
5
6
TO
8
9
19
11
CCNTINUE
CONTINUE
Cf iMT lMt,iF
DO fi I*ltlAMl
pir> 70 KsistrA
AR CMft, IAP1 1 -AR (HOt I API I-AR * 75»PROGRAH SIZE »
•STATISTICS* NO DIAGNOSTICS GBJFRATEO
2316
-------
FORTRAN iv G uvft
CVINT
CAIt * 74C«S
PACE 0001
§
oooi
0002
0003
0004
CGI 5
OOOb
0007
OOOB .
0009
CO 10
COll
0012
0013
0015
0016
0018
OG19
0021
foil
««-*
0026
CO27
SUBROUTINE CVIMTtN&C, C,NC»KC,V,NV.NV,CV1.VX)
DIMENSION C(ll,11),V(li.Ill
CVI=0.
Vl*0.
IFINCCJ 100,100.200
1OO DO 14O 1*1,tt
CO 130 J=4,MC
CO 120 K*1*HV
CO 110 L=1,KV
110
1ZO CONTINUE
130 COM INUt
l«> COHTINt'E • • .
CV1«2.*CVI
t«Q UO 1»1
CO 1M> J=1,MV
D1*J*U+1>
150 VI^V1«V( J.I 1/01
160 .COMT IMUt ,_
VI-2.*V1
200 DO 2*0 f»l«NC
CO 230 J«1,HC
00 220 K»1,NV
.00 .
N«
Cvil
6032
UC33
0034
O03S
If (2*/01
210 COM INUt
220 CONTIMUE__
230. CCNTINUt
240 CONTINUE
DO £(.
00 25O J*1,NV
IF (2*fN/2l-m 245.250,245
O043
250 CONTINUE
26O CONTINUE
Rt TURN
tNC
-------
fOKIkAN IV G LtVIL 21
CVINT
DATE = 74065
13/06/37
PAGE 0002
SCALAR HAP
CYMcOL LOCATION SYMBOL LOCATION
cv: FO vi ^4
j 104 MC 108
HV lie 01 nc
SYMBOL
NGC
K
N
LOCATION
FB
IOC
170
SYMBOL
I
NV
LOCATION J
fC t
110 I
. ._ . . . . _
SYMPOL LOCATION
10O
ro
o
ARRAY HAP
SYMu-.L LOCATION SYMLOL LOCATION
C 124 V 128
SYMBOL
LOCATION
SYMBOL
LOCATION
SYMBOL LOCATION
*r;PTIONS IN tFFECl* ID,E6CDIC,SOURCE,N(JLIST,NOO£CK,LOAD,HAP
•^OPTIONS Iti EFFECT* NAME B CVINT , LlNECNT = SO
*$TAT1J.T1CS* SOURCE SlATfcNEHTS « 4A,PROGRAM SIZE •=
STATISTICS* NO DIAGNOSTICS GENERATED
1442
-------
KihlKAK IV G UVU.
PRtAtJi
tATE
13/C6/37
PAGE 0001
o
tn
Ot-ni
CC02
0003
000
C-C 40 l=-l,NPR
J-lARSEb(Z)
40 IUSEG«J1«1USEG(J)»1
ICZ=0
...00 50 l = l.NASfcG
IF (lUStG(I)l
so corniNUE
00 1O I*-
Ct=Pl-UCcNtC ,NC tNCiXORK < I) »YORT( 1) I
VC«PPL&rN(V»HV,NVtXORRCI)«YORTtIlt
00i9
.10 CV6AR*.CVJ>AR*CB*V6/F_
Ct>AfC*CBAR/DA
VBAp.=VbAF/CA
CV6AR&CVBAR/OA
RETURN
END
-------
fCKTKAH IV C LlVtL 21
PkbANS
DATE =
13/06/37
PAGE 0002
SUBPROGRAMS CALLED
: YrtiOL
••LiLGtfJ
SY.-il-.5L
c;-*i»
!ir";
Li,
NV
S?~.i C-4.
