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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 i n p a r t i a l fulfilment of the  requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available for referenceand study.  I further agree that permission for ex-  tensive copying of this thesis for scholarly purposes may by the Head of my  Department or by his representatives.  be granted It i s  understood that copying or publications of this thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission.  Department of Mechanical Engineering, The University of B r i t i s h Columbia, Vancouver 8, Canada.  May,  1963.  ii ABSTRACT The turbulent flow of a i r through the annular gap between two tubes was studied experimentally,  both with the tubes concentric  and with the inner tube at e c c e n t r i c i t i e s of 50% and 1007.. A i r v e l o c i t i e s 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 v e l o c i t y p r o f i l e s were obtained at Reynolds numbers around 55,000 and the v a r i a t i o n of average f r i c t i o n factor with Reynolds number was studied i n the Reynolds number range 20,000 55,000.  The v a r i a t i o n of l o c a l shear stress aKound the surface of  the inner tube was obtained  for the eccentric annuli.  The r e s u l t s for the concentric annulus agree well with previous  investigations.  For the eccentric annuli the r e s u l t s are  compared q u a l i t a t i v e l y with Deissler and Taylor's investigation.  semi-theoretical  The agreement i s not good and this i s thought 'to  show that the Deissler-Taylor method i s not applicable to annuli.  It  is concluded that the study of v e l o c i t y p r o f i l e s i n non-symmetrical ducts i s of l i t t l e help i n obtaining quantitative heat transfer data.  Iii TABLE  OF  CONTENTS  PAGE _1  Chapter 1.1.  Introduction  1  1.2.  B a s i c Theory  2  1.3.  P r e v i o u s Work on Flow i n A n n u l a r Ducts  6  1.4.  Entrance E f f e c t s  9  1.5.  Measurements i n t h e Laminar Sublayer  Chapter  i n Annuli  9  II  II.1.  Apparatus  13  II.2..  Preliminary Calibrations  17  111.1.  E x p e r i m e n t a l Technique  30  111.2.  R e s u l t s and D i s c u s s i o n o f R e s u l t s  35  IV.1.  Summary o f R e s u l t s  62  IV.2.  Conclusion  63  Dimensional A n a l y s i s o f t h e Shear Probe  66  The Laminar Flow S o l u t i o n i n an E c c e n t r i c Annulus  68  Chapter  III  Chapter  APPENDIX  APPENDIX  BIBLIOGRAPHY  IV  Ij_  II.  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 C a l i b r a t i o n Curve  27  9  C a l i b r a t i o n Curve of Boundary Layer Probe  28  10  C a l i b r a t i o n Curve of Shear Probe  29  11  S t a t i c Pressure Gradients i n the Annuli  42  12  V e l o c i t y P r o f i l e i n the Concentric Annulus Re = 53,000  43  13  V e l o c i t y P r o f i l e i n the Concentric Annulus  44  Re -= 38,000 14 .15  Laminar Flow P r o f i l e i n the Concentric Annulus  45  Concentric Annular Data Plotted i n V v s  46  +  Y  +  co-ordinates .16  P r o f i l e i n the 50% Eccentric Annulus at 0 = 0  47  17  P r o f i l e i n the 50% Eccentric Annulus at 9=  180°  48  18  P r o f i l e s Perpendicular to the Inner Wall i n the 50% Eccentric Annulus P r o f i l e s Perpendicular to the Outer Wall i n the  49  1.9  50  50% Eccentric Annulus 20 .21  V e l o c i t y Contours i n the 50% Eccentric Annulus  51  Laminar Flow V e l o c i t y Contours i n the 50% .Eccentric Annulus  52  FIG  PAGE  22  P r o f i l e i n the 100% Eccentric Annulus at9  23  P r o f i l e s Perpendicular to the Inner Wall of the 100% Eccentric Annulus  54  24 .  P r o f i l e s Perpendicular to the Outer Wall of the  55  = 0  53  100% Eccentric Annulus 25  V e l o c i t y Contours i n the 100% Eccentric Annulus  26  F r i c t i o n Factors i n the 50%  27  F r i c t i o n Factors i n the 50% Eccentric Annulus  58  28  F r i c t i o n Factors i n the 100% Eccentric Annulus  59  29  Shear Stresses on the Inner Wall of the 50% Eccentric Annulus Shear Stresses on the Inner Wall of the 100%  60  30  Eccentric Annulus  56 . 57  61  Eccentric Annulus 31  Idealized Cross Section of Shear Probe  66  32  Geometry of an Eccentric Annulus  68  VI  LIST  OF  SYMBOLS  SYMBOL p jX  UNITS Density  lb/ft  Dynamic v i s c o s i t y  l b / f t sec  D  Diameter  \l  Velocity  T  Radius  ft  U  Distance from a boundary  ft  \ j  3  ft ft/sec  Angle measured from plane of e c c e n t r i c i t y  De  Equivalent diameter,  Re  Reynolds number,  U  Average f r i c t i o n factor,  degrees  t^Dj.— D|)  ft  pVt-i De/yCA  Shear stress  l b / f t sec  Shear stress acting on a wall  l b / f t sec  ^tftv&  Average shear stress over perimeter  l b / f t sec  V  Dimensionless wall distance, ^ / ^ Y w p / j J .  HTW  +  \/  +  Dimensionless v e l o c i t y ,  R  Equivalent radius,  ft  V  Equivalent wall distance, ^~TL ^/MAX j — R  ft  V~L  Eddy v i s c o s i t y  EL  E c c e n t r i c i t y parameter /core displacementNxlOO  P  S t a t i c pressure  l b / f t sec  P  Total pressure  lb/ft sec  P  Pressure gradient along annulus  lb/ft  M  A; B  Constants  l b / f t sec Percent  2  2  sec  2  SYMBOL  UNITS  Q_  Fan discharge  ft /sec  |-j  Fan pressure head  f t of a i r  3  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 v e l o c i t y  1 CHAPTER 1.1  I  INTRODUCTION Flow of f l u i d s i n non-circular ducts i s a phenomenon often  encountered i n engineering p r a c t i c e .  Heat exchangers i n p a r t i c u l a r  are often based on annular or rectangular passages, or on tube bundles enclosed i n cylindrical, s h e l l s .  An example of such a flow system may  be taken from the CANDU nuclear reactor.  This i s a heavy water cooled  and moderated reactor i n which the coolant flows l o n g i t u d i n a l l y through a c y l i n d r i c a l tube of 3.25  inches inside diameter i n which are  nineteen c y l i n d r i c a l fuel element containers each of 0.6  arranged  inches outside  diameter. The power used to.pump the coolant i n a nuclear power plant may,be an appreciable proportion of the power generated. system should be designed unit "of pumping power.  