fanning friction factor

Flow in Pipes Exploring the Fanning Friction Factor and Velocity Profiles

Team #1 3/15/2002

Anastasia Gribik Queenelle Ogirri

Sara M. Royce Earl Osman P. Solis

TABLE OF CONTENTS Abstract 3 Introduction 3 Background and Theory 3 Experimental: Equipment 6 Experimental: Procedure 8 Results and Discussion 9 Conclusions & Recommendations 13 Nomenclature 15 References 17 Appendix 18

ABSTRACT

This experiment analyzes the fanning friction factor (f) of brass and Plexiglas pipes as a

function of Reynolds number (NRe) for water in both laminar and turbulent flow regimes.

Documented comparisons of the theoretical correlations present in literature, and the

experimental values obtained provided the basis for this research. The calculated function of f

for laminar flow in pipes was found to be f =19.2/NRe or, expressed in power form, f =

9.73/NRe0.90. For turbulent flow, the experimental data yielded f = 0.029/NRe

0.14. Finally, the

correlation of the turbulent flow proved to be bimodal. Analyzing the velocity profile, the data

exhibited a value of n =14 for the power law equation. Integrating this equation, the

experimental volumetric flow rate was calculated and compared to the flow rate determined by

the equipment manufacturers’ correlation yielding a 9.3% error.

INTRODUCTION

The objective of this experiment was to analyze the fanning friction factor (f) of brass and

Plexiglas pipes as a function of Reynolds number (NRe) for water in both laminar and turbulent

flow regimes, and to examine the velocity profile of turbulent flow. Exploring the flow of water

through pipes has many practical applications. An early example of water flow, on a large scale,

traces back to the Roman aqueducts, which transported water to farms and cities through a

system of pipes. Today, transportation of water all over the world for drinking, irrigation, and

mechanical cooling involves the knowledge of fluid flow and friction. Other fluids transported

by way of pipe systems, include oil, natural gas and steam. Pipe flow is not limited to large-

scale transportation; it is also an imperative part of many experimental setups and product

manufacturing processes. Understanding f is an important because it describes frictional losses

in flow through a system of pipes. Collection of pressure drop and volumetric flow rate data

used to analyze flow and calculate f required the use of the piping apparatus available in the

Rothfus Laboratory.

BACKGROUND & THEORY

Fluid flow is characterized by the dimensionless constant known as the Reynolds

number.

µρ⋅><

=DvNRe (1)

As defined here, <v>, D, ρ, and µ represent the bulk fluid velocity, inner diameter of the pipe,

fluid density, and fluid viscosity, respectively. The Reynolds number differentiates fluid flow

into three major categories. Flows with NRe less than 2100 are considered laminar, while flows

with NRe greater than 4000 are considered fully turbulent. Flows with NRe values between these

two extremes are considered transitional, which has characteristics of both flows, thus, making it

harder to analyze and predict (Geankoplis 49).

In this experiment, D, ρ, and µ are known and <v> must be calculated using the

relationship:

AQv = (2)

where Q is the volumetric flow rate and A is the cross sectional area of the pipe. For laminar

flow, Q was determined by measuring the change of liquid height in a graduated cylinder over a

measured amount of time. For turbulent flow, the pressure drop across the orifice plate was

measured using a mercury manometer. The equipment manufacturer, Engineering Laboratory

Design, Inc., provides us with the equation:

21

hkQ ⋅= (3)

where k represents the orifice coefficient (also provided by the manufacturer), and h represents

the pressure drop across the orifice plate.

As stated in the objective, the purpose of this experiment was to express f as a function of

NRe. Friction factor is defined as the drag force per wetted surface unit area divided by density

times the velocity head (0.5ρ<v>2). For laminar flow in SI units, f is defined as:

22 20.5 ><⋅⋅∆

=><

=vlDP

vf s

ρρτ

(4)

where τs, ∆P, and represent shear stress at the surface, total pressure drop through the pipe, and

pipe length, respectively (Geankoplis 86).

l

The Reynolds number is related to f for Newtonian fluids by the Hagen-Poiseuille

equation:

232D

vdxdP ><

=− µ (5)

This first order differential equation can be integrated:

232D

lvP ><=∆

µ (6)

Using the definition of head loss (hL) in equation (7) and equating equations (4) and (6), one can

find the equality:

⋅⋅><

=⋅><

=⋅∆

= 2

2

322Dg

lvgD

vlfg

PhL ρµ

ρ (7)

This relationship gives f as a function of NRe for laminar flow of Newtonian fluids, which can be

used as a theoretical basis for experimental results (Welty et. al. 185).

Re

1616NvD

f =><

=ρ

µ (8)

For turbulent flow, f can still be calculated using (4). Although f still depends on NRe, the

Hagen-Poiseuille equation (5) no longer applies. One available equation, derived from an

analysis of the velocity profile for turbulent flow, is the Von Karman equation (Welty et. al.

