Dr. Maillacheruvu

1

POTENTIALLY USEFUL INFORMATION 1. UNITS AND CONVERSIONS

Length

1 m = 103 cm = 103 mm = 106 µm = 109 nanometer = 1012 picometer 1 light year = 9.461 x 1015 m

1 km = 0.6292 mi 1 mi = 1589 m

1 yard = 3 ft 1 in = 2.54 cm 1 foot = 12 in 1 mi = 5280 ft = 1760 yards

Area 1 ft2 = 0.09290 m2

1 ft2 = 144 in2

1 acre = 43560 ft2

1 square mile = 2.7878 x 107 ft2 1 hectare = 104 m2

Volume 1 ft3 = 7.48 gal

1 ft3 = 0.02832 m3

1000 L = 1 m3

1 gal = 3.785 L = 3.785 x 10-3 m3

1 liter = 0.2642 US gallon = 61.02 in3 = 0.03531 ft3 = 0.001308 yard3

Time 1 day = 1440 min = 86400 s 1 year = 365 days = 8760 hrs = 5.256 x 106 min = 3.1536 x 106 s

Discharge “Q”

1 cfs = 86400 ft3/day

1 cms = 1 m3/s = 35.31 cfs 1 gal/min = 1 gpm = 0.06369 L/sec = 0.002228 ft3/s = 63.09 x 10-6 m3/s

Velocity 1 knot = 1.152 mph = 0.508

ms 1

€

milehour = 1.589

€

kmhour = 0.441

ms 3.6

€

kmhour = 1

ms = 2.25 mph

1 ft/day = 12.70 x 10-6 km/h = 11.57 x 10-6 ft/s = 7.891 x 10-6 mph = 3.528 x 10-6 m/s 1 ft/s = 86400 ft/day = 1.097 km/hour = 0.6818 miles/hour = 0.3048 m/s

Mass & mass loading

1 slug = 32.2 lbm 1

€

kg

m2−day = 0.205

€

lbmft2−day

1 slug = 14.59 kg

Force 1N = 236.47 lbf 1 lbf = 1 slug-ft

sec2 = 32.2 lbm-ft

sec2 1 N = 1000 dynes = 1 kg m

sec2

Energy & Power 1 calorie = 4.2 Joule 1 hp = 0.747 kW = 550

ft-lbfs 1 watt = 1

€

Joules

Pressure 1psi = 144 psf = 6.871 kPa 1 atm = 101 kPa = 14.7 psia = 760 mm of Hg = 29.9 inches of Hg

1 psi = 2.4 inches of Hg; 1 psf = 47.88 Pa

Temperature T in oR = T in oF + 460oR T in oF = 1.8*(T in oC) + 32oF T in oK = T in oC + 273oK

