Discharge Calculator - Flowrates, Jet Properties and Outfall Design The following spreadsheet was developed for the estimation of effluent flowrates and their characteristics of a reverse osmosis (RO) desalination plant. The procedere is based on the technical note "Improved Discharge Configurations for Brine Effluents from Desalination Plants" by Jirka (2006). First, the flowrates of the desalination plant are estimated related to the drinking water flowrate Q drink . Additionally the effluent properties (e.g. density, salinity) are calculated. Second, the required design of the outfall (geometry) and its characteristics (e.g. discharge velocity U o , momentum flux M o , length scales L Q & L M ) are calculated for a discharge into a stagnant water body. Third, the properties of the negatively buoyant jet are estimated (Dilution S , maximum level of rise z max , x max , impingement point z i , x i ) and a recommendation for the outfall location is given (distance to shore). In the summary all important parameters are listed. The density & viscosity calculator is attachted and can be used separately. The figures for the calculation of the jet trajectory and the dilution are also attached. Note that the calculations are only valid for dense discharges (= effluent is more dense than the ambient)! For the calculation following parameters are required: - ambient characteristics: temperature T a , salinity Sal a , bottom slope qB - plant characteristics: flow rate Q drink , recovery rate r , brine temperature T desal - if existent blended effluent char.: flow rate Q effl,ex , temperature T effl,ex , salinity Sal effl,ex - outfall geometry: discharge angle qo , port openenings n Please enter the values of these parameters into the highlighted cells on the following three worksheets. Declaration of the colors of the highlighted cells: yellow fixed parameters (ambient characteristics, plant/effluent properties) blue freely selectable (outfall geometry) green calculation not yet implemented
Flowrates & Effluent Characteristics RO annotations/limitations:
- ambient characteristics (seawater) ambient temperature ambient salinity ambient density ambient kin. viscosity
Ta Sal a ra na
= = = =
22.00 33.00 1022.44 9.98E-07
°C ppt kg/m 2 m /s
T = 10 to 180°C Sal = 0 to 160 ppt (ppt = g/kg) 3
allowed ranges for viscosity calculation: Sal = 0 to 130 ppt, T = 10 to 180°C (El-Dessouky, Ettouny (2002))
- fresh water (permeate) flowrate recovery rate intake flowrate
Q drink = r = Q in =
1.00 m³/s 50 % 2.00 m³/s
recovery rate: percentage of intake water converted into permeate; plant characteristic; following Lattemann: r = 40-65%
- brine characteristics (effluent from desalination process) plant effluent flowrate temperature salinity density substance concentration
Q desal T desal Sal desal rdesal cdesal
= = = = =
1.00 22.00 66.00 1047.49 20.00
3
m /s °C ppt kg/m3 ppm
usually ambient or 1°C above with Sal drink = 0 ppt e.g. coagulants, anti-scalants, .... (has no effect on density or mixing characteristics)
- blended effluent - external flowrate temperature salinity density
Q effl,ex T effl,ex Sal effl,ex reffl,ex
= = = =
(e.g. waste water or others) 1.00 m3/s 20.00 °C 0.00 ppt Sal = 0 to 160 ppt, T = 10 to 180 °C 998.40 kg/m3
Final effluent characteristics: flowrate effluent temperature effluent salinity effluent density buoyant acceleration -> negatively buoyant, ok! kin. viscosity
Qo To Sal o ro go'
= = = = =
2.00 21.00 33.00 1022.73 -0.00281
m3/s °C ppt kg/m3 m/s2
mean average mean average g o ' = g ∙(|ra-ro|)/ra g o' < 0: negatively buoyant, g o' > 0: positively buoyant
no =
1.02E-06 m2/s
co =
10.00 ppm
allowed ranges for viscosity calculation: Sal = 0 to 130 ppt, T = 10 to 180°C (El-Dessouky, Ettouny (2002))
substance concentration
Discharge Characteristics RO annotations:
- ambient characteristics ambient density buoyant acceleration offshore slope
ra = g' o = qB =
1022.44 kg/m3 -0.00281 m/s2 10 °
Qo = ro = no =
2.00 m3/s 1022.73 kg/m3 1.02E-06 m2/s
- effluent characteristics flowrate discharge density kin. viscosity
- discharge characteristics Choose a discharge angle (recommended: 45°) : qo =
discharge angle discharge angle ok! port discharge velocity number of openings port diameter
Uo = n = D =
dens. Froude Number Reynolds Number
Fro = Reo =
45 15 °
5.00 m/s 6 0.29 m 174.75 1.43E+06
[0°qo 90°] only integer! recommended: U o = 4-6 m/s start with one opening!
