Lab2_SachinDasu_A1

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Laboratory 2: Oil Flow in a Pipe CEE 4200: Hydraulic Engineering Section A1 Written By: Sachin Dasu Experiment Date: February 6, 2024 Submitted On: February 20, 2024

2 Table of Contents Extended Abstract……………………………………………………….3 Experiment………………………………………………………...3 Results………………………………………………….………….4 Discussion…………………………………………………………4 Conclusion………………………………………………………...8 Appendix………………………………………………………………...9

3 Extended Abstract Experiment In the Oil Flow in a Pipe laboratory, the experiment performed observes and quantifies the frictional loss for oil moving through a brass pipe. Fluid flow in a pipe is important in many engineering applications including domestic water supply, sewage removal, blood flow in arteries, and industrial oil pipelines. Fluid flow through a straight pipe in these applications loses energy because of friction. The frictional loss depends on several parameters including the fluid viscosity and flow rate. Dimensional analysis reveals that for a smooth pipe wall, the Reynolds number and pipe geometry characterize the frictional losses. The objectives of this laboratory experiment are to measure the streamwise pressure distribution for several Reynolds numbers, use the pressure drop in the fully developed region to calculate the resistance coefficient, and to compare the resistance coefficient with known theoretical and empirical relationships. Fluid flow through a pipe loses energy due to shear in both the laminar and turbulent regimes. The conservation of energy equation can be used, and the shaft work is0 and the velocity are elevation are the same at the inlet and outlet. Thus, the head loss in the pipe flow can be observed as a pressure drop in the flow direction. In the laminar case, the shear stress is equal to the velocity gradient times the molecular viscosity, thus, relating the velocity and the pressure drop. For turbulent flow, the shear stress is due to both molecular viscosity and the fluctuating fluid motion. For both the laminar and turbulent regimes, the resistance coefficient (or Darcy friction factor), is used to relate the pressure drop and velocity. Thus, If the resistance coefficient is known for a flow, the pressure drop and velocity are related. In the laboratory experiment, oil is pumped from the reservoir to the test pipe. After passing through a smooth contraction, the flow enters the 264 ¾ - in long brass pipe. The flow rate is controlled with a gate-valve, the flow rate in the test section increases. At the end of the pipe, oil flows into a weigh tank. The weigh tank will be used to measure the length of time required for a known weight to pass through the system. The specific weight of the fluid is then used to calculate the volumetric flow rate. The weigh tank drains into the reservoir. Nine static pressure taps are located at the following locations measured from the start of the pipe. Rubber tubing connects each tap to an individual manometer attached to the wall. A surveying stick mounted on the manometer board allows for elevation measurements for each meniscus. The static pressure is the distance from the datum location multiplied by the specific weight of oil. Pressure tap #9 is very close to the end of the pipe, and it is assumed that the pressure is 0 gage at that location, thus h datum is the elevation of the oil in manometer tube #9. The specific gravity of the red oil is 0.87 and the viscosity is a function of temperature. The oil temperature is measured with the thermometer mounted on the reservoir. The pump motor is started, and a flow rate is set by adjusting the gate valve. The weighing tank is used to measure the volumetric flow rate. With the tank drain valve open, the scale is balanced to set a datum. The drain valve is closed on the weighing tank and a known weight is added to the scale. A stopwatch is used to measure the time required to balance the additional weight. From the weight difference and measured time, the weight flow rate can be calculated as well as the volumetric flow rate. Once the manometer board has reached equilibrium,

