USER'S GUIDE AND DOCUMENTATION
FOR OUTFALL PLUME MODEL
VWATEBX
ENVIRONMENTAL
PROTECTION
AGENCY
REGION X
PACIFIC NORTHWEST
WATER LABORATORY
-------
ERRATA: "User's Guide- end Documentation for Outfall Plume Model,"
by D. J. Baurr.gartner, D. S. Trent, and K. V. Byram.
Working Paper ,780, EPA, Pacific Northwest Water Laboratory.
May 1971.
A program modification has been made to the above publication
which improves its response for plumes which exhibit both low
initial densimetric Froude numbers and small but non-zero angles of
inclination. Under this modification, the angular orientation of
the plume axis at the end of the zone of flow establishment is de-
termined by a linear interpolation process. This replaces the
assumption in the original program that the plume axis maintains
the port orientation angle throughout the zone of flow establish-
ment for all angles other than zero. The effects of this modifi-
cation are represented by a small decrease in the predicted
uilutiuii Over that of the original proyrarn, fostered, of course,
by the shorter travel distances associated with the improved
specification of the angle of the plume.
The revised program is useful for all der.simetric Froude
numbers between 10"^ and 10+^ and for all angular orientation
between and including horizontal and vertical. The program
changes are included on revised pages 25 and 26.
W. F. Rittall
D. J. Baumgartner
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c
C PROGS*AH PLUME» VERSION OF <-/6/7"2.
C
COMMON G»KKtFH»COSTM»SINTH*COSTHC,DS9Cl»C2»E13»FLAGf6«AV
DIMENSION ZD(SO) tOG<6wi »fcHO(50> ~n?{50i
LOGICAL N'dC* TkAPPLi * FLAG? CHGDElMiC TZRS
NOCASE. = C
20 READ(5» 731 END = 7M NDC~ MtTERS *NPT5»ANGLE sDIA »DEP I'M? RHO Jt RFD» U» F,*J? P:
73 FORM A n2Ll«I3^5.t)«7ri0.0)
FO=AbS(KFD)
DO 102 1 = 1 •riPTS
READ(5?75)DP(IJ?SAL
IF(SAL•EG•G•)GO TO lo2
RHO(I> --1 • ~ « G 01 * 31 CM AT(SAL*RHO(I))
102 CONTINUE
IF(NPTS.NE.1)GO TO 76
NPTS=2
DP(2)-DEPTH
RHO(2)-WHO(1)
76 NOCASONOCASEM
75 FOiWAl ( BP 10 c 0 )
GRAV=32.172
IF(METERS)GHAV=9. 80665
DO 55 1 = 1* .\'PT
If (Df»(I) oGE.DEPrjDGO TO 56
55 CONTINUE
WRITE(6*59)NCCASE
59 FOUHaT('-NO OLN'SITV INFORM,*7 IOr* F03 JET LEVEL. EXECUTION fOtf*.
• • CASE NO. ' f12 ? * DELETED.')
GO TO 20
56 NP=I
NM=I-1
RHG3=(DEPTH-DP(NM) i >/OIA
54 DG (I) - C-'.HO (J* i > ••titi'j iJ) )'>DIA/(DI5P*(DP(J*1)-0P(J) ) )
IF(NDC)GO TO 53
U0=O/ (FN<* . 7tl539ri2f"01 A»U IA)
RFD='J0»J33333
IF(FD.GT.10. )b-S.6-FD/SQRl (F0^FD->10.)
62 TEKP=ATAN(I. <,16667* S/FU)
T H £ T A 0 = . 0 U 1 A i; G. E « (1. 3 7 0 8- T £ M P> «¦ T E M P
C0STHE=C03(THE TAG)
SINThE = S IN(ThETAO >
-------
£6 (Rgv.)
