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At the outlet, four varieties of pipe assemblies were designed and assembled; each assembly had at least one flowmeter, and a manometer was created to find the head loss between various points of each pipe assembly (it was ensured that the top of the manometer tubes were higher than the water level of the constant head tank). The inlet pipe, outlet pipe and overflow pipe all led back to the water-filled reservoir in which the pump was placed.

Figure 2: Constant Head tank

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Leaks in the apparatus were occasionally present throughout trials, which called the assumption of volumetric flow being constant at the input and output into question. The flowmeter itself may also have provided inconsistent frequencies (and, conversely, volumetric flow rates) - in some instances, rotation would stop or slow down before abruptly resuming as normal. Future experimentation would necessitate stronger connections between pipe components - especially around flanged fittings, which tended to pop off between trials.

It should also be mentioned that due to the complexity of the experiment, time constraints prevented more trials from being executed - a larger sample size may have resulted in more valid results. Future experimentation would call for samples done at various flow rates, and thus various velocities - data collected with laminar or transitional flow would be valuable, as well as data with turbulent flow simply for comparison.

Figure 10: Evident leakage in the apparatus

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It should be noted that two loss coefficients were missing from the table - the flowmeter and the flanged union. For the former, Eq. 6 was used in conjunction with the velocity and the experimentally calculated pressure drop from Assembly 1 to find a loss coefficient (Kflowmeter). For the latter, the loss coefficient was deemed to be negligible; the loss coefficient table lists a K-value of 0.08 for the threaded union, and flanged components notably have only a fraction of the loss coefficient of their threaded counterparts - it was predicted that a loss coefficient for a flanged union would thus be too small to affect calculations.

Furthermore, for Assembly 3 and 4, the minor head losses found by summing the minor losses of their respective top/bottom subassemblies and their left/right subassemblies for a total minor head loss throughout.

The total piping length (L) of each assembly and subassembly was found by summing the lengths of the inidividual straight pipes used (lengths of pipe components were assumed to be negligible compared to the straight pipes).

Using the Reynolds number derived earlier, along with  a roughness value of ε = 0.0015, the friction factors (f) of each subassembly were found through the Haaland equation:

Eq. 7

It should be noted that this equation only applies to turbulent flow (Re > 2600); all previously derived Reynolds numbers proved turbulent flow throughout the experiment.

These, in turn, were used to find the major head loss (Pf = γhf) using another equation derived from Experiment 5:

Eq. 8

Note that for for Assembly 3 and 4, as was done for the minor head losses, the major head losses throughout their subassemblies were again summed.

The sum of the total major and minor pressure drops in each subassembly led to a theoretical total pressure drop across the system:

Eq. 9

This was compared to the experimental pressure drop found earlier.

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For Assembly 3 and 4, the top and bottom sections of the assembly do not have their own flow meter - however, assuming continuity (i.e. Qin=Qout)it can be proven that:

Eq. 4

Using this velocity, along with density, diameter (D) and viscosity (μ), the Reynolds number (Re) for each assembly and subassembly was found:

Eq. 5

At the same time, the total loss coefficients (KL)of each assembly and subassembly were found by summing the loss coefficients of each pipe component - these were sourced from Experiment 5 - Pressure Drop in Piping Systems’ Table 1. From the same source, it is shown that the minor head loss (Pm = γhm) due to these components can be found using the total loss coefficient:

Eq. 6

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The theoretical pressure drop was as follows. In the first experiment, the following relationship was found between volumetric flow rate (Q) and flowmeter frequency (F):

Eq. 2

The validity of this relationship was tested with a similar procedure as in the first experiment; the relationship applied reasonably to each assembly, and it was used to find the flow rate of each flow meter. The velocity (v) of the fluid through the flow meter was then found using the flow rate and the cross-sectional area (A) of the pipes (which in turn is found using the radius R) :

Eq. 3

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The experiment was divided into two sections - the first of which led to an empirical pressure drop, and the second led to a theoretical pressure drop. The empirical pressure drop (∆Pexp) was found simply as follows, using water density (⍴), gravitational constant (g = 9.81m/s2) and the experimental height difference in the ‘manometer’ from each assembly:

