MEE 3506 FLUIDS LABORATORY IMPACT FORCE OF A JET I. OBJECTIVES 1. Measure the force exerted by a stream of water on vanes of various shapes using a beam balance. 2. Compare the measured impact forces with those determined theoretically by the momentum equation. 3. Study the effects of flow rate and vane shape on the impact forces. 4. Apply conservation principles II. THEORY The conservation of mass principle states that the time rate of change of the system mass is zero, or 𝜕 ⃗ ⋅𝒏 ̂ 𝑑𝐴 (∫ 𝜌𝑑𝑉 ) + ∫ 𝜌 ⃗𝑽 𝜕𝑡 𝑐𝑣 𝑐𝑠 1 where ρ is the material’s density, V is volume, V is the velocity vector, and A is the cross-sectional area. The first term in equation 1 is the time rate of change of mass within the control volume of the study. The second term is the next rate of mass flow through the boundary of the control volume. For a system operating at steady state, the first term in equation 1 is zero and the second can be expressed as ∑ 𝑚̇𝑜𝑢𝑡 − ∑ 𝑚̇𝑖𝑛 = 0 2 where 𝑚̇ is the mass flow rate. For a one inlet one exit system, the mass flow rate is constant. The mass flow rate can be defined in terms of the volumetric flow rate, Q, by 𝑚̇ = 𝜌𝑄 = 𝜌𝑉𝐴 3 A simple way to determine the mass flow rate of a fluid stream is to record the amount of time it takes to accumulate a known mass, or 𝑚̇ = 𝑚 Δ𝑡 4 where m is the known mass of the fluid, Δt is the amount of time it took to accumulate the known mass of fluid. Newton’s second law states that the time rate of change of the linear momentum of a system is equal to the sum of the external forces acting on the system, Fsys, 𝜕 ⃗⃗ 𝜌𝑑𝑉 ) + ∫ 𝑽 ⃗ 𝜌𝑽 ⃗ ⋅𝒏 ⃗ 𝑠𝑦𝑠 ̂ 𝑑𝐴 = ∑ 𝑭 (∫ 𝑽 𝜕𝑡 𝑐𝑣 𝑐𝑠 5 The first term in 5 is the time rate of change of the linear momentum in the system which reduces to zero for a system operating at steady state. The second term is the net rate of change of linear momentum flow through the system. Figure 1 – Impact of a water jet on a vane, (a) general, (b) flat vane, and (c) cup vane. Consider a vertical jet of water exiting a nozzle and impacting a vane as seen in Figure 1a. The jet first strikes the vane with a uniform velocity V1 the water is then turned through an angle θ before exiting the system. If the system is symmetric the sum of the forces in the x-direction will the zero. In the vertical direction, equation 5 becomes −𝑉1 𝜌𝑉1 𝐴1 + 𝑉2 𝑐𝑜𝑠𝜃𝜌𝑉2 𝐴2 = 𝐹𝑦 6 The only force in the y-direction is the reaction force of the water impacting the vane if the effects of gravity and viscosity are ignored. The pressure force can also be neglected as there is a uniform pressure acting on all sides of the system. Using equations 2 and 3 the product V2A2 can be replaced by V1A1. For steady, incompressible, inviscid flow, the velocities along a streamline are related by Bernoulli’s equation 𝑉12 𝑃1 𝑉22 𝑃2 + 𝑔𝑧1 + = + 𝑔𝑧2 + 2 𝜌 2 𝜌 7 By neglecting the effects of gravity and since P1 = P2, V1 = V2, therefore, equation 6 further reduces to 𝑉1 (𝑐𝑜𝑠𝜃 − 1)𝜌𝑉1 𝐴1 = 𝐹𝑦 8 For a flat vane, Figure 1b, θ is 90° and equation 8 becomes −𝑉1 𝜌𝑉1 𝐴1 = 𝐹𝑦 9 Note that Fy is negative as the vane exerts a downward (negative y-direction) force on the fluid flow while the jet exerts an equal force in the positive y-direction on the vane. For the cup vane Figure 1c, θ is 180° and equation 8 becomes −2𝑉1 𝜌𝑉1 𝐴1 = 𝐹𝑦 10 Lastly, the velocity V1 impacting the jet is lower than the velocity exiting the jet as it has to travel vertically upward. Using equation 7, the impact velocity can be found by 𝑉1 = √𝑉02 − 2𝑔𝐻 11 Note that the force gauge is attached to a moment arm so the reading on the gauge needs to be correct based on the diagram below. For additional theory please refer to the materials posted on Canvas and the following sections of the 6th edition of Fundamentals of Fluid Mechanics by Munson et al. • Chapter 5 Sections 1 and 2 and example 5.10 III. EXPERIMENTAL APPARATUS AND PROCEDURE A schematic of the experimental test rig is shown in Figure 2. Key components include: • • • • • • Jet-Impact Measuring Bench Flat and Cup Vanes Stopwatch Force gauge Gravimetric flow rate system Standard weight Required Constants • • • Nozzle diameter: Dj = 10 mm Distance from the nozzle to vane: H = 35 mm Distance from vane to pivot center: R = 150 mm Figure 2 – Impact jet experimental setup Experimental Procedure: 1. Record the ambient temperature and barometric pressure. 2. Connect the force gauge such that the chain is taught 3. Initially balance the beam prior to turning on the pump by adjusting the tension spring 4. Adjust the flow rate until the force meters read either 1.5 N for the flat vane or 3 N for the cup vane 5. Record the force gauge reading three times 6. Close the weight tank and begin timing how long it takes to record 8 kg of water 7. Lap the timer, add a 10 lbm mass to the hanging weight and record the time it takes to collect an additional 13.62 kg. 8. Repeat steps 5-8 two times twice for a total of three flow rate trials and 6 total flow rate measurements 9. Repeat steps 4 to 8 for the forces listed below: a. Flat vane 1.5, 1.25, 1.0, and 0.75 b. Cup vane 3.0, 2.50, 2.0, and 1.50 VI. PRESENTATION AND DISCUSSION OF RESULTS Your data analysis must include 1. A table that lists the following for all flowrate investigated a. The actual flow rate in m3/s b. The actual force measured for the flat vane c. The actual force measured for the cup vane d. The theoretical forces for the flat and cup vanes e. The percent difference between your actual and theoretical results The following discussion question must be answered in your report. Do not number your answers however you may highlight them. 1. Compare the theoretical and actual forces between the flat and cup vanes. 2. How much did the velocity decrease between the nozzle and the vane? 3. Plot the theoretical and actual forces versus the velocity of the jet (V1) on the same chart. 4. Perform a curve fit of your experimental data using a power fit, F=aVn where a and n are constants. 5. Discuss the validity of assuming that V1 = V2 for each vane

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