Chapter 6: Fundamentals of Fluid Flow

Learning outcomes By the end of this chapter students should be able to: Understand the terms & concepts of velocity, average velocity and discharge. Understand the principle of conservation of mass and able to apply the continuity equation. Understand the principle of conservation of energy (Bernoulli s principle) and able to apply the energy equation. UiTMKS/ FCE/ BCBidaun/ ECW

Velocity & discharge Velocity: A parameter that tells how fast and in what direction the fluid flows. Unit : m/s Discharge: A parameter that tells the total quantity of fluid flowing in a unit time past any particular cross-section of a stream. Unit: m 3 /s UiTMKS/ FCE/ BCBidaun/ ECW 3

Velocity profiles UiTMKS/ FCE/ BCBidaun/ ECW 4

When fluid is in motion, velocity for each fluid particle will varies throughout the flow. However in most engineering problem, velocity variation over the cross-section can be ignored, the velocity being assumed to be constant and equal to the mean velocity v. UiTMKS/ FCE/ BCBidaun/ ECW 5

True & average velocities Consider an annular area at radius, r, with thickness, dr, as shown in figure below. The elemental area through which the flow passes, da, can be given by the following expressions, da rdr UiTMKS/ FCE/ BCBidaun/ ECW 6

The fluid velocity, v, which passes through this area is fairly constant if dr is kept small. This velocity is the true velocity of the fluid at radius r from the centre of the pipe. The elemental discharge, dq, through this area is given by; dq vda v rdr UiTMKS/ FCE/ BCBidaun/ ECW 7

The total discharge Q can be obtained by integrating dq over the whole cross sectional area of the flow; Q R O vrdr The average or mean velocity can then be expressed as; v average Q A UiTMKS/ FCE/ BCBidaun/ ECW 8

Discharge Can be expressed as: Volume: Volume flowrate, unit : m 3/ s Mass :Mass flowrate, unit : kg/s Weight: Weight flowrate, unit : N/s Q M Q Av W gq UiTMKS/ FCE/ BCBidaun/ ECW 9

Example 6. Benzene flows through a 00 mm diameter pipe. The mean velocity of flow is 3 m/s. Find the volumetric rate, weight of flow rate and mass flow rate. Mass density of benzene is 879 kg/ m 3. UiTMKS/ FCE/ BCBidaun/ ECW 0

Conservation of mass Matter is neither created nor destroyed. In steady flow, the mass of fluid in the control volume remains constant, therefore Mass of fluid entering per unit time = Mass of fluid leaving per unit time UiTMKS/ FCE/ BCBidaun/ ECW

Applying this principle to steady flow in streamtube with constant cross-sectional area small enough for velocity to be considered constant, Mass of fluid entering per unit time at section = Mass of fluid leaving per unit time at section UiTMKS/ FCE/ BCBidaun/ ECW

Massentering per unit time at A u M assleaving per unit time at Au For steady flow, A u Using mean velocity, Consider incompressible Or A u A u Q A Q A u u A u Q constant m fluid, is known as continuity equation UiTMKS/ FCE/ BCBidaun/ ECW 3

Continuity equation can also be applied to determine the relation between the flows into and out of a junction. Totalinflow to junction Q Q A v Totaloutflow from junction Q Q A v Q Q 3 3 3 A v 3 3 ( incompressible fluid) UiTMKS/ FCE/ BCBidaun/ ECW 4

Example 6. Water is flowing in a 5 mm diameter pipe at a velocity of 0.6 m/s. What is the velocity of water coming out of the nozzle which is attached to the pipe? The nozzle has a diameter of 3 mm. UiTMKS/ FCE/ BCBidaun/ ECW 5

Example 6.3 Water flows through a pipeline in which the diameter reduces from 500 mm at A to 300 mm at B as shown in Figure. The pipe then forks, one branch has a diameter of 50 mm discharging at C, while the other branch with diameter of 00 mm discharges at D. Given that the velocity at A is.0 m/s and the velocity at D is 3.6 m/s, find discharges at C and D and the velocities at B and C. UiTMKS/ FCE/ BCBidaun/ ECW 6

Example 6.4 (Douglas, 006) Water flows from A to D and E through the series pipeline shown in Fig. Given the pipe diameters, velocities and flow rates below, complete the tabular data for this system. UiTMKS/ FCE/ BCBidaun/ ECW 7

Energy equation To analyze flow problems, consider 3 forms of energy i.e. potential energy, kinetic energy & pressure energy. Consider an element of fluid as shown in the next figure. UiTMKS/ FCE/ BCBidaun/ ECW 8

