Asian Power Systems Review Center

FLUID MECHANICS (HYDRODYNAMICS) I. THEORY A. Continuity Flow Equation

Q1 = Q2 A1V1 = A2V2 Where: Q = volume flow rate A = cross sectional area of the flow V = mean velocity ρ = density m = mass B. Energy Equation (Bernoulli’s Equation) The energy of the flowing fluid per unit time passing any upstream section is the same as the energy per unit time passing any downstream section plus the loss of head between two sections.

Bernoulli’s Equation 2

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V1 P V P + 1 + Z 1 = 2 + 2 + Z 2 + H L (1⊗2) 2g γ 2g γ Where:

V2 = velocity head 2g P

γ

= pressure head

Z = elevation HL = head loss C. Types of Steady Flow 1. Laminar Flow (Re ≤ 2000) 2. Turbulent Flow (Re > 4000) D. Reynolds Number (Re) Reynolds Number is a dimensionless parameter equal to the ratio of the inertia forces to the friction forces.

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Re = Re =

VD

µk VDρ

µd

Re =

VDγ µd g

Ss =

fρV 2 8

Where: V = mean velocity D = diameter of the pipe μd = dynamic viscosity μk = kinematic viscosity ρ = density γ = specific weight g = acceleration due to gravity

E. Major Head Losses in Pipes 1. Laminar Flow (Hagen-Poiseuille Equation)

hf =

32 µ d LV γD 2

2. Turbulent Flow a. Darcy-Weisbach Equation

hf =

fLV 2 2 Dg

hf =

0.083 fLQ 2 D5

b. Manny-Chezy Equation

6.35n 2 LV 2 hf = D4/3

10.3n 2 LQ 2 hf = D 16 / 3

c. Hazen-Williams Formula

hf =

10.67 LQ 1.85 C 1.85 D 4.87

Where: Q = discharge rate (m3/s) n = coefficient of roughness f = coefficient of friction V = mean velocity D = diameter of the pipe μd = dynamic viscosity γ = specific weight L = pipe length C = Hazen-William’s Coefficient F. Minor Head Losses in Pipes

h=k

V2 2g

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Where: h = minor head loss k = coefficient of minor loss V = mean velocity g = acceleration due to gravity

G. Pipes in Series

Q1 = Q2 = Q3 = Q4 = Q H L = hL1 + hL 2 + hL 3 + hL 4 H. Pipes in Parallel

Q = Q1 + Q2 + Q3 + Q4 hL1 = hL 2 = hL 3 = hL 4 = H L `

Where: Q = discharge h = head losses

ORIFICES AND WEIRS Orifice – is a small opening from which the liquid flows. The actual velocity through an orifice:

V = C v 2 gH The actual discharge through an orifice

Q = C d Ao 2 gH Where: Cv = coefficient of velocity Cd = coefficient of discharge Ao = area of the orifice H = total head loss

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Note:

For theoretical velocity, Cv = 1.0 For theoretical discharge, Cd = 1.0

Weirs – is any obstruction of stream flow over which water flows.

t=∫

h2

h1

dV Qw

Where: dV = AdH Qv = CdLH3/2 (rectangular weir) Qd = CdH5/2 (triangular weir) Sharp-Crested and Free Flowing Weir

Theoretical Flow 3/ 2 2 3/ 2   V1 2   V1  2    − Qt = 2 gL  H +  2g   3 2 g      

Actual Flow 3/ 2 2 3/ 2   V1 2   V1       Qa = C d L  H + − 2 g  2 g       2

Qa = CdLH 3 / 2

Where:

V1 =0 2g

Francis Formula (Cf = 0.622)

Qa = 1.84 LH 3 / 2 Where: C d = C f Cf L Wc H P

2 2g 3

= correction factor (≈ 0.60) = length of the crest = width of the channel = head over the crest = height of the weir

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V1 d

= velocity approach = depth of upstream

HYDRODYNAMICS A. Force on Fixed Objects

1. Horizontal Component of Force

Fh =

γQ

(Voh − V fh )

g

2. Vertical Component of Force

Fv =

γQ

(Vov − V fv )

g

3. Resultant 2

R = Fv + Fh

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B. Force on Moving Objects

1. Horizontal Component of Force

Fh =

γQ ' g

(Voh − V fh )

2. Vertical Component of Force

Fv =

γQ' g

(Vov − V fv )

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C. Drag Force Drag Force – it is the resistance caused by friction in the direction opposite to that of the motion of the center of gravity of a moving body in a fluid.

