SURFACE WATER PATHWAY - DEUkisi.deu.edu.tr/orhan.gunduz/turkce/diger/8_SW_Pollution.pdf ·...

7_Surface Water Pollution

SURFACE WATER PATHWAY

Mustafa M. Aral

MESL @CEE,GThttp://mesl.ce.gatech.edu/[email protected]


Surface Water Regulations

For centuries, fecal waste and other pollutants were dumped in rivers, with “dilution is considered to be the solution to pollution.”

In the mid-twentieth century, many rivers in USA and streams were open sewers, choking on everything from human waste to highly toxic industrial discharges.


Surface Water Regulations: 1

� Public Health Service Act of 1912

� U.S. first water-quality standards created


Surface Water Regulations:

� Clean Water Act (CWA) of 1972� Jurisdiction over water quality in rivers,

lakes, estuaries, and wetlands

� Regulations for wastewater effluents

� Goals of “fishable and swimmable” waters throughout the U.S, by 1983

� By late-1990s, 40 percent of surface water in U.S. not suitable for fishing, swimming, or other designated uses


� Safe Drinking Water Act (SWDA) of 1974

� Regulates tap water quality

� All community water systems serving 15 or more outlets or 25 or more customers

� EPA required to set national standards for drinking water quality

� Safe Drinking Water Act (SWDA) of 1986

� Use of “lead-free” pipe, solder, flux




� National Pollutant Discharge Elimination System (NPDES)

� Permit issued under Clean Water Act

� Requires discharger to meet certain technology-based effluent limits and perform effluent monitoring


� 1990s: Water Quality Inventory found almost 40 percent of U.S. rivers and 45 percent of lakes are polluted.

� More than 95 percent of water tested near four population centers in Great Lakes between 2001 and 2002 contained unsafe levels of mercury and pesticides according to a National Wildlife Federation report.


Since late 1970s

� Billions of dollars have been invested since 1972 building and upgrading sewage treatment facilities.

� Nationally, more than thirty thousand major industrial dischargers pre-treat their wastewater before it enters local sewers.

� By 2000, some 75 percent of toxic discharges, including heavy metals and PCBs, were being prevented.


Drinking-Water Standards

� Primary Standards� Specify Maximum Contaminant Levels (MCLs) based on

health-related criteria

� Secondary Standards� Unenforceable guidelines based on aesthetics (taste, odor,

color) and non-aesthetics (corrosivity, hardness)

� Maximum Contaminant Level Goals (MCLGs)� Unenforceable goals set at levels that present no known or

anticipated health effects, regardless of technological feasibility or cost


Open Channels Review

Characterized by flows with a free surface:a. Simplification � P = 0 at the surfaceb. Complication � Free surface configuration is not known

Datumz1

y1Pz

γ+

2

1

2

V

g

oS

1

EL

2

2

2

V

g

y2

z2

Lh

HGL


FLOW CLASSIFICATIONS

Laminar FlowTurbulent Flow

Uniform FlowVaried Flow

Gradually varied FlowRapidly Varied Flow

Sub-Critical FlowSuper-Critical Flow

µ

ε

→

→

,y P→

,V c→


GVFRVF GVF Uniform GVF GVFRVF

Uniform Flow: Weight Forces are balanced with shear forces, pressure distribution is hydrostatic,

depth is constant, EL� HGL� Channel slope parallel

c

v

ypFFp


UNIFORM FLOW EQUATIONS:

1F

2F

W

ny

wFτ

sinW θ

θ

1 2

0F M a

F F

= =

=

∑��

sin

sin

w w

Lo

F PL

W AL

hS

L

τ τ

γ θ

θ

=

=

= =

L

ny

AP


1 2sin 0

sin

w

Lw o h o h

F W F PL

AL A hS R S R

PL P L

θ τ

γ θτ γ γ γ

+ − − =

= = = =

( )

( ) ( )

2

2

2

1/ 2

1/ 2 1/ 2

8

8

8 8

8

w

Lw h

Lh h o

h o

fV

f V hR

L

g h gV R R S

f L f

gV R S

f

τ

ρ

ρτ γ

=

= =

= =

=

1F

2F

W

ny

wFτ

sinW θ

θ

L

( ) ( )

1/ 2

1/ 2 1/ 2

8Chezy constant

h o

gC

f

V C R S

=

=

( ) ( )

( ) ( )

1/6 1/6

2/3 1/2

2/3 1/2

1.49 1

1.49

1.0

h h

h o

h o

C R or Rn n

V R Sn

V R Sn

= = =

=

=

BU

SI

Mannings eqn


Hydraulic Radius Plays an important role in open channel computations:hR

2h

A ByR

P y B= =

+B

y

Constant B � Deep Channels

0 0

2

h

h

y R

By R

= =

→ ∞ =22

h

B BR

y B

y y

= =

+

y

hR

2

B


Hydraulic Radius Plays an important role in open channel computations:hR

2h

A ByR

P y B= =

+B

y

Constant Depth y � Wide Channels

0 0h

h

B R

B R y

= =

→ ∞ = 2h

yR y

y B

B B

= =

+

B

hR

y


Shear Velocity:*

u

L

V

wτ

*u

*u

*u

*uB

yz

V(z)

y*

u

max Largest eddy sized y=


The eddies cause vertical / lateral or longitudinal diffusion which proceed at a rate given by vertical/lateral diffusivity proportional to the product of the orbital velocity u* and the maximum length scale of an eddy

*

verticalD u y=

But what is u*?

