1

Hydrodynamic interaction between suspended sediments and the transport

of contaminants in open channel flow

Project for

Environmental Fluid Mechanics, CE380S

The University of Texas at Austin, Civil Engineering Department

Environmental and Water Resources Engineering

By

Eusebio Ingol Blanco

This paper presents a research about the interaction between suspended solids and

the pollution transport reactions in streams. Since that now there is a great interest

of preserving the environment of the aquatic life, as result of the contamination,

the study of transport processes in streams is very important in fluid mechanics.

This issue involves the study of the interaction between sediments, reactive

chemical substance and stream. For which, the hydrodynamic processes need be

modeled including the advection, turbulent diffusion, sedimentation, and the

effect of kinetics. The main concern is to know how suspended solids can

influence in the pollutant concentrations and how they are able of absorbing part

of the chemical substances, which generally follow decay processes (Revelli and

Ridolfi, 2002). Interaction between suspended solids and chemical reactions in

open channel flow has been studied. Mathematical models described in this

document take account the advection, turbulent diffusion, and the exchange

kinetics between the solid phase and dissolved phase.

2

Contents 1. Introduction ................................................................................................ 1

1.1 Research objectives ...................................................................................... 1

2. Concepts and definitions ............................................................................ 1

3. Problem statement ...................................................................................... 2

3.1 Transport of suspend sediments .............................................................. 3

3.2 Transport of chemical species ................................................................. 4

4. Decay process .............................................................................................. 6

5. Scales ........................................................................................................... 7

5.1 Spatial Scales ............................................................................................ 7

5.2 Time Scales ............................................................................................. 10

6. Discussion.................................................................................................. 11

7. Conclusions ............................................................................................... 16

Required reading: ............................................................................................ 16

Supplementary reading: .................................................................................. 17

1

1. Introduction

The hydrodynamic of suspended solids and its effect on the transport of

contaminants in a stream is a relevant issue in Environmental Fluid Mechanics. It

is known that the transport of dissolved matter in a stream is affected by physical,

biological, and chemical processes. Physical processes include the advection,

turbulent diffusion and dispersion. When the contaminants are discharged to the

stream, the chemical species is partitioned in two aqueous (dissolved) and sorbed

(solid) phases with permanent exchange regulated by local gradient

concentrations between both phases (NG and Yip 2001, and Revelli and Ridolfi

2002). Because of the shear process, the dissolved phase follows the advection,

turbulent diffusion and dispersion while the sorbed phase is linked to the sediment

following the sedimentation process (Revelli and Ridolfi, 2002). Here the

question is how the solids in suspension can alter the transport of chemical

substances in a river or channel and how is the interaction between both. For

answer this question, the main objective of this research is to investigate the effect

and interaction of suspended solids on substance contaminants in open channel

flows. For this end, this research reviews three major papers related with this

issue. Equations described in this paper take account the advection, turbulent

diffusion, and the exchange kinetics between the sorbed and dissolved phase.

Additionally, decay process for these two phases, in the chemical pollutant, is

included showing the influence of the decay and suspend solids on the alteration

of the transport mechanism.

1.1 Research objectives

To answer the questions formulated in this research, two main objectives are

addressed:

1. Investigate the interaction between suspended solids and contaminant in

open channel flows. For this end, three (3) papers related of this issue are

reviewed.

2. Understand the advection- diffusion process in the transport of the suspend

solids; for which mathematical models are described.

2. Concepts and definitions

To understand the problem to be studied is fundamental defining some

concepts of the processes involved in the phenomena transport.

2

Advection

Advection is a transport mechanism of a substance in which the flow is

unidirectional and whose property of the substance transported does not change

with the fluid in motion. Mathematically, the fluid motion by advection is

described as a vector field, and the substance transported is generally described as

a scalar concentration. Transport of pollutants in a river is an example of this

process.

Diffusion

This process is referred to the movement of the mass by random water motion. In

small scales, molecular diffusion result from de random Brownian motion of

water molecules. In large scales a random motion also occur due to the presence

of eddies which is called turbulent diffusion. In both processes the mass is

transported by difference of gradients moving from sections of high to low

concentrations.