Thus the  to obtain the maximum heat transfer per  In order to do this the-average heat, transfer  c o e f f i c i e n t and the average f r i c t i o n factor, i n the complex duct described above, must be accurately known.  As high a coolant exit  temperature as-possible i s desireable, but l o c a l b o i l i n g of the coolant must be prevented fuel elements.  as this leads to.high surface temperatures of the  The l o c a l surface temperatures which control the  incidence of b o i l i n g are dependent, on the l o c a l heat transfer c o e f f i c i ents, so.that these  must be known i n 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 l o c a l values at any point i n the duct.  These l o c a l 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 c o e f f i c i e n t s in a duct are uniquely determined by the v e l o c i t y d i s t r i b u t i o n and by the thermal boundary conditions. usually known, and  The  latter  are  so an exact knowledge of the v e l o c i t y p r o f i l e  should s u f f i c e to determine a l l the quantities of interest.  In  practice the v e l o c i t y d i s t r i b u t i o n i s seldom known in s u f f i c i e n t d e t a i l to enable accurate calculations to be made, and  the heat  transfer c o e f f i c i e n t s and:.friction factors are obtained.by experiment. In c i r c u l a r tubes the relationship between f l u i d flow.and heat transfer has been extensively  studied using analogies between  the transfer of heat and momentum. concentric  The  same methods are applicable  to  annuli, but. i n practice the v e l o c i t y p r o f i l e s available -are  not s u f f i c i e n t l y detailed to allow accurate calculations to be made. In ducts not possessing c y l i n d r i c a l symmetry the problem i s complicated by the fact that l i n e s of heat flux and of momentum flux need no longer coincide and no r e l i a b l e method i s known of r e l a t i n g the heat transfer to the v e l o c i t y p r o f i l e in such ducts. .This i s the case in eccentric  annuli. In t h i s investigation a study i s made,of f l u i d flow in  concentric  and eccentric annuli.  The work is part of a j o i n t project  for the study of f l 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 i n a duct i s the  Reynolds number.  U n t i l recently there was  disagreement as to the best  3 means of defining the c h a r a c t e r i s t i c length dimensions which the Reynolds number i s based. heat transfer experiments one boundary.  i n an annulus, on  This arose largely,as a result of  i n which the heat transfer took place at only  For use i n flow measurements there i s l i t t l e doubt that  i t i s preferable to use the equivalent diameter defined by, Cross Sectional Area Wetted Perimeter  De = 4 x  On this basis the equivalent diameter of an annulus i s given by,  De  -  (Dx-D.)  Using this dimension the Reynolds number i s ,  R  =  s  P ^ D ,  ( 1 )  In concentric annuli i t i s found that the flow i s turbulent for values of Re>2,000. Turbulent shear flow i s at present too complex.a phenomenon even i n c i r c u l a r tubes, to be amenable to mathematical semi-empirical approach i s usually  analysis.  A  adopted.  For two dimensional laminar flow i n the x - y plane, with the v e l o c i t y V i n the X d i r e c t i o n being a function o f ^ only, the viscosity/^, i s defined by,  Where  1ax  i s the shear stress acting i n the f l u i d on a plane normal to the is.  By.analogy an eddy v i s c o s i t y E ^ i s defined for turbulent flow  by,  T"  x  dV  — ( ^1 +- Ew,) "JT^  . (2)  The value of E i s i n general a function of both V and of l^.. M  Flow i n a duct i s said to be established when the pressure gradient i s constant and the v e l o c i t y p r o f i l e i s independent  of the  a x i a l position.  When established flow exists i n tubes and between  p a r a l l e l plates i t i s possible to calculate the l o c a l shear stresses from the geometry and from the pressure gradient.  In addition i t i s  possible to predict on dimensional grounds (Ref. [7] ) * that Ew>/p i s proportional to u ., the distance from the wall, for points close to the walls, and proportional t o ^ - g ^ J  / ^-j J^J i n the region remote from the  walls. Using these facts equation (2) may be integrated to obtain an expression for V i n terms of ^ p j i ^ T w C t h e and unknown constants.  shear stress at the wall),  This expression together with  experimental  values of the constants, forms the well known 'universal v e l o c i t y profile  1  r e l a t i n g the dimensionless v e l o c i t y  to the ldi intension-  less distance ^x/Tw py^U- The relationship i s plotted i n F i g . 15. It is found that the 'universal v e l o c i t y p r o f i l e  1  as derived above i s  applicable-to both flow i n tubes and flow between p a r a l l e l plates with the same values of the experimental constants.  There i s no theoretical  j u s t i f i c a t i o n for i t s application to other types of ducts. For values of V less than 5 the 'universal v e l o c i t y p r o f i l e ' 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 e x i s t .  This region  is known as the laminar sublayer and i s 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 v e l o c i t y ^MAX  i s known.  From equation (2) i t can be seen that this i s also  * Numbers i n square brackets refer to the Bibliography  the radius at which no shear stress exists i n the f l u i d ,  A force  balance can be applied to the region between this radius and the inner /  wall to give,  I.  =  x  %\  x\  V,  /  (3a)  S i m i l airrlly for the outer wall, (3b) • Thus i f Y~nt\%  i s known, the shear stress at any point i n a  concentric annulus can be found i n terms of the pressure gradient. In an eccentric annulus, or i n any.other duct not possessing c y l i n d r i c a l symmetry, the shear stress d i s t r i b u t i o n cannot.