L

V

wτ

*u

*u

*u

*uB

yMass Acceleration Force× =

*

w

uLBy LBρ τ

τ× =

*

yeddy turn around time

uτ ≈ −

( )2

*

wuρ τ=


L

V

wτ

*u

*u

*u

*uB

y

( )2

*

wuρ τ=

* wu friction velocityτ

ρ=

Lw h o h

hR S R

Lτ γ γ= =

* h oh o

R Su gR S

γ

ρ= =


2

-9 2

2

9

(units)

Molecular diffusion coefficient of salt (NaCL) = 1.5x10 m /s

Given a cup of depth 5cm how long will it take

for diffusive concentration to reach the top of the cup:

0.0519.29

1.5 10

LD

t

t dayx

−

=

= = s

DEFINITION OF DIFFUSION COEFFICIENT:

This is not going to be useful for environmental dilution


VERTICAL DIFFUSION:

( )*

verticalD f u y=

*0.067 0.067vertical h o

D u y y gR S= =

Empirical constant � 0.067 ~ 0.05 – 0.07

Remember the mechanical mixing molecular mixing concept in GW applications

Mechanical mixing component due to turbulence

0.1H p m p m

D V D LV Dα= + = +


For a contaminant discharge at mid depth

*0.067vertical

D u y=

( )

2

**

*

1

0.2684 0.067

3.73

v

v

y y

uu y

y

u

τ

τ

= =

=

*3.73

v v

yx V V

uτ= =

xv

VERTICAL DIFFUSION:

y/2

z

x

2

1

v

y

τ=

2

4 v

y

τ=


For a contaminant discharge at the base

*0.067vertical

D u y=

( )

2

**

*

1

0.0670.067

14.9

v

v

y y

uu y

y

u

τ

τ

= =

=

*14.9

v v

yx V V

uτ= =

VERTICAL DIFFUSION:

xv

y

z

x

2

v

y

τ=


TRANSVERSE DIFFUSION:

( )*

transverseD f u y=

*0.15 0.15transverse h o

D u y y gR S= =

Empirical � 0.15 ~ 0.1 – 0.4

For a contaminant discharge on one side:

B

Width

x

xtransverse

Plan view


TRANSVERSE DIFFUSION: For a contaminant discharge on one side:

*0.15transverseD u y=

2

*6.66

t

B

u yτ =

2

*6.66t t

Bx V V

u yτ= =

B

B

x

xtransverse

B

2

t

B

τ=


Example: An industrial plant discharges a conservative substance at a point along the side of a river which is 2m deep 500m wide. River slope 0.02%. What is the down stream distance the pollutant starts to affect the other bank.

B

Width

x

xtransverse

B


for annings n 0.035 0.61 /M V m s= =

( ) ( ) ( )* 49.81 1.85 2 10 0.06 /h ou gR S m s−= = × =

2 501.85

2 50 2h

AR m

P

×= = =

+ +

Example: An industrial plant discharges a conservative substanceat a point along the side of a river which is 2m deep 50m wide.River slope 0.02%. What is the down stream distance the pollutant starts to affect the other bank.

( ) ( )2 / 3 1/ 21.0

h oV R Sn

=

( ) ( )* 20.15 0.15 0.06 2 0.018 /transverse

D u y m s= = =

( )

2 2

*

506.66 6.66 138750 38.54 1.6

2 0.06t

Bs days days

u yτ = = = = =

( )138750 0.61 / 84637.5 84.6transverse

x s m s m km= = =

B

B

x

xtransverse

B


LONGITUDINAL DIFFUSION:

( )*

longitudinalD f u y=

*5.93 5.93longitudinal h o

D u y y gR S= =

Shallow and wide rivers

Deep and narrow rivers where bank shear is important

2 2

*0.011longitudinal

V BD

u y=

Actually there is more…


SUMMARY:

( ) ( )

( ) ( )

2 / 3 1/ 2

2 / 3 1/ 2

1.49

1.0

h o

h o

V R Sn

V R Sn

=

=

Mannings Eqn:

( )*D f u y=

*

h ou gR S=

*

yeddy turn around time

uτ ≈ −

h

AR

P=

ny

AP


*0.067vertical

D u y=

*0.15transverseD u y=

*5.93longitudinalD u y=

* h oh o

R Su gR S

γ

ρ= =y = flow depth

Empirical � 0.15 ~ 0.1 – 0.4

Empirical � 0.067 ~ 0.05 – 0.07

2 2

*0.011longitudinal

V BD

u y=

Shallow and wide rivers Deep and narrow rivers where bank shear is important

SUMMARY:


*

*

0.152

0.067

transverse

vertical

D u y

D u y= ≈

*

*

5.9390

0.067

longitudinal

vertical

D u y

D u y= ≈

*

*

5.9340

0.15

longitudinal

transverse

D u y

D u y= ≈

SUMMARY:

0.01vertical longitudinal

D D=

0.025transverse longitudinalD D=


Stages of Model Complexity:

1.CSTR models

2.Turbulent Mixing analysis

3.Plug flow models (no Turbulent Mixing)

4.Empirical Models: Near Field, Far Field Mixing models.

5.Advection + diffusion models (Conservation of Mass).

6.Advection + diffusion + reaction models (Conservation of Mass).