Fick’s Fisrt Law

The Fick’s Law states that mass flux of solute is proportional to the gradient of

the solute concentration. In x direction, the linear transport equation is:

x

CDqx

Where qx is the mass flux in the x direction (ML-2

T-1

), D is the diffusion

coefficient (L2T

-1), and C is the concentration (ML

-3).

Peclet Number(Pe)

This number is a measure of the rate of advection of a flow to its rate of diffusion.

It measures the importance of the advection to diffusion; higher Peclet number

means that the advection is more important.

diffusiveofrat

transportadvectiveofrate

E

LuPe

__

___ , u is the velocity, L is the length, and E is the

turbulent diffusion.

3. Problem statement

From the articles reviewed, a 2-dimensional steady and uniform turbulent

flow, with mild slope in the channel are considered to formulate the advection-

diffusion equation that governs the transport of suspend sediments and chemical

substances in the domain considered ( in this case 0 < z < h). Figure 1 shows the

reference system used for this end, with x-coordinate along the river and a z-

3

coordinated vertical normal to the bed. Also, h is the normal depth, u(z) is the

time average stream velocity, q is flow in the river, and is the concentration of

suspend sediments. Likewise, it is assumed that the channel is carrying a steady

uniform suspend load and that the solid particles behave as those of the fluids,

with difference that solid particles tend to settle with a fall velocity fw . In

addition to this, mass exchange with the bed is not considered which means the

interaction between the groundwater and the river flow is ignored.

3.1 Transport of suspend sediments

By the conservation mass, the transport equation for the suspended

sediments is (Ng and Yip 2001):

zE

zxE

xzw

xu

tzxf

Where ),,( tzx is the suspend particle concentration, t is time, u(z) is time local

flow velocity (advection velocity), Ex(z) and Ez(z) are the horizontal and vertical

eddy diffusivity. For this case, the boundary conditions are following:

0z

Ew zf

To derive equation 2, steady uniform suspended sediment is considered which

implies that the rate of deposition is equal along the depth. A steady uniform load

indicates that the local rate of change of the suspend concentration at a given

point and the settle gradient horizontal of the particle concentration (the advective

derivative, advection in this direction), and the horizontal diffusion are neglected.

Where )(z is the suspend sediment concentration, wf is the settling velocity.

In equation 2, boundary conditions mean that net flux on the bottom and top of

the river or channel is zero.

Figure1. Reach of the river showing the profiles of local average

velocity and the suspend sediment concentration (Revelli and

Ridolfi 2002).

(1)

at z = 0,h (2)

4

From equation above, it is possible to derivate the vertical profile of the sediment

concentration, solving, integrating, and using the boundary conditions of the

following way:

z

z

fz

z

f

zz

z

fz

z

f

z

f

zf

zE

wzz

E

w

zz

E

wz

E

w

E

zw

zEw

00

0

00

exp)0(

)(

))0(

)(ln(ln0

Then, the equilibrium vertical profile for the suspend sediment concentration is

given by the following equation:

z

z

fz

zE

wz

0

'

'0)(

exp)( (3)

Where 0 is a reference sediment concentration at the bed level z = 0.

3.2 Transport of chemical species

When the pollutant is discharged to the stream, the chemical specie is

partitioned in dissolved (aqueous) and sorbed (solid) phases; therefore, the total

mass concentration (mass of chemical per bulk volume) Ctot (x,z,t) is:

stot CCC (4)

Where C(x, z, t) is the mass of aqueous phase per volume of water, and Cs (x, z, t)

is the mass of the sorbed concentration per mass of solid. The, the mathematical

model that represents the chemical transport in the domain 0 < z < h is the

following:

z

CE

zx

CE

xz

Cw

x

Cu

t

C totz

totx

sf

tottot

The firs term represent the local rate of change of the total mass concentration at a

given location. The second term is the change (gradient) of totC in the x direction

as result of the advection of the particles from one location to another where the

value of chemical mass is different and where u is the advection velocity. The

terms of right side are the horizontal and vertical diffusion of the total mass

concentration.

At the free surface and the bottom of channel, the vertical net flow will be zero;

then the boundary condition is:

(5)

5

0z

CECw tot

zsf at z = 0, h (6)

Using condition (2), the equation (4) can be written the following way:

z

C

z

CE

zx

CE

xx

Cu

t

C sz

totx

tottot (7)

For 0 < z < h, the boundary condition is:

hzz

C

z

CE s

z ,00 (8)

It is noted that vertical diffusion has been breakdown in two components: vertical

gradient of the aqueous and sorbed phases respectively.

