URBAN DRAINAGE Elements of Hydraulics Butler,D., Davies,J.D. Urban - - PDF document

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Dec 16, 2022 41 likes •155 views

URBAN DRAINAGE Elements of Hydraulics Butler,D., Davies,J.D. Urban Drainage, 2 nd edition: Spon Press, 2004 Chapter 8: Hydraulics Chapter 9: Hydraulic features Chapter 14: Pumped systems Mays,L.W. (ed) Hydraulic Design Handbook: McGraw-Hill, 1999 a

SLIDE 1

URBAN DRAINAGE Elements of Hydraulics

Butler,D., Davies,J.D. Urban Drainage, 2nd edition: Spon Press, 2004 Chapter 8: Hydraulics Chapter 9: Hydraulic features Chapter 14: Pumped systems Mays,L.W. (ed) Hydraulic Design Handbook: McGraw-Hill, 1999a Chapter 14: Hydraulic Design of Urban Drainage Systems (14.1-14.6,14.9)

aAvailable online (accessible from UCL computers):

http://www.knovel.com/web/portal/browse/display? EXT KNOVEL DISPLAY bookid=730.

HYDRAULICS 1

SLIDE 2

TYPICAL PROBLEMS

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HYDRAULICS 2

CONSERVATION LAWS

Steady (quasi-steady) flows

1. Mass conservation (continuity):

Q = U/A = const

2. Energy conservation:

Energy per unit weight [Joule/Newton] ⇒ Head [meter] H2 − H1 = Power ρ g Q − hf where hf is the head loss and H = P ρ g + U 2 2 g + z is the total head.

3. Momentum conservation (forces)

HYDRAULICS 3

SLIDE 3

Part I: Pipe Flow

Massey,B.S. Mechanics of Fluids, 8th edition: Taylor & Francis, 2006 Chapter 7: Flow and losses in pipes and fittings Chapter 13: Fluid machines ( 13.4.7, 13.4.8 ) Mays,L.W. (ed) Hydraulic Design Handbook: McGraw-Hill, 1999a Chapter 2: Hydraulics of Pressurized Flow (2.1-2.7)

aAvailable online (accessible from UCL computers):

http://www.knovel.com/web/portal/browse/display? EXT KNOVEL DISPLAY bookid=730.

HYDRAULICS 4

FORCE/ENERGY BALANCE IN A PIPE

d U V(r) x R r L τ

Energy: ∆H = ∆P ρ g = hf Force: π d2 4 ∆P = π d L τw Friction factor (non-dimensional friction): f = τw ρ U 2/2; f = f(Re, k/d) Darcy equation: hf = 4 f L d U 2 2 g

HYDRAULICS 5

SLIDE 4

MOODY DIAGRAM

HYDRAULICS 6

LOCAL LOSSES

hi = Ki U 2 g

HYDRAULICS 7

SLIDE 5

PIPES IN SERIES

z2 z1

1 2

. .

hf hf = Q2

κi

HYDRAULICS 8

PIPES IN PARALLEL

2 1

B A C

Q = Q2

Qi; κi Q2

i = hf HYDRAULICS 9

SLIDE 6

PIPE BRANCHES

z1 z2 z3 z0 Q

Q

. . .

3 1 2

Each node i:

j Qij = 0

Each branch j: ∆Hj = κj Q2

j HYDRAULICS 10

TANK EMPTYING TIME

Q ( t ) = − A dZ / dt

Z ( t )

1

A

2 3

a

. 1 2

−AdZ(t) dt = Q(t) = Q(Z(t)) ⇒ dt = − A Q(z) dZ

HYDRAULICS 11

SLIDE 7

PUMPING

Q H

. .

H

∆

P

Water power: Pw = ρ g q∆H; Efficiency: η = Pw/Ps ∆H = H0 + κ Q2 ⇒ Q = . . .