Mass balance models based on a PLUG-FLOW SYSTEM: (no Turb. diffusion)

ReactionsAccumulation Inputs Outputs= − ±

ny

θ

QC ( )Q C C+ ∆

x∆

d V( )( )

CQC Q C C RV

dt= − + ∆ ±

Constant flowsConstant x-section area A

V A x= ∆

( )Q C CdC QCR

dt A x A x

+ ∆= − ±

∆ ∆

( ) ( )Q C Q CdC QCR

dt A x A x A x

∆= − − ±

∆ ∆ ∆


ny

θ

QC ( )Q C C+ ∆

x∆

( ) ( )Q C Q CdC QCR

dt A x A x A x

∆= − − ±

∆ ∆ ∆

( )Q CC CR V R

t A x x

C CV R

t x

∆∂ ∂= − ± = − ±

∂ ∆ ∂

∂ ∂+ = ±

∂ ∂Time dependent

CV R

x

∂= ±

∂Steady state


CV R

x

∂= ±

∂Steady state

if reaction is first order decay dC R k

C R kCdx V V

= ± = − = −

0

0

ln

o

o

C x

C

xC

C

dC kdx

C V

dC kdx

C V

kxC

V

= −

= −

= −

∫ ∫( )

ln

exp /

o

o

C kx

C V

C C kx V

−=

= −


Variable flows and Variable x-section area A

( )

( )

exp

exp

o

o

Q Q qx

A A ax

=

=,

o oA Q

d V( )( )

CQC Q C C RV

dt= − + ∆ ±

d V( )( ) ( )

CQC Q Q C C RV

dt= − + ∆ + ∆ ±

Previous model

d VC V

dt+ ( ) ( )

dCQC Q Q C C RV

dt= − + ∆ + ∆ ±

V ( ) ( )dC

QC Q Q C C RVdt

= − + ∆ + ∆ ±

Volume is not changing as a function of time


V ( ) ( )dC

QC Q Q C C RVdt

= − + ∆ + ∆ ±

VdC

QC QC C Q Q C Q C RVdt

= − − ∆ − ∆ − ∆ ∆ ±

dC C Q Q C Q CR

dt A x A x A x

∆ ∆ ∆ ∆= − − − ±

∆ ∆ ∆

C C Q Q C Q CR

t A x A x A x

∂ ∂ ∂ ∂ ∂= − − − ±

∂ ∂ ∂ ∂

Product of two changesAre small relative to other terms

( )1 CQCR

t A x

∂∂= − ±

∂ ∂Q,A and R can be variable


( )1 CQCR

t A x

∂∂+ = ±

∂ ∂Time dependent

Steady State( )1 d CQ

RA dx

= −

( )

( )

( )

exp

exp

exp

o

o

o

Q Q qx

A A ax

k k Rx

=

=

= −

( )( )

( )exp

a q xo oo

o o

RA kAC C qx e

a q Q a q Q

− = − − +

− −

Steady State solution:


2

2

C C CV D kC

t x x

∂ ∂ ∂+ = −

∂ ∂ ∂Time dependent with mixing

We will not discuss the solutions of this differential equation!! May get too complicated.

ACTS uses 1D, 2D and 3D solutions of this model for very complex cases.

Advection and Mixing in rivers (Conservation of Mass based):


LAKES AND RESERVOIRS:

As opposed to rivers: Low advective velocity

Thus they impound water for long time.

An important parameter to consider is the Residency Time (or retention time).

Retention time:

Average time spent for a particle in the lake from inflowto outflow.

r

Volt

Q=


A reservoir has surface area 107 m2 and average depth 30m,

Through flow rate to the lake is 5 m3/s

7730(10 )

6 10 sec 25

r

Volt yrs

Q= = = × =


ReactionsAccumulation Inputs Outputs= − ±

d V( )( )

CQC Q C C RV

dt= − + ∆ ±

Vol.

Q

Lakes:


V ( )C

Q C RVt

∂= − ∆ ±

∂

VC V

t

∂= −

∂( )

r

C kC Vt

−

VC

Vt

∂+

∂

1

r

C k Wt

+ =

Q � volume/time∆C � point concentration

( )W

C tVβ

= ( ) 11 t t

o

r

e C e kt

β β β− − − + = +

Constant W source input over time

Co = background conc.

Vol.

Q


( )W

C tVβ

= ( ) 11 t t

o

r

e C e kt

β β β− − − + = +

If the background concentration is negligible:

( )W

C tVβ

= ( ) 11 t

r

e kt

β β− − = +

Equilibrium concentration t→∞

( )W

C tVβ

= ∞ =1

r

kt

β

= +

If there is backgroundconcentration but no contaminant source input:

1( ) t

o

r

C t C e kt

β β− = = +

What if the contaminant is conservative? k = 0


Dispersion in a lake …… low velocity

2

2

C CD kC

t x

∂ ∂= −

∂ ∂

Again we will not spend time in discussing the solutions to this PDE!! ACTS uses these models and more complex cases.

We will define other models for lakes and reservoirs later.

Q

D=?

*u

( )*D f u y= But there are many other issues.


Empirical Turbulent Mixing Models for Surface Water applications


Mixing Processes and Applications in Surface Waters

� Advective and diffusive transport in water bodies and transport with sediment movement

� Intermedia transfer characterized by adsorption, desorption, precipitation, dissolution, and volatilization

� Degradation and decay

� Chemical transformation which may yield daughter products


Source Conditions:

� Direct discharge from point sources

� Dry and wet deposition from the atmosphere

� Runoff and soil erosion from land surfaces

� Seepage to and from groundwater


Classification of Empirical Models

� Near-field mixing

� Far-field mixing

� Sediment models


H

Near field mixing

2.5 H

Vr

Far Field Mixing

Uo

Diffuser


Vr

Pure Plume(buoyancy flux)

Buoyant jet or Forced Plume(buoyancy flux + momentum flux)

Pure Jet(momentum flux)

Near-Field Mixing Models

1_Prerequisites and introduction

ACTS:

1_Prerequisites and introduction

Surface Water:



� Surface point discharges� Stagnant water� Weak cross-currents

� Submerged point discharges� Stagnant water� Weak cross-currents

� Submerged multi-port diffusers� Deep receiving water� Shallow receiving water


Near-Field Mixing--continued

When the quantity of effluent is small and the receiving water

body is relatively large, rapid initial mixing by means of a

properly designed discharge structure is an effective process of

reducing the concentrations.

