TURBULENT FLOW IN CONCENTRIC

AND ECCENTRIC ANNULI

by

JOHN DOUGLAS DENTON

B.A. Cantab. 1961

A THESIS SUBMITTED IN PARTIAL FULFILMENT OF

THE REQUIREMENTS FOR THE DEGREE OF

MASTER OF APPLIED SCIENCE

in the Department

of

Mechanical Engineering

We accept this thesis as conforming to the

required standard

THE UNIVERSITY OF BRITISH COLUMBIA

MAY, 1963

In presenting this thesis in partial fulfilment of the

requirements for an advanced degree at the University of British

Columbia, I agree that the Library shall make i t freely available

for referenceand study. I further agree that permission for ex­

tensive copying of this thesis for scholarly purposes may be granted

by the Head of my Department or by his representatives. It is

understood that copying or publications of this thesis for financial

gain shall not be allowed without my written permission.

Department of Mechanical Engineering,

The University of British Columbia,

Vancouver 8, Canada.

May, 1963.

i i

ABSTRACT

The turbulent flow of air through the annular gap between

two tubes was studied experimentally, both with the tubes concentric

and with the inner tube at eccentricities of 50% and 1007..

Air velocities were measured using small traversable impact

tubes. The shear stresses on the boundaries were studied both by

measuring the pressure gradient and:by means of a calibrated shear

probe attached to the inner tube. For a l l three annuli complete

nondimensional velocity profiles were obtained at Reynolds numbers

around 55,000 and the variation of average f r i c t i o n factor with

Reynolds number was studied in the Reynolds number range 20,000 -

55,000. The variation of local shear stress aKound the surface of

the inner tube was obtained for the eccentric annuli.

The results for the concentric annulus agree well with

previous investigations. For the eccentric annuli the results are

compared qualitatively with Deissler and Taylor's semi-theoretical

investigation. The agreement is not good and this is thought 'to

show that the Deissler-Taylor method is not applicable to annuli. It

is concluded that the study of velocity profiles in non-symmetrical

ducts is of l i t t l e help in obtaining quantitative heat transfer data.

I i i

TABLE OF CONTENTS

PAGE

Chapter _1

1.1. Introduction 1

1.2. Basic Theory 2

1.3. Previous Work on Flow i n Annular Ducts 6

1.4. Entrance E f f e c t s i n Annuli 9

1.5. Measurements i n the Laminar Sublayer 9

Chapter II

II.1. Apparatus 13

II.2.. Preliminary C a l i b r a t i o n s 17

Chapter III

111.1. Experimental Technique 30

111.2. Results and Discussion of Results 35

Chapter IV

IV.1. Summary of Results 62

IV.2. Conclusion 63

APPENDIX Ij_

Dimensional Analysis of the Shear Probe 66

APPENDIX I I .

The Laminar Flow Solution i n an 68 Eccentric Annulus

BIBLIOGRAPHY 73

LIST OF FIGURES

FIG PAGE

1 Schematic. Layout of Flow System .20

2 Details of--Annulus. and Extension Tube 21

3 Details of Boundary Layer Probe 22

-4 Details of Inner Probe 23

5 Details of Shear Probe 24

6 Photograph of Entire Assembly 25

7 Photograph of Probes 26

8 Fan Calibration Curve 27

9 Calibration Curve of Boundary Layer Probe 28

10 Calibration Curve of Shear Probe 29

11 Static Pressure Gradients in the Annuli 42

12 Velocity Profile in the Concentric Annulus 43 Re = 53,000

13 Velocity Profile in the Concentric Annulus 44

Re -= 38,000

14 Laminar Flow Profile in the Concentric Annulus 45

.15 Concentric Annular Data Plotted in V +vs Y + 46

co-ordinates

.16 Profile in the 50% Eccentric Annulus at 0 = 0 47

17 Profile in the 50% Eccentric Annulus at 9= 180° 48

18 Profiles Perpendicular to the Inner Wall in the 49 50% Eccentric Annulus

1.9 Profiles Perpendicular to the Outer Wall in the 50

50% Eccentric Annulus

20 Velocity Contours in the 50% Eccentric Annulus 51

.21 Laminar Flow Velocity Contours in the 50% 52 .Eccentric Annulus

FIG PAGE

22 Profile in the 100% Eccentric Annulus at9 = 0 53

23 Profiles Perpendicular to the Inner Wall of the 54 100% Eccentric Annulus

24 . Profiles Perpendicular to the Outer Wall of the 55

100% Eccentric Annulus

25 Velocity Contours in the 100% Eccentric Annulus 56

26 Friction Factors in the 50% Eccentric Annulus . 57

27 Friction Factors in the 50% Eccentric Annulus 58

28 Friction Factors in the 100% Eccentric Annulus 59

29 Shear Stresses on the Inner Wall of the 50% 60 Eccentric Annulus

30 Shear Stresses on the Inner Wall of the 100% 61

Eccentric Annulus

31 Idealized Cross Section of Shear Probe 66

32 Geometry of an Eccentric Annulus 68

VI

LIST OF SYMBOLS

SYMBOL UNITS

p Density l b / f t 3

jX Dynamic viscosity lb/f t sec

D Diameter ft

\l Velocity ft/sec

T Radius ft

U Distance from a boundary ft \ j Angle measured from plane of eccentricity degrees

De Equivalent diameter, t^Dj.— D|) ft

Re Reynolds number, pVt-i De/yCA

U Average f r i c t i o n factor,

Shear stress l b / f t sec

HTW Shear stress acting on a wall l b / f t sec

tftv& Average shear stress over perimeter lb / f t sec

V+ Dimensionless wall distance, ^ /^Yw p/jJ.

\/ + Dimensionless velocity,

R Equivalent radius, ft

V Equivalent wall distance, ~TL /MAX j — R ft

V~LM Eddy viscosity l b / f t sec

EL Eccentricity parameter /core displacementNxlOO Percent

P Static pressure lb / f t sec

P Total pressure lb / f t sec 2

P Pressure gradient along annulus l b / f t 2 sec 2

A; B Constants

SYMBOL UNITS

Q_ Fan discharge ft 3/sec

|-j Fan pressure head ft of a i r

S U B S C R I P T S

SUBSCRIPT . MEANING

1 Value at inner wall of annulus

2 Value at outer wall of annulus

m Mean or average value

max Value at position of maximum velocity

1

CHAPTER I

1.1 INTRODUCTION

Flow of fluids in non-circular ducts is a phenomenon often

encountered in engineering practice. Heat exchangers in particular

are often based on annular or rectangular passages, or on tube bundles

enclosed in cylindrical, shells. An example of such a flow system may

be taken from the CANDU nuclear reactor. This is a heavy water cooled

and moderated reactor in which the coolant flows longitudinally through

a cylindrical tube of 3.25 inches inside diameter in which are arranged

nineteen cylindrical fuel element containers each of 0.6 inches outside

diameter.

The power used to.pump the coolant in a nuclear power plant

may,be an appreciable proportion of the power generated. Thus the

system should be designed to obtain the maximum heat transfer per

unit "of pumping power. In order to do this the-average heat, transfer

coefficient and the average f r i c t i o n factor, in the complex duct

described above, must be accurately known. As high a coolant exit

temperature as-possible is desireable, but local boiling of the coolant

must be prevented as this leads to.high surface temperatures of the

fuel elements. The local surface temperatures which control the

incidence of boiling are dependent, on the local heat transfer coeff i c i ­

ents, so.that these must be known in addition to the average value. ..

Thus the example shows that the important properties of

the flow include not only the average values of the.heat transfer

coefficient, and f r i c t i o n factor, but also the local values at any

point in the duct. These local properties are much more d i f f i c u l t to

2 obtain than are the average values.

In general the f r i c t i o n a l losses and heat transfer

coefficients in a duct are uniquely determined by the velocity

distribution and by the thermal boundary conditions. The latter are

usually known, and so an exact knowledge of the velocity profile

should suffice to determine a l l the quantities of interest. In

practice the velocity distribution is seldom known in sufficient

detail to enable accurate calculations to be made, and the heat

transfer coefficients and:.friction factors are obtained.by experiment.

In circular tubes the relationship between flu i d flow.and

heat transfer has been extensively studied using analogies between

the transfer of heat and momentum. The same methods are applicable to

concentric annuli, but. in practice the velocity profiles available -are

not sufficiently detailed to allow accurate calculations to be made.