Fro = U o /(g' o ∙D )1/2 = Q o /(D ²p/4)/(g' o ∙D )1/2 Reo = U o ∙D /no
Checking of characteristic properties: Diameter D : in required range, ok! Froude Number Fo: huge value, please check openings! Reynolds Number Reo: in required range, ok!
required: 0.1 D 1.0 Fro 10, recommended: Fro=20–25 Reo > 4000
Choose an appropriate port diameter (DN according to ISO standard): port diameter
D =
0.30 m
Final discharge characteristics: port diameter number of openings discharge angle flowrate (individual) port discharge velocity dens. Froude Number Reynolds Number momentum flux buoyancy flux discharge length scale momentum length scale
0.30 m D = 6 n = qo = 15 ° Q o,ind. = 0.33 m3/s Uo = 4.72 m/s Fro = 162.42 Reo = 1.39E+06 Mo = 1.57 m4/s2 Jo = -0.001 m4/s3 LQ = 0.27 m LM = 45.87 m
M o = U o ∙Q o J o = g' o ∙Q o (<0: negatively buoyant) L Q = (D ²∙p/4) 1/2 = Q o / M o 1/2 L M = M o 3/4 / J o 1/2
Jet Properties RO annotations:
- discharge & ambient characteristics qo = ho =
discharge angle port height in required range, ok! offshore slope imaginary offshore slope
qB = qB * =
momentum length scale dens. Froude Number
LM = Fro =
15 ° 0.50 m 10 ° 10 °
h o = 0m or h o = 0.5-1.0m
due to port height, not yet implemented
45.87 m 162.42
- geometric jet properties
(for discharge angles that are not a multiple of 15°: linear iterpolation!) (c /c max = 3%) Z max /L M (3%) = 0.564 (c /c max = 25%) Z max /L M (25%) = 0.447 (taken from Fig. 2(a)) z max /L M = 0.232 x max /L M = 1.241 z i /L M = -0.487 (taken from Fig. 4(a)) x i /L M = !port height not considered! 2.700 Z max (3%) = upper jet boundary 25.86 m Z max (25%) = 20.51 m z max = maximum jet centerline 10.64 m x max = position 56.94 m zi = jet centerline position at -22.34 m x i = 123.85 m the impingement point x H ao
869.62 m 15.38 m
Choose an appropriate outfall location: offshore location x = H ao = local water depth
870.0 m 15.39 m
offshore location local water depth
- dilutions & concentration minimum centerline dilution at z max bulk dilution at impingement point
H ao 0.75∙Z max (25%)
in required range, offshore location ok!
(for bottom slopes that are not a multiple of 10°: linear iterpolation!) S m /Fr o = Fig. 2(b) 0.21 Sm = 34.8 S i /Fr o = Si =
substance concentration at the centerline of max. level of rise (z max ) cm = impingement point (z i ) ci =
1.00 162.5 0.29 ppm 0.06 ppm
Fig. 4(b)
S = c o /c c -> c c = c o /S
erpolation!)