IF (DO (1) .EU.U.JGO TO 77
DGT£/-'P~. 0 1/OG ( i )
1F (DGTEfIP. LE.0.)DG 7 EMP=-DGTEfSF
OSI = . 12* i .6*"* (ALOGIO (Fu/lO.) ) "2or"4' {ALOGIO (OGTEMP) >
77 tJPO=iriX(PI>/ 10SI»DIAJ ~•S)
N=0
Z=S*S I NT HE
X=S*C0S1KE
E-»» INITIAL CONDITIONS ')
IF (NOC) WRITE <6* 104)
104 FOKMAT ( • 0 UNITSS NON DIMENSIONAL ')
IF ( « NOT«NDC . AND® MET ENS )I.R1TE(6»105)
1G5 FCaMAT < * 0 UNITS: V"3^j£.R c • « • • • » • • « • »F/• 1/
«40H LENGTH FO'-i FLO.i ESTABLISHMENT . o . »F0c2/
°40H INTEGRATION STtP LENGTH . . . * . .»F9,3/
«40H PRINTOUT INTERVAL »F8.2/
»40h( X0.«o «•«•••» e»T8,2/
«40H ZO .»FtJ.2/
«4C»H DISCHARGE DENSITY »F11.5/
ft^0H PORT DlPiH • • • • • • • • o • • • • »F8« 2)
IF («NOTeNDC) W^ITE (6* e>0) U«F.W»UO pDI A
60 FORMAT(
C40H FLO^RATE E 15» 5/
»40H NUHSEK OF POrtTS iF5o0/
-------
USER'S GUIDE AND DOCUMENTATION FOR OUTFALL PLUME MODEL
Prepared by
D. J. Baumgartner, D. S. Trent, arid K. V. Byram
Library
Pficiflc Northwest Water Laboratory
200 South 35th Strsw
Corvsllls, Oregon 97330
Working Paper #80
Environmental Protection Agency
Pacific Northwest Water Laboratory
200 S.W. 35th Street
Corvallis, Oregon 97330
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CONTENTS
Page
INTRODUCTION 1
DESCRIPTION OF PROGRAM PLUME 2
INPUT DESCRIPTION (IBM 360) 4
Initial Conditions 4
Density (s) Profile Cards 4
OUTPUT DESCRIPTION 7
Example Format 7
Explanation 8
Case a 9
Case b 9
Case c 10
Case d 10
EXAMPLES 12
Case Number One 12
Case Number Two 12
Case Number Three 14
Case Number Four 18
Case Number Five 18
Case Number Six 21
REFERENCES 23
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LIST OF FIGURES
Page
Figure 1. Definition Sketch for Input Specification
and Output Interpretation 5
Figure 2. Four Possible Plume Configurations Analyzed
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INTRODUCTION
Many EPA offices have need for a computational orogram for
analysis of pipeline discharges into lakes, reservoirs, estuaries,
or the ocean. The enclosed plume program, P L U M E, is offered
as a standard procedure for analysis of industrial waste, thermal,
and sewage streams, incorporating the most recent knowledge of
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DESCRIPTION OF PROGRAM PLUME
PLUME is designed to solve for the geometric and dynamic
behavior of a buoyant round plume issuing from a port in stagnant,
density stratified surroundings. The port may be oriented at any
angle from 0° (horizontal) to 90° (vertical). The governing differ-
ential equations used are based on similarity principles and are
similar to those presented by Baumgartner and Trent (1, pp. 64-71).
Solution is carried out by a fourth-order Runge-Kutta technique.
Additional information concerning the exact equations used and
detailed theory will be given by Trent, Baumgartner, and Byram (2).
Calculation of the potential core length is based on Abraham's
method and is an integral part of the results. Computed results
include coordinates for the plume centerline, centerline dilution as
a function of position, and the maximum height of rise if the receiving
water is stratified. Calculation of centerline dilution is terminated
should the centerline buoyancy become neutral. The reason for this
is that similarity assumptions become invalid above this point and
dilution results may be largely inaccurate. The dilution at the
point of neutral buoyancy may be used as a conservative (minimum)
estimate of the trap level dilution.
As in the case with all plume similarity solutions, the program
should not be applied to cases where the receiving water is very
-------
3
densimetric Froude numbers may range from values smaller than 1 to
greater than 10,000. Calculations may be carried out in either MKS
or FPS units. Flow rate inputs must be specified for the entire
diffuser section, along with the number of ports and a constant
port diameter. Hence, the program assumes equal flow conditions at
each diffuser port.*
* No difficulty is encountered if actual conditions are not consistent
with this assumption. Flow distribution between unequal diameter
ports may be approximated (see also Rawn, Bowerman, and Brooks (4))
-------
4
INPUT DESCRIPTION (IBM 360)
Data input for program PLUME consists of the following
records (see Figure 1):
Initial Conditions
Format: (2L1, 13, F5.0, 7F10.0)
Field Column Description
1 1 Leave blank-
2 2 Logical: T (true)=MKS units; Blank=FPS units,
3 3-5 Number of ambient density points to be
entered, £ 50, right justify (do not use a
decimal point).
4 6-10 Angle of port orientation from horizontal,
degrees.
5 11-20 Port diameter.
6 21-30 Vertical distance between water surface and
outfall port centerline (port depth).
7 31-40 Density of effluent in grams per cubic
centimeter.
8 41-50 Leave blank.
9 51-60 Total volumetric flow rate.
10 61-70 Number of ports.
11 71-80 Desired data printout interval along plume
centerline.
Density (s) Profile Cards
The program needs a profile of either density or temperature
-------
Origin -SL
Sea Surface
Trap Level
Depth, Z |
Port Depth
Elev
rt Angle
Seabed
Port Diameter
FIGURE 1. Definition Sketch for Input Specification and
-------
6
One set of paired depth and density values is entered on each card,
starting at the surface. If ambient density is constant with depth,
use only one card.
Format: (3F10.0)
Field Column Description
1 1-10 Depth (distance measured from surface=0)
2 11-20 Density (or temperature, in degrees Celsius)
3 21-30 Salinity (in parts per thousand) but entered
only if temperature is entered in columns
11-20. Otherwise, blank. You must use a
non-zero salinity if temperature is used.
