Lecture #37 Toxics: Rivers

(Chapra, L44)

David Reckhow CEE 577 #37 1

Updated: 17 April 2013 Print version

Review: Toxics Model- CSTR with sediments Internal Transport Processes (between

compartments) dissolved: diffusion particulate: settling, resuspension & burial

Expressed as velocities (e.g., m/yr)

David Reckhow CEE 577 #37 2

Water (1)

Mixed Sediments (2)

Deep Sediments (3) Burial (vb)

Settling (vs) Resuspension (vr) Diffusion (vd)

Solids: Mass Balance

In water column

in mixed sediments

David Reckhow CEE 577 #37 3

2111

1 AmvAmvQmWdt

dmV rsm +−−=

2212

2 AmvAmvAmvdt

dmV brs −−=

Can be expressed as: Qmin

Solids: Steady State Solution Fixed ratio of solids concentration in water column

and mixed sediments

David Reckhow CEE 577 #37 4

( ) 12 )1( mvv

vmbr

s

+=−= ρφ

Toxicant Mass Balance

In the water column

In the mixed sediments

David Reckhow CEE 577 #37 5

( ) 21111111112211

1 AcvcAfvcAfvcVkcfcfAvQcWdtdcV rpsdvTdddT +−−−−+−=

Can be expressed as: Qcin

( ) 221122222112

2 AcvAcvcAfvcVkcfcfAvdt

dcV brpsTddd −−+−−=

New Chapra approach

Steady State Solution Water column

Mixed Sediments

David Reckhow CEE 577 #37 6

1222

112 c

fHkff

cddbr

ddps

ννννν

+++

+=

AffFAfVkQQcc

ddpsrdv

in

))(1( 111111 ννν +′−+++=

Steady State Solution Formulate total loss velocity

which can be in the form of a total rate

At SS

So this reduces to:

David Reckhow CEE 577 #37 7

( )( )rddpsdvT FfvfvfvHkv ′−+++= 111111

1HvK T

T =

11 VKQ

QccT

in

+=

AffFAfVkQQcc

ddpsrdv

in

))(1( 111111 ννν +′−+++=

𝐻1 = 𝑉1𝐴� And since:

𝐾𝑇𝑉1

𝑐1 =𝑊𝜆𝑉1

𝜆 =𝑄𝑉1

+ 𝐾𝑇 or Where:

Sediment Feedback & k The ratio of sediment feedback to total sediment

purging

The first-order constants k1 and k2 incorporate various decay processes Biodegradation Hydrolysis photolysis

David Reckhow CEE 577 #37 8

222

2

HkffF

ddbr

ddrr +++

+=′

ννννν

Toxics Model: CSTR to a PRF Internal Transport Processes (between

compartments) dissolved: diffusion particulate: settling, resuspension & burial

Expressed as velocities (e.g., m/yr)

David Reckhow CEE 577 #37 9

Water (1)

Mixed Sediments (2)

Deep Sediments (3) Burial (vb)

Settling (vs) Resuspension (vr) Diffusion (vd)

Combine with advective flow

David Reckhow CEE 577 #37 10

Water (1)

Mixed Sediments (2) Deep Sediments (3) Burial (vb)

Settling (vs) Resuspension (vr) Diffusion (vd)

Solids: Mass Balance In water column

in mixed sediments

David Reckhow CEE 577 #37 11

2111

1 AmvAmvQmWdt

dmV rsm +−−=

2212

2 AmvAmvAmvdt

dmV brs −−=

21

11

11 mHvm

Hv

dxdmU

dtdm rs +−−=

0

2212

2 mvmvmvdt

dmH brs −−=0

Divide by V, and replace loadings with advective term

Earlier lake model

Divide by A

Solids: Steady State Solution Assuming that the sediments do not move

downstream mass balance on m2 and therefore relationship

between m2 and m1 are identical for lake and river

David Reckhow CEE 577 #37 12

( ) 12 )1( mvv

vmbr

s

+=−= ρφ

Toxicant Mass Balance

In the water column

In the mixed sediments

David Reckhow CEE 577 #37 13

( ) 21111111112211

1 AcvcAfvcAfvcVkcfcfAvQcWdtdcV rpsdvTdddT +−−−−+−=

( ) 221122222112

2 cvcvcfvcHkcfcfvdt

dcH brpsTddd −−+−−=

New Chapra approach

Divide by V, and replace loadings with advective term

( ) 21

111

111

1111221

11 cHvcf

Hvcf

Hvckcfcf

Hv

dxdcU

dtdc r

ps

dv

Tddd +−−−−+−=

Earlier lake model

( ) 221122222112

2 AcvAcvcAfvcVkcfcfAvdt

dcV brpsTddd −−+−−=

Divide by A

Steady State Solution

Formulate total loss velocity

which can be in the form of a total rate

and the ratio of sediment feedback to total

sediment purging remains:

David Reckhow CEE 577 #37 14

( )( )rddpsdvT FfvfvfvHkv ′−+++= 111111

′ =+

+ + +F

FF k Hr

r d d

r b d d

ν νν ν ν

2

2 2 2

1HvK T

T =

Steady State Solution

The water column solution is:

and as before the sediment solution is:

David Reckhow CEE 577 #37 15

1222

112 c

FHkFF

cddbr

ddps

ννννν

+++

+=

cQc

Q k V AF F F F Ain

v d r s p d d1

1 1 1 1 11=

+ + + − ′ +ν ν ν( )( )

UxK

o

T

ecc−

= 111

1 VKQQcc

T

in

+=

R21

Earlier lake model

Simplified Versions Level 0 no movement from sediment to water column vr = 0, vd = 0 therefore, F’r = 0

Now c2 has no effect on c1 and we return to a one compartment model all time-variable solutions can be applied

David Reckhow CEE 577 #37 16

K kvH

fvH

fTv

ds

p1 11

11

1= + +

Simplified Versions (cont.)

Level 1 no volatilization, no decomposition vv = 0, k1 = 0, k2 = 0

Useful for modeling toxic metals

David Reckhow CEE 577 #37 17

( )K kvH

f FvH

fvH

fTv

d rs

pd

d1 11

11

11

11= + + − ′ +

Solutions (cont.)

David Reckhow CEE 577 #37 18

( )K kvH

f FvH

fvH

fTv

d rs

pd

d1 11

11

11

11= + + − ′ +

Where: ′ =+

+ + +F

FF k Hr

r d d

r b d d

ν νν ν ν

2

2 2 2

k k f k fd d p p1 1 1 1 1= +

k k f k fd d p p2 2 2 2 2= +

The ratio of sediment feedback to total sediment purging

Overview of solutions to toxic model

David Reckhow CEE 577 #37 19

CWVT1 = λ

C C eT T oKT

xU

1 11= −

λ = +QV

KT1

For Lakes:

For Rivers:

Where:

QW

To next lecture

David Reckhow CEE 577 #37 20