-be determined u n t i l the complete v e l o c i t y p r o f i l e is.known.  From the  v e l o c i t y p r o f i l e i t i s possible to construct v e l o c i t y gradient lines which are everywhere orthogonal to the lines of constant v e l o c i t y . Such lines are shown i n F i g . 21.  The v e l o c i t y gradient i s zero i n  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 l o c a l shear stress at the walls may be obtained i n terms of the pressure gradient. Shear stresses at a boundary, i n f l u i d flow are usually presented non-dimensionally as f r i c t i o n factors defined by,  f  -  where V i s a characteristic-^velocity, which i n a duct i s usually the average or bulk v e l o c i t y . o f 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, i s usually used. given by,  This i s  pv n  <4>  r —  w  Local f r i c t i o n factors based on the l o c a l shear stress may be defined similarly. annulus  Dimensional analysis shows that the f r i c t i o n factor i n an  i s a function of Reynolds number, diameter r a t i o , and surface  roughness. 1.3.  PREVIOUS WORK ON FLOW IN ANNULAR DUCTS Most of the experimental work on flow i n annuli has been  directed towards heat t r a n s f e r measurements.  However i t i s 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 s i d e l i n e .  These data are extremely  scattered and.no general c o r r e l a t i o n was made u n t i l Davis  [ l l ] i n 1943  reviewed a l l the availablje data and obtained the equation, D  " & D , '  )"°"'  ~  O.0  5 5  Re'*  as the best f i t to the data for smooth concentric annuli.  This i s  shown plotted i n F i g . 26 for D 2 / D 1 - 3.0. The v e l o c i t y d i s t r i b u t i o n for laminar flow i n a concentric annulus was obtained, t h e o r e t i c a l l y by. Lamb [21J , who showed that.,  .  '  ^  '= ( \L^/<-,) R  (5B)  . The v e l o c i t y prof i l e so obtained i n an annulus with "Tx/'fi = 3.0 i s shown plotted non-dimensionally i n Fig.14. The v e l o c i t y p r o f i l e i n turbulent flow i n an annulus was f i r s t investigated by Rothfus jjQ i n 1948.  He obtained v e l o c i t y  p r o f i l e s for turbulent flow of a i r through two concentric annuli of radius r a t i o 6.17 and 1.54.  In each case, he found that the radius of  maximum velocity, was very c l o s e l y 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 v e l o c i t y , prof i l e s for water flowing i n a concentric annulus of radius r a t i o 3.60.  Their  results agreed with Rothfus' on the p o s i t i o n of maximum v e l o c i t y . same workers t r i e d several methods of c o r r e l a t i n g t h e i r own Rothfus' data on v e l o c i t y p r o f i l e s , including 'universal v e l o c i t y p r o f i l e .  and  plots as i n the  None-of the methods t r i e d gave a  1  s a t i s f a c t o r y c o r r e l a t i o n of a l l the p r o f i l e s available. they showed that the unmodified applicable to annuli.  The  In p a r t i c u l a r  'universal velocity, p r o f i l e ' i s not  Knudsen and Katz  |jQ  also t r i e d to use their  p r o f i l e s to calculate heat transfer c o e f f i c i e n t s , but found that the p r o f i l e s were not s u f f i c i e n t l y detailed i n the region of the walls. Rothfus, et a L Qf] i n a concentric annulus the group R  observed that the laminar flow solution  (equation (.5)) i s parabolic with respect to  , where, R * — (C^  -(\***) — .T^  +  + X<»**  to-<^  The laminar flow solution i n tubes and between p a r a l l e l plates i s parabolic with respect to the wall d i s t a n c e , U . defined ah equivalent wall distance for annuli, V equivalent wall shear stress,LL  .  Thus they  =. \ -C™**]—R,  The equivalent shear stress  be shown to equal'Yx at both inner and outer walls.  and an may  They, observed  that equal values of V on the inner and outer halves of the p r o f i l e gave equalvalues of v e l o c i t y , i n turbulent as well as i n laminar flow. It follows that the dimensionless parameters  V/fVp a ndY/STp/^ w i l l f i t the laminar p r o f i l e i n a concentric annulus of any radius  8  r a t i o , to the  'universal v e l o c i t y •profile',.and w i l l also give a single  curve for the inner and outer portions of the p r o f i l e in turbulent, flow. Rothfus' results plotted in this way.indicated that the curve obtained in turbulent  flow agreed c l o s e l y with the  p r o f i l e , for any radius 1  'universal v e l o c i t y  ratio.  Barrow J8] obtained v e l o c i t y p r o f i l e s for a i r flowing annulus of radius r a t i o 2.25  in an  and.found that his results agreed f a i r l y  well with the above c o r r e l a t i o n . The above three investigations seem to constitute the previous experimental work on v e l o c i t y p r o f i l e s in concentric  only  annuli.  In eccentric annuli no experimental v e l o c i t y p r o f i l e s 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 i s lowered by e c c e n t r i c i t y . On the theoretical side Deissler and Taylor  £7}  applied  their method of obtaining v e l o c i t y . p r o f i l e s in ducts, to an annulus of radius r a t i o 3.5:1  at various  method consists i n assuming the along perpendiculars  e c c e n t r i c i t i e s . The  Deissler-Taylor  'universal v e l o c i t y p r o f i l e ' to hold  to the walls of any duct.  By applying  this to an  eccentric annulus Deissler and'Taylor obtained a l i n e on which the same v e l o c i t y resulted from applying the p r o f i l e to either w a l l .  They  called this a l i n e of zero shear stress and proceeded by an i t e r a t i v e process to construct  the v e l o c i t y p r o f i l e .  They also obtained average  f r i c t i o n f a c t o r s , , l o c a l shear stresses and heat transfer c o e f f i c i e n t s . However the analysis must be considered untrustworthy as Knudsen and Katz [5] had shown that the unmodified 'universal v e l o c i t y p r o f i l e ' does not apply even to concentric  annuli.  9 1.4.  