# Fluid Flow Analysis Penn State Chemical Engineering

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1 Fluid Flow Analysis Penn State Chemical Engineering Revised Spring 2015

2 Table of Contents LEARNING OBJECTIVES... 1 EXPERIMENTAL OBJECTIVES AND OVERVIEW... 1 PRE-LAB STUDY... 2 EXPERIMENTS IN THE LAB... 2 THEORY... 3 BACKGROUND... 3 ADDITIONAL THEORY TOPICS: (These are important learning points for prelab, prelab quiz, conducting the experiment and for writing the report. Please download the electronic reserve materials for this lab and review. Also make sure to watch the video.)... 4 PRE-LAB QUESTIONS (to be completed before coming to lab)... 5 DATA PROCESSING PREPARATION (Excel spreadsheet to be used for data processing in the lab must be prepared before coming to the lab for the experiment)... 9 DATA PROCESSING KEY POINTS FOR REPORT EXPERIMENTAL SETUP EXPERIMENTAL PROCEDURE REFERENCES Appendix A: Fanning Friction Factor Chart LEARNING OBJECTIVES 1. Understand the engineering Bernoulli equation and use it to calculate the pressure drop ( P tubing ) in a pipe due to the skin friction loss in the pipe. 2. Learn correlation between fluid flow rate (Q) and tube diameter (D) at a given P tubing. 3. Assess the effect of tube coiling on fluid flow rate and pressure drop. 4. Utilize the student s t-test and understand the statistical significance. EXPERIMENTAL OBJECTIVES AND OVERVIEW In this experiment, you will use both a straight pipe model and a coiled pipe model to estimate the pressure drop given the flow rate and also to estimate the flow rate given the pressure drop. 1

3 You will then determine which model better represents the piping system in the lab. The piping system consists of a flow meter, a pressure gauge, and a series of elbows, valves, and 50 ft lengths (=L) of ½ in, ¼ in, and 4 mm internal diameter smooth plastic tubing. The tubing is coiled around an 8 inch diameter spool and experiences approximately 5 feet of elevation drop. The end of the tubing is open to the atmosphere and drains to the floor drain. We want to find the relationship Q = f(d) at a fixed ΔP tubing. But, ΔP tubing cannot be read directly from the gauge because the elbows and valves are between the gauge and the start of the tubing. So, we must find Q iteratively until ΔP tubing reaches the set value for each tube with different diameter. PRE-LAB STUDY: 1) Calculate the pressure drop through the tubing ( P tubing ) at Q = 4.5 gpm for a tube with D = 0.5 and L = 50 ft using a straight pipe model. This will be the tubing pressure drop set straight point, P tubing, for the remainder of the experiment. straight 2) Predict Q straight for tubes with D = 0.25 and 4mm at the same P tubing using the straight straight pipe model and the same P tubing. 3) Re-process the data using a coiled tube model. a. Using a coiled model and the predicted Q straight values, calculate P coiled tubing. b. Recalculate the flow rate for a coiled pipe model (Q coiled ) using P straight tubing. 4) Q is proportional to D x when ΔP tubing and elevation change are held constant. Derive an equation describing how Q varies with D, Q = f(d), for the straight tube. It should have a form Q = a D x where a and x are constants. EXPERIMENTS IN THE LAB: exp 5) Measure P tubing when you set the flow rate to the predicted Q straight value in the pre- straight lab calculation. Check if your initial guess for Q gives the target P tubing value. 6) Adjust Q exp exp until P tubing becomes equal to the P straight exp tubing value. Note that P tubing is not read directly, so Q exp must be adjusted iteratively until (P gauge ΔP conduit ) = P straight tubing. Note that ΔP conduit is also a function of Q exp. CALCULATIONS IN THE LAB: 2