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4 the elevation of the oil in each tube is noted by aligning the sliding reference plate with the meniscus. The flow rate is changes and the experiment is repeated for 8 flow rates. Results The table below, Table 1, contains the data collected in the laboratory. Case Weight (lb) Time (s) P1 P2 P3 P4 P5 P6 P7 P8 P9 1 50 61.16 6.35 6 5.18 4.33 3.45 2.61 1.79 0.94 0.12 2 50 67.49 5.45 5.3 4.56 3.83 3.03 2.3 1.51 0.82 0.11 3 50 72.62 5.05 4.79 4.19 3.5 2.8 2.11 1.4 0.79 0.11 4 45 71.72 4.53 4.29 4.29 3.12 2.5 1.9 1.26 0.7 0.1 5 45 82.47 4 3.77 3.77 2.75 2.2 1.67 1.12 0.64 0.1 6 45 96.78 3.48 3.29 3.29 2.4 1.93 1.47 0.98 0.56 0.1 7 40 100.01 3 2.85 2.85 2.09 1.68 1.29 0.87 0.5 0.11 8 40 122.25 2.5 2.37 2.07 1.74 1.4 1.08 0.74 0.44 0.11 Table 1. Oil Flow in a Pipe Laboratory Data. Discussion 1. Dimensional analysis can be used to show that the Reynolds number and non-dimensional pipe length characterize the non-dimensional pressure drop. The functional form can be rearranged to show this. Calculations are shown below. If the resistance coefficient is a function of Re, this agrees with the dimensional analysis. 𝛥𝛥𝛥𝛥 = 𝑓𝑓 ( 𝐷𝐷 , 𝐿𝐿 , 𝜌𝜌 , 𝜇𝜇 , 𝑉𝑉 ) 𝛥𝛥𝛥𝛥 = 𝑘𝑘𝐿𝐿 𝑎𝑎 𝐷𝐷 𝑏𝑏 𝜌𝜌 𝑐𝑐 𝜇𝜇 𝑑𝑑 𝑉𝑉 𝑒𝑒 𝑀𝑀𝐿𝐿 −1 𝑇𝑇 −2 = 𝐿𝐿 𝑎𝑎 𝐷𝐷 𝑏𝑏 ( 𝑀𝑀𝐿𝐿 −3 ) 𝑐𝑐 ( 𝑀𝑀𝐿𝐿 −1 𝑇𝑇 −1 ) 𝑑𝑑 ( 𝐿𝐿𝑇𝑇 −1 ) 𝑒𝑒 𝑀𝑀 : 1 = 𝑐𝑐 + 𝑑𝑑 𝐿𝐿 : − 1 = 𝑎𝑎 + 𝑏𝑏 − 3 𝑐𝑐 − 𝑑𝑑 + 𝑒𝑒 𝑇𝑇 : − 2 = −𝑑𝑑 − 𝑒𝑒 𝑏𝑏 = − 1 − 𝑎𝑎 + 3 𝑐𝑐 + 𝑑𝑑 − 𝑒𝑒 = 𝑎𝑎 − 𝑑𝑑 𝛥𝛥𝛥𝛥 = 𝑘𝑘𝐿𝐿 𝑎𝑎 𝐷𝐷 −𝑎𝑎−𝑑𝑑 𝜌𝜌 1−𝑑𝑑 𝜇𝜇 𝑑𝑑 𝑉𝑉 2−𝑑𝑑 𝛥𝛥𝛥𝛥 = 𝑘𝑘 � 𝐿𝐿 𝐷𝐷 � 𝑎𝑎 � 𝜇𝜇 𝜌𝜌𝐷𝐷𝑉𝑉 � 𝑑𝑑 𝜌𝜌𝑉𝑉 2

