ERKEL Daniel Laboratory Report Thermofluids1

7/29/2019 ERKEL Daniel Laboratory Report Thermofluids1

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DEPARTMENT OF MECHANICAL ENGINEERING

CODE AND TITLE OF COURSEWORKCourse code:

MECH2004:

Title:

Turbulent Flow in a Circular Cross-section Long Pipe;

1/nth Power Law

STUDENT NAME: ERKEL, DANIEL

DEGREE AND YEAR: EBF, 3rd YEAR

LAB GROUP: -

DATE OF LAB. SESSION: -

DATE COURSEWORK DUE FOR SUBMISSION: 30/11/2012

ACTUAL DATE OF SUBMISSION: 30/11/2012

LECTURERS NAME: Dr Pavlos Aleiferis

PERSONAL TUTORS NAME: DR KEVIN DRAKE

RECEIVED DATE AND INITIALS:

I confirm that this is all my own work (if submitted electronically, submission will be taken asconfirmation that this is your own work, and will also act as student signature)

Signed: Daniel Erkel


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Contents

1 Introduction 21.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.1 Laminar and Turbulent Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2.2 Reynolds Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2.3 Power Law and Other Approximations to Turbulent Flow . . . . . . . . . . . . 31.2.4 Friction Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2.5 Relative Roughness and Pipe Roughness . . . . . . . . . . . . . . . . . . . . . . 41.2.6 Viscous Sub-Layer in Turbulent Flows . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Methodology and Apparatus Used . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Results 52.1 Initial Recordings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 First Part of the Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 Second Part of the Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.3.1 Graph Plotted for u against y . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.3.2 Finding the Exponent of a Suitable Power Law Expression . . . . . . . . . . . 72.3.3 Plotting Various Graphs to Represent the Velocity Profile . . . . . . . . . . . . 72.3.4 Determining u/U and Comparing to Previous Results, Finding the Reynolds

Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3.5 Finding the Friction Factor and the Sub-Layer Thickness . . . . . . . . . . . . 9

3 Discussion 10

4 Conclusion 10

5 Appendices 12

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Mechanics of Fluids and Thermodynamics - Laboratory Report

Turbulent Flow in a Circular Cross-section Long Pipe; 1n

th Power Law

Daniel Erkel, 3rd year, EBF

Abstract

The laboratory experiment presented in the followings aimed at the demonstration of turbulentflows in pipes. By turning on a suction pump connected to a long horizontal pipe, air was drawn tothe pipe. Static and dynamic pressures were measure using two types of manometers: an inclinedand a Betz manometer. Through a set of calculations it was determined from the recorded datathat the flow is turbulent and the pipe is close to, but not exactly hydraulically smooth. Resultsobtained through various calculations were compared to formulae commonly found in textbooks

with good results achieved in each case, demonstrating the validity of these models.

1 Introduction

1.1 Objectives

Liquid is transported in closed conduits or pipes in numerous daily applications, from oil pipes towater pipes. The phenomenon is essential not only in man made but also in natural systems too, asthe blood vessels carrying in many animals are pipe flows as well. Hence the understanding of howfluids behave being transported in a pipe is crucial in many aspects. Based on Osborne Reynoldsfundamental experiments, laminar and turbulent flows are distinguished in pipe flows, terms to be

explained later in this report [1]. The experiment discussed in the present report aims at the betterunderstanding of the latter, through observing air flowing through a long pipe section. The air beingsucked through the pipe using a centrifugal pump attached to the outlet section its static and dynamicpressure were recorded at different stations using two types of manometers. From recorded resultscertain characteristics of the flow were observed and compared to theory.

1.2 Theory

1.2.1 Laminar and Turbulent Flow

Two types of flow were described by Reynolds in his famous dye experiments [1]: laminar and turbu-lent. Using the velocity of the fluid at which they occur to differentiate between these is not correct,

but instead a dimensionless number taking velocity, viscosity, density and dimensional characteristics,the Reynolds number (Re) should be used. Based on this, laminar flows in a pipe are those that areperceived at lower Re (below 2100-2300) and turbulent flows are those appearing at Re above approx-imately 4000 (the region between these is usually termed as transitional) [2]. Difference is presentbetween the prevalence of viscous forces in laminar flows and greater inertial forces and vibrationcaused by the turbulence in turbulent flows, all of which are captured by the Reynolds number, intro-duced later in this section [2]. In most applications, flows tend to be turbulent and their complexityresulting from random fluctuation makes accurate mathematical description and modelling difficult.Most efforts, even advanced ones, depend on experimentally obtained data [3] and are valid withcertain restrictions or errors. This report presents a few of these comparing them to experimentallyacquired results.