Any number of additional cases may be run by providing additional
-------
OUTPUT DESCRIPTION
Example Format
BUOYANT PLUME IN A DENSITY STRATIFIED MEDIA********
CASE NO. n INITIAL CONDITIONS
UNITS: aaa
PORT ANGLE •••••••• nnn
FROUDE NUMBER . . . . . . . . . nnn
LENGTH FOR FLOW ESTABLISHMENT ... nnn
INTEGRATION STEP LENGTH nnn
PRINTOUT INTERVAL nnn
XO } nnn
ZO nnn
DISCHARGE DENSITY nnn
PORT DEPTH nnn
"FLOWRATE nnn
NUMBER OF^ PORTS . ^ nnn
DISCHARGE VELOCI'TY'V V " " . ". ". ". ' nnn
PORT DIAMETER nnn
density"stratTfica riON DEPTH " RHO
nnn nnn
nnn nnn
nnn nnn
S_ X ELE V THETA DILN
nnn nnn nnn nnn nnn nnn
nnn nnn nnn nnn nnn nnn
nnn nnn nnn nnn nnn nnn
PLUME HITS SURFACE
nnn nnn nnn nnn_ nnn
LAST LINE ABOVE IS FOR MAXIMUM HEIGHT OF RISE.
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8
Explanation
The "FROUDE NUMBER" is an initial densimetric Froude number
based on velocity squared.
The "LENGTH FOR FLOW ESTABLISHMENT" is the length of the
plume potential core.
"INTEGRATION STEP LENGTH" is the length of the increment along
the plume centerline used for numerical calculation. If there
is no stratification, the increment is 1/177 of the depth. If
there is stratification the increment is computed as a function
of the Froude number and the density gradient at the outfall port.
This function was arrived at arbitrarily based on operating
experience with the model.
"XO" is the horizontal coordinate of the point for flow
establishment measured from the origin at the port centerline.
"ZO" is the depth of the point for flow establishment.
"DENSITY STRATIFICATION" is a read out of the density distri-
bution input. If salinity and temperature data are input, the
output will be given in terms of density.
Each line of the final tabular listing above corresponds to
a point along the plume centerline defined by coordinates X and Z.
Heading Description
S Distance along plume centerline
.X Horizontal distance to point on plume
centerline
-------
9
ELEV
Distance above outfall port to point
on plume centerline (i.e., the differ-
ence between port depth and Z)
THETA
Inclination of plume trajectory
above horizontal at point.
DILN
Dilution at point on the plume cen-
terline.
No dilution values (DILN) are printed after the plume first
reaches neutral buoyancy.
Four plume configurations are described by the model (see Figure
If the plume reaches the surface with residual buoyancy the
messages PLUME HITS SURFACE and TRAPPING LEVEL NOT REACHED will be
printed along with a message announcing that the last line in the
table describes conditions at the surface.
Case b
If a heavy fluid is discharged upward the plume may proceed
toward the surface for some distance before falling back toward the
bottom. In this case the computations cease as soon as the centerline
angle, TIIETA, becomes negative, and the last data printout will
describe the dilution at the highest level of the plume centerline.
The message TRAPPING LEVEL NOT REACHED will be printed to advise that
the fluid retains residual negative buoyancy.
2):
-------
10
Case c
If fluid near the region of the plume centerline becomes
sufficiently diluted so that it is neutrally buoyant, the message
TRAP LEVEL IS NNN WITH DILUTION OF NNN
will be printed to serve as an estimate of the lower level of the
horizontally spreading lens of diluted waste.
Case d
The message PLUME HITS SURFACE will be printed when the case c
condition is met and if the fluid has sufficient momentum to reach
-------
PLUME HITS SURFACE
Case a
TRAP LEVEL
NOT REACHED
"W
Case b
MAXIMUM RISE
\
TRAP LEVEL
NOT REACHED
rw"
PLUME HITS SURFACE
Case C
MAXIMUM RISE
TRAP LEVEL
IS
^777"
Case d
TRAP LEVEL
IS
^77/ ^//y ^77? g/y
-------
12
EXAMPLES
Case Number One
An industrial waste consisting of 0,7 cfs of heated, essentially
fresh process water (specific gravity 0.9933) is discharged through
a single 4" horizontal port 36.7 feet below the surface of the ocean.
A pycnocline occurs between 8.6 feet and 14.1 feet and a slight
increase in density occurs with depth below the pycnocline.
It is desired to determine if the plume will reach the surface
or not, and what minimum dilution can be expected. Results are
desired at 4 feet increments along the plume path. The density
gradient data are:
Depth Density
0 1.02272
8.6 1.02272
14.1 1.02452
36.7 1.02500
Results are shown on the computer printout. Neutral buoyancy
was first reached between the 8th and 9th increments (trapping
level at Z=13.4 feet). This is a "case c" example.