Viscosity 1 centi-poise = 10-4 N-s

m2 1 N-s/m2 = 21.424 lbf-s/ft2 1 m2/s = 11.037 ft2/s

Concentration 1

mgL ≡ 8.34

€

lbmmilliongallons

= 1 x 10-3 kg

m3 ; 1 kg/m3 = 1.94 x 10-3 slug/ ft3 = 6.24 x 10-2 lbm/ft3

1 m/s2 = 3.281 ft/s2 1 m2/s = 11.037 ft2/s 1 N/m3 = 6.70 lbf/ft3

2. STANDARD CONSTANTS FOR WATER

p = 1 atm; T = 10 oC γw = Specific Wt

9810

€

Nm3

= 62.4

€

lbfft3

p = 1 atm; T = 10 oC ρw= Density

103

€

kg

m3= 1.94

€

slug

ft3= 62.4

€

lbmft3

T = 10 oC µw = Absolute Viscosity

1.31 x 10-4 N-s

m2 = 1.31 cp

T = 25 oC µw = Absolute Viscosity

0.89 x 10-4 N-s

m2 = 0.89 cp

p = 1 atm; T = 10 oC Bulk Modulus

Ev (water) = 2.2 x 109 N

m2

p = 1 atm; T = 10 oC Surface Tension

σwater-air = 0.073

€

Nm

T = 10 oC Kinematic Viscosity

νw= 1.31 x 10-6

€

m2s

T = 25 oC Kinematic Viscosity

νw= 0.89 x 10-6

€

m2s

Vapor pressure at 10oC pv= 1,230 Paabs

Vapor pressure at 20oC pv= 2,340 Paabs

Vapor pressure at 100oC pv= 101,300 Paabs

Dr. Maillacheruvu

2

3. SELECTED STANDARD CONSTANTS

Standard constants

g = 9.81 m

s2 = 32.2 ft

s2 α = Adiabatic lapse rate

5.87 x 10-3 oK

km = 3.221 x 10

-3 oRft

k = Boltzmann's constant

1.38 x 10-16

€

ergoK

4. GAS LAW EQUATIONS AND GAS CONSTANT EQUATIONS Universal Gas Law PV = n RuT

P = pressure; V = volume; n = number of moles Ru = Univ. gas const

T = Absolute Temperature ONE mole of any gas occupies approximately 24.05 liters at

298oK and 1 atm.

Value of Ru

8.314 J

oK-mol

0.0821 L atm

g-mol oK

82.05 cm3 atm

g-mol oK

3.141 ft-lbf

slug-oR

1.99 cal

mol-oK

Gas Constant Unit Conversions

2.647 J

oK-mol = 1

ft-lbf

slug-oR

101.267 J

oK-mol = 1

L atm

g-mol oK

To get gas constant “R” for any gas as in p = ρRT

R =

€

RuMolecularWt.of Gas

5. SELECTED INTEGRALS, DERIVATIVES AND EXPRESSIONS

⌡⌠dXX = ln (X) + Constant

⌡⌠Co

C

dCC = ln {

CCo

} ⌡⎮⌠dX

X2 = 1X + Constant

⌡⎮⌠

A

Bdx

(1-x2) =

1B -

1A

⌡⎮⌠

Xo

XfdX

X2 = {1

Xo -

1Xf

} ⌡⌠ro

r

r dr = {r2

2 - ro

2

2 } ea

eb = e(a-b) ln(en) = n

6. ATOMIC NUMBER AND ATOMIC WEIGHTS OF ELEMENTS

Element Symbol At. No. Mol. Wt. Element Symbol At. No. Mol. Wt. Actinium Ac 89 227.0 Molybdenum Mo 42 95.9 Aluminium (Aluminum) Al 13 27.0 Neodymium Nd 60 144.2 Americium Am 95 243.0 Neon Ne 10 20.2 Antimony (Stibium) Sb 51 121.8 Neptunium Np 93 237.0 Argon Ar 18 39.9 Nickel Ni 28 58.7 Arsenic As 33 74.9 Niobium Nb 41 92.9 Astatine At 85 210.0 Nitrogen N 7 14.0 Barium Ba 56 137.3 Nobelium No 102 259.0 Berkelium Bk 97 247.0 Osmium Os 76 190.2 Beryllium Be 4 9.0 Oxygen O 8 16.0 Bismuth Bi 83 209.0 Palladium Pd 46 106.4 Bohrium Bh 107 264.0 Phosphorus P 15 31.0 Boron B 5 10.8 Platinum Pt 78 195.1 Bromine Br 35 79.9 Plutonium Pu 94 244.0 Cadmium Cd 48 112.4 Polonium Po 84 210.0 Caesium (Cesium) Cs 55 132.9 Potassium (Kalium) K 19 39.1 Calcium Ca 20 40.1 Praseodymium Pr 59 140.9 Californium Cf 98 251.0 Promethium Pm 61 145.0 Carbon C 6 12.0 Protactinium Pa 91 231.0 Cerium Ce 58 140.1 Radium Ra 88 226.0 Chlorine Cl 17 35.5 Radon Rn 86 220.0 Chromium Cr 24 52.0 Rhenium Re 75 186.2 Cobalt Co 27 58.9 Rhodium Rh 45 102.9 Copper (Cuprum) Cu 29 63.5 Roentgenium Rg 111 272.0