187):

(9)60.0)log(06.41Re −= fN

f

Experimental data can also be analyzed by using empirically calculated correlations. Three

different equations, which are accurate in specific NRe ranges, have been experimentally

developed to help simplify the correlation. These equations can be found in the form:

nNbafRe

+= (10)

Table 1 Empirical equation coefficients for turbulent flow with NRe ranges (Jhon Fluid

Mechanics 9-2)

Equation Name a b n NRe range for equation use

Blasius 0 0.079 0.25 4x103 < NRe < 105

Colburn 0 0.046 0.20 105 < NRe < 106

Koo 0.0014 0.125 0.32 4x103 < NRe < 3x106

The turbulent correlations described thus far are specific to smooth pipes, but most pipes

have a surface roughness (e) that can be associated with the material. For turbulent flow, f

depends not only on NRe, but also on the relative roughness (e/D) of the pipe. A plot of f versus

NRe was presented by Moody (Figure C.1), which can be used to find f taking e/D into

consideration. Churchill derived another approximation where (Perry 6-10):

+−=

9.0

Re

727.0log41ND

ef

(11)

“Most models of turbulent flow include three regions: a boundary layer near the pipe wall

which can be either laminar or turbulent, a buffer zone where viscous forces are overcome by the

fluid’s inertia, and a turbulent core which is assumed to be the central region.” However, since

the buffer region and the boundary layer are negligible, they can be neglected in the analysis for

this experiment and only the turbulent core is used when constructing the velocity profile (Jhon

Velocity Profiles in Turbulent Pipe Flow 1).

Using the pressure drops and local velocities (v), measured by the pitot tube, a velocity

profile can be derived. The local velocity for the turbulent flow can be calculated from the

equation:

ρtP

Cv∆

=2

(12)

where C is a constant for the pitot tube ranging from 0.98 to 1.0 and ∆Pt is the pressure drop

from the location of the pitot tube to the inner pipe wall (Perry 10-8).

After obtaining a value for v, n can be found by plotting the log of the left hand side

versus the log of the right hand side of (13) and taking the inverse of the slope:

n

Ry

vv

1

max

= (13)

where y is the distance from the pitot tube to the closest pipe wall, vmax is the maximum velocity,

R is the radius of the pipe, and n is a slowly varying function of NRe, which increases from 6 to

10 as NRe increases from 4000 to 3,200,000 (Welty et al, 177). To determine the validity of v,

(13) is integrated to determine <v>, which can be used to calculate Q (see A.1 Derivation of Q

Equation). The volumetric flow rate calculated is then compared to Q found by using (3).

EXPERIMENTAL

Equipment:

Using the piping equipment setup located in the Rothfus Laboratory, we were able to

obtain data for this experiment. The apparatus consists of two pumps (one for laminar and the

other for turbulent flow) and four pipes, see Figure C.2. For this experiment, the 0.125” brass

pipe was associated with the laminar flow pump and the 2.00” Plexiglas pipe with the turbulent

flow pump. Below is a complete diagram of the apparatus.

Figure 1 Piping apparatus. Adapted from Lab Manual.

Two manometers were used for this experiment. One, with an attached ruler, was used to

measure the pressure drop across the entire pipe, and the other was used to measure the pressure

drop across the orifice plate in the 2.00” Plexiglas pipe. Since this manometer had no attached

measurement apparatus, we used a tape measure to acquire accurate pressure drop

measurements. A graduated cylinder was included in the laminar flow setup to measure the

volumetric flow rate.

The velocity profile of the turbulent flow was obtained using a simple pitot tube (impact

tube). The depth at which the pitot tube measured the velocity profile could be altered using an

attached dial that indicated depth. Shown below is a diagram of the pitot tube used. Pressure

difference was measured between the flow on the pipe wall surface and the location of the pitot

tube.

Figure 2 Simple pitot tube. Adapted from Lab Manual.

A temperature transmitter is located in the water reservoir tank with an electronic reader

that displayed the temperature of the water. A series of recycle pipes with control valves were

located at the back of the apparatus, which assisted in varying the flow rate (Figure C.2).

Procedure:

To analyze f in laminar flow, measurement of pressure drop and volumetric flow rates

allowed for calculations of NRe and f. The two ends of the manometer were connected to the

pipe 10’ away from each other, and a nozzle at the end of the pipe allowed the water to empty

into the graduated cylinder. The laminar pump was turned on, and the control valve on the pipe

was fully opened so that the maximum flow was achieved, while the valve connected to the

manometer was closed to disconnect it from the water flow. This closed configuration provided

us with a zero measurement of the pressure drop (Po) for calibration purposes. When the

manometer valve was opened, ∆P could be measured. The valve at the end of the tube was

closed at small increments and ∆P was measured for an acceptable range of data points. The

temperature was also measured so that fluid properties dependant on temperature could later be

used in the calculations. The volumetric flow rate was measured at each flow by closing the

drainage on the graduated cylinder and recording the amount of time it took for the cylinder to

fill to a certain height.

The second part of the experiment measured the pressure drop and volumetric flow rate

for turbulent flow. The manometer was once again connected to the pipe with a distance of 10’

between the two ends. The pressure difference through the orifice plate was measured by

connecting the two ends of a separate mercury monometer to the pipe just before and after the

orifice plate. The turbulent pump is turned on and the valve is opened so that maximum flow is

achieved. Measurements for Po and ∆P were measured from both manometers at several flow

rates. Temperature readings were also taken for each flow rate.

The last task was to measure the velocity profile. The mercury manometer was

connected to the pitot tube so that the ∆Pt could be measured. Once again, ∆P was measured

across 10’ of the pipe. A maximum flow was achieved by closing the recycle valve and fully

opening the control valve on the pipe. Since the flow will stay constant throughout the pitot tube

measurements, ∆P across the orifice plate is only measured once. Beginning at the bottom of the

pipe, ∆Pt is measured, using the dial to move the pitot tube by increments of 0.1” to the top of

the tube. Temperatures are once again measured.

To ensure that the only variable in the experiment was the flow rate, other sources of

error were carefully considered. Specific group members were assigned specific tasks to

perform throughout the experiment, e.g. manometer, graduated cylinder height, and time

measurements. This eliminated error associated with technique variation, which could

potentially add error to the data.