In some cases, it is the only feasible way to meet regulatory

requirements.

The near-field mixing process is based on a high level of

turbulence produced by means of the discharge momentum (jet

action) and/or discharge buoyancy (plume action).

Large dilutions on the order of 10 to 100 can be achieved.


Near-Field Mixing

H

hmax

2.5H

hmax to be computed


Characteristic Parameters

Density Deficit

oρρρ −=∆

where: ρo = ambient density, M/L3

ρ = density of discharging fluid, M/L3

∆ρ = density deficit, M/L3



oo

o

og

UF

�)/( ρρ∆=

Uo = is the (mean) discharge velocity (m/s) (effluent discharge),

ρρρρo = is the ambient density (kg/m3),

∆∆∆∆ρρρρ = is the discharge density deficit (kg/m3),

g = is the acceleration of gravity (m/s),

lo = is a characteristic length scale (m) of the discharge, which is related to its cross-

sectional area Ao by,

� o

oA=

2

Densimetric Froude Number, Fo



Example: A Rectangular Discharge Channel

oo bho=�

ho

2bo


Near-Field Mixing--continued

The contaminant concentration is computed by:

S

CC o=

Co is the initial concentration,

S is the dilution, and

C is the concentration at some point of interest, most likely

to be at the end of the near-field mixing zone.


Surface Point Discharges: 1

� Stagnant and Weak Cross-Currents� Deep receiving water

� Stagnant and Weak Cross-Currents� Shallow receiving water

� Strong cross-flow� Shoreline attached jets

� Zero or negative buoyancy


Stagnant and Weak Cross-Currents


H

hmax

hmax / H > 0.75Shallow Receiving water

hmax / H < 0.75Deep Receiving water

ooFlh 42.0max =



S F

x F

o

t o o

=

=

14

15

.

�

Deep Receiving Water

A deep receiving water condition exists when the

vertical extent of the buoyant jet is sufficiently less

than the existing water depth H

S = dilution factor; xt = transition distance



ooFlh 42.0max =


Maximum vertical penetration of the surface jet:

Occurs at an approximate distance of 5.5loFo from the outfall

H

hmax

5.5loFo



75.0max >H

h

Shallow Receiving Water

If the bottom of the water body affects the behavior of the

jet, the receiving water can be identified as a shallow water

body. Virtually most river outfalls can be grouped under

this category. A criterion for shallow water conditions

obtained from experimental and field data is:

where H is the depth a the point of maximum plume depth, hmax




An example:

hmax/H = 4/5 = 0.8 > 0.75

H = 5m

hmax = 4mH

hmax




SrS s='

An empirical correction factor, rs can be applied to the

deep-water equations for dilution to account for the

inhibiting effect of a shallow receiving water. Bulk dilution

under shallow water conditions S’ is estimated by:

The empirical factor, rs, is given by:

75.0

max

75.0

=

Hh

rs


Strong Cross Flows

For strong cross flows the plume maybe pinned to the shoreline which restricts mixing. Thus corrections to dilution factor must be implemented.

If:

3/ 2

max0.05

a

R

UR

U

hR

H

−

=

>

Than strong cross flow case should be used:

URUa

Surface Point Discharge: 9

Ua= Cross flow velocityUR = Longitudinal velocity


Strong Cross Flows

'1 1

2 2attached sS S r S= =

For Shallow water:

The extent of near field zone for this case is the smallest of:

2 15oc t o o

x or x FR

= =�

�



Zero or Negative Buoyancy

All previous models discussed are valid for buoyant discharges. Most jets of effluents are buoyant due to presence of density differences. If the jet does not exhibit density differences (∆ρ→0) than the dilution factor is calculated based on the geometry of the discharge vessel:

0.32x

SD

=

x = downstream distanceD = Diameter of the round discharge point




ACTS Interface



� Surface point discharges� Stagnant water� Weak cross-currents

� Submerged point discharges� Stagnant water� Weak cross-currents

� Submerged multi-port diffusers� Deep receiving water� Shallow receiving water


Stagnant Water

Weak Cross-Currents

Surface Point

Discharges

Stagnant Water

Weak Cross-Currents

Submerged Point

Discharges



Submerged Multiport

Discharges

Near Field Mixing Far Field Mixing Sediments

Pathway Models

Surface Water Pathway

Models

ACTS Surface Water Models


Submerged Point Discharges: 1

� Deep receiving water condition

� Buoyant jet rises to the surface and dilution occurs because of turbulent jet entrainment up to surface level

� Shallow receiving water condition

� Discharge momentum is sufficiently strong to cause a dynamic breakdown of the buoyant jet motion and to create a local circulation zone



Hz Stratified

Counter FlowDo

Uo

∇∇∇∇

H

z

Bouyant JetDo

Uo

∇∇∇∇Deep discharge with

buoyant jet

Shallow discharge with

circulation


Distinction between deep and shallow water conditions is important.

( / )

oo

o o

UF

gDρ ρ=

∆

0.22o

o

HF

D>

Do diameter of the outfall

Implies deep water condition



For deep water conditions buoyant jet models can be used.