In.-OtG
V
LOCATION
BO
LOCATION
CC
fO
10*.
lie
LOCATION
11C
50C
SYMfcOL
SY.4EGL
VfcAR
J
MC
LOCATION
SCALAR MAP
L tK AT I ON
to
F4
ica
ARRAY MAP
SYKiCL LOCATION
XQRK
120
SYMBOL
SYME9L,
CVf-AR
ICZ
NC
SYMEOL
YQRT
LOCATION
LOCATION-
E*
F8
IOC
LOCATION
124
SYMBOL
_ . . ..
SYMEOL
I
DA
Vb
SYMfcOL
1USFG
LOCATION
•
- -.- • -
LnCATfnm
(8
FC
110
_
LOCATION
128
- -•- — - -
SYMBOL
- -
SYMBOL
NASEG
F
MV
-
SYM&OL-
C
LOCATION
.- . - .
- -
\ nr AT (CM
FC
100 ...
114
—
LOCATION
508
£ IN EF«CT» 10,EtC01C,ICURCE,NCLIST,NOD9CK,LOAO,MAP
*i.i»rj;.M i% EFFECT*' NAME * PREANS , LII^CCNT * sc
*ilAT::,TICS* SCURCfc tTATiKtMTS » 29,PROGRAM SIZE *
*iTAUi11t£* NO DIAGNOSTICS
2388
-------
ro
o
••J
l.KltvAK l\i G LtVLL
OUOi
OC-02
ocoe
CC(J<>
CO 1C
ten
0012
.(.C13
0016
CO 17
UU18
0019
PULGEN
PUNCH ON POLGbN(CORV(H,N,X,YJ
_D1*FNSIOM CQRVtll,]
PClLttN=0.
DO 2 I=l,N
P&=0.
tn=i-i
en i J=I,M
DATE = 7«0b5
13/06/37
PACfc 0001
POWtR'l.
If-(N2) 1,1,2
3 DO 4 L=1,N2
IHMl 2,2.i
5 DO 6 L=1,N1
6 POWER=PCWER*X
2 priLG£N
Rf-TUkN
-------
fORTKAN IV C LEVEL 21
POLGFN
DATE * 74085
13/06/37
PAGE 0002
SYMblL
POL&IN
LOCATION
CO
EQUIVALENCE DATA HAP _
SYHBOL LOCATION SYMBOL
LOCATION
SYMBOL
LOCATION
SYMBOL
LOCATION
SYHfcOL
I
n
x
SYKbUL
CORV
LOCATION.
C*
OB
EC
LOCATION
fO
N
N2
SCALAR HAP
LDCATIOt!
C6
. OC.
ARRAY HAP
SYHbOL LOCATION
LOCATIOJL
JYRBOL
POWER
SYNbOL
Nl
L
DO
ft
04
LOCATION
SYHBOt LOCATION
SYHBOL
LOCATION
•OPTIONS IN EFFECT* ID,E&COK,SOURCE,NOLIST,NODCCKfLOAD,HAP
•OPTIONS IN £F«CT* NAME « POLGfN , LINECNT « 50
*STATiaiICS* SOURCF STATEMENTS » 20.PROOAAH SIZE «
•STATISTICS* NO DIAGNOSTICS GENERATED
732
•STATISTICS* NO DIAGNOSTICS THIS STEP
-------
OS/360 LOADLR
i
'f IONS USED - PRiNT,mAP.LET,CALL»KORES,NOTERH.SIZF-l80224,NAKF»**GO
NAHfc ... TYPE
MAIN SD
PULGEN SO
IHCtCONH* SO
SEODASU * LR
IHttFNTH* SO
FlUCSbtP* LR
FCVLOUTP* LR
1MT6SHCH* LR
TOTAL LEN&TH
ENTRY ADDRESS
ADDR ,
124810
139178
JL195CO
13A6AO
13B4BO
13B9Ffc
13CF8A
130E8B
1AOA8
124910
NAME TYPE AODR
COEFG SO 137410
IHC&SQRT* SO 1394S8
ItCMM * l» 11»«5ro
IHCFRRM *
AR1TH* *
1HCFIOS2*
FCVZOUTP*
IHCUATBL*
SD 13AB88
LR 13B4&0
SO 13C920
LR 13DODA
SO 13DFFO
NAME TYPE
CROUT . SO
SCtRT * LR
pni nr.^i * t a
FRKMON *
ADJSHTCH*
1HCFCVTH*
FCVIOUTP*
IHCETRCH*
LR
LR
SO
LR
SO
ADOR
137968 - - -
139458
13B84C
13CE»0
130488...