Case Number Two
Same conditions as example one except specific gravity is 1.040
and port angle is 30°. Anticipating that the plume will not rise
-------
BUOYANT PLUME IN A. DEN5ITY STRA11 F I ED MEDIA******** 13
CASE NO. 1 INITIAL CONDITIONS
UNITS! FPS
PORT ANGLE 0
FROUDE NUMBER 197.7
LENGTH FOR FLOW ESTABLISHMENT ... 1.85
INTEGRATION STEP LENGTH .229
PRINTOUT INTERVAL 4.00
XO 1*8 5
ZO * 36.63
DISCHARGE DENSITY .99330
PORT DEPTH 36.70
FLOWRATE 7.00000E-01
NUMBER OF PORTS 1
DISCHARGE VELOCITY b«18
PORT DIAMETER 3.30000E-01
DENSITY STRATIFICATION DEPTH RHO
0 1.02272
8.60 1.02272
14*10 1.02452
36.70 1.02500
s
X
5.52
5.48
9.42
9.02
13*32
1 1 .64
17.22
13*94
21 .12
15.55
25*02
16.85
28*91
17*93
32.81
18.87
36.71
19.85
39.07
21.14
Z ELEV
36.16 .54
34.58 2.12
31*90 4.60
26*63 8.07
25.08 11.62
21.40 15.30
17.65 19.05
13*88 22.82
10.10 26.60
8.28 28.42
THETA DILN
14.2 2.9710
34.4 5.4971
51.5 8.9473
62.1 13.3521
68.5 Id.5041
72.5 24.1900
75.1 30.1967
76.7 36.2566
71 .8
0
LAST LINE ABOVE IS FOR MAXIMUM HEIGHT OF RIbE.
-------
Results (following) show that the plume rose 4.16 feet before
turning downward due to a residual negative buoyancy in the fluid.
This program does not compute the additional dilution achieved in
the downward travel, because the models available to describe this
portion of the flow field are less reliable and are still under
development. This is a "case b" example.
Case Number Three
A nuclear power plant intends to use ocean water for cooling
i
and discharge 2500 cfs of heated water (specific gravity 1.02081)
through 93 horizontal ports 1.5 feet in diameter. The water depth
at the proposed discharge site is 30 feet. Determine centerline
coordinates and dilutions at two-foot intervals when the ambient
salinity is 33 parts per thousand and the temperature distribution
is as foilows:
Depth
Temperature
0
20°C
5
19.7
10
19.4
15
18.9
20
18.7
25
18.6
30
18.3
Results (following) show that plume spreads below the surface
after achieving a minimum dilution of 5.1. This also ic a "case c"
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BUOYANT PLUME IN A DEiMil il ^rtiAHHED MEDIA «* + **~**
IS
CASE NO. - INITIAL CONDITIONS
UNITS: FPS
POUT ANGLE 30.0
FROUDE NUMBER 437.4
LENGTH FOR FLOW ESTABLISHMENT . . . 1.83
INTEGRATION STEP LENGTH .215
PRIN10UT INTERVAL 1.00
XO 1.59
£0 35.78
DISCHARGE DENSITY 1.04000
PORT DEPTH 36.70
FLOWRATE 7.00000E-01
NUMBER OF PORTS 1
DISCHARGE VELOCITY 8.16
PORT DIAMETER 3.30000E-01
DENSITY STRATIFICATION DEPTH RHO
0 1.02272
8.60 1.02272
14.10 1.02452
36.70 1.02500
s
X
L
ELEV
1HETA
DIL^O
2.69
2.33
35.36
1 .34
29.3
1.4362
3.77
3.28
34.84
1 .86
26. 1
1•9984
4.85
4.24
34.35
2. 35
26. 3
2.5492
5.92
5.21
33.89
2.61
24.1
3.0864
7.00
6.20
33.47
3.23
21 . 1
3.6091
8*08
7.22
33. 12
3. 58
17.5
4.1183
9.15
8.26
32.83
3.87
1 3- 1
4.6183
10.23
9.31
32.63
4.07
7.8
5. 1 178
11*31
10.39
32.54
4.16
1. 7
5.6313
11 .52
10.60
32.54
4. 16
0
b.7370
LAST LINE ABOVE IS FOR MAXIMUM HEIGHT OF hIi>E
-------
I()
~~~~~~~~A BUOYANT plume in a density stratified media********
CASE NO. 3 INITIAL CONDITIONS
UNITS: FPS
PORT ANGLE 0
FROUDE NUMBER 1696*0
LENGTH FOR FLOW ESTABLISHMENT ... 8.40
INTEGRATION STEP LENGTH .555
PRINTOUT INTERVAL 2.00
6 .40
ZO 29.96
DISCHARGE DENSITY 1.02081
PORT DEPTH 30.00
FLOURATE 2.50000E
NUMBER OF PORTS 93
DISCHARGE VELOCITY 15.21
PORT DIAMETER 1.50000E
03
00
DENSITY STRATIFICATION DEPTH RHO
0
1.02326
5.00
1.02334
10.00
1.02342
15.00
1.02354
20.00
1.02359
25.00
1.02362
30*00
1 .02369
S
X
Z
ELEV
THETA
DILN
10.07
10.07
29*95
*05
.3
1.1743
12.29
12.29
29*94
*06
.5
1.4334
14.51
14*51
29*92
*08
• 6
1 .6925
16.73
16.73
29.89
*11
*8
1.9517
18.95
18*95
29.86
*14
1.0
2.2108
21 .17
21*17
29*81
*19
1 . 1
2.4700
23.39
23*39
29.77
*23
1.3
2.7292
25.61
25.61
29.71
*29
1.5
2.9884
27.83
27.83
29.65
*35
1 .7
3*2477
30.05
30.05
29*58
*42
1.9
3.5071
32.27
32.27
29.50
.50
2.1
3.7664
34.49
34.49
29*41
.59
2.3
4.0258
36.71
36-71
29*32
* 68
2.4
4.2853
38 .94
38.92
29.22
*78
2.5
-------
17
41.16
41 .14
29.12
43.38
43.36
29*02
45*60
45.58
28.92
47.82
47 .80
28*82
50.04
50*02
28*72
52.26
52.23
28*63
54.48
54.45
28*54
56.70
56.67
28*48
58.92
58.89
28*42
61.14
61 .1 1
28*39
62.81
a
to
.
