Typical DO Sag Curve David Reckhow CEE 577 #16 2 1 CEE 577 - PDF document

CEE 577 Lecture #16 10/23/2017 Updated: 23 October 2017 Print version Lecture #16 Streeter ‐ Phelps: Reaeration & Dams (Chapra, L22 & L23) David Reckhow CEE 577 #16 1 Typical DO Sag Curve David Reckhow CEE 577 #16 2 1

CEE 577 Lecture #16 10/23/2017 Streeter Phelps Equation V dD   k VL k VD d a dt   k t L L e d o 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 = D o . This yields the DO sag   k L       k t d o k t k t D D e e e a r a  o k k a r where = stream deficit at time t, [mg/L] D = initial oxygen deficit (@ t = 0), [mg/L] D o David Reckhow CEE 577 #16 3 And recognizing that: t=x/U   k L  k xU  k xU  k xU    d o D D e a e r e a  o k k a r David Reckhow CEE 577 #16 4 2

CEE 577 Lecture #16 10/23/2017 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,      ( 1 D k k k a  1   ln  o a r  t = crit   k - k  k L  k a r r d o = time at which maximum deficit (minimum DO) t crit occurs, [days] David Reckhow CEE 577 #16 5 Special Case for D and t c  When k a and k r are equal:    1 D    0 1 t   c   k L r o      k t D L k t D e r o r o From Davis & Masten, page 290-291 David Reckhow CEE 577 #16 6 3

CEE 577 Lecture #16 10/23/2017 Critical concentration  Once the critical time is known, you can calculate the c min k    k r t d c c L e crit min s o k a  this is an abbreviated form of the full equation, which is only valid for t crit This differs from Chapra’s equation 21.14 on page 397, Why?????? David Reckhow CEE 577 #16 7 Estimating Reaeration Rates (d ‐1 ) (U in ft/s; H in ft)  O’Connor ‐ Dobbins formula 0 . 5 U k a  12 . 9  based on theory 1 . 5 H  verified with some deep waters  Churchill formula  Tennessee Valley U k a  11 . 6 1 . 67  deep, fast moving streams H  Owens formula  British 0 . 67 U k a   shallow streams 21 . 6 1 . 85 H David Reckhow CEE 577 #16 8 4

CEE 577 Lecture #16 10/23/2017 The method of Covar (1976) Values for k a are in units of d -1 David Reckhow CEE 577 #16 9 Using the Covar approach  Determine proper Domain H (ft) U (ft/s) Formula <2 Any Owens <1.2H 0.34 >2 O’Connor- Dobbins >1.2H 0.34 >2 Churchill David Reckhow CEE 577 #16 10 5

CEE 577 Lecture #16 10/23/2017 More on k a estimation  Tsivoglou & Wallace (1972) method  k a = 0.88US, for Q = 10 ‐ 300 cfs  k a = 1.8US, for Q = 1 ‐ 10 cfs  Temperature correction   =1.024    o 20 T C k k T 20 o C David Reckhow CEE 577 #16 11 Dam Reaeration Temperature ( o C)  Butts and Evans (1983):     1 0 38 . ( 1 011 . )( 1 0 046 . ) r abH H T Ratio of deficit Difference in water above and Empirical elevation below the dam coefficients which relate to water quality and dam type (Table 20.2) David Reckhow CEE 577 #16 12 6

CEE 577 Lecture #16 10/23/2017 Wind dependent reaeration formulas David Reckhow CEE 577 #16 13  To next lecture David Reckhow CEE 577 #16 14 7