RESULTS & DISCUSSION

Using ∆P, temperature, and Q, values of NRe and f were calculated for laminar and

turbulent flow; see A.1 for sample calculations and B.1 and B.2 for data tables. From the

experimental data for laminar flow, the equation of f as a function of NRe for laminar flow is:

Re

2.19N

f = or (expressed in power form) 90.0Re

73.9N

f = (14)

Figure 3 compares a graphical representation of our data, with error bars representing one

standard deviation, to the theoretical equation (8).

Laminar Flow

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0 200 400 600 800 1000 1200 1400 1600

NRe

f

Experimental

16/ Re( h i l)

f = 19.2/ Nre -or-f = 9.73N Re

-0.90

Figure 3 Graphical comparison between theoretical f and experimental data for laminar flow.

Overall, the calculated values of f were off by an average factor of 1.3 from the

theoretical correlation. Several sources of error attributed to the deviation from the theorized

correlation. For instance, impurities in the water could have affected the analysis, in that the

fluid properties used in the calculations assumed that pure water was flowing through the pipes.

This can help explain why the experimental friction factors were higher in value. Another major

source of error could be the inaccuracy of physical measurements. Impurities were present in the

manometers, which blocked accurate measurement of manometer fluid heights. This would

affect ∆P calculations. In addition, human error becomes a factor when taking any experimental

measurements.

Figure 3 shows that the standard deviation error bar ranges increased as the NRe values

decreased. One can attribute this to errors in manometer readings. The smaller the NRe value, the

lower ∆P is through the pipe. Due to manometer fluid fluctuations with time, the fluid heights

were difficult to measure. At small pressure drops, these fluctuations played a more significant

role in error because as ∆P decreases the fluctuations become a larger percentage of the total

height.

The experimental data for turbulent flow showed that the power form equation for f as a

function of NRe was equal to:

14.0Re

029.0N

f = (15)

Omission of the two outliers, which did not fit the trend for turbulent flow, allowed for a more

accurate correlation between f and NRe (see Figure 4 and Table B.2). These points represented

low ∆P values, which presented the maximum amount of error for these measurements, as

mentioned above. The standard deviation error bars show that our measurements were accurate,

but the experimental correlation is visibly shifted upward compared to the known turbulent flow

equations found in literature. This effect is also visible in Moody’s friction factor plot (Figure

C.1). As noted in Background & Theory, turbulent flow f is also dependant on roughness of the

pipe, accounted for by the relative roughness factor (e/D). The Plexiglas pipe can be placed in

the category of drawn tubing (e = 1.526E-6 m) (Perry 6-10). This value is relatively small and

has little effect on the flow, as can be seen in the line derived from Churchill’s equation (Figure

4). Therefore, the calculated f values for turbulent flow should coincide more with the known

correlations for smooth pipes. The apparent shift cannot be the result of the roughness of the

Plexiglas, but other aspects of the experiment. Again, the assumption that pure water was in the

pipes would affect the calculations through the fluid properties. The water visibly contained rust,

which deposited onto the inner pipe wall. The presence of such deposits would increase relative

roughness, which would increase the friction factor values. Using the Churchill equation for f,

the amount of increase in surface roughness would have to be about 0.002” (Figure 4) to derive

the friction factor correlation found in the experiment. This would be the about the same

roughness of concrete or wood stave (Welty et al. 189).

Turbulent Flow

0.0035

0.004

0.0045

0.005

0.0055

0.006

0.0065

0.007

0 20000 40000 60000 80000 100000 120000 140000 160000 180000

NRe

f

Exp

Blasius

Koo

Colburn

von Karmen

Churchill

Churchill w / rough0.002"

f = 0.0294N Re-0.14

Figure 4 Graphical comparison between theoretical f and experimental data for turbulent flow.

Also noticeable in Figure 4 is the fact that the experimental data are closest to the

Colburn correlation for smooth pipes. This is because the derived NRe values were found in the

range specific to the Colburn correlation (105 < NRe < 106). The Koo and Von Karmen equation

values, with broader NRe ranges, deviate more from the experimental data. The Blasius equation,

specific to an NRe range between 4x103 and 105, is significantly lower than the Colburn equation

in the experimental data range. Therefore, in making comparisons, the Colburn equation was

used because it was specific to the range of the data. The average factor of deviation for the

experimental results compared to the correlation was 1.3, which can be attributed to the sources

of error discussed above. Other sources of error, similar to that of laminar flow, were human

measurement errors, manometer fluctuation, and manometer fluid impurities.

Analyzing the velocity profile for turbulent flow using the pitot tube the data exhibited a

bimodal distribution (see Figures 5 and 6).

Velocity Profile

2.95

3.05

3.15

3.25

3.35

3.45

3.55

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05

Position in Pipe

Flui

d Ve

loci

ty

Figure 5 Graphical representation of the velocity profile with error bars.

ln(V/Vmax) vs ln(y/R)

-0.07

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0-1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0

ln(y/R)

ln(V

/Vm

ax)

bottomtop

Figure 6 Graphical representation of the logarithm of the power equation.

The local velocity was calculated using (12) and compared to the distance of the pitot tube to the

closest pipe wall. The local velocity ranged from 2.95 to 3.28 m/s with a standard deviation of

.17 m/s. To further analyze the velocity profile, n was solved for using equation (13). The value

for n was found to be 17.0 for the bottom half of the pipe and 11.7 for the top half of the pipe,

yielding an overall value of n equal to 14.3. The value found is higher than the literature value;

consequently, we feel that this is due to rust particles in the water and deposits of rust on the pipe

wall as mentioned above in the analysis of turbulent flow. As for the bimodal distribution, the

local velocities for the bottom half of the pipe were higher than those of the top half. This

phenomenon is not well understood and merits further investigation.