GO
28*39
.88 2.6 4.6040
.98 2.7 5.0633
1.08 2.7
1.18 2.6
1.28 2.5
1.37 2.3
1.46 2.0
1.52 1.6
1.58 1.1
1.61 .5
1 .61 0
LAST LINE ABOVE IS FOR MAXIMUM HEIGHT OF RISE.
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18
Case Number Four
3
A municipality discharges 0.126 m /sec of raw sewage (s.g.=1.000)
through a single 0.76 m diameter port oriented at 15° in 27 meters of
water off Florida's coast. Calculate dilutions and coordinates every
three meters. The ambient salinity is a constant 36 °/O0 over the
depth. The temperature is constant at 85°F from zero to 12 meters and
83°F below 13 meters.
Results (following) show the plume spreading at the surface with
residual buoyancy and a minimum dilution of 52. This is a "case a"
example.
If the temperature data are entered erroneously in °F rather than
the °C units as specified in the INPUT DESCRIPTION (page 3), the
computed ambient densities would be unrealistically low, and the
computed plume results would be erroneous.
Case Number Five
A pulp mill discharges 10.8 cfs of process water (s.g.=1.000)
through 26 horizontal 3" ports in 40 feet of water. Surface density
is 1.0245 grams/cc while density at 40 feet is 1.0250. Plume calcu-
lations are desired at an interval of 3.5 feet.
Results (following) show that the plume hits the surface and
also that a trapping level is reached 8.7 feet below the surface.
The minimum dilution expected is 64. This is a "case d" example.
The horizontally spreading layer may be exposed at the surface
-------
19
BUOYANT PLUME IN A DENSITY STRATIFIED MEDIA********
CASE NO. 4 INITIAL CONDITIONS
UN ITS I MKS
PORT ANGLE 15*0
FROUDE NUMBER .4
LENGTH FOR FLOW ESTABLISHMENT ... 1*56
INTEGRATION STEP LENGTH .153
PRINTOUT INTERVAL 3.00
XO 1 . SO
ZO 26 • 60
DISCHARGE DENSITY t.00000
PORT DEPTH 27.00
FLOWRATE 1.26000E-01
NUMBER OF PORTS 1
DISCHARGE VELOCITY .28
PORT DIAMETER 7.60000E-01
DENSITY STRATIFICATION DEPTH RHO
0
1.02267
12.00
1.02267
13.00
1.02307
27.00
1.02307
S
X
Z
ELEV
THE! A
DILN
4.45
4.08
25.33
I .67
39.4
3.1353
7.51
5.98
22.96
4.04
60.8
6.6508
10*56
7.19
20*16
6.84
71.3
11.5495
13*61
8.01
17.23
9.77
76.8
17.6167
16.66
8.62
14.24
12.76
80.0
24.7170
19.54
9.07
11 .40
15.60
81.3
31.1603
22.59
9.53
8*38
18.62
81 .6
36.4967
25.64
9.97
5*36
21 .64
61.8
42.0214
28*69
10.40
2*34
24.66
82. 1
47.7505
PLUME HITS SURFACE
31*05 10*72 .00 27.00 82.2 52.3366
LAST LINE ABOVE IS FOR MAXIMUM HEIGHT OF RISE.