Dr. Maillacheruvu

3

Curium Cm 96 247.0 Rubidium Rb 37 85.5 Darmstadtium Ds 110 271.0 Ruthenium Ru 44 101.1 Dubnium Db 105 262.0 Rutherfordium Rf 104 261.0 Dysprosium Dy 66 162.5 Samarium Sm 62 150.4 Einsteinium Es 99 2521.0 Scandium Sc 21 45.0 Erbium Er 68 167.3 Seaborgium Sg 106 266.0 Europium Eu 63 152.0 Selenium Se 34 79.0 Fermium Fm 100 257.0 Silicon Si 14 28.1 Fluorine F 9 19.0 Silver (Argentum) Ag 47 107.9 Francium Fr 87 223.0 Sodium (Natrium) Na 11 23.0 Gadolinium Gd 64 157.3 Strontium Sr 38 87.6 Gallium Ga 31 69.7 Sulfur S 16 32.1 Germanium Ge 32 72.6 Tantalum Ta 73 180.9 Gold (Aurum) Au 79 197.0 Technetium Tc 43 98.0 Hafnium Hf 72 178.5 Tellurium Te 52 127.6 Hassium Hs 108 277.0 Terbium Tb 65 158.9 Helium He 2 4.0 Thallium Tl 81 204.4 Holmium Ho 67 164.9 Thorium Th 90 232.0 Hydrogen H 1 1.0 Thulium Tm 69 168.9 Indium In 49 114.8 Tin (Stannum) Sn 50 118.7 Iodine I 53 126.9 Titanium Ti 22 47.9 Iridium Ir 77 192.2 Tungsten (Wolfram) W 74 183.8 Iron (Ferrum) Fe 26 55.8 Ununbium Uub 112 285.0 Krypton Kr 36 83.8 Ununhexium Uuh 116 292.0 Lanthanum La 57 138.9 Ununpentium Uup 115 288.0 Lawrencium Lr 103 262.0 Ununquadium Uuq 114 289.0 Lead (Plumbum) Pb 82 207.2 Ununtrium Uut 113 284.0 Lithium Li 3 6.9 Uranium U 92 238.0 Lutetium Lu 71 175.0 Vanadium V 23 50.9 Magnesium Mg 12 24.3 Xenon Xe 54 131.3 Manganese Mn 25 54.9 Ytterbium Yb 70 173.0 Meitnerium Mt 109 268.0 Yttrium Y 39 88.9 Mendelevium Md 101 258.0 Zinc Zn 30 65.4 Mercury (Hydrargyrum) Hg 80 200.6 Zirconium Zr 40 91.2

1. In an oxidation-reduction reaction, the equivalent weight of a compound/element will depend on the electrons transferred

2. Molecular weight is commonly expressed as

€

gramgmole

3. Equivalent weight is commonly expressed as

€

gramequivalent

4. When chemical concentrations are expressed in units of mg/L as CaCO3, to convert the “mg/L as CaCO3” to units of milli-equivalents per liter (meq/L), divide by 50 (1 meq/L ≈ 50 mg/L as CaCO3)

8. COMMON VALENCE STATES OF SELECTED ELEMENTS Element & Form Oxidation State(s) Element & Form Oxidation State(s) H+ +1 H2 0

O2- (in combined form) -2 O2 0

Ca++, Mg++ +2 Ca, Mg 0

K+, Na+ +1 K, Na 0 Mn +7, +6, +4, +3, +2, 0 C +4, +3, +2, +1, 0, -1, -2, -3, -4 Fe +3, +2, 0 N +5, +4, +3, +2, +1, 0, -3 Cu +2, +1, 0 P +5, +4, +3, 0, -3 Cr +6, +3, 0 S +6, +4, +2, 0, -2 Ag +1, 0 Cl +7, +5, +4, +3, +1, 0, -1 Hg +2, +1, 0 I +7, +5, +1, 0, -1 Br +5, +1, 0, -1 1. All gases in native (molecular) state have an oxidation state of 0 (e.g. such as O2, H2, N2, Cl2) 2. All elements in native form have an oxidation state of 0 (e.g. Mn, Fe, Cu, Cr, Ag, Hg, C, N, P, S) 3. The oxidation of an element “releases” electrons and the valence state of the oxidized element increases by number of electrons

transferred (e.g. when hydrogen (in H2 form) is oxidized to H+”, the oxidation change of hydrogen increases from 0 to +1)

Dr. Maillacheruvu

4

9. CARBONATE CHEMISTRY

Ka1 = [HCO3-][H+]

[H2CO3*] Ka1 = 10-6.3

Ka2 = [CO32-][H+]