HYDRAULICS 12

PUMP CHARACTERISTICS

κ1 H0 κ HYDRAULICS 13

SLIDE 8

Part II: Open Channel Flow

Massey,B.S. Mechanics of Fluids, 8th edition: Taylor & Francis, 2006 Chapter 10: Flow with a Free Surface (10.1-10.12) Mays,L.W. (ed) Hydraulic Design Handbook: McGraw-Hill, 1999a Chapter 3: Hydraulics of Open Channel Flow Chow,V.T. Open-Channel Hydraulics: McGraw-Hill, 1959 Henderson,F.M. Open Channel Flow: The Macmillan Company, 1966

aAvailable online (accessible from UCL computers):

http://www.knovel.com/web/portal/browse/display? EXT KNOVEL DISPLAY bookid=730.

HYDRAULICS 14

FLOW PARAMETERS

Steady (quasi-steady) flow: Q = const

xxxxxxxxxxxxxx xxxxxxxxxxxxxx xxxxxxxxxxxxxx xxxxxxxxxxxxxx xxxxxxxxxxxxxx xxxxxxxxxxxxxx xxxxxxxxxxxxxx xxxxxxxxxxxxxx xxxxxxxxxxxxxx xxxxxxxxxxxxxx xxxxxxxxxxxxxx xxxxxxxxxxxxxx xxxxxxxxxxxxxx

h1 h 2 U1

P

U2

α z z

1 2

Q A T

A = A(h); P = P(h); T = T(h) Hydraulic radius: R = A/P Hydraulic depth: D = A/T

HYDRAULICS 15

SLIDE 9

CONSERVATION LAWS

1. Mass (continuity):

∆Q = 0

2. Energy:

∆E = −hf − ∆z, where E = Q2 2 g A2 + h is the specific energy, energy measured from channel’s bed

3. Momentum:

∆M = F ρ g, where M = Q2 g A + A h is the momentum function and h is the depth of the centroid

HYDRAULICS 16

SPECIFIC ENERGY

Q 1<Q Q E c h c Q 2>Q h E

Subcritical Supercritical

E=const h c Q max H h Q

HYDRAULICS 17

SLIDE 10

CRITICAL FLOW

Minimal specific energy: dE dh = d dh

2 g A(h)2 + h

= − Q2

g A3 dA dh + 1 = 1 − U 2 g D = 0 Critical conditions: 1 − Fr2 = 0; ⇒ Fr = 1 Froude number: Fr = U √g D = Q A √g D Another meaning: Fr = U C , where C−celerity Fr > 1– supercritical flow; Fr < 1– subcritical flow Upstreal/Downstream controls!!!

HYDRAULICS 18

UNIFORM FLOW

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P

L 1

m g S

τ A S

Energy: ∆z + hf = 0 ⇒ hf = −∆z = L sin(α) = S L Force: ρ g A L S = τ P L; τ = f ρ U 2/2 ⇒ U 2 ∼ R S Chezy equation: U = C √ R S; C = R1/6/n Maning coefficient: n– property of bed material

HYDRAULICS 19

SLIDE 11

UNIFORM FLOW

Normal depth: hn– depth of an uniform flow on a slope Mild slope: hn > hc– uniform flow

n the slope is subcritical

Steep slope: hn < hc– uniform flow on the slope is supercritical

HYDRAULICS 20

GRADUALLY VARIED FLOWa

Specific energy gradient: dE dx = d dx

2 g A(h)2 + h

dh

2 g A(h)2 + h dh dx = (1 − Fr2) dh dx On the other hand dE dx = d dx(−∆z − hf) = S − Sf , where Sf– frictional slope, Sf : U = C

R Sf

Governing differential equation (for a given Q): dh dx = S − Sf(h) 1 − Fr(h)2 With specified B.C. (upstream/downstream ⇔ super/subcritical) allows to calculate flow profile.

aLocally uniform, quasi-uniform

HYDRAULICS 21