SUMMARY RO - ambient characteristics ra = qB =
ambient density offshore slope
1022.44 kg/m3 10 °
- effluent characteristics Qo ro no g' o
flowrate discharge density kin. viscosity buoyant acceleration
= = = =
2.00 1022.73 1.02E-06 -0.00281
m3/s 3 kg/m 2 m /s 2 m/s
(<0: negatively buoyant)
- dicharge characteristics & jet properties flowrate (individual) port discharge velocity dens. Froude Number Reynolds Number momentum flux buoyancy flux discharge length scale momentum length scale max. jet centerline position jet centerline position at the impingement point min. centerline dilution @ z max dilution @ impingement point substance concentrat. @ z max substance concentration @ z i
Q o,ind. Uo Fro Reo Mo Jo LQ LM z max x max zi xi Sm Si cm ci
= 0.33 m3/s = 4.72 m/s = 162.42 = 1.39E+06 = 1.57 m4/s2 = 0.00 m4/s3 = 0.27 m = 45.87 m = = = = = = = =
10.64 56.94 -22.34 123.85 34.8 162.5 0.06 0.22
m m m m
= = = = = =
0.30 6 15 0.50 870.0 15.39
m
ppm ppm
- recommended outfall geometries port diameter number of openings discharge angle port height offshore location local water depth
D n qo ho x H ao
° m m m
SEAWATER DENSITY & VISCOSITY CALCULATOR Please enter the values of salinity and temperature of the effluent or ambient water (p=1atm):
33.00
Salinity: Sal =
ppt
for r: 0 Sal 160 ppt for m: 0 Sal 130 ppt 10 T 180 °C
T=
20.00
Density:
r=
1023.019
Dynamic Viscosity:
m=
1.069
*10-3 kg/m s
Kinematic Viscosity:
n=
1.045
*10-6 m²/s
Temperature: Temperature:
°C
kg/m³
source: El-Dessouky, Ettouny (2002): Fundamentals of Sea Water Desalination (Appendix A: Themodynamic Properties)
Density of sea water as a function of salinity and temperature Salinity: 0 to 160 ppt, Temperature: 10 to 180 °C, Pressure: p = 1 atm The density correlation of sea water is given by:
r = (A1F1 + A2F2 + A3F3 + A4F4)*10³
[kg/m³]
where: F1 = F2 = F3 = F4 =
0.5 A 2A² - 1 4A³ - 3A
A = (2T-200)/160
G1 = 0.5 G2 = B G3 = 2B² - 1
A1 = 4.032219 G1 + 0.115313 G2 + 3.26*10-4 G3 A2 = -0.108199 G1 + 1.571*10-3 G2 - 4.23 * 10-4 G3 A3 = -0.012247 G1 + 1.74*10-3 G2 - 9.0*10-6 G3 A4 = 6.92*10-4 G1 - 8.7*10-5 G2 - 5.3*10-5 G3
B = (2Sal-150)/150
Viscosity of sea water as a function of salinity and temperature Salinity: 0 to 130 ppt, Temperature: 10 to 180 °C, Pressure: p = 1 atm The dynamic viscosity correlation of sea water is given by:
m = (mW)(mR)*10-3
[kg/m s]
n = m/ r where: Ln(mW) = mR = A= B=
[m²/s]
-3.79418 + 604.129/(139.18+T) 1 + A*Sal + B*Sal² 1.474*10-3 + 1.5*10-5 T - 3.927*10-8 T2 1.0734*10-5 - 8.5*10-8 T +2.23*10-10 T2
Fig. 2 (a)
Geometric jet properties at maximum level of rise 2.5
Zmax/LM zmax/LM xmax/LM
2.0
1.5
1.0
0.5
0.0
0
10
20
x_max
