Hydraulic Transients

Objectives

Measure the pressure trace after a sudden valve closure. Determine the pressure fluctuation period and determine if the system is best modeled as gradually varied (lumped) or rapidly varied (distributed). Develop a model to describe the velocity in the pipe given the measured pressure fluctuations.

Theory

Gradually Varied vs. Rapidly Varied

A system can be modeled as gradually varied when the pressure fluctuation period (T) is much greater than the time for a sound wave to travel back and forth the length of the pipe twice.
                                                           3.1
where the speed of sound in a pipe is given by
                                                   3.2
where the terms are defined in Table 3-1.

If the system is gradually varied then Newton’s 2nd law can be applied to describe the acceleration of water in the pipe. In finite difference form we have
                                                3.3

where H is the difference in piezometric head between the ends of the pipe. The head loss is given by
                                               3.4
Given an intial flow rate (before closure of the valve) and pressure as a function of time at both ends of the pipe equations 3.3 and 3.4 can be used to predict the velocity of water in the pipe as a function of time using a finite difference solution.
If the system is rapidly varied then the magnitude of the pressure wave is predicted to be a function of the instantaneous change in velocity.
                                                        3.5
                                                        3.5

Procedure

• Create the experimental setup using Figure 3-1 as a guide.
• Monitor the pressure sensors using Easy Data  software. Configure the software to monitor the 200-kPa pressure sensors with the physical units as either centimeters of water or Pascals. Configure the 7-kPa pressure sensor as a flow sensor (it will have physical units of mL/min). Plug the solenoid valve into the first port (1-24 V) of the Stamp box (not the data acquisition system.)
• Set the data acquisition rate to the maximum possible (500 Hz).
• Open the State controller software. The software allows the user to control the 24 V outputs on the Stamp box.
• Turn on the first 24 V switch to open the solenoid valve. The valve should make a clicking noise as it opens.
• Turn on the tap water supply.
• Make sure there is no air in the system. Turn the pressure regulator upside down and make sure no air comes out!
• Reduce the pressure at pressure regulating valve by turning the screw out until the flow stops.
• Zero all 3 pressure sensors.
• Empirically determine the maximum flow rate that will cause a pressure spike at the valve that is approximately 200 kPa. Do this carefully by gradually increasing the flow rate (by turning the screw in on the pressure regulating valve) while opening and closing the solenoid valve. If the flow is too great the resulting high pressure will rupture the silicone diaphragm that is used to sense the pressure change.
• Measure the flow rate using the beaker and stopwatch method to get a more accurate measurement.
• Record a transient pressure trace to file (save this file on your local hard drive!) by logging data while the solenoid valve is closed and for several seconds while the pressure fluctuations decrease in magnitude.
• Decrease the flow rate significantly and record another set of measurements.

Data Analysis

• Determine the pressure fluctuation period and determine if the system is best modeled as gradually varied (lumped) or rapidly varied (distributed).
• Develop a model to describe the velocity in the pipe given the measured pressure fluctuations. Describe your assumption concerning head loss as a function of flow rate and verify your assumptions! What model will you use to estimate head loss as the flow rate is varying? Plot the pressure fluctuations and the velocity as a function of time on the same graph using different y axis.
• Calculate (and plot) the distance that the water traveled after the valve closed as a function of time. What was the maximum distance that the water traveled and where did the water go?
• For the brave… Estimate the volume of air in the system and then use the ideal gas law to estimate the downstream pressure based on the compression of the air as the water slows down. Note that you can estimate the volume of air in the system using the initial downstream pressure, the maximum downstream pressure, and the calculated cumulative flow at the time corresponding to the maximum downstream pressure. The two equations in two unknowns are P1V1=P2V2 and V2-V1=DV where the initial state is before the valve closed and the final state is at the maximum pressure. Plot the calculated downstream pressure and compare with the measured downstream pressure.

Note that an offset error in the pressure sensor readings will cause a continuous increase in cumulative error in the model.

Here are some ideas to get your started…

• You know the flow rate at the time when the valve was closed based on the stopwatch and beaker measurements
• You can calculate the initial head loss in the tube based on the pressure and flow measurements before the valve closed.
• The pressure peaks and valleys occur when the velocity in the tube is zero. (Can you explain why this is the case?)

Here is an idea for those of you who learn by playing. If you figured out where the water went when you closed the valve, try to modify your experimental apparatus so there was less available volume (or much more available volume) for the water to fill. Repeat the experiment (taking care not to rupture the pressure sensor!) and see how the system changes!

Lab Setup

Acquire data at 500 Hz (Change the data frequency at the Data Server)
Save data locally to avoid clogging the intranet.