In ducts not possessing cylindrical symmetry the problem is complicated

by the fact that lines of heat flux and of momentum flux need no

longer coincide and no reliable method is known of relating the heat

transfer to the velocity profile in such ducts. .This is the case in

eccentric annuli.

In this investigation a study is made,of flu i d flow in

concentric and eccentric annuli. The work is part of a joint project

for the study of fl u i d flow and heat transfer in annuli, and of the

relationship between the two.in non-symmetrical ducts.

1.2. BASIC THEORY

The most important parameter of the flow in a duct is the

Reynolds number. Until recently there was disagreement as to the best

3

means of defining the characteristic length dimensions in an annulus, on

which the Reynolds number is based. This arose largely,as a result of

heat transfer experiments in which the heat transfer took place at only

one boundary. For use in flow measurements there is l i t t l e doubt that

i t is preferable to use the equivalent diameter defined by,

De = 4 x Cross Sectional Area Wetted Perimeter

On this basis the equivalent diameter of an annulus is given by,

De - ( D x - D . ) Using this dimension the Reynolds number is,

R s = P ^ D , ( 1 )

In concentric annuli i t is found that the flow is turbulent for values

of Re>2,000.

Turbulent shear flow is at present too complex.a phenomenon

even in circular tubes, to be amenable to mathematical analysis. A

semi-empirical approach is usually adopted.

For two dimensional laminar flow in the x - y plane, with

the velocity V in the X direction being a function o f ^ only, the

viscosity/^, is defined by,

Where is the shear stress acting in the fluid on a plane normal to the

ax i s . By.analogy an eddy viscosity E^is defined for turbulent flow

x dV . (2)

1 by,

T " — ( 1 +- Ew,) "JT^

The value of E M i s in general a function of both V and of l^..

Flow in a duct is said to be established when the pressure

gradient is constant and the velocity profile is independent of the

axial position. When established flow exists in tubes and between

parallel plates i t is possible to calculate the local shear stresses

from the geometry and from the pressure gradient. In addition i t is

possible to predict on dimensional grounds (Ref. [7] )* that Ew>/p is

proportional to u ., the distance from the wall, for points close to the

walls, and proportional to^ - g ^ J / ^-j J^J in the region remote from the

walls.

Using these facts equation (2) may be integrated to obtain an

expression for V in terms of pji^TwCthe shear stress at the wall),

and unknown constants. This expression together with experimental

values of the constants, forms the well known 'universal velocity

l i n t e n s i o n -p r o f i l e 1 relating the dimensionless velocity to the di

less distance ^x/Tw py^U- The relationship is plotted in Fig. 15. It

is found that the 'universal velocity p r o f i l e 1 as derived above is

applicable-to both flow in tubes and flow between parallel plates with

the same values of the experimental constants. There is no theoretical

justification for i t s application to other types of ducts.

For values of V less than 5 the 'universal velocity profile'

reduces to the curve V = Y , or, returning to more usual variables, V—

Thus very close to the walls there seems to exist a region

of laminar flow where no turbulent shear stresses exist. This region

is known as the laminar sublayer and is usually only a few thousandths

of an inch thick.

In a concentric annulus'the wall shear stresses can be

obtained from the pressure gradient i f the radius of maximum velocity

^ M A X is known. From equation (2) i t can be seen that this is also

* Numbers in square brackets refer to the Bibliography

irly for the outer wall,

the radius at which no shear stress exists in the f l u i d , A force

balance can be applied to the region between this radius and the inner

wall to give, / x x\

I. = %\ V, / (3a)

Similarly for the outer wall,

(3b)

• Thus i f Y~nt\% is known, the shear stress at any point in a

concentric annulus can be found in terms of the pressure gradient.

In an eccentric annulus, or in any.other duct not possessing

cylindrical symmetry, the shear stress distribution cannot.-be

determined un t i l the complete velocity profile is.known. From the

velocity profile i t is possible to construct velocity gradient lines

which are everywhere orthogonal to the lines of constant velocity.

Such lines are shown in Fig. 21. The velocity gradient is zero in

directions normal to these lines and i t follows from equation (2),that

they are lines of zero shear stress. By a force balance over the area

between adjacent velocity, gradient lines the local shear stress at the

walls may be obtained in terms of the pressure gradient.

Shear stresses at a boundary, in fl u i d flow are usually

presented non-dimensionally as f r i c t i o n factors defined by,

f -where V is a characteristic-^velocity, which in a duct is usually

the average or bulk velocity.of the f l u i d .

In annuli the average f r i c t i o n factor based on the

average shear stress over the perimeter, is usually used. This is given by,

p v w n r — <4>

Local f r i c t i o n factors based on the local shear stress may be defined

similarly. Dimensional analysis shows that the fr i c t i o n factor in an

annulus is a function of Reynolds number, diameter ratio, and surface

roughness.

1.3. PREVIOUS WORK ON FLOW IN ANNULAR DUCTS

Most of the experimental work on flow in annuli has been

directed towards heat transfer measurements. However i t is easy to

measure average f r i c t i o n factors at the same time and a large amount

of data on these was produced as a sideline. These data are extremely

scattered and.no general correlation was made until Davis [ll] in 1943

reviewed a l l the availablje data and obtained the equation,

D " & D , ' )"°"' ~ O . 0 5 5 R e ' *

as the best f i t to the data for smooth concentric annuli. This is

shown plotted in Fig. 26 for D 2 / D 1 - 3.0.

The velocity distribution for laminar flow in a concentric

annulus was obtained, theoretically by. Lamb [21J , who showed that.,

. ' ^ '= ( R\L^/<-,) ( 5 B )

. The velocity prof i l e so obtained in an annulus with "Tx/'fi = 3.0 is

shown plotted non-dimensionally in Fig.14.

The velocity profile in turbulent flow in an annulus was

f i r s t investigated by Rothfus jjQ in 1948. He obtained velocity

profiles for turbulent flow of air through two concentric annuli of

radius ratio 6.17 and 1.54. In each case, he found that the radius of

maximum velocity, was very closely the same as that obtained from

equation (5b) for laminar flow. Using this fact he washable to obtain

the shear stresses and f r i c t i o n factors at both inner and outer walls

from equations, (3a), (3b) and (4).

Knudsen and Katz 5 ., \$\ measured velocity, prof iles for

water flowing in a concentric annulus of radius ratio 3.60. Their

results agreed with Rothfus' on the position of maximum velocity. The

same workers tried several methods of correlating their own and

Rothfus' data on velocity profiles, including plots as in the

'universal velocity p r o f i l e 1 . None-of the methods tried gave a

satisfactory correlation of a l l the profiles available. In particular

they showed that the unmodified 'universal velocity, profile' is not

applicable to annuli. Knudsen and Katz |jQ also tried to use their

profiles to calculate heat transfer coefficients, but found that the

profiles were not sufficiently detailed in the region of the walls.

Rothfus, et aL Qf] observed that the laminar flow solution

in a concentric annulus (equation (.5)) is parabolic with respect to

the group R , where, R * — (C^ -(\***) — .T^ + + X<»** to-<^

The laminar flow solution in tubes and between parallel

plates is parabolic with respect to the wall distance,U . Thus they

defined ah equivalent wall distance for annuli, V =. \ -C™**]—R, and an

equivalent wall shear stress,LL . The equivalent shear stress may

be shown to equal'Yx at both inner and outer walls. They, observed

that equal values of V on the inner and outer halves of the profile

gave equalvalues of velocity, in turbulent as well as in laminar flow.

It follows that the dimensionless parameters

V / f V p andY/STp/ w i l l f i t the laminar profile in a concentric annulus of any radius

8

ratio, to the 'universal velocity •profile',.and w i l l also give a single

curve for the inner and outer portions of the profile in turbulent, flow.

Rothfus' results plotted in this way.indicated that the curve

obtained in turbulent flow agreed closely with the 'universal velocity

p r o f i l e 1 , for any radius ratio.

Barrow J8] obtained velocity profiles for air flowing in an

annulus of radius ratio 2.25 and.found that his results agreed f a i r l y

well with the above correlation.

The above three investigations seem to constitute the only

previous experimental work on velocity profiles in concentric annuli.

In eccentric annuli no experimental velocity profiles are available,

and the results of Stein et a l . {26J do l i t t l e more than show that the

f r i c t i o n factor is lowered by eccentricity.

