Exploring the Relationship of Small Tube Diameters and Laminar Flow Nicole Kowtko Harvey Mudd College Physics 22 Section 1 20 April 2012 Abstract The flow rate of water through 4 ± 0.03 m tubes at 2 ± 0.01 psi and four diameters (1/8, 8/47, 3/16, 1/4") was measured by recording the times to reach 150 ± 10 ml in order to explore whether smaller diameter tubes cause a water circuit to stray into laminar flow. The flow rate was found to be proportional to the tube diameter to the 2.95 ± 0.14 power, with the dominant error from the resolution uncertainty of the volume readings. This exponent was greater than the previous experiment s result of 2.43 ± 0.04, and closer to the laminar flow value of 4. The 8/47 diameter tube still demonstrated a lower flow rate than expected, and is suspected to have an incorrectly labeled diameter. Introduction This experiment deals with water circuits, and the relationship between flow rate and small tube diameters in a transitional regime between turbulent and laminar flow. The water circuit contains a pump that provides pressure P (psi) to a connected tube with length L (cm) and an inner diameter d (in). The rate at which the water exits the tube is the flow rate Q (ml/sec). The pressure of the water exiting the tube was considered negligible and the input pressure was assumed to be approximately equal to the pressure drop across the tube. In a previous experiment comparing the relationship between flow rate and tube diameter of 1 ±.013 m tubes at 1.5 ± 0.01 psi, the smallest tube used, 8/47 diameter, exhibited a lower flow rate than expected. When this point was included in the data set, it increased the slope of ln(q) vs. ln(d) from 2.43 ± 0.04 to 2.72 ± 0.029 and increased the reduced chi-square value from 4.92 to 57.2, demonstrating how the inclusion of this point did not produce the best relationship [2]. The current experiment arose from interest in whether this increased slope was due to the system falling into laminar flow, which is when the fluid moves along in perfectly smooth layers and exhibits proportionality to pressure, tube length, and tube diameter according to the following equation [1]: Q Pd 4 L (1) 1
Laminar flow does not occur in everyday water circuits, though lower flow rates can approach laminar flow. Because water circuits vary depending on the specific tube length, diameter, and pressure being used, knowing the limits of laminar flow would be useful to help engineers know when the laminar flow proportionalities can be used in water circuit design. These limits would allow an increased predictability of water circuits. When dealing with power-law relationships such as Q = Adα (2) where A is a constant and α is the unknown exponent, the data can be revisualized as: ln(q) = ln(a) + α ln(d) (3) or as a log-log plot [1]. On a log-log plot the exponent is simply the slope of the graph. Based on this context, the outlier point in the previous experiment suggested a possible stray into laminar flow because it increased the slope of the graph, and thus increased the exponent α relating flow rate to tube diameter in Eq. 2. The larger slope was closer to the laminar flow exponent of 4. In order to test this hypothesis, it was attempted to induce laminar flow by decreasing the flow rate by using longer 4 ± 0.03 m tubes, smaller diameter tubes, and the relatively low pressure of 2 ± 0.01 psi in order to observe if the smaller tube diameter of 8/47 was straying into laminar flow. Experiment The apparatus for the experiment is shown in Fig. 1. Fig. 1 Apparatus used to measure the flow rate of a water circuit. A pump is connected to a pressure-regulating valve that is attached to a pressure gauge. The shutoff valve turns the flow of water on or off. A tube adapter connects the tube to the apparatus, which is placed in a smaller tub filled with water. The graduated beaker is placed in the encompassing larger tub, and a stopwatch is used to measure time. 2
