Chapra, L14 (cont.) David A. Reckhow CEE 577 #10 1 Longitudinal - - PowerPoint PPT Presentation

Lecture #10 (Rivers & Streams, cont)

Chapra, L14 (cont.)

David A. Reckhow CEE 577 #10 1

Updated: 23 October 2017

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Longitudinal Dispersion

 From Fischer et al., 1979

David A. Reckhow CEE 577 #10 2

E U B HU = 0 011

2 2

.

*

Where the Shear Velocity is:

U gHS

* =

m2s-1 m/s Width (m) Mean depth (m)

Lateral Mixing

 Lateral or transverse dispersion coefficient for a

stream:

 Length required for complete mixing:

David A. Reckhow CEE 577 #10 3

E HU

lat = 0 6

.

*

L U B E

m lat

= 0 40

2

. L U B E

m lat

= 010

2

.

Side discharge: Center discharge: Mean depth Shear velocity Width

General Stream Geometry

 Chapra’s nomenclature for discharge coefficients

 Velocity  Depth  Width

David A. Reckhow CEE 577 #10 4

U aQ H Q B cQ

b f

= = = α

β

Where: b

f + + = β 1

Because Q=UHB

Sample Problem

David A. Reckhow CEE 577 #10 5

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 (Ac) were measured for the given flows:

Q (cfs) 500 750 1300 2200 3400 Ac (ft

2)

680 950 1100 1600 2200

MP 74.2 MP 64.7 Estimate travel time through this reach for flows of 600 and 3000 cfs

Thomann & Mueller, problem 2.1

David A. Reckhow CEE 577 #10 6

) ( f f

cQ BH A cQ B Q H

+

= = = =

β β

α α

Flow (cfs)

500 600 700 800 2000 3000 4000 5000 1000

Area, sq. ft.

500 600 700 800 2000 1000 b[0]=2.9235296469 b[1]=0.581536813 r ²=0.9788480936

589 .

3 . 18 Q A =

Manning Equation

 Derived from the momentum balance

 relates velocity to channel characteristics including

slope

David A. Reckhow CEE 577 #10 7

U n R Se = 1486

23 12

.

ft/s Manning’s roughness coefficient 0.012-0.100 see Table 14.3 Hydraulic Radius (ft) =Ac/wetted perimenter ≈Ac/(B+2H) Slope of energy grade line = slope of stream bed for constant H & U

Manning Equation adapted to a Trapezoidal section

 Area, perimeter and

hydraulic radius can all be expressed as a function of depth

 substitute these into the

Manning Equation and calculate “y” from known “Q

David A. Reckhow CEE 577 #10 8 2 1 3 2

486 . 1

e c

S R A n Q =

y

( ) ( )

1 2 1 2

2 2

+ + + = = + + = + = s y B y sy B P A R s y B P y sy B A

• c
• c

( )

[ ]

( )

2 / 1 3 / 2 2 3 / 5

1 2 486 . 1

e

• S

s y B y sy B n Q + + + =

1 s Bo

Distributed Systems

 Lecture #9 in Chapra’s book

 systems that have spatial resolution

 Ideal Reactors

David A. Reckhow CEE 577 #10 9

∆x

H B Jin Jout

∆V c t J A J A reaction

in c

• ut

c

∂ ∂ = − ±

Plug-Flow Reactors (PRF)

David A. Reckhow CEE 577 #10 10

∆V c t J A J A reaction

in c

• ut

c

∂ ∂ = − ±

∆ ∆ ∆ V c t UcA U c c x x A k Vc

c c

∂ ∂ ∂ ∂ = − +       −

Combining and taking the limit as ∆x→0

∂ ∂ ∂ ∂ c t U c x kc = − −

Which at steady state is:

0 = − − U c x kc ∂ ∂

And for c=co at x=0:

c c e

• k x u

=

−

Plug Flow vs CSTR

 First order reactions  Mixed Flow: intermediate

 read section 9.1.3

David A. Reckhow CEE 577 #10 11

c c e

• k x u

=

−

c c Q Q kV

• =

+

Mixed Flow

David A. Reckhow CEE 577 #10 12

Mixed Flow

 Peclet Number

David A. Reckhow CEE 577 #10 13

∆V c t J A J A reaction

in c

• ut

c

∂ ∂ = − ±

∆ ∆ ∆ ∆ V c t Uc E c x A U c c x x E c x x c x x A k Vc

c c

∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ = −       − +       − +                   −

Consider mixing in the longitudinal direction

P LU E

e =

= rate of advective transport rate of dispersive transport

Pe > 10, PFR-like Pe < 0.1, CSTR-like

Application of PRF to streams

 Point sources  Mass balance:

 Water Flow  Concentration

David A. Reckhow CEE 577 #10 14

Outfall: Qwcw Qrcr Qco

Q Q Q

w r

= +

c Q c Q c Q Q

• w

w r r w r

= + +

Assumptions

David A. Reckhow CEE 577 #10 15

Chloride Problem

 Determine the required industrial reduction in

chlorides to maintain a desired chloride concentration

• f 250 mg/L at the intake

David A. Reckhow CEE 577 #10 16

Q=25 cfs c=30 mg/L Qw=6.5 MGD cw = 1500 mg/L QT= 5 cfs cT = 30 mg/L Water intake

1.55 cfs/MGD

David A. Reckhow CEE 577 #10 17

Q s

W

x x

Q s

x+ x x+ x x+ x x

 To next lecture

David A. Reckhow CEE 577 #10 18