On the theoretical side Deissler and Taylor £7} applied

their method of obtaining velocity.profiles in ducts, to an annulus

of radius ratio 3.5:1 at various eccentricities. The Deissler-Taylor

method consists in assuming the 'universal velocity profile' to hold

along perpendiculars to the walls of any duct. By applying this to an

eccentric annulus Deissler and'Taylor obtained a line on which the same

velocity resulted from applying the profile to either wall. They

called this a line of zero shear stress and proceeded by an iterative

process to construct the velocity profile. They also obtained average

fri c t i o n factors,,local shear stresses and heat transfer coefficients.

However the analysis must be considered untrustworthy as Knudsen and

Katz [5] had shown that the unmodified 'universal velocity profile'

does not apply even to concentric annuli.

9 1.4. ENTRANCE EFFECTS IN ANNULI

As has been previously mentioned,.flow in a duct is said to

be f u l l y established when the static pressure gradient is constant and

the velocity profile is independent of axial position. Near the entrance

to a duct these conditions are not f u l f i l l e d and a certain entrance

length is needed for the flow to become established.

The entrance lengths for pressure gradients and for velocity

profiles are not in general the same. In circular tubes the pressure

gradient becomes constant within a few diameters of the entrance but

the velocity profile requires approximately-fifty diameters, the exact

values being dependent on the Reynolds number. Rothfus .et a l . [u]

found that in a concentric annulus the pressure gradient was not quite

constant after as many as 250 diameters. No data are available on the

development of velocity profiles in annuli. However, by comparison

with tube flow i t seems unlikely that the velocity profile would

become established before the pressure gradient.

Rothfus took velocity profiles at 93 and 223 equivalent

diameterscdownstream from the inlet. Knudsen and Katz allowed 63

equivalent diameters. In the present experiment only 33 equivalent

diameters were allowed for the flow to stabilize. It therefore seems

unlikely that the velocity profile was f u l l y developed. However large

entrance lengths are seldom used in practical applications and the

results are no less useful than i f they were for established flow.

.1.5. MEASUREMENTS IN THE LAMINAR SUBLAYER

Considering i t s importance in theories of fluid f r i c t i o n

and heat transfer, l i t t l e work has been done to investigate the laminar

10

sublayer. In fact Miller [17] considered that its very existence had

not been conclusively demonstrated as recently as 1949.

The layer is usually only a few thousandths of an inch thick

and in i t the velocity increases linearly with distance from the wall,

the fl u i d in contact with the wall being assumed at rest. The shear

stress acting on the boundary and in the sublayer is given by, I —jJ^~j~~

Stanton [li l was the f i r s t to investigate this region.

Using a pitot tube of approximately 0.004 inch thickness, he was able

to measure velocities to within 0.002 inch of a wall. The velocity

profiles he obtained in this way did not extrapolate to zero velocity

at the wall and did not link up with the straight "lines V — /j^-

calculated from the wall shear stress assuming a laminar sublayer to

exist. He continued his observations using a very small probe, one

boundary of which was formed by the tube wall. In this way he was

able to obtain openings of the..order of 0.001 inch. The profiles

obtained on the assumption that the probe measured the velocity at i t s

geometric centre were incompatible with the existence of a laminar

sublayer.

Stanton decided that his results could only be explained by

a displacement effect, whereby,a pitot tube near a boundary picked up

the velocity at a point appreciably.further from the wall than its

geometric-centre, in cases even outside the area covered by the probe

mouth. He verified this effect by installing his probes in a circular

tube through which air was passing in laminar flow. The velocity

distribution near the wall could be calculated theoretically and by

comparing i t with the observed profile he obtained a graph of 'probe

opening' vs 'effective distance from the wall'. Using this calibration

he replotted his previous results and obtained good agreement with the

laminar sublayer hypothesis.

Stanton's results awakened interest in the behaviour of pitot

tubes at low Reynolds numbers. Barker |l4] showed that for Reynolds

-numbers less than 30 the pressure picked up by a pitot tube was

approximately given by, P^="^pV -t^i where "C is the radius of the

pitot tube. The second term represents a viscosity effect which is

negligible at high Reynolds numbers, but which may be the dominant

term for slow flow of viscous fluids.

G. I. Taylor |l3] considered probes of the type used by

Stanton in which one wall is formed by a boundary of the f l u i d . He

showed by dimensional analysis that i f such a probe is placed in a

region of constant velocity gradient, at very low Reynolds numbers so

that the dynamic pressure term, "-JlpV ., is negligible, i t w i l l pick up

•up a pressure directly proportional t o t h e ' S h e a r stress on the wall.

He performed experiments to show that the constant of proportionality

was closely 1.2. This pressure proportional to the viscous shear

stress is the cause of the displacement effect observed by Stanton.

Stanton had assumed for convenience that the magnitude of

the displacement effect was independent of Reynolds number. Fage and

Falkner [l6] using similar probes "obtained curves of 'displacement

effect' vs 'velocity picked up', and successfully used the probes and

the laminar sublayer hypothesis to measure the skin f r i c t i o n on a n

aerofoil. Rothfus \l] used similar probes in a concentric annulus but

did not allow for the variation of the displacement effect with

Reynolds number and was unable to use his profiles near the walls to

calculate shear stresses.

12

The probes used in the present experiment are similar to thos

used by Fage and Falkner. The ..method of calibration is d i f f e r e n t . N o

attempt was made to determine the exact size of the probes,or to use

them to measure velocities. Instead dimensional analysis was used to

calibrate the probes so that shear stress could be measured directly

from them ( see Appendix I).

CHAPTER II

II. I APPARATUS

The arrangement of apparatus used in this investigation is

shown schematically in Fig. 1 and photographically in Fig. 6.

Atmospheric air is supplied to the mixing box by a 1/3 Hp.

centrifugal fan. An extension to the outer tube.of the annulus f i t s

into the mixing box through an airtight rubber ring seal. The inner,

or core, tube of the annulus passes through a glass fibre flow screen

into the test section of the duct. The flow is unobstructed for 74

inches, or 37 equivalent diameters, after which the duct discharges

to atmosphere. The core tube extends beyond the outer tube and is

supported externally, so that there is no disturbance to flow caused

by.supports in-the test section.

More details of the annulus and the extension tube are shown

in Fig. 2. The outer tube is made of clear plastic with inside

diameter 3 inches and outside diameter 3% inches and has flanges at

each end. The outer tube extensions are of the same material and have

a flange at one end. 'Three different extensions were used, one for

each eccentricity, the flow screen and attached sleeves being glued

to each at the correct eccentricity. Each extension tube contained a

support tube of 1 inch inside diameter through which the core tube

could slide. The support tubes were traversable within the extension

tubes to allow exact setting of the eccentricity. The holes in .the

flanges of the extension tubes and main outer tube were carefully

aligned so that when bolted together the two tubes were always

concentric.

14

The main outer tube was supported by two semicircular bearings

made from clear plastic tubing of 3% inches inside diameter. Thus the

main outer tube could be rotated about its axis without lateral

displacement of the axis.

The inner tube of the annulus was a 1 inch outside diameter

by 7/8 inch inside diameter aluminium tube 9 feet long. She outside

diameter was 1.000 $ 0.0005 inches as measured by micrometer at random

points along the length. It was straightened so that its axis deviated

from a straight line by a maximum of 0.005 inches and weights were

added to the overhanging tube ends to minimise sag due to self weight.

The sag was thereby reduced to a calculated maximum of 0.0035 inches,

so that the maximum possible deviation of the tube axis from linearity

was 0.0085 inches, less than 1% of the mean annular gap width. The

core was supported from the extension tube as previously described

and also externally in a semicircular wooden bearing which was lined

with paper and glued rigidly.to the frame of the apparatus. Different

supports were used for each eccentricity. Thus the annulus core could

be rotated freely about its axis and could also slide parallel to its

axis. The rotation of the core was measured by attaching a pointer

to i t and allowing the pointer to move over a protractor attached to

the external support in such a way that i t was concentric with the

core tube.

The eccentricity was set at the required values using

accurately machined templates. Two of these were made for each

eccentricity. -The - templates :St1ld.;Oiiercttoe.-;-tq0e-vt.ube and fitted inside

the Inner tube, thus supporting the core in exact position whilst •'••=

the supports were adjusted to hold the core in the same position.

15 In this way i t was estimated that the core could be placed within

0.005 inches of the required position.

Static pressure tappings were located on the surface of the

«pre at distances of; 1,24,43,57 and 66 inches beyond the flow screen.