Eq. 1

Figure 9: Measuring height difference

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Figure 8: Assembly 4

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Figure 7: Assembly 3

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Figure 6: Assembly 2

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Figure 5: Assembly 1

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To begin, the analysis of Assembly 1’s experimental pressure drop (as calculated from the head loss) of 313.920 Pa led to a loss coefficient of 7.580. Though it was difficult to judge the accuracy of this without a literature value, the coefficient seems reasonable when compared to the coefficients of other pipe components - its loss coefficient was somewhat more than that of a ⅓ closed ball valve. It should be noted that since the theoretical pressure drops for all assemblies relied on the flowmeter K-value derived from Assembly 1, and the only component in Assembly 1 is the flowmeter, the theoretical pressure drop inevitably equaled the experimental pressure drop (as the former was derived from the latter); thus, the result was considered to be irrelevant.

Assembly 2, with a theoretical and experimental pressure drop of 782.692 Pa and 372.780 Pa respectively, started off a trend of large discrepancies between the experimental and theoretical values - these values had a percent error of 52.372%. This trend continued with Assembly 3 (13262.810 Pa compared to 2746.800 Pa) and Assembly 4 (22473.420 Pa compared to 4414.500 Pa) - the former had a percent error of 79.290%, and the latter had a slightly larger percent error of 80.357%. It should be noted that, despite these large differences, both the theoretical and experimental pressure drops tended to increase with the complexity of the assemblies. Furthermore, though the large difference in complexity between Assembly 2 and Assembly 3 easily explained the much larger experimental and theoretical pressure drop of the latter, it was surprising to see that Assembly 4 also had a similarly large increase in pressure drop despite only receiving a relatively small alteration on only one subassembly.

Saying that, with correlations being weak and inconclusive throughout the experiment, a number of sources of error may have resulted in these discrepancies.

Perhaps most significantly, theoretical pressure values were calculated using the K-value for when the valve was closed ⅓ of the way - in this experiment, however, the pressure valve may not have been precisely ⅓ closed. Considering that a ⅔ closed valve was only about a 20 degree turn away, and its K-value was 210 (as opposed to the much smaller 5.5 of a ⅓ closed valve), even a small fault in the estimation of ⅓ may have resulted in a much higher or lower K-value than assumed. Future experimentation should incorporate ball valves with markings indicating what amount they are closed to.

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The experiment was started off by reconstructing the constant head tank from Experiment 1 - a K’nex structure that could sustain the weight of water filled pipes, along with a tank with an inlet pipe, an outlet pipe and an overflow pipe. At the inlet, a pump was installed.
Figure 1: Initial K'nex structure

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Similar to Assembly 1 - however, two ball valves closed by ⅓ were placed on the straight pipe, and the flow meter was moved to the bottom of the assembly. The manometer was placed to find the head loss between the inlet and outlet of the whole assembly.

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Built based on assembly three; however, a long, straight pipe was removed from the left subassembly and replaced with a more convoluted set of pipe components (thus making the subassemblies asymmetrical).

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Composed of two parallel pipe subassemblies branching from a top pipe and converging into a bottom T-section. The left and right subassemblies were symmetrical and composed of various pipe components (including a flowmeter each); the manometer was placed to find the head loss between the inlet and outlet of the whole assembly.

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Consisted of a straight pipe assembly with a flowmeter at the top; the manometer was placed to find the head loss between the inlet and outlet of the flowmeter.

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As in Experiment 1, a calibration trial was taken for each assembly by timing the filling of a predetermined volume after turning the pump on, and recording the frequency of the flowmeter(s). The frequency was applied to an equation determined from Experiment 1, and the flow rate derived was compared to the calibration trial’s flow rate to ensure that the Experiment 1 calibration was still valid.

Figure 3: Timing the filling of a predetermined volume for calibration

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Once the configurations were set and the pump was turned on, it sometimes occurred that that the pipes - whether they were tygon tubes of the manometer or the PVC pipes of the assemblies - would be partially filled with air. In order to eliminate this, a back pressure was created by adjusting a ball valve at the assembly to take out any air bubbles, and an appropriate voltage (equal to or less than the pump’s 12V rating) was controlled to keep the water level just above the overflow in the constant head tank.

For the main experiment, a set of values were recorded per assembly - namely, the manometer height difference and the flow rates from each of the flowmeters.

Figure 4: Purging Air bubbles