4 7) Process the experimental data and compare them with straight tube model and coiled tube model calculations. 8) Calculate exponential values (x) for Q = a D x from the plot of Q vs. D at a given exp pressure drop ( P tubing = P straight tubing ) and compare the experimental results with the straight and coiled pipe models while considering statistical significances. THEORY BACKGROUND There are two basic problems that are encountered in the industry dealing with fluid flow mechanics -- determining the pressure drop at a given flow rate and determining the flow rate at a given pressure drop. In the first case, when the flow rate is given, the Reynolds number may be directly determined to classify the flow regime so that the appropriate relations between the Fanning friction factor, f, and Reynolds number, Re, can be used. In the second case, the velocity is unknown, and the Reynolds number and flow regime cannot be immediately determined. In this case, it is necessary to assume the flow regime, apply the necessary calculations, and verify the Re afterwards to determine if the equations used are applicable. Symbols used for the variables are shown in the following table: Variables associated with flow in closed conduits Variable Symbol Dimension pressure drop ΔP mass/(length*time 2 ) Fluid velocity v length/time Volumetric flow rate Q length 3 /time conduit diameter D length conduit length L length conduit roughness length Fluid viscosity mass/(length*time) Fluid density mass/length 3 3

5 ADDITIONAL THEORY TOPICS: (These are important learning points for prelab, prelab quiz, conducting the experiment and for writing the report. Please download the electronic reserve materials for this lab and review. Also make sure to watch the video.) Reynold number, Re (laminar vs turbulent flows) Fanning friction factor (1) Bernoulli equation without friction and then with skin friction (h fs ) [neglect the kinetic energy correction factors these are minor] (2) relationship between skin friction parameters in a straight pipe Fanning friction factor as a function of pressure difference, pipe diameter & length, fluid velocity (length/time) and density (3) the Bernoulli equation with the skin friction term substituted in and no velocity change P g h 2 2 fv L D (4) The Bernoulli equation when gravity is neglected (no or negligible elevation change) and no velocity change. Friction factor parameter in a straight pipe from Hagen-Poiseuille equation in the laminar flow regime (friction factor equation as a function of Reynold s number) Effects of roughness on the friction parameter in the turbulent flow regime Friction factor chart Empirical equation for the friction factor for a smooth pipe in the turbulent flow regime as a function of Reynold s number use a simpler single-term equation there are many correlations. McCabe, Smith, and Harriott has one, Perry s handbook has another called the Blasius Equation. Pressure drop across fitting and valves Dean effect and critical Reynolds number for the coiled tubing Effect of tube coiling on friction factor (Perry s handbook section 6-18) Student s t-test (you can find and learn this subject from internet and examples on ANGEL) They can be found in: 4

6 1. McCabe, Smith, & Harriott, Unit Operations of Chemical Engineering (end of chapter 4 and champter 5) In 7 th Edition: pp (TP155.7.M3 2005) In 6 th Edition: pp (TP155.7.M3 2001) In 5 th Edition: pp (TP155.7.M ) In 4 th Edition: pp (TP155.7.M ) 2. Perry s Handbook, 7 th edition 3. S. Ali, Fluid Dynamics Research 28, (2001) [on ANGEL] PRE-LAB QUESTIONS (to be completed before coming to lab) Note: The theoretical calculations involved in determining the flow rates for the three tubing sizes are based on the assumption that water behaves as a nearly incompressible Newtonian fluid, in which the physical properties may be estimated as ρ = 1 gm/cm 3 and μ = 0.01 poise for the observed water temperature range of 32 F - 60 F. All of the theoretically calculated results correspond to the pressure drop across the 50 ft lengths of tubing only. However, preceding elbows and valves introduce a ΔP in addition to the 50 ft of tubing. Therefore, when setting up the pressure gauge for the flow measurement, you must account for the pressure drop through the elbows and valves for the corresponding flow rates. The pressure drops through the valves and elbows for given flow rates can be found in Appendix B. This is an extremely important detail to keep in mind. A flow rate of 4.5 gal/min is assumed for the ½ in tubing. The following questions help you to understand the governing principles and how to set up the experiment and handle data. 1. For flow through a horizontal pipe or tube (no elevation change) with constant diameter D and length L, a) What is the pressure drop, ΔP, if the friction factor, f, is zero (no friction loss)? ΔP<0, ΔP=0, or ΔP>0 b) What is the pressure drop if f > 0? Justify your answer. 5