Fd = C d γA p

V2 2g

Where: Cd = drag coefficient V = mean velocity Ap = projected area of the moving object Fd = drag force

II. REVIEW PROBLEMS 1. Find the mass flow rate of a liquid (ρ = 0.690 g/cm3) flowing through a 5 cm (inside diameter) at 8.3 m/s. a. 69 kg/s b. 450 kg/s c. 11.24 kg/s d. 430 kg/s 2. Water is flowing through a pipe a 0.60 m in diameter and 2 km length. If the average velocity of water is 2 m/s, what is the loss of head due to friction considering that the friction factor is 0.005? a. 13.6 b. 12.7 c. 11.7 d. 9.7 3. A sphere of radius 5 x 10 to the negative two cm and density of 1.1 g/cc falls at a constant velocity through a liquid of density 1.0 g/cc and viscosity of 1.00 poise. What is the velocity of the falling sphere in cm/s? a. -0.0544 b. 0.4 c. -0.744 d. 1.53 4. A 12 inch inside diameter circular duct has been flowing at given suction with an average velocity of 1000 feet per minute. Determine the quantity of air flowing in cfm. a. 785.4 b. 685.4 c. 585.4 d. none of these 5. What is the static head corresponding to a flow of 5 m/sec? a. 3.05 m b. 3.055 m c. 3.52 m

d. 1.275 m

6. What is the expected head loss per mile of closed circular pipe (17-in inside diameter, friction factor of 0.03) when 3300 gal/min of water flow under pressure? a. 38 ft b. 0.007 ft c. 3580 ft d. 64 ft 7. What is the rate of flow of water passing through a pipe with a diameter of 20 mm and speed of 0.5 m/sec? a. 1.24 x 10-4 m3/s c. 1.57 x 10-4 m3/s -4 3 b. 2.51 x 10 m /s d. 1.87 x 10-4 m3/s 8. An orifice has a coefficient of discharge of 0.62 and a coefficient of contraction of 0.63. Determine the coefficient of velocity for the orifice. a. 0.98 b. 0.99 c. 0.97 d. 0.96 9. The theoretical velocity of flow through an orifice 3 m below the surface of water in a tall tank is: a. 8.63 m/s b. 9.85 m/s c. 5.21 m/s d. 7.67 m/s

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10. Water having kinematic viscosity v = 1.3 x 10-6 m2/s flows in a 100-mm diameter pipe at a velocity of 4.5 m/s. The Reynolds Number is: a. 346,150 b. 258,250 c. 387.450 d. 298.750 11. Oil having specific gravity of 0.869 and dynamic viscosity of 0.0814 Pa-s flows through a cast iron pipe at a velocity of 1 m/s. The pipe is 50 m long and 150 mm in diameter. Find the head lost due to friction. a. 0.73 m b. 0.45 m c. 0.68 m d. 1.25 m 12. What commercial size of new cast iron pipe shall be used to carry 4490 gpm with a lost of head of 10.56 feet per mile? Assume f = 0.019. a. 625 mm b. 576 mm c. 479 mm d. 352 mm 13. Assume that 57 liters per second of oil (ρ = 860 kg/m3) is pumped through a 300 mm diameter pipeline of cast iron. If each pump produces 685 kPa, how far apart can they be placed? (Assume f = 0.031) a. 23.7 m b. 32.2 m c. 12.6 m d. 19.8 m 14. A 20-mm diameter commercial steel pipe, 30 m long is used to drain an oil tank. Determine the discharge when the oil level in the tank is 3 m above the exit of the pipe. Neglect minor losses and assume f = 0.12. b. 0.000179 m3/s c. 0.000113 m3/s d. 0.000869 m3/s a. 0.000256 m3/s 15. Water leaves a faucet with a downward velocity of 3 m/s. As the water falls below the faucet it accelerates with acceleration g. The cross sectional area of the water stream leaving the faucet is 1.0 cm2. What is the cross-sectional area of the stream 0.50 m below the faucet? a. 0.96 cm2 b. 0.45 cm2 c. 0.69 cm2 d. 0.25 cm2 16. A submarine is 100 m long. The shape of its hull is roughly cylindrical with a diameter of 15 m. When it is submerged, it cruises at a speed of about 40 knots or approximately 20 m/s. Compute the Reynolds number, if viscosity is 1.0 x 10-3 Pa-s. a. 6 x 108 b. 4 x 108 c. 3 x 108 d. 5 x 108 17. Find the speed of sound in 32°F water if the compressibility is 3.4 x 10-6 psi? a. 4674 ft/s b. 4584 ft/s c. 4764 ft/s d. 4854 ft/s 18. A jet of water issues from 100 mm diameter nozzle with a velocity of 20 m/s. Find the HP of the jet. a. 42.1 HP b. 34.2 HP c. 43.2 HP d. 32.1 HP 19. A waterfall is 60 meters high. It discharges at a constant rate of 1.0 cu. m/sec. A mini-hydroelectric plant is to be constructed below the waterfalls. If the turbine efficiency is 80% and the generator efficiency is 95%, calculate the kW output of the generator. a. 865 kW b. 845 kW c. 850 kW d. 860 kW 20. Theoretical horsepower required to pump water at 100 gallons per minute from a large reservoir to the surface of another large reservoir 400 ft. higher is nearest to: a. 10 HP b. 35 HP c. 18 HP d. 6 HP