Throughout the entire experiment the pump was kept submerged underwater. After attaching the tube the valve was turned on to start water flow through the tube. Because water pressure varies with height, the pressure was adjusted to 2 ± 0.1 psi while the tube was raised to the height of the beaker. Then the tube was suspended over the graduated beaker, and the timer started. The timer was stopped when the graduated beaker read 150 ± 10 ml through the side of the large tube at eye level. This volume was chosen because at small diameters the flow was so low that any greater volume would not have allowed enough time to collect an appropriate data set during the one Physics Lab period appropriated for data taking. Each time the beaker was emptied as well as possible. Before taking any measurements, the length of the four tubes (1/8, 8/47, 3/16, 1/4") was measured with a tape measure by taping down one end to the edge of the measure, and stretching out the tube to its full extent so no bumps remained in the tube. Each was measured four times, and the lengths were averaged to give a final value. The tube length is defined in Fig. 2. Fig. 2 The tube length is measured from the end of the tube adapter within the tube to the end of the tube. The tube adapter was used to normalize the tube lengths by inserting or removing it as necessary to make each tube within ±0.03 m of 4 meters. Results The data was collected through 8 trials per diameter with the exception of 1/8 with only 6 trials. The pressure was measured as the constant volume 150 ± 10 ml, divided by the time it took to reach this volume based on the stopwatch, which varied for each trial. Because of the assumption that the diameter readings for the tubes were accurate, there are no diameter uncertainties and therefore the flow rate was plotted on the y-axis with its corresponding uncertainties. The volume uncertainty δv was simply the resolution error of 10 ml, which was calculated based on the graduated beaker s tick marks. The time uncertainty δt was calculated through adding the resolution error from the stopwatch 0.005 sec and the standard error in quadrature as follows: δt = std.err 2 + res.err 2 3 (4)
δq and δln(q) were calculated through error propagation of δv and δt values according to these equations: δq = δv 1 2 $ ' & ) + δt V 2 $ ' & % T ( T 2 ) % ( δ ln(q) = δq Q (5) (6) The following are log-log graphs of flow rate versus tube diameter for the old and new data: (a) Old Flow Rate vs. Tube Diameter Log-Log Plot (b) New Flow Rate vs. Tube Diameter Log-Log Plot lnq = 7.26 ± 0.05 + ( 2.43 ± 0.04) ln(d) χ 2 = 4.92(4.92 /DoF) P(>) = 0.027 Fig. 3: Natural log of flow rate (Q) as a function of the natural log of the tube diameter (d). These are linear fits where the slope (the number preceding ln(d)), represents the exponent of proportionality between Q and d or. (a) This graph demonstrates the relationship for the previous experiment s data, in which 8/47 was excluded. The upper panel shows the residuals, which appear to be random. (b) This graph demonstrates the relationship from the new experiment s data, showing the flow rate for 8/47 is below the fitted line as in the previous experiment; however, when this point was removed the slope was not affected. The data from the new experiment, graphed in Fig. 3b, suggests that the natural log of flow rate and tube diameter for Q measured in ml/sec, d measured in inches, a tube length of 4 ± 0.03 m, and a pressure of 2 ± 0.01psi are related in the following manner: lnq = 7.87 ± 0.25 + ( 2.95 ± 0.14) ln(d) (7) The new data exhibits an increased slope, 2.95 ± 0.14, compared to the old experiment, shown in Fig. 3a, with slope 2.43 ± 0.04 [2]. This exponent is closer to the expected exponent of 4 for laminar flow compared to the old data. These slopes differ by 3.57 standard deviations, demonstrating that they are statistically different. σ was calculated according to the following equation: σ new&old data = α new α old δα new δα old = lnq = 7.87 ± 0.25 + ( 2.95 ± 0.14) ln(d) χ 2 =10.1(5.06 /DoF) P(>) = 0.0064 2.95 2.43 0.14 2 + 0.04 2 = 3.57σ With a χ 2 of 5.06, this relationship seems reasonable, suggesting that at a reduced flow rate, Q d 2.95±0.14 (9) Q d α (8) 4
The tube diameter of 8/47 still exhibits a lower flow rate than the trend, as shown in Fig. 3b, similar to the previous experiment. The data was plotted excluding this point, and the corresponding fit was used to calculate an expected tube diameter for the corresponding flow rate in order to determine if the tube diameter of 8/47 could have been labeled incorrectly (Fig. 4). Flow Rate vs. Tube Diameter Log-Log Plot Excluding 8/47 lnq = 7.88 ± 0.25 + ( 2.92 ± 0.14) ln(d) χ 2 = 0.174(0.174 /DoF) P(>) = 0.68 Fig. 4: Natural log of flow rate (Q) as a function of the natural log of the tube diameter (d) of the new data, excluding the data point for 8/47 tube diameter. The fit for this graph was used to calculate the expected tube diameter for a flow rate of 11.6801 ml/sec, which was assumed to correspond to a diameter of 