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Lecture #37 Toxics: Rivers (Chapra, L44) David Reckhow CEE 577 #37 1 Updated: 17 April 2013 Print version

CEE 577: Surface Water Quality Modeling · 1 W Qc A f kV Afcv Ac dt dc = T d d d T v d− s p + r Can be expressed as: Qc in ( 1 1 2 2 ) 2 2 2 1 1 2 2 2 2 A f kV Afcv Ac dt dc = d

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Text of CEE 577: Surface Water Quality Modeling · 1 W Qc A f kV Afcv Ac dt dc = T d d d T v d− s p + r...

  • Lecture #37 Toxics: Rivers

    (Chapra, L44)

    David Reckhow CEE 577 #37 1

    Updated: 17 April 2013 Print version

    http://www.ecs.umass.edu/cee/reckhow/courses/577/slides/577l37p.pdf

  • 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 AmvAmvQmWdtdmV rsm +−−=

    2212

    2 AmvAmvAmvdtdmV 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( mvvvm

    br

    s

    +=−= ρφ

  • Toxicant Mass Balance

    In the water column

    In the mixed sediments

    David Reckhow CEE 577 #37 5

    ( ) 211111111122111 AcvcAfvcAfvcVkcfcfAvQcWdtdcV rpsdvTdddT +−−−−+−=

    Can be expressed as: Qcin

    ( ) 2211222221122 AcvAcvcAfvcVkcfcfAvdtdcV brpsTddd −−+−−=

    New Chapra approach

  • Steady State Solution Water column

    Mixed Sediments

    David Reckhow CEE 577 #37 6

    1222

    112 cfHk

    ffc

    ddbr

    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 TT =

    11 VKQ

    QccT

    in

    +=

    AffFAfVkQQcc

    ddpsrdv

    in

    ))(1( 111111 ννν +′−+++=

    𝐻1 =𝑉1

    𝐴� 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 AmvAmvQmWdtdmV rsm +−−=

    2212

    2 AmvAmvAmvdtdmV brs −−=

    21

    11

    11 mHvm

    Hv

    dxdmU

    dtdm rs +−−=

    0

    2212

    2 mvmvmvdtdmH 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( mvvvm

    br

    s

    +=−= ρφ

  • Toxicant Mass Balance

    In the water column

    In the mixed sediments

    David Reckhow CEE 577 #37 13

    ( ) 211111111122111 AcvcAfvcAfvcVkcfcfAvQcWdtdcV rpsdvTdddT +−−−−+−=

    ( ) 2211222221122 cvcvcfvcHkcfcfvdtdcH 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

    ( ) 2211222221122 AcvAcvcAfvcVkcfcfAvdtdcV 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 TT =

  • Steady State Solution

    The water column solution is:

    and as before the sediment solution is:

    David Reckhow CEE 577 #37 15

    1222

    112 cFHk

    FFc

    ddbr

    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−

    = 11 11 VKQ

    QccT

    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 k vH f FvH

    fvH

    fTv

    d rs

    pd

    d1 11

    11

    11

    11= + + − ′ +

  • Solutions (cont.)

    David Reckhow CEE 577 #37 18

    ( )K k vH 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

    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

    http://www.ecs.umass.edu/cee/reckhow/courses/577/slides/577l38.pdf

    CEE 577: Surface Water Quality ModelingReview: Toxics Model- CSTR with sedimentsSolids: Mass BalanceSolids: Steady State SolutionToxicant Mass BalanceSteady State SolutionSteady State SolutionSediment Feedback & kToxics Model: CSTR to a PRFSlide Number 10Solids: Mass BalanceSolids: Steady State SolutionToxicant Mass BalanceSteady State SolutionSteady State SolutionSimplified VersionsSimplified Versions (cont.)Solutions (cont.)Overview of solutions to toxic modelSlide Number 20