The propagation of error was calculated for laminar and turbulent flows to determine the

standard deviation of the final results due to fluctuations in manometer readings (see A.1 Error

Analysis for sample calculations). In the laminar example, the standard deviation of the

measured height was very small; therefore, our results can be reported with high accuracy. For

values of f, the relative error is approximately 1% except at the very lowest flow rates where it

increases to 8 % due to the extremely low bulk velocities (see error bars in red, Figure 3).

In the turbulent regime, the relative error was found to be quite low; values of 0.02%

were consistently reported (see error bars in red, Figure 4). Such a decrease is associated with

the increased flow rate and pressure drop. Relative error for the fluid velocity is approximately

5% (see error bars, Figure 5), which remains relatively consistent with the standard deviations

calculated for the turbulent data. However, because the scope of the fluid velocity is within the

error range, these results are highly uncertain.

CONCLUSIONS & RECOMMENDATIONS

In analyzing the experimental results, it is clear that the resulting correlations for f exhibit

the same trends as the known correlations for laminar and turbulent flows. Both the laminar and

the turbulent results exhibited high precision; however, the velocity profile did not. To help

improve results, modification and/or repetition of certain parts of the experiment is imperative.

For instance, calibration of the thermocouple is essential. Since, the fluid properties of water are

dependent on temperature, errors in the results could stem from variations in the fluid properties,

due to changes in temperature. Replication of data would also have been beneficial in the data

analysis and would have provided further feedback on the accuracy and precision of our results.

Using either pure water or a known fluid would diminish the errors introduced by the impurities

in the water. In addition, investigation of the effects of the rust on inner pipe diameter, fluid

properties, and contamination of the manometer are vital. Given that these impurities deposit

crud onto the pipe wall, the roughness changes; as a result, further investigation of these effects

should allow the experimental results to resemble the theoretical equations provided in literature.

NOMENCLATURE

e Surface rougness [=] m

f Fanning friction factor [=] unitless

g Acceleration of gravity 9.8 [=] 2sm

h Pressure drop across orifice plate [=] 2mN

hL Head loss [=] m

k Orifice coefficient [=] 0.5

3.5

kgm

l Length of pipe [=] m

n Power law exponent variable [=] unitless

v local velocity [=]sm

<v> Average bulk velocity [=]sm

y Distance of pitot tube to closest pipe wall [=] m

A Cross-sectional area [=] m2

D Inner diameter of pipe [=] m

NRe Reynolds Number [=] unitless

Po Zero pressure reading for calibration [=] 2mN

∆P Pressure drop [=] 2mN

∆Pt Pressure drop across pitot tube and inner wall [=] 2mN

Q Volumetric flow rate [=]s

m3

R Radius of pipe [=] m

T Temperature [=] °C

µ Viscosity [=]secm

kg⋅

ρ Density [=] 3mkg

τs Shear stress at the wall surface [=] 2mN

REFERENCES Adomi, Scott; Benedict, Jason; Stroup, Gregory; Drag Reduction in Turbulent Flow, Carnegie

Mellon University, 1997. Brown, Link; Kimmel, Mark; Wagner, Mike; Wieloch, Kelan; Fluid Flow in Pipes,

Carnegie Mellon University, 1998. Geankoplis, Christie J. Transport and Unit Operations, Prentice Hall, Englewood Cliffs, New

Jersey, 1993. Jhon, Myung. Velocity Profiles in Turbulent Pipe Flow, Carnegie Mellon University. Jhon, Myung, Fluid Mechanics, Carnegie Mellon University, 2001. Perry, R. H.; Green, D.W. Perry's Chemical Engineers' Handbook (7th Edition), McGraw-Hill,

1997.

Welty, James R., Charles E. Wicks, Robert E. Wilson, and Gregory Rorrer, Fundamentals of Momentum, Heat, and Mass Transfer, John Wiley and Sons, New York, NY, 2001.

