Conventional Pollutants in Rivers and Estuaries

OXYGEN

CARBONDIOXIDE

ORGANICMATTER

INORGANICNUTRIENTS

DECOMPOSITION(bacteria/animals)

PRODUCTION(plants)

Chemicalenergy

Solarenergy

THE DISSOLVED OXYGEN SAG

WWTP

Distance

DO(mgL-1)

Critical concentration(decomposition=reaeration)

Decompositiondominates

Rearetiondominates

River

BIOCHEMICAL OXYGEN DEMAND

• Experiment– decomposition of carbonaceous matter

– C6H12O6+6O2 6CO2+6H2O

– Mass balance

– General solution g=g0e -k1t

– Oxygen mass balance

– General solution for oxygen

Vgkdt

dgV 1

Vgkrdt

doV 1og

)e1(groo tk0g00

1

BIOCHEMICAL OXYGEN DEMAND(ctd.)

L=rog g

L=L0 exp(-k1t)

BOD=L0-L

Lkdt

dL1

tk01

1eVLkdt

doV

)e1(Loo tk00

1

Streeter-Phelps equation

• Steady state for ultimate BOD

• Steady state for DO

• Deficit D=Cs-C

• Solutions:

– BOD :

– D:

Lkx

LU d

)CC(kLkx

CU sad

DkLkx

DU ad

U

xk

0

d

eLL

U

xk

0da

d

eLkDkx

DU

)ee(kk

LkeDD U/xkU/xk

da

0dU/xk0

ada

Streeter-Phelps Equation (Example)

• L0=10 mg L-1

• ka=2.0 d-1

• kd=0.6 d-1

• D0 = 0 mgL-1

• U=16.4 mi d-1

Streeter-Phelps Equation (Code)• xspan=0:100;• %parameter definition• global ka kd U• ka=2.0;• kd=0.6;• U=16.4• y0=[10 0]';• %initial concentrations are given in

mg/L• [x,y] = ODE45('dydx_sp',xspan,y0) ;• plot(x, y(:,1),'linewidth',1.25)• hold on• plot(x,

y(:,2),'r','linewidth',1.25);• do_an=y0(2)*exp(-ka*x/U)+y0(1)*...• kr/(ka-kd)*(exp(-kd*x/U)-exp(-

ka*x/U));• ylabel('mg L^{-1}')• xlabel('Distance (mi)')• legend('BOD', 'DO')• plot(x,do_an,'r+')

•function dy=dydx_sp(x, y)•global ka kd U•L=y(1);•D=y(2);•dy=y;•dy(1)=-kd/U*L;•dy(2)=+kd/U*L-ka/U*D;

Critical Deficit and Distance

• Dc , xc dDc/dxc=0

•

•

•

0

0

d

da

d

a

rac L

D

k

kk1

k

kln

kk

Ux

U/xk

a

0rc

crek

LkD

sw

sswwo QQ

CQCQL

DC’S DEPENDENCE ON VARIOUS FACTORS

•Increases with W=LQ

•Increases with T

•Increases with D0

•Decreases with Q

Stream Re-aeration Formulas

• ka = re-aeration constant, d-1

• U = mean stream velocity, ft-1

• H = mean stream depth, ft

• Q = flow-rate, ft3s-1

• t = travel time, d

5.1

5.0

a H

U29.1k • O’Connor-Dobbins

• Owens-Edwards-GibbsH=1-2.5; U=0.1-0.5; Q=4-36

• USGS

75.1

73.0

a H

U23k

33.1a H

U6.7k

Sedimentation of BOD

• Steady state for ultimate BOD

• Deficit D=Cs-C

• Solution

L D 0kd ka

ks

Lkx

LU r

DkLkx

DU ar

)ee(kk

LkeDD U/xkU/xk

ra

0dU/xk0

ara

Estimation of kr and kd in a stream

Principle of Superposition

• Mass balance for DO deficit

• In terms of L and N

)PR(H

SNk

LkDkx

DU

t

D

n

da

)PR(H

SeNk

eLkDkx

DU

t

D

U/xk0n

U/xk0da

n

r

Diurnal Variations

Sensitivity Analysis

• First order analysis– y=f(x)

– y0=f(x0)

0

0

x

2

22

0

x00

x

f)xx(

x

f)xx()x(f)x(f

2

x

20

20

0x

fxxyy

Sensitivity Analysis

• Monte Carlo Analysis– 1. Generate dx0 = N(0,x)

– 2. Determine y=f(x0+dx0)

