Hydrostatics and Manometry

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Dec 6, 2023

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Hydrostatics and Manometry TAM 335 Block 1 Full Report Rahul Jayachandran Pillai Section ABK, Thursday 1-3pm TA: Yuvam Kulkarni October 1, 2023

I. INTRODUCTION The objective of Lab 2, Hydrostatics and Manometry, is to measure the static pressure and static pressure differences within a fluid system using a differential manometer. Understanding static pressure is fundamental in Fluid Mechanics. Furthermore, correctly measuring and analyzing static pressure differences is crucial in any field involving fluids. This report also evaluates relative errors in measuring pressure differences by different methods. Hydrostatic pressure is defined as the pressure exerted by a fluid at equilibrium at any point of time due to the force of gravity [1] . Hence, it is the pressure measured when the fluid is at rest relative to the measurement. Static fluid pressure can be calculated with the following equation [2] : P s = ρgh = γh (1) Where ρ ( kg m 3 ¿ is the density of the fluid, g is the acceleration of gravity, and h is the depth of fluid. Additionally, γ is the specific weight of the fluid and is equivalent to ρg . This equation is fundamental in Fluid Mechanics and will be used through the report. Eqn. 1. holds true assuming that the liquid is incompressible ( ρ = constant). The differential manometer used in this experiment consists of three manometer fluids: water, which is clear; mercury, which is silver; and bromoform, which is colorless but is rendered visible by adding a purple or brownish-yellow dye. The set-up of the apparatus is shown in Fig. 1 in the appendix. Additionally, the value of the specific gravity of water, bromoform, and mercury is 1, 2.95, 13.55 respectively. Specific gravity is dimensionless as it is a ratio of the density of a substance to the density of water at 4 °C. II. EXPERIMENTAL METHODS

The pressures p A and p B at junction A and B, and the pressure difference p A – p B are calculated for ten different static configurations. p A and p B can be measured using Eqn. 1. Initially, the differential manometer set-up is prepared by opening the supply valve with the drain valve closed until a controlled maximum height of water (b 3 ) above junction B is obtained. With respect to the common datum line, the 0 mark of the meter stick, measure the fluid heights for a 1 , a 2 , a 3 , a 4 , a A , b B , b 1 , b 2 , b 3 , b 4 . Then, open and close the drain valve to obtain 10 incremental changes of the pressure difference p A – p B for the next setting. Repeat the previous steps for further settings and record the date. To ensure accurate and consistent data, use the level on the slider to keep it horizontal, and align the two parallel marker lines on the front and rear of the plexiglass slider with a consistent part of each liquid’s meniscus and with the meter stick. III.RESULTS AND DISCUSSION In this experiment, three different methods are used to determine the pressure difference between junction A and B. The differential manometer allows for three different methods to calculate p A – p B. First by measuring the heights of the water columns (Piezometric data), data from the heights of the mercury columns, and data from the heights of the bromoform columns. The Piezometric p A – p B can be calculated using the following equation: p A − p B = γ w ( a 3 − b 3 ) + γ w ( b B − a A ) (2) Where γ w is the specific weight of water which is 9810 N m 3 at 4 ° C. This equation is derived by calculating p A and p B using Eqn. 1 as seen below. p A = γ w ( a 3 − a A ) + p 0 p B = γ w ( b 3 − b B ) + p 0 p A − p B = γ w ( a 3 − a A ) − γ w ( b 3 − b B )

