Environmental Engineering Science . PART # 16

8/10/2019 Environmental Engineering Science . PART # 16

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Principles of WaterQuality Modeling

Saumyen Guha



Outline

Models for small ponds and lakes: SingleCompartment Model

BOD-DO

General Mass-Balance Equation in 3-D

1-D models for rivers: with and withoutdispersion

Different source/sink scenarios in river



Single Compartment Model

Key Assumption:Volume is Completely

mixed with uniform

concentration

C (t ): concentration, afunction of time (ML-3)

M i: Source (MT-1)

S i: Sink (MT-1

)

V : Volume (L3), may be

a function of timeV, C

Q in, C in Q out, C



Single Compartment Model: Water Balance

Volume Balance:

Qin: inflow to the lake/pond (L3T-1)

Qout : outflow from the lake/pond (L3T-1)G: exchange with groundwater flow (L3T-1)

p: rate of precipitation (LT-1)

e: rate of evaporation (LT-1)

A: surface area of the lake/pond (L2)

An initial condition required: V =V 0 at t =t 0

Ae pGQQdt

dV out in



Single Compartment Model: Mass Balance

An initial condition is required: C =C 0 at t =t 0

Solution Scenarios:Constant Volume: A reasonable assumption. Although volumedoes not remain constant throughout the year, once the volumeequation is solved for an annual cycle, the component mass

balance equation can be solved for a few critical cases ofvolumes.

Steady State: inflow concentration, source and sink arereasonably constant for long period of time. Appropriate forlarge time scales and for engineered systems.

Unsteady State: inflow concentration, source and sink has largevariability in the chosen time scale. Appropriate for mostnatural systems.

iiout inin S M C QC Q

dt

VC d

Each Term has a dimension of MT-1!



Single Compartment Model: Source/Sink

All processes have to be parameterized. Final terms in

the equation should have a dimension of MT-1.



Single Compartment Model: BOD in a

Small Lake/Pond

Steady State BOD for long-term steady input/output:

Time of dissipation for accidental spill in a clean lake,

initially at steady state:

Q

V

k

C C CV k QC QC

dt

VC d

l

l in

1 0 0

t k

in

l

t k

in

l

l in

l l

eV

M C

k eC C

V

M C

C k dt

dC

C C t CV k M QC QC dt

VC d

11

0

0

11

1

,0at

Always Check for Dimensional Consistency of each term!



Single Compartment Model: DO in a Small

Lake/Pond

Dissolved Oxygen:

Initial Condition: t =0, DO = DO0

k a = re-aeration mass transfer coefficient (LT-1)

k l = first order rate constant for BOD decay (LT-1)

A s = top surface area at air-water interface (L2)

Ab

= bottom surface area at sediment-water interface (L2)

k sod = sediment oxygen demand (ML-2T-1)

DO s = saturation dissolved oxygen concentration (ML-3)

Can be solved analytically for steady as well as

unsteady cases

b sod s sal in Ak A DO DOk CV k QDOQDO

dt

VDOd



Single Compartment Model: Nitrogen

constants.tricstoichiomeare and . ,

, ,.. ,0at:ConditionsInitial

.

....

.

033022

0440

2342

23333

2342222

421444

1

.

324

3

2

4

321

niam

niam

b sod s sal in

NO

out inin

NO

out inin

NH

out inin

N Org

out inin

k k k

f f NO NO NO NO

NH NH N Org N Org t

V NOk f V NH k f

Ak A DO DOk CV k QDOQDOdt

VDOd

V NOk M NOQ NOQdt

NOV d

V NOk V NH k M NOQ NOQdt NOV d

V NH k V N Org k M NH Q NH Qdt

NH V d

V N Org k M N Org Q N Org Qdt

N Org V d

NO NO NH N Org



Single Compartment Model: Time Varying Input

and Attenuation with Constant Flow and Volume

C in can have periodic variation, diurnal, seasonal or annual

M can be step input or periodic in nature, weekly, seasonal,

etc. (examples)

Time

M

Time

M



Single Compartment Model: General Time

Varying Flow/Source/Sink

Flow:

Time varying source/sink:

Analytical solution does not exist for food-chain model and timevarying input. Use numerical method for coupled system of ODEs.

