EXMPLE SHEET FOR TOPI 2 UTUMN 2013 Q1. What is the significance of the Reynolds number Re for the flow of fluid in a circular pipe? If the friction factor for a pipe is given by λ = 64/Re for laminar flow, and λ = 0.32/Re 1/4 for turbulent flow, find the mean velocity and discharge of water through a smooth pipe of length 2 m and diameter 6 mm if the head available to overcome pipe friction is: 30 mm; 1 m. Q2. 0.75 m diameter pipe carries 0.6 cumec. t point, elevation 40 m, a ourdon gauge fitted to the pipe records 1.75 bar, while at point, elevation 34 m and 1.5 km along the pipe a similar gauge reads 2.1 bar. Determine the flow direction and calculate the friction factor. Q3. Reservoir feeds two lower reservoirs through a pipe (length 10 km, diameter 0.6 m and average downward slope 0.25%) which branches at its end into two pipes leading to reservoirs and. The branch to reservoir has length 1.5 km and average downward slope 0.5%. The branch to reservoir has length 5 km and downward slope 0.1%. friction factor of 0.02 may be assumed in all pipes and losses other than friction may be neglected. alculate the diameter of each branch pipe in order that the flow in each branch is 150 L s 1 when the water level in each reservoir is 4 m above the corresponding pipe entry. Hydraulics 2 E2-1 David psley
Single-Pipeline alculations Q4. (Examination, January 2012) Water is pumped from a slow-running river (water level 120 m OD) to a storage tank (water level 150 m OD). The head across the pump is 40 m. The pipeline is of total length 800 m and diameter 0.2 m, with lining of roughness k s = 0.1 mm. Minor losses due to pipe fittings, entry and exit losses can be neglected. Find: the frictional head loss along the pipe; the friction factor and volumetric flow rate through the pipe; (c) the dynamic head in the pipe. (d) If the intake to the pipe is 2 m below the water level in the river and the pipe runs straight and horizontal to the pump, what is the maximum distance between river and pump if sub-atmospheric pressure is to be avoided in the pipeline? Q5. (Examination, January 2003) pipeline is to be constructed to bring water from an upland storage reservoir to a town 30 km away, at an elevation 150 m below the water level of the reservoir. In summer the pipeline must be able to convey up to 5000 cubic metres per day. If the pipe is fabricated from material of roughness 0.3 mm, find the required diameter. During the winter, water requirements fall to only 3000 cubic metres per day and the excess head available can be used to drive a small turbine. If the turbine has an efficiency of 75% find the maximum power output. Q6. (Examination, January 2008) storage reservoir is to be used to supply a factory with water. The pipeline connecting the reservoir and factory has length 3 km and diameter 300 mm and its lining has effective roughness 0.1 mm. Minor losses may be neglected. If the head difference between reservoir and factory is 40 m, calculate the maximum quantity of flow in the pipeline. For short periods it is necessary to increase the quantity of flow by 100 L s 1. alculate the additional pumping head required. (You should calculate a new friction factor). Q7. (Examination, January 2006) Water is to be pumped from a reservoir to a housing development 30 m above the top-water level of the reservoir. The pipe is of length 5 km and diameter 0.3 m and is fabricated from material of roughness 0.1 mm. If water is to be supplied at 80 L s 1, find: the head which must be developed across the pump; the input power to the pump, if it is operating at an efficiency of 40%; (c) the maximum gauge pressure in the pipeline if the pump is located at its entry, some 10 m below the water level in the reservoir and at the lowest point in the system. Minor losses may be neglected. Hydraulics 2 E2-2 David psley
