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Memo Report: Fluid Flow Circuit

TO: Professor Andrew R. Teixiera, PhD.

FROM: Michael Bodanza ____________________

(Project Leader)

Martin Burkardt _____________________

Brandon Clark ______________________

Sotirios Filippou ____________________

Marcus Lundgren ___________________

DATE: September 19th, 2017

SUBJECT: Relationship of Reynolds Number and Fanning Friction Factor for Various Piping

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Introduction

Different forms of piping and pipe fittings are used ubiquitously for effective fluid transfer in

operating chemical processes. Fluid flow is induced by a pressure drop across the pipe, with fluid

flowing from high to low pressure. If the pressure fluctuates significantly within a pipe, damage

to the pipe and fixtures can result from cavitation. Fluids can flow in two characteristic flow

regimes – turbulent and laminar. In laminar flow, the fluid forms parallel sheets that do not mix

and remain steady in the absence of disturbances. However, in turbulent flow the fluid motion is

chaotic, forming swirling eddies, leading to higher skin friction losses. Figure 1 below provides

general velocity profiles for both regimes.

Figure 1: Velocity profiles. For fully developed laminar flow (left), the velocity profile is a smooth parabola,

increasing to a maximum at the centerline and tending to zero at the walls. However, for turbulent flow (right) the

bulk fluid velocity is nearly constant throughout the pipe while approaching zero at the walls.

These flow regimes are characterized by three dimensionless groups: the Reynolds number, the

Fanning friction factor, and the relative roughness. Several empirically derived correlations, such

as the Colebrook and Hagen-Poiseuille equations, are available that relate these variables. These

variables and equations are described as follows:

𝑅𝑒 = 𝐷𝑉𝜌

𝜇 (1)

𝑓 =Δ𝑃𝐷

2𝐿𝜌𝑉2 (2)

𝑟 =𝜀

𝐷 (3)

Δ𝑃 =32𝐿𝑉𝜇

𝐷2 (4)

1

√𝑓𝑑

= −2.0 (𝑟

3.7+

2.51

𝑅𝑒√𝑓𝑑

) (5)*

* Note the colebrook equation equates the Darcy friction factor, not the Fanning friction factor. The Darcy is equal to

four times the Fanning. This is accounted for in later calculations.

Within this experiment, we assessed the validity of these equations under varying piping

conditions with water employed as a test fluid. Additionally, as pipe fittings lead to large

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pressure drops and friction losses, we validated the equations (Equation 6) that relate the

pressure drop to velocity of various fittings, including a venturi meter, an orifice meter, and an

elbow joint. An example of a venturi and orifice meter is shown below in Figure 2.

𝑢𝑜 =𝐶𝑜

√1−𝛽4√

2Δ𝑃

𝜌 (6)

Figure 2: Internal configuration of orifice and venturi meters.[5][6] Both meter types measure the flow rate

primarily by restricting the flow, which induces a pressure drop. However, venturi meters create smaller pressure

drops, allowing the feed to maintain more of its internal energy.

Methodology

Fluid flow was tested for various flow rates across three cylindrical pipes of different diameters

and materials, specifically two L-type hard copper tubes with ID of 0.430 inches and 0.315

inches, and a PVC tube with an ID of 0.525 inches. For each flow rate, the pressure drop was

measured using the respective high and low pressure taps (Figure 3). Multiple flow rates in

laminar and turbulent flow were chosen. The data for each regime was plotted against the

respective governing correlation. The same procedure was followed for three typical pipe

fittings, including an orifice meter, a venturi meter and an elbow joint. The following tables

(Tables 1 & 2) summarize the parameters of the laminar and turbulent flow tests in pipes.

Corresponding charts for pipe fittings are not shown.

Figure 3: PFD for fluid flow circuit and PVC piping. The fluid circuit is constructed from type L copper tubing

with nominal diameters ranging from ½” to ¼”. The PVC (ID = 0.525 in.) was a separate apparatus. The needle

valve (left) was used to control the fluid flow, and pressure drop was subsequently measured. All pressure taps (both

high, PH and low, PL) are sent to a pressure manifold. The pressure manifold consisted of four differential pressure

gauges with maximum ΔP of 40, 20, 5 and 2 inH2O.