APPENDIX A

Nre =

NreρH2O v⋅ Dp⋅

µH2O:=

v =

vQ

Ap:=

Q =

Volumetric flow rateQAgc hgc⋅

t:=

Height of water in graduated cylinderhgc .4ft:=

Time it took graduated cylinder to fillt 161s:=

∆P 4.63 103× Pa=

Calculated pressure drop across the pipe∆P sg ρH2O⋅ ρH2O−( )g ∆h⋅:=

Change in height read off manometer∆h .631m:=

Specific gravity of blue stuffsg 1.75:=

g 9.8m

s2:=

Length of brass pipeLp 3.048m:=

Agc 3.14159Dgc

2

2⋅:=

Ap 3.14159Dp2

2⋅:=

Diameter of graduated cylinderDgc .07m:=

Diameter of brass pipeDp18

in:=

µH2O 0.001017725kgm s⋅

:=

ρH2O 998.305kg

m3:=

Sample CalculationsLaminar Flow: Trail #1

v =

vQ

Ap:=

Q =

Equation for volumetric flow rate as reported on apparatusQ k ∆hH2O

1

2⋅:=

∆P =



Change in height of water used in Q equation∆hH2OρHg ∆hHg⋅

ρH2O:=

g 9.8m

s 2:=

Length of plexiglass pipeLp 3.048m:=

ρHg 13554.823707kg

m3:=

Ap 3.14159Dp2

2⋅:=

Change in height of mercury manometer∆hHg 13.31250 in:=

k value as reported on apparatusk .044ft3 s 1−

⋅

ft⋅:=

Diameter of plexiglass pipeDp 2in:=

µH2O 0.000801583kg

m s⋅:=

ρH2O 995.694kg

m3:=

Sample CalculationsTurbulent Flow: Trial #1

f Nre⋅ =

f =

Calculated Fanning friction factorf∆P Dp⋅

2 ρH2O⋅ Lp⋅ v2⋅

:=

Bulk Velocityvb =

vbQ

Ap:=

Q =

Equation for volumetric flow rate as reported on apparatusQ k ∆hH2O

1

2⋅:=

∆P =


Change in height read off manometer∆h 0.593m:=

Change in height of water used in Q equation∆hH2OρHg ∆hHg⋅

ρH2O:=

g 9.8m

s2:=

Length of plexiglass pipeLp 3.048m:=

ρHg 13554.823707kg

m3:=

Ap 3.14159Dp2

2⋅:=

Change in height of mercury manometer∆hHg 17.12500in:=

k value as reported on apparatusk .044ft3 s 1−

⋅

ft⋅:=

Diameter of plexiglass pipeDp 2in:=

ρH2O 993.399kg

m3:=

Sample CalculationsVelocity Profile: Trial #1

f =



:=

Nre =

NreρH2O v⋅ Dp⋅

µH2O:=

jh

vbvmax

2π

πR2R

0

y−( )yR

1

nR y−( )⋅

⌠⌡

d⋅:=

jh dr dy−:=r R y−:=

jkhafkjdhjhafjh jh

vbvmax

1

π R2⋅ 0

R

θ

0

2π

ryR

1

nr⋅

⌠⌡

d

⌠⌡

d⋅:=

jkhafkjdhjhafjh

Derivation of Q equation

Q =

Q 21

1n

1+

11n

2+

−

⋅ vmax⋅ Ap⋅:=

n =

n

1slopeb

1slopet

+

2:=

slopet 0.0857:=

slopeb 0.0587:=

yDp2

=

Distance of pitot tube from pipe wally 0.0002032m:=

lnvl

vmax

=

vmax 3.282544745ms

:=

Local Velocityvl =

vl2 ∆P⋅

ρH2O:=

sd∆h =

Standard deviation of the height readingsd∆h sdm12 sdm22+ sdm32

+ sdm42+( ).5:=

sdm4 .1cm:=

sdm3 .1cm:=

sdm2 .1cm:=

sdm1 .1cm:=Standard deviation of manometer readingsas approximated by lab member reading the manometer


Specific gravity of blue stuffsg 1.75:=

g 9.8m

s2:=

Length of brass pipeLp 3.048m:=

Agc 3.14159Dgc

2

2⋅:=

Ap 3.14159Dp2

2⋅:=

Diameter of graduated cylinderDgc .07m:=

Diameter of brass pipeDp18

in:=

µH2O 0.001017725kgm s⋅

:=

ρH2O 998.305kg

m3:=

Sample Calculations of Error AnalysisLaminar Flow: Trail #1

vb vmax 2⋅1

1n

1+

11n

2+

−

⋅:=

vbvmax

21

1n

1+

11n

2+

−

⋅:=jh

vnvmax

2

0

1

yR

yR

1

n1

yR

−

⋅

⌠⌡

d⋅:=

jh

R



:=

sdNre =Calculated standard deviation of Nre

sdNreρH2O sdv⋅ Dp⋅

µH2O:=

Nre =

NreρH2O v⋅ Dp⋅

µH2O:=

sdv =Calculated standard deviation of velocity

sdvsdQAp

:=

v =

vQ

Ap:=

sdQ =Calculated standard deviation of volumetric flow rate

sdQAgc sdhgc⋅

t:=

Q =

Volumetric flow rateQAgc hgc⋅

t:=

Observed uncertainty in reading graduated cylindersdhgc .1ft:=

Height of water in graduated cylinderhgc .4ft:=

Time it took graduated cylinder to fillt 161s:=

sd∆P =Calculated standard deviation of ∆P

sd∆P sg ρH2O⋅ ρH2O−( )g sd∆h⋅:=

∆P =


f =

f Nre⋅ =

deltaf∆PDp


:= Since f is dependent on multipe errors, it is calculated using CITE

deltaf∆P =

deltafvDp− ∆P⋅

ρH2O Lp⋅ v3⋅

:=

deltafv =

sdf f deltaf∆P( )2 sd∆P( )2⋅ deltafv( )2 sdv( )2⋅+ deltafv( )2 sdv( )2

⋅+

1

2⋅:=

sdf =

APPENDIX B

Table B.1 Laminar Flow Datasheet

Laminar Flow

Pipe Diameter (m): 0.00318Tube Diameter (m): 0.07Area of Pipe (m):

7.9E-06 g (m/s^2): 9.8

Trial Temp. Delta h SD of

h Delta P SD of delta

P Time HeightSD of Height Q SD of Q

No.