The tappings were made by passing lengths of 0.075 inch diameter

polyethylene tubing through holes d r i l l e d in the c*re. The tubes were

glued in position with epoxy glue and then cut off.flush with the

surface. The area was fi n a l l y polished with fine emery cloth. The

other ends of the plastic tubes were led out of the ends of the core

tube. A static pressure tapping was also taken from the mixing box.

Velocity measurements were made with impact probes located

66 inches downstream from the flow screen. Three impact probes were

used, two attached to the core and one to the outer tube. Details of

these probes and their traversing mechanisms are shown in Fig. 2,

Fig. 3, Fig. 4 and Fig.7. The tips of both probes used for traversing

the mainstream were made of hypodermic tubing with external diameter

0.016 inch. The one attached to the outer tube, which w i l l in fujtftire

be referred to as the outer probe, was traversed by a micrometer head

and could be positioned to within 0.001 inch at distances of up to 0.85

inch from the outer wall. The other mainstream probe w i l l be referred

to as the inner probe. This was traversed by.a 40 threads per inch

screw and its distance from the inner wall measured directly by

micrometer to 0.001 inch. Two heads were used on this inner probe,

one at distances of less than 0.75 inch from the inner wall and the

other for distances of 0.75 inch to 1.25 inch from the wall.

The third probe was used to measure velocities very close to

the core surface. Its construction is shown in Fig. 3 and Fig. 7.

16 It was traversed by a 40 threads per inch screw with a small pointer ,

attached to the screw head. This pointer moved over a scale formed by

scratching lines on the core surface at intervals of 22% degrees

around the axis of the screw. The probe could therefore be traversed

inwards in steps of 0.0016 inches until i t touched the surface, at

which point i t s centre was 0.003 inch from the surface. To move the

probe outwards the screw was slackened and the probe pushed out.

Traverses could only be made with the probe being moved continually

towards the surface.

The shear probe was also located 66 inches downstream from

the flow screen. Its construction is shown i n Fig. .5. A static

pressure tapping was made as previously described. -A strip .of 0.0015

inch thick feeler gauge material 0.040 inch wide was laid over the

tapping parallel to the axis of the core tube, with one end just

covering the hole of the tapping. A strip of aluminium f o i l 0.0007

inch thick was then laid over the tapping and the spacer and glued

to the core tube surface by a thin coating of epoxy glue. The f o i l

was pressed down so that i t was glued to the tube everywhere except

where the latter was covered by the spacer. The spacer was then

carefully.removed and excess f o i l trimmed away when the glue set.

This l e f t the pressure tapping connected to the annular gap by an

approximately rectangular passage 0.0015 inches deep, 0.040 inches

wide and approximately.0.10 inch in length. The entire probe projected

only.about 0.0025 inches from the surface and was estimated to be

completely within the laminar sublayer at -all'.but the highest flow

rates.

Four micromanometers were used to measure the pressures

17 picked up by the static pressure taps and by the probes. Three of these

were variable slope Lambrecht micromanometers containing fl u i d of

density 0.800 and having a scale graduated in millimeters. When set

at a slope of 25:1 these could be read to 0.5 millimeters thus allowing

pressures to be measured to approximately.0.0007 inch of water. The

other manometer used was an E.V. H i l l type 'C' micromanometer reading

directly in inches of water with an accuracy of 0.001 inches.

Air temperature was measured by a mercury in glass.A.S.T.M.

precision test thermometer which was inserted through a hole in the

wall of the mixing box. It was -observed that the air temperature

at outlet from the annulus never differed appreciably from that in

the mixing box and so:the thermometer was assumed to give the air

temperature in the annulus.

Atmospheric pressure was obtained from an aneroid barometer

located in the same room as the apparatus.

II.2 PRELIMINARY CALIBRATIONS

Fan Calibration

The air flow rate at f u l l flow was obtained from the 'head'

vs 'flow' curve for the fan. The fan speed was found to be very

closely constant and was therefore not brought into the calibtation.

The cal ib r a t i am. curve was obtained by replacing the annulus by a 100

inch length of 3% inch inside diameter plastic tubing and discharging

to atmosphere through an A.S.M.E. long radius flow nozzle of diameter

ratio 0.695.

The pressure drop across the nozzle was obtained from a static

pressure tapping one diameter upstream of the nozzle. The pressure

head and air temperature in the mixing box were also measured,together

•18 with atmospheric pressure. Discharge coefficients and properties of air

were obtained from A.S.M.E. Power Test Codes, Chapter 4, Part 5 .2} .

The flow was varied by using different combinations of flow

screens at exit from the mixing box. For each combination the fan

head, H , expressed as feet of air at. its density in the mixing box,

and the volumetric flow rate,Q ., in cubic feet per second, also at

mixing box conditions, were calculated. The resulting curve is shown

in Fig. 8 which applies to the fan with its inlet unrestricted.

Manometer Comparisons

The manometers used were checked by connecting them to the

same pressure source. A l l the Lambrecht manometers were found to give

the same reading within experimental accuracy (T 0.5 m.m.) and the

E. V. H i l l type 'C' manometer also agreed with the result obtained by

converting the Lambrecht readings to inches of water. Accordingly a l l

manometers were assumed to read accurately.

Impact Probes

The impact probes were compared with a larger manufactured

pitot static tube by using them to measure velocities at the exit

of the nozzle previously described. The velocity profile at. the

nozzle exit is very f l a t and the probes could be spaced about \ inch

apart to reduce interference,with negligible variation of the velocity

at the probe mouth.

It was found that the two mainstream probes picked up the same

total head as the manufactured probe and they were.:therefore assumed

to measure the velocity head exactly. The boundary layer probe picked

up an appreciably lower pressure and a calibration curve of, 'indicated

velocity head' vs 'true velocity head' (as recorded by the other

probes), was obtained for i t . When installing this probe i t was found

necessary to give i t a slight inclination to the wall,so that when

traversed towards the wall the lower l i p of the opening touched the

wall f i r s t . This seemed to change the calibration of the probe,as the

profiles obtained with i t did not join smoothly with the profiles

obtained from the other probes. Accordingly the boundary layer probe

was recalibrated in situ by placing i t at the radius of maximum

velocity in a concentric annulus and comparing i t with the outer probe

placed at the same radius. The resulting calibration curve is shown

in Fig. 9.

The Shear Probe Calibration

As mentioned in Section 1.2 the shear stress on the' inner walls

of a concentric annulus can be calculated from equation (3). For an

annulus of radius ration 3:1, with I^MAX assumed to be the same as

in laminar flow, this equation reduces to, T, =0.&^,9 tlP.This

relationship was used to calibrate the shear probe.

The annulus was set up with the core concentric and the flow

varied by restricting the fan inlet. The pressure picked up by the

shear probe was very closely independent of its angular position. For

each flow rate the pressure gradient, air temperature and pressure,and

shear probe head,.were recorded. The groups obtained in Appendix I

were calculated and are plotted in Fig. 10.

FIG. I SCHEMATIC L A Y O U T O F FLOW S Y S T E M

M I C R O M E T E R H E R D

PLftSTIC B L O C K

S U P P O R T

T U B E

T R R V E R S I N &

S C R E W S

INNER P R O B E

STATIC P R E S S U R E T A P P I N G

FIG. 2 D E T A I L S O F A N N U L U S A N D E X T E N S I O N T U B E

F I G . B DETAILS O F B O U N D A R Y LAYER P R O B E

N3

A L U M I N I U M F O I L

i^O-OU-0 INS

F R O N T V I E W

T O MANOMETER

CROSS SECTION PROM smr

S DETAILS OF SHEAR PROBE ( NOT TO SCALE)

F I G - . 6 P H O T O G R A P H OF ENTIRE ASSEMBLY

FIG-. 4 PHOTOGRAPH OF PROBES

O U T E R P R O B E V E L O C I T Y H E A b (INCHES OF WATE.R">

F I G . 1 C A L I B R A T I O N C U R V E OF BOUNPflRy LAYER PROBE OO

CHAPTER III

,111.1 EXPERIMENTAL TECHNIQUE

Velocity Profiles

Preliminary tests showed that the fan discharge, and

consequently the velocities in the annulus, varied appreciably.from

day to day as a result of variations in air temperature and barometric

pressure. Tests of several days duration were needed to obtain a

complete veloicty profile in an eccentric annulus and so the long

period variations in velocity had to be allowed for in the results.