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Reynolds Number 2300 1800 1500 500

Pipe Diameter

(in)

Flow Rate 1

(gpm)

Flow Rate 2

(gpm)

Flow Rate 3

(gpm)

Flow Rate 4

(gpm)

0.315 0.23 0.18 0.15 0.05

0.430 0.32 0.25 0.21 0.07

(PVC) 0.525 0.39 0.31 0.25 - *

Table 1: Testing matrix for laminar flow. For a given Reynolds number a flow rate was calculated from the

properties of the pipe and water. *Note for low Reynolds number in the large PVC pipe, the given flow rate was

excluded for inability to detect a pressure drop.

Reynolds Number 14000 10000 7000 3500

Pipe Diameter

(in)

Flow Rate 1

(gpm)

Flow Rate 2

(gpm)

Flow Rate 3

(gpm)

Flow Rate 4

(gpm)

0.315 1.42 1.02 0.71 0.36

0.430 1.95 1.39 0.97 0.49

0.525 1.87

(Re =11000)*

1.70 1.19 0.59

Table 2: Testing matrix for turbulent flow. As before, the Reynolds number was used to calculate the given flow

rate from given parameters. *Note the corresponding run for large Reynolds numbers in the PVC pipe were not

obtainable, as the rotameters on this stream only measure to a given limit.

Additionally, the properties of the fluid were assumed to be constant throughout the apparatus

for the given feed temperature, 20oC. The density was assumed at 998.2 kg/m3 and the viscosity

was assumed to be 1.002 cP from tabulated data. The absolute roughness of the L-type copper

and PVC piping were to be that of brand new copper tubing, 0.0015 mm, and PVC piping,

0.0035 mm, from tabulated data. [3]

Results and Discussion

Correlation between Reynolds Number and Fanning Friction Factor

Based on the Moody charts below (Figure 4), experimental data for the PVC pipe seems to

correlate well with theoretical data, barring some error with experiments running in the copper

tubing. To find a plausible explanation for this deviation, the absolute roughness of the copper

tubes was revaluated. As the tabulated data is for new copper pipes, it is possible to conceive that

this may be an inaccurate assumption, given the age of the system and corrosion effects of water

on copper pipes. Consequently, the copper pipe roughness in the theoretical model was

incrementally increased from 0.0015 mm until the average experimental deviation from the

model (shown as %deviation on the graphs) was minimized. The percent deviation of the graphs

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were calculated using all theoretical values (xi) corresponding to experimental values (yi) at

similar Reynolds numbers through the following equation.

𝐴𝑣𝑒𝑟𝑎𝑔𝑒 %𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛 = (∑

|𝑦𝑖 − 𝑥𝑖|𝑥𝑖

𝑖) ∗ 100

The final copper pipe roughness was found to be 0.06 mm. This suggests that either the

experimental data points were consistently too high or the copper pipes are not as smooth as

brand new copper pipes due to wear and tear. The former assumption seems less likely,

considering the laminar regime experimental points, which do not depend on pipe roughness,

were not too high for the smooth model like the turbulent points. The y-axis is spaced linearly

instead of logarithmically to show the error between data sets more clearly.

The final turbulent flow experiences a Fanning friction factor that gradually decreases as

Reynolds number increases. In contrast, laminar flow experiences a Fanning friction factor that

decreases much more rapidly as Reynolds number increases. The gap in data seen on the graphs

illustrates the transition region between laminar and turbulent flow. Furthermore, the

experimental laminar regions seem to follow the laminar relationship, f = 16/Re, as exhibited by

the theoretical lines below a Reynolds number of 2300. The overlap in rough and smooth

theoretical lines in the laminar regime show that laminar flow is not a function of pipe

roughness. The sum total of this data proves the validity of the Hagan-Poiseuille equation for

relating Reynolds number, pressure drop, and Fanning friction factor correctly. All experimental

data regarding Part 1 measurements can be found in Appendix B, and percent deviation data can

be found in Appendix C.

