Fluid Mechanics Lab Report

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Curtin University *

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Oct 30, 2023

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1. Introduction The following laboratory looks to develop one’s understanding with the study of flow in pipes by applying the Fluid Mechanics principles and theories learnt from the unit ENGR2000. Flow in pipes can vary from laminar, transitional and turbulent. This experiment looks to examine the differences between three flow regimes, and in order to do so, differentiate the variations between these types of flows. As well as this, the following experiment will look at the Hydraulic Gradient in the pipe. From this, the results will be used to evaluate Reynolds Number ( R e ¿ and the relative roughness of the pipe wall ( ϵ / D ¿ . This in order to determine the different flow conditions using a Moody Chart by looking at the pipe’s Friction Factor ( f ¿ . 2. Background Reynolds Number ( R e ) is a dimensionless number which is used to indicate the state of a fluid flow as it passes a body. By examining this number, a fluid can be determined to be either laminar, transitional or turbulent. R e = ρ ∗ U m ∗ L μ R e = Reynolds Number ( non- dimensional ) ρ = Density of fluid ( kg / m 3 ) U m = Average velocity of flow ( m / s ) L = Length scale of flow perpendicular to surface ( m ) μ = Viscosity of fluid ( N s / m 2 ) From the following equation, we can see Reynolds Number ( R e ) is the ratio of inertial forces to viscous forces. The distinctive flow regimes in a pipe can be seen in Figure 1, and are as follows: Figure 1: Flow Differences Between Laminar, Transitional and Turbulent. 1 Laminar: R e < 2100 Transitional: 2100 ¿ R e < 4000 Turbulent: R e > 4000

The relative roughness of the pipe wall is its roughness ( ϵ ¿ divided by the inner diameter ( D ). This can then be used on a Moody Chart, to find the corresponding predicted values of Friction Factor ( f ¿ . The Moody chart relates Reynolds Number, relative roughness and the Darcy-Weisbach equation all collectively. The Darcy-Weisbach equation is as follows: h L = f ∗ L ∗ V 2 2 ∗ g ∗ D h L = ¿ Friction head loss ( m ¿ f = ¿ Darcy resistance factor ( non- dimensional ) L = Length of the pipe ( m ) D = Pipe Diameter ( m ) V = Mean velocity ( m / s ) g = Acceleration due to gravity ( m / s 2 ¿ Hydraulic Gradient due to friction ( i ) is the ratio of head loss to the length. The equation is as follows: i = h L L Darcy’s Friction Factor ( f ¿ is used to calculate the friction factors for the flow regimes. It is dependent on Reynolds Number R ¿ ¿ ¿ ) and to the relative roughness ( ϵ / D ) . Whereby: Laminar Flow: f = 64 R e Turbulent Flow: f = 0.316 R e 1 4 As mentioned, these formulae are all related to the Moody Chart, which is a graph in a non- dimensional form. It is conventionally used to find the estimated friction factor for the flow regimes. There are several factors that can affect the fluid flow in a pipe, and these needed to be taken into consideration whilst the experiment was conducted. They are as follows:  The viscosity, velocity and density of the fluid  The inner diameter and length on a pipe  The temperature of the fluid 2

 The inner roughness of the pipe  Entrance and the exit conditions of the pipe (Lekner 2007) 3. Objective The main objectives of the laboratory are to identify the specific values of Reynolds Number ( R e ), in relation to the laminar, transition and turbulent regimes of flow in a pipe with a circular cross-section, to understand the concept of friction factor whilst it is applied in pipe flow, and to understand the principle of the Darcy-Weisbach equation. By the end of the experiment, one should sufficiently understand the theory relative to flow through pipes. 4. Apparatus I. Small horizontal pipe with a nominal diameter 3 mm II. Water tank III. Flexible hose IV. Bell-mouthed entrance leading to a straight pipe V. Piezometer tapping’s, upstream approximately 45 pipe diameters away from pipe exit VI. Piezometer tapping’s connected to an inverted U-tube manometer VII. Water Manometer VIII. Mercury Manometer IX. Stop watch X. Water XI. Plunger 3