ENTRANCE EFFECTS IN ANNULI As has been previously mentioned,.flow i n a duct i s said to  be f u l l y established when the s t a t i c pressure gradient i s constant and the v e l o c i t y p r o f i l e i s independent  of a x i a l p o s i t i o n .  Near the entrance  to a duct these conditions are not f u l f i l l e d and a certain entrance length i s needed for the flow to become established. The entrance lengths for pressure gradients and for v e l o c i t y p r o f i l e s are not i n general the same.  In c i r c u l a r tubes the pressure  gradient becomes constant within a few diameters of the entrance but the v e l o c i t y p r o f i l e requires approximately-fifty diameters, the exact values being dependent on the Reynolds number.  Rothfus .et a l . [u]  found that i n a concentric annulus the pressure gradient was not quite constant after as many as 250 diameters.  No data are available on the  development of v e l o c i t y p r o f i l e s i n annuli.  However, by comparison  with tube flow i t seems u n l i k e l y that the v e l o c i t y p r o f i l e would become established before the pressure gradient. Rothfus took v e l o c i t y p r o f i l e s at 93 and 223 equivalent diameterscdownstream from the i n l e t . equivalent diameters.  Knudsen and Katz allowed 63  In the present experiment  diameters were allowed for the flow to s t a b i l i z e . u n l i k e l y that the v e l o c i t y p r o f i l e was  only 33 equivalent It therefore seems  f u l l y developed.  However large  entrance lengths are seldom used i n p r a c t i c a l 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  i n theories of f l u i d  friction  and heat transfer, l i t t l e work has been done to investigate the laminar  10 sublayer.  In fact M i l l e r  [17]  considered that i t s very existence had  not been conclusively demonstrated  as recently as  1949.  The layer i s usually only a few thousandths of an inch thick and i n i t the v e l o c i t y increases l i n e a r l y with distance from the wall, the f l u i d i n contact with the wall being assumed at rest.  The shear  stress acting on the boundary and i n the sublayer i s given by, Stanton  I —jJ^~j~~  [ l i l was the f i r s t to investigate this region.  Using a p i t o t tube of approximately 0.004 inch thickness, he was able to measure v e l o c i t i e s to within 0.002 inch of a wall.  The v e l o c i t y  p r o f i l e s he obtained i n this way did not extrapolate to zero v e l o c i t y at the wall and did not l i n k 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,  boundary of which was  formed by the tube wall.  able to obtain openings of the..order of 0.001 obtained on the assumption  In this way he inch.  one was  The p r o f i l e s  that the probe measured the v e l o c i t y 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 e f f e c t , whereby,a p i t o t tube near a boundary picked up the v e l o c i t y at a point appreciably.further from the wall than i t s geometric-centre, i n cases even outside the area covered by the probe mouth.  He v e r i f i e d this effect by i n s t a l l i n g his probes in a c i r c u l a r  tube through which a i r was passing i n laminar flow.  The v e l o c i t y  d i s t r i b u t i o n near the wall could be calculated t h e o r e t i c a l l y and by comparing i t with the observed p r o f i l e he obtained a graph of 'probe opening' vs 'effective distance from the w a l l ' .  Using this c a l i b r a t i o n  he replotted  his previous results and obtained good agreement with the  laminar sublayer hypothesis. Stanton's results awakened interest i n the behaviour of p i t o t tubes at low Reynolds numbers.  Barker  |l4] showed that for Reynolds  -numbers less than 30 the pressure picked up by a p i t o t tube approximately given by, P^="^pV -t^i p i t o t tube.  was  where "C i s the radius of the  The second term represents a v i s c o s i t y effect which i s  n e g l i g i b l e at high Reynolds numbers, but which may  be the dominant  term for slow flow of viscous f l u i d s . G. I. Taylor  |l3] considered probes of the type used by  Stanton in which one wall i s formed by a boundary of the f l u i d .  He  showed by dimensional analysis that i f such a probe i s placed i n a region of constant v e l o c i t y gradient, at very low Reynolds numbers so that the dynamic pressure term, "-JlpV ., i s n e g l i g i b l e , i t w i l l pick up •up a pressure d i r e c t l y proportional t o t h e ' S h e a r stress on the wall. He performed  experiments  was closely 1.2.  to show that the constant of proportionality  This pressure proportional to the viscous shear  stress i s the cause of the displacement effect observed by Stanton. Stanton had assumed for convenience that the magnitude of the displacement effect was Falkner  independent  of Reynolds number. Fage and  [l6] using similar probes "obtained curves of  'displacement  e f f e c t ' 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 i n a concentric annulus but  did not allow for the v a r i a t i o n of the displacement effect with Reynolds number and was unable to use his p r o f i l e s near the walls to calculate shear stresses.  12 The probes used i n the present experiment are similar to thos used by Fage and Falkner.  The ..method of c a l i b r a t i o n i s 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 v e l o c i t i e s .  Instead dimensional analysis was used to  c a l i b r a t e the probes so that shear stress could be measured d i r e c t l y from them ( see Appendix I ) .  CHAPTER  II. I  II  APPARATUS The arrangement of apparatus used i n t h i s investigation i s  shown schematically in F i g . 1 and photographically i n F i g . 6. Atmospheric a i r i s supplied to the mixing box by a 1/3 centrifugal fan.  An extension to the outer tube.of the  into the mixing box through an a i r t i g h t rubber ring seal.  Hp.  annulus f i t s The inner,  or core, tube of the annulus passes through a glass f i b r e flow screen into the test section of the duct.  