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# Relationship Between Friction Factor and Reynolds Number

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The nature of fluid flow through pipes is an important industrial process as well as any engineering process that may require piping. In fluid mechanics, the flow of fluids is classified as external or internal however this report will be concerned with the latter to meet the objective of the study. For a liquid to flow through a pipe, there must exist a pressure drop which is mainly affected by friction as well as kinetic energy loss. In any flow system, two primary type of friction exists, and they include form friction and skin friction. The latter is generated when the piping system results in two unseparated boundary layer, and this occasionally occurs in straight pipes. This can be explained when the transfer of momentum during the flow of the fluid brings about tangential drag or stress on the surface of the pipe that is parallel l to the direction of the fluid flow. Form friction on the other hand results when wakes are formed due to the separation of boundary layers, in other words, it is energy dissipation. Such can result when a fluid flows through a fitting, valve or obstructions which leads to enlargement and contraction of the pipes cross-sectional area (Haaland, 89-90).

During its flow, it has been known that fluids lose energy to heat resulting from the friction drag and this is referred to as friction loss thus a contributing variable when it comes to the discussion of the friction factor. The primary purpose for this experiment, therefore, is to find the relationship that exists between the friction factor itself and the Reynolds number using two pipes of different diameters. The variation of these two attributes will be explored over the turbulent flow as well as the laminar transition then the rests of the two compared to meet the objective of the study. This report will be divided into different sections but will start by providing the necessary theory about the experiment, then discuss the procedure used in the collection of data, the results will then be presented with their summary, and eventually the same will be discussed before a conclusion is provided. Both attributes that are Reynold number and friction factor can be calculated using the equation below respectively (Haaland, 89-90):

Re

= Dʋρ / η       f = │∆P│/ 2 ρʋ2 * D / L

Theory

Friction factor cannot be explained in details without mentioning the relationship between the friction in the pipe as well as the head loss or pressure loss. The relationship between the two can be described using the Darcy equation (h푓= 푓퐿푣2/2푔퐷) and from the equation the f,,riction factor is normally plotted as a function of NR of any piping system in a log-log scale. In a case where the value of NR is less than 2100 then it would mean that the flow is laminar (Chen, 296). This type of relationship can be illustrated or described using h푓= 32푣퐿푣/푔퐷2

(Poiseuille’s equation) which results in a new equation f = 64/NR.

In a case where NR is above 2100 then the type of flow illustrated by the latter is turbulent flow. The head loss however for this type of flow is accurately proportional to 푣n

where n represent a range of values from 1.75 to 2.0. In the previous studies therefore it is important to note that for laminar friction factor is proportional to the inverse of the Reynold number resulting in a linear relationship between the two ((Badeer and Costas, 601)). For turbulent however the relationship cannot be classified to be negatively linear has it has been reported by previous authors since it has been suggested that it can be determined only experimentally. In a nutshell therefore this experiment will determine whether the relationship reported in the laminar exists and whether for turbulent it can be determined experimentally as suggested by previous studies.

Methodology

In this experiment, various apparatus was used and set up as illustrated by diagram 1 below. The diagram comprises a reservoir where water is stored and will be released to flow in the piping system with different cross-sectional. Other apparatus include a gear pump, temperature sensor (Pt 100 ohm), differential pressure transmitter, data acquisition system as well as a flow meter. The length of the pipes was 1.635m while the cross sections ranged from ¼ inches to 3/8 inches. The two fluids used in this experiment were water and Ethylene Glycol which were released into the system simultaneously to measure the two attributes to come up with conclusive results.

Diagram 1: Apparatus Set-up

Results

This section comprises all the data recorded from the experiment for both fluids to determine the relationship between friction factor and Reynold’s number for analysis purpose. Table one and two as illustrated below indicate the reading collected from the experiment for water under two different conditions.