– 3. Save Y={Y | y}

– 4. i=i+1

– 5. If i < imax go to 1

– 6. Analyze statistically Y

xspan=0:100;%parameter definitionLr=zeros(100,101);Dr=zeros(100,101);global ka kd UU=16.4;y0=[10 0]';%initial concentrations are given in mg/Lfor i=1:100, ka=2.0+0.3*randn; kd=0.6+0.1*randn; while ka < 0 | kr < 0,

ka=2.0+0.3*randn; kd=0.6+0.1*randn;

end [x,y] = ODE45('dydx_sp',xspan,y0) ;

Lr(i,:)=y(:,1)'; Dr(i,:)=y(:,2)'; endsubplot 211plot(x, mean(Lr,1),'linewidth',1.25)hold onplot(x, mean(Lr,1)+std(Lr,0,1),'--', …

'linewidth',1.25)plot(x, mean(Lr,1)-std(Lr,0,1),'--', …

'linewidth',1.25)ylabel('mg L^{-1}')title('BOD vs. distance')subplot 212plot(x, mean(Dr,1),'r','linewidth',1.25);hold onplot(x, mean(Dr,1)+std(Dr,0,1),'r--', …

'linewidth',1.25);plot(x, mean(Dr,1)-std(Dr,0,1),'r--', …

'linewidth',1.25);xlabel('Distance (mi)')title('DO Deficit vs. distance')print -djpeg bod_mc.jpeg

DYNAMIC APPROACH

• Routing water (St. Venant equations)

– Continuity equation

– Momentum equation (Local acceleration + Convective acceleration+pressure + gravity + friction = 0

0t

A

x

Q c

0SSgx

yg

A

Q

xA

1

t

Q

A

1fo

c

2

cc

Kinematic wave

Diffusion wave

Dynamic wave

KINEMATIC ROUTING

• Geometric slope = Friction slope

• Manning’s equation

• Express cross section area as a function of flow

0SS fo

2/1f3/2

3/5c S

P

A

n

1Q

QAc

5/3

S

nP5/3

0

3/2

KINEMATIC ROUTING (ctd)

• Express the continuity equation exclusively as a function of Q

• Discretize continuity equation and solve it numerically

0t

QQ

x

Q 1

1 2 3 4 n n-1 n5

k

k+1

x

t

KINEMATIC ROUTING (ctd)

• Discretize continuity equation and solve it numerically

• Example– Q=2.5m3s-1; S0=0.004

– B=15m; n0=0.07

– Qe=2.5+2.5sin(wt); w=2pi(0.5d)-1

0t

QQ

2

QQ

x

QQ ki

1ki

1k1i

ki

k1i

ki

)1(ki

1k1i

)1(ki

1k1i1k

i1k

1i1k

i

2QQ

xt

2QQ

QQxt

Q

S0=0.004;B=15;n0=0.07;n=80;Q=zeros(2,n)+2.5;dx=1000.; %metersdt=700.; %secondsalpha=(n0*B^(2./3.)/sqrt(S0))^(3./5.)beta=3./5.;for it=1:150 if it*dt/24/3600 < 0.25 Q(2,1)=2.5+2.5* …

sin(2.*pi*it*dt/(0.5*24*3600)); else Q(2,1)=2.5; end for i=2:n Q(2,i)=(dt/dx*Q(2,i-1)* …

((Q(1,i)+Q(2,i-1))/2.)^(1-beta)... +alpha*beta*Q(1,i))/…

(dt/dx*((Q(1,i)+Q(2,i-1))/2.)^(1-beta)... +alpha*beta); end Q(1,:)=Q(2,:); if floor(it/40)*40==it x=1:n; plot(x,Q(1,:)); hold on end

end

ROUTING POLLUTANTS

• Mass conservation

• Discretized mass balance equation

0t

)cA(

x

)Qc( c

1 2 3 4 n n-1 n5

k

k+1

x

t

0t

cAcA

x

cQcQ ki

kci

1ki

1kci

k1i

k1i

ki

ki

tQQV

tcQcQcVc

ki

k1i

ki

ki

ki

k1i

k1i

ki

ki1k

i

ROUTING POLLUTANTS (ctd)

• Alternate formulation

• Example – u = 1 ms-1

x = 1000 m t = 500 m

ki

k1i

ki

1ki cc

x

tucc

u=1.; %m/sdx=1000; %mdt=500; %sn=100;x=1:100;y=x-20;c0=exp(-0.015*y.*y);c1=c0;plot(x,c0);hold onfor it=1:120 for i=2:n-1 c1(i)=c0(i)+u*dt/dx* …

(c0(i-1)-c0(i)); end c0=c1; if fix(it/40)*40==it plot(x,c0); endendxlabel('x (km)');ylabel('C mgL^{-1}');

ROUTING POLLUTANTSNumerical Example

ROUTING POLLUTANTS (ctd)

• More accurate (second order both time and space) formulation

k1i

ki

k1i2

22

k1i

k1i

ki

1ki

cc2cx2

tu

ccx2

tucc