Abstract
The experiment was done to investigate the relative roughness of pipe three though a plot of the friction factor verses the Reynolds number. A comparison of the obtain results was made against a Moody diagram. Also, additional comparison of the experimental results obtained from pipe 1 and 2 were made against theoretical values obtained through Prandtl and Blasius equations. This process entailed plotting the theoretical and the experimental values on the same comparative roughness against Reynolds number chart. Additional results of different types of flow, exhibited by the flowing fluid, was done and represented in a graph of head loss versus velocity.
Pipe 3
Experimental readings in bold
Pipe 2
Pipe 1
Formulae and sample calculations used for pipes 1, 2, and 3
Static pressure head hF
hF=Δp⍴g (Haaland 90).
Flow rate Q
Q=ΔVΔt (Haaland 90).
Mean flow velocity V
V=QA (Haaland 90).
Reynolds number Re
Re=⍴VDµ (Hooper, 89).
Darcy friction factor λ
λ=2gDhF(LV)2 (Hooper, 89).
Pipe 3 analysis
Figure 1: Moody Diagram
It is imperative to note that relative pipe roughness is a ration of mean internal height roughness of the pipe and its diameter. Relative roughness is obtained by the following equation: Relative roughness=kD (Bagarello, V., et al. 316). This ration can be obtained through the use of a Moody chart. A comparative analysis is done on the experimental results using the Moody chart as a basis for comparison (Zielke 115). The relative roughness of the pipe was estimated to be between 0.05 and 0.06 according to the Moody chart.
Pipe 1 & 2 analysis
Blasius : λ=0.316Re0.25
Blasius equation was used to obtain the values represented in the above table (Von 298). The values were later used to display a Blasius curve on Figure 2. Blasius curve present a theoretical graph of the theoretical values which were used to investigate the experimental results (Vardy and Brown 1036).According to the comparative analysis performed, it is apparent that the experimental results fall within close proximity to the theoretical curve (Vardy and Kuo 548). The small discrepancy between the two outcomes is due to random error given that a systematic error was already accounted for at the start of the experiment.
Prandtl
1λ=2log10(Reλ2.51)
Reynolds number Re was obtained by rearranging the Prandtl equation as follows;
Re=10[12λ+log102.51-log10λ]
Given a set of Re values, the λ values were obtained through the use of goal seek feature on Microsoft excel. The values were tabulated as follows;
The values outlined in the above table were used to generate a Prandtl curve displayed in Figure 2. Similarly, this curve was used to interpret the experimental results. It is apparent that the results obtained from the experiment fall within a close proximity to the theoretical curve (Vardy, Alan and Jim 456). Since systematic error was already accounted for at the start of the experiment, the discrepancy that exists between the experimental results and the theoretical values is as a result of random error (Romeo, Carlo and Antonio 374).
Log10 Graph analysis
The graph in Figure 3 indicates how the head loss, hF varies with the increase in velocity, V. According to the curve, it is apparent that the head loss and velocity exhibit a relatively linear relationship (Ito 140). According to Houghtalen and Ned (16), as the head loss increases there is an increase in velocity of the flow. However, it is imperative to take note of the middle section where there is a slight deviation from the relative linear relationship. There are three distinct regions in the curve. The first region (blue) indicates a linear relationship that exists between the head loss and the mean flow velocity (hF α V). Hooper (89), asserts that a relatively linear relationship exists between the head loss and the mean flow velocity for a laminar flow. The gradient of the curve at this region is determined to be 0.9335, implying that the flow in this section exhibits laminar attributes. According to Haaland (90), a fluid with laminar flow travels in a constant path whereby the pressure, velocity and other properties of the fluid remain constant at all points of the flow.
On the other hand, the fluid exhibits a different type of flow after the disturbance as demonstrated by the green part of the curve. Duan et al. (362), suggest that the head loss value assumes a proportional relationship to the square of the mean flow velocity; hF α V2. The gradient at this region was determined to be 1.8049. The gradient suggests that the region after the disturbance assumes the characteristics of a turbulent flow (Bernuth and Tonya 192). A fluid in turbulent flow flows haphazardly, which is damaging to the volume flow rate give that it generates huge amounts of resistance (Brunone et al. 244). The tremendous resistance generated by this type of flow results into an increase in pressure to balance the flow.
The red part of the curve was determined to be the transition flow. The transition flow is a region between the laminar flow and turbulent flow. In most cases, it is an incorporation of the two types of flows (Bergant, Angus and John 257). It is imperative to point out that the flow at the edge of the pie assumes a laminar flow while the fluid at the mid-section of the pipe assumes a turbulent flow. The data used to plot the graph in Figure 3 can be found in Appendix A
Experimental limitations
Systematic error
The difference in pressure was outlined as one of the systematic errors in this experiment. It was accounted for at the initial stage of the experiment by recording zero error presented by the instrument. The recorded value was later subtracted from the results obtained.
Random error
Fluctuation in the readings and human error constituted random errors experienced in this experiment. Human error exhibited in the experiment was due to the reaction speed of the person in charge of controlling the stop watch. Its effects on the flow rate and Reynolds number are undeniable. A fluctuation in the reading was also outlined; the oscillation in the readings will interpret into a lack of precision in the results.
Discussion
Pipe 3
The value of relative pipe roughness obtained for pipe 3 is concurrent with the statement made in the experiment briefing. It support the original hypothesis made about the relative pipe roughness of pipe 3. Pipe three was originally described as a rough turbulent flow pipe. The experimental results obtained support this claim (relative pipe roughness between 0.05 and 0.06).
Pipe 2 and 1
Similarly, the experimental data obtained for pipe 1 and pipe 2 are concurrent with the initial hypothesis made at the beginning of the experiment. The pipes were described as smooth and the experimental results obtained support that claim. The experimental results of pipe 1 and pipe 2 were plotted on the same curve as Blasius and Prandtl curves. This process enabled comparison between experimental data and theoretical values. It is imperative to point out that the outcome fall extremely close to the two theoretical curves. Apart from the experimental errors, the deficiencies in the Prandtl and Blasius equations explain the small discrepancy between the theoretical values and the obtained readings.
Graph of head loss against mean flow velocity (Logs graph)
The graph generated indicated a transition from laminar to turbulent flow with the mid-section showing transition flow characteristics. The velocity, V value for the laminar flow region was 0.9335, while the V value for the turbulent flow region was 1.8049. These two values refer to the gradient of the lie in those regions.
Appendix A
Log graph data
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