This was done by. using the fact that the dimensionless ratio of the

velocities at any two points, was expected, by analogy with tube flow,

to vary.only very slowly with Reynolds number. The day, to day variation

of Reynolds number was-of the order of 2%. This would cause very

significant variations in a profile based on actual velocities, but

negligible variations in a dimensionless profile.

For each eccentricity the point of maximum velocity.on the

line0 =0was f i r s t gound by a rapid traverse using the outer ppobe.

This point was then used as the positionuof a reference velocity by

which a l l other velocities were divided. In practice the velocity

heads were divided and the square root of the result taken as the

velocity ratio. This procedure eliminated the necessity.of calculating

the air density for each velocity measurement. Velocity profiles were

taken by-keeping one probe fixed at this reference position and

traversing one of the other probes over its f u l l range. At each point

of the traverse the local and the reference velocity heads were

measured within a short time interval,to give a velocity ratio which

was independent of the day to day changes in conditions.

31 The distances of the probes from the walls were measured in

several ways. The boundary layer and inner probes were accessible by

sliding out the annulus core parallel to its axis until the probes were

no longer within the outer tube. The traversing screws could then be

turned to move the probe heads. The distance of the inner probe head

from the inner wall was measured directly using a micrometer. The

boundary layer probe was however too fragile for its position to be

determined similarly, so for each traverse the position of the boundary

layer probe was determined by experiment.

I n i t i a l l y the probe was set at an unknown distance from the

wall. It was traversed towards the wall in steps of known magnitude

as obtained from the rotation of the head of its -traversing screw.

The velocity was measured at each point of the traverse. This was

continued until the probe was touching the wall, and,as the probe

was mounted flexibly,further rotation of the screw produced no motion

of the head. A plot of measured velocity against screw displacement

showed an abrupt discontinuity of slope when the probe touched the wall.

This discontinuity located the wall position on the scale of screw

movement and hence the distance from the wall of a l l previous points on

the traverse was obtainable. As the boundary layer probe was 0.006 inch

in thickness i t was assumed to measure the velocity at 0.003 inch from

the wall when in contact with the wall.

The outer probe was traversed by the micrometer head and could

be positioned to 0.001 inch on the micrometer scale. Its actual

distance from the outer wall was obtained in a similar way to that

of the boundary layer probe. This probe was 0.016 inch in diameter

and so when touching the wall i t was assumed to read the velocity at a

32

point 0.008 inch from the wall. The wall position was found to be

reproducible t o ' t 0.001 inch on the micrometer scale and so only.on

the liore accurate traverses was this obtained directly.

A l l velocity measurement in the -concentric annulus were made

on,or close to,a vertical radius of the annulus{Q —O)• . In the

eccentric annuli the inner and outer tubes with attached probes could

be rotated independently to set the probes at the required angular

positions. The angular position of the inner tube was obtained from

the position of the pointer on. the protractor. The outer tube was

rotated in steps of 30 degrees by aligning the holes in the flange

connecting i t to the fixed extension tube. Thus the velocity could

be .measured at almost any point in the cross section. To save time

symmetry was assumed about the vertical plane(9 = C>) and measurements

made over only.half the cross section. A few spot checks showed that

this was justifiable.

A l l three impact tubes were used for the profiles at zero and

50% eccentricity. At 100% eccentricity/the boundary layer probe could

not be used as i t interfered with the rotation of the inner tube. In

this case the inner probe was bent inwards to obtain readings close

to the.inner wall.

The Ratio of Mean Maximum Velocity

The ratio A*, in each annulus was measured for f u l l flow,

using the fan calibration to obtain the mean velocity. The,outer probe

was placed at the position of maximum velocity»as deduced from the

previously .obtained velocity proflie, and a series of readings of, fan

head, air temperature, barometric pressure and maximum velocity head,

was taken. The volumetric flow rate was obtained from Fig. 8 and

adjusted to allow for the slight expansion between the mixing box and

the annulus. Dividing the result by the cross sectional area of the

duct gave the mean velocity. The .maximum velocity was obtained from

the probe velocity head as usual.

Friction Factor Measurements

To determine the mean fr i c t i o n factor the static pressure

-gradient along the-annulus had to be measured, together with the mean

air velocity, the air temperature and barometric pressure. Plots of

static pressure against axial distance from the flow screen, Fig. 11,

showed that the pressure gradient was constant at distances :of more

than ten diameters beyond the flow screen. This is-contrary/to.the

observations of Rothfus et a l . The static pressure gradient

was therefore obtained from the -difference in .the pressures picked up

by the second and the f i f t h tappings,^divided 'by their separation,

which was 42 inches..

The mean air velocity could be measured directly only at f u l l

flow since the fan calibration was valid only with the inlet

unrestricted. To obtain the mean air velocity for the f r i c t i o n factor

runs,the maximum velocity.over the cross section was measured using

the outer probe,and the ratio of mean velocity to maximum velocity

.obtained at f u l l flow, as described-in the previous section, was

assumed to be independent of Reynolds number. The two profiles

obtained in the concentric annulus showed that this was very closely

true. Results for circular tubes [63 show a change.of approximately

27o in the ratio VH /VHAX over the range of Reynolds number covered.

The air temperature and pressure were measured as before and the air

density and viscosity were obtained from reference [if]. Friction

34 factors and Reynolds numbers were calculated from equations (1) and

(4).

Shear Probe Measurements

The maximum pressures picked up-by , the shear probe were of

the order of 0.025 inches of water. To obtain satisfactory accuracy

at such low pressures several precautions had to be taken. A

Lambrecht micromanometer was used at a slope of 25:1 so.that each

millimeter scale division was equivalent to 0.00126 inches of water.

The manometer was placed.in its case to shield.it from short period

temperature fluctuations, which would cause fluctuations in reading

because of volume changes of the air above the reservoir. When

readings were taken the liquid level was always allowed to increase

from its zero position to its •equilibrium position and the zero

position was always returned.to,and read,between readings. In this

way readings were reproducible to.;-f 0.1 divisions, i.e. to T 0.00013

inches of water. Thus giving an accuracy of about 1% on the measure­

ments of shear probe -pressure head.

When measuring the shear stress distribution, a l l probes

were removed except the outer probe and this was fu l l y withdrawn.

This was because the presence of the probes caused variations in

static pressure around the perimeter, which were -negligible in the

previous.tests,but significant at the low pressures involved in shear

stress measurement. The shear probe was moved by rotation of the

inner tube and readings taken at 10 degree intervals in the range

0 = 0 to 180 degrees.

I I I . 2 RESULTS AND DISCUSSION OF RESULTS

The most important numerical r e s u l t s are presented i n Table 1

below. Other r e s u l t s are plo t t e d i n F i g s . 1 2 to 3 0 .

Table 1 . Numerical Results for a l l E c c e n t r i c i t i e s

. E c c e n t r i c i t y 07o 507c 1007c

Reynolds number at f u l l flow 5 3 , 0 0 0 5 4 , 0 0 0 5 6 , 0 0 0

Average VM«* at f u l l flow (ft/sec") 57.7 61 .5 fid .A Postion of reference v e l o c i t y (inches from inner wall) 0 . 4 5 4 0 . 7 7 5 - 1 . 2 5

Average value of V i i / VM*X 0 . 8 9 1 0 . 8 5 4 0 . 8 4 9

V e l o c i t y P r o f i l e s i n Concentric Annuli

Detailed v e l o c i t y p r o f i l e s i n the concentric annulus at

two values of Reynolds number are shown i n F i g s . 1 2 and 1 3 . It can

be seen that both p r o f i l e s are very s i m i l a r , the p r o f i l e at the lower

Reynolds number has s l i g h t l y lower values of MM i n the region

outside the point of maximum v e l o c i t y . The p o s i t i o n o f maximum

vel o c i t y , i n both p r o f i l e s i s i n very close'agreement with the p o s i t i o n

of maximum veloci t y , i n laminar flow as -calculated from equation (5b).

This r e s u l t has been found by a l l previous workers and can be

considered well established. The t h e o r e t i c a l laminar flow p r o f i l e

.in the annulus i s pl o t t e d i n F i g . 1 4 for comparison.

The average value of VM X/M** obtained for the concentric

annulus at f u l l flow was 0 . 8 9 1 . The values obtained by graphical

inte g r a t i o n of F i g s . 1 2 and 1 3 are 0 . 8 7 8 and 0 . 8 7 1 r e s p e c t i v e l y .

