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