Also, assuming that the sorption rate can is described by a first- order linear

kinetics equation:

sfs kCCk

t

C (9)

Where the kf and k are the forward and reverse rate constants for the sorption

reaction. Dissolved and solid phases will be in chemical equilibrium when the

reaction is fast compared with other processes; therefore the ratio of their

concentration is given by:

d

fs Kk

k

C

C (10)

Where Kd is the sorption partition coefficient which depends on the properties of

both sorbed (solid) and dissolved phases. When the concentration of the dissolved

phase tends to be small, Kd is bigger.

Now substituting (10) into (9), the first- order linear kinetics can be expressed as:

)( sds CCKk

t

C (11)

Where, the reverse reaction rate k can be interpreted as a sorption rate constant.

Furthermore, eq. (11) means that the rate of change of the sorbate concentration

(mass solid phase of chemical sorbed) regard the change of time is linearly

proportional to the deviation from local equilibrium.

6

4. Decay process

Decay process of chemical substances in a river, as result of transport

phenomenon, is very important in the analysis and chemical reactions of suspend

solids. From articles studied, Revelli and Ridolfi (2002) considered this aspect,

apart from the effect of sorption –desorption kinetics. Decay for dissolved and

sorbed phases with corresponding reaction rates is pointed in this project. Using

power laws, reactions can be modeled of the following way:

m

sss

n CRCR , (12)

Where R and Rs are the reactions for the dissolved and sorbed phases. s, are

constant values of decay rate for the dissolved and solid (sorbed) phases,

respectively. In the equation 12, if the power values n and m are equal to zero, the

respective reaction will be equal to a constant decay rate which indicate an

irreversible degradation of a reactant independent of its concentration. On the

other hand, the reactant can be represented as a variation of its concentration

regard to the time:

m

ssssn CR

t

CCR

t

C, (13)

For m =n = 0, the solution will be:

tCtCCt

C0

0 )( and tCC sss 0 (14)

Where C0 is the initial concentration at t =0. Equation above represents a zero-

order reaction.

Introducing (7) in (6), we have the following equation (Revelli and Ridolfi 2002):

m

ss

nsz

totx

tottot CCz

C

z

CE

zx

CE

xx

Cu

t

C (15)

Equation (15) is a two dimensional mathematical model for the transport and fate

of the pollutant in which decay processes for the dissolved and solid phase are

considered. On the other hand, the concentration of the solid phase (Cs) is

governed by the kinetics of the sorption exchange with the dissolved phase and

the decay reaction in the solid phase.

In the domain 0 < z < H, considering decay process, the behavior of the sorbate

concentration may be described through of mathematical model:

7

m

sssds CCCKk

t

C)( (16)

Equation above indicates that rate of change of the sorbate concentration is linear

proportional to deviation of the equilibrium subtracted by its process of decay.

The boundary conditions for (15) are the same of those pointed in the expression

(8) which considers zero flux for the free surface and the bed of the channel or

river.

The mathematical model (15), the sediment profile equation, and the boundary

conditions pointed respectively, are the expressions that describe the interaction

between the advection-diffusion-decay process and the suspend load.

Additionally to terms described in (5), in model (15), it can be noted that the right

side terms, apart from horizontal and vertical diffusion, the decay processes

(subtracting) for the dissolved and solid phases have been considered (third term).

Also, it should be observed that in the third term of the right-hand side , and s

will decrease with an increase of the nonlinearity and an increase the initial

concentration (Revelli and Ridolfi 2002).

5. Scales

5.1 Spatial Scales

Ratio of the longitudinal dispersion coefficient to Eddy diffusivity

It should be noted in equation (15) that longitudinal diffusion is controlled by the

eddy diffusivity Ex, but its effective value after a long time of transport, the shear

dispersion coefficient D is expected to dominate in controlling the longitudinal

spreading of the solute and sediments. This behavior is as resulted from the

velocity variation associated with turbulent diffusion in the z-direction (Ng,

2000). For a steady two dimensional turbulent open channel flow, the following

relationship has been established:

huDhuEE zx **

____

86.5,07.0 (17)

Where the over bar denotes the depth average, h is the channel depth, D is the

dispersion coefficient, u* is the shear or friction velocity given by the following

expression:

2/12/1

* )sin()/( ghu b (18)

8

Where b is the bottom shear stress, is the fluid density, g is the gravity, and the

is the slope of the river or channel. In (17), h = D/5.86u*, substituting h in Ez,

it is found that:

1)10(86.5/07.0/ 2__

ODEz (19)

which was used as a perturbation parameter.

