FLUIDS MECHANICS LABORATORY THE HYDRAULICS BENCH The Hydraulics Bench is a service module designed to accommodate a range of accessories, each of which demonstrates an aspect of hydraulic theory. A self-priming centrifugal pump draws water from the sump tank and delivers into a vertical transparent pipe. A panel mounted bench control valve is used to regulate the flow in the delivery pipe which terminates in a quick release pipe connector situated in the bed of a channel. Water discharging from the accessory on test is collected in the volumetric measuring tank. This tank is stepped to accommodate low or high flow rates and incorporates a stilling baffle to reduce turbulence. MEASUREMENT OF VOLUMETRIC FLOW RATE The molded bench top incorporates a volumetric measuring tank which is stepped to accommodate low or high flow rates. A remote sight gage, consisting of a sight tube and scale is connected to a tap in the base of the tank, and gives an instantaneous indication of water level. The scale is divided into two zones corresponding to the volume above and below the step in the tank. To make a volumetric flow reading, the ball of the dump valve is lowered, retaining the water in the tank. Timings are taken as the water level rises in the tank. Low flow rates are monitored on the lower portion of the scale corresponding to the small volume beneath the step. Larger flow rates are monitored on the upper scale corresponding to the main tank. When not used for volumetric determinations, the water drains back to the sump by lifting the dump valve. F1-10 Hydraulics Bench Drain Valve 0 Water Inlet Stilling Baffle Sight Tube and Scale Location of Hook & Point H Centrifugal Service Pump Bench Control Valve Weir Carrier J Stilling Baffle Dump Valve RCD Pump ON/OFF Switch E1: THE IMPULSE MOMENTUM PRINCIPLE (Impact of a Jet) LEARNING OBJECTIVES Demonstrate experimentally the impulse-momentum principle (linear-momentum equation). Calculate the force generated by a jet of water as it strikes a flat plate or a hemispherical cup and compare with the momentum change of the jet. THEORY OF THE EXPERIMENT For a given control volume (c.v) with a control surface (c.s) that completely surrounds the control volume, the Reynolds transport theorem for steady flow is: days = √nph-VdA dt where the integral is over the control surface and the symbols and variables mean: Nsys = Extensive quantity (any property of the flow, e. g., mass, velocity, etc.) t = time (s) n = Intensive quantity: refers to Nsys per unit of mass [kg] p = Density of the flow [1] f = unit direction vector V = Velocity vector and V = Velocity [] A = Area [m²] Newton's second law, referred to as the momentum equation, states that the resultant force acting on a system equals the rate of change of momentum of the system when measured in an inertial reference frame. Replacing with V, this becomes: ΣF=SU = sum of all forces acting on the control volume = [pV(V-n)dA C.S Vout J Figure 1: A jet of water impinging on a flat plate V₁lak Z = H Z=O When applying this general equation to the conservation of linear momentum for the case of a jet impinging on a flat plate (Figure 1), we get: where N is the number of flow exit/entrance areas and fi is the unit vector normal to the control area. For the entrance to the control volume, f = -1 relative to the flow of the jet of fluid. It then follows for a vertically upward jet: F(-1)= PQin (1) Vin (1)(-1) F = PQin Vin where Qin = Q = Volumetric flowrate [T Ĵ= unit direction vector p= Density of the flow Vin = Vplate = ? Σ F₁ = Σ P₁AiV₁ (V₁-A) R=Σ PiA¡Vi The velocity of the jet as it strikes the plate, Vplate, is less than the exit velocity at the nozzle, Vjat, because of the deceleration due to gravity. We can apply Bernoulli's equation to determine Vplate- PV² For Bernoulli's Equation + P 2 + gh = constant (along a streamline) the following assumptions apply: P Y Vi _ Vi V² = [kg] Assumptions of Bernoulli's Equation for the impact of a jet No shaft work • No heat transfer Steady flow Incompressible flow (p = constant) Frictionless flow Applying Bernoulli's equation yields: P₁V² + 2 = P₂²_V₂² P, Y2g Y 2g = - Pressure Head 2g 2g Vi _ V₁ + H 2g 2g +2₂ V₂ = √V²-2gH V₁ =VjV₂ = Vplate or V₂ cup V₂ 2g . - Velocity Head Flow along the streamline z = Elevation Head Hence, F=PQV=PQ √V-2gH Vat Figure 2: A jet of water impinging on a hemispherical cup. Qin = -20o and Vin = -Vout F = PQin (Vin +Vout) = 2pQinVin 2n = Q = volumetric flowrate p = density of fluid [kg ʼn always points out of the volume = -1 at the entrance and exit. F(-ĵ) = pQ_(†)V_(†)(−1) + pQout (-Ĵ)V out (-Ĵ)(−1) + µQ…(−Ĵ)V…(-Ĵ)(−1) -F=-pQV₁ -2pQmVant == וון CV V₂ - V - √V²-28 H = = g in F=2pQ√ √V-2 g H Z-H When applying this general equation to the conservation of linear momentum for the case of a jet impinging on a hemispheric cup (Figure 2), we get: ΣF₁ = [p,4,V, (V₁.ñ) 3. S Neur 'cup V₂-p Ajel Z=H Experimental Procedure Figure 3 presents the experimental apparatus. The inlet nozzle is housed in a transparent cylinder, the interior of which remains at atmospheric pressure due to air vents in the top plate. The pump draws water from the sump tank and this water issues from the nozzle at a flow rate that is regulated by the bench control valve. The water strikes the target (flat plate or hemispherical cup) and the mass loaded in the weight pan is used to determine the force needed to balance that from the water jet. The deflected water then exits through outlet holes into the tank where the time taken to collect a certain volume is used in the calculation of the volumetric flow rate. flow control valve weight hanger Spring motor Air Vent Knurled nuts Top Plate A tank pivot CG- pump weigh tank plug sump tank Weight Pan Level Gauge Spirt Level Target Plate Nozzle Tank Figure 3: Impact of a jet apparatus. (Images from http://www.armfield.co.uk/images/fl 16 rgb.jpg)