Model Streeter - Phelps

Effect of Organic Wastes on Stream Ecosystems StreeterStreeter--Phelps Model Phelps Model DO sag curveDO sag curve

Many equations and computer programs are available Many equations and computer programs are available today to describe the quality of water in streams, rivers today to describe the quality of water in streams, rivers and lakesand lakesand lakesand lakes

The most prevalent is the Streeter Phelps equation. The most prevalent is the Streeter Phelps equation. Addition of wastewater (BOD) typically causes a slow Addition of wastewater (BOD) typically causes a slow

decrease in Odecrease in O22, followed by a gradual increase close to , followed by a gradual increase close to the dissolved oxygen (D.O.) saturation concentrationthe dissolved oxygen (D.O.) saturation concentration

Streeter-Phelps Model Assumptions of the ModelAssumptions of the Model

stream is an ideal plug flow reactorstream is an ideal plug flow reactor steadysteady--state flow and BOD and DO reaction state flow and BOD and DO reaction steadysteady--state flow and BOD and DO reaction state flow and BOD and DO reaction

conditionsconditions The only reactions of interest are BOD The only reactions of interest are BOD

exertion and transfer of oxygen from air to exertion and transfer of oxygen from air to water across airwater across air--water interfacewater interface

Streeter-Phelps Model Mass Balance for the ModelMass Balance for the Model

Not a SteadyNot a Steady--state situationstate situationrate Orate O2 2 accum. = rate Oaccum. = rate O22 in in rate Orate O22 out + prod. out + prod. cons.cons.rate Orate O2 2 accum. = rate Oaccum. = rate O22 in in 0 + 0 0 + 0 rate Orate O22 cons.cons.

Both reoxygenation and deoxygenation are 1Both reoxygenation and deoxygenation are 1stst orderorderrate of deoxygenation = rate of deoxygenation = --kk11CCkk11= deoxygenation constant, function of waste type and = deoxygenation constant, function of waste type and

temperaturetemperature

Streeter-Phelps Modelrate of reoxygenation = krate of reoxygenation = k22DDD = deficit in D.O. or difference between D = deficit in D.O. or difference between

saturation and current D.O.saturation and current D.O.kk22 = reoxygenation constant= reoxygenation constant

[ ]( )2

3

21)20(21

2025.19.3

H

vkT

=

Streeter-Phelps Model

WhereWhereT = temperature of water, T = temperature of water, CCH= average depth of flow, mH= average depth of flow, mH= average depth of flow, mH= average depth of flow, mv = mean stream velocity, m/sv = mean stream velocity, m/s

Oxygen DeficitOxygen DeficitD = S D = S CC

D.O. deficit = saturation D.O. D.O. deficit = saturation D.O. D.O. in the waterD.O. in the water

Streeter-Phelps Model

Streeter-Phelps Model

Deoxygenation rate is equivalent to BOD of Deoxygenation rate is equivalent to BOD of wastewaste

rro o = k= k11 LLtt rro o = k= k11 LLttLLtt = L= Looee--KtKt

LLoo or L = ultimate BOD of the wastewater and or L = ultimate BOD of the wastewater and stream water mixturestream water mixture

Streeter-Phelps Model

In terms of the deficit with timeIn terms of the deficit with time

tkLez

DkzkdtdD

1

21

=

=

Streeter-Phelps Model Substiting and integrating yields the following Substiting and integrating yields the following

equationsequationstk

o

tktko eDeekk

LkD 221 )(1 +

=

kk 12

tkoc

tko

eLkkD

DkeLkdtdD

1

1

2

1

21 0

=

==

=

o

oc Lk

kkDkk

kkt

1

12

1

2

12

)(1ln1

Streeter-Phelps Model Example:Example:

Wastewater mixes with a river resulting in aWastewater mixes with a river resulting in aBOD = 10.9 mg/L, BOD = 10.9 mg/L, DO = 7.6 mg/LDO = 7.6 mg/LThe mixture has a temp. = 20 The mixture has a temp. = 20 CCThe mixture has a temp. = 20 The mixture has a temp. = 20 CCDeoxygenation const.= 0.2 dayDeoxygenation const.= 0.2 day--11Average flow = 0.3 m/s, Average flow = 0.3 m/s, Average depth = 3.0 mAverage depth = 3.0 mDO saturated = 9.1 mg/LDO saturated = 9.1 mg/L Find the time and distance downstream at which the Find the time and distance downstream at which the

oxygen deficit is a maximum?oxygen deficit is a maximum? Find the minimum value of DO?Find the minimum value of DO?

Streeter-Phelps Model Initial DeficitInitial DeficitDDoo = 9.1 = 9.1 7.6 = 1.5 mg/L7.6 = 1.5 mg/L Estimate the reaeration constantEstimate the reaeration constantEstimate the reaeration constantEstimate the reaeration constant

[ ]( )( ) 12/3

2120202/1

2/3

2/1

2 day 14.0)0.3(025.1)/3.0(9.39.3

===

m

sm

Hvk

Streeter-Phelps Model

Calculate the time at which the maximum deficit Calculate the time at which the maximum deficit is reached, with tis reached, with tcc::

kkDOkt o

)(1ln1 122

=

mdaysdayssmvtxdays

LkkkDO

kk

kkt

cc

o

oc

300,6967.2/400,86/3.067.2

9.102.0)2.041.0(5.11

2.041.0ln)2.041.0(

1

)(1ln11

12

1

2

12

===

=

=

=

Streeter-Phelps Model

The maximum DO deficit is:

1 1=

tkoc eLk

kD

mg/L 1.3

e mg/L) 9.10(41.02.0 ))(2.67days(0.2day

2

1

=

=

oc k