Ration of longitudinal to vertical length scales

The focus is on the mass transport after long time of produced of the discharge. In

this condition, the longitudinal scale “L” for the chemical spreading substances

will be much longer than the flow depth “h” of the river. It is assumed that the

transport rate due to longitudinal dispersion is two orders smaller than that due the

vertical diffusion.

2

z

CE

zx

CD

xz

(20)

Also, it is assumed that vertical diffusion is one order of magnitude greater than

the longitudinal advection, then:

)(Oz

CE

zx

Cu z

(21)

Taking account the expressions (17) and (19), the following relationship can be

defined:

)(Ox

CD

xx

CE

xx

(22)

Substituting the longitudinal dispersion value from (22) into (20), we get the

relationship:

)( 3Oz

CE

zx

CE

xzx

(23)

After that, and using (20), we get:

)()( 2/33

2

OL

h

L

h (24)

9

From (19), the value was approximated to 10-2

, then substituting this value in

(24), we get that length scales is of order 10-3

, where h is of order of meters and L

of order kilometers.

Ration advection to vertical diffusion rates

The relation of the advection to the rate of vertical diffusion is a measure of the

Peclet number and can be estimated using (17).

*

____

07.0 u

uO

E

huP

z

e

(25)

Where __

u is the depth average velocity. Now using (21) and length scale (24) we

can get an order estimate for the Peclet number:

)( 2/1

2/3OPe

(26)

Physically, expression (26) means that longitudinal dispersion time-scale would

be one-half order of magnitude slower than advection (Chiu-On Ng 2003).

Ratio of desorption to vertical diffusion rates

This ratio is expressed through of Damkohler number and using the relation (17),

it can be

)07.0

(*

2

u

khO

E

khD

z

a (27)

Where k is the first- order sorption rate constant. The sorption kinetic is often

modeled with rate limiting diffusion into a spherical particle due to the aggregate

nature of the suspend solids and sediments. In that sense, assuming a value of

order 10-8

cm2s

-1 as an effective diffusion coefficient, the desorption rate was

estimated as k = O(10-4

s-1

). Likewise, if we assume typical values for u*=0.01ms-1

and h = 5m, Da is more or less of the order of unity and which means that the

sorption rate is comparable to the vertical turbulent diffusion.

On the other hand, it should be noted that the fall velocity wf of sedimentation

depends of shape and size of particle. A typical value of the stream velocity is of

order of 1-0.1 ms-1

; consequently, it is assumed that:

)(/ 2/1__

Ouwf (28)

Ratio of dissolved phase to the advection

10

This ratio can be estimated choosing a half-life time of the chemical, t1/2, as a

feature scale of the decay and considering the reaction in (12):

xCu

C

nT

C

xCu

C nnn

t

n

/)1(

)12(

/ 2/1

11

0 (29)

Also, the following dimensional quantities are considered:

0

^__^^

/,/,/ tCCCuuuLxx (30)

5.2 Time Scales

Three times scales, T0, T1, and T2 are important in the process and which

correspond to the times scales for the vertical diffusion of flow, longitudinal

advection and longitudinal dispersion. T0=h2/Ez, T1=L/u, with L is the

characteristic distance and u is the average velocity, and T2 = L2/D. From these

magnitudes, we can get:

2210

2

__

2 1:

1:1:::: TTT

D

L

u

L

E

h

z

(31)

Or

)(//),(//__

21

__2

10 OLuDTTOLEuhTT z (32 a,b)

Finally, with the scaling described above, the order of magnitude of terms of the

main equations and boundary conditions can be estimated. In other words, the

order of parameter is introduced in the governing equations. Then, the sediment

transport equation is:

zE

zxE

xzw

xu

tzxf

2 (33)

m

ss

nsz

totx

tottot CCz

C

z

CE

zx

CE

xx

Cu

t

C 2 (34)

The sorption kinetic equation is:

m

sssds CCCKk

t

C)( (35)

Being the boundary condition the same that described in equation (8).