[HCO3-] Ka2 = 10-10.3

CTCO3 = [HCO3-] + [CO32-] + [H2CO3-]

Alkalinity ≈ [HCO3-] + [CO32-] + [OH-] - [H+] -------- all in meq/L (or in mg/L as CaCO3) 10. REACTOR ENGINEERING EQUATIONS

τ = θ = VQ =

CAo - CAf(-rA)f

(rA)f = reaction rate for substance "A" at the endpoint

CA = concentration of substance "A" in the reactor at any time

CAo = concentration of substance "A" in the reactor at time t = 0

CAf = concentration of substance "A" in the reactor at end

θ = Hydraulic residence time = Volume of reactor "V"

Flow rate "Q"

Cn(t) = Co

(n-1)! (t/τ)n-1 e

[t/τ]n! = n(n-1)(n-2)...!

CssCin

= 1

{1 + kτn }

n

CCin

= 1

{1 + kτ} C

Cin =

1

{1 + kτn }n

CCss

= (1 - e-(1/τ +k)t)

C = Coe-kτ CssCin

= 1

{1 + kτ} C

Css = (1 - e-t/τ)

Accumulation = Input - Output - Decrease due to consumption reaction + Increase due to a production reaction ∂C∂t = - Advective flux + Disperive flux ± Reaction flux

Pe = DuL or

EuL Da =

kLu

_______________________________________________________________________________________

C = Co2 exp

(ux - v)x2Dx erfc{

Rx - vt2 DxRt

} + Co2 exp

(ux + v)x2Dx erfc{

Rx + vt2 DxRt

}

where

erfc (y) = 1 - 2π

∑n = 0

∞(-1)n y2n+1

n! (2n+1) Dx = αux + D* v = ux 1 + 4kDxux2

R = retardation factor, dimensionless t = time, T Dx = longitudinal dispersion coefficient, LT-2 k = first-order degradation rate constant, T-1

D* = molecular diffusion coefficient, LT-2 α = dispersivity, L ux = longitudinal actual velocity of water, LT-1 _______________________________________________________________________________________

Dr. Maillacheruvu

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11. SOME NUMERICAL METHODS Euler method:

Ci+1 = Ci + [W(t)

V - λCi][ti+1 - ti]

where, Ci = Old value of concentration (at time step = ti); Ci+1 = New value of concentration (at time step = ti+1); [ti+1 - ti] = step size = h

Heun method:

Ci+1 = Ci + [W(t)

V - λCi][ti+1 - ti]

where, Ci = Old value of concentration (at time step = ti) Ci+1 = New value of concentration (at time step = ti+1) [ti+1 - ti] = step size = h

Predictor for initial value of concentration at the next time step (Ci+1o)

Ci+1o = Ci + f(ti,Ci)h

Corrector for "actual" value of concentration at the next time step (Ci+1)

Ci+1 = Ci + f(ti,Ci) + f(ti+1,Ci+1)

2 h

Runge-Kutta 4th order method:

Ci+1 = Ci + [16 {k1 - 2 k2 + 2k3 + k4}]h

k1 = f(ti, Ci)

k2 = f{ti + 12 h, Ci +

12 hk1}

k3 = f{ti + 12 h, Ci +

12 hk2}

k4 = f{ti + h, Ci + hk3}

where, h is the step size or time interval; Ci is the value of the concentration at ith interval;

Ci+1 is the predicted value of the concentration at the (i+1)th interval

______________________________________________________________________________ 12. PRESENT WORTH CALCULATIONS ______________________________________________________________________________

Present worth factor = [(1 + i)n - 1]

i[1 + i]n

PW = F

(1 + i)n = A x PWF = A

[(1 + i)n - 1]i[1 + i]n

where “i” is expressed as a decimal not %

______________________________________________________________________________

Dr. Maillacheruvu

6

13. WATER QUALITY MODELING AND STREETER-PHELPS EQUATIONS BOD AND STREAM WATER QUALITY EQUATIONS

BODt = L0 (1 - e-kt)

L = L0e-k1(x/U)

BODt = BOD exerted at any time “t” (when t = 5, it is called BOD5) --- units are mg/L

BODu = ultimate BOD--- units are mg/L; L = ultimate BOD at any point in the stream at a distance of “x” from the point of waste

discharge; L0 = ultimate BOD at any point in the stream at a distance of “x” from the point of waste discharge

k1 = BOD rate constant (day-1); k2 = stream reaeration rate constant (day-1);

xU( ) is the travel time t; x is distance downstream; U is stream speed or stream velocity

D = k1L0k2 −k1

"

#$$

%

&'' [exp −

k1xU

"

#$

%

&' - exp −

k2xU

"

#$

%

&' ] + Doexp −

k2xU

"

#$

%

&'

Dc = k1L0k2!