Simple methods such as Euler Implicit or Runge-Kutta (2nd

or 4th

order) should be good enough.

Watch the stability limit for time step if you are using explicit methodslike R-K!

If there are large differences in the kinetic constants of different processes, you may land up with stiff-system . Use BDF’s or Gear’smethods.

0

1

,0at C C t

CV t k t M C t Qt C t Qdt VC d

n

i

iout inin

t At et pt Gt Qt Qdt

dV out in



Single Compartment Model: Extension to

multiple compartments

Q 1 , C 1 V 1 , C 1 Q 01, C 01

Q 2 , C 2 V 2 , C 2 Q 02 , C 02

Mass Flux (ML-2T-1) from

compartment i to j due to exchange

at the interface = - K (C i-C j)

This leads to continuum approximation of the state variables as Δx → 0.



Outline

Models for small ponds and lakes: SingleCompartment Model

BOD-DO

General Mass-Balance Equation in 3-D

1-D models for rivers: with and withoutdispersion

Different source/sink scenarios in river



General Mass Balance: Representative Elementary Volume

x

y

z C ( x, y, z, t ) is concentration

of contaminant (ML-3)

u = (u, v, w) is the velocity

vector (LT-1)

q=(q x , q y , q z ) is the net mass-

flux vector (ML-2

T-1

) D = Dispersion Coefficient

(L2T-1)

Source (mi) and Sink ( si)

inside the control volume

The system is monitored

for infinitesimal time

length t in the interval (t ,

t+ t )

D x

D y

D z

q x x+D xq x

x

q y y

q y y+D y

q z z

q z z +D z



Mass Flux, Source and Sink

Advection Flux (ML-2T-1): uC = (uC, vC, wC )

Dipersion Flux (ML-2T-1): Fick’s Law

Net Mass Flux (ML-2T-1): q = uC – D

C

Source (ML-3

T-1

): mi , external addition, transfer fromanother phase, generation of secondary compounds, etc.

Sink (ML-3T-1): si , attenuation through physical, chemical, photochemical and biological pathways.

z

C D

y

C D

x

C DC D ,,

z

C DwC q

y

C DvC q

x

C DuC q z y x

, ,



General Mass Balance Equation

C t + t

x

y

z –

C t

x

y

z =q x x y z t – q x x + x y z t +

q y y x z t – q y y + y x z t +

q z z x y t – q z z + z x y t +

m i x y z t - s i x y z t

Dimension of each term is M. Divide both sides

by D x D y D z Dt

ii

z

z

z

z z

y

y

y

y y x

x

x

x xt t t sm

z

qq

y

qq

x

qq

t

C C

D

D

D

D

DDDD

x

f

x

f f

x

Lt x x x

D

DD

0Let D x→0, D y→0, D z →0, Dt →0 and recognize

ii

z y x

sm z

q

y

q

x

q

t

C

Dimension of each term is ML-3T-1



General Mass Balance Equation

iiii

z y x

smt C sm

z q

yq

xq

t C

qor

iiii smC DC C smC DC t

C

uuu

Continuity Equation for incompressible flow q = 0

ii

ii

sm z

C

y

C

x

C D

z

C w

y

C v

x

C u

t

C

smC DC t

C

2

2

2

2

2

2

2 or u

Each Term has a dimension of ML-3

T-1

!



A Comparison with single compartment model

V, C Q in, C in Q out, C

Cross Section A

Δx

V = AΔ x

Use 1-D approximation

Difference approximation of

space derivative.

Steady Flow: Qin = Qout = QQ = uA

Completely mixed. Constant

concentration everywhere in

the box. Concentration

gradient is zero. Therefore,

Difffusion Flux is Zero due

to Fick’s Law.