11

6. Discussion

The equations described above come from three main papers reviewed to

achieve the objective of this project (Ng 2000, Ng and Yip 2001, and Revelli and

Ridolfi 2002). Ng 2000 developed a set of mathematical equations to analyze the

effect and the sorptive exchange on the transport of chemical in a sediment open

channel flow. For this end, a method scale of homogenization is used to derivate

the effective transport equation in 1D for chemical species where the sorptive

phase (solid) exchange is produced between the water depth and a steady uniform

distribution of suspend solids. Using power series, perturbation equations of the

parameter are obtained, as well as for the time scales:

2122

2

11 //....,..........),(),(),(),( tttandCCCCCCCC sssoos (36)

According to (35) expression without includes the decay process, it must be noted

that Co and Cso are independent of z coordinate whose relation is: ),(),( txCKtxC odso

meaning that for smaller concentration of the aqueous phase the sorption partition

coefficient increases (see equation 10). For that, two assumptions are taken

account that concentrations are uniform across of the channel depth and the solid-

water partition is in equilibrium; which means total concentration is:

otot RCC 0 (37)

where R is the retardation factor defined as the ratio of the total concentration to

the dissolved phase (aqueous). In addition R can be determined as a function of

the sorption partition coefficient and the variation of the vertical profile of the

sediment. Introducing this consideration into (34) and taking the depth average

with zero flux boundary conditions, a first-order effective transport equation was

established for which is assumed that this transport only depend on the advection.

Likewise, developing the scale expansion for the sorption kinetic equation

described in (35) and the first order effective transport equation, Cs1 and Ctot1 are

estimated as a function of the sorption partition coefficient and the change of

concentration in the horizontal direction. Equations established for Ctot0 and Ctot1

and substituted into (34) followed by the depth averaging and using the first-order

effective transport equation, the second-order mathematical model is established.

Finally, combining the first and second order effective transport equation and

neglecting the order of the parameter , the following effective transport equation

in 1D was built:

2

2_

/x

CDDRRE

x

Cu

t

C osTx

oe

o (38)

Where the first and second term of the left side are the local rate of change of the

concentration and the advection in the x direction (ue is the effective velocity of

the advection). The term of the right side is the dispersion in the same direction

12

where Dt and Ds are the effective dispersion coefficients which are composed by a

Taylor dispersion coefficient and a component due to a finite rate of sorption

exchange between dissolved and solid phase on suspend sediments (Ng 2000). __

/ RRExis the effective longitudinal eddy diffusivity. Assumptions such as vertical

scale shorter than the longitudinal length scale, a much longer time scale for

longitudinal advection than the vertical diffusion, first order kinetic for the

sorption-desorption, and other described in previous section. Also it is assumed

that transport is governed by the advection at an effective velocity ue which Is

proportional to the ratio between the average velocity implied by a retardation

factor and the mean retardation factor.

On the hand, Ng and Yip 2001 taking as base the work developed by Ng 2000,

formulated a theory for the transport of chemical species in open channel flow. As

in the first paper reviewed, chemical species is partitioned in two phases: a

dissolved phase in the water depth and a sorbed phase into suspend solid particles

whose behavior is due to the sorption kinetic in which the advection and

dispersion of the solute cloud are functions are the particle cloud (Ng and Yip

2001). Most equations (1-36) are used in this paper except those that include

decay processes which are material of discussion of the third paper studied. Using

the homogenization method to derivate the effective mass transport equations for

Taylor dispersion, two very important mathematical expressions are established.

This technique is a multi scale procedure that allows derivate the coefficient of

dispersion at long time for any Taylor dispersion problems (Ng 2003). The

effective transport equation for the sediment concentration in 1D is written as

follow:

xxD

xu

t

bs

bs

b (39)

Where ),( tx is the sediment concentration and it will be zero at the bed level

(z=0), first and second terms of the left side of this equation were defined in

previous sections, being us the effective advection velocity for the sediment (Ng

and Yip 2002) and defined as: ffuus / which is a depth average speed,

weighted by the sediment distribution factor f which results from a equilibrium

between the sedimentation and the turbulent mixing and can be derived from

expression (2) with the respective boundary conditions. On the right side, the only

one term is the horizontal diffusion where Ds is the diffusivity coefficient which is

set up as Txs DffED / . The first term of this expression is depth-averaged eddy

diffusivity in the horizontal direction and weighted by the sediment factor f, the

second one is the Taylor dispersion coefficient.