The flow i s unobstructed for 74  inches, or 37 equivalent diameters, after which the duct discharges to atmosphere.  The core tube extends beyond the outer tube and i s  supported externally, so that there i s no disturbance to flow caused by.supports  in-the test section.  More d e t a i l s of the annulus and the extension tube are shown in F i g . 2.  The outer tube i s made of clear p l a s t i c 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 e c c e n t r i c i t y , the flow screen and attached sleeves being to each at the correct e c c e n t r i c i t y .  glued  Each extension tube contained a  support tube of 1 inch inside diameter through which the core tube could s l i d e .  The support tubes were traversable within the extension  tubes to allow exact setting of the e c c e n t r i c i t y .  The holes i n .the  flanges of the extension tubes and main outer tube were c a r e f u l l y 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 p l a s t i c tubing of 3% inches inside diameter.  Thus the  main outer tube could be rotated about i t s axis without l a t e r a l displacement of the axis. The by 7/8  inner tube of the annulus was  a 1 inch outside diameter  inch inside diameter aluminium tube 9 feet long.  diameter was  1.000  She  outside  $ 0.0005 inches as measured by micrometer at random  points along the length.  It was  straightened  so that i t s axis  deviated  from a straight l i n e by a maximum of 0.005 inches and weights were added to the overhanging tube ends to minimise sag due to s e l f 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 l i n e a r i t y was  0.0085 inches, less than 1% of the mean annular gap width.  core was  supported from the extension  tube as previously  The  described  and also externally in a semicircular wooden bearing which was with paper and glued r i g i d l y . t o the frame of the apparatus.  lined  Different  supports were used for each e c c e n t r i c i t y . Thus the annulus core could be rotated f r e e l y about i t s axis and could also s l i d e p a r a l l e l to i t s axis.  The r o t a t i o n of the core was measured by attaching a pointer  to i t and allowing the pointer to move over a protractor attached the external support i n such a way  that i t was  to  concentric with the  core tube. The e c c e n t r i c i t y was  set at the required values using  accurately machined templates. eccentricity.  Two  of these were made for each  -The - templates :St1ld.;Oiiercttoe.-;-tq0e-vt.ube and  the Inner tube, thus supporting the supports were adjusted  f i t t e d inside  the core i n exact p o s i t i o n whilst • ' • • =  to hold the core in the same p o s i t i o n .  15 In this way i t was estimated that the core could be placed within 0.005 inches of the required p o s i t i o n . S t a t i c 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 i n the c*re.  The tubes were  glued i n position with epoxy glue and then cut o f f . f l u s h with the surface.  The area was f i n a l l y polished with fine emery cloth.  The  other ends of the p l a s t i c tubes were led out of the ends of the core tube.  A s t a t i c pressure tapping was also taken from the mixing box. V e l o c i t y 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 i n F i g . 2, Fig. 3, F i g . 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 i n 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 w a l l . to as the inner probe.  The other mainstream probe w i l l be referred  This was traversed by.a 40 threads per inch  screw and i t s distance from the inner wall measured d i r e c t l y 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 w a l l . The t h i r d probe was used to measure v e l o c i t i e s very close to the core surface.  Its construction i s shown i n F i g . 3 and F i g . 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 i n steps of 0.0016 inches u n t i l 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 i s shown i n F i g . .5.  A static  pressure tapping was made as previously described. -A s t r i p .of 0.0015 inch thick feeler gauge material 0.040 inch wide was l a i d over the tapping p a r a l l e l to the axis of the core tube, with one end just covering the hole of the tapping.  A s t r i p of aluminium f o i l 0.0007  inch thick was then l a i d 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 l a t t e r 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 i n 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 s t a t i c pressure taps and by the probes.  Three of these  were v a r i a b l e slope Lambrecht micromanometers containing f l u i d of density 0.800 and having a scale graduated i n 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 d i r e c t l y i n inches of water with an accuracy of 0.001 Air  inches.  temperature was measured by a mercury i n glass.A.S.T.M.  precision test thermometer which was inserted through a hole i n the wall of the mixing box.  It was -observed that the a i r temperature  at outlet from the annulus never d i f f e r e d appreciably from that in the  mixing box and so:the thermometer was assumed to give the a i r  temperature in the annulus. Atmospheric pressure was obtained from an aneroid barometer located i n the same room as the apparatus. II.2  PRELIMINARY CALIBRATIONS Fan C a l i b r a t i o n The a i r 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 c a l i b t a t i o n . 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 p l a s t i c tubing and discharging to atmosphere  through an A.S.M.E. long radius flow nozzle of diameter  r a t i o 0.695. The pressure drop across the nozzle was obtained from a s t a t i c pressure tapping one diameter upstream of the nozzle.  The pressure  head and a i r temperature i n the mixing box were also measured,together  •18 with atmospheric pressure.  Discharge c o e f f i c i e n t s and properties of a i r  were obtained from A.S.M.E. Power Test Codes, Chapter 4, Part 5 ^.2} . The flow was varied by using d i f f e r e n t combinations of flow screens at exit from the mixing box.  