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Figure 4: Comparison of experimental data with empirical Moody chart. For laminar regimes the Hagen-

Poiseuille equation was used for comparison. For turbulent flow regimes the Colebrook equation is used. Some error

bars are provided for errors derived from pressure gauges during experiments with the smallest diameter pipe.

Three flow rates in the 0.315 inch diameter pipe were ran in triplicate in order to calculate a

standard deviation for experimental error. The standard deviations are, at most, ±3% of the data

point values, which is extremely reliable data for all intents and purposes. It can be argued that

this error may be extrapolated to all other data points. This is because all fluid flows in this

portion of the experiment were in straight, horizontal pipes with no fittings, both laminar and

turbulent regimes were studied in triplicate, and the same four pressure gauges were used for

each pipe involved. Therefore, the same fluid dynamics properties were studied in all three pipes.

Also, any inconsistencies in pipe flow at any specific point in time are trivial, considering that

pressure gauges were allowed to settle over a long period of time before data points were taken.

Sample calculations for the standard deviation can be found in Appendix C.

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Determining Fitting Discharge and Loss Coefficients

Three different fittings – an orifice, venturi, and elbow joint – were analyzed in this process by

measuring the pressure drop across each unit. With the pressure drops, measured flow rates and

given characteristic data, such as entrance area (Sb) and β, experimental orifice and venturi

coefficients (Cexp) were found for each run (Table 3 & 4).

Run Re V (m3/s) β ΔP (N/m2) ⍴ (kg/m3) Sb (m2) Cexp

1 14000 8.99x10-5 0.315 4044 998.2 5.03x10-5 0.59

2 7000 4.49x10-5 0.315 1369 998.2 5.03x10-5 0.51

3 3500 2.25x10-5 0.315 348 998.2 5.03x10-5 0.50

Table 3: Experimental venturi coefficients with equation variables. From equation (6) the discharge coefficient

can be determined for the venturi meter.

Run Re V (m3/s) β ΔP (N/m2) ⍴ (kg/m3) Sb (m2) Cexp

1 14000 8.90x10-5 0.312 8087 998.2 4.93x10-5 0.42

2 7000 4.45x10-5 0.312 2488 998.2 4.93x10-5 0.38

3 3500 2.23x10-5 0.312 547 998.2 4.93x10-5 0.41

Table 4: Experimental orifice coefficients with equation variables. From equation (6) the discharge coefficient

can be determined for the orifice meter.

Based on the variation between Cexp (SDv = 0.049, SDo = 0.021, respectively), a better approach

was chosen to determine the orifice and venturi coefficients. Using the coefficient expression

seen in Equation 4, the variables were rearranged to represent a linear correlation (Equation 5).

The Ko value is a variable made up of the fitting factors (Equation 6) to represent the slope of

this linear expression.

𝛥𝑃

⍴= 𝐾𝑜 ⋅

𝑢𝑜2

2 (7)

𝐾𝑜 =(1−𝛽4)

𝐶𝑜2 (8)

After plotting 𝛥𝑃

⍴versus

𝑢𝑜2

2 (Figure 6 & 7), a linear regression was performed to estimate the

slope, or Ko. From Ko, the fitting discharge coefficient was calculated. This method is a better

engineering approach for determining the coefficient because determines one value for the fitting

using every run’s data. The orifice and venturi coefficients were found to be 0.59 and 0.42

respectively. When comparing these values with theoretical coefficients found in the literature

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there is some discrepancy. In general, the discharge coefficient for both venturi and orifice

valves will vary within a small range depending on the speed of the inlet flow. For a venturi

valve with a β ratio of 0.315, the discharge coefficient should theoretically fall within 0.955 -

0.995. From the experimental findings, the venturi coefficient falls about 0.3 short of that

prediction. A similar trend was found for the orifice valve (should fall within 0.594 - 0.61),

which falls about 0.17 short. This is most likely due to the flow speeds used to measure pressure

drop across the two valves. The literature has Reynolds numbers (and corresponding

coefficients) in the range of 104 - 107 whereas our chosen flow rates gave Reynolds number from

103 - 104. It is important to note the orifice coefficient also did not correlate with the provided

characteristic chart (Appendix E). A similar problem is faced where the highest Reynolds

number used in the experiment is the lowest on the orifice characteristic chart. An expected

discharge coefficient for a Re of 104, or turbulent flow, would be about 0.55. Although closer

than the literature estimation, it is still significantly lower than expected.

