TD 3 “Bernoulli equation” 1. Water is flowing through a pipe having diameters 600 mm and 400 mm at the bottom and upper end respectively. The intensity of pressure at the bottom end is 350 kN/m2 and the pressure at the upper end is 100 kN/m2. Determine the difference in datum head if the rate of flow through the pipe is 60 litres/s. 2. We want to accelerate the circulation of a perfect fluid in a pipe in such a way that its speed is multiplied by 4. For this, the driving includes a convergent characterized by the angle α (diagram below). a. Calculate the ratio of the radii (R1 / R2). b. Calculate (R1 - R2) as a function of L and α. Deduce the length L. (R1 = 50 mm, α = 15°). 3. We consider a tank filled with water at a height H = 3 m, equipped with a small orifice at its base with a diameter of d = 10 mm. a. By specifying the assumptions taken into account, apply the theorem of Bernoulli to calculate the water flow velocity V2 b. Determine the volume flow Qv in (l/s) at the outlet of the orifice. Assume that g = 9.81 m/s. 4. A perfect incompressible fluid flows from a circular orifice located on the side of a tank with a volume flow qv=0.4 L/s. The diameter of the orifice is d=10 mm. a. Determine the flow velocity at the orifice. b. State Bernoulli's theorem c. How far from the free surface to the orifice? 5. We consider a cylindrical tank with an internal diameter D = 2 m filled with water up to a height H = 3 m. The bottom of the tank is provided with an orifice of diameter d = 10 mm allowing the water to evacuate. If a very small time dt is allowed to pass, the water level H of the reservoir drops of a quantity dH. We notice V1=dH/dt the rate of descent of the water level, and V2 the flow velocity in the orifice. a. Write the continuity equation. Deduce the expression of V1 as a function of V2, D and d. b. Write Bernoulli's equation. We assume that the fluid is perfect and incompressible. c. From the answers to questions (a) and (b) establish the expression of the speed of flow V2 as a function of g, H, D and d. d. Calculate the speed V2. We assume that the diameter d is negligible compared to D. That is to say d/D<<1. e. Determine the volume flow qV 6. The cylindrical tank shown above, open to the air, has a section SA of diameter DA = 2 m. It is equipped, at its base, with an SB section emptying orifice and of diameter DB = 14 mm. The tank is full up to a height H= (ZA –ZB) = 2.5 m of fuel oil, liquid considered as perfect fluid, of density ρ=817 kg/m3. We note α=(SB/SA). Part 1: The orifice is closed with a plug a. By applying the RFH, determine the PB pressure at point B b. Determine the value of the pressure force FB which is exerted on the cork. Part 2: The orifice is opened The tank is emptied. Fuel oil flows from the tank. Its average flow velocity at point A is denoted VA, and its flow velocity at the orifice is denoted VB. a. Write the continuity equation. Determine VA as a function of VB and α. b. By applying Bernoulli's theorem between A and B, establish the literal expression of the speed VB as a function of g, H and α. C. Calculate the value of α. The hypothesis of considering a level H of the fluid varies slowly is it true? Justify your answer. d. Calculate VB considering the assumption that α<<1. e. Determine the volume flow QV of the fluid flowing through the orifice. (In liter per second) f. What would be the duration T of the emptying if this flow rate remained constant? 7. In a horizontal pipe of diameter D = 9 cm, we want to measure the flow of water. A Venturi tube is inserted (D=9 cm, d=3 cm). The height difference h of the mercury in a U-tube can be measured accurately. We have: - the density of water: ρwater = 1000 kg/m3 - the density of mercury: ρmercury = 13600 kg/m3 a. Write the continuity equation. Determine the average flow velocity VB at the pass in the SB section as a function of the speed VA in the SA section. b. By applying the fundamental relationship of hydrostatics (RFH) between the points A' and B' relative to mercury equilibrium, determine the pressure difference: (PA' - PB') as a function of g, ρmercury, ZA' and ZB'. c. Similarly, determine the expression of the pressure difference (PA-PA') in function of g , ρwater, ZA' and ZA. d. Similarly, determine the expression of the pressure difference (PA'-PB) in function of g , ρwater, ZB' and ZB e. Using the equations established in questions (b), (c) and (d) give the relationship between (PA-PB) as a function of ρmercury, ρwater, g and h. f. Assuming that water is a perfect fluid, and applying the Bernoulli's theorem between A and B, give the expression of the flow velocity VA as a function of pressure difference (PAPB), ρwater. g. Determine the expression of the volume flow Qv as a function of D, ρmercury, ρwater, g, h. h. Make a numerical application for a difference in level h= 4 mm.
