Bernoulli and the Free Jet
Objectives
In this laboratory you will measure fluid velocities in a free jet using a stagnation tube. You will confirm that the Bernoulli equation can be used to measure fluid velocities using a simple stagnation tube.
Theory
The conservation of mechanical energy will be verified using the Bernoulli equation.
1.1
A stagnation tube will be connected to a pressure sensor. The stagnation tube will be filled with water prior to connecting to the pressure sensor and the pressure sensor output will be zeroed with the stagnation tube held vertically (in the same orientation used for taking measurements.) Thus the pressure sensor will measure the pressure at the stagnation point (Figure 1-2). From the Bernoulli equation we can obtain the following relationship.
1.2
where point 1 is on a horizontal line in the jet away from the stagnation point and point 2 is the stagnation point. The stagnation pressure, , will be measured using a pressure transducer. Equation 1.2 requires the assumption that the streamlines are straight and parallel (allowing us to cross streamlines from point 1 to point 2). Since and are zero and we can obtain
1.3
Solving for the velocity at point one next to the tip of the stagnation tube
1.4
Thus by measuring the stagnation pressure we can calculate the velocity of the jet (Figure 1-1).
Experimental Methods and Analysis
Set up a small jet powered by a centrifugal pump. Fill the 10 cm diameter column with 15 cm of water. Fill the stagnation tube completely with water before connecting the pressure sensor. Monitor the pressure sensor with the Easy Data software and apply scaling so the output is measured in Pascals. Zero the output of the pressure sensor with the stagnation tube held vertically in air.
Install a valve between the effluent of the centrifugal pump and the elbow. Set the flow rate using the valve so the stagnation pressure at z = 0 is approximately 300 Pa.
Make the following measurements and calculations.
1)What is the stagnation pressure at z = 0?
2)What is the velocity at z = 0?
3)What is the jet flow rate?
Connect a 7-kPa pressure sensor to the volumetric detector and monitor the sensor with the Easy Data software and apply scaling so the output is measured in mL. Log the data to file and with the pump running turn the jet so it discharges into a different container. You will use the initial slope of the resulting data to determine the flow rate out of the volumetric detector.
4)What is the flow rate that you obtain from the initial slope and how does it compare with the flow rate calculated using the stagnation tube?
5)Fill the volumetric detector to 15 cm again and measure the stagnation tube pressure at various elevations in the jet. Use Excel to plot the velocity in the jet as a function of elevation. On the same graph plot the prediction based on Bernoulli’s equation.
1.7
where points one and two are any two points in a free jet. Make sure that the theoretical model is plotted as a smooth curve. Format the graphs (from 3 and 5) correctly ( ). Insert a textbox or Word object into Excel and summarize what you learned and your suggestions for making this laboratory exercise better. Email the Excel file to the TA and to the Instructor.