Theta0.9715 0 2.6160 4.5568 7.2432 10.3725 13.6449 18.2490 22.9980 28.3379 34.1195 39.7502 43.3051 45.3786 48.6357 51.2990 55.4410 58.6937 61.3548 64.9021 68.3006 71.6981 74.6517 78.1945 80.5566 83.6556 86.4593 88.0818 89.7043
xmax/LM zmax/LM Zmax(3%)/LM Zmax(25%)/LM xmax*/LM
x_max 0 0.1151 0.3157 0.5162 0.7835 1.0285 1.2065 1.3621 1.4730 1.5615 1.6129 1.6270 1.6192 1.6115 1.5813 1.5364 1.4541 1.3645 1.2898 1.1778 1.0585 0.9242 0.7974 0.6260 0.5142 0.3502 0.2012 0.1043 0.0075
30 z_max
Theta 0 4.6697 10.6027 17.1311 23.2161 28.8572 34.3508 40.1412 45.1897 48.6048 54.8402 60.6289 66.1192 71.0146 75.6121 83.9150 89.8436
z_max 0 0.0403 0.1362 0.2693 0.4172 0.5726 0.7355 0.9058 1.0613 1.1650 1.3426 1.4906 1.6089 1.6975 1.7639 1.8521 1.8885
40 q [] 50 Z_max_3%
Theta 0.5342 5.7276 8.9923 13.0000 15.3753 20.7202 24.2840 28.2938 31.8582 34.9781 38.5429 42.8513 46.8623 49.6848 52.9523 56.2192 59.4851 63.3435 66.9045 70.6126 75.3588 79.6589 84.8479 87.6643 89.7390
Z_max 0.2118 3% 0.3227 0.3967 0.5002 0.5669 0.7223 0.8334 0.9668 1.0853 1.2038 1.3298 1.4928 1.6410 1.7448 1.8559 1.9596 2.0484 2.1371 2.2110 2.2701 2.3438 2.3953 2.4467 2.4687 2.4759
60
70 Z_max_25%
Theta 0.6768 3.3468 9.5783 15.3665 18.1868 21.7495 26.7980 30.8074 34.6691 37.7879 40.0156 44.1747 46.5511 49.3725 53.6787 57.8356 61.6946 64.8106 70.1516 73.7110 77.5667 80.5318 84.5340 87.6462 89.5726
Z_max 25% 0.1375 0.1817 0.3074 0.4479 0.5219 0.6181 0.7736 0.8995 1.0329 1.1366 1.2107 1.3588 1.4403 1.5292 1.6625 1.7809 1.8771 1.9436 2.0471 2.0987 2.1502 2.1796 2.2089 2.2233 2.2305
= -3.87340930091997E-11*theta^6 + 1.35855657125514E-08*theta^5 - 1.88466399633769E-06*theta^4 + 1.32370099986 = 8.96034580956262E-12*theta^6 - 1.96811134631301E-09*theta^5 + 1.61940812676287E-07*theta^4 - 1.104171015597 = -1.90212551452414E-12*theta^6 + 2.10387511821525E-09*theta^5 - 3.86272303589499E-07*theta^4 + 2.18413065660 = -4.15536063691431E-12*theta^6 + 2.07657972657878E-09*theta^5 - 3.13673259007529E-07*theta^4 + 1.51073246639 = 5.84507682255264E-08*theta^6 - 2.48546607631953E-06*theta^5 + 4.05917264968633E-05*theta^4 - 3.322326912037
80
90 x_max*
Theta 0.081158673 1.418344437 3.201626854 4.687879569 6.174132283 8.404063455 10.18789797 12.26931429 13.31002245 14.05370091 15.09440907 15.98715448
x_max* 1.096654275 1.104089219 1.118959108 1.133828996 1.148698885 1.178438662 1.200743494 1.230483271 1.24535316 1.260223048 1.275092937 1.29739777
769E-06*theta^4 + 1.32370099986678E-04*theta^3 - 5.59539047637969E-03*theta^2 + 1.42606256975711E-01*theta 87E-07*theta^4 - 1.1041710155979E-05*theta^3 + 6.91284581577634E-04*theta^2 + 6.3922033613153E-03*theta 499E-07*theta^4 + 2.18413065660464E-05*theta^3 - 2.2331771852952E-04*theta^2 + 2.34174001868774E-02*theta + 1.96157411179688E-01 529E-07*theta^4 + 1.51073246639019E-05*theta^3 - 5.80846006220177E-06*theta^2 + 1.85734633977261E-02*theta + 1.22231246044606E-01 33E-05*theta^4 - 3.32232691203778E-04*theta^3 + 1.84266719305346E-03*theta^2 + 3.45977712866332E-03*theta + 1.0963449609836E+00