V, C

Q in, C in Q out, C



A Comparison with single compartment model

ii sm x

C D

x

C u

t

C

2

2

ii sm z

C

y

C

x

C D z

C w y

C v x

C ut

C

2

2

2

2

2

2

x A s x AmC C Q x Adt

dC iiin DDD )(

x A s x Am x

C x Au x Adt

dC ii DD

DDDD

iiin S M QC QC

dt

VC d

Source/Sink terms in the continuum equation are same when divided by volume.

Can you identify approximation involved in each step?



Models for small rivers or streams

ii sm x

C u

t

C

Approximations:

• One dimensional advection: v = 0, w = 0

• Longitudinal Dispersion is negligible: D = 0

The Equation becomes:

Point Source

Point Source

Area Source



Point Source

Point Source: Qin (L3T-1), C in (ML-3)

No entry in the governing equation. A point has zero

volume. The point has to be a physical boundary and

the boundary condition has to be specified. If the

location of the point discharge is ( x0), and the

discharge and concentration in the river at that point

are Qr0 and C r0, the boundary condition is given by:

0

000000 and ,At

QC QC QC C QQQQ x x ininr r

inr



Line Source

Line Source: qin (L2T-1), C in (ML-3)

If the volume addition is considered, u becomes

function of x. Often ignored unless substantial volume

is added.

Source Strength: sin = qin C in (ML-1T-1)

A

C q

A

s

t x A

t x sm inininin

i DD

DD

Cross Section A

Δx

Line Source s in



Area Source

Line Source: pin (LT-1), C in (ML-3)

If the volume addition is considered, u becomes

function of x. Often ignored unless substantial volume

is added.

Source Strength: ain = pin C in (ML-2T-1)

h

C p

h

a

t xbh

t xbam inininin

i DDDD

Δx

Area Source a in

b

h



BOD-DO Model: Streeter-Phelps Equation

Assumptions: 1-D, Steady State, No Dispersion, First

order BOD decay, Re-aeration through air-water interface.

BOD Equation:

DO Equation:

Boundary Conditions:

DO DOk C k dx

dDOu

DO DO

h

k C k

dx

dDOu

sal

sa

l

or

C k dxdC u l

00 and ,0at DO DOC C x



BOD-DO Model: Streeter-Phelps Equation

BOD Equation:

DO Equation:

Many packages exist to simulate Streeter-Phelps. Some

of these are: SIREM, MACRIV, QUAL2K

U

xk l

eC C

0

U

xk

U

xk

l a

l U

xk

s s

al a

ee

k k

C k e DO DO DO DO 0

0



Models for Large Rivers and Estuary

Approximations: 1-D advection, v = 0, w = 0

General Equation: requires one initial and two boundary

conditions

Steady State: requires two boundary conditions

Typical BC’s:

ii sm x

C D

x

C u

t

C

2

2

ii smdx

C d D

dx

dC u 2

2

boundedis , as and ,0at0

C xC C x



Modeling Rivers: Other Applications

The “continuum model” can be used to simulate all the

processes that was done using the “single compartment model”

The source/sink terms M i and S i used to represent each process

in the “single compartment model” has to be replaced by mi

and si in the “continuum model”

These terms in the the “single compartment model” has

dimension of (MT-1) but in continuum model, they have

dimension of (ML-3T-1).

Each of these terms in the “single compartment model” has to

be divided by the volume (V ) of the compartment in order to

obtain the equivalent term in the “continuum model”.



Modeling Rivers: Example Nitrogen

2342

2333

234222

42144

1

.

324

3

2

4

321

.

...

.

NOk f NH k f h

k DO DOk C k

t

DOu

t

DO

NOk mt

NOu

t

NO

NOk NH k mt

NOu

t

NO

NH k N Org k mt

NH u

t

NH

N Org k mt

N Org u

t

N Org

NO NO NH N Org

niam sod

sal

NO

NO

NH

N Org

k k k



Thank You

Environmental Engineering Science . PART # 16

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