( C) (m) (m) (N/m^2) (N/m^2) (sec) (m) (m) (m^3/sec) (m^3/sec) 8 19.4 0.161 0.002 1177.693 14.68 210 0.05 0.03048 8.37862053E-07 0.000000569

19.4 0.220 0.002 1614.283 14.68 152 0.05 0.03048 1.15757257E-06 0.000000777 19.4 0.266 0.002 1951.815 14.68 263 0.09 0.03048 1.33803065E-06 0.000000456 19.4 0.353 0.002 2586.522 14.68 186 0.09 0.03048 1.89194657E-06 0.00000063

10 19.4 0.413 0.002 3026.781 14.68 230 0.12 0.03048 2.04001196E-06 0.000000515 19.4 0.438 0.002 3213.892 14.68 216 0.12 0.03048 2.17223495E-06 0.000000544 19.4 0.525 0.002 3848.598 14.68 184 0.12 0.03048 2.55001494E-06 0.000000643 19.4 0.580 0.002 4255.838 14.68 170 0.12 0.03048 2.76001617E-06 0.000000691 19.5 0.631 0.002 4629.989 14.68 161 0.121920 0.03048 2.91430279E-06 0.000000732 19.5 0.707 0.002 5187.642 14.68 148 0.12 0.03048 3.17028885E-06 0.00000079

11 19.4 0.738 0.002 5411.518 14.68 135 0.12 0.03048 3.47557592E-06 0.00000087

Table B.1 (continued)

Velocity SD of

Velocity Density of

H20 Viscosity of

H20 Nre Uncertainty of

Nre Length f(m/s) (m/s) kg/m^3 (kg/m*sec) (m)

0.10582677 0.07055118 998.320 0.00102027 328.77133 219.18088 3.048 0.0548620.14620804 0.09747203 998.320 0.00102027 454.22354 302.81569 3.048 0.0393970.16900093 0.05633364 998.320 0.00102027 525.03406 175.01135 3.048 0.0356520.23896368 0.07965456 998.320 0.00102027 742.38686 247.46229 3.048 0.0236310.25766518 0.06441630 998.320 0.00102027 800.48670 200.12168 3.048 0.0237850.27436570 0.06859143 998.320 0.00102027 852.37010 213.09253 3.048 0.0222740.32208148 0.08052037 998.320 0.00102027 1000.60838 250.15210 3.048 0.0193550.34860584 0.08715146 998.320 0.00102027 1083.01142 270.75286 3.048 0.0182700.36809312 0.09202328 998.305 0.001017725 1146.39486 286.59872 3.048 0.0178280.40042562 0.10010641 998.305 0.001017725 1247.09171 311.77293 3.048 0.0168800.43898513 0.10974628 998.320 0.00102027 1363.79216 340.94804 3.048 0.014650


df/ddeltaP df/dv SD of f Theoretical f/Nre f (calculated)

0.000047 -1.036823 0.004487 0.04866604 18.036986650.000024

-0.538921 0.002314 0.03522495 17.895173820.000018 -0.421920 0.000947 0.03047421 18.718758520.000009 -0.197779 0.000416 0.02155211 17.543317460.000008 -0.184617 0.000316 0.01998784 19.039375930.000007 -0.162368 0.000277 0.01877119 18.985794760.000005 -0.120189 0.000209 0.01599027 19.367083970.000004 -0.104819 0.000187 0.01477362 19.786903470.000004 -0.096866 0.000178 0.01395680 20.437807480.000003 -0.084308 0.000159 0.01282985 21.050392490.000003 -0.066747 0.000120 0.01173199 19.98005656

Table B.2 Turbulent Flow Datasheet

Turbulent Flow

Pipe Diameter (m): 0.0508 Area of Pipe (m):

0.0020

g (m/s^2):

9.8

Trial Temp. (initial) Temp. (final) Tavg

Delta height std dev h Delta P

std dev of delta P h

std dev of h

No. ( C) ( C) ( C) (m) (m) (N/m^2) (N/m^2)

(in Hg) (in Hg) 7 33.6 33.9 33.75 0.0650 0.002 475.07 14.62 2.18750 0.12508

34 34.3 34.15 0.0115 0.002 84.04 14.62 0.81250 0.12504 32.1 32.4 32.25 0.1340 0.002 979.88 14.63 3.25000 0.12503 31.5 31.9 31.7 0.2300 0.002 1682.21 14.63 5.68750 0.12505 32.7 32.9 32.8 0.2690 0.002 1966.71 14.62 6.81250 0.12506 33.1 33.3 33.2 0.3485 0.002 2547.60 14.62 9.18750 0.12502 31.2 31.3 31.25 0.4510 0.002 3299.11 14.63 11.75000 0.12501 29.6 30.3 29.95 0.5050 0.002 3695.77 14.64 13.31250 0.1250

10 35 35.1 35.05 0.4910 0.002 3587.01 14.61 13.37500 0.12509 34.5 34.7 34.6 0.5320 0.002 3887.14 14.61 14.18750 0.1250


density Hg h std dev h Q std dev Q Velocity std dev v Density of

H20 Viscosity of

H20 Nre(kg/m^3)

(ft H20) (ft H20) (m^3/sec) (m^3/sec) (m/s) (m/s) kg/m^3 (kg/m*sec)

13554.8237 2.48486 0.14199 0.00196403 0.00046949 0.96901773 0.23163954 994.394 7.4095000E-04 6.60640E+0413554.8237

0.92308 0.14201 0.00119706 0.00046953 0.59060796 0.23165550 994.257 7.3495500E-04 4.05883E+0413554.8237 3.68989 0.14192 0.00239334 0.00046937 1.18082940 0.23157970 994.908 7.6405000E-04 7.81110E+04 13554.8237 6.45608 0.14189 0.00316579 0.00046933 1.56194212 0.23155771 995.097 7.7282000E-04 1.02168E+05 13554.8237 7.73603 0.14195 0.00346543 0.00046942 1.70977819 0.23160158 994.720 7.5558000E-04 1.14347E+05 13554.8237 10.43445 0.14197 0.00402469 0.00046945 1.98570769 0.23161765 994.582 7.4942000E-04 1.33873E+05 13554.8237 13.33576 0.14187 0.00454995 0.00046929 2.24486155 0.23153979 995.251 7.8020000E-04 1.45472E+05 13554.8237 15.10241 0.14181 0.00484195 0.00046919 2.38893209 0.23148828 995.694 8.0158300E-04 1.50746E+05 13554.8237 15.19997 0.14206 0.00485757 0.00046960 2.39663558 0.23169151 993.948 7.2208500E-04 1.67587E+05 13554.8237 16.12084 0.14203 0.00500255 0.00046956 2.46816621 0.23167356 994.102 7.2852000E-04 1.71091E+05


