Fluid Flow in Pipes

Fanning Friction Factor and Velocity Profiles

Team #11

Greg Allen

Vishal Berry

Ameet Kulkani

Victor Lee

Luke Schwartz

March 2003

Abstract:

The characteristics of water flow in pipes were analyzed in the turbulent and laminar

regimes. Friction factor was calculated as a function of Reynolds number for pipes of various

diameter and composition. In a 0.125 ID brass pipe the function, f(NRe) = 22.8 / NRe was fit to

the experimentally collected data for laminar flow. For turbulent flow in a 1.025 ID copper

pipe f(NRe) = 0.16 / NRe0.20, was determined from the experimental data. For turbulent flow in a

2 ID Plexiglas pipe f(NRe) = 0.38 / NRe0.28, from experimental data. Two velocity profiles at

different turbulent flow rates were created from local velocity measurements in a 2 ID Plexiglas

tube. The integrated velocity profile of the bottom half of the pipe matched the volumetric flow

rate measured by an orifice plate within a 5% error. Power law fits of the velocity profile

provided n = 13.67 0.1 for NRe = 1.41 x 106 9 x 104 and n = 12.85 0.2 for NRe = 1.83 x 106

6 x 104. Additionally log-law correlations were made between dimensionless velocity, u+, and

dimensionless position, y+. For NRe = 1.41 x 106 9 x 104, A was 1.6 0.3 and B was 3.3. For

NRe = 1.83 x 106 6 x 104, A was 1.9 0.3 and B was 1.6

Comment: What is A & B?

ByAu += ++ ln... [1]

Table of Contents

INTRODUCTION 1

BACKGROUND AND THEORY 1

EXPERIMENTAL PROCEDURE 4

RESULTS 6

DISCUSSION 9

CONCLUSIONS 10

NOMENLATURE 11 REFRENCES 12

-1-

Introduction

A fundamental understanding of fluid flow is essential to almost every industry related

with chemical engineering. In the chemical and manufacturing industries, large flow networks

are necessary to achieve continuous transport of products and raw materials from different

processing units. This requires a detailed understanding of fluid flow in pipes, which is still

actively researched 1. In pipe flow substantial energy is lost due to frictional resistances; to study

this overall pressure drops, volumetric flow rates and local velocity profiles were determined.

Correlations were made between the fanning friction factor and the Reynolds number for both

laminar and turbulent flow in a variety of pipes.

Background and Theory

Flow in pipes is most commonly classified by the Reynolds number, a dimensionless

relationship between inertial and viscous forces defined as:

bReD vN

= (1)

were D is the inner diameter, vb the bulk velocity, and is the kinematic viscosity. When inertial

forces dominate, (NRe < 2100), flow is smooth as adjacent layers of fluid slide past each other 2.

Flow in this range is labeled as laminar. As Reynolds number increase (NRe > 4000) disturbance

eddies form and the flow becomes random and chaotic in nature. Flow in this range is labeled as

turbulent. The Reynolds number range between 2100 and 4000 represents transitional flow

between turbulent and laminar. An accurate description of transitional flow is beyond the scope

of this paper.

Frictional losses are characterized by the Fanning friction factor3:

22

22

( )2

s cv

H O b

P g Dfv L

= =

(2)

were s is the drag force per wetted surface unit area, (P) is the pressure drop across the pipe, gc

is the gravitational correction factor, is the fluid density, and L is the pipe length.

Comment: Its good to explain the physical meaning of Re number, but its better give the expression of initial force

Comment: This part can organize better.

-2-

The purpose of this analysis was to relate Reynolds number to Fanning friction factor for

a variety of pipes and flow conditions. For laminar flow, first principles can be used to develop a

relation between Reynolds number and friction factor. The Hagen-Poiseuille equation relates the

frictional pressure drop to fluid velocity, viscosity, and pipe dimension:

232( ) bf

c

v Lpg D

=

(3)

were ( ) fp is the pressure drop due to frictional losses, and is the fluid viscosity. Equating

the pressure drop due to friction in the Hagen-Poiseuille equation (3) with the overall pressure

drop across the pipe, and combining with the Fanning equation (2) results in this relation for

laminar flow 3:

16 16

b Re

fD v N

= =

(4)

A first principle analysis can not be used to develop a relation between friction factor and

Reynolds numbers for turbulent flow. A variety of 3-parameter empirical correlations exist for a

smooth pipe3:

+= nNbafRe

(5)

However, if the pipe has a characteristic roughness f will deviate from these empirical

correlations. As Reynolds number increases f approaches a constant value defined by the

relative roughness8.

For this experiment an orifice meter was used to measure volumetric flow rate, Q. An

orifice meter is a thin plate with a concentric hole cut in the center; more information is available

in Perrys Handbook4. The orifice meter manufacturer (Engineering Laboratory Design Inc.,

Lake City, MN) provided the following relationship to determine the volumetric flow rate:

1

2Q k h= (6)

where k is the orifice coefficient and h is pressure drop in feet of water across the plate.

Table 1: Theoretical Turbulent Flow correlations Equation a b n NRe range Blasius 0 0.079 0.25 4 x 103 < NRe < 105 Colburn 0 0.046 0.20 105 < NRe < 106

Koo 0.0014 0.125 0.32 4 x 103 < NRe < 3 x 106

Comment: For Turbulence

-3-

To determine the velocity profile pitot tube measurements were taken at a variety of

cross-sectional positions, focused on the boundary layer near the pipe wall. A pitot tube

provides a local velocity through an impact tube that faces the flow of water. The local velocity

can be related to the differential pressure using a dimensionless pitot tube coefficient, C.

tPCV = 2

(7)

C typically lies between 0.98 and 1 5. For a more thorough discussion of pitot tubes the reader is

referred to Perrys Handbook4.

The velocity profile of fully developed turbulent flow in a circular pipe can be

superficially devolved into three regions7; a viscous sublayer very near the pipe wall, an overlap

region, and an outer turbulent core through the center of the pipe. The viscous sublayer is

characterized by the dominance of a viscous shear stress to turbulent stress and thus the lack of

turbulent eddies. The turbulent core is characterized by the dominance of Reynolds stress and

thus highly random flow structures.

Empirical methods are essential to deriving velocity profiles in turbulent flow, as clean-

cut methods for calculating turbulent velocity profiles are not at our disposal7. Since the local

fluid velocity fluctuates with time it is necessary to use time-averaged equations of change.

From previous experimental work a commonly derived relationship is the power-law velocity

profile proposed by Prandtl6. n

xC

x

Rr

Vu /1

,

1

=

(8)

Here the time-smoothed local velocity, xu , and the average centerline velocity xCV , , are

related to the radial distance from the pipe wall and a constant n that is empirically derived and

depends on Reynolds Number. This matches results from dimensional analysis as velocity

depends solely on the radial coordinate in the cylindrical coordinate system8. The power law is

generally applicable over the entire range of the velocity profile, except at the centerline where

the derivative of the local velocity with respect to radial distance is non-zero. Additionally a

universal velocity distribution can be developed, the log-law. From Berliner et al. (2002)5, we

see that:

ByAu += ++ ln (9) where u+ is a dimensionless time averaged velocity, and y+ is the friction distance parameter .

-4-

( ) 5.02/fvuu x

>