13E628
. NAME - TY
CV1NT - ....
IHCFEXIT*
IHCLRRE *
IHCtPlOS*
ADCbNH *
FCVtOUTP»«
IHCTRCH *
PF
SD
IP
LR
^D-
LR
LR.
LR
>PW M^f Tvar *nrtD
1)8278— PRBANS
1395AO EXIT *
13ABAO
13&9F3
13CE50
13098A
13E628
1HCUOPT *
Floes* .*
FCVAOUTP*
FCVCOUTP*
ERRTRA *
- $0-138820—
LR 1395AO
SD 13B160
-LR--13B9F8 -
LR 13CEFA
LR 13DAA4
LR 13E63O
-------
TECHNICAL REPORT DATA
(fltmtt rnd AuonrMMU on the rntnt btfort eompuHagf
1. REPORT NO.
EPA-650/2-74-086-a and
3. RECIPIENT? ACCESSlON>NO.
». TITLl AWO »USTITLI
Procedures for Measurement in Stratified Gases,
Volumes I and II (Appendices)
I. REPORT DATE
September 1974
ft, PERFORMING ORGANIZATION CODE
7.AUTHORc*.A>Zakak, R.Siegel, J. McCoy, S.Arab-femali
J. Porter, L.Harris, L. Forney, and R. Llsk
t. PERFORMING ORGANIZATION REPORT NO
I. PERFORMING ORG \NIZATION NAME AND ADDRESS
Walden Research Division of Abcor, Inc.
201 Vassar Street
Cambridge, Mass. 02139
tO. PROGRAM ELEMENT Mb. '
1AB013; ROAP 21ACX-092
11. CONTRACT/GRANT NO.
68-02-1306
12. SPONSORING AGENCY NAME AND ADDRESS
EPA, Office of Research and Development
I NERC-RTP, Control Systems Laboratory
Research Triangle Park, NC 27711
13. TYPE OF REPORT AND
Final; 6/73-5/74
IO PERIOD COVERED
14. SPONSORING AGENCV CODE
IS. SUPPLEMENTARY NOTES
AMT(IACT The report gives results of a program to develop methods for the continuous
extraction of representative gas samples from gas streams that exhibit compositional
stratification. The program considered available data in the literature, as well as
field data generated during the program. Wind tunnel tests and. mathematical model-
ing were used to develop sampling methodologies which are recommended. Data from
the literature, as well as program data, indicate that stratification exists, although
it is unlikely that gas stratification is as widespread or as severe as particulate
stratification. Depending on conditions, two different methods are recommended.
The first method involves monitoring the ratio of SO2, NQx, etc. to CO2 at a single
location. Then, from the measured fuel flow and chemistry of the process, the mass
flow of CO2 can be predicted. The product of the measured ratio and the predicted
mass flow of CO2 is the mass flow of the pollutant. Where conditions do not permit
using this method, it is recommended that a schedule of manual surveys be conducted
followed by installation of a multi-element proportional sampler and gas velocity
array.
KEY WORDS AND DOCUMENT ANALYSIS
DESCRIPTORS
b.IDENTIFIERS/OPEN ENDED TERMS
c, COSATt Field/Group
Air Pollution
Measurement
Gases
Sampling
Stratification
Wind Tunnels
Mathematical Models
Data
Carbon Dioxide
Air Pollution Control
Stationary Sources
Proportional Sampler
Grfti Velocity Array
13B , 12A
14B
07D, 07B
1C. DISTRIBUTION STATEMENT
Unlimited
IS. SECURITY CLASS
Unclassified
21. NO. OF PAGES
213
M. SECURITY CLASS (Thltptft)
Unclassified
22. PRICE
•PA Perm IJM-1 (t-71)
2KT
------- |