"#

$

%& exp −

k1xU

"

#$

%

&'

Cmin = Cs - D

tc = 1k2 −k1"

#$

%

&' ln{ k2

k1!

"#

$

%& [1 -

D0 k2 − k1( )k1L0

"

#$

%

&' ]}

xc = U 1k2 −k1"

#$

%

&' ln { k2

k1!

"#

$

%& [1 -

D0 k2 − k1( )k1L0

"

#$

%

&' ]}

D = DO deficit = (Cs - C)

Cs = saturation dissolved oxygen conc. (mg/L) at the temperature of the stream; C = conc. of dissolved oxygen at any point (mg/L); Do is

the initial DO deficit (at point of waste discharge) = (Cs - Co)

14. DIMENSIONLESS NUMBERS IN WATER QUALITY MODELING

1. Dispersion number "d” = D

UL or E

UL

2. Peclet number "Pe" = ULD or

ULE

3. Damkohler number = Da = kLUL

4. Estuary number = η = kE

U2

5. Reynolds Number, NRe = U*Diameter*ρ

µ = U*Diameter

ν

15. NON IDEAL PLUG FLOW MODEL 1. Point Source Load

The general solution is of the form: C = F*exp(λ

1x) + G*exp(λ

2x)

C = WQ

1

[1 + 4η] exp{

U x2E (1 + [1 + 4η] }x ≤ 0

C = WQ

1

[1 + 4η] exp{

U x2E (1 - [1 + 4η] x}x ≥ 0

η = kE

U2 is the estuary number

Dr. Maillacheruvu

7

2. Non-Point Source (or Distributed Load) a. x ≤ 0

C = {Sdk }

[1 + 4η] - 1

2 [1 + 4η] ) (1 - exp{

-U a2E (1 + [1 + 4η] }) (exp{

U x2E (1 + [1 + 4η] })

b. 0 ≤ x ≤ a

C = {Sdk }{1 -

[1 + 4η] - 1

2 [1 + 4η] ) exp[

U(x-a)2E (1 + [1 + 4η]] -

[1 + 4η] + 1

2 [1 + 4η] exp{

U(x-a)2E (1 - [1 + 4η] })

c. x ≥ a

C = {Sdk }

[1 + 4η] + 1

2 [1 + 4η] ) (1 - exp{

-Ua2E (1 - [1 + 4η] }) (exp{

U(x - a)2E (1 - [1 + 4η] })

η = kE

U2 is the estuary number

16. ANALYTICAL SOLUTIONS FOR VARIOUS CASES Instantaneous Spill No mixing

∂C∂t = - U

∂C∂x - kC

C(x or t) = Co exp[- kxU ] = Co exp[- kt]

Instantaneous Spill with Dispersion

∂C∂t = E

∂2C

∂x2 - U ∂C∂x - kC

C(x,t) = mp

2 πET exp[-

k[x - Ut]2

4Et - kt]

where mp is the mass of the spike (or impulse input or instantaneous spill) for a case where the mass is initially concentrated at x =

0 Continuous Spill with Dispersion

∂C∂t = E

∂2C

∂x2 - U ∂C∂x - kC

C = Co2 exp

Ux2E [1 - Γ] erfc{

x - UtΓ2 Et

} + Co2 exp

Ux2E [1 + Γ] erfc{

x + UtΓ2 Et

}

where Γ = 1 + 4η , η = kE

U2 and erfc(y) is the complementary error function for "y"

Spill that occurs for a finite time interval "τ" with Dispersion

∂C∂t = E

∂2C

∂x2 - U ∂C∂x - kC

C = Co2 exp

Ux2E [1 - Γ] [erfc{

x - UtΓ2 Et

} - erfc{x - U(t - τ)Γ

2 E(t - τ) } ]

+ Co2 exp

Ux2E [1 + Γ] [erfc{

x + UtΓ2 Et

} - erfc{x + U(t - τ)Γ

2 E(t - τ) }]

where Γ = 1 + 4η , η = kE

U2 is the estuary number, and erfc(y) is the complementary error function for "y"