Table 1: Turbulent Flow of Water

Fluid: H2O, Turbulent flow, O.D.: 3/8 in

Data #

1

2

3

4

5

flow rate [mL/min]

1488.9

1486.1

1484.3

1476.6

1487

differential pressure [pa]

129.8

128.2

128.6

128.3

128.6

Data #

1

2

3

4

5

flow rate [mL/min]

2040.8

2041

2019.3

2044.5

2031.9

differential pressure [pa]

211.5

210.9

211

211.6

211.4

Data #

1

2

3

4

5

flow rate [mL/min]

2492.6

2495.1

2500.6

2506.2

2480.4

differential pressure [pa]

282.4

280.1

282.5

284

284

Data #

1

2

3

4

5

flow rate [mL/min]

2989.4

2986.9

3005.5

2970.3

2998.1

differential pressure [pa]

395.3

391.6

394.6

392.6

392.8

Table 2: Laminar Flow of Water

Fluid: H2O, Laminar flow, O.D.: 1/4 in

Data #

1

2

3

4

5

flow rate [mL/min]

50.9

61.5

62.9

59.4

62.2

differential pressure [pa]

16.2

17.1

18.7

18.9

18.5

Data #

1

2

3

4

5

flow rate [mL/min]

186.5

186.3

186.6

179.5

186.3

differential pressure [pa]

53.5

50.8

51.7

49.9

52.5

Data #

1

2

3

4

5

flow rate [mL/min]

290

289.2

290.8

289.3

288.2

differential pressure [pa]

85.5

84.5

85.5

80.55

89

Data #

1

2

3

4

5

flow rate [mL/min]

3930

395.4

391.5

394.2

392.4

differential pressure [pa]

165

164.5

157

156.5

163

From the above results, the following three figures were obtained to illustrate the association between friction factor and Reynolds number and whose interpretation will be discussed in the result analysis section.

Figure 1: Friction Factor vs. Reynold’s Number for Water (Turbulent Flow)

From the above figure, the relationship between the two variables is linear and thus proportional to one another throughout the experiment.

Figure 2: Friction Factor vs. Reynold’s Number for Water (Laminar Flow)

The same results from figure 1 are observed for figure 2 however after sometimes the relationship shifts or deflects to the right.

Friction Factor vs. Reynold’s Number for Ethylene Glycol (Laminar Flow)

The report on figure 2 is also observed in the figure above for the case of Ethylene Glycol.

Discussion

The results represented project a different picture of both laminar and turbulent flow but what can be noted from table 1 and 2 is that the flow rate in the turbulent experiment is higher than those recorded for the laminar flow. This is because the fluid faces more resistant or friction between its layers and that of the pipe resulting in a higher pressure drop. However, for the same, it is observed that the relationship between the Reynold number and friction factor remains proportional throughout the experiment. The association between the two is positively linear, and no bend can be seen on the proportionality line as it is the case with the other two. Despite resembling each other, it is observed that the Reynold number, as well as friction factor, is higher in water than Ethylene Glycol and this could be attributed to the difference in density which is determined by the mass of the fluid and its volume. In other words, water is heavier than it. The two figures, however, resemble each other because they both have laminar flow, so they are subjected to the same conditions. The relationship between the two variables is positive however it can be seen that it is not perfect in linearity as in figure 1.

Unlike it is in figure one at some point the line deflects upwards and towards the Reynold, number sides to signify a reduction in friction between the fluid and the surface of the pipe. This occurs because fluids have a lubricating effect which reduces friction over time thus the sudden change however the relationship remains linear. Therefore, in a nutshell, we can conclude that the association that exists between the two variables is positively linear in all flow conditions especially turbulent and laminar as well as similar for all the fluids exposed to the same conditions.

Conclusion

The experiment was concluded successfully with the relationship between Reynold number and friction factor being linear and positive. The experiment was however not free of errors, and the possible sources of error were time measurement in regards to flowrate as well as the volumetric analysis of the two fluids. The other potential source of error was from the reading of temperatures.

Work Cited

Badeer, Henry S., and Costas E. Synolakis. “The Bernoulli‐Poiseuille equation.” The Physics Teacher 27.8 (1989): 598-601.

Chen, Ning Hsing. “An explicit equation for friction factor in pipe.” Industrial & Engineering Chemistry Fundamentals 18.3 (1979): 296-297.

Haaland, Skjalg E. “Simple and explicit formulas for the friction factor in turbulent pipe flow.” Journal of Fluids Engineering 105.1 (1983): 89-90.

October 05, 2023
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