MECH152-L22-1 (1.0) - 1 Powder Metallurgy. MECH152-L22-1 (1.0) - 2 Typical Parts.
(Chapra, L22 & L23)
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Lecture #16 Streeter-Phelps: Reaeration & Dams (Chapra, L22 & L23) David Reckhow CEE 577 #16 1 Updated: 23 October 2017 Print version
Transcript of (Chapra, L22 & L23)
CEE 577: Surface Water Quality Modeling(Chapra, L22 &
L23)
Streeter Phelps Equation
David Reckhow CEE 577 #16 3
This equation can be solved by separation of variables and integration, or by use of an integrating factor. The boundary condition is t = 0 @ D = Do. This yields the DO sag
where D = stream deficit at time t, [mg/L] Do = initial oxygen deficit (@ t = 0), [mg/L]
V dD dt
k td= −
k k e eo
k t d o
−− − −
( )D D e k L
k k e eo
a r
Ua r a= + −
And recognizing that: t=x/U
Critical Time The most stress is placed on the aquatic life in a stream when
the DO is at a minimum, or the deficit, D, is a maximum. This occurs when dD/dt = 0. We can obtain the time at which the deficit is a maximum by taking the derivative of the DO sag equation with respect to t and setting it equal to zero, then solving for t. This yields,
David Reckhow CEE 577 #16 5
tcrit = time at which maximum deficit (minimum DO) occurs, [days]
crit a r
ln (
1− −
Special Case for D and tc When ka and kr are
equal:
From Davis & Masten, page 290-291
Critical concentration
Once the critical time is known, you can calculate the cmin
this is an abbreviated form of the full equation, which is only valid for tcrit
David Reckhow CEE 577 #16 7
critrtk
a
d os e
k kLcc −−=min
This differs from Chapra’s equation 21.14 on page 397, Why??????
Estimating Reaeration Rates (d-1) O’Connor-Dobbins formula
based on theory verified with some deep waters
Churchill formula Tennessee Valley deep, fast moving streams
Owens formula British shallow streams
David Reckhow CEE 577 #16 8
5.1
5.0
67.16.11 H Uka =
David Reckhow CEE 577 #16 9
Values for ka are in units of d-1
Using the Covar approach
David Reckhow CEE 577 #16 10
H (ft) U (ft/s) Formula <2 Any Owens >2 <1.2H0.34 O’Connor-
Dobbins >2 >1.2H0.34 Churchill
More on ka estimation Tsivoglou & Wallace (1972) method
ka = 0.88US, for Q = 10-300 cfs ka = 1.8US, for Q = 1-10 cfs
Temperature correction θ=1.024
k kT C T C
o
o
= − 20
20θ
Dam Reaeration Butts and Evans (1983):
David Reckhow CEE 577 #16 12
r abH H T= + − +1 0 38 1 011 1 0 046. ( . )( . )
Ratio of deficit above and below the dam
Difference in water elevation
Temperature (oC)
Empirical coefficients which relate to water quality and dam type (Table 20.2)
Wind dependent reaeration formulas
To next lecture
Typical DO Sag Curve
Critical concentration
Using the Covar approach
More on ka estimation
Streeter Phelps Equation
David Reckhow CEE 577 #16 3
This equation can be solved by separation of variables and integration, or by use of an integrating factor. The boundary condition is t = 0 @ D = Do. This yields the DO sag
where D = stream deficit at time t, [mg/L] Do = initial oxygen deficit (@ t = 0), [mg/L]
V dD dt
k td= −
k k e eo
k t d o
−− − −
( )D D e k L
k k e eo
a r
Ua r a= + −
And recognizing that: t=x/U
Critical Time The most stress is placed on the aquatic life in a stream when
the DO is at a minimum, or the deficit, D, is a maximum. This occurs when dD/dt = 0. We can obtain the time at which the deficit is a maximum by taking the derivative of the DO sag equation with respect to t and setting it equal to zero, then solving for t. This yields,
David Reckhow CEE 577 #16 5
tcrit = time at which maximum deficit (minimum DO) occurs, [days]
crit a r
ln (
1− −
Special Case for D and tc When ka and kr are
equal:
From Davis & Masten, page 290-291
Critical concentration
Once the critical time is known, you can calculate the cmin
this is an abbreviated form of the full equation, which is only valid for tcrit
David Reckhow CEE 577 #16 7
critrtk
a
d os e
k kLcc −−=min
This differs from Chapra’s equation 21.14 on page 397, Why??????
Estimating Reaeration Rates (d-1) O’Connor-Dobbins formula
based on theory verified with some deep waters
Churchill formula Tennessee Valley deep, fast moving streams
Owens formula British shallow streams
David Reckhow CEE 577 #16 8
5.1
5.0
67.16.11 H Uka =
David Reckhow CEE 577 #16 9
Values for ka are in units of d-1
Using the Covar approach
David Reckhow CEE 577 #16 10
H (ft) U (ft/s) Formula <2 Any Owens >2 <1.2H0.34 O’Connor-
Dobbins >2 >1.2H0.34 Churchill
More on ka estimation Tsivoglou & Wallace (1972) method
ka = 0.88US, for Q = 10-300 cfs ka = 1.8US, for Q = 1-10 cfs
Temperature correction θ=1.024
k kT C T C
o
o
= − 20
20θ
Dam Reaeration Butts and Evans (1983):
David Reckhow CEE 577 #16 12
r abH H T= + − +1 0 38 1 011 1 0 046. ( . )( . )
Ratio of deficit above and below the dam
Difference in water elevation
Temperature (oC)
Empirical coefficients which relate to water quality and dam type (Table 20.2)
Wind dependent reaeration formulas
To next lecture
Typical DO Sag Curve
Critical concentration
Using the Covar approach
More on ka estimation