Fig. 4 (b)
02*theta + 1.96157411179688E-01 E-02*theta + 1.22231246044606E-01 -03*theta + 1.0963449609836E+00
Fig. 2 (b)
Minimum centerline dilution 0.30
0.25
Sm/Fo
0.20
0.15
0.10
0.05
0.00 0
10
Theta 1.2691 1.7086 2.2937 3.4710 4.2061 5.0901 5.8268 6.8580 8.4814 9.6643 10.8460 12.9178 14.6951 17.3624 19.8836 23.2975 26.1192 28.7943 31.9160 35.4860 40.5455 43.3738 46.6494 50.8188 54.5432 58.2677 62.5887 67.6559 71.9781 84.9463 87.9280 90.1641
20
30
40
q []
50
60
70
S_m/F_o 0.0466 0.0562 0.0703 0.0887 0.1013 0.1138 0.1242 0.1389 0.1581 0.1692 0.1817 0.1986 0.2111 0.2280 0.2412 0.2552 0.2646 0.2712 0.2777 0.2820 0.2854 0.2860 0.2859 0.2849 0.2818 0.2786 0.2740 0.2670 0.2609 0.2403 0.2349 0.2311
Sm/Fo = -6.6480900894465E-12*x^6 + 2.13443627515762E-09*x^5 - 2.78481177335259E-07*x^4 + 1.94758862414959E-05* qo =
Sm/Fo =
90.00 ° 0.231
S_m/F_o
80
90
x^4 + 1.94758862414959E-05*x^3 - 8.33299633928484E-04*x^2 + 2.17919728705891E-02*x + 2.19626824040669E-02
Fig. 4 (b)
Fig.4 (a)
Jet trajectories for variable offshore slope 2.5 2.0
1.77
1.89
90°
1.5
1.47
qo = 75°
qo = 60°
1.05
1.0
45°
0.61
z/LM
0.5
30°
0.23
15°
0.0 0°
-0.5
qB = 10°
-1.0 qB = 20°
-1.5 -2.0
qB = 30°
-2.5 -3.0
-0.5
0.0
0.5
1.0
1.5
2.0
x/LM
2.5
3.0
3.5
4.0
4.5
x z_max (0°) x z_max (15°) x z_max (30°) x z_max (45°) x z_max (60°) 0.1095 x 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0324 -0.0016 0.1176 0.0325 0.0834 0.0383 0.0605 0.0440 0.0659 0.0783 0.0488 0.3175 -0.0022 0.2428 0.0722 0.2025 0.1180 0.1736 0.1694 0.1502 0.2380 0.5685 -0.0026 0.3679 0.1119 0.3273 0.1977 0.3206 0.3233 0.2907 0.4946 0.1370 0.6142 -0.0084 0.5957 0.1629 0.4692 0.2773 0.4394 0.4430 0.3526 0.5916 0.1868 0.7055 -0.0143 0.7323 0.1912 0.5942 0.3456 0.5583 0.5455 0.4483 0.7456 0.2200 0.8825 -0.0375 0.8576 0.2081 0.7249 0.4025 0.6943 0.6594 0.5329 0.8596 0.2701 1.0995 -0.0779 1.0856 0.2248 0.9182 0.4763 0.7963 0.7335 0.6345 0.9964 0.3536 1.2481 -0.1124 1.2339 0.2245 1.0376 0.5161 1.0177 0.8701 0.7475 1.1218 0.4039 1.3796 -0.1526 1.4052 0.2070 1.1857 0.5500 1.1938 0.9554 0.8890 1.2586 0.4654 1.4997 -0.1985 1.5879 0.1781 1.4306 0.5838 1.3360 1.0008 1.0534 1.3782 0.5667 1.6543 -0.2673 1.7194 0.1436 1.5561 0.5836 1.5752 1.0346 1.2240 1.4464 0.6685 1.7803 -0.3304 1.9367 0.0690 1.7045 0.5719 1.7464 1.0229 1.3151 1.4633 0.7253 1.9064 -0.4106 2.1314 -0.0284 1.8416 0.5488 1.8837 0.9826 1.4807 1.4402 0.7594 2.0325 -0.4907 