7 2. For the same length and diameter of tube, with the same flow rate, would the friction factor be larger for a straight tube or a coiled tube? Why? 3. Flow meter ΔP conduit = 12 psi P tubing The right-side (boxed with dashed line) is the system you should consider in theoretical calculation: P friction = pressure due to frictional loss in the 50ft tube. P tubing = head pressure applied to the tube. P h = pressure due to the elevation change between the entrance and exit of the tube. P gauge P gauge = pressure you need to know or will measure during the experiment. Do not copy ΔP h = 4 psi ΔP friction = 20 psi P exit =0 psig D coil = 8 ΔP h is the pressure change due to elevation change, ΔP conduit is the pressure drop due to valves and elbows and fittings, and ΔP friction is the pressure loss due to friction in the tube (which is denoted as p s in McCabe, Smith, & Harriott book). In the figure, the thick line is the tube part of interest. a) If ΔP friction = 20 psi and ΔP h = 4 psi, what is the pressure at point P tubing? Write the balance equation for ΔP friction, ΔP h and ΔP tubing [P tubing is the applied pressure through the tube, which you will be using in the following theoretical calculations.] b) If ΔP conduit = 12 psi, what is the pressure gauge value (P gauge ) that you should read to get ΔP friction = 20 psi. Write the balance equation for P gauge, ΔP conduit, ΔP h and ΔP friction. The overall procedure of this experiment is described in the video lecture (on Angel). Briefly, you will perform the following 2 experiments. 1) Set Q to predicted value and measure P tubing You will set Q to the value determined from the straight pipe model (for each tubing size) and record the resulting P gauge. You will then calculate the corresponding P tubing and compare it to the predicted straight model and coiled model P tubing. 2) Set P tubing and measure Q. You will do the reverse experiment. You will set P tubing to the straight model predicted value. 6

8 You will then read the flow rate that gives you this P tubing. However, you cannot read P tubing directly; instead, you can set Q and measure P gauge. P tubing is then calculated using ΔP conduit at the set flow rate (from the chart in the Appendix B). Thus, you need to know a rough estimate of the flow rate that will give the target P tubing value. You will then iterate through different flow rates until you obtain a P tubing that matches the target straight model predicted P tubing. Note that ΔP conduit is dependent on flow rate. The following pre-lab questions guide you through the necessary calculations. 4. Write an equation for the skin friction loss in pipes that relates ΔP friction, friction factor (f), velocity (V), pipe length (L), pipe diameter (D), and density (ρ). 5. The friction factor (f) can often be estimated using empirical equations. Find friction factor estimation equations for laminar flow and turbulent straight pipe flow. Note: pick an equation that has a single term (i.e. no addition or subtraction). 6. How does the skin friction factor (f) vary with the fluid velocity (V) or Reynolds number (Re)? Sketch a plot of log(f) vs log (V) or log(f) vs log (Re). Mark the different fluid flow regimes and the (approximate) slopes of the lines. 7. Use Excel (using the Solver function), Mathematica ( FindRoot command), or any other program (you can use Matlab or Mathcad) to simultaneously solve three equations. Note that debugging help is most easily available for Excel format work. (1) the Bernoulli equation applicable to the 50 ft tube (thick line part in figure for #3) [It should include the pressure drop (P tubing ), elevation change (h), and skin friction factor (f)]. (2) the friction factor (f) equation for turbulent flow. (3) the Reynold number (Re) equation. Note that lb m (pound mass) and lb f (pound force) are different quantities. Set up these calculations so that you can easily repeat them a number of times. You need these calculations to answer #8 and #9. The calculation results of questions 8 and 9 should be summarized in Table 1. Refer to the diagram in question #3 for a schematic of the system (the pressure values in this diagram should not be used) and Appendix B for the pressure loss due to valves and elbows (conduit). Bring your program to the lab. 7

9 8. For the ½ inch tube: Determine whether the flow rate is laminar or turbulent for a flow rate of 4.5 gpm. Determine the theoretical pressure drop through the tubing, P straight tubing (in psi), for a flow rate of 4.5 gpm and h = 5 ft. 9. Now, we are repeating the experiment with ¼ inch and 4mm tubing, instead of ½ inch tube. Calculate the velocity and flow rate using the same P straight tubing at the entrance of the tube as calculated for the ½ inch tubing. Determine whether the flow rate is laminar or turbulent for the flow rates found. Adjust your previous calculation if the Reynold s number is not in the regime you assumed. Table 1. Theoretically calculated pressures and flow rates in the experimental setup for a specific P straight tubing utililizing a straight pipe model D Predicted Q straight (gpm) 0.5 inch 4.5 Re Friction factor f straight tube Theoretical P friction straight P tubing Estimated P conduit Predicted P gauge 0.25 inch 4mm (using turbulent flow equation) (using laminar flow equation) 10. The fluid flow (Q) through a pipe depends on the tubing diameter (D) if P tubing, elevation change, and tube length are held constant. Derive the relationship Q = f(d) for the straight pipe model (L = 50 ft, no valves or fittings). It should take the form Q = a D x where a and x are constant. Determine the theoretical x value for turbulent flow. (you do not need to find a value for a) 11. Explain the Dean Effect observed in coiled tube flow. 12. The friction factor (f) can often be estimated using empirical equations. Find friction factor estimation equations for laminar and turbulent coiled pipe flow (D coil = 8 inches). Use 8