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Figure 2: Mercury Manometer Figure 3: Water Manometer 5. Experimental Procedure I. The experimental procedure was read to understand the objectives of the laboratory. II. Discussion between group members so everyone understood the experiment. III. Apparatus level was checked. This is so the manometers stood vertically. IV. Visual inspection to see the connection of the manometer to the pipe being tested. V. Pressure lowered to zero to eliminate the air bubbles in the water. VI. Reads were taken in two sets, one using a water manometer and other using mercury manometer. VII. Water manometer (readings 1-12): The bench supply valve was opened, until a steady flow is obtained at the supple tank overflow pipe. VIII. From this, the isolation tap is turned to read the values on the water manometer. This was done by looking at the needle valve. IX. Readings of the water manometer were noted ( h 1 and h 2 ) as well as time ( s ). X. After this, the flow’s rates were regulated quarterly through the pipe for the 12 readings. XI. All results were noted XII. Mercury manometer (readings 13-30): higher flow rate needed, so the supply pipe is from the bench supply, not the elevated tank. 4

XIII. Water manometer was isolated to read the mercury manometer. This was done by the isolating tap. XIV. Needle value was turned off the stop the flow, and to check that the mercury levels in the two sections of the manometer were level. XV. The needle value was used to regulate the flow rate in the pipe, similar to the water manometer readings. Turned quarterly. XVI. Small differences in the flow were observed and noted as this corresponded to the transitional flow. XVII. Readings of the mercury manometer were noted ( h 1 and h 2 ) as well as time ( s ) XVIII. After all the values for readings (13-30) were obtained, flow rates were calculated. XIX. All results were summarised in the instruction sheet. 6. Results Diameter of Pipe D = 3 mm Length of Pipe between Piezometer Tapping’s L = 524 mm Roughness height of Pipe-Wall Material ϵ = 0.0015 mm Density of Water ρ H 2 O = 1,000 kg / m 3 Dynamic Viscosity of Water μ = 1.002 ∗ 10 − 3 N . s / m 2 Density of Mercury ρ Hg = ¿ 13,500 kg / m 3 Table 1: Properties of Pipe, Water and Mercury Test No. Water Quantity W (ml) Time Taken ∆t (s) Flow Rate Q ( m 3 /s) Manometer Reading h 1 (mm) Manometer Reading h 2 (mm) 1 20 11.37 1.75901 * 10 − 6 331 310 2 20 10.91 1.83318 * 10 − 6 344 292 3 20 6.53 3.06279 * 10 − 6 356 275 4 50 11.63 4.29923 * 10 − 6 368 261 5 50 10.18 4.91159 * 10 − 6 380 243 6 50 8.56 5.84112 * 10 − 6 395 224 7 50 7.66 6.52742 * 406 209 5

10 − 6 8 50 7.44 6.72043 * 10 − 6 429 181 9 50 6.85 7.29927 * 10 − 6 466 136 10 100 13.38 7.47384 * 10 − 6 500 95 11 100 12.41 8.05802 * 10 − 6 516 76 12 100 11.50 8.69565 * 10 − 6 528 59 Table 2: Water Manometer Tests Test No. Water Quantity W (ml) Time Taken ∆t (s) Flow Rate Q ( m 3 /s) Manometer Reading h 1 (mm) Manometer Reading h 2 (mm) 13 50 12.44 4.01929 * 10 − 6 142 135 14 50 7.94 6.29723 * 10 − 6 144 132 15 50 6.75 7.40741 * 10 − 6 147 130 16 50 6.91 8.07754 * 10 − 6 148 128 17 50 5.29 9.46972 * 10 − 6 152 125 18 50 4.75 1.05263 * 10 − 5 155 122 19 50 4.56 1.09649 * 10 − 5 157 120 20 100 7.87 1.27065 * 10 − 5 161 117 21 100 7.16 1.39665 * 10 − 5 165 113 22 100 6.87 1.45562 * 10 − 5 168 110 23 100 6.40 1.56253 * 10 − 5 171 108 24 150 8.91 1.68355 * 10 − 5 174 107 25 150 8.78 1.70843 * 10 − 5 178 104 26 150 8.60 1.74419 * 10 − 5 179 102 27 150 8.03 1.86821 * 10 − 5 180 100 28 150 7.84 1.91327 * 10 − 5 183 98 29 150 7.78 1.92802 * 10 − 5 187 95 30 200 9.88 2.02429 * 10 − 5 190 92 Table 3: Mercury Manometer Tests Test No. Mean Velocity V ( m / s ) Hydraulic Gradient − ¿ i ¿ Reynolds Number − ¿ ℜ ¿ Flow Regime Friction Factor − ¿ f ¿ 1 0.249 0.0401 745 Laminar 0.0859 2 0.259 0.0992 777 Laminar 0.0824 3 0.433 0.1546 1297 Laminar 0.0493 4 0.608 0.2042 1821 Laminar 0.0351 6

