Basic Hydraulics. Rabi H. Mohtar ABE 325

Transcription

1 Basic Hydraulics Rabi H. Mohtar ABE 35 The river continues on its way to the sea, broken the wheel of the mill or not. Khalil Gibran The forces on moving body of fluid mass are:. Inertial due to mass (ρ density). Gravitational associated with specific gravity 3. Viscous related to viscosity (μ) 4. Pressure to water/fluid Steady flow vs. transient flow Uniform vs. varied flow Pipeflow velocity changes with CS area Open channel velocity V changes with slope (s), CS area (A) and roughness (n) Shear stress:velocity relationship dv τ = μ dy Pipeflow VD ρ inertial force R e = = μ viscous force R <000 laminar flow e R >4500 turbulent e Open Channel v speed of flow Fr#= gy = celerity (speed of wave) Fr#> supercritical Fr#< subcritical Mass conservation: Q=A V =A V

2 Energy conservation: Momentum conservation: Examples AV=AV AV V= A In the above network, what are the flows from the two reservoirs: Q i = 0 Q reservoirs ( )=0 Q reservoirs = 700 gpm Energy conservation. kinetic energy mass flowing at a certain velocity = mv. potential energy mass of a body situated at a distance above a datum = Zg 3. energy due to pressure P*A=Force*L=energy PAL PAL P Unit mass = ρal ρ

3 Bernoulli s Energy Equations: Energy = constant P v + +gz=constant ρ v P + +z=constant g ρg 4. Mechanical energy (h m ) pump (+ve) or turbine (-ve) add or remove energy Friction forces or losses: 0.9 n L Manning: hf = 5.33 D Darcy Weisbach: L(ft) Q h f (ft)=f D(ft) ga Q.85 K 0.44Q L n C Hazen-Williams = = KQ or h f = 4.87 CHW D D \ Bernoulli s can be written now between any sections to (see image below) P V P V +Z + ±h m= +Z + +hl g g h=0 f (around a loop) Flow will split such that flow and will have the same friction loss..85 Momentum Conservation = momentum fluxes leaving - momentum entering the system F F=Q ρ( V - V ) Hydraulic Grade Line (HGL) =Z + P Energy Line (EL) =Z+ P V + g

4 v V g p P g h f h w So V g p Z P Z v g S o So h f =h w =S o for open channel P Hazen William Friction Formula Q K C h f = 4.87 D.85 m h f = friction 00 m pipe length C = Hazen William (H-W) resistance coefficient D = diameter in mm Q = flow rate in l sec K =. * 0 for SI units C unlike n (Manning coef. ) is Inversely proportional to roughness (friction) α Pipe C values - New coated steel 0 aluminum w/ - Asbestos-cement 40 - Plastic 50 Minor losses Friction due to Bend, Valves, other fittings v hf = KL g K = coefficient of head losses Valves of K L for each of the components are available in the literature.

5 Orifice flow Q = CA gh C = for sharp-edged orifices = for smooth orifice A = area of orifice h = head Uses of this in open-channel in pipe flow Open Channel Hydraulics: Continuity Equation: Q = AV Q = flow rate cfs or cms V = flow velocity mps or fps A = area of flow ft or m Velocity profile Energy equation V P V P + Y + Z + = + Y + Z + + h g g velocity elevation pressure=0 in open channel kinetic energy potential energy Y = flow depth Z = channel elevation (bottom) P = pressure g = unit weight of water h L = energy loss if h L = negligible v Y z constant g + + = if Z=0 (datum = channel bottom) L

6 v + Y = constant=e=specific energy for rectangular channel at depth y g q=vy q=flow per unit width q y E gy + = specific energy diagram critical depth flow with min E, i.e. y c is de 0 dy = de = q 3 + = 0 dy gy or Y 3 / c = q g or v gy = c v Froude number gy = c F = critical flow F> supercritical flow F< subcritical flow For non-rectangular channel V Area F= dh=hydraulic depth= gdh top width 5 3 Life had stunned my eyes and left them so confused, they wanted to keep looking. Danté Manning Equation 3 R S V = n n = surface roughness coefficient R = hydraulic radius (m) = Area/wetted perimeter S = slope of energy line º slope of bed V = velocity in m/sec ft Insert a factor m = To convert to ft/sec 3.486R S v = n substitute A Q=AV and R= then P c A S Q= np 3 c = m 3 /sec c =.486 ft 3 /sec A = m (ft )