10 Perry s Handbook Section 6-18 and associated text and/or the on-line video (all found on ANGEL). 13. What is the critical Reynold s number (definition)? What is the critical Reynold s number for straight pipe flow and for coiled pipe flow? 14. What are the objectives for this experiment? 15. Explain what data you will collect, how you will collect it, and what you will use it for. DATA PROCESSING PREPARATION (Excel spreadsheet to be used for data processing in the lab must be prepared before coming to the lab for the experiment) 1. Prepare a data processing excel spreadsheet to be used for the data processing in the lab. All calculations will be done in excel. a) Prepare a header section with your names and group ID. b) Prepare a units section where you show unit conversions. Make it so that you can reference the appropriate cell when a certain conversion is needed in later calculations. Refer to the pre-lab calculations for the unit conversions used. c) Show all needed formulas from the pre-lab calculations clearly explained in text boxes d) When using your spreadsheet in the lab, make sure that you use cell references when using previously calculated values or constants (instead of copying them); this will update the entire spreadsheet if/when a mistake is found early in the spreadsheet. (no work required for 1.d) 2. Using the calculated flow rates for each diameter tubing for the straight pipe model, calculate the tubing pressure drop for coiled pipe flow, and fill out table 2. Make sure to check the flow regime and adjust the model if necessary. Note that the only difference with the straight pipe model calculations is in the friction factor. Table 2. Theoretically calculated pressures for set flow rates in the experimental setup utililizing a coiled pipe model. The set flow rates match the calculated flow rates for the straight pipe model with a specified straight pipe P straight friction. 9

11 D Predicted Q straight (gpm) Re Crictical Re Friction factor f coiled tube P friction coiled P tubing Estimated P conduit Predicted P gauge 0.5 inch inch From Table 1 4mm From Table 1 (using turbulent flow equation) (using laminar flow equation, if necessary) 3. Repeat the coiled tube calculations, but this time hold P coiled tubing constant to the same value used in the straight tube model calculations. Again, check the flow regime and adjust models if necessary. Fill out Table 3. Table 3. Theoretically calculated pressures and flow rates in the experimental setup for a specific P coiled tubing utililizing a coiled pipe model D Predicted Q coiled (gpm) Re Critical Re Friction factor f straight tube Theoretical P friction coiled P tubing Estimated P conduit Predicted P gauge 0.5 inch From Table inch 4mm (using turbulent flow equation) (using laminar flow equation) 4. Set up tables for data collection. a. Part 1: you will set Q and measure P gauge. You will need to calculate P conduit from Q and appendix data (use an equation in your spreadsheet so you can repeat the 10

12 calculation easily), then calculate P tubing. Test your spreadsheet by using Q = 5 gal/min and P gauge = 19 psi. b. Part 2: you will set P gauge and measure Q. You will need to calculate P tubing as in part 1. You will adjust P gauge until P tubing matches the theoretical. You will record every iteration. DATA PROCESSING Overview Part 1. Let s check if the flow rate that you calculated using the straight tube model works well. Adjust P gauge until the actual flow rate is the same as the predicted Q straight in Table 1. Record the experimental P gauge value. Repeat this step for D = 0.5 inch, 0.25 inch, and 4 mm. Part 2. You may see discrepancy between the theoretical prediction and the experimental data. Could that be due to the tube coiling? Adjust Q until the (P gauge P conduit ) value becomes set equal to P tubing. Record Q exp for D = 0.5 inch, 0.25 inch, and 4 mm. Note that P conduit changes each time the flow rate changes. Calculations 1. Find P conduit from the chart given in the Appendix and calculate the actual P actual tubing. Fill out Table 4. (you will have 3 data points for each condition) actual Table 4. Calculating P tubing when Q is set to the predicted Q straight. Tube diameter Q straight (from pre-lab) Experimental P gauge Experimental P conduit actual P tubing 0.5 inch 4.5 gpm 0.25inch 4 mm 11