Figure 6: Venturi linear regression to determine the fitting factor Ko. Difficulties in determining a linear fit for

these values may be difficult as many published values correspond to Reynolds number on the order of 104-107.

Figure 7: Orifice linear regression to determine the fitting factor Ko. Again, difficulties in a linear fit are due to

the range of typical values corresponding to Reynolds numbers on the order of 104-107, according to the provided

characteristic chart (Appendix E).

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Even though the theoretical discharge coefficients did not correspond with the experimental data,

the loss coefficient, KL found for the elbow fitting held true to its theoretical estimation. To

calculate KL the pressure drop was measured as well as flow rate. Using Equation 7, KL was

determined to be about 1.4. According to the literature, the theoretical loss coefficient for a 90º

threaded elbow was 1.5, which is relatively close to the experimentally determined factor, 1.4.

𝐾𝐿 =𝛥𝑃

0.5⋅⍴⋅𝑢𝑜2 (9)

Conclusions and Recommendations

Process design is intricate, and many factors must be considered during design. Friction within a

pipe can cause inadequate flow rates that result in cavitation and process irregularity or failure.

The friction factor is determined with correlations to the physical properties of the pipe (length,

diameter, and roughness), the velocity of the fluid within the pipe, and the pressure drop across

the pipe. In order to achieve desired flows, the magnitude of the pressure drop will dictate what

type of a pumping system is appropriate for our design.

After comparing experimental to theoretical values, we can confirm the relationship between

Reynold’s number and friction factor; as stated on the Moody chart. Our experimental values

were very close (RSD <10%) compared to the theoretical values. That being said, challenges and

shortcomings were still faced in light of the roughness of the copper pipes. For further studies

using this fluid system, a study is recommended for determining the wear on the copper tubing

and the resulting absolute roughness.

For the fitting experiments it was difficult to reliably determine the discharge coefficients, as the

Reynolds numbers were out of the linear range. For further studies it is recommended that higher

flow rates be used for the orifice and venturi meters to easily achieve higher Reynolds numbers

within the linear range of values for the discharge coefficients.

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References

1. Welty, J. R., Wicks, C. E., Wilson, R. E., and Rorrer, G., Fundamentals of Momentum,

Heat, and Mass Transfer, 4th Ed., Wiley, New York (2001)

2. McCabe, W.L., Smith, J.C., and Harriott, P., Unit Operations of Chemical Engineering,

7th Ed., McGraw Hill, New York (2005)

3. “Major loss in Ducts, Tubes and Pipes.” The Engineering ToolBox. Accessed 18 Sept.

2017.

4. Perry, R. H. and Green, D. W., Perry’s Chemical Engineer’s Handbook, 7th Ed.,

McGraw Hill, New York (1997)

5. “Venturi meter calculations with Spreadsheets.” The Engineering Toolbox. Accessed 18

Sept. 2017.

6. “Orifice Plate Flow Meter Calculations Spreadsheet.” The Engineering Toolbox.

Accessed 18 Sept. 2017.

Appendix A: Theoretical Fanning Factor Iterative Calculation Using Colebrook Equation 0.315 inch copper pipe smooth

0.315 inch copper pipe rough

0.43 inch copper pipe smooth

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0.43 inch copper pipe rough

0.525 inch PVC pipe

Appendix B: Experimental Fanning Friction Factors Based on Reynolds Number and Pressure Drop

Appendix C: Average Experimental Deviation from Theoretical Models

0.315 inch copper pipe

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0.43 inch copper pipe

0.525 inch PVC pipe

Experimental error calculations for 0.315 inch copper pipe

Appendix D: Pipe Fitting Discharge and Loss Coefficients

Elbow (90º - Threaded)

Venturi & Orifice

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Appendix E: Images of the Actual venturi and orifice meters in Goddard Hall.

Characteristic chart for orifice meter used.