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20
********A BUOYANT PLUME IN A DENSITY STRATIFIED MEDIA*#****"**
CASE NO. 5 INITIAL CONDITIONS
UNITS! FPS
PORT ANGLE ...... 0
FROUDE NUMBER 356.1
LENGTH FOR FLOW ESTABLISHMENT ... 1•40
INTEGRATION STEP LENGTH .233
PRINTOUT INTERVAL 3.50
1 .40
£0 39.97
DISCHARGE DENSITY 1.00000
PORT DEPTH 40.00
FLOWRATE 1.08000E 01
NUMBER OF PORTS 26
DISCHARGE VELOCITY 8>46
PORT DIAMETER 2.50000E-01
DENSITY STRATIFICATION DEPTH RHO
0 1.02450
40.00 1.02500
s
X
Z
ELEV
THETA
DILN
4.66
4*64
39*70
*30
9.8
3.2838
8*15
7.95
38.63
1 *37
26.7
6.0331
11 .64
10.78
36*61
3*39
44*0
9.6010
15*13
12.99
33.92
6.08
56.3
14.2168
18.62
14.70
30.88
9.12
64*2
19.7461
22*11
16*07
27*67
12*33
69*2
25.9852
25.60
17.21
24.37
15.63
72*6
32*7390
29*09
18*16
21 *02
18*96
74*8
39*8085
32.58
19.05
17*64
22*36
76*4
46*9702
36.07
19.84
14*24
25.76
77*4
53*9555
39. 56
20.59
10.83
29*17
77.9
60* 4244
43.05
21 .31
7*42
32.58
77*9
46. 54
22.06
4.01
35.99
77*4
50.03
22.85
.61
39.39
75.9
PLUME HITS SURFACE
50.66 23.01 -0.00 40-00 75.5
LAST LINE ABOVE IS FOR MAXIMUM HEIGHT OF RISE.
TRAPPING LEVEL IS
-------
21
because the mechanics of mixing in the transition region are less
well understood, and the models which exist are not sufficiently
well varied.
Case Number Six
Same case as number five except the density at the surface is
increase to 1.02499.
Results (following) show that the centerline dilution calculation
is very sensitive to the value of the density gradient. In this case -
essentially no gradient - the dilution is increased to 107, and no
trapping level is reached. Since the plume reaches the surface with
residual buoyancy, the spreading lens is considerably thinner, but
no estimate of thickness is generally available. This is a "case a"
-------
22
BUOYANT PLUME IN A DENSITY STRATIFIED MEDIA********
CASE NO. j INITIAL CONDITIONS
UN ITS J FPS
PORT ANGLE 0
FHOUDE NUMBER 356.1
LENGTH FOR FLOW ESTABLISHMENT ... 1.40
INTEGRATION STEP LENGTH ...... .755
PRINTOUT INTERVAL 3.50
1.40
ZO 39.97
DISCHARGE DENSITY 1.00000
PORT DEPTH 40.00
FLOWRATE 1.08000E 01
NUMBER OF PORTS 26
DISCHARGE VELOCITY 8.46
PORT DIAMETER 2.50000E-01
DENSITY STRATIFICATION DEPTH RHO
0
40.00
s
X
Z
4.42
4.41
39.74
8.20
8.00
38.62
11 .98
11 .01
36.36
15.75
13.28
33.36
19.53
14.98
29*99
23*31
16.30
26.45
27.08
17.35
22*82
30.86
18.22
19.15
34.64
18.96
15*44
38.41
19.59
11 .72
42.19
20.14
7.99
45.97
20.63
4.24
49.74
21 .07
*49
PLUME HITS SURFACE
50.24 21.12 >00
1.02499
1.02500
ELEV
THETA
DILN
.26
8.8
3.1131
1.38
27. 1
6.0814
3.64
45.9
10*0493
6.64
58.9
15*3324
10.01
66.9
21*8086
13*55
72.0
29*3221
17*16
75*4
37*7670
20*85
77.8
47 .0715
24.56
79*6
57*1827
28.28
81.0
68*0580
32.01
82* 1
79*6614
35.76
83.0
91*9613
39.51
83.7
104*9265
40*00
83.8
106*6648
LAST LINE ABOVE IS FOR MAXIMUM HEIGHT OF RISE.
-------
23
REFERENCES
1. Baumgartner, D. J. arid D. S. Trent, "Ocean Outfall Design:
Part I, Literature Review and Theoretical Development." FWPCA,
April 1970.
2. Trent, D. S., D. J. Baumgartner, and K. V. Byram, "Forced Plumes
in Stratified, Quiescent Media." To be published.
3. Abraham, G., "Jet Diffusion in Stagnant Ambient Fluid." Delft
Hydraulics Publication. November 29, 1963.
4. Rawn, A. M., F. R. Bowerman, and Norman H. Brooks, "Diffusers
for Disposal of Sewage in Sea Water." Proceedings of the
American Society for Civil Engineers, Journal of the Sanitary
-------
24
APPENDIX I
-------
C PROGRAM PL)Mf_, V".SION OF 1/6/71
C
COMMON , FK, cM,COSTH,SINTH,COSTHlI,DS,Cl,C?, F1 3, PL AG
DT MEN5: TIM 7 n (50) , OG ( 50 ) , RH 0 ( 5 Q ) ,DP(50)
LOGICAL MDC,r"flPPO,FLAG,CHGPEN,METERS
NOC A Sr = 1
20 RFA0(5,71,^n = 7<*) NQH , MFTFRS,NPTS,ANGLE,OTA ,n<-pT<,9HDJ,^Fn,o,Fn,Jc
73 FORMAT(?L1,T3»F5. G , 7 F1Q . 0)
FD=A9S RFD)
DO in? l=l,NPTS
READ (5, rr,) np (i) ,RHO( I) ,SAL
IF(S AL.TO.0.)GO TO 102
RHO ( I) = L . + .0 HI'SIGMAT(SAL,RHO(I) )
102 CONTINUE
IF(NOTS.MC.i)G0 TO 76
NP TS-?