13 2. Calculate the average P actual tubing and the standard deviation for each tubing size. 3. Fill out Table 5 and make a graph of P straight tubing, P coiled tubing, and P actual tubing versus tube diameter (D) showing all 3 cases. Table 5. Comparison of predicted P straight tubing and P coiled actual tubing to P tubing when Q is set to the predicted value for the straight pipe. Tube diameter Predicted Q straight (from pre-lab) straight P tubing (from pre-lab) coiled P tubing (from lab-prep) actual P tubing actual P tubing Standard Deviation 0.5 inch 0.25inch 4 mm 4. Now let s look at the data where you set the flow rate to match a certain pressure drop. exp You adjusted the flow rate, Q exp, until you got P tubing = P straight tubing in the lab. a. Calculate the average Q and standard deviation for each diameter studied. b. Make a table comparing the actual Q values with the predicted Q values for straight and coiled tubing. Fill out Table 6. Table 6. Comparing flow rate and Reynolds numbers Tube diameter 1.5 inch 0.25inch 4 mm exp P tubing = P straight tubing = (write your theoretical value) Straight tube model Coiled tube model Q straight Re straight Q coiled Re coil Re critical Q Exp Experimental S.D. Q Exp Re Exp 12

14 5. Using the data from Table 6, calculate the exponent x of the model Q = a D x for the experimentally measured data as well as for the theoretical values predicted for the straight and coiled tubing. Make sure to calculate the error on x. The error can be calculated in Excel. To do so, you can linearize the data by taking logarithm: i.e. plotting lnq versus lnd. Then, you can perform statistical analysis on the linearized data and obtain the error of the slope. Determine the statistical significance of the difference between the two models and of each model with the experimental data by performing a t- test. Information on statistical analysis can be found on the class ANGEL site. KEY POINTS FOR REPORT If report requires a theory section, do further research on subjects included in relevant theory section on page Include some industrial examples for which the material studied in this experiment is applicable. 2. Include the process diagram or schematics of the fluid flow experimental system (identify essential parts and connect them in a simple and easy-to-follow way). Place this in the appendix or experimental section of the report, as appropriate. 3. [from part 1] Consider the plot of experimental P tubing and theoretical P tubing calculated with the straight and coil tube models for the ½, ¼, and 4mm diameter tubing. Discuss any trends or differences that you observe. 4. [from part 2] Consider the plot of the experimental Q values and Q theoretical (calculated for both straight and coiled tubes) vs. the tube diameter at constant P tubing. Discuss why the deviation between the straight and coiled models is larger at higher tube diameters. Which model, straight or coiled, would you recommend for the experimental set-up? Why? 5. Compare the experimental Reynolds number with the theoretical values calculated for both straight and coiled tubing as well as the critical Reynolds number for the transition in the coiled tubing. (all comparisons based on the same ΔP tubing ) 13

15 6. Discuss how the exponent x of the experimental data compares with the theoretical values. Discuss the statistical significance of the differences from the theoretical models. Based on the x values, which model would you recommend? If your recommendation of models is different here from Q#3, reconcile the difference. 14

16 EXPERIMENTAL SETUP The following figure is a picture of the fluid mechanics apparatus. The system consists of a pressure gauge to measure the total pressure of the system, three valves to route the flow through a specific tube, a rotameter to measure the flow rate in the system, and three sizes ( 1 / 2 in, ¼ in, and 4mm) of 50 ft tubing, which are arranged in an 8 in diameter coil. Figure 1: Fluid Mechanics System Pressure Gauge ½ in valve Rotameter ¼ in valve air valve 4mm valve Do not copy Elevation change of tubing Tubing coil 15

