APPLICATION OF BERNOULLI EQUATION 1. The Problem The student is asked to investigate the parameters that influence the flow of water through a convergent/divergent pipe, see illustration below. Steady flow is assumed to exist in the pipe. The static and total pressure heads in the pipe are to be measured at different locations and for a range of volume flow rates. The variables that might affect the volume flow rate in the pipe are: (i) static pressure p; (ii) velocity v (iii) height z. a 14° b 61.08 76.08 15.8 2.9 10 Convergent/Divergent Pipe 40.46 65.46 a = 25.0 b= 13.9 c = 11.8 d=10.7 e=10.0 F= 25.0 All dimensions in mm Between any two pressure tappings (e.g. 1 and 2), Bernoulli's equation gives: 2 P₁ + v₁² + Z₁ = P1 V1 + pg 2g 2 P2 v2 + pg 2g +22 (m) If the pipe is horizontal then: 2 2 P1 + pg 2g V₁ P2 V2' = + pg 2g (m) Here Р is the static pressure head, and pg v2 is the velocity or dynamic pressure head. 2g The total pressure head, which is found by adding the static pressure head to the dynamic pressure head, can be measured using the steel probe supplied (called a pitot tube). The probe must be used with the end hole facing into the flow so that it brings the flow to rest locally at the probe end. Hence, you can use this probe tube to measure the total pressure head at any axial location along the centre line of the pipe. Flow in the tube is also governed by the continuity equation, which for incompressible flow gives A₁₁ = A2V2 (m³/s) 2. The Apparatus The apparatus consists of a hydraulic bench and convergent/divergent pipe with static pressure tappings, that also includes: • Flow measurement by timed volume collection. • Stopwatch • Spirit Level • Manometer • Pitot tube 3. The Procedure Set up the apparatus on the hydraulic bench so that its base is horizontal; this is necessary for accurate height measurement from the manometers. First, ensure that the test section has the 14° tapered section converging in the direction of flow. Ensure that the test rig outflow tube is positioned above the volumetric tank, in order to facilitate timed collections. Connect the test rig inlet to the bench flow supply; open the bench valve and the apparatus flow control valve, and start the pump to fill the test rig with water. Before running, determine the maximum permissible volume flow rate by determining the maximum height (h₁) and the minimum height (h) of the manometer readings that fit on the manometer scale. If required, the manometer levels can be adjusted by using the air bleed screw and the hand pump supplied. The air bleed screw controls the air flow through the air valve so, when using the hand pump, the bleed screw must be open. To retain the hand pump pressure in the system, the screw must be closed after pumping. The following measurements should be carried out: 1. At the maximum flow rate, take readings of all manometers when the levels have steadied, having ensured that the total pressure probe (pitot tube) is retracted from the test-section. 2. Carry out a timed volume collection, using the volumetric tank, in order to determine the volume flow rate at the maximum flow rate. This is achieved by closing the ball valve and measuring (with a stopwatch) the time taken to accumulate a known volume of fluid in the tank, which is read from the sight glass. Collect fluid for at least one minute in order to minimise timing errors. Repeat these measurements at least twice and check for repeatability of results. Use the average reading calculated over a number of measurements in order to determine the volume flow rate. 3. Now return the total pressure probe into the test section and measure the total pressure at appropriate locations along the test section. 4. Repeat the three measurements described above for a range of different volume flow rates (you should study at least two volume flow rates). For the laboratory report submission. For two different volume flow rates: (i) Determine the fluid velocity in the centre of the pipe at axial locations below each pressure tapping. (ii) Determine the velocity head in the centre of the pipe at axial locations below each pressure tapping. (iii) Add the measured velocity head to the measured static pressure head at the same location. Hence, determine the total pressure head at axial locations below each pressure tapping. (iv) Plot the static pressure head, and the total head, as a function of axial position along the test- section. (v) Compare the measured total head against a total head calculated theoretically using Bernoulli's equation. With reference to the assumptions made in deriving Bernoulli's equation, discuss the validity of the equation for the water flow through the convergent/divergent pipe. Below are results from the lab handwritten in cells. Experiment Run 1 Volume of water collected (ml) 430 Time taken to collect water (s) 5.23 420 4.76 450 5.18 435 440 5.28 5.26 Volumetric flow rate (m³/s) Location A B C D E F Diameter (mm) 25 13.9 11.8 10.7 10 25 Area of cross-section (m²) Velocity v (m/s) Manometer identifier 1 2 Pressure Head h (mm) 203 140 3 145 4 5 6 150 165 180 2 h+ (mm) 2g Total Head (mm) measured 205 203 201 200 200 185 Experiment Run 2 Volume of water collected (ml) 410 Time taken to collect water (s) 3.26 425 3.36 460 3.68 455 3.53 490 3.65 Volumetric flow rate (m³/s) Location A B C D E F Diameter (mm) 25 13.9 11.8 10.7 10 25 Area of cross-section (m²) Velocity v (m/s) Manometer identifier 1 2 3 4 5 6 Pressure Head h (mm) 235 90 103 125 166 196 22 h+: (mm) 2g Total Head (mm) measured 245 241 235 235 234 201 1. Describe experimental work carried out in a clear and logical manner with the inclusion of appropriate diagrams and figures if necessary an appropriate description of the apparatus and procedure followed should be provided. 2. Present calculations carried out in accordance with the Application of Bernoulli Equation laboratory worksheet equations and graphs will need to be word processed and graphs will need to be appropriately labelled with accompanying explanations in the text. 3. Present collected experimental data (measurements) and calculated results using appropriate tables and plots if necessary tables and plots will need to be word processed with appropriate headings and labels and their contents explained. 4. Provide an evaluation of experimental and calculated results and draw appropriate conclusions observations made during the course of the work carried out should be noted in the text and suitable justification for conclusions made should be provided. Bernoulli Apparatus 240 220 200 100 160 100 80 armfield BERNOULLI ARATUS