Q8. (Examination, January 2007) rude oil (specific gravity 0.86, kinematic viscosity 9.0 10 5 m 2 s 1 ) is to be pumped from a barge to a large storage tank. The pipeline is horizontal and of diameter 250 mm, length 400 m and roughness 0.1 mm. It enters the tank 8 m below the level of oil in the tank. When the control valve is fully open the static pressure at pump delivery is 3 10 5 Pa gauge. Ignore minor losses due to pipe fittings, entrance/exit losses etc. Pump arge ontrol valve 8 m Storage tank Find: (c) (using hydrostatics) the gauge pressure where the pipe enters the tank; (from the pressures at the two ends) the head loss along the pipeline; the volumetric flow rate in the pipeline. If the pump delivery pressure remains the same but a valve reduces the flow by half, find: (d) the head loss at the valve; (e) the power loss at the valve. Q9. (Examination, January 2011) pipeline is to be constructed to convey water under gravity from a reservoir to a water-treatment plant at a rate of 0.2 m 3 s 1. The difference in water levels is 25 m. If the pipeline is of length 5 km and is lined with material of roughness k s = 0.1 mm determine the diameter of pipe required. (Losses other than pipe friction may be neglected.) Some time later it is found necessary to boost the flow rate to 0.4 m 3 s 1 for short periods. Find the new friction factor and the pumping head required. Q10. (Examination, January 2010) Water flows from a reservoir through a rough-walled pipe 3 km long and discharges to a lower tank where the water level is 25 m below that in the reservoir. The equivalent sand roughness of the pipe is k s = 1.0 mm. The Darcy friction factor λ is given here by 1 3.7D 2.0log 10( ) λ k s where D is the diameter of the pipe. If the flow rate is 50 L s 1, find the pipe diameter. Flows of 10 L s 1 and 20 L s 1 are tapped off along the pipe at distances 1 km and 2 km respectively from the upper reservoir. Find the new discharge to the lower tank. Hydraulics 2 E2-3 David psley
Pipes in Series and Parallel Q11. pipeline carrying oil with specific gravity 0.9 is made up of two horizontal pipes, both of length 50 m, joined end to end. For this fluid the first pipe has friction factor 0.014 and diameter 0.2 m; the second pipe has friction factor 0.012 and diameter 0.1 m. The head loss at 2 the junction is 0.5V 1 /(2g). If the pressure drop along the combined system is 2 kpa, find the quantity of flow. Q12. Flow passes through a branched system of 3 1 parallel pipes as shown. The resistance law for 2 2 each pipe is of the form h αq, where, if h is in m and Q in m 3 s 1, the numerical values of α 3 for each pipe are: α 1 = 100, α 2 = 150, α 3 = 300 If the total flow is 0.4 m 3 s 1, find the head loss across the branched system. Pipe Junctions and Networks Q13. In the pipe network shown, head losses along each pipe can be expressed as 2 h αq where, if h is in m and Q is in m 3 s 1, the numerical values of α for each pipe are: α D = 100, α D = 100, α = 300, α = 200 There is an inflow of 0.6 m 3 s 1 to the network at and outflows at and D. If the outflow at D is 0.4 m 3 s 1 what is the outflow at? Write expressions for the quantity of flow in each pipe in terms of the quantity of flow Q in pipe. (c) y equating the net head change from to along alternative paths and D, find Q and hence the quantity of flow in each pipe. (d) Repeat part (c) if α D is increased to 900. D 0.6 m s 3-1? Q 0.4 m s 3-1 Hydraulics 2 E2-4 David psley
Q14. Reservoirs, and are connected by pipes via a single junction J, as shown in the figure. The water levels in and are, respectively, 240 m and 160 m OD. Pipe lengths, diameters and friction factors are given in the table below. Minor losses may be neglected. 240 m 160 m J Pipe Length (km) Diameter (mm) Friction factor, λ J 1.5 250 0.016 J 2.0 300 0.016 J 2.5 400 0.016 What minimum water level must be maintained in reservoir to ensure that any flow is from to J? If the water level in is actually 220 m OD, calculate the discharge in all pipes. Q15. (Examination, January 2010) Three reservoirs, and are connected by a network of pipes with a common junction J as shown. The water levels OD in the three reservoirs are marked on the figure, whilst the pipe parameters are given in the table below. Minor losses may be neglected. 300 m 220 m J 280 m Pipe Length (km) Diameter (mm) Friction factor, λ J 3.5 300 0.02 J 4.0 200 0.03 J 2.5 200 0.02 alculate the discharge in each pipe and indicate whether the flow is to or from reservoir. n additional controlled discharge of 40 L s 1 is abstracted directly from junction J. alculate the flow in each pipe now, again indicating whether there is flow to or from reservoir. Hydraulics 2 E2-5 David psley