Dr. Maillacheruvu

8

17. NUMERICAL DISCRETIZATION SCHEMES IN WATER QUALITY MODELING 1. Discretization scheme for the general equation in water quality modeling:

∂C∂t = E

∂2C

∂x2

2. Explicit forward time centered space method (FTCS)

∂C∂t =

Cl+1i - C

li

∆t

∂2C

∂x2 = Cl

i+1 - 2Cli + Cl

i-1

∆x2

Cil+1 = Ci

l + E { Cl

i+1 - 2Cli + Cl

i-1

∆x2 } ∆t

3. Criterion for limitation on value for time step to obtain stable solutions

∆t < 12 ∆x2

E

4. Explicit Solution for control-volume approach using weighted differences:

VdCdt = W(t) - QinCin- QoutC - kVC

V{Cl+1

i - Cli

∆t } = Wil + Qi-1,i{αi-1,iC

li-1 - ßi-1,iC

li} - Qi,i+1{αi,i+1Cl

i - ßi,i+1Cli+1} + E'

i-1,i(Cli-1 - Cl

i) + E'i,i+1(Cl

i+1

- Cli) - kiViC

li

Cl+1i =

Wil

Vi ∆t +

∆tVi

{- Qi-1,iαi-1,i - E'i-1,i} Cl

i-1 + {1 - ∆tVi

[- Qi-1,ißi-1,i + Qi,i+1ßi,i+1 + E'i-1,i + E'

i,i+1 + kiVi]} Cl

- ∆tVi

[Qi,i+1ßi,i+1 - E'i,i+1] Cl

i+1

∆t < ∆xU

5. Numerical Dispersion

En = {U ∆x [α - 1]} - {U2 ∆t

2 }

6. Simple Implicit or backward-time centered-difference approximation (BTCS):

For centered difference approximation, α = β = 0.5 and the expression simplifies to:

Cli +

Wil+1

Vi ∆t = {-

U2 ∆t∆x -

E ∆t

(∆x)2 } Cl+1

i-1 + {1 + 2E ∆t

(∆x)2 + ki∆t} Cl+1

i

+ {U∆t2∆x +

E ∆t

(∆x)2 } Cl+1

i+1

Dr. Maillacheruvu

9

18. WATER QUALITY MODELING IN RIVERS AND STREAMS 1. Formulation of water-quality models follows the same general pattern as that for hydrologic models. 2. Both types of models include continuity considerations combined with equations of motion and reaction. 3. Important factors include: estimation of stream hydrology and geometry (bathymetry sometimes), point estimates for velocity and

depth, Gauss quadrature basis for selection of sampling points, and estimation of stream reach. 4. Low-flow conditions (7Q10) are usually a basis for design.

Cumulative probability of occurrence = p = m

N +1 where m is the rank

Recurrence interval = T = 1p

5. Dispersion and Mixing: Estimation of lateral and longitudinal dispersion coefficients

Longitudinal dispersion: Fisher (1979):

Elong = 0.011 U2B2

HU*

U* = shear velocity = gHS McQuivey and Keefer (1974):

Elong = 0.05937 Q

SB

Lateral dispersion:

Side discharge vs. central discharge

Elat = 0.60 HU*

6. Flow depth and velocity calculations: Manning's equation. 19. WATER QUALITY MODELING IN ESTUARIES Net flow in estuaries can be computed using the following equation.

Qnet = 2π

qeTe - qfTf

Te+Tf

DO and BOD in Streams and Rivers

[dDdx ] -

kdLoe-kr(

x/U)

U + kaD

U = 0

D =

€

kd Loka− kr{exp(-k rxU ) − exp(

-kaxU )}+ Do

€

Do{exp(-k axU )}

Dr. Maillacheruvu

10

20. NEAR-SHORE MODELS FOR LAKES AND IMPOUNDMENTS The general equation for advective dispersive 2-dimensional model for a lake receiving a point-source input which decays at a rate of "k" is:

∂C∂t = Ex

∂2C

∂x2 + Ey ∂2C

∂y2 - Ux ∂C∂x - Uy

∂C∂y - kC

These models are useful if we want to analyze what happens due to discharge of waste into lakes, particularly bacterial counts in lakes due to specific point discharges (which may then lead to closing of beaches etc.). Selected Cases for Lakes and Impoundments 1. Steady-state case in an infinite fluid with no advection. Further, we assume that dispersion is same in all directions.