The effective transport equation for chemical pollutants is:

x

CD

xx

Cuu

t

C oc

occ

o )( ' (40)

13

Where uc is the effective advection velocity for the solute concentration which is

given by RRutxuc /),( which is a velocity weighted by the retardation factor R

for the chemical partitioning. This factor is given by odKR 1 . Where the

sorption partition coefficient can be computed by (9). Co is the solute

concentration and Dc is the effective dispersion coefficient which is a depth

average longitudinal eddy diffusivity weighted by the retardation factor R ( Ng

2000) and it is defined by kcTcxc DDRRED / , where Dkc is a dispersion coefficient

due to the influence of the kinetics of the solid water sorptive exchange. DTc is

inversely proportional to the eddy diffusion in the vertical direction weighted by a

retardation factor and proportional to the variance of the advective velocity for the

chemical pollutant. In addition, from (39) and (40), we can say that the sediment

and the solute are advected to different velocities.

Revelli and Ridolfi (2002) used the same theory developed by Ng and Yip (2001)

but they introduced the effect of decay processes and the kinetics with the phase

sorbed by suspended sediments (16) as it was discussed in previous papers. In two

dimensional, the decay process is considered in (15) and in terms of the parameter

through equation (35). The one-dimensional depth-averaged transport equation

for chemical pollutant which includes decay processes is established as follow:

)'''( 1

0

12

0

12

00

**

0

*

2

0

2*0**0*00 nmmnmn

mn

e CCCCCx

CD

x

Cu

x

Cu

x

Cu

t

C (41)

Where ue is the effective advective velocity for the concentration of the chemical

which is define by RRuue / . As it is noted, this velocity is proportional to the

average velocity weighted by the retardation factor R and inversely proportional

to the same average factor. R/* and RK m

ds /** being ***and the

effective no linear decay coefficient, u* and u

** are the effective coefficients for

two pseudo advective no linear terms (Revelli and Ridolfi 2002). D*

is the

effective coefficient for dispersion in the horizontal direction which can be

estimated with the velocity and effective velocity, retardation factor R, sorption

partition coefficient Kd and the concentration of the sediments. '''' ,'', and are

the coefficient of three no linear terms which are expressed as a function of the

effective no linear decay coefficients, Kd, R, and variance of the concentration of

sediment. Values of the exponent n and m are used to evaluate the hydro dynamic

of the nonlinearities of the chemical pollutant.

Now, let me to illustrate the importance of the bulk solid-water distribution ratio

0dK in regulating the influence of the suspend load on the chemical dynamics.

For this end, the effective velocity us give by equation below:

14

3

2^

2

)1(2)2(26.5

3

26.5 e

kR

K

kRfu d

be

Where α is a suspension number proportional to the depth of channel weighted by

the settling velocity and inversely proportional to the vertical eddy diffusion. The

term fb is the slip velocity relative to shear velocity (Ng 2000) and can be

computed through equation:

kE

h

kf

s

b

63.25.8ln

1

Also the depth average velocity is given by:

kfu b

3

26.5_

Where k is the Karman constant equal to 0.4, h/Es =1000, for α = 0.1, 1, 5, and 10,

and 0< 0dK <5. Excel was used to calculate the effect of these parameters. Figure

2 shows that the effective advection velocity decreases when 0dK increases

whose effect is more notable for higher suspension number. Also, when 0dK

trends to zero the sediment transport loses influence and the transport of the

dissolved phase is given. On the other hand, higher values of this parameter

indicate that the sediments are more important in the chemical dispersion. In

addition, the bulk solid water distribution ratio expresses how the chemical is

distributed in solid and aqueous phase (Rivelli and Ridolfi 2002).

Figure 2. Effective advection velocity as function of the bulk solid-water

distribution and suspension number α.

On other hand to study of the effect of decay process, the linear decay is

addressed using the following equation:

0.84

0.87

0.90

0.93

0.96

0.99

1.02

0 1 2 3 4 5

α=0.1

α=1

α=5

α=10

dK

__^

/ uu e

(42)

(43)

(44)

15

0

0***

)1(

)1(

d

de

Ke

Kee (45)

Where e is the effective coefficient of reaction and β is defined as the ratio

between the rate decay of the sorbed and dissolved phase. Figures 3 and 4 show

the effects of these parameters. It is noted that the ration of the decay constant

decrease with small β values. Also it decrease when the suspension number α

increment.