Theoretical

std dev Nre Length f df/ddeltap df/dv std dev f Koo Blasius Colburnvon

Karmen (m) f

f

1.57923E+04 3.048 4.2399E-03 8.9248E-06 -6.4198E-04 1.0493E-06 4.9853E-03 4.9276E-03 4.9976E-03 4.9034E-031.59200E+04

3.048 2.0193E-03 2.4028E-05 -5.0158E-04 7.8295E-07 5.5901E-03 5.5658E-03 5.5091E-03 5.4719E-031.53188E+04 3.048 5.8862E-03 6.0071E-06 -7.3176E-04 1.5025E-06 4.7981E-03 4.7255E-03 4.8330E-03 4.7276E-031.51464E+04 3.048 5.7744E-03 3.4326E-06 -5.4280E-04 1.0666E-06 4.5184E-03 4.4187E-03 4.5803E-03 4.4642E-031.54891E+04 3.048 5.6361E-03 2.8658E-06 -4.8381E-04 9.2382E-07 4.4080E-03 4.2961E-03 4.4783E-03 4.3600E-031.56153E+04 3.048 5.4135E-03 2.1249E-06 -4.0007E-04 7.2909E-07 4.2600E-03 4.1300E-03 4.3393E-03 4.2200E-031.50043E+04 3.048 5.4816E-03 1.6615E-06 -3.5858E-04 6.5726E-07 4.1850E-03 4.0451E-03 4.2678E-03 4.1488E-031.46073E+04 3.048 5.4199E-03 1.4665E-06 -3.3331E-04 6.0273E-07 4.1534E-03 4.0093E-03 4.2375E-03 4.1188E-031.62013E+04 3.048 5.2358E-03 1.4597E-06 -3.2039E-04 5.6088E-07 4.0617E-03 3.9045E-03 4.1487E-03 4.0315E-031.60594E+04 3.048 5.3490E-03 1.3761E-06 -3.1788E-04 5.6737E-07 4.0441E-03 3.8844E-03 4.1315E-03 4.0147E-03


e/D (e/D)+.002in f (rough)f(rough) w/.002

(mm) 2.9921E-05 1.0299E-03 4.9196E-03 5.8613E-03 OUTLIER 2.9921E-05

1.0299E-03 5.4756E-03 6.2502E-03 OUTLIER 2.9921E-05 1.0299E-03 4.7492E-03 5.7527E-03 2.9921E-05 1.0299E-03 4.4956E-03 5.6017E-032.9921E-05 1.0299E-03 4.3960E-03 5.5461E-032.9921E-05 1.0299E-03 4.2627E-03 5.4753E-032.9921E-05 1.0299E-03 4.1953E-03 5.4411E-032.9921E-05 1.0299E-03 4.1670E-03 5.4270E-032.9921E-05 1.0299E-03 4.0848E-03 5.3873E-032.9921E-05 1.0299E-03 4.0690E-03 5.3799E-03

Table B.3 Velocity Profile Datasheet Turbulent Velocity Profile

Pipe Diameter (m): 0.0508 Area of Pipe (m):

0.002026828

g (m/s^2):

9.8

Trial Temp. (initial) Temp. (final) Tavg Delta height

std dev height Delta P

std delta P

k (orifice coef) h

std dev h

No. ( C) ( C) ( C) (m) (N/m^2) (N/m^2) (in Hg) (in Hg) 0 36.6 36.7 36.65 0.593 0.002 4329.78 14.60 0.044 17.12500 0.125001

36.7 36.8 36.75 0.6275 0.002 4581.52 14.60 0.044 17.12500 0.125002 36.8 36.9 36.85 0.6495 0.002 4741.98 14.60 0.044 17.12500 0.125003 36.9 37 36.95 0.6875 0.002 5019.25 14.60 0.044 17.12500 0.125004 37 37.1 37.05 0.7065 0.002 5157.79 14.60 0.044 17.12500 0.125005 37.1 37.2 37.15 0.7215 0.002 5267.11 14.60 0.044 17.12500 0.125006 37.2 37.4 37.3 0.7280 0.002 5314.29 14.60 0.044 17.12500 0.125007 37.4 37.5 37.45 0.7330 0.002 5350.51 14.60 0.044 17.12500 0.125008 37.6 37.7 37.65 0.7300 0.002 5328.24 14.60 0.044 17.12500 0.125009 37.7 37.8 37.75 0.7275 0.002 5309.81 14.60 0.044 17.12500 0.12500