CCo

= Ko{

kr2

E }

Ko{kro

2

E }

where Ko{y}is the modified Bessel function of the second kind (Appendix F, Table F-2 on p. 819)

For a boundary condition at r = 0, C (0) = WH

C = WπHE Ko{

kr2

E }

2. Steady-state case in an infinite fluid with advection in one direction ("x" direction). Further, we assume that dispersion is same in

all directions.

C = W {exp[

Ux*x

2E ]}

πHE Ko{r kE + [

Ux2E]

2 }

3. Steady-state case in a bounded fluid with advection in one direction ("x" direction). Further, we assume that dispersion is same in

all directions.

C = W {exp[

Ux*x

2E ]}

πHE Ko{r kE + [

Ux2E]

2 }

Dr. Maillacheruvu

11

21. STREETER PHELPS MODELS

Streeter-Phelps model for point sources

kd20 in day-1 = 0.3[

8H ] for depth ≤ 8 ft

kd20 in day-1 = 0.3 for depth ≥ 8 ft

kdT = kd

20 x 1.024(T - 20)

ka = DLU

H3

DL = diffusivity of oxygen in water (0.000081 ft2

hour at 20oC)

U = velocity of stream, ft

hour

H = depth of stream, feet

ka,T = ka,20 x 1.047(T - 20)

D = D0e-(kax/U) + kdL0

ka - kr {e -(kr

x/U) - e -(kax/U))}

Dc = kdka

L0 e - krtc

tc = 1

ka - kr ln{

kakr

[1 - D0(ka - kr)

kdL0 ]}

Streeter-Phelps model for point & distributed sources, photosynthesis, respiration, & SOD

L = L0 exp(-krt) + SLkr

[1 - exp(-krt)]

D = D0 exp(-krt) + kdLo

ka - kr [e -(kr

t) - e -(kat)] + - P + R +

SB'

Hka

[1 - exp(-kat)]

+ kdSLkrka

[1 - exp(-kat)] - kdSL

kr[ka - kr] [exp(-krt) - exp(-kat)]

Streeter-Phelps model for point sources in an advective-dispersive system

L = L0exp(UX2E [1 ± α1])

D = kdW

[ka - kd]Q {1α1

(exp{UX2E [1 ± α1]} -

1α2

(exp{UX2E [1 ± α2]}}

α1 = 1 + 4krE

U2 α2 = 1 + 4kaE

U2 L0 = 1α1

WQ

Dr. Maillacheruvu

12

22. TOXICS AND PARTITIONING

C = FdC + FpC

Fd = 1

1 + Kdm Fp = Kdm

1 + Kdm

where m = suspended solids concentration and Kd = partition coefficient

V1

dC1dt = QCin

- QC1 - V1k1C1

- AvvFd1C1 - AvsFp1C1 + vrAC2

+ AvdFd2C2 - AvdFd1C1

V2

dC2dt = -V2k2C2

+ AvvFp1C1 - vrAC2

- AvbC2 + AvdFd1C1 - AvdFd2C2

C2 = {VsFp1 + VdFd1

Vr + k2H2 + Vb + VbFd2 } C1

C1 = ß Cin

ß = Q

{Q + V1k1 + AvvFd1 + (1 - Fr')A{vsFp1 + AvdFd1}}

Fr' = vr + vdFd2

vr + vb + vbFd2 + k2H2

Fd1 = 1

1 + Kd1[mass of suspended solids] Fp1 = 1 - Fd1

Fd2 = 1

φ + Kd2(1 - φ)ρ Fp2 = 1 - Fd2

If sediment feedback is almost zero (Fr' ≈ 0),

ß = Q

{Q + V1k1 + AvvFd1 + AvsFp1}

If sediment feedback is overwhelming (Fr' ≈ 1),

ß = Q

{Q + V1k1 + AvvFd1}