17 Using this setup, the pressure drop for a flow rate of 4.5 gal/min through the ½ inch tubing is measured. Compare this experimental value to the theoretical pressure drop through the 50 ft of tubing. Make sure to account for the pressure drop in the elbows and valves. Then measure the flow rate achieved in the smaller tubing with the same theoretical pressure drop over the tubing, again making sure to account for the pressure drop in the elbows and valves, and compare the flow rates to the theoretically calculated values. Warnings: After adjusting the rotameter, it takes a minute or two for the float to settle into position. Therefore, after making flow adjustments, wait for the float to attain its position before taking flow and pressure readings. Rotameter readings are generally taken at the top of the widest part of the float. The rotameter is a flow measuring/regulating device and should not be used to stop flow completely. To stop flow completely, use the ball valve upstream. Also note that the lowest flow rate on the rotameter is not zero. All of the valves associated with the tubing system must be completely opened. If the valves are only partially opened this will introduce a restriction in your system and produce a larger pressure drop than expected, resulting in poor experimental data. 16

18 EXPERIMENTAL PROCEDURE 1.) Observe the piping system on the blue mounting board. Close the three yellow ball valves to the off position. Also, make sure that the three tee valve knobs have the pointed end downward so that the flow is directed to the tubing coil. 2.) Open the main water valve on the back wall by turning it a ¼ turn and ensure that it is fully opened. 3.) Open the yellow ball valve for the ½ in tubing system. 4.) Twist the rotameter dial to the left to allow the water to flow through the 0.5 inch tubing. Make sure that the end of the tubing is in the drain. 5.) [part 1 of data processing.] Set the flowrate at 4.5 gpm for 0.5 inch tubing and record P gauge. The experimental value might be different from the theoretically calculated P gauge in Table 1. Reduce the flow rate and again bring it back up to 4.5 gpm in order to obtain a total of 3 data points at 4.5 gpm. 6.) Repeat step 5.) for the 0.25 inch and 4 mm tubing using the predicted Q straight from Table 1. Note that the rotameter cannot read flow rates below 0.5 gpm. A graduated cylinder and stopwatch must be used to determine the flow rate for 4 mm tubing. 7.) [part 2 of data processing] Adjust the flow on the rotameter until P gauge matches your theoretical P gauge from Table 1 and record the flow rate (Q). Calculate P conduit from this Q and then P tubing. P tubing (=P gauge - P conduit ) Adjust the flow on the rotameter until P tubing becomes equal to your theoretical P tubing in Table 1. Record the experimental Q value that gives the theoretical P tubing. Note that P conduit varies with the flow rate and your flowrate may not be the same as Q theoretical in Table 1. It is likely that ΔP conduit will also vary from the ΔP conduit in Table 1. You will need to use trial and error for several iterations from the theoretical starting point. 8.) Reduce the flow rate and redo step 7.) in order to obtain 3 data points. 9.) Reduce flow to less than 0.5 gpm using the rotameter. Open the yellow ball valve for the next tubing size to be studied before closing the yellow ball valve for the ½ in tubing. 17

19 10.) Repeat steps 7.) - 9.) for both 0.25 inch and 4 mm tubing using the predicted P tubing from Table 1 Note than the rotameter is incapable of reading low flow rate. A graduated cylinder and stopwatch must be used to determine the flow rate for 4 mm tubing. 11.) Reduce the flow rate to <0.5 gpm using the rotameter, and close the main water line. 18

20 REFERENCES 1 Welty, Wicks, Wilson, and Rorrer. Fundamentals of Momentum, Heat, and Mass Transfer. 4 th ed. John Wiley & Sons, Inc. New York, Perry. Perry s Chemical Engineering Handbook. 7 th ed. McGraw-Hill, New York, Bird, Stewart, and Lightfoot. Transport Phenomena. 2 nd ed. John Wiley& Sons Inc. New York, McCabe, Smith, and Harriott. Unit Operations of Chemical Engineering. 7 th ed. McGraw-Hill, New York,

21 Appendix A: Fanning Friction Factor Chart 20

22 pressure drop (psig) pressure drop (psig) pressure drop (psig) APPENDIX B: Pressure drops through elbows and valves (all combined in experimental set-up) 1/2 inch ID tube 1/4 inch ID tube 4 mm ID tube Q (gpm) ΔP (psig) Q (gpm) ΔP (psig) Q (gpm) ΔP (psig) NA NA NA NA NA NA NA 1/2" ID tube 1/4" ID tube 4mm ID tube y = x 3-18x x R² = y = 4.0x 2-4.0x R² = y = x 3-10x x - 3E-12 R² = flow rate (gpm) flow rate (gpm) flow rate (gpm) The equations given in the graphs are the best fit. You can use these equations to calculate the pressure drop at the flow rate that you measured in the lab. 21

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