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The Bromoform p A – p B can be calculated using the following equation: p A − p B = γ w [ S Br ( a 1 − a 2 − b 1 + b 2 ) + ( a 2 − b 2 ) ] + γ w ( b B − a A ) (3) Where S Br is the specific gravity of Bromoform which is 2.95. This equation is derived by equating the pressures at a 2 to find p A and equating the pressures at b 2 to find p B as seen below. p a 2 = γ Br ( a 1 − a 2 ) + p 0 p a 2 = γ w ( a A − a 2 ) + p A p A = γ Br ( a 1 − a 2 ) + p 0 − γ w ( a A − a 2 ) p b 2 = γ Br ( b 1 − b 2 ) + p 0 p b 2 = γ w ( b B − b 2 ) + p B p B = γ Br ( b 1 − b 2 ) + p 0 − γ w ( b B − b 2 ) p A − p B = γ Br ( a 1 − a 2 − b 1 + b 2 ) + γ w ( a 2 − b 2 ) + γ w ( b B − a A ) The Mercury p A – p B can be calculated using the following equation: p a − p b = γ w ( S Hg − 1 )( b 4 − a 4 ) + γ w ( b B − a A ) (4) Where S Hg is the specific gravity of Bromoform which is 13.55. This equation is derived by equating the pressures at a 4 to find p A – p B as seen below. p a 4 = γ w ( a A − a 4 ) + p A p a 4 = γ Hg ( b 4 − a 4 ) + γ w ( b B − b 4 ) + p B p A − p B = γ Hg ( b 4 − a 4 ) + γ w ( b B − b 4 ) − γ w ( a A − a 4 ) p A − p B = γ Hg ( b 4 − a 4 ) + γ w ( a 4 − b 4 ) + γ w ( b B − a A ) p A − p B = γ Hg ( b 4 − a 4 ) − γ w ( b 4 − a 4 ) + γ w ( b B − a A ) γ ( ¿¿ Hg − γ w ) ( b 4 − a 4 ) + γ w ( b B − a A ) p A − p B = ¿ Table 1 shows the measured height data for settings 1-10. Using Eqn. 2, 3, and 4 the p A – p B - values were calculated for each of the methods. Table 2 shows the calculated pressure differences for the 3 methods Piezometric p A – p B , Bromoform p A – p B , and Mercury p A – p B . Figure 2 shows the values of p A – p B for the 3 different methods as a function of Piezometric p A – p B . Due to the relative precision of the measurements being limited to the least count of the meter stick, in this case 1mm, causes a type of random error that has a restricted size. [3] Additionally, the measurements are made more precise by aligning the flat edge slider to the lower level of the

meniscus and the rule height. Figure 2 provides a better understand of the recorded data and the precision of the three methods used. Bromoform displays a higher precision to the piezometric method as the slope of its trendline is equal to .9876. Mercury demonstrates a larger difference from the piezometric method as the slope of its linear trendline is equal to 1.0423. Mercury exhibits lower height values when there is a change in the water level because it is a heavy substance. Therefore, an error in the reading of mercury’s height can cause for a larger recorded pressure difference between A and B. There are also possibilities of random errors due to observational errors and measurement errors as the recorded needed to squat and reading the values at an uncomfortable position causing the slight fluctuation in points as seen in Figure 2. Systematic errors due to uncalibrated measuring instruments such as the 0 mark in the meter stick not being correctly aligned with the reference datum line could cause the recorded data to be shifted to one direction. The possible systematic percentage error of the piezometric pressure difference is calculated when a a is increased by 0.001 m for the first trial. The original pressure difference is equal to -412.02 Pa, but with the increase of two measurements, the new pressure difference is -421.83 Pa. This corresponds to a 2.38% increase. The equation below determines the calculation of the percentage error: % difference = | new − exact exact | × 100 (5) Similarly, the percentage difference of the pressure difference for the bromoform measurement is 2.16% and for the mercury measurement is 0.7%. As noticed, the error is greater for the piezometric, followed by the bromoform method, and then the mercury method. This could be due to mercury having a higher specific density compared to the other 2 fluids, making it more