10 37.8 37.9 37.85 0.7230 0.002 5276.78 14.60 0.044 17.12500 0.1250011 37.9 38.1 38 0.7240 0.002 5283.81 14.60 0.044 17.12500 0.1250012 38.1 38.2 38.15 0.7170 0.002 5232.45 14.60 0.044 17.12500 0.1250013 38.2 38.2 38.2 0.7155 0.002 5221.41 14.60 0.044 17.12500 0.1250014 38.2 38.3 38.25 0.6950 0.002 5071.73 14.59 0.044 17.12500 0.1250015 38.3 38.4 38.35 0.6730 0.002 4911.01 14.59 0.044 17.12500 0.1250016 38.4 38.4 38.4 0.6490 0.002 4735.80 14.59 0.044 17.12500 0.1250017 38.4 38.5 38.45 0.6045 0.002 4411.00 14.59 0.044 17.12500 0.12500


density Hg h std dev

h Q std dev Q Velocity std dev v Density of

H20 Pipe

Position Pipe

Position (kg/m^3)

(ft H20) (ft H20) (m^3/sec) (m^3/sec) (m/s) (m/s) kg/m^3 (m) (m)

13554.8237 19.47240 0.14213 0.00549823 0.00046975 2.71272777 0.23176397 993.399 0.0002032 0.000203213554.8237

19.47307 0.14214 0.00549833 0.00046975 2.71277419 0.23176794 993.365 0.0027432 0.002743213554.8237 19.47375 0.14214 0.00549842 0.00046976 2.71282199 0.23177202 993.33 0.0052832 0.005283213554.8237 19.47442 0.14215 0.00549852 0.00046977 2.71286841 0.23177599 993.296 0.0078232 0.007823213554.8237 19.47509 0.14215 0.00549861 0.00046978 2.71291485 0.23177996 993.262 0.0103632 0.010363213554.8237 19.47575 0.14216 0.00549871 0.00046979 2.71296128 0.23178392 993.228 0.0129032 0.012903213554.8237 19.47677 0.14217 0.00549885 0.00046980 2.71303230 0.23178999 993.176 0.0154432 0.015443213554.8237 19.47777 0.14217 0.00549899 0.00046981 2.71310196 0.23179594 993.125 0.0179832 0.017983213554.8237 19.47913 0.14218 0.00549918 0.00046983 2.71319621 0.23180400 993.056 0.0205232 0.020523213554.8237 19.47979 0.14219 0.00549928 0.00046983 2.71324266 0.23180796 993.022 0.0230632 0.023063213554.8237 19.48048 0.14219 0.00549937 0.00046984 2.71329048 0.23181205 992.987 0.0256032 0.025603213554.8237 19.48148 0.14220 0.00549951 0.00046986 2.71336016 0.23181800 992.936 0.0281432 0.028143213554.8237 19.48248 0.14221 0.00549966 0.00046987 2.71342985 0.23182396 992.885 0.0306832 0.030683213554.8237 19.48283 0.14221 0.00549971 0.00046987 2.71345444 0.23182606 992.867 0.0332232 0.033223213554.8237 19.48317 0.14221 0.00549975 0.00046988 2.71347767 0.23182804 992.85 0.0357632 0.035763213554.8237 19.48383 0.14222 0.00549985 0.00046988 2.71352413 0.23183201 992.816 0.0383032 0.038303213554.8237 19.48417 0.14222 0.00549989 0.00046989 2.71354737 0.23183400 992.799 0.0408432 0.040843213554.8237 19.48450 0.14222 0.00549994 0.00046989 2.71357060 0.23183598 992.782 0.0433832 0.0433832


y Fluid

Velocity std dev fluid

velocity LN(v/vo) d/dv d/dvo sd(v/vo) y/ro (m/s) (m/s)

0.0002032 2.952473539 0.171464282 -0.10598 0.304642 -0.27401 -0.005338444 -4.828310.0027432

3.037145041 0.171464282 -0.0777 0.304642 -0.28187 -0.003895373 -2.225620.0052832 3.089927184 0.171464282 -0.06047 0.304642 -0.28677 -0.003038774 -1.570220.0078232 3.179032872 0.171464282 -0.03204 0.304642 -0.29504 -0.001630468 -1.177660.0103632 3.222661943 0.171464282 -0.01841 0.304642 -0.29908 -0.000945958 -0.896490.0129032 3.256693108 0.171464282 -0.00791 0.304642 -0.30224 -0.000409893 -0.677270.0154432 3.271330005 0.171464282 -0.00342 0.304642 -0.3036 -0.000178166 -0.497580.0179832 3.282544745 0.171464282 0 0.304642 -0.30464 0 -0.345310.0205232 3.275820508 0.171464282 -0.00205 0.304642 -0.30402 -0.000106896 -0.213190.0230632 3.270206416 0.171464282 -0.00377 0.304642 -0.3035 -0.000195986 -0.096510.0251968 3.260076686 0.171464282 -0.00687 0.304642 -0.30256 -0.0003564 -0.008030.0226568 3.262330455 0.171464282 -0.00618 0.304642 -0.30277 -0.000320744 -0.114290.0201168 3.246521215 0.171464282 -0.01103 0.304642 -0.3013 -0.000570468 -0.233190.0175768 3.243123494 0.171464282 -0.01208 0.304642 -0.30098 -0.000624031 -0.368170.0150368 3.196326016 0.171464282 -0.02662 0.304642 -0.29664 -0.00135926 -0.524250.0124968 3.145329871 0.171464282 -0.0427 0.304642 -0.29191 -0.00215996 -0.709280.0099568 3.088737606 0.171464282 -0.06086 0.304642 -0.28666 -0.003057829 -0.936490.0074168 2.980964609 0.171464282 -0.09637 0.304642 -0.27665 -0.004840946 -1.231

APPENDIX C

Figure C.1 Moody graph of f as a function of NRe and e/D (Welty et. al. 188).

Courtesy of Anastasia Gribik and her G2

Figure C.2 Pump view of “Big Blue” Apparatus.

fanning friction factor

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