Abstract
This experiment looked at the possibility of deviation from the Hagen-Poiseuille equation for calculation of flow rate through a tube. The new flow control apparatus was used to measure flow rates given head loss. When plotted against the expected values of the Hagen-Poiseuille a strong divergence was noted. Variables such as tube length and tube diameter were varied/tested to find possible causes of deviation.
Introduction and Objectives
The problem of dosing is an important one that composes part of the corner stone for the entirety of the plant. Dosing with chlorine and alum are critical for purification of relatively clean water and the flocculation process. However, due to the inability of the equation to properly model the dosing process, head losses which should produce certain flow rates produce substantially lower flow rates. This experiment will quantify the deviation of flow rate data collected from the expected results as calculated by the Hagen-Poiseuille equation.
Procedures
The tested apparatus was constructed as follows.The constant head tank was constructed using two elevated buckets. The storage was significantly larger and with a larger diameter than the actual constant head bottle. They were connected by a single tube which went from the base of the stock tank and into the top portion of the bottle where the float was located. From the bottom of the bottle was a from which the dosing tube was attached. This tube is adjustable to allow for different lengths. A PVC pipe was procured and sealed on the bottom and placed in a wooden base to stabilize it. In this, along the top 26 centimeters, 26 7 mm holes were drilled. The top hole had its center line leveled with the water level in the constant head bottle so that ever additional hole below was another centimeter of head loss.
To cut off additional flow into the constant head tank, the force exerted on the float, i.e. the buoyant force, must be equal to the force exerted by the water attempting to leave, i.e. the force of the water column in the stock tank. The buoyant force is the force exerted by the amount of water displaced by the float. As such, there is a relationship between the height of water in the stock tank and the amount of water displaced by the float in the constant head tank. Since the water level in the constant head tank is affected by the water level in the stock tank, before each iteration of the experiment, the water collected from the previous test is replaced to keep the water constant. Data is collected via the EasyData program which collects voltage output for a 7 kPa pressure sensor.
At first, increments of 25 mL were placed into the tube of the experimental apparatus to obtain a direct conversion value between voltage and volume of water. This allowed for a direct change of voltage to volume. Finding a direct conversion between the voltage and flow rate allowed the graph of voltage vs. time to be converted directly to volume vs. time. The derivative of this would then be the flow rate through the tube.
The flow rate was measured for a series of tubing lengths and head losses. The stock tank was cleaned with distilled water and then filled with distilled water and attached to the constant head tank. The distribution tube, the tube exiting the constant head tank and entering the tube of the testing apparatus, was placed in a hole on the apparatus such that a known head loss was obtained. Using Easy Data, a plot of voltage vs. time was constructed. Transforming voltage to volume produced volume vs time. The procedure was varied using differing head losses and different tubing lengths.
Results and discussion
The test was run for two different tube lengths, 1 m and 44.5 cm. The test was run at many different flow rates from head loss of 3 cm to 25 cm. The plants in Honduras use, at the most, a head loss of 20 cm. The testing of up to 25 cm head loss allowed for an analysis of even higher flow rates, in case there was little to no deviation at the lower flow rates
Figure 1: Flow rate vs. Headloss for 44.5 cm Tube
Figure 2: Flow rate vs. Headloss for 1 m Tube
The head losses were taken from the experiment such that one hole below the base hole was only 1 cm of head loss. The length of the tubes, 44.5 cm and 1 m, were measured. The inner diameter was 1/8 in as given by the manufacturer for the experimental. The correction was applied to this diameter.
As can be noted from the graphs, calculated flow rates was consistently lower than the estimated low rate using the Hagen-Poiseuille equation. However, once the correction was added to the diameter of the tubing in the equation, the data matched much better.
Conclusions
This experiment confirmed that the Hagen-Poiseuille equation overestimated the actual flow rate and thus was providing an inaccurate estimate. However, the extent of this over estimation was previously difficult to determine. However, what is most interesting is the validity of the tubing inner diameter correction. Using the values calculated from the experiments regarding the tube diameter provided a drastic improvement which makes the equation a better model. Also, because the inner diameter of the tube was 1/8th of an inch, small variation in inner diameter produce large changes in terms of particle movement within the tube and thus head loss and the Hagen-Poiseuille equation. Getting a more accurate image of the inside of the tube allows for the equation to operate more predicatively. Varying tube diameters should be tested to find if these minor inner diameter variations are negligible at larger diameters. Varying tube lengths should be tested to consider the possibility of non-negligible minor head losses.