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5 0.695 0.2615 2080 Laminar 0.0308 6 0.826 0.3263 2474 Transitional 0.0259 7 0.923 0.3759 2765 Transitional 0.0231 8 0.951 0.4542 2847 Transitional 0.0225 9 1.033 0.6298 3092 Transitional 0.0207 10 1.057 0.7729 3166 Transitional 0.0202 11 1.139 0.8397 3413 Transitional 0.0188 12 1.231 0.8951 3683 Transitional 0.0174 Table 4: Calculations and Results (Water Manometer Tests) Test No. Mean Velocity V ( m / s ) Hydraulic Gradient − ¿ i ¿ Reynolds Number − ¿ ℜ ¿ Flow Regime Friction Factor − ¿ f ¿ 13 0.569 0.1669 1702 Laminar 0.0376 14 0.891 0.2863 2667 Transitional 0.0239 15 1.048 0.4055 3138 Transitional 0.0204 16 1.143 0.4771 3421 Transitional 0.0187 17 1.339 0.6441 4011 Turbulent 0.0397 18 1.489 0.7872 4459 Turbulent 0.0387 19 1.551 0.8826 4644 Turbulent 0.0383 20 1.798 1.0496 5382 Turbulent 0.0369 21 1.976 1.2405 5916 Turbulent 0.0361 22 2.059 1.3836 6165 Turbulent 0.0357 23 2.211 1.5029 6618 Turbulent 0.0351 24 2.382 1.5983 7131 Turbulent 0.0344 25 2.417 1.7653 7236 Turbulent 0.0343 26 2.468 1.8368 7388 Turbulent 0.0341 27 2.643 1.9084 7912 Turbulent 0.0335 28 2.707 2.0277 8104 Turbulent 0.0333 29 2.728 2.1947 8166 Turbulent 0.0332 30 2.864 2.3378 8574 Turbulent 0.0328 Table 5: Calculations and Results (Mercury Manometer Tests) Figure 4: Hydraulic Gradient against Mean Velocity 7

Figure 5: Experimental Friction Factor plotted with Reynolds Number (Moody Chart) 7. Analysis The theoretical Friction Factors were obtained by plotting the Reynold Number ( ℜ ) values for laminar and transitional flow. Whereas relative roughness of pipe wall ( ϵ / D ) was only used for turbulent flow on the Moody Chart. This can be seen in Appendix A. Test No. Reynolds Number − ¿ ℜ ¿ Friction Factor f E (Experimental) Friction Factor f T (Theoretical) 1 745 0.0859 0.085 2 777 0.0824 0.082 3 1297 0.0493 0.048 4 1821 0.0351 0.035 5 2080 0.0308 0.030 6 2474 0.0259 0.026 7 2765 0.0231 0.0235 8 2847 0.0225 0.022 8