OP(2)=OE pTM
RHO ( ?) ==?HO (1 )
76 NOCASE = vJOn ASE+1
75 FORMAT( ^ r 1 0 ,0)
GRAV =32.172
IF(MFTF^s)GRAV=9.80665
DO 55 I=1,NPTS
IF(OP(T).GF.DEPTH)GO TO 56
55 CONTINUE
WRIT E(6,59)NOOASF
59 FORM AT f £-NO OFNSITY INFORMATION FOR JET LrVFL. "XFrilTION FOR/,
* t CASE NO.#,I?,# 9FLETF0.t)
GO TO 21
56 NP=I
NM=I-1
RHOR = (DEPTH-DP(NM))*(RHO(NP)-RHO(NM))/(DP(NP)-DP(MM))+RHO(NM)
DISP=RHOJ-RHOT
DO 5k 1=1, N»«
J=NP-I
70 (I ) = ( )EPTH-DP(J))/DIA
5k DG(I) = (R '0(J + 1)-RHO(J) ) *OIA/(OISP*(DP(J*-l) -RD(J) ) )
IF (NOD jO TO 58
U0=Q/(FvJ4. 7953982*DIA*0IA)
RFD = UO*'JQ*RHOJ/(-OISD*niA*GRAV)
FO = A BS (-?FO )
58 IF(ANGLE.NF.0.>GO TO 66
IF (FD.L-.1+.r)l.OR.F0.GE.9.99)G0 TO 61
S=.117*Fn + t+.
GO TO 6?
61 IF (FD.L" .«~ . n li S = 2. 8*FD»». 333 3 33
IF (FO.GT.10.)S=5.6»F0/SQRT(FD»F0+18.)
62 THFTA0 = ATAN( 1.<*16667»S/F0)
SI NTHF = '1 .
GO TO 63
66 NIT= 0
S=2.
6<+ IF(NIT.GF.1Q0)STOP
SO=S
-------
26 NIT=NIT«-1
IF(i\3S<3-SP).GT.. 000 1)GO TO 6*t
THETaO = '\MGLr/^7. 2957 3
63 COSTHf-r.nSi (THrTflO)
SINTH^IN ITHETAO)
OSI=D'"PrH/ (177.*DIA)
IF(DG(1>.EO.0.)GO TO 77
DGTE MP=.01/OG(1)
IF(HGTE 1D.LE.H.)DGTEHP=-DGTEMP
0SI=.1?*1.6**(AL0G1Q(FO/10.))*2.**(ALOGIO(OGTEMPU_
77 NPO=tFI<(PS/(nSI»DI4»*.5) "
N= 0
Z=S*SINTMf
X = S*COSTHtT
E= C+ ./S)
R=.?5
C1=E**.*666667
C?=.
TPTS=1
G=OG(T°TS)
ZLIH=7H( IPTS)
CHGDEN=.FALSE.
FLAG=.F\LST.
TRAP°D=.PALSE.
XP = X*PT'\
7PrQtpT 1-Z^nifl
DSIP=DSI*DIA
sp=s*nn
c*»
C WRIT17 I NT TI 3 L CONDITIONS
C#»
WRITE<6, *»¦?) MOCASE
21 FOPHAT*BUOYANT PLUME IN A OENSITY STRATIFIED MEDIA****
*****#/*-CASE NO.#,12,# INITIAL CONDITIONS *)
IP (NQC) •JPTTE(6,lCi4)
10k f o RM AT ("* 0 UNITS* NON DIMENSIONAL#)
IF(.NOT.NDO.AND.METERS)WRITE(6,10 5)
105 FORM AT(£ 1 UNITS! MKS#)
IF(.NOT.NPf.AND..NOT.METERS)WRITE(6,106)
106 FORM AT ( < 0 UMITS» FPS#)
WRITE(6, 10**) ANGLE,PFD,SP,'0*SIP",PS,XP,ZP,RHOJ,DEPTH
108 FORMAT(
*i+0H- =»OPT ANGLF ,F7.1/
* i+0 H 0 cROUOF NUMRCR ,F7.1/
*<+ OH LENGTH FOR ^LOW ESTABLISHMENT . . .,F8.2/
*V)H INTEGRATION STEP LENGTH ..... .,F9.3/
*£*0H PRINTOUT INTERVAL . . . . . . . . .,F8.2/
*<+0H XT ,F8.2/
*^DH 7ni ,F8.2/
*<40H HSCHARGE DENSITY ,F11.5/
*^+0H '"'OPT D E p T H . * . . • . « • . • • • . , F ft. 2)
IF(. NOT. NDC) WRITE (6,60) Q, FN, U0,DIA
60 FORMflT(
*i|QH FLOWRATT ,E15.5/
*i»0H J'J^PER OF P0PT9 « .,F5.0/
-------
27
»ifOH <*ORT DIAMETER ,F15.5)
WRIT E ( 6 i ?<*) ((OP(I) j RHO ( I) ) ,I=1,NPTS)
7U, FORMAT ( '-DENSITY STRATIFICATION DEPTH RH 01// 1 0 ( ? U X Ffi . 2 , F 13 . 5 / ) )
c**
C SET INITIAL CONDITIONS
C**
DS=DSI
WRITE(6,109)
109 FORMAT(* 1 S X Z ELEV THETA*
* ,t DILN *)
GO TO 16
11 DS=DSI
kS DELX =C. OSTH*DS
OELZ=ST'UH»OS
OELE=FK
OELT=FM
SP0S2 = S«-ns/2.