The agreement i s good considering that the flow was not measured

directly but only by the fan calibration.

The profiles in the v i c i n i t i e s of the walls are plotted

on the same graphs on an enlarged distance -scale. Also shown are

the velocity gradients at the wall as obtained from equation (3).and

the laminar sublayer hypothesis. It can be seen that -the measured

profiles l i e above these -lines.''. This effect was observed'by Stanton

[J5] , and by Rothfus |jQ ,, and is explained by. the displacement effect

on an impact tube near a wall. No attempt is made to allow for the

displacement and the profiles must be considered inaccurate at

distances of less than 0.010 inch from the walls.

Both concentric profiles are shown plotted in V vs V

coordinates as suggested;by Rothfus in Fig. 15. This is the most

general of the correlations suggested for velocity profiles in

concentric annuli and was used by both Rothfus and Barrow with fair

success. Both profiles obtained show fa i r agreement with the . 'universal

velocity profile'. Exact agreement was not expected since equal

velocities did not occur at exactly equal values of Y on the profil.es

inside and outside of the radius of maximum velocity. This results in

slightly-different curves for the inner and outer half profiles when

plotted : in coordinates. At low values of Y the results l i e

above the 'universal velocity profile', this is probably a result of

the displacement effect previously referred to.

Velocity Profiles in Eccentric Annuli

Profiles in the plane -of eccentricity(^0 =0) are shown in

Figs.16 and 17 for a 50% eccentric annulus and in Fig. 22 for a 100%

eccentric annulus, a l l are at f u l l flow and the respective values of

Reynolds number are shown on the graphs.

It can be seen that the dimensionless profiles are not similar

to those obtained for a concentric annulus, most noticeably the point

of maximum velocity on the line 0=0moves considerably towards the

outer wall with increasing eccentricity. It can also be seen that the

point of maximum velocity on the line0 = O at an eccentricity of 50% ,

does not coincide with that :in laminar flow in the same annulus as

obtained from Appendix II and Fig. 21. This indicates that the .laminar

flow solution cannot be used as a basis for calculating the shear

stress distribution in turbulent flow, as i t can in a concentric

annulus.

Less detailed profiles were taken along perpendiculars to

both inner and outer walls at selected values of 8 ., both for 507o and

for 100% eccentricity. These profiles are shown in Figs. 18,.19,,23

and 24. The same results were also plotted as graphs of V/VMB* VS 0

at constant distances from the walls. These curves are not shown as

they have l i t t l e physical significance,,but together with the other

curves they were used to.prepare contours joining points with equal

values of in the eccentric annuli. These contours are shown

in Figs. 20 and 25 for 50% and 100% eccentricity respectively. A

large number of points were used to draw each contour, and as these

were somewhat scattered and confused only the smoothed contours are

shown. The 50% eccentricity contours may be compared with the laminar

flow solution shown in Fig. 21.

The contours can be compared on a qualitative basis with

Deissler's [f\ semi-theoretical results in an annulus of radius ratio

•3.5:1. As was mentioned previously these results are somewhat

questionable. Comparison of the contours shows that the locus of the

38

position of maximum velocity as obtained by Deissler, is much closer to

the inner wall than that obtained by experiment. As the whole semi-

theoretical analysis is based on the calculated position of this line

of zero shear stress Deissler's profiles w i l l be similarly in error.

In general Deissler's contours are much more rounded than the

experimental ones.

A further comparison is possible on the basis, of the values

of VM/VMAX obtained by Deissler and by experiment. These are

functions of radius ratio and of Reynolds number but should vary only

slowly with either of these. At a Reynolds number of 20,000 and radius

ratio of 3.5:1 Deissler obtained values of Mftxof approximately,

0.84, 0.72, 0.75 and 0.76 at eccentricities of 0%,.60%,.80% and 100%.

The experimental values for Reynolds numbers of around 55,000 at radius

ratio 3:1, are, 0.891, 0.854, 0.849 at eccentricities of 0%, 50% and

100% respectively. It can be seen that the experimental values are

much higher than Deissler's values and vary much less with eccentricity.

No further comparisons of the profiles are possible because of the

different radius ratios and different eccentricities used.

Friction Factor Results

Figures 26, 27 and 28 show the average f r i c t i o n factors

obtained in the three annuli plotted against Reynolds number. Davis'

equation jjl] for concentric annuli is plotted in Fig. 26 for comparison

with the experimental data. The agreement is good by comparison with

the scattered data.of other investigations from which Davis' equation

was obtained.

The experimental data may be represented by the equations,

= O.I6 5 KG at 0% eccentricity,

39

— O- lSS We _ _ _ at 50% eccentricity,

— — — — '-at. 100% eccentricity.

These"lines are plotted in Figs. 26, .27 and 28. The

equations must be regarded as approximate .because .of the scatter

of the data, and are only valid in the Reynolds number range 20,000 -

55,000 for an annulus of radius ratio 3:1.

Numerically the 50% eccentric annulus gave f r i c t i o n factors

which scarcely differed from those in the concentric annulus, being

very slightly lower over most of the range. The 100% eccentric

annulus gave f r i c t i o n factors about 20% lower than the concentric one.

This is qualitatively the same type of variation as predicted by

Deissler \f[ whose f r i c t i o n factors were 10% below the concentric ones

at 607. eccentricity and 30% below at 100% eccentricity. It therefore

appears that eccentricities up to 50% have l i t t l e effect on the-average

fr i c t i o n factor, but greater eccentricities cause i t to decrease

considerably from i t s concentric value.

.Shear Stress Measurements

The shear probe calibration curve, Fig. 10, is i t s e l f of

interest. Assuming that the dynamic pressure obtained from Bernoulli's

equation, and the pressure directly proportional to shear stress as

predicted by Taylor [li] , are additive, the curve of shear stress T

to pressure head r should be of the form:-

P - P = A + B T " (6>

The factor ^/jJi varied by only a few percent and so the calibration

curve should also be of this form. This is seen to be the case, in

Fig. 10. Taylor predicted that the constant. A in equation (6) would

be independent of the probe dimensions. This is supported by the fact

40

that a calibration curve obtained for a different sized shear probe,

which was not subsequently used, differed considerably from Fig, 10

at large values of shear stress where the second term of equation (6)

was significant, but coincided very closely to Fig. 10 for.values O f

-8

'less than 1.5 x 10 ., Further investigation of these curves

would have required a more sensitive manometer.

The shear stress variations around the inner surface o f the

annulus at eccentricities of 50% and.100% are shown in Figs. 29 and

30, The curve for 100% eccentricity is particularly.interesting. It

shows that the shear stress is .not a maximum at the point 9 =0 as would

be expected, but is:a maximum in the region 6 =40 degrees. This effect

is,also apparent in Fig. 23 where the velocity profiles at 9= 45 degrees

and 0 = 60 degrees have steeper velocity gradients near to the .wall than

those -for other values of 0 .. Also in Fig, 30 i t can be seen that the

shear stress tends to zero at 0= 180 degrees, where the tubes touch,

this is as expected since the velocities must also be zero at that

point.

In Fig, 29 for the-50% eccentric annulus the variation of

shear stress around the perimeter is much less than would be expected

intuitively. The shear stress decreases'by only about 25% as. 9 increases

from 0 to 180 degrees, whereas the width of the annular gap decreases

by 67%. The slight dip shown in the curve of Fig. 29 at0 = 150 degr ees

is also associated with lower velocity gradients near the wall for

9= 120 and 150 degrees as shown in Fig. 18.

A comparison is possible between the average measured shear

stress on the inner surface,.and the average deduced from the velocity

contours. The lines of zero shear stress can be drawn reasonably

41

accurately.on the v e l o c i t y contours as shown i n F i g s . 20 and 25. By

a force balance over the area inside t h i s l i n e the mean shear stress

on the inner surface can be obtained as a f r a c t i o n of the o v e r a l l mean

shear s t r e s s . This can then be compared with the mean heights of the

curves i n F i g s . 29 and 30. The results" of th i s procedure are tabulated

i n Table 2 below.

Table 2. Comparison of Measured and Calculated Shear Stresses

E c c e n t r i c i t y . 0% 507» 100% F r a c t i o n a l area in s i d e l i n e of zero shear stress 0.330 0.328 0.320

AVG-/ TftVG-From countours 1 .320 1 ,312 1 .288 "YTAVG- / YftVfr From probe 1.320 1.190 1.003

The exact agreement of the two values of T[AV6./T V6- A T

zero e c c e n t r i c i t y i s assumed i n the c a l i b r a t i o n of the probe and i s

not an experimental r e s u l t . The values ofAVC-/TAX*obtained i n the

eccentric annuli by the two methods are not i n s a t i s f a c t o r y agreement.