Energy equation Potential energy, PE (energy possessed due to elevation) PE per unit weight z mgz Kinetic energy, KE (energy possessed due to velocity) PE KE KE per unit weight mv v g UiTMKS/ FCE/ BCBidaun/ ECW 9

Energy equation Work Energy (amount of work needed to move from AB to A B ) Force exerted on AB pa When section moved to A B, Volume passing AB, W V W V mg g m UiTMKS/ FCE/ BCBidaun/ ECW 0

Energy equation Distance AA m A WE Force x Distance AA pa m A WE per unit weight p g UiTMKS/ FCE/ BCBidaun/ ECW

Energy equation Total Energy, H WE KE WE H p g v g z Where, p g pressurehead (m) v velocity head (m) g z potentialhead (m) UiTMKS/ FCE/ BCBidaun/ ECW

Bernoulli s equation States, for a steady flow of frictionless fluid along a streamline, the total energy per unit weight remains constant from point to point although its division between the three forms of energy may vary. H p g v g z UiTMKS/ FCE/ BCBidaun/ ECW 3

Bernoulli s equation Assumptions: Velocity on the flow cross section is uniform, i.e. average velocity is used. Effect of viscous forces are very small compared to the gravitational forces, therefore there are no losses of energy due to friction Energy of flow is not converted into any other form apart from kinetic, potential and pressure energies. UiTMKS/ FCE/ BCBidaun/ ECW 4

Bernoulli s equation between points H H p g v g z p g v g z UiTMKS/ FCE/ BCBidaun/ ECW 5

Energy losses & gains Energy losses between two points, Including energy losses by turbine (h q ) and additional energy by pump (h p ) UiTMKS/ FCE/ BCBidaun/ ECW 6 h L z g v g p z g v g p p q L h h h z g v g p z g v g p

Kinetic Energy Correction Factor As correction to kinetic energy (v ave ) Consider a small elemental fluid mass dm moving with velocity v. True kinetic energy, But the KE across the section per unit time, where M is the mass flow rate. Integrating, KE KE KE v dmv 3 dmv da UiTMKS/ FCE/ BCBidaun/ ECW 7

Considering KE True KE v 3 Qv 3 v 3 da A vave Uniform velocity,.0 v da ave Turbulent flow,.0.5 Laminar flow,, ave Av ave Average KEx KEcorrection Av.0 ave 3 3 factor UiTMKS/ FCE/ BCBidaun/ ECW 8

Example 6.3 A pipe conveying water tapers from a cross sectional area of 0.5 m at A to 0. m at B. The pressure at A is 0 kn/m and the velocity is.0 m/s. Assuming no energy losses, determine the pressure at B, which is 4.0 m above the level of A. A A A = 0.5 m v A =. 0 m/s p A = 0 kn/ m B A B = 0. m UiTMKS/ FCE/ BCBidaun/ ECW 9

Example 6.4 A siphon has a uniform circular section of 70 mm diameter and consists of a bent pipe with its crest.6 m above the water level as shown in figure. The siphon discharges into the atmosphere at a level 3.0 m below the water level. i. Calculate the velocity and the discharge. ii. The pressure head at the end of the siphon is equivalent to 0 m head of water. Calculate the pressure head at the crest. Neglect all losses. A B.6 m C 3 m UiTMKS/ FCE/ BCBidaun/ ECW 30

Example 6.5 An elevated water tank as in figure is being drained to an underground storage through a 300 mm diameter pipe. The flow rate is 0. m 3 /s and the head loss is 3.0 m. If the underground pipe is located at.5 m below ground level, determine the water surface elevation in the tank. UiTMKS/ FCE/ BCBidaun/ ECW 3

Example 6.7 (Douglas, 006) A fire engine pump develops a head of 50 m, i.e. it increases the energy per unit weight of water passing through it by 50 N m N -. The pump draws water from a sump at A through a 50 mm diameter pipe in which there is a loss of energy per unit weight due to friction h = 5u / g varying with the mean velocity u in the pipe, and discharges it through a 75 mm nozzle at C, 30 m above the pump, at the end of a 00 mm diameter delivery pipe in which there is a loss of energy per unit weight h = u / g. Calculate, a. The velocity of the jet issuing from the nozzle at C and b. The pressure in the suction pipe at the inlet to the pump at B. UiTMKS/ FCE/ BCBidaun/ ECW 3

UiTMKS/ FCE/ BCBidaun/ ECW 33

Energy line Change of energy of head from one form to another can be represented by: Total Energy Line (TEL) : total head Hydraulic Grade Line (HGL): elevation + pressure head UiTMKS/ FCE/ BCBidaun/ ECW 34