For each combination the fan  head, H , expressed as feet of a i r at. i t s density i n the mixing box, and the volumetric flow r a t e , Q . , i n cubic feet per second, also at mixing box conditions, were calculated.  The r e s u l t i n g curve i s shown  in F i g . 8 which applies to the fan with i t s i n l e t 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 p i t o t s t a t i c tube by using them to measure v e l o c i t i e s at the exit of the nozzle previously described.  The v e l o c i t y p r o f i l e at. the  nozzle exit i s very f l a t and the probes could be spaced about \ inch apart to reduce interference,with n e g l i g i b l e v a r i a t i o n of the v e l o c i t y at the probe mouth. It was found that the two mainstream probes picked up the same t o t a l head as the manufactured probe and they were.:therefore assumed to measure the v e l o c i t y head exactly.  The boundary layer probe picked  up an appreciably lower pressure and a c a l i b r a t i o n curve of, 'indicated v e l o c i t y head' vs 'true v e l o c i t y head' (as recorded by the other  probes), was obtained for i t .  When i n s t a l l i n g this probe i t was found  necessary to give i t a s l i g h t i n c l i n a t i o n 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 c a l i b r a t i o n of the probe,as the  p r o f i l e s obtained with i t did not j o i n smoothly with the p r o f i l e s obtained from the other probes.  Accordingly the boundary layer probe  was recalibrated i n s i t u by placing i t at the radius of maximum v e l o c i t y i n a concentric annulus and comparing i t with the outer probe placed at the same radius.  The r e s u l t i n g c a l i b r a t i o n curve i s shown  in F i g . 9. The Shear Probe Calibration As mentioned i n Section 1.2 the shear stress on the' inner walls of a concentric annulus can be calculated from equation (3). annulus of radius ration 3:1, with  For an  assumed to be the same as  I^MAX  in laminar flow, this equation reduces to, T, =0.&^,9 lP.This t  relationship was used to c a l i b r a t e the shear probe. The annulus was set up with the core concentric and the flow varied by r e s t r i c t i n g the fan i n l e t .  The pressure picked up by the  shear probe was very closely independent  of i t s angular p o s i t i o n .  each flow rate the pressure gradient, a i r temperature shear probe head,.were recorded.  and pressure,and  The groups obtained i n Appendix I  were calculated and are plotted i n F i g . 10.  For  FIG.  I  SCHEMATIC  LAYOUT  OF  FLOW  SYSTEM  MICROMETER HERD  PLftSTIC  BLOCK  SUPPORT TUBE  TRRVERSIN& SCR  INNER  PROBE  EXTENSION  TUBE  STATIC PRESSURE TAPPING  FIG.  2  DETAILS  OF  ANNULUS  AND  EWS  FIG.  B  DETAILS  OF  BOUNDARY  LAYER  PROBE  N3  ALUMINIUM  FOIL  i^O-OU-0 INS  FRONT  VIEW  TO  CROSS  S  DETAILS  OF  SHEAR  PROBE  ( NOT  MANOMETER  SECTION  TO  PROM  SCALE)  smr  FIG-.  6  PHOTOGRAPH  OF  ENTIRE  ASSEMBLY  FIG-. 4  PHOTOGRAPH  OF  PROBES  OUTER  FIG. 1  CALIBRATION  CURVE  OF  PROBE  VELOCITY  BOUNPflRy  HEAb  (INCHES  LAYER  OF  WATE.R">  PROBE  OO  CHAPTER I I I ,111.1  EXPERIMENTAL TECHNIQUE Velocity Profiles Preliminary tests showed that the fan discharge, and  consequently the v e l o c i t i e s i n the annulus, varied  appreciably.from  day to day as a result of variations i n a i r temperature and barometric pressure.  Tests of several days duration were needed to obtain a  complete v e l o i c t y p r o f i l e i n an eccentric annulus and so the long period variations i n v e l o c i t y had to be allowed  for i n the r e s u l t s .  This was done by. using the fact that the dimensionless  r a t i o of the  v e l o c i t i e s at any two points, was expected, by analogy with tube flow, to vary.only very slowly with Reynolds number. of Reynolds number was-of the order of 2%.  The day, to day v a r i a t i o n  This would cause very  s i g n i f i c a n t v a r i a t i o n s i n a p r o f i l e based on actual v e l o c i t i e s , but n e g l i g i b l e v a r i a t i o n s i n a dimensionless  profile.  For each e c c e n t r i c i t y the point of maximum velocity.on the l i n e 0 =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 v e l o c i t y by which a l l other v e l o c i t i e s were divided.  In practice the v e l o c i t y  heads were divided and the square root of the result taken as the velocity ratio.  This procedure eliminated the necessity.of c a l c u l a t i n g  the a i r density for each v e l o c i t y measurement.  V e l o c i t y p r o f i l e s were  taken by-keeping one probe fixed at this reference p o s i t i o n and traversing one of the other probes over i t s f u l l range.  At each point  of the traverse the l o c a l and the reference v e l o c i t y heads were measured within a short time i n t e r v a l , t o give a v e l o c i t y r a t i o which was independent of the day to day changes i n conditions.  31 The distances of the probes from the walls were measured i n several ways.  The boundary layer and inner probes were accessible by  s l i d i n g out the annulus core p a r a l l e l to i t s axis u n t i l the probes were no longer within the outer tube. turned to move the probe heads.  The traversing screws could then be The distance of the inner probe head  from the inner wall was measured d i r e c t l y using a micrometer.  The  boundary layer probe was however too f r a g i l e for i t s p o s i t i o n to be determined s i m i l a r l y , so for each traverse the p o s i t i o n 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 i n steps of known magnitude  as obtained  from the rotation of the head of i t s -traversing screw.  The v e l o c i t y was measured at each point of the traverse. continued  This was  u n t i l the probe was touching the wall, and,as the probe  was mounted f l e x i b l y , f u r t h e r rotation of the screw produced no motion of the head.  