No explanation of t h i s could be found, as both the shear stresses

measured by.the probe and the l i n e s of zero shear stress were thought

to be reasonably accurate.

Despite t h i s disagreement i t i s thought that the method used

to measured l o c a l shear stresses i s v a l i d , and gives at least a

q u a l i t a t i v e p i c t u r e of the shear stress v a r i a t i o n around the perimeter,

which could not have been obtained ei t h e r by i n t u i t i o n or from the

v e l o c i t y p r o f i l e s .

Ol : L. o 1 0

FIG-. II S T A T I C

3 . 0 3 0 1+0 So 6o 1o

DISTANCE PFtoM FLOW S C R E E N (.INCHES)

P R E S S U R E &RRDI E NTS IN T H E ANNULI

LU

U-o

r

et V-

z

>

•ooS •OIO o)S WALL DISTANCE (.INCHES) -QiS •OIO •oo5

POSITI ON OF MA* IMOH VELO CITY

(

1N I . H H I N H R Ft r O W ~ ~ -

/ A - — - P F ,OFI L E NE( R INNER WALL

PROF I L E N E A 1 I O U T E R ­1/ WALL

FOR 4 Al N STREAM PROFILE I K E L-U. AND Lov, '£R SCALE 'S

FOR BOO NDARN LRVEPf P« S.OFIL&?; U SE R-H- A N D UPPER SCALES

Lftk_CjULATl i D S L O P E

/ / 1 .ALCULPITEC > SLOPE AT AT O U T E R WfiLL- '

1 INNER V* J B L U

1

•h-

V> 01 _ J

H o a

cc c r 0 z o o OD

>

u« \ i i n u v- c r «N VJ i i i ctv, w n i-1_ ( . I N L H C ^

FIG-TO. VELOCITY PROFILE. IN THE CONCELMTRtC ANNULUS. Re = 53 ,poo

DISTANCE FROM INNER WALL (iNCHEi)

FIG. 18 PROFILES PERPENDICULAR TO INNER WALL AT S0°/o ECCENTRICITY

io

^ o •/ -7. -B -k- '6 "6 "1 -8 -9 l-o DISTANCE FROM OUTER V f t L L (iNCHes)

FIG-. P R O F I L E S P E R P E N D I C U L A R T O O U T E R W A L L A T So'/* ECCENTRIC IT7

FIG. Z O VELOCIT 7 CONTOURS IN THE 50 % ECCENTRIC ANNULUS

FIG-. 2Ll THEORETICAL LAMINAR FLoW VELOCITY CONTOURS IN THE 5 0 % ECCENTRIC ANNULUS

> 6

DISTANCE FROM INNER W R L L I. INCHES]

FI&23 PROFILES P E R P E N D I C U L A R T O INNER W A L L A T l O O ^ E C C E N T R I C I T Y 4>-

I-O DISTANCE F-RoM OUTER VfiLL ^INCHES)

FIG. %l+ PROFILES PERPENDICULAR To OUTER WALL AT lOO°/o ECCENTRICITY

R e =1 5 6 , 0 0 0

F\G. 1 5 VELOCITY CONTOURS IN T H E 1 0 0 ° / ECCENTRIC A N N U L U S

57

REYNOLDS NUMBER

FIG. X b FRICTION FACTORS IN THE

CONCENTRIC ANNULUS

58

lo

6w6

o H u a.

z 0

p

^=o.lS5 Re o

z*/tr id" i-sW1-

REYNOLDS NUMBER

FIG-. I 1 ! FRICTION FACTORS IN THE S 0 %

ECCENTRIC ANNULUS

59

cc o \-o <t a

r o P i u ff u.

r

REYNOLDS N U M B E R Sx40V

PI&. 1 8 FRICTION F A C T O R S IN T H E I O O %

E C C E N T R I C A N N U L U S

IS

10

FIG-- X I

io So \xo Q ( DEG-REEs)

ISO

S H E A R STRESSES ON THE INNER WALL OF THE ECCENTRIC ANNULUS

\8D

5 0 %

o

FIG. 3 0 SHEAR STRESSES ON INNER WALL O F T H E IOO°/>

E C C E N T R I C A N N U L U S

CHAPTER IV

IV. 1. SUMMARY OF RESULTS

The results obtained for flow in a concentric annulus show

good agreement with the results of previous investigations. This is

a useful check on the accuracy of the whole experiment as the probes

and experimental technique used for eccentric annuli were the same

as for the concentric case. The results contribute nothing new to the

study of flow in concentric annuli, but serve to substantiate the few

previous investigations.

.No previous experimental results in eccentric annuli are

available for comparison. The profiles obtained by Deissler and

Taylor \j\ in semi theoretical analysis do not agree well with the

experimental profiles. This is thought to show that the Deissler-

Taylor method is not applicable to annuli.

Complete velocity profiles were obtained in annuli of 507.

and 1007. eccentricity and are plotted as contours of equal velocity.

Themost striking feature of these contours is the relatively,small

variation, of mainstream velocity around the annular gap. This is in

contrast to the laminar flow solution in eccentric annuli which has

been obtained for the 507. eccentric annulus.

Average f r i c t i o n factors in the annuli decrease with

increasing eccentricity, but the change is :small for eccentricities

of less than 507.. Friction factors in the concentric annulus agree

well with Davis' equation.

The local shear stress varies much less around the inner

wall of an eccentric annulus than intuition would suggest. This is

63 an important result as the local heat transfer coefficient would vary

in a similar manner. The fact that the maximum shear stress in an

eccentric annulus need not occur at the position of greatest wall

spacing is an interesting result which could not have been predicted

on simple theoretical grounds.

IV. 11. CONCLUSION

As was stated in the introduction the i n i t i a l purpose of

this work was to study the relationship between flow and heat

transfer in non-symmetrical ducts. The magnitude of this problem was

not realised when the project was started. The d i f f i c u l t i e s

encountered can now be discussed.

The velocity contours obtained in eccentric annuli may be

considered reasonably accurate, yet they do not permit the accurate

drawing of velocity gradient lines in the annuli. From such lines i t

would be possible to calculate the local shear stress at any point in

the fluid, dividing this by the local velocity gradient and density,

would give the eddy dif f u s i v i t y of momentum. Using the analogy

between the transfer of heat and momentum to equate this to the eddy

diff u s i v i t y of heat, a l l the information needed to obtain the heat

transfer from the velocity distribution and the thermal boundary

conditions would be available. In practice extremely accurate and

detailed profiles are needed for this to be done, the basic d i f f i c u l t y

being that both shear stresses and velocity gradients are very small

over most of the area of flow. In particular on the line of zero

shear stress the eddy dif f u s i v i t y of momentum is not defined.

\) If. the eddy dif f u s i v i t y of heat could be found accurately

at any point i t would s t i l l be extremely d i f f i c u l t to calculate heat

64 transfer coefficients. -As an example consider the case of heat

transfer from the inner tube of an eccentric annulus to a fl u i d flowing

through the annular gap, the.inner tube being assumed to have a

uniform surface temperature,and the outer tube being thermally insulated.

This situation may be visualized by considering the annular space to

be f i l l e d with a medium whose thermal conductivity varies with position

in the same manner as the eddy diffusivity, and which in addition,

acts as a non-uniformly distributed heat sink, with strength proport­

ional to the local velocity in the annulus. If subjected to the same

thermal boundary conditions:the above model would have the same heat

transfer characteristics as the actual flow system. Even this model

is somewhat simplified as the conductivity would probably need to be

anisotropic.

Thus to calculate the heat transfer from the experimental

velocity profiles seems to;be impracticable. A great simplification

would be possible i f a mathematical description of the profile could

be found, similar to the 'universal velocity -profile' in circular

tubes. Considering the complexity of the solution in laminar flow,

there seems to be l i t t l e likelihood of a simple equation serving as

even a very approximate description of the profile. .Another possibility

would be the construction of an analogue system on the lines of the

model described above.

The most practical method of obtaining heat transfer data

from a velocity profile would be to use a large di g i t a l computor.