2.3149 -0.1430 2.0932 0.4684 2.0268 0.9196 1.6124 1.3771 0.8165 2.1760 -0.5995 2.4756 -0.2632 2.2765 0.3710 2.1700 0.8336 1.7387 1.2741 0.9084 2.4059 -0.8112 2.8494 -0.6522 2.5462 0.1649 2.4748 0.5247 1.9289 1.0282 1.0473 2.4864 -0.8970 2.9820 -0.8180 2.7360 -0.0296 2.6191 0.3246 1.9984 0.9025 1.1056 2.7514 -1.2229 3.2653 -1.2867 3.0186 -0.4127 2.7811 0.0388 2.1668 0.5310 1.2171 2.8611 -1.3830 3.4044 -1.5668 3.0822 -0.5156 2.8971 -0.2012 2.2250 0.3882 1.2585 2.9998 -1.6117 3.5148 -1.8182 3.2038 -0.7442 3.0018 -0.4526 2.2777 0.2225 1.3119 3.0867 -1.7717 3.5730 -1.9553 3.3489 -1.0528 3.0833 -0.6526 2.3479 -0.0003 1.3657 3.1389 -1.8746 3.6313 -2.1096 3.4071 -1.1956 3.1767 -0.9211 2.4360 -0.3145 1.4137 -2.0517 3.6722 -2.2353 - 1.2344505827059E-02*x^5 3.4887 -1.4070 3.2529 -1.1725 2.5011 -1.8167 -0.5944 1.4559 3.7189 -2.3610 3.5997 -1.7384 3.3350 -1.4467 2.5545 -0.8514 1.4801 3.6876 -2.0184 3.3763 -1.6124 2.6138 -1.1256 1.4985 3.7288 -2.1726 3.4528 -1.9094 2.6676 -1.4283 1.5166 3.7698 -2.3154 3.4941 -2.0694 2.7034 -1.6282 1.5350 3.5297 -2.2407 2.7335 1.5593 z/L 3.2259 (0°) = 5.39080486760213E-04*x^6 + 6.27249541430501E-02*x^4 - 1.77953956910642E-01*x^3 + 8.34397
z/L (0°) z/L M (15°) z/L (15°) = M (30°) MM (30°) M (45°) M (45°) (60°) z/Lz/L M (60°) M (75°)= MM (75°) M M bottom slope [°]: 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
-0.075 -3.54761572680218E-04*x^6 3.1475920441153E-03*x^5 ++3.02414455945836E-02*x^4 - -1.20827495396952E-01*x^3 7.2658 0.225 ===-1.04388586385085E-03*x^6 +-- 1.01641578112321E-03*x^5 + 1.3786 0.533 5.470577491450970E-04*x^6 1.721598829499270E-02*x^5 +2.47415202684351E-02*x^4 9.609741519670930E-02*x^4 - 3.034449287806640E-01*x^3 + 8+ 0.916 1.1117107795144E-02*x^6 - 1.516790444657090E-01*x^5 + 6.031069714954360E-01*x^4 - 1.4149195193022E-01*x^3 1.233453456967250*x^3 + 5.11257 1.406 1.83778805987095*x^6 - 9.5838303292112*x^5 + 19.0983936369884*x^4 - 19.1190879476489*x^3 + 6.65051415114431*x^2 1.298 0° 15° 30° 45° 60° x_i/L_M z_i/L_M x_i/L_M z_i/L_M x_i/L_M z_i/L_M x_i/L_M z_i/L_M x_i/L_M z_i/L_M 0.000 0.000 2.079 0.000 2.712 0.000 2.802 0.000 2.348 0.000 0.658 -0.011 2.148 -0.037 2.755 -0.048 2.827 -0.049 2.360 -0.041 0.838 -0.029 2.215 -0.077 2.796 -0.098 2.851 -0.100 2.372 -0.083 0.984 -0.052 2.281 -0.120 2.836 -0.149 2.875 -0.151 2.383 -0.125 1.112 -0.078 2.345 -0.164 2.876 -0.201 2.899 -0.203 2.395 -0.167 1.230 -0.108 2.408 -0.211 2.915 -0.255 2.923 -0.256 2.406 -0.211 1.339 -0.141 2.469 -0.259 2.953 -0.310 2.946 -0.310 2.418 -0.254 1.443 -0.177 2.529 -0.310 2.991 -0.367 2.969 -0.365 2.429 -0.298 1.542 -0.217 2.587 -0.364 3.027 -0.425 2.992 -0.421 2.440 -0.343 1.637 -0.259 2.644 -0.419 3.063 -0.485 