A plot of measured v e l o c i t y against screw displacement  showed an abrupt discontinuity of slope when the probe touched the w a l l . This d i s c o n t i n u i t y located the wall p o s i t i o n 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 v e l o c i t y at 0.003 inch from the wall when i n 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.  distance from the outer wall was obtained of the boundary layer probe.  Its actual  i n a similar way to that  This probe was 0.016  inch i n diameter  and so when touching the wall i t was assumed t o read the v e l o c i t y at a  32 point 0.008 inch from the wall.  The wall p o s i t i o n 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 v e l o c i t y measurement i n the -concentric annulus were made on,or close to,a v e r t i c a l radius of the annulus{Q —O)• .  In the  eccentric annuli the inner and outer tubes with attached probes could be rotated independently positions.  to set the probes at the required angular  The angular p o s i t i o n of the inner tube was obtained from  the p o s i t i o n of the pointer on. the protractor.  The outer tube was  rotated i n steps of 30 degrees by a l i g n i n g the holes i n the flange connecting i t to the fixed extension tube.  Thus the v e l o c i t y could  be .measured at almost any point i n the cross section.  To save time  symmetry was assumed about the v e r t i c a l plane(9 = C>) and measurements made over only.half the cross section.  A few spot checks showed that  this was j u s t i f i a b l e . A l l three impact tubes were used for the p r o f i l e s at zero and 50% e c c e n t r i c i t y .  At 100% e c c e n t r i c i t y / t h e 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 w a l l . The Ratio of Mean Maximum V e l o c i t y The r a t i o  A*, i n each annulus was measured for f u l l  using the fan c a l i b r a t i o n to obtain the mean v e l o c i t y .  flow,  The,outer probe  was placed at the p o s i t i o n of maximum velocity»as deduced from the previously .obtained v e l o c i t y p r o f l i e , and a series of readings of, fan head, a i r temperature, barometric pressure and maximum v e l o c i t y head, was  taken.  The volumetric flow rate was obtained from F i g . 8 and  adjusted to allow for the s l i g h t expansion between the mixing box the annulus.  and  Dividing the result by the cross sectional area of the  duct gave the mean v e l o c i t y .  The .maximum v e l o c i t y was obtained from  the probe v e l o c i t y head as usual. F r i c t i o n Factor Measurements To determine the mean f r i c t i o n factor the s t a t i c pressure -gradient along the-annulus had to be measured, together with the mean a i r v e l o c i t y , the a i r temperature and barometric pressure.  Plots of  s t a t i c pressure against a x i a l distance from the flow screen, F i g . 11, showed that the pressure gradient was  constant at d i s t a n c e s o f more :  than ten diameters beyond the flow screen. observations of Rothfus et a l . was  This is-contrary/to.the  The s t a t i c pressure gradient  therefore obtained from the -difference i n .the pressures picked up  by the second and the f i f t h  t a p p i n g s , ^ d i v i d e d 'by t h e i r s e p a r a t i o n ,  which was 42 inches.. The mean a i r v e l o c i t y could be measured d i r e c t l y only at f u l l flow since the fan c a l i b r a t i o n was v a l i d only with the i n l e t unrestricted.  To obtain the mean a i r v e l o c i t y 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 r a t i o of mean v e l o c i t y to maximum v e l o c i t y .obtained at f u l l flow, as described-in the previous section, was assumed to be independent of Reynolds number.  The two p r o f i l e s  obtained i n the concentric annulus showed that this was very closely true.  Results for c i r c u l a r tubes  [63  show a change.of approximately  27o i n the r a t i o VH /VHAX over the range of Reynolds number covered. The a i r temperature and pressure were measured as before and the a i r density and v i s c o s i t y were obtained from reference [if]. F r i c t i o n  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 s a t i s f a c t o r y 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 d i v i s i o n was equivalent to 0.00126 inches of water. The manometer was placed.in i t s case t o s h i e l d . i t from short period temperature fluctuations, which would cause fluctuations i n reading because of volume changes of the a i r above the reservoir.  When  readings were taken the l i q u i d l e v e l was always allowed to increase from i t s zero position to i t s •equilibrium position and the zero position was always returned.to,and read,between readings.  In this  way readings were reproducible to.;-f 0.1 d i v i s i o n s , 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 d i s t r i b u t i o n , a l l probes were removed except the outer probe and this was f u l l y withdrawn. This was because the presence of the probes caused variations i n s t a t i c pressure around the perimeter, which were -negligible i n the previous.tests,but s i g n i f i c a n t at the low pressures involved i n shear stress measurement.  The shear probe was moved by rotation of the  inner tube and readings taken at 10 degree intervals i n the range 0=0  to 180 degrees.  III.2  RESULTS AND DISCUSSION OF RESULTS The most important  below.  Other  numerical  i n F i g s . 1 2 to 3 0 .  results are plotted  Table 1 .  Numerical  Results f o r a l l E c c e n t r i c i t i e s  .Eccentricity  07o  Reynolds number a t f u l l Average V M « * a t f u l l  flow  flow  507c  53,000  (ft/sec")  1007c  54,000  57.7  56,000  61 .5  Postion of reference v e l o c i t y ( i n c h e s from i n n e r w a l l )  0 . 4 5 4  0.775  Average v a l u e o f V i i / V M * X  0.891  0 . 8 5 4  Velocity Profiles  i n Table 1  r e s u l t s a r e presented  fid .A -1.25  0.849  i n Concentric Annuli  Detailed velocity profiles  i n t h e c o n c e n t r i c annulus a t  two v a l u e s o f Reynolds number a r e shown i n F i g s . 