00 10 1=1,2
CALL SDERIV(SP0S2,E+.5»FKtR+.5*FM)
0ELX=DPLX+2»*C0STH*0S
nELZ=PELZ+?.*SINTH*OS
OELE=DFLE+2.*FK
DcLT=DELT+2.*FM
10 CONTINUE
CALL SOc:RIV(S + OS,E + FK,R + Ffi)
ZL AST = Z
ZINCR=(3FL7+SINTH*0S)/6.
z=z+zin,:r
IF (CHGOEN) GO TO £+1
IF(Z.GT.7LIM)GO TO ^0
*~3 X=X + (nFLX + COSTH»DS)/ft.
E=E+(HEL^+FK)/6.
R=R+(DELT+FM)/6.
s=s+ns
IF(F.LE.O.)FLAG=.TRUE.
C
C THIS STOPPING CRITERIA IS BASED ON VELOCITY GOING TO ZERO
C
IF(FLAG)GO TO 13
16 CALL SDIRIVtSjEtR)
IF ( TRAor,D) GO TO 1<*
IF(R.GT.G.)GO TO 15
RAT=R/(RQ-R)
E13TRP-E13-ME13-E130)
ZT RA °=OE DT H-(Z + ZINCR'RAT)*01A
SMTRAP=. 2<+5»(S+DS*RAT)»E13TRP
tRA~PPD = • TRUF.
GO TO i<+
15 R0=R
E130 =E15
14 N=N+1
IF(N-(N/NPO)»NPO.NF.O)GO TO 11
13 XP=X*OIA
ELEV = Z*r)IA
ZP = DEPTH-Euc-
-------
nTLN=.r»r>»s*Fn
T4FTA=A>COS(COSTH)»57.?9(58
IF {.NOT. rp^on)G0 TO 72
MRIT'(fj.ini) S^,XP,ZP,ELEV,THETA
GO TH 7 1
WR IT r ( £), 101) SP,X P» 7P» ELEV» THETfl ,DILN
rORMAT('(Fin.2,F10.1,F10.UI
IF ( .NOT ,rLi1r)G0 jq ^
WRITr(l"3, e,5)
FORMAT Ct-LAST LI'lE ABOVE IS FOR MAX IMUM_ HEIGHT OF RISE_._*).
FORMAT ( f H T ° A po T N G LEVFL IS/F1Q.1,* WITH DILUTION OF #F10 • <~)
TF ( TRA ao j » < ITE ( 61 *7 > 7TR AP , SMTRAP
IF(.NnT.T^Ar>pn)WRiTP(f,,6R>
FO»MA1 <* IT"A^PTNG IFVEL NOT REACHED#)
GO TO '1
C FTMO NEXT -TKMTIfication AND RECOMPUTE LAST STEP IF NECF.SSV
C
1*0 OS=OSM !LI'WLAST> /(Z-ZLAST)
CALL Sn'PTVtStEtR)
CHGOFN=. Toijc.
Z=ZLAST
GO TO «»<5
<~1 CHGnE^= . CALSE.
IPTS=I°TS+1
TF(I°TS.r,T.NM)GO TO k?
G = OG (IDTS)
ZL I ^ =7 0( IPTS)
GO TO m
h2 WRIT"(6,
i*t* FORM AT ( / QPUJT HITS SURFACE*)
FL AG=. TR'ir.
GO TO U X
7STOP
END
SUOROUT T *Jr SDF3 3
COSTH=nSTHrv?,l/ (E 13 * E1 3)
IF (CCST KLT.1.»GO TO 1
C
C THIS STOPPING C°ITERIA IS BASED ON PLUME BECOMING HORIZONTAL
C AGAIN
C
FL AG=. T
SI NTH=(].
CO S T H=1.
GO to ^
1 SINTH=STWT(l,-OOSTH*COSTH)
3 FK=C?*S*r;*STMT1H»DS
FH = . lpq»S*Fl7*G*0S
RETURN
FMO
28
72
m
71
f><5
67
-------
FUNCTION STGMAT(SAL,T)
SIGO = ( (eS.RE-6»SAl)-^.82E-»(T-3.98>MT + 283.)/(503.57*(T+67.26)
SIGMAT = (SIG0 + .132«t)» (1.-A+.BMSIGJL-. 132>> >-SUMT
RETURN
------- |