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21. What is the force acting on one side of the tank moving upward with an acceleration of 4.7 m/s2 when it is filled with oil whose specific gravity is 0.752 and a height of 1.3 m for the bottom of the tank? a. 11.986 kN b. 14.572 kN c. 15.723 kN d. 17.457 kN 22. Which of the following is NOT correct? a. Steady flow do not change with time at any point b. Bernoulli’s equation only holds on the same streamline c. The Reynolds number is the ratio of the viscous force to the inertial force d. For a fluid at rest, the pressure is equal in all directions. 23. The negative sign for a gage reading is called a. Transmission on pressure b. Vapor pressure

c. Vacuum pressure d. Pressure head

24. An instrument used to measure small pressure a. Venturi b. Orifice

c. Aneroid

d. Manometer

III. SELF-TEST 1. A centrifugal pump discharges water through 160 mm horizontal pipe floating full at the outlet. The jet striking the ground at a horizontal distance of 3.5 meters and a vertical distance of 1.2 from the end of the pipe. Find the discharge rate of the pump in cubic meters per sec. b. 0.707 m3/sec c. 0.142 m3/sec d. 5 m3/sec a. 7.07 m3/sec 2.

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The theoretical velocity generated by a 10 ft hydraulic head is a. 12.2 ft/sec b. 17.9 ft/sec c. 25.4 ft/sec

d. 29.2 ft/sec

What is the static head corresponding to a flow velocity of 10 ft/sec? a. 1.55 ft b. 1.75 ft c. 2.05 ft

d. 2.5 ft

4.

A centrifugal pump discharges water through a 180 mm diameter pipe flowing full at the outlet. The jet striking the ground at a horizontal distance of 4 m and a vertical distance of 1.5 m from the end of the pipe. The capacity of the pump in GPM is nearest to: a. 5,100 b. 2,900 c. 10,000 d. 6,200

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What is the theoretical horsepower to pump water at 100 gallons per minute from a large reservoir to the surface of another reservoir 400 ft higher. Ignoring friction losses, pump efficiency, etc.? a. 100 HP b. 101 kW c. 10.11 HP d. 10.11 kW

6.

A steel box rectangular pan flows with a draft of 4 ft. If the box is 20 ft long and 40 ft wide and 6 feet deep. Compute the time necessary to sink to its top edge by opening a standard orifice 6 inches in diameter in its bottom. Neglect thickness of vertical sides and assume C = 0.6. a. 135 sec b. 211 sec c. 309 sec d. 405 sec

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The diameter of a pipe carrying water changes gradually from 150 mm at A to 450 mm at B. A is 4.5 m lower than B. What will be the difference in pressure in kPa, between A and B, when 25 liters per sec is flowing, loss of energy being neglected? a. 41.36 b. 36.14 c. 64.41 d. 43.16

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8.

An orifice 2 inches in diameter discharges fluid from a tank with a head of 15 feet. Discharge rate, Q, is measured at 0.5 cfs. Actual velocity at the vena contract, v.c. is 29.0 feet/sec. The coefficient of discharge, CD, is nearest to: a. 0.62 b. 0.74 c. 0.79 d. 0.86

9.

What HP is required to pump water to a reservoir 60 ft high with a discharge velocity of 20 ft per second? If the total loss is 13 ft and the pipe is 2 inches in diameter.

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