7 P = m(ft) S, Q, and n are solved explicitly S= Q.np ca ca S n = QP 3 A & P require an implicit solution, similarly flow depth, top width. This requires trial and error or Newton Raphson/regula falsi Design Velocity Table 7. of Soil and Water Conservation Engineering, Schwab 4 th ed m/sec Roughness (n) for small channels Retardance Class Class Length of grass Good stand Fair stand >30" A B - 4" B C 6 0" C D - 6" D D <" E E To find n Orifice flow equation Q=AC gh 3 Q=ft / s design flow A=cross sectional Area ft C=constant = 0.6 g=gravitational acceleration 3 ft/s h=height of water above orifice in ft Example: Runoff =.5 in Terrace = 600 ft long with spacing of 00 ft If h=4 ft (ridge height) Find the diameter (D) for 48h max time of storage. 3 Q=.5 in in = 7500 ft ft 3 3 q = 7500 ft / 8 hr/3600 s/h=0.043 ft / s 9 A= = ft C gl π D A= = D=0.9 in=3 mm 4 Weir and Orifice Flow Equation from computer applications in hydraulic engineering Orifice flow equation across - ( inside flow, orifice location)

8 P V P V g g + Z+ + HG = + Z+ + HL Z =Z ;V = g(h-h ) L V = 0; Q=AV = g(h-h L) g P = 0; Q=AV = A g(h-h L) P = H; Q=CA gh Weir equation: V = g(h-h ) L AV = Q=LH g(h-h ) L C=weir coefficient that includes headloss and g Q=CLH 3 Steady, uniform flow Design criteria for open channel:. flow velocity is not serious scouring or sedimentation occurs. sufficient capacity to carry the design runoff 3. adequate depth 4. stable sideslopes Manning s formula:.49 3 v= R S n v = average velocity (ft/sec) n = roughness coefficient, measure of channel lining resistance R = hydraulic radius (ft=area/wetted perimeter) S = hydraulic gradient (ft/ft) = headloss due to friction divided by channel length or sinθ, θ=angle between channel bottom as horizontal = slope of channel R is a measure of slope resistance of a channel.49 a 3 q= a S n p q is determined using design flow for T a and P are typically unknown n varies from is typical of grass-lined channels Θ Most efficient section is close to a circular b=d tan d=depth b=bottom width θ=side slope angle with the horizontal. solve quadratic equation root. trial and error

9 variation of n and other uncertainties. stage and discharge. size and shape of channel affects R and V 3. vegetal cover seasonal changes in vegetal cover and the resistance due to flow depth 4. channel irregularities. They may change by and therefore affect is at small values. 5. seasonal changes ice, vegetal, obstruction all change n n values table are available in all books. Table 6. Flow classification: Steady: does not vary in time Non-steady: varies in time Uniform: does not vary in space Non-uniform: varies in space Steady flow:. uniform. non-uniform a. gradually varied b. rapidly varie Unsteady flow. unsteady uniform flow (rare). unsteady varied flow (unsteady flow) a. gradually varied b. rapidly varied Open channel canal designs.49 3 v= R S n Types of open channel problems encountered:. Z, S, b, d, characteristic of channel Find Q Solution direct from Manning s Equation. Z, S, b, n, Q known Find y Solution A, P are functions of y Substitute all parameters (polynomial) Solve for y by trial and error

10 3. Z, S, Q, soil type (erodible) find b, d? V max in erodible channel Solution: Pick V max, n Manning s equation; solve for R equations, unknowns quadratic equation with only one acceptable solution 4. Vegetated channel (n=f(d)) usually Q is known (storm), S, grass type and management known. Find depth of flow d, D, & T, t Solution: a. Find retardance class for cut and uncut grass b. Select V max for mowed c. Determine A from continuity d. From V max, slope determine R trial or error Manning s or graphical solution for retardance class Fig. 4.5 (Soil and Water Conservation 4 th ed., Schwab) e. equations and unknowns, solve for t and d f. the above solutions carries the flow during short grass period. How about when grass is tall? Add depth to carry capacity when grass is long. Pick D, determine T and R and A g. use retardance class chart, R, S to find v. h. Check Q for V and A i. If not adequate increase D until satisfied.

11 Channel Example Given: Short grass S = 0.5% = Q = 50 ft 3 /s d F.B.(Free Board) = 6" 0 d =? and total depth (D) n = 0.05 A = bd + Zd 6 ft Z=0 P = b + d Z + d(ft) A(ft ) P(ft) R(ft) V(ft/s) q(ft 3 /s) Too shallow Too deep Too deep Close enough