Representation of energy changes in a fluid system UiTMKS/ FCE/ BCBidaun/ ECW 35

Energy line UiTMKS/ FCE/ BCBidaun/ ECW 36

Exercise 6. When 0.3 m3/s of water flows through a 75 mm constriction in a 350 mm horizontal pipeline, the pressure at a point in the pipe is 300 kpa and the head loss between this point and the constriction is m. Calculate the pressure in the constriction. UiTMKS/ FCE/ BCBidaun/ ECW 37

Exercise 6. Water flows in a pipeline. At a point in the line where the diameter is 00 mm the velocity is 3.6 m/s and the pressure is 345 kpa. At a point m away the diameter reduces to 00 mm. calculate the pressure here when the pipe is a) Horizontal b) Vertical with flow downward UiTMKS/ FCE/ BCBidaun/ ECW 38

Exercise 6.3 A pump draws water from a reservoir through a pipe 0.5 m in diameter. When 300 L/s is being pumped, calculate the pressure in the pipe at a point.4 m above the reservoir surface, in Pa and in meter of water. UiTMKS/ FCE/ BCBidaun/ ECW 39

Exercise 6.4 Water flows from one reservoir in a 00 mm diameter pipe, while from a second reservoir, water flows in a 50 mm diameter pipe as shown in figure. The two pipes meet in a tee junction with a 300 mm diameter pipe that discharges to the atmosphere at an elevation of 5 m. If the water surface in both reservoirs are 35 m elevation, calculate the flowrate in the 300 mm pipe. UiTMKS/ FCE/ BCBidaun/ ECW 40

Apr 00 For the frictionless siphon shown in Figure Q4(b), determine the discharge and the pressure heads at A and B, given that the pipe diameter is 00 mm and the nozzle exit diameter is 50 mm. UiTMKS/ FCE/ BCBidaun/ ECW 4

Review of past semesters final exam questions UiTMKS/ FCE/ BCBidaun/ ECW 4

Apr 00 For the frictionless siphon shown in Figure Q4(b), determine the discharge and the pressure heads at A and B, given that the pipe diameter is 00 mm and the nozzle exit diameter is 50 mm. UiTMKS/ FCE/ BCBidaun/ ECW 43

Oct 009 Water flows into a large tank at a rate of 0.05 m 3 /s, as shown in Figure Q4 (b). The water leaves the tank through 8 holes at the bottom of the tank, each of which produces a jet of mm diameter. Neglecting losses, determine the height, for steady flow condition using Bernoulli Equation. UiTMKS/ FCE/ BCBidaun/ ECW 44

Apr 009 Water flows in the pipe system as shown in Figure Q4(a). The diameters of the pipes at inlet and outlet are 30cm and 00cm respectively. Apply the continuity equations to find the following: i) Velocity at the mid length of pipe ii) Velocity at the end of 00cm diameter iii) Weight flowrate (p=808 kg/m3) iv) Mass flowrate UiTMKS/ FCE/ BCBidaun/ ECW 45

Apr 009 A siphon has a circular bore of 60mm and consists of a bent pipe with its crest.5m above the water surface and discharging into the atmosphere 3.5 m below the water surface. Use the energy equation to calculate the flow velocity, discharge and absolute pressure at the crest level for an atmospheric pressure of 00 kpa. State all assumptions made. UiTMKS/ FCE/ BCBidaun/ ECW 46

Oct 008 The closed tank of a fire engine is partly filled with water, the air space above being under pressure as shown in Figure Q4(b). A 60 mm bore connected to the tank discharges on the roof of a building.5 m above the level of water in the tank. Given the friction losses are 450 mm of water. Determine the air pressure which must be maintained in the tank to deliver 0 litres/s on the roof. UiTMKS/ FCE/ BCBidaun/ ECW 47

Apr 008 Water flows in the pipe system as shown in Figure Q4(a). The diameters of the pipes at points, and 4 are 30 mm, 75 mm and 50 mm respectively. The velocity at points and 3 are.5 m/s and.5 m/s respectively. If the flow rate at point 3 is twice that at point 4, determine: i) Flow rates at point,,3, and 4 ii) Velocities at points and 4 iii) Diameter of pipe at point 3 UiTMKS/ FCE/ BCBidaun/ ECW 49

Apr 008 A 50 mm diameter pipe is used to deliver oil from tank A to tank B with the help of a pump, P, as shown in Figure Q4(b). The pressure at point S in the suction pipe is a vacuum of 40 mm of mercury. If the flowrate is 0.05 m 3 /s of oil (S.G = 0.7), find the total energy head at point S with respect to a datum at the pump. UiTMKS/ FCE/ BCBidaun/ ECW 5

End UiTMKS/ FCE/ BCBidaun/ ECW 53