resistant to a 0.001m change in the measurement. Additionally, an advantage of using mercury in manometers is that small pressure fluctuations causes the pressure reading to not move much and hence providing a more accurate reading. However, the random error caused by observational errors in measurement of the values a 3 and b 3 for the piezometric pressure difference by increasing a 3 by 0.001m causes a percentage difference of 2.38%. Similarly, for the bromoform pressure difference by increasing a 3 by 0.001m causes a percentage difference of 2.17% and for the mercury pressure difference by increasing a 4 by 0.001m causes a percentage difference of 17.62%. This is due to the measured height of mercury being small, a 0.001m difference caused a grater change in the output value. This is also represented by mercury’s higher variability or R value in Figure 2. Heavier liquids have lower heights on the manometer, so making a small incorrect readings largely impacts the calculated pressure difference because the specific gravity increases. IV. CONCLUSION AND RECOMMENDATIONS This lab focuses on determining the pressure differences by using 3 different methods (piezometric, bromoform, mercury) for 10 decremental height settings. Due to the small height range, mercury displays the largest deviation from the most accurate method (piezometric). The heavy fluid provides large room for error in finding accurate readings of the pressure difference between A and B. Due to the large range of the piezometric measurements, there was lesser variation and hence can be said to be the more accurate measurement of the 3. Additionally, to improve the accuracy of the lab, more number of readings and trials could be performed to minimize the random errors in the experiment.

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V. REFERENCES [1] Admin. “Hydrostatic Pressure - Definition, Formula, Derivation, Problems, Video and Faqs.” BYJUS , BYJU’S, 29 July 2022, byjus.com/physics/hydrostatic. Accessed 2 Oct. 2023. [2] “Static Fluid Pressure.” Pressure , hyperphysics.phy-astr.gsu.edu/hbase/pflu.html. Accessed 2 Oct. 2023. [3] Vernier Caliper: Least Count, Diagram, Uses of Vernier Calipers , collegedunia.com/exams/vernier-caliper-physics-articleid-876. Accessed 2 Oct. 2023. APPENDICES Table 1. Height Measurements for Each Setting Elevatio n a 1 a 2 a 3 a 4 a A b B b 4 b 3 b 2 b 1 (m) (m) (m) (m) (m) (m) (m) (m) (m) (m) Setting 1 0.554 0.064 1.518 0.105 0.455 0.300 0.106 1.405 0.023 0.490 2 0.560 0.064 1.518 0.093 0.455 0.300 0.111 1.279 0.051 0.470 3 0.559 0.064 1.518 0.088 0.455 0.300 0.111 1.196 0.064 0.442 4 0.553 0.066 1.518 0.088 0.455 0.300 0.116 1.100 0.081 0.427 5 0.552 0.064 1.518 0.085 0.455 0.300 0.123 1.031 0.104 0.420 6 0.557 0.064 1.518 0.078 0.455 0.300 0.126 0.902 0.125 0.384 7 0.558 0.069 1.518 0.074 0.455 0.300 0.131 0.792 0.146 0.368 8 0.555 0.066 1.518 0.072 0.455 0.300 0.133 0.703 0.165 0.345 9 0.550 0.065 1.518 0.065 0.455 0.300 0.137 0.593 0.186 0.321 10 0.554 0.064 1.518 0.063 0.455 0.300 0.142 0.500 0.206 0.305

Figure 1. Manometer System Table 2. Calculated Pressure Differences for 3 Different Methods P A P B P A -P B P A P B P A -P B P A -P B PIEZOMETER PIEZOMETER PIEZOMETER LEFT BROMO RT. BROMO BROMO HG (Pa) (Pa) (Pa) (Pa) (Pa) (Pa) (Pa) Settin g 1 10428.03 10840.05 -412.02 10344.6 5 10797.3 8 -452.73 - 1397.43 2 10428.03 9603.99 824.04 10518.2 8 9682.96 835.32 695.53 3 10428.03 8789.76 1638.27 10489.3 4 8623.97 1865.37 1311.11 4 10428.03 7848.00 2580.03 10277.4 5 7864.68 2412.77 1926.68

-1000 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 -2000 0 2000 4000 6000 8000 10000 f(x) = 1.04 x − 510.22 R² = 0.99 f(x) = 0.99 x + 9.94 R² = 1 Piezomet- ric Bromo- form Linear (Bromo- form) Pa-Pb (water) Pa-Pb Figure 2. Piezometric, Bromoform, Mercury Pressure Difference (p a -p b ) vs Piezometric p a -p b

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