3 044394606 1 3 02 04 002 0025 003 0035 004

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࠵?(࠵?3) = 0.44394606 (࠵?࠵?࠵?࠵?࠵?)13[࠵?]00.050.10.150.20.250.30.350.40.450.50.0150.020.0250.030.0350.040.0450.05Height (m)Volumetric Flow Rate (m3/s)Height vs Volumetric Flow Rate (Air)Air, High M&DAir, moderate M&DAir, low M&D00.10.20.30.40.50.60.0150.020.0250.030.0350.040.0450.05h (m)Volumetric Flow Rate (m3/s)Calculated and Real hagainst Q (Air)Real hCalculated h
Table 3: Percentage Relative Error (ℎ−ℎ࠵?࠵?࠵?࠵?) × 100% (Using Equation [4]) Having ensured that the curvature is a near-exact fit to the original curve, substituting equation [4]into equation [3]yields the final empirical relationship: ℎ = −338.533314 × ࠵?࠵?× (࠵?࠵?࠵?2) × (࠵?࠵?࠵?࠵?࠵?)−2+ 0.44394606 × ࠵?࠵?× (࠵?࠵?࠵?࠵?࠵?)13+ 3.8023࠵? [࠵?] All raw data for all plots are presented in Appendices with appropriate titles. Error Analysis:All parameters in the virtual platform excluding the dependent variable hwere fixed values without possible error; thus, all the experimental error of the collected data can be attributed to misreading of h. However, in a real-life scenario, each of the variables would have associated errors. To emulate this scenario, a full-scale error analysis of the potential errors of all the ࠵?groups was conducted. The assumed uncertainties of each variable are presented in Table 4, and the derivations of the probable error are shown in Figure 9. Using these calculations, the final probable error for his presented in Table 5. In all derivations−338.533314 = ࠵?and 0.44394606 = β.Δℎ =(࠵?ℎ࠵?࠵?࠵?Δ࠵?࠵?)2+ (࠵?ℎ࠵?࠵?Δ࠵?)2+ (࠵?ℎ࠵?࠵?Δ࠵?)2+ (࠵?ℎ࠵?࠵?Δ࠵?)2+ (࠵?ℎ࠵?࠵?Δ࠵?)2+ (࠵?ℎ࠵?࠵?Δ࠵?)2࠵?ℎ࠵?࠵?࠵?= [3࠵? ×࠵?࠵?࠵?2࠵?2࠵?] + [࠵? ((࠵?࠵?࠵?࠵?࠵?)13࠵?࠵?࠵?3࠵?࠵?× (࠵?࠵?࠵?࠵?࠵?)43)]࠵?ℎ࠵?࠵?= ࠵? × (࠵?࠵?࠵?࠵?2) × (࠵?࠵?࠵?࠵?࠵?)2= ࠵? ×࠵?࠵?3࠵?2࠵?࠵?ℎ࠵?࠵?= [−࠵? ×࠵?࠵?࠵?3࠵?2࠵?] + [࠵? ×࠵?3࠵?× (࠵?࠵?࠵?࠵?࠵?)23]࠵?ℎ࠵?࠵?= −࠵? × (࠵?࠵?23࠵?࠵?) × (࠵?࠵?࠵?࠵?࠵?)43Real ࠵?Calculated ࠵?Percentage Relative Error0.067 0.0616722 7.95194% 0.21 0.2115047 0.716524% 0.296 0.3013493 1.807186% 0.356 0.3620269 1.692949% 0.401 0.4065956 1.395411% 0.438 0.4414015 0.776598% 0.47 0.4698554 0.030766% 0.498 0.4939373 0.815803% 0.524 0.5148701 1.742347%
Figure 10: hUncertainty Derivation Table 4: AssumedVariable Uncertainties Table 5: Uncertainty of hfor Real hin Figure 9 (Using Equations in Figure 10) ࠵?ℎ࠵?࠵?= [−2࠵? ×࠵?࠵?࠵?3࠵?3࠵?] + [࠵?࠵?3࠵?× (࠵?࠵?࠵?࠵?࠵?)23]࠵?ℎ࠵?࠵?= 3.8023Thus, assuming the relatively accurate measurements are taken to ensure that the assumed variable uncertainties are valid, the uncertainty of his reasonably low for all measurements. Discussion and Conclusions The Buckingham Pi theorem was integral to finding a functional relationship between hand other system parameters due to the relatively large number of variables considered. Reducing the variables from 7 to 4, of which one was the dependent, limited the amount of experiments necessary to develop the final function. Furthermore, it assists in making experiments more feasible, as changing parameters such as viscosity and density independently for different experiments presents significant logistical challenges. Instead, one can form dimensionless groups dependent upon variables that are reasonably easy to change so to expedite the experimental process required to create the empirical correlation of the variables in the system. This empirical correlation is applicable to any system so long as the other system is similar geometrically and dynamically to the original system, so that similitude is always maintained. Variable Uncertainty:࠵?࠵?±0.0005 m࠵?±0.0005 N࠵?±0.01 kg/m3࠵?±0.1 × 10−5Pa ∙ s࠵?±0.001 m3/s࠵?±0.001 m࠵?(m) Uncertainty (%) ࠵?(m) ࠵?(N) Q (m3/s) ࠵?(kg/m ) ࠵?࠵?(m) ࠵?(࠵?࠵? ∙ ࠵?)0.067 5.521% 0.02 0.0491 0.016 1.29 0.019 1.8× 10−50.21 3.137% 0.02 0.0491 0.02 1.29 0.019 1.8× 10−50.296 2.069% 0.02 0.0491 0.024 1.29 0.019 1.8× 10−50.356 1.557% 0.02 0.0491 0.028 1.29 0.019 1.8× 10−50.401 1.318% 0.02 0.0491 0.032 1.29 0.019 1.8× 10−50.438 1.218% 0.02 0.0491 0.036 1.29 0.019 1.8× 10−50.47 1.190% 0.02 0.0491 0.04 1.29 0.019 1.8× 10−50.498 1.195% 0.02 0.0491 0.044 1.29 0.019 1.8× 10−50.524 1.217% 0.02 0.0491 0.048 1.29 0.019 1.8× 10−5
The final empirical relationship derived (equation [5]

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