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1.2.2 Reynolds Number

Earlier it was mentioned that it was Osborne Reynolds (1842-1912) who, through countless experi-ments tried finding a measure to describe the behaviour of flow in closed conduits and express capturethe main characteristics of the fluid under certain conditions [1]. Based on his experiments, a non-dimensional number named after him, the Reynolds number was defined as (in one of its forms usingkinematic viscosity):

Re = ud

In another (perhaps more common form) it is given as:

Re =vd

This dimensionless number can be translated as a ratio between the following

|Intertia force|

|Net viscous force|

and its value, gives a good description of different types of flows relative to each other [3]. The numberwill be used later in the report.

1.2.3 Power Law and Other Approximations to Turbulent Flow

There are several different mathematical models and approximations to describe the velocity of flowwithin a pipe. Several are used in the report later, the one introduced here is the one appearing inthe title of the experiment, the power law. The power law attempts to recreate the velocity profile offlow in a pipe using an exponential relationship derived empirically, which is relatively easy to use:

u

U

= y

a

1

n

Where n in the exponential is dependent on the Reynolds number in the following distribution:

Figure 1: Relationship between n and the Reynolds number (from [1])

The power law profile is not valid next to the wall or at the centreline (in the first case it would giveinfinite velocity gradient, whereas in the second case it should give 0 and this condition is not met),however it well represents the velocity profile at other points as it will be seen later [1].

1.2.4 Friction Factor

Another term of great importance, relevant in this experiment, is the friction factor. The frictionfactor, f is a value characterising friction in the pipe, which depends on the relative roughness of the

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pipe and determines the losses arising due to friction and is also the ratio of wall stress to dynamicpressure:

f =p

L

a

u2=

w1

2U2

where

w =p

L

a

2

Two approximations to the value of friction factor are used in this study, one given by Blasius andone from Lees [3], the former being

fB = 0.0791Re0.25d

and the latterfL = 0.0018 + 0.153Re

0.35d

These are used in the discussion to evaluate experimental findings.

1.2.5 Relative Roughness and Pipe Roughness

A non-dimensional term, the relative roughness is defined to account for a parameter, roughness,

affecting fluid flow and generating losses in pipes. No pipe is perfectly smooth, small it may be, thereis always an unevenness of the walls. Roughness is denoted by k here and relative roughness is theratio of kd [1].

1.2.6 Viscous Sub-Layer in Turbulent Flows

In turbulent flows there is a region very close to the wall, where the viscous forces and shear aredominant, this realm is termed viscous sub-layer. The velocity given for this region is denoted by u+

or y+ and is understood to be between 0 < y+ < 5 8.

y+ =yU

where U is the shear velocity (equal to (w/)

1/2) and y is the distance from the wall. Based on this,and on the friction factors definition from before, the thickness of the viscous sub-layer is given as

Ld

=5

Red

2

f

When k < L, or in other words the roughness is smaller than the viscous sub-layer the flow becomeshydraulically smooth [2].

1.3 Methodology and Apparatus Used

In the experiment air was drawn through a long horizontal pipe (fixed at columns in the laboratory)using a suction pump attached to its outlet section. The pipe had 6 holes with manometers attachedto measure static pressure (using an inclined manometer) and a 7th with another device, the Betzmanometer attached to it. The latter enables the measurement of small changes in dynamic pressureand is connected to a pitot tube probing into the pipe parallel to the flow. Turning a knob the pitottube can traverse the pipes cross section by turning the knob and thus driving the tube further in.At the start of the experiment ambient pressure (using the inclined manometer) and temperaturewere measured. This was then followed by starting the suction pump and recording two readings foreach of the 6 holes, once the water in the manometer stopped oscillating. In the second part of theexperiment, with the pump turned on again, measurements were taken for 29 full turns of the knobmoving the pitot tube inside the pipe, with the first few readings taken at quarter turns. These resultsare presented in the next section. Illustrations showing the equipment are presented in the appendix.

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2 Results

2.1 Initial Recordings

The readings made preceding the main parts of the experiment are presented below

Table 1: Initial measurements

Initial recordings

Ambient Temperature (C) 20.5Ambient Temperature (K) 293.65Ambient pressure (mBar) 1016Ambient pressure (P a) 101600Position of vent sleeve closedInclined manometer angle () 45Initial static pressure (manometer inclined at 45 ) (cmH2O) 31.4Betz manometer zero error (mmH2O) 0Air specific gas constan(Jkg1K1)t 287

Air density (kgm

3) 1.205541a (inch) 1a (mm) 25.4

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2.2 First Part of the Experiment

In the first part of the experiment, static pressure was recorded for the 6 holes, to which the inclinedmanometer was connected. These readings are presented below:

Table 2: Readings taken in the first part of the experiment

Static pressure on the inclined manometer (cmH2O) Static pressure (cmH2O)Hole 1st reading 2nd reading Average value

1 32.2 32.2 32.2 0.56572 33.6 33.6 33.6 1.55563 34.6 34.6 34.6 2.26274 35.4 35.4 35.4 2.82845 36.4 36.4 36.4 3.53556 37.3 37.3 37.3 4.1719

The last column here shows the static pressure converted to a vertical value, with the initial manometer

readings for the pump switched off deducted.