3.015 -0.478 2.451 -0.388 1.728 -0.305 2.700 -0.476 3.099 -0.546 3.038 -0.536 2.463 -0.434 1.817 -0.353 2.755 -0.535 3.134 -0.609 3.060 -0.595 2.474 -0.481 1.902 -0.404 2.808 -0.597 3.168 -0.673 3.082 -0.655 2.485 -0.528 1.985 -0.458 2.860 -0.660 3.202 -0.739 3.104 -0.717 2.496 -0.576 2.065 -0.515 2.912 -0.726 3.236 -0.807 3.127 -0.780 2.507 -0.625 2.143 -0.574 2.962 -0.794 3.269 -0.876 3.149 -0.844 2.518 -0.675 2.218 -0.636 3.011 -0.863 3.301 -0.947 3.170 -0.909 2.530 -0.725 2.291 -0.701 3.059 -0.935 3.333 -1.019 3.192 -0.976 2.541 -0.777 2.363 -0.768 3.107 -1.009 3.365 -1.093 3.214 -1.044 2.552 -0.829 2.432 -0.837 3.153 -1.086 3.396 -1.169 3.236 -1.114 2.563 -0.883 2.500 -0.910 3.199 -1.164 3.427 -1.247 3.258 -1.186 2.574 -0.937 2.566 -0.985 3.244 -1.245 3.458 -1.327 3.279 -1.259 2.586 -0.992
22 23 24 25 26 27 28 29 30
2.630 2.692 2.754 2.813 2.872 2.929 2.986 3.041 3.095
-1.063 -1.143 -1.226 -1.312 -1.401 -1.493 -1.587 -1.686 -1.787
3.289 3.332 3.375 3.418 3.460 3.501 3.542 3.583 3.623
-1.329 -1.414 -1.503 -1.594 -1.687 -1.784 -1.883 -1.986 -2.092
3.488 3.518 3.548 3.578 3.608 3.637 3.666 3.695 3.724
-1.409 -1.493 -1.580 -1.668 -1.760 -1.853 -1.949 -2.048 -2.150
3.301 3.323 3.345 3.367 3.389 3.411 3.433 3.455 3.477
-1.334 -1.410 -1.489 -1.570 -1.653 -1.738 -1.825 -1.915 -2.008
2.597 2.608 2.620 2.631 2.643 2.654 2.666 2.678 2.690
-1.049 -1.107 -1.166 -1.227 -1.289 -1.352 -1.418 -1.484 -1.553
4.5
z_max (75°) 0.020 x z_max (90°) x z_max x 0.0000 0.0000 0.0219 1.89 0.000 0.000 10° 0.000 15° 0.000 0.0840 -0.2814 0.7732 1.77 4 5° -0.3499547 -0.7053079 -1.0717968 0.3294 -0.1957 1.2842 0.23 0.4607 0.1868 1.3648 1.47 0.6433 0.6265 1.6035 0.61 0.7631 1.2602 1.6360 1.05 0.9172 0.020 1.8900 0.5000 0 1.1683 1.2995 1.4478 1.6189 1.7272 1.7556 1.7613 1.7555 1.6811 1.4296 1.2753 0.8868 0.7040 0.4527 0.1557 -0.1356 -0.4211 -0.5867 -0.7409 -0.8666 -1.0265 -1.2035 1E-02*x^4 ++8.34397081634961E-02*x^2 - +1.09367819386534E-02*x 36E-02*x^4- 1.77953956910642E-01*x^3 - -1.20827495396952E-01*x^3 7.26586495875381E-03*x^2 3.07851264689816E-01*x 351E-02*x^4 1.37863402342191E-02*x^2 6.08062166546006E-01*x 70930E-02*x^4 - 3.034449287806640E-01*x^3 + 8.137753393384630E-02*x^2 + 1.00714068002344*x 360E-01*x^4 - 1.4149195193022E-01*x^3 1.233453456967250*x^3 + 5.112572979705870E-01*x^2 9.1190879476489*x^3 + 6.65051415114431*x^2 + 2.65092721076326*x+ 1.609496010525620*x 75° x_i/L_M z_i/L_M 1.391 0.000 1.395 -0.024 1.399 -0.049 1.403 -0.074 1.407 -0.098 1.411 -0.123 1.415 -0.149 1.419 -0.174 1.423 -0.200 1.427 -0.226 1.430 -0.252 1.434 -0.279 1.438 -0.306 1.442 -0.333 1.446 -0.361 1.451 -0.389 1.455 -0.417 1.459 -0.446 1.463 -0.475 1.467 -0.505 1.471 -0.536 1.476 -0.566