5/ 3

2 / 30.11c o

o

zS F

D

− =

z is the distance above the nozzle to the water surfaceSc is the centerline dilution

Dilution at the boundaries of the flow field at z:

1.4 cS S=H

z

Bouyant JetDo

Uo

∇∇∇∇

θ



For shallow water conditions:

5/3

2/30.9c o

o

zS F

D

− =

Hz Stratified

Counter FlowDo

Uo

∇∇∇∇


Dilution at the boundaries of the flow field at z:

1.4 cS S=



ACTS Interface



� Surface point discharges� Stagnant water

� Weak cross-currents

� Submerged point discharges� Stagnant water

� Weak cross-currents

� Submerged multi-port diffusers� Deep receiving water

� Shallow receiving water


Stagnant Water

Weak Cross-Currents

Surface Point

Discharges

Stagnant Water

Weak Cross-Currents

Submerged Point

Discharges



Submerged Multiport

Discharges

Near Field Mixing Far Field Mixing Sediments

Pathway Models


Models

ACTS Surface Water Models


Multiport diffusers: 1

Typical behavior of wastewater discharged from an ocean outfall

Singh and Hager, 1966, Fig. 3.1


2

4

dB

s

π=

( / )

oo

o

UF

gBρ ρ=

∆

Diffuser details

Singh and Hager, 1966, Fig. 3.2


( )2

4/3 21.84 1 coso o

HF

Bθ> + Deep water condition


2 / 30.27 s

HS F

B

−=

0.58 a D

o

U L HS

Q=

2 / 30.44s

HS F

B

−=Stagnant conditions:

Ambient Cross flow:

Strong Cross flow:

Deep water condition



1/ 22

1 12

2 2

a D a D

o o

U L H U L H HS

Q Q B

= + +

0.67H

SB

=Stagnant conditions:

Ambient Cross flow:

Shallow water condition



Submerged Multiport Diffusers

ACTS Interface


General Classification of Models



� Sediment models


Far-Field Mixing Models

� Rivers

� Transverse mixing

� Longitudinal advection and dispersion

� Estuaries

� Small lakes and estuaries

� Oceans and great lakes


Rivers – Transverse Mixing and

Longitudinal Advection and Dispersion

Near Field Mixing

Transverse Mixing

Longitudinal

Advection and

Dispersion

Rivers Estuaries Small Lakes and

Reservoirs

Oceans and

Great Lakes

Far Field Mixing Sediments

Pathway Models


Models


Mixing in Rivers

Sewage

A

A

B C


Zones of Mixing in Rivers

� Zone A� Initial momentum and buoyancy of the discharge

is important – Mechanics of jets and plumes

� Zone B� Mixing is primarily due to river turbulence and

currents –Transverse mixing

� Zone C� Cross-sectional mixing is complete. Longitudinal

dispersion erases longitudinal gradients –Longitudinal dispersion


Transverse Mixing in Rivers: 1

x

yC(x,y)

σσσσy

Plume

boundary

U

Point

Source

U

W

C(x,y)

Line source

Line

Source



Two-Dimensional Contaminant Concentration

2

( , ) exp44

o o

yy

Q C y U xC x y

D x UH D Ux

λ

π

= − −

Co = initial concentration, M/L2

Qo = initial effluent flow rate, L3/T

x = longitudinal distance, L

y = transverse distance, L

H = depth of river, L

U = ambient river velocity, L/T

λ = radioactive decay (λ = ln2/T1/2), 1/T

Dy= transverse diffusion coefficient, L2/T

Qo and Co may represent variables determined by near-field mixing models

Assuming plume width is much less than river width,



2

( , ) exp44

o o

yy

Q C y U xC x y

D x UH D Ux

λ

π

= − −

Interpretation of two-dimensional contaminant

concentration equation

Variation over

width of river

Peak

Concentration



2

( , ) exp44

o o

yy

Q C y U xC x y

D x UH D Ux

λ

π

= − −

What is the expression for the concentration of a

conservative substance at any distance downstream of the

source, along the centerline of the river?

0

( , )4

o o

y

Q CC x y

H D Uxπ=

Example

0



Two-Dimensional Concentration--continued

*y yD u Hβ=

Dy= transverse diffusion coefficient, L2/T


u* = shear velocity, L/T

βy = 0.6 +/- 0.3



Values for ββββy

Description ββββy

Straight, uniform rectangular

channel 0.13 (0.1 0.2)

Natural rivers > 0.4

Slowly meandering rivers 0.4 – 0.8

Fischer (1979) recommends 0.6 +/- 0.3

*0.15transverse

D u y= Constant= 0.1 -0.4

ββββy = [Ky/(u*H)] Ky = transverse diffusion coefficient, L2/T



Two-Dimensional Concentration (wide rivers)

gHSu =*

u* = shear velocity, L/T

g = gravitational constant, L/T2


S = channel slope, L/L



Two-Dimensional Concentration--continued

2 /y yD x Uσ =

σy = standard deviation of the lateral Gaussian

concentration distribution, L

Ky = transverse diffusion coefficient, L2/T

x = longitudinal distance, L

U = ambient river velocity, L/T


Transverse Mixing in Rivers: 9Two-Dimensional Concentration--continued

2 2

2 1 1 2

21 2 1

( , ) exp

2 ( ) ( )1 2 exp sin cos cos

( )

o o

r

y

n r

Q C xC x y

Q U

n D x W y y y y yn n n

Q n y y W W W

λ

ππ π π

∞

=

= −

− + × + × −

∑

Whenever the initial source dimensions is significant and/or

the plume interacts with the river banks, the concentration

distribution is given by:

Qr = river flow rate = UHW, L3/T

UW

C(x,y)

Line source

Line

Source



ACTS Interface


Zones of Mixing in Rivers

� Zone A� Initial momentum and buoyancy of the discharge

is important – Mechanics of jets and plumes

� Zone B� Mixing is primarily due to river turbulence and

currents –Transverse mixing

� Zone C� Cross-sectional mixing is complete. Longitudinal

dispersion erases longitudinal gradients –Longitudinal dispersion


Longitudinal Advection and Dispersion in Rivers: 1

+−

+

=2

2

4112

exp

41

)(U

D

D

xU

U

DQ

QCxC L

LLR

oo λ

λ

x = downstream distance from point of release, L

DL = longitudinal dispersion coefficient, L2/t

Concentration for Steady-State Release



Longitudinal Dispersion Coefficient

According to Fischer et al. (1979), DL can be computed according to:

2 2

*

0.01L

U WD

Hu=

*5.93longitudinal

D u y=2 2

*0.011longitudinal

V BD

u y=

Shallow and wide rivers Deep and narrow rivers where bank shear is important



−

−−= t

tD

Utx

tDWH

MtxC

LL

o λπ 4

)(exp

4),(

2

For an instantaneous accidental release of a chemical mass Mo,

the time- and space-dependent concentration distribution can be

given as:

This equation is a useful first-order expression for estimating exposure

levels downstream of accidental releases.