2.3 Second Part of the Experiment

Readings taken in the second part of the experiment are presented in tables provided in the Appendix(placed there due to their length).

2.3.1 Graph Plotted for u against y

The first graph plotted for Table 3 is shown below:

Figure 2: Graph plotted for u against y

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From this, the maximum velocity is found to be U = 13.858ms1 occurring at y = 24.99mm.

2.3.2 Finding the Exponent of a Suitable Power Law Expression

Plotting results for y/a and u/U where U is found from the previous part (as U = 13.858ms1) on alogarithmic scale, the following results graph is obtained:

Figure 3: Graph plotted for y/a against u/U

The trendline fitted gives an exponent equal to n, as

ya

=

uU

n

Thus n = 7.669.

2.3.3 Plotting Various Graphs to Represent the Velocity Profile

Results from experimental values and others from theoretical formulae are superimposed on the samegraph for comparison:

The parabolic profile is given as:u

U = 1 a y

a2

2.3.4 Determining u/U and Comparing to Previous Results, Finding the Reynolds Num-ber

Plotting u

U

r

a

againstr

afor 0 r

a

1 gives the following graph, which can then be fitted with a trendline to perform anapproximate integration and find the area under the curve

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Figure 4: Graphs plotted for y/a against u/U obtained in different ways

Figure 5: Last graph plotted to obtain

u

/U

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The area shown in grey is for which an integration was performed. Using a sixth order polynomialtrendline, the curvature of the plot is almost perfectly captured. Integrating this function gives:

10

(7.6269x6 + 19.897x5 19.651x4 + 8.8163x3 1.9867x2 + 1.1655x 0.0031)dx = 0.417901

As the graph was obtained by plotting values ofuU multiplied by

ra , it has to be divided by the mean

value of the latter. Since it was integrated between 0 and 1, the mean is 0.5. Dividing the valueobtained from the integration by this gives

u

U=

0.417901

0.5= 0.835802

Comparing this to the value obtained from the power law using the expression (from [2]):

u

U=

2n2

(n + 1)(2n + 1)=

2 7.6692

(7.669 + 1)(2 7.669 + 1)= 0.8305

yields a difference of 0.64%, a very small error, which can result from the fact that in the approximateintegration method, limits of 1 and 0 were used instead of the actual limits, which would have been0.016 an 0.964. The Reynolds number can be obtained using

Re =ud

where = 1.511 105 (obtained through simple interpolation using the temperature measured at thestart and the table [4]), the diameter of the pipe is (twice the radius) d = 2 25.4 = 50.8mm and thevelocity is u = 0.835802 13.9 as obtained previously through the approximate integration. Thus theReynolds number is:

Re =ud

=

0.835802 13.9 50.8 103

1.511 105= 39058.67

Showing that the flow is turbulent.

2.3.5 Finding the Friction Factor and the Sub-Layer Thickness

The friction factor can be found using the pressure gradient, which can in turn be calculated as

p

L= 32.8175

where p is the pressure difference between holes 1 and 6 (values from the first measurement wereconverted to P a) and L is the distance between hole 1 and 6, equal to L = 10.81 0.03 = 10.78m.Using this and the equation defined previously for the friction factor:

f =p

L

a

u2= 32.8175

25.4 103

1.205540503 (0.835802 13.9)2= 0.005123

Comparing these to values obtained using formulae from Blasius and Lees:

fB = 0.0791Re0.25d = 0.0791 39058.67

0.25 = 0.00562661

and the latter

fL = 0.0018 + 0.153Re0.35d = 0.0018 + 0.153 39058.67

0.35 = 0.00558086

gives an error of 9% and 8% respectively, again very small. Finally, the sub-layer can be calculated as

L = d 5

Red

2

f= 50.8 103

5

39058.67

2

0.005123= 1.2849 104m

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The relative roughness of the pipe can be obtained using a Moodys chart [3], already having theReynolds number and f. Hence

k

d 0.028

hencek 0.028 50.8 103 = 1.4224 103m

Therefore as k > L the flow is not smooth hydraulically.