5 2. The specific gravity can be used to calculate the density and the specific weight of the oil. The table given in the laboratory handout, Table: Viscosity of the Red Oil, is used to determine the viscosity, μ, at the measured temperature. Calculations are shown below. Density = ρ (oil) = SG * ρ( H20) = 0.87 * (1.94 slugs/ft3) = 1.688 slugs/ft3 Specific Weight = γ( oil) = ρ( oil) * g = 1.689 * (32.2 ft/s2) = 54.347 lbs/ft3 Viscosity (Oil at 19⁰C) = 26.07 cP / 47880 cP = 5.44*10 -4 lbs/ft2 3. The volumetric flow rate, mean velocity and Reynolds number for each test are calculated below. All Reynolds numbers are in the laminar regime. The table, Table 2, below provides the values for the flow rate, mean velocity, and Reynolds number. Sample calculations are shown below. Case Time (s) Q (ft3/s) V (ft/s) Re 1 61.16 0.0150419 4.0519837 863.88628 2 67.49 0.0136311 3.6719414 782.86094 3 72.62 0.0126682 3.4125492 727.55832 4 71.72 0.0115444 3.1098354 663.01947 5 82.47 0.0100396 2.704467 576.5946 6 96.78 0.0085551 2.3045815 491.33867 7 100.01 0.007359 1.9823564 422.64002 8 122.25 0.0060202 1.6217216 345.75238 Table 2. Volumetric Flow Rate, Mean Velocity, and Reynolds Number Flow Rate = Q = 𝑤𝑤 𝛾𝛾 ( 𝑜𝑜𝑜𝑜𝑜𝑜 ) ∗𝑡𝑡 = (50 𝑙𝑙𝑏𝑏𝑙𝑙 )/( � 54.32 𝑜𝑜𝑏𝑏𝑙𝑙 𝑓𝑓𝑡𝑡 3 � ∗ ( 61.16s) = 0.015 ft3/2 Mean Velocity = 𝑄𝑄 𝐴𝐴 = ( 0 . 015� 𝑓𝑓𝑓𝑓 3 𝑠𝑠 � � 𝜋𝜋 4 �∗� 0 . 825𝑖𝑖𝑖𝑖 12𝑖𝑖𝑖𝑖 𝑓𝑓𝑓𝑓 � ^ 2 ) = 4.052 ft/s Reynolds Number = 𝜌𝜌𝜌𝜌𝜌𝜌 𝜇𝜇 = (1.688 slugs/ft3 * 4.052 ft/s * 0.06875 ft)/( 5.44*10 -4 lbs/ft2) = 863.89 4. The plot below, Figure 1, shows the pressure (in psf) versus tap location data for each Reynolds number. Each data set is labeled with the appropriate Reynolds number. The pressure gradient is decreasing with Reynolds Numbers.

6 Figure 1. Pressure Vs. Tap location for Each Reynolds Number. 5. The method of least squares is used to calculate the slope in the fully developed regime for each curve. The table below, Table 3, provides the slopes for each corresponding curve. Reynolds Number Slope dp/dx 863.88628 -15.63664 782.86094 -13.67655 727.55832 -12.47188 663.01947 -11.55847 576.5946 -10.13309 491.33867 -8.807439 422.64002 -7.553231 345.75238 -6.009862 Table 3. Slope of Each Curve in the Fully Developed Regime. 6. The slope is used to calculate the wall shear stress. The pressure gradient is constant in the fully developed regime. The shear stress decreases with the Reyolds number. The shear stress is calculated below: τ = -(dp/dx)*D/4 = -(-15.637)*0.06875/4 = 0.269 lb/ft2 Reynolds Number Shear Stress lb/ft2 863.88628 0.2687547 782.86094 0.2350657 727.55832 0.2143605 663.01947 0.1986612 576.5946 0.1741625 491.33867 0.1513779 0 50 100 150 200 250 300 350 400 0 5 10 15 20 25 Pressure (psf) Tap Location (ft) 864 784 728 663 576 491 423 346