9 3092 0.0207 0.020 10 3166 0.0202 0.019 11 3413 0.0188 0.018 12 3683 0.0174 0.017 13 1702 0.0376 0.037 14 2667 0.0239 0.024 15 3138 0.0204 0.019 16 3421 0.0187 0.0185 17 4011 0.0397 0.039 18 4459 0.0387 0.0385 19 4644 0.0383 0.038 20 5382 0.0369 0.037 21 5916 0.0361 0.036 22 6165 0.0357 0.0355 23 6618 0.0351 0.035 24 7131 0.0344 0.034 25 7236 0.0343 0.034 26 7388 0.0341 0.0335 27 7912 0.0335 0.033 28 8104 0.0333 0.033 29 8166 0.0332 0.0325 30 8574 0.0328 0.0325 Table 6: Comparison of the Experimental and Theoretical Friction Factor 8. Discussion Reviewing the results that were obtained, it is distinctly shown that there is a relationship between the Friction Factor ( f ) and the Hydraulic Gradient ( i ). As well as this, there is also a relationship between Reynolds Number ( R e ) and the Flow Rate ( Q ). The readings 1-12 from the Water Manometer show a laminar and transitional flow regime. However, when plotting the Hydraulic Gradient ( i ) and the mean velocity of the fluid ( V ), the relationship is not linear. The readings 13-30 from the Mercury Manometer show all three flow regimes, being laminar, transitional and turbulent. The experimental values had similar discrepancies to readings 1-12 when compared to the theoretical values obtained from the Moody Chart. This would be due to human error, or systematic errors when rounding values before plotting. At low velocity, turbulent flow will turn into transitional and finally laminar flow. However, if there is an increase to the velocity, it will turn back to transitional and ultimately turbulent flow. During the testing, when using small volumes of water, such as 50 ml , it was found to be difficult to accurately time the flow rate. In order to accurately test any readings, larger volumes of water are needed. This allows more time to precisely measure the time. By doing this, it may reduce the errors within the experiment. As can be seen on the Moody Chart Figure 6, turbulent flow is disordered and unpredictable. Comparing to linear flow, which is consistent with a predictable gradient. 9

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Inconsistencies with the experimental and theoretical friction factors can be due to the amount of friction loss that occurred in the pipe during the experiment. As well as this, the dynamic viscosities of Water and Mercury would also affect Reynolds Number ( R e ). It must be noted that temperature and apparatuses used would also alter the experimental results from the theoretical results. From observation, it can be suggested that Reynolds Number ( R e ) and Friction Factor ( f ) are inversely proportional. This observation is consistent with the theory, as a large friction factor will generate high a viscous effect. The experiment could be improved from increasing the number of tests on each reading. This would improve the accuracy in results, leading to a more accurate laboratory. Average times could be also calculated during the water to fill up the cylinder to increase the level of accuracy, as it was only recorded once per reading. Overall, repeating the experiment multiple times and comparing results each time would lead to a precise experiment. 9. Conclusion Overall, the results that were acquired during the experiment are similar to what was theoretically expected. This led to a successful laboratory, with all of the objectives being achieved. By conducting this experiment, it allows one to understand the concept of fluid flow through pipes and how it relates to Fluid Mechanics. The theory learnt throughout the semester from ENGR2000 assisted the finalisation of the report, as it gave a good foundation of knowledge in order to effectively complete this laboratory. As mentioned in the discussion, the experiment contains errors, with the main error being from human analysis. However, the results level of accuracy is relatively high, therefore the minimal errors suggested are deemed negligible. To improve the laboratory the next time it is conducted, it is recommended that the number of tests for each reading is increased, with an overall average calculated before summarising results. 10

10. References Lekner, John. 2007. “Viscous Flow Through Pipes of Various Cross-Sections”. European Journal of Physics 28 (#): 521-527. doi:10.1088/0143-0807/28/3/014 11. Appendix Appendix A - Predicted values of friction factors 11

Appendix B - Formula’s used Hydraulic gradient for the water manometer: i water = ( h 1 − h 2 ) L ( non- dimensional ) Hydraulic gradient for the mercury manometer: i mercury = ( h 1 − h 2 ) ∗( 13.5 − 1 ) L ( non- dimensional ) Hydraulic gradient due to friction: i = h L L = f ∗ V 2 2 ∗ g ∗ D ( non-dimensional ) 12

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Cross-section area of pipe: A = π ∗( 0.003 ) 2 4 = 7.06858 * 10 − 6 m 2 Relative roughness of pipe wall: ϵ / D = 0.0015 3 = 5 ¿ 10 − 4 Flow Rate Q ( m 3 / s ): = Volume ( V ) Time ( s ) Mean Velocity V = Flow Rate ( Q ) Area ( A ) Friction Factor for Laminar Flow: f = 64 R e Friction Factor for Turbulent Flow: f = 0.316 R e 1 4 13