1.480 1.484 1.489 1.493 1.498 1.503 1.508 1.512 1.517
-0.598 -0.630 -0.663 -0.696 -0.731 -0.766 -0.802 -0.838 -0.876
20° 0.000 25° 0.000 30° 0.000 -1.4558809 -1.8652306 -2.3094011
Fig. 4 (b)
Bulk dilutions for variable offshore slope 2.5
2.0
Si/Fo
1.5
1.0
0.5
0.0 0 Theta 0.4865 2.9514 4.9242 8.7093 12.4962 18.0975 22.7130 26.3401 30.4628 35.9057 42.5055 49.6026 56.0408 62.9764 68.5917 73.5476 78.5040 85.1141 89.5767
Si/Fo (0°) Si/Fo (10°) Si/Fo (20°) Si/Fo (30°) qo =
Si/Fo (0°) = Si/Fo (10°) = Si/Fo (20°) = Si/Fo (30°) =
10 Si/F0.1014 o (0°) 0.2207 0.3071 0.4388 0.5541 0.6941 0.7848 0.8507 0.9167 0.9951 1.0694 1.1274 1.1688 1.1938 1.2065 1.2067 1.2029 1.1827 1.1624
20 Theta 1.4180 2.7350 5.3705 8.3354 10.9705 14.1011 18.7157 23.6612 28.7733 34.2171 39.8275 45.6048 51.8796 57.8263 63.9404 69.7259 75.6789 81.3028 86.1008 88.9141
30
40
Si/Fo0.6439 (10°) 0.6851 0.7551 0.8333 0.9074 0.9816 1.0805 1.1752 1.2577 1.3279 1.3857 1.4271 1.4521 1.4566 1.4405 1.4079 1.3548 1.2894 1.2239 1.1788
q []
Theta0.5490 2.8545 5.8190 9.4425 14.0553 18.3392 23.1190 27.7355 33.3431 39.1187 44.7327 48.8622 52.4979 56.1354 62.4206 67.0533 73.8405 78.1455 80.6298 84.7706 88.4154 89.7401
50
60
Si/Fo1.0343 (20°) 1.1001 1.1825 1.2813 1.3966 1.4996 1.5985 1.6809 1.7634 1.8213 1.8463 1.8506 1.8385 1.8099 1.7404 1.6749 1.5479 1.4577 1.4003 1.3019 1.2076 1.1789
70 Theta 0.6499 3.4479 6.2458 10.6908 15.4670 19.5861 24.3655 29.6432 33.6034 37.0703 40.7037 44.3389 49.2989 53.1034 55.5859 59.5593 63.2031 66.1866 70.4971 74.9747 78.9557 82.4393 86.2551 88.5783 89.7397
80 Si/Fo1.6178 (30°) 1.7125 1.8072 1.9471 2.0789 2.1778 2.2808 2.3591 2.4004 2.4212 2.4296 2.4216 2.3849 2.3399 2.2989 2.2210 2.1349 2.0447 1.9052 1.7493 1.6016 1.4702 1.3225 1.2281 1.1830
= -1.5278595890795E-11*x^6 + 4.75779358979899E-09*x^5 - 5.93044679357057E-07*x^4 + 3.83986481204819E-05* = 5.4329867258437E-12*x^6 - 1.45882071146886E-09*x^5 + 1.48914754346514E-07*x^4 - 0.0000070172645695533 = -8.09073720783789E-12*x^6 + 2.24448932774251E-09*x^5 - 1.95454256787158E-07*x^4 + 4.26443939539355E-06 = 2.93122121847444E-12*x^6 - 4.02975262671317E-10*x^5 + 4.28138973616809E-08*x^4 - 5.47863676698675E-06* 10.0 ° 0.48 0.880 1.295 1.925
S_i/F_o 0°
S_i/F_o 10°
S_i/F_o 20°
S_i/F_o 30°
80
90
x^4 + 3.83986481204819E-05*x^3 - 1.5726647486815E-03*x^2 + 5.29165999419988E-02*x + 7.7166165750381E-02 ^4 - 0.0000070172645695533*x^3 - 0.000127066526488538*x^2 + 0.0295722168120381*x + 0.602753425771075 *x^4 + 4.26443939539355E-06*x^3 - 2.17177084384487E-04*x^2 + 2.96971296063456E-02*x + 1.01742812853606E+00 x^4 - 5.47863676698675E-06*x^3 - 2.00376622634102E-04*x^2 + 3.56861260665765E-02*x + 1.59361796727542E+00
Fig. 4 (b)