Q16. Water flows from a header tank to a junction J, where it splits to supply two fountains sent vertically, without nozzles, from points F and G. The water level in is 30 m above F, whilst outlet G is 4 m above F. Pipe properties are given below. F J G Pipe Length (m) Diameter (mm) Friction factor, λ J 50 100 0.02 FJ 15 50 0.01 GJ 10 50 0.01 Find: (c) Write down the (piezometric) head loss vs discharge relations for each pipe, including numerical values. (Note: there is significant dynamic head at the exits F and G.) the flow rate and height of the water fountain at F if the pipe to G is shut off. the flow rate in each pipe and height of the water fountain at F if all pipes are open. Q17. (Examination, January 2012) In a water-storage scheme three reservoirs, and are connected by a single junction J as shown. The water levels in, and are 300 m, 200 m and 140 m respectively. The pipeline properties are given below. Friction factors may be assumed constant and minor losses may be neglected. Pipeline J J J Length L (m) 5000 3000 4000 Diameter D (m) 0.4 0.25 0.3 Friction factor λ 0.015 0.03 0.02 200 m J 300 m 140 m alculate the total flow in each pipe and the direction of flow in pipe J if: there is a valve-regulated flow of 50 L s 1 to reservoir but water flows freely under gravity in the other pipes; water flows freely under gravity in all pipes. Hydraulics 2 E2-6 David psley
omplex Pipe Networks (Optional) 3 L s -1 Q18. In the pipe network shown the resistance law (h = αq 2 ) has the same numerical value of α for each pipe. Use the Hardy-ross method to find the flow in each pipe. 4 L s -1 3 L s -1 2 L s -1 Q19. y calculating the heads at junctions and D, find the flows in all the pipelines connecting the reservoirs shown in the figure. The head loss (m) in a pipe is given by h = αq 2 and the value of α for each pipe given below. You should start your calculation with an initial estimate for the head at of 300 m and for D of 150 m. Pipe α (s 2 m 5 ) 10 000 12 000 D 5 000 DE 8 000 DF 10 000 E 400 m 100 m D F 350 m 50 m Hydraulics 2 E2-7 David psley
Open-hannel Flow Q20. V-shaped channel with sides sloping at 30º to the horizontal has a gradient of 1 in 100 and an estimated Manning s n of 0.012 m 1/3 s. alculate: the discharge for a depth of 0.5 m; the depth when the discharge is 2 m 3 s 1. Q21. concrete pipe 750 mm in diameter is laid to a gradient of 1 in 200. The estimated value of Manning s n is 0.012 m 1/3 s. alculate the discharge when: the pipe is full; the depth is 90% of maximum. Explain why the answer in exceeds that in. Q22. circular pipe of diameter 0.3 m carries a discharge of 30 L s 1 down a slope of 1 in 400. Manning s n may be taken as 0.015 m 1/3 s. Find the depth of water in the pipe. Q23. concrete-lined channel has the cross section shown, comprising a semicircular lower part (diameter D) with vertical walls above. Write expressions for the cross-sectional area of flow and the wetted perimeter when the maximum depth of water in the channel is h. (onsider both cases h D/2 and h D/2). For a particular channel of this shape, D = 0.6 m and the streamwise slope is 1 in 100. Taking Manning s n to be 0.013 m 1/3 s, use Manning s formula to calculate: the discharge when the depth is 0.8 m; (c) the depth of water in the channel when the discharge is 2 m 3 s 1. D h Q24. n earth drainage ditch may be approximated as a trapezoidal channel with base 0.6 m and side slopes (vertical:horizontal) of 1:2. The streamwise bed slope is 1 in 400 and Manning s n can be assumed to have the value 0.025 m 1/3 s. If the depth of water is 0.4 m calculate the discharge. In the design storm the channel is required to carry away floodwater at a rate of 0.8 m 3 s 1. Find the depth of water. Q25. (Examination, January 2008) culvert used to divert run-off has a rectangular cross section with base width 0.4 m and side heights of 0.3 m. Manning s coefficient may be taken as n = 0.012 m 1/3 s. Find the minimum slope S necessary to carry a discharge Q = 0.3 m 3 s 1. If the slope from part is doubled for the same discharge, calculate depth of flow. Hydraulics 2 E2-8 David psley