3 Discussion

The power law gave a good approximation to the velocity profile, with n = 7.668, calculated using thelog-graph on Figure 3. As observed on the figure with superimposed graphs (Figure 4, the velocityprofile is closer in shape to that obtained using the power law than to that using the parabolicprofile, mainly suitable for laminar flow [3], which suggests not only that the power law is a goodrepresentation for this flow but also that it is turbulent, as a model used for describing turbulent flowwas closer to the experimental results, than the one used for laminar flows. To obtain the mean valueof velocity for the flow, an approximate integration was used, integrating the area under the curve on

Figure 5, as this was very close (0.64% error) to the mean velocity obtained using the correspondingformula for the power law [2] it was again shown that the flow is turbulent and the power law is a goodapproximation to its behaviour. Further proof to the flow being turbulent was the found Reynoldsnumber, which was Re = 39058.67, far above the critical value given for pipes (2100-2300) [1]. Thetwo empirical formulae used in comparison with the friction factor calculated from the experimentalresults showed little difference (both being less than 10%) with that of Lees slightly closer to theexperimental. The best results would be obtained using the Haaland formula [3]. The roughness ofthe pipe compared to the sub-layer thickness shows that the pipe is very close, but not hydraulicallysmooth [1]. Errors in the calculations were present due to two main factors:

1. Human errors

2. Instrumental errors

Examples of the former included reading the manometers, or turning the control knob traversingthe pitot tube attached to the Betz manometer, whereas the latter meant errors resulting from pipeleakages in pipes connecting the manometers. Further error sources are the inherent errors in theaccuracy of the manometers or the pipe not having the same cross-section at every part. Smalldifferences experienced in calculations may result partly from this. With better manometers, and ascaled knob used to traverse the pitot tube (with clear markings for quarter and full turns) couldimprove the results.

4 ConclusionThe experiment well demonstrated certain characteristics of turbulent flows in pipe and proved thatempirical formulae used to approximate these can be reasonably accurate (at least for a flow with aReynolds number close to this, e.g. having the same order of magnitude). It was proven that theflow generated in the pipe by the suction pump is definitely turbulent and the pipe is close to, butnot exactly hydraulically smooth. The experiment was successful in all aspects, well demonstratingand proving the fundamental theory of turbulent pipe flows, the understanding of which is essentialin many engineering applications.

References

[1] B. Munson, D. Young, T. Okiishi, and W. Huebsch, Fundamentals of fluid mechanics. Wiley,2009.

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[2] R. Fox, A. McDonald, and P. Pritchard, Introduction to fluid mechanics. Wiley internationaledition, Wiley, 2004.

[3] B. Massey and J. Smith, Mechanics of Fluids. No. v. 1 in Mechanics of Series, Spon Press, 1998.

[4] G. Rogers and Y. Mayhew, Thermodynamic and Transport Properties of Fluids. Wiley, 1995.

List of Figures

1 Relationship between n and the Reynolds number (from [1]) . . . . . . . . . . . . . . . 32 Graph plotted for u against y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 Graph plotted for y/a against u/U . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 Graphs plotted for y/a against u/U obtained in different ways . . . . . . . . . . . . . 85 Last graph plotted to obtain u/U . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

List of Tables

1 Initial measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 Readings taken in the first part of the experiment . . . . . . . . . . . . . . . . . . . . 63 Second measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

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5 Appendices

Table 3: Second measurements

No. of turns y (mm) Betz reading (mmH2O) u (ms1)

0 0.93 5.2 9.1990.25 1.26 5.6 9.5470.5 1.59 5.9 9.799

0.75 1.93 6.2 10.0451 2.26 6.3 10.126

1.25 2.60 6.5 10.2851.5 2.93 6.7 10.442

1.75 3.26 6.8 10.5202 3.60 7.0 10.6743 4.94 7.5 11.0484 6.27 8.0 11.4105 7.61 8.5 11.7626 8.95 8.9 12.0357 10.28 9.3 12.3038 11.62 9.6 12.5009 12.96 10.0 12.757

10 14.29 10.3 12.94711 15.63 10.6 13.13412 16.97 10.9 13.31913 18.30 11.1 13.44114 19.64 11.3 13.56115 20.98 11.5 13.68116 22.31 11.6 13.74017 23.65 11.7 13.79918 24.99 11.8 13.85819 26.33 11.7 13.79920 27.66 11.6 13.74021 29.00 11.6 13.74022 30.34 11.4 13.62123 31.67 11.2 13.50124 33.01 11.0 13.38025 34.35 10.7 13.19626 35.68 10.4 13.010

27 37.02 10.1 12.82128 38.36 9.8 12.62929 39.69 9.5 12.434

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ERKEL Daniel Laboratory Report Thermofluids1