2L LD tσ =

A useful measure of the longitudinal extent of the dispersing

pulse is its standard deviation, given by:



Cx

CD

x

CU

t

Cλ−

∂

∂+

∂

∂−=

∂

∂2

2

For a continuous source of infinite duration, the one-

dimensional advective-diffusive transport model is given as:



Continuous source of infinite duration:

C x tC Ux

Derfc

x Ut

Dt

Ux

Derfc

x Ut

Dt

o( , ) exp ( ) exp ( )= −

−

+ +

+

2 2

12 2

12

ΓΓ

ΓΓ

2/2

21

UDH

H

λ=

+=Γ



Continuous source of finite duration:

C x tC x

Uerfc

x Ut H

Dterfc

x U t H

D t

o( , ) exp

( ) ( )( )

( )=

−

− +

+

− − +

−

2

1

2

1

2

λ τ

τ



ACTS Interface



� Rivers



� Estuaries




Far-Field Mixing – Small Lakes and Reservoirs

Near Field Mixing

Transverse Mixing

Longitudinal

Advection and

Dispersion


Reservoirs

Oceans and

Great Lakes


Pathway Models


Models


Small Lakes and Reservoirs: 1

� Small natural, man-made impoundments, and cooling ponds represent an extreme situation of geometric constraints and limitedadvective transport

� Half-life of chemicals considerably longerthan impoundment residence time (through flow on the order of a few days to weeks)

� Chemical concentration is essentially uniform within impoundment



Contaminant concentration, which is a function of

time only, can be computed by:

o

o

CC

Q / Q Vλ=

+ o

Q1 exp t

/ Q Vλ

− − +



0

0/

CC

Q Q Vλ=

+ 0/ Q



Contaminant

Source

)(

)(

11

22

darcy

darcyo

VdLQ

VdLQ

=

=

Depth

d1

Lake Volume

V

Groundwater

flow

Contaminant

Plume

Depth

d2

L2

L1

x

yz



ACTS Interface


Far-Field Mixing -- Estuaries

Near Field Mixing

Transverse Mixing

Longitudinal

Advection and

Dispersion


Reservoirs

Oceans and

Great Lakes


Pathway Models


Models


Estuaries: 1

Transport and dispersion processes

in estuaries are considerably more

complicated than in non-tidal rivers


Estuaries: 2

� Hydrodynamic conditions� Oscillatory tidal motion with cyclic variations in

velocity and elevation

� Vertical (baroclinic) circulations� Density differences between freshwater and

saltwater

� Wind currents in wide, shallow (bay like) estuaries

Processes to consider:


Estuaries: 3

The longitudinal distribution C(x) of any pollutant that is released in

a steady-state fashion, at a distance L upstream of the estuary

mouth, is given by (Stonmel, 1953):

++−

+−

++

−−

+−

−

+=

)11(2

exp)11(2

exp

)11(2

)(exp)11(

2

)(exp

1)(

αα

αα

α

TK

fLU

TK

fLU

TK

fULx

TK

fULx

rQ

oC

oQ

xC

Qr = fresh river flow, L3/T

α = 4λKT/Uf2

Uf = freshwater velocity, L/T

KT = tidal dispersion coefficient, L2/T


Estuaries: 4

ACTS Interface



� Rivers



� Estuaries




Far-Field Mixing – Oceans and Great Lakes

Near Field Mixing

Transverse Mixing

Longitudinal

Advection and

Dispersion


Reservoirs

Oceans and

Great Lakes


Pathway Models


Models


Oceans and Great Lakes: 1

� Main feature of pollutant dispersion is unlimited extent

� Pollution analysis� Determine velocity field

� Compute dispersion of release (instantaneous or continuous)

� Neglect dynamic coupling between two phases of analysis if mass of chemical is small and buoyancy negligible



� Eulerian approach for solution of advection-diffusion equation with decay term. Solution presented by Brooks (1960):

� Steady-state solution

� Uniform source of finite extent

� Steady, uniform flow


Ocean and Great Lakes: 3

Advection-Diffusion Equation

Cy

CK

yx

CU y λ−

∂

∂

∂

∂=

∂

∂

Ky = eddy diffusivity

λ = first order decay coefficient defined in terms of the

half-life (τ) of of a chemical

τλ

2ln=



Definition Sketch for

Brooks Modelh

Z

U

Source

H

x

y

bUo

Co

C(x,y)

b



Solution for Brooks (1960) model:

CC x

HUerf

y b U

K xerf

y b U

K x

o

y y

= −

+

−−

2

2

2

2

2exp

( / ) ( / )λ

H = vertical extent of water column, L

U = velocity of water, L/T

Ky = eddy diffusivity (constant), L2/t

λ = Radioactive decay constant, 1/T



ACTS Interface


SURFACE WATER POLLUTION

Sediment Pathway Models


General Classification of Models



� Sediment models


Sediment Models

� Rivers

� Fletcher-Dotson model

� Onishi mixing-tank model

� Estuaries

� USNRC estuarine model

� Lakes

� USNRC two-layer lake model


Sediment Pathway Model -- Rivers

Near Field Mixing Far Field Mixing

Fletcher & Dotson Model

Onishi Model

Rivers

NRC Model

Estuaries Coastal waters

and Oceans

NRC Model

Lakes

Sediments

Pathway Models


Models


Sediment Pathway Models Rivers: 1

� Fletcher and Dotson Model (1971)