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7 422.64002 0.1298212 345.75238 0.1032945 Table 4. Wall Shear Stress of Each Curve in the Fully Developed Regime. 7. The slope is used to calculate the resistance coefficient. The figure below, Figure 2, is a log-log plot of the resistance coefficient versus Reynolds number. The plot also shows the theoretical relationship for the laminar regime. The observed and theoretical relationship is calculated below. A table of values is also provided in Table 5. Sample calculations are also shown below. Figure 2. Resistance Coefficient Vs. Reynolds Number Re F (observed) F (Theoretical) 863.88628 0.0775871 0.0740838 782.86094 0.0826355 0.0817514 727.55832 0.0872481 0.0879655 663.01947 0.097366 0.0965281 576.5946 0.1128653 0.1109965 491.33867 0.1350975 0.1302564 422.64002 0.1565853 0.1514291 345.75238 0.1861631 0.1851036 Table 5. Reynolds Number and Resistance Coefficients Values 𝑓𝑓 𝑜𝑜𝑏𝑏𝑙𝑙𝑒𝑒𝑜𝑜𝑜𝑜𝑒𝑒𝑑𝑑 = 2�− 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 � ( 𝜌𝜌 ) 𝜌𝜌 2 𝜌𝜌 = 2 ( −−15 . 64 )(. 06875 ) 4 . 05 2 ∗1 . 687 = 0.078 𝑓𝑓 𝑡𝑡ℎ𝑒𝑒𝑜𝑜𝑜𝑜𝑒𝑒𝑡𝑡𝑜𝑜𝑐𝑐𝑎𝑎𝑜𝑜 = 64 𝑅𝑅𝑒𝑒 = 64 863.89 = 0.074 0.01 0.1 1 100 1000 Resistance Coefficient Reynolds Number Observed Theoretical

8 8. The uncertainty of the pressure measurements is estimated below. All assumptions and steps are shown below. Standard Deviation of Ruler: 𝜎𝜎𝛥𝛥 = 0 . 01 2 = 0.005 ft Uncertainty of Static Pressure: 𝛥𝛥 = 𝛾𝛾𝛾𝛾 𝜎𝜎𝛥𝛥 = � � 𝑑𝑑𝛥𝛥 𝑑𝑑𝛾𝛾 � 2 𝜎𝜎 𝑧𝑧 2 𝑑𝑑𝛥𝛥 𝑑𝑑𝛾𝛾 = 𝛾𝛾 𝜎𝜎𝛥𝛥 = �𝛾𝛾 2 (0.005) 2 = � 54.35 2 (0.005) 2 𝜎𝜎𝛥𝛥 = 0.272 𝑙𝑙𝑏𝑏 𝑓𝑓𝑡𝑡 2 Conclusion In the Oil Flow in a Pipe laboratory, the experiment performed observes and quantifies the frictional loss for oil moving through a brass pipe. Fluid flow through a straight pipe in these applications loses energy because of friction. The frictional loss depends on several parameters including the fluid viscosity and flow rate. Dimensional analysis reveals that for a smooth pipe wall, the Reynolds number and pipe geometry characterize the frictional losses. The objectives of this laboratory experiment are to measure the streamwise pressure distribution for several Reynolds numbers, use the pressure drop in the fully developed region to calculate the resistance coefficient, and to compare the resistance coefficient with known theoretical and empirical relationships. The uncertainty of the pressure measurements is also estimated in this laboratory. Calculations are shown above in the Test Results section of this laboratory report. The uncertainty of the static pressure measurement was calculated to be 0.272 lb/ft2.

9 Appendix The raw data collected in this laboratory experiment is provided below. Case Weight (lb) Time (s) P1 P2 P3 P4 P5 P6 P7 P8 P9 1 50 61.16 6.35 6 5.18 4.33 3.45 2.61 1.79 0.94 0.12 2 50 67.49 5.45 5.3 4.56 3.83 3.03 2.3 1.51 0.82 0.11 3 50 72.62 5.05 4.79 4.19 3.5 2.8 2.11 1.4 0.79 0.11 4 45 71.72 4.53 4.29 4.29 3.12 2.5 1.9 1.26 0.7 0.1 5 45 82.47 4 3.77 3.77 2.75 2.2 1.67 1.12 0.64 0.1 6 45 96.78 3.48 3.29 3.29 2.4 1.93 1.47 0.98 0.56 0.1 7 40 100.01 3 2.85 2.85 2.09 1.68 1.29 0.87 0.5 0.11 8 40 122.25 2.5 2.37 2.07 1.74 1.4 1.08 0.74 0.44 0.11

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