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Turbulent flow in concentric and eccentric annuli Denton, John Douglas 1963

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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 t^ftv& 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 d  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) Similarl(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 . > 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 \ cr h _ 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 bet-ween 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 Knudsen, J.G., Katz, D.L. "Fluid Dynamics and Heat Transfer" McGraw-Hill Book Co. Inc. 1958 Deissler, R. G., Taylor, M.F. "Analysis of Fully.Developed Turbulent Heat Transfer and Flow in an Annulus with Various Eccentricities.." N.A.C.A. T.N. 3451, 1955 Barrow, H. "Fluid Flow and Heat Transfer in an Annulus with a Heated Core Tube." Proc I.M.E.. Vol. 169, 1955, pp 1113 Barrow, H. "A Semi-theoretrical Solution of Asymmetric Heat Transfer in Annular Flow" Journal Mech. Eng. Sc. Vol 2, 1960, pp 331 Ower, E. "The Measurement of Airflow" Chapman and Hall, Ltd.!1927 Davis, E.S. N "Heat Transfer and Pressure Drop in Annuli." Trans. A.S.M.E. Vol 65, 1943, pp 755 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 Barker, M. "On the Use of Very Small Pitot Tubes for Measuring Wind Velocity." Proc. Roy. Soc. London. Sec. A. Vol 101, .1922 Stanton, T.-E., Marshall, D., Bryant, C.N. "On the Conditions at the Boundary of a Fluid in Turbulent Motion". Proc. Roy. Soc. London, Sec. A. Vol. 97, 1920 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 Mizushina, T. "Analogy Between Fluid Friction and Heat Transfer in Annuli" A.S.M.E. - I.M.E..General Discussion on Heat Transfer 1951, pp 191 Hartnett, H.P., Koh, J.C.Y., McComas, S.T. "A Comparison of Predicted and Measured Friction Factors for Turbulent Flow Through Rectangular Ducts1.' A.S.M.E. Jour. Heat Transfer Feb. 1962, pp 82. Stein, R.P., Hooper, J.W., Markets, M., Selke, W.A., Bemdler, A.J., Bonilla, C.F. "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 Milne - Thomson, L.M. "Theoretical Hydrodynamics" 4.th Edition Macmillan and Co. Ltd..1960 Parkinson, G.V., Denton, J.D. "Laminar Flow Through an Eccentric Annular Pipe" Aeronautical Research Council. A.R.C. 24,326. F.M. 3263, 1962 

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