Longitudinal Dispersion From Fischer et al., 1979 m/s m 2 s -1 - PDF document

CEE 577 Lecture #10 10/23/2017 Updated: 23 October 2017 Print version Lecture #10 (Rivers & Streams, cont) Chapra, L14 (cont.) David A. Reckhow CEE 577 #10 1 Longitudinal Dispersion  From Fischer et al., 1979 m/s m 2 s -1 Width (m) 2 2 U B  0 011 . E * HU Mean depth (m) Where the Shear Velocity is: *  U gHS David A. Reckhow CEE 577 #10 2 1

CEE 577 Lecture #10 10/23/2017 Lateral Mixing  Lateral or transverse dispersion coefficient for a stream: Mean depth lat  0 6 * Shear velocity . E HU  Length required for complete mixing: Side discharge: Center discharge: 2 2 U B U B  010  0 40 . . L L m m E E lat lat Width David A. Reckhow CEE 577 #10 3 General Stream Geometry  Chapra’s nomenclature for discharge coefficients   Velocity b U aQ   Depth   H Q  Width  f B cQ     Where: b 1 Because Q=UHB f David A. Reckhow CEE 577 #10 4 2

CEE 577 Lecture #10 10/23/2017 Thomann & Mueller, problem 2.1 Sample Problem The Black River, NY between MP 74.2 and MP 64.7 is to be characterized as a constant flow - constant area reach. Assume the following cross-sectional area (A c ) were measured for the given flows: Q 5 0 0 7 5 0 1 3 0 0 2 2 0 0 3 4 0 0 (c fs ) A 6 8 0 9 5 0 1 1 0 0 1 6 0 0 2 2 0 0 c MP 74.2 (ft 2 ) MP 64.7 Estimate travel time through this reach for flows of 600 and 3000 cfs David A. Reckhow CEE 577 #10 5    H Q  f B cQ      ( ) f A BH cQ 2000 A  0 . 589 18 Q . 3 Area, sq. ft. 1000 800 700 b[0]=2.9235296469 b[1]=0.581536813 600 r ²=0.9788480936 500 500 600 700 800 1000 2000 3000 4000 5000 Flow (cfs) David A. Reckhow CEE 577 #10 6 3

CEE 577 Lecture #10 10/23/2017 Manning Equation  Derived from the momentum balance  relates velocity to channel characteristics including slope Slope of energy grade  1486 . line = slope of stream 23 12 ft/s U R S e bed for constant H & U n Manning’s roughness Hydraulic Radius (ft) coefficient 0.012-0.100 =A c /wetted perimenter see Table 14.3  A c /(B+2H) David A. Reckhow CEE 577 #10 7 Manning Equation adapted to a Trapezoidal section 1 . 486 Q  2 1 3 A R S 2 c e n  Area, perimeter and hydraulic radius can all be expressed as a function of depth      5 / 3 1 . 486 B sy y  substitute these into the  1 / 2   o Q S Manning Equation and 2 / 3 e n   2 2 1 calculate “y” from known B y s o “Q     A B sy y 1 y c o   2  2 1 P B y s s o   B o  A B sy y   c o R  2  P 2 1 B y s David A. Reckhow CEE 577 #10 8 o 4

CEE 577 Lecture #10 10/23/2017 Distributed Systems  Lecture #9 in Chapra’s book  systems that have spatial resolution  Ideal Reactors  c  V     J A J A reaction in c out c t B J in J out H  x David A. Reckhow CEE 577 #10 9 Plug‐Flow Reactors (PRF)  c  V     J A J A reaction in c out c t     c c          V UcA U c x A k Vc     c c t x Combining and taking the limit as  x  0   c c    U kc   t x And for c=c o at x=0: Which at steady state is:   k x u c c e  o c 0    U kc  x David A. Reckhow CEE 577 #10 10 5

CEE 577 Lecture #10 10/23/2017 Plug Flow vs CSTR  First order reactions Q   k x u  c c e c c  o o Q kV  Mixed Flow: intermediate  read section 9.1.3 David A. Reckhow CEE 577 #10 11 Mixed Flow David A. Reckhow CEE 577 #10 12 6

CEE 577 Lecture #10 10/23/2017 Mixed Flow  c  V     J A J A reaction in c out c t                 c E c c c c                    V  Uc x A  U c x E x A k Vc               c c   t x x x x Consider mixing in the longitudinal direction  Peclet Number  rate of advective transport P e > 10, PFR-like LU e  P P e < 0.1, CSTR-like rate of dispersive transport E David A. Reckhow CEE 577 #10 13 Application of PRF to streams  Point sources Q r c r Qc o  Mass balance: Outfall: Q w c w  Water Flow    Concentration Q Q Q w r  Q c Q c  w w r r c  o Q Q w r David A. Reckhow CEE 577 #10 14 7

CEE 577 Lecture #10 10/23/2017 Assumptions David A. Reckhow CEE 577 #10 15 Chloride Problem 1.55 cfs/MGD  Determine the required industrial reduction in chlorides to maintain a desired chloride concentration of 250 mg/L at the intake Water intake Q w =6.5 MGD c w = 1500 mg/L Q=25 cfs c=30 mg/L Q T = 5 cfs c T = 30 mg/L David A. Reckhow CEE 577 #10 16 8

CEE 577 Lecture #10 10/23/2017 W Q Q x x+ x s x s x+ x x x+ x David A. Reckhow CEE 577 #10 17  To next lecture David A. Reckhow CEE 577 #10 18 9