� Unsteady, one-dimensional, liquid pathway

� Calculates temporal and longitudinal distributions of dissolved radio-nuclide concentrations

� Calculates concentrations of chemicals attached suspended and bottom sediments



CQ

Q C e Q Cx t

x t

x x t t x x t t

t

i i

n

,

,

( , ) ( , )= +

− − − −

− ∑1

1∆ ∆ ∆ ∆

λ∆

Fletcher and Dotson Model (1971)

Cx,t = dissolved radio-nuclide concentration at location x, and time t

Ci = dissolved radio-nuclide concentration of the tributary

Qx,t = flow rate at location x, and time t

Qi = tributary flow rate

λ = decay coefficient



ACTS Interface



� Onishi et al. (1981) Mixing-Tank Model with Sediment Transport� River reaches are divided into segments and are

represented by a series of tanks; sediments and chemicals are concentrations completely mixed

� Chemicals and sediment contributions from point and non-point sources are treated as lateral influx that is uniformly distributed along the river reach for each segment

� Dissolved and particulate chemicals are linearly related by a distribution coefficient



� Onishi et al. (1981) Mixing-Tank Model with Sediment Transport--continued

� Dissolved and particulate chemicals reach their equilibrium conditions within one time step

� Particulate chemical deposition to the riverbed and re-suspension from the riverbed does not occur



CL1CL2

CLn

V1

C1

V2

C2

Vn

Cn

Q0

C0

Q1

C1

Q2

C2

Qn

Cn

Qn-1

Cn-1

Onishi et al. (1981) mixing tank model




Mass conservation of sediment in the nth tank

Qn = flow discharge from the nth tank

Sn = Sediment concentration in the nth tank

SLn = lateral influx of sediment into the nth tank

Vn = water volume in the nth tank

t = time

∂

∂

∂

∂

S

tS

V

V

tQ

Q S SL

V

n

n

n

n n n

n

= − +

+

+− −1 1 1




Mass balance of dissolved and particulate chemical in nth tank

{ }∂

∂ λ∂

∂

C

t V S K

S K Q C CL C L S K Q C

V C S K Ct

V S K

n

n n d

n d n n n p n n d n n

n n n d n n n d

=+

+ + + − +

− + − +

− − −1

1

1 1

1 1

1 1 1

( )

( ) ( ) ( )

( ) ( )

C K CPn d n=


Sediment Models

� Rivers



� Estuaries


� Lakes



Sediment Pathway Model -- Estuaries



Onishi Model

Rivers

NRC Model


and Oceans

NRC Model

Lakes

Sediments

Pathway Models


Models


Sediment Pathway ModelsEstuaries: 1

NRC Estuarine Model with Sedimentation

Sedimentation

Velocity Net Downstream

Velocity U

Water

Layer

Interface

Bed Velocity UB

d1

d2

Unmovable sediment layer

Movable sediment layer


Sediment Pathway ModelsEstuaries: 2

� Water layer is moving with a net tidally averaged downstream velocity of U

� Erodible bed is moving with a net downstream velocity, Ub

� Diffusive transport from tidal oscillations in water and sediment layers assumed to be constant with the longitudinal dispersion coefficients, Ddx and Dxb

� Sedimentation and burial occur uniformly at vertical velocity v

� dissolved and particulate chemicals are in equilibrium

Assumptions for the

NRC Estuarine Model with Sedimentation


Sediment Models

� Rivers



� Estuaries


� Lakes



Sediment Pathway Model -- Lakes



Onishi Model

Rivers

NRC Model


and Oceans

NRC Model

Lakes

Sediments

Pathway Models


Models


Sediment Pathway Model Lakes: 1

� Flow conditions

� Stratification and seasonal turnover

� Sedimentation interaction

� Biotic interaction

Major Processes Affecting Chemical Movement



NRC Two-Layer Lake Model

Sediment Layer

Decay

Decay

Water In Water Out

InterfaceSedimentation Direct Exchange Direct Exchange

Burial



� Water inflow and outflow are constant

� Sediment rate is constant

� Dissolved and particulate chemicals undergo decay

� The thickness of the sediment layer remains constant. (If sedimentation occurs, it is assumed that the affected portion of the original bed layer becomes inactive, and it is eliminated from the analysis)

NRC Two-Layer Lake Model Assumptions


Sediment Pathway Models Lakes: 4

ACTS Interface


Lake Pollution Application:


Problem:

A small “lake” is receiving a radioactive contaminant with concentration of 100 kg/m3. The circulating flow rate (qo) is 1.0 m3/day, the net flow through in the lake is 5.0 m3/day. The volume of the lake is 1000 m3. What is the concentration in the lake at 60, 180, 240, and 360 days if the radioactive decay constant has a value of?

Radioactive decay• 0.00/day (conservative) • 0.00265/day? • 0.005/day?• 0.0075/day?• 0.01/day?• 0.025/day?