8/47 during experimentation. With random residuals and a χ 2 of 0.174, somewhat close to the ideal value of 1, the fit is reasonable, though it seems that the errors were overestimated. The dominant error was the volume uncertainty; the first half of Eq. 5 was at least three times larger than the second portion for each of the four data points. This error could have been reduced through the use of a graduated beaker with finer tick marks or by allowing the water to flow for a longer period of time, though an increase in time would decrease the second half of δq twice as fast as the first half. However, the fit from Fig. 4 was good enough to be used to predict a tube diameter for the flow rate corresponding to the assumed diameter of 8/47 or 0.17. The fit gave a tube diameter of 0.156 for the flow rate of 11.6801 ml/sec. The fit from Fig. 3a was also used to predict a tube diameter for the old data s flow rate corresponding to the assumed diameter of 8/47 or 0.17 in order to determine whether the predicted diameter from the new data could be realistic. Fig. 3a predicted a tube diameter of 0.149 for the flow rate of 14.0028 ml/sec. Fig. 3a and 4 predict an average tube diameter of 0.1525 as opposed to 8/47 or 0.17, but when included in the new 5
data s graph, the corrected version did not exhibit a significant change in slope. For this reason the corrected data point was not included in Fig. 3b. Conclusion The slope of the graph for 4 ± 0.03 m tubes at 2 ± 0.01 psi and small diameters is greater than 1 ±.013 m tubes at 1.5 ± 0.01 psi, which supports the hypothesis that smaller tube diameters and overall lower flow rates will stray into laminar flow, for the exponent in Eq. 9 approached the laminar relationship of d 4 in Eq. 1. However, the basis for the hypothesis is still not resolved. The tube diameter of 8/47 still exhibited a lower flow rate than predicted by data from diameters above and below it, so the old data was not exhibiting a progression into laminar flow. However, the fitted lines from Fig. 3a and Fig. 4 were used to calculate predicted tube diameters for the flow rates corresponding to the supposed tube diameter of 8/47. The result was 0.149 and 0.156 respectively, both of which are lower than the listed value of 8/47 or 0.17. The data confirms that smaller tube diameters do tend towards laminar flow, and that the flow rate for 4 ± 0.03 m long tubes at 2 ± 0.01 psi is proportional to d 2.95 ± 0.14. One systematic error could be an incorrectly high tube diameter labeling for 8/47, causing a lower than expected flow rate. A more prominent systematic error could be the level at which the tube was raised when filling the graduated beaker. The pressure was adjusted when the tube was held at a height level to the lip of the beaker, but when filling the beaker the tube tended to be held slightly lower than the lip, causing a decreased pressure gauge reading. However, even though the gauge s pressure reading decreases, the water pressure due to gravity increases more, causing a higher flow rate. Ambient pressure is another systematic error; if the pressure in the room were higher during this experiment than previously, the flow rate would likely be slightly lower because there would be a lower pressure difference between the ambient air and the water in the tube. However, the assumption was made that the change in pressure due to water exiting the tube was insignificant. This experiment applies to the real world because it demonstrates that water circuits do tend to approach idealized laws as they approach laminar flow. With more experimentation it could possibly be determined if this is asymptotic, or if there is some threshold pressure after which the circuit truly exhibits laminar flow. These results already suggest a general predictability of a water circuit s flow based on tube diameter: the lower the flow, the closer the exponent will be to 4. More experimentation could also help reveal if there is an equation that will predict the exponent of proportionality between flow rate and tube 6
diameter, which would allow further predictability of water circuits. This could have practical applications in robotics by informing engineers of the minimum tube diameter allowed for a certain flow rate in hydraulics systems. References [1] Connolly E., A. Esin, S. Gerbode, E. Henriksen, Physics 22 Introduction to Experimental Physics: Laboratory Course Manual. Spring 2012. P. 12-14. [2] Kowtko, Nicole. Physics 22 Lab Notebook. P. 29-39 7