This would be woEthwhile only i f an extremely detailed profile were

available. The solution thereby obtained would,however^not be a

general solution but would apply to one particular annulus at one

flow rate.

For practical purposes.it would seem to be better to rely.on

direct experiment to obtain values of the heat transfer coefficients

in non-symmetrical ducts. Further research on flow in such ducts is

needed, however, to obtain a f u l l understanding of the mechanism of

heat transfer and f r i c t i o n which could lead to improved design of heat

exchangers.

66 APPENDIX I

Dimensional Analysis of theShear Probe

P

F i g . 31 Idealized Cross Section of Shear Probe

An i d e a l i z e d cross section of the shear probe i s shown

above. It i s assumed to be completely surrounded by f l u i d i n laminar

flow, the f l u i d i n contact with the walls being at r e s t . This w i l l

be the case i f the probe i s within a laminar sub layer and i t s

thickness,b , i s small compared to the thickness of the layer.

The shear stress i n the f l u i d and on the w a l l . i s given by,

—yU.C. The pressure head picked up by,the probe i s and the f l

around the probe i s completely s p e c i f i e d by,the values of the

variables ,0 tjx , b andC...

Thus i t follows that.,

(p*_p) - (p,jUL,t,c)

Choosingp>|CA and t as independent v a r i a b l e s and applying

the methods of dimensional analysis i t can be predicted that,

Replacing C by ^ J }

If a s i n g l e probe i s considered b i s constant and mdy be

ow

dropped from the r e l a t i o n s h i p leaving,

67

Thus there exists a unique relationship between the flu i d

properties, the shear stress on the wall and the pressure head picked

up by,the probe. If a means-of calculating the shear stress can be

found this relationship can be plotted and the-curve so obtained used to

measure shear stresses where they cannot be calculated..

An annulus with a traversable core tube provides an ideal

situation for the use of such a probe, as i t can be calibrated in

the concentric case and then used to measure shear stresses with the

core tube eccentric.

It should be pointed out that the validity.of the method does !

not depend on the velocity profile being linear over the probe mouth.

It is sufficient that the profile should be determined entirely by

the shear stress at the wall and by the fl u i d properties. This should

be very closely true even at distances from the wall of several times

the sublayer thickness.

APPENDIX II

The Laminar Flow Solution i n an

Eccentric Annulus

The symbols used i n th i s discussion are defined i n F i g . 32.

1

0

^ -<3^ ^

yV/ \

c ^ > c \ c J >

1 •

s . >

1 •

Fife. 32 Geometry.of an Eccentric Annulus

For established laminar flow of an incompressible newtonian

f l u i d i n a duct the a x i a l v e l o c i t y , V , obeys the equation (Ref. [_22] )

/x. VXV = - P ( i )

with boundary conditionsV= 0 on a l l s o l i d boundaries.

Let V = T ^ ( X * + L £ ) (2)

Equation (1) reduces to, V ^ U ^ — O .

* For a more d e t a i l e d discussion see Ref.

and the boundary conditions to, l|i - ^-/^ i * ^ * * ^ ) ° n

a l l solid boundaries.

Apply the conformal transformation (Ref. \22\ )

"Z - i c Cot (§/>0 - - (3)

where, "Z. - X +• L

§ = 1 +• CY}

Under this transformation an annular region in the 2. plane

transforms into a rectangle in the ? plane bounded by the lines;

= YJ = ^ , | - O and | = XTT. Geometrically the value of Y at any point in theZplane is

equal to the logarithm of the ratio of its distances from the points

(+C, o) and ( -C ; . The value of f at a point is the value of the

angle subtended at the point.by the line C C .

It may easily be shown that,

-y — r .^rJl

and ^ L . , i ^ ^ n ^ j ^ ? ^ (4)

U - c ° (ccr yli Y) _ c e o | )

In the new coordinate system the funct ionsat i s f i es the a.

equationV^-0 j with boundary conditions,

( Co^vjn » c e o ^ — V \ Ccrvh _ c e o | / o n a 1 1 boundaries.

Assume a solution of the form

Substituting this in the equation V y — O and solving gives.

70

where ^ 5 ^ , Cw, and O w are arbitrary constants.

Since ^ is an even function of \ , Aw, ~ O . A

solution is therefore ^ = CoOrr\ (Cm61- Dw£ ^).The most general

solution is therefore, CO

f (|,*)) - D ^ e ^ ) Co-^ro^ + CoY) + IX - -(5) where vn is now a positive integer, andCr^and CCare functions of w .

This may be rewritten as, . OO

= Y L CU(v j ) C o ^ w ^ * C v j f U 0 (6)

where CUA*)) is an unknown function oft(\ a n d .

j ^ i } a n d j ^ ( ^ 5 ^ ^ A R E known from the boundary

conditions, therefore for =^iandYj=T| zequation (6) may be treated

as a Fourier series.

Multiplying byCotim^and integrating from^ ~ O to — Tf

withiTjzv^i equation (6) becomes,

-TT

TT r

The value of M/l' G^YY^cK may be shown by

using the theory of residues to be equal to WT £. C rOn Y" (Ref.jji})

P. C> - y o / ) ' L|

P . C * -*»\*)x I I V _ ( 8 )

Similarly, CU l< ) x ) - /X € CO^Gh <~j

From equations (5) and (6)

and c u (^)x) = Cw, e ' +- Dm e

71

and

Hence - JJT \ ^ _ ])

For the special case of vr\ = O ,

t p ( Y ) 7 |) — C o / j ^ - D o (9)

PuttingTjr-fj | and integrating over the range^ = O to — TT

Co V], + fco = ^ ^ ( x a r t h (io)

Similarly Co + Tj„ - fyj ^1 C<rtk fy- |) '(10)

r P.CX ( Go-tin V). - c c ^ q n x \ Hence C 0 ^ X ^ ^ j

Substituting the values obtained for Cw\, D»v^Co> L\>,back into

equation (5) the complete solution for is obtained.

Converting back fromty* to V using equations (2) and (4) the

solution for the axial velocity is obtained as,

72

For any eccentric annulus the values of f| ( > Y^and C may be

calculated from the geometry, and the velocity distribution computed

from the above equation.

This was done for the annulus used in the experimental work

at an eccentricity of 50%. The calculations were performed on the IBM

1620 computer at this University and the results are plotted as velocity

contours in Fig. 21.

BIBLIOGRAPHY

Rothfus, R. R, "Velocity Distribution and Fluid Friction in Concentric -Annuli" D. Sc. Thesis, Carnegie Institute of Technology. 1948

Rothfus, R. R. Monrad, C. C , Sehecal, V. E. "Velocity Distribution and Fluid Flow in Smooth Concentric Annuli" Ind. Eng. Chem. Vol. 42,.1950 pp 2511

Rothfus, R. R. Walker, J. E., Whan, G.A. "Correlation of Local Velocities in Tubes, Annuli and bet­ween Parallel Plates" A.I. Ch. E. Journal Vol. 4, ,1958, pp 240

Rothfus, R. R., Monrad, G.C., Sikchi, K.G., Heideger, W.J. "Isothermal Skin Friction in Flow Through Annular Sections." Ind. Eng. Chem. Vol. 47, 1955, pp 913

Knudsen, J.G., Katz, D.L. "Velocity Profiles in Annuli.V Proc. Midest Conf. Fluid Dynamics. 1st Qonf., .1950, pp 175

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Barrow, H. "Fluid Flow and Heat Transfer in an Annulus with a Heated Core Tube." Proc I.M.E.. Vol. 169, 1955, pp 1113

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Ower, E. "The Measurement of Airflow" Chapman and Hall, Ltd.!1927

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A.S.M.E. Power Test Codes Chap. 4, Pt 5.

Taylor, G. I. "Measurements with a Half Pitot Tube." Proc. Roy. Soc. London. Sec. A. Vol 166, 1938

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Fage, A., Falkner, V.M. "An Experimental Determination of the Intensity of Friction on the Surface of an Aerofoil". Proc. Roy Soc, London. Sec. A. Vol 129, 1930

Miller, B. "The Laminar Film Hypothesis" Trans. A.S.M.E. Vol. 71, 1949, pp 357

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"Pressure Drop and Heat Transfer to Non Boiling and Boiling Water in Turbulent. Flow in an Internally Heated Annulus". .Chem. Eng. Prog. Symposium Series. No. 11, 1954, pp 115

Lamb, H. "Hydrodynamics", 5th Edition pp 555 Cambridge University,Press

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