Figure 3. Linear Decay. Influence of the bulk solid-water distribution ratio, β, and

the ratio between decay constants on the effective decay, for α=0.1

Figure 4. Linear Decay. Influence of the bulk solid-water distribution ratio, β, and

the ratio between decay constants on the effective decay, for α=10.

0.00

0.30

0.60

0.90

1.20

1.50

1.80

0 2 4 6 8 10

β=1.5

β=1.1

β=0.9

β=0.5

0.00

0.30

0.60

0.90

1.20

1.50

0 2 4 6 8 10

β=1.5

β=1.1

β=0.9

β=0.5

/e

dK

/e

dK

16

7. Conclusions

Interaction between the suspend solids and chemical pollutant in open channel

flow has been studied through the revision of three major papers. Mathematical

equations that governs the transport of suspend solids and chemical species, and

boundary conditions were described. These models take account the advection,

turbulent diffusion, and the exchange kinetics between the sorbed and dissolved

phases. Additionally to this, decay processes for the same phases were included in

equation (15), for instance, which describe the interaction between the advection-

difussion-decay processes and the suspend sediment. Likewise, longitudinal and

vertical scales for the dispersion, vertical diffusion and longitudinal advection are

pointed for the perturbation analysis, and time scales for the diffusion across the

flow, and longitudinal dispersion and advection.

From the papers reviewed and from equations described in previous sections, it is

shown that exchange kinetics given between the dissolved and solid phase can

influence on the advection and dispersion of chemical substances in an open

channel flow; therefore, the advection velocity and dispersion coefficients for the

chemical species are shown to depend on the sediment concentration via effect of

sorption partitioning and kinetic phase exchange (Ng and Yip 200).

Effective transport equations in 1D are discussed and understood for the sediment

concentration and chemical including the linear and nonlinear decay. The

Mathematical model in (41) shows the influence of the suspended sediments and

decay process on the alteration of the mechanism of transport. Additionally, the

influence of the bulk solid-water distribution ratio 0dK on the advective velocity

and on the ration of the rate decay between the effective coefficient of reaction

and the coefficient constant coefficient decay for the dissolved phase has been

addressed. It is noted that when 0dK increases the effective advective velocity

decrease and it increases when 0dK tends to zero.

Required reading:

Rivelli, R., and Didolfi, L. (2002). Influence of suspended sediment on the

transport processes of nonlinear reactive substances in turbulent streams.

Journal Fluid Mechanics, vol. 472, pp. 307-331.

Ng, C.O., and Yip, T.L. (2001). Effects of kinetic sorptive exchange on solute

transport in open-channel flow. Journal Fluid Mechanics, vol. 446, pp. 321-

345.

Ng, C.O. (2000). Dispersion in sediment –laden stream flow. Journal of

Engineering Mechanics. Vol. 126, No. 8, pp. 779-786.

17

Supplementary reading:

Bombardelli, F.A., and Jha, S.K. (2009). Hierarchical modeling of the dilute

transport of suspended sediment in open channels. Environmental Fluid

Mechanic, 9:207–235. DOI 10.1007/s10652-008-9091-6.

Chapra, S.C. 1997. Surface Water Quality Modeling. McGraw-Hill. United States

of America.

Kundu, P.K., and Cohen, I. 2008. Fluid Mechanics. Elsevier-AP. Fourth Edition,

United States of America.

Lick, W. 2009. Sediment and Contaminant Transport in Surface Waters. CRC-

Press. United States of America.

Liu, Q.Q., Shu, A.P., and Singh, V.P. (2007). Analysis of vertical profile of

concentration in sediment –laden flows. Journal of Engineering Mechanics.

Vol. 133, No. 6, pp 601-607.

Ng, C.O. (2003). Homogenization technique applied to problems of Taylor

Dispersion. Department of Mechanical Engineering, The university of

Hong Kong.

Partheniades, E. 2009. Cohesive Sediment in Open Channels: Properties,

Transport, and Applications. Elsevier-BH. United States of America.