V =1000m3

λ

1.0m3/d

5.0m3/d


The earlier CSTR solution :

Vo o

dCQ C QC C V

dtλ= − −

Steady state solution: 0dC

dt=

o oQ C QC C Vλ− − 0

o oQ CC

Q Vλ

=

=+

o

o o

C

Q Q V Qλ=

+


o oQ CC

Q Vλ=

+o

o

C

Q Q Vλ=

+3

;

100kg/m ;

o

o

Q

C V=

( )( )

( )( )

( )( )

3 3 3

3

3

3

1000m ; 5m /d; 1m /d

10020kg/m

5 1 0 1000 1

with decay = 0.00265 per day

10013.07kg/m

5 1 0.00265 1000 1

with decay = 0.025 per day

1003.3kg/m

5 1 0.025 1000 1

oQ Q

C

C

C

∞

∞

∞

= = =

= =+

= =+

= =+



� Small natural, man-made impoundments, and cooling ponds represent an extreme situation of geometric constraints and limitedadvective transport

� Chemical concentration is essentially uniform within impoundment

� Time dependent analysis maybe necessary for exposure analysis and also wrt the relative values of residence time and decay.


Residence time:

1000200days 0.55

5r

Volt yrs

Q= = = =


Half Life• ∞ (conservative) • 261 day? • 138 day?• 92.4 day?• 69.3 day?• 27.7 day?



Contaminant concentration, which is a function of

time only, can be computed by:

o

o

CC

Q / Q Vλ=

+ o

Q1 exp

/ Q V− − tλ

+



0

0/

CC

Q Q Vλ∞ =

+ 0/ Q


Half Life• ∞ (conservative) • 261 day? • 138 day?• 92.4 day?• 69.3 day?• 27.7 day?

200r

t d=


o

o

CC

Q / Q Vλ=

+ o

Q1 exp

/ Q V− − tλ

+

Radioactive

Decay Rate

Constant

(1/day)

60

Days

180

Days

240

Days

360

Days

ACTS

Steady-State

Time,

Concentration

Materials

Mass

Balance

Steady-

State

Conc.

0.00 5.2 11.9 14.0 16.71,500 days

19.99 kg/m3 20.00

0.00265 4.8 9.8 11.0 12.21,000 days

13.07 kg/m3 13.07

0.0050 4.5 8.3 9.1 9.7720 days

9.99 kg/m3 10.00

0.0075 4.2 7.2 7.6 7.9600 days

7.99 kg/m3 8.00

0.0100 4.0 6.2 6.5 6.66480days

6.66 kg/m3 6.67

0.0250 2.8 3.3 3.3 3.3180 days

3.33 kg/m3 3.33


0 60 120 180 240 300 360TIME, IN DAYS

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20N

OR

MA

LIZ

ED

CO

NC

EN

TR

AT

ION

(C

/CO)

SURFACE WATER ANALYSIS: FAR-FIELD MIXING -- SMALL LAKES AND RESERVOIRS

EXPLANATION

λ = 0.00 d-1

λ = 0.00265 d-1

λ = 0.005 d-1

λ = 0.0075 d-1

λ = 0.01 d-1

λ = 0.025 d-1


Use the same input parameters as given earlier. However, this time assume that there is a probability distribution function (PDF) for the Decay Rate Constant that is described by an EXPONENTIAL distribution. Assume the PDF has the following properties:

Minimum: 0.00265/dayMaximum: 0.025/dayMean: 0.008/dayVariance: 0.025/day2

V =1000m3

λ

1.0m3/d

5.0m3/d


Conduct a simulation of the lake problem using the Monte Carlo option. What is the probability of contaminant levels in the lake exceeding a concentration value of 5 kg/m3 at a time of 180 days.


Radioactive Decay Rate

Constant

(1/day)

180

Days

Materials Mass

Balance Steady-

State Conc.

0.00 11.9 20.00

0.00265 9.8 13.07

0.0050 8.3 10.00

0.0075 7.2 8.00

0.0100 6.2 6.67

0.0250 3.3 3.33


Complementary Cumulative Probability Function

Conduct a simulation of the lake problem using the Monte Carlo option. What is the probability of contaminant levels in the lake exceeding a concentration value of 5 kg/m3 at a time of 180 days.

81%


Small Lakes and Reservoirs:

Contaminant

Source

2 2

1 1

( )

( )

o darcy

darcy

Q L d V

Q L d V

=

=

Depth

d1

Lake Volume

V

Groundwater

flow

Contaminant

Plume

Depth

d2

L2

L1

x

yz


Transverse diffusion application


2

( , ) exp44

o o

yy

Q C y U xC x y

D x UH D Ux

λ

π

= − −

TRANSVERSE DIFFUSION: For a contaminant discharge on one side:

*0.15transverse

D u y=

2

*6.66

t

B

u yτ =

2

*6.66t t

Bx V V

u yτ= =

B

B

x

xtransverse

B

2

t

B

τ=

But also an analytical sol.:


Example: An industrial plant discharges a conservative substance at a point along the side of a river which is 2m deep 50m wide. River slope 0.02%. What is the down stream distance the pollutant starts to affect the other bank.

B

Width

x

xtransverse

B


for annings n 0.035 0.61 /M V m s= =

( ) ( ) ( )* 49.81 1.85 2 10 0.06 /h ou gR S m s−= = × =

2 501.85

2 50 2h

AR m

P

×= = =

+ +

Example: An industrial plant discharges a conservative substanceat a point along the side of a river which is 2m deep 50m wide.River slope 0.02%. What is the down stream distance the pollutant starts to affect the other bank.

( ) ( )2 / 3 1/ 21.0

h oV R Sn

=

( ) ( )* 20.15 0.15 0.06 2 0.018 /transverse

D u y m s= = =

( )

2 2

*

506.66 6.66 138750 38.54 1.6

2 0.06t

Bs days days

u yτ = = = = =

( )138750 0.61 / 84637.5 84.6transverse

x s m s m km= = =

B

B

x

xtransverse

B


2

( , ) exp44

o o

yy

Q C y U xC x y

D x UH D Ux

λ

π

= − −

Transverse diffusion model:


Diffusion only analysis ignores effluent discharge value it also ignores the initial effluent concentration.

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