1 2 and 1 3 . be  seen t h a t b o t h p r o f i l e s a r e v e r y s i m i l a r ,  Reynolds number has s l i g h t l y  v e l o c i t y , i n both p r o f i l e s of  t h e p r o f i l e a t t h e lower MM i n the region  lower v a l u e s o f  o u t s i d e t h e p o i n t o f maximum v e l o c i t y .  The p o s i t i o n o f maximum  i s i n v e r y close'agreement  maximum v e l o c i t y , i n laminar  I t can  flow as - c a l c u l a t e d  with the p o s i t i o n  from e q u a t i o n ( 5 b ) .  T h i s r e s u l t has been found by a l l p r e v i o u s workers and can be considered w e l l e s t a b l i s h e d . .in  t h e annulus The  annulus  i s plotted  laminar  flow  profile  i n F i g . 1 4 f o r comparison.  average v a l u e o f VM^X/M** o b t a i n e d f o r t h e c o n c e n t r i c  at f u l l  f l o w was  integration of Figs. The  The t h e o r e t i c a l  12  0 . 8 9 1 .  and  13  The v a l u e s o b t a i n e d by g r a p h i c a l are  0.878  and  0.871  respectively.  agreement i s good c o n s i d e r i n g t h a t t h e flow was n o t measured  d i r e c t l y but only by the fan c a l i b r a t i o n . The p r o f i l e s i n 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 v e l o c i t y gradients at the wall as obtained from equation (3).and the laminar sublayer hypothesis.  It can be seen that -the measured  p r o f i l e s l i e above these -lines.''. This effect was observed'by  Stanton  [J5] , and by Rothfus |jQ ,, and i s explained by. the displacement e f f e c t on an impact tube near a wall.  No attempt  i s made to allow for the  displacement and the p r o f i l e s must be considered inaccurate at distances of less than 0.010  inch from the walls.  Both concentric p r o f i l e s are shown plotted i n V vs V coordinates as suggested;by Rothfus i n F i g . 15.  This i s the most  general of the correlations suggested for v e l o c i t y p r o f i l e s i n concentric annuli and was used by both Rothfus and Barrow with f a i r success.  Both p r o f i l e s obtained show f a i r agreement with the . 'universal  velocity profile'.  Exact agreement was not expected since equal  v e l o c i t i e s d i d not occur at exactly equal values of Y on the profil.es inside and outside of the radius of maximum v e l o c i t y .  This results i n  s l i g h t l y - d i f f e r e n t curves for the inner and outer h a l f p r o f i l e s when plotted : in  coordinates. At low values of Y  the results l i e  above the 'universal v e l o c i t y p r o f i l e ' , this i s probably a result of the displacement effect previously referred t o . V e l o c i t y P r o f i l e s i n Eccentric Annuli P r o f i l e s i n the plane -of e c c e n t r i c i t y ( ^ 0 =0) are shown i n Figs.16 and 17 for a 50% eccentric annulus and i n F i g . 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 p r o f i l e s are not similar to those obtained for a concentric annulus, most noticeably the point of maximum v e l o c i t y on the l i n e 0=0moves considerably towards the outer wall with increasing e c c e n t r i c i t y .  It can also be seen that the  point of maximum v e l o c i t y on the l i n e 0 = O at an e c c e n t r i c i t y of 50% , does not coincide with that :in laminar flow i n the same annulus as obtained from Appendix II and F i g . 21. This indicates that the .laminar flow solution cannot be used as a basis for calculating the shear stress d i s t r i b u t i o n i n turbulent flow, as i t can i n a concentric annulus. Less detailed p r o f i l e s were taken along perpendiculars to both inner and outer walls at selected values of 8 ., both for 507 and o  for 100% e c c e n t r i c i t y .  These p r o f i l e s are shown i n F i g s . 18,.19,,23  and 24. The same results were also plotted as graphs of V/VMB* at constant distances from the walls.  VS  0  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  i n the eccentric a n n u l i .  These contours are shown  in Figs. 20 and 25 for 50% and 100% e c c e n t r i c i t y 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% e c c e n t r i c i t y contours may be compared with the laminar  flow solution shown i n F i g . 21. The contours can be compared on a q u a l i t a t i v e basis with Deissler's [f\ semi-theoretical results i n an annulus of radius r a t i o •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 v e l o c i t y as obtained by Deissler, i s much closer to the inner wall than that obtained by experiment.  As the whole semi-  theoretical analysis i s based on the calculated p o s i t i o n of this l i n e of zero shear stress Deissler's p r o f i l e s w i l l be s i m i l a r l y i n error. In general Deissler's contours are much more rounded than the experimental ones. A further comparison i s possible on the basis, of the values of VM/VMAX obtained by Deissler and by experiment.  These are  functions of radius r a t i o and of Reynolds number but should vary only slowly with either of these.  At a Reynolds number of 20,000 and radius  r a t i o of 3.5:1 Deissler obtained values of  Mftxof  approximately,  0.84, 0.72, 0.75 and 0.76 at e c c e n t r i c i t i e s of 0%,.60%,.80% and 100%. The experimental values for Reynolds numbers of around 55,000 at radius r a t i o 3:1, are, 0.891, 0.854, 0.849 at e c c e n t r i c i t i e s 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 e c c e n t r i c i t y . No further comparisons of the p r o f i l e s are possible because of the d i f f e r e n t radius r a t i o s and d i f f e r e n t e c c e n t r i c i t i e s used. F r i c t i o n Factor Results Figures 26, 27 and 28 show the average f r i c t i o n factors obtained i n the three annuli plotted against Reynolds number. equation  Davis'  jjl] f o r concentric annuli i s plotted i n F i g . 26 for comparison  with the experimental data.  The agreement i s 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% e c c e n t r i c i t y ,  39  O - l S S We  — —  _  _  _  at  50% eccentricity,  —  —  — '-at. 100% e c c e n t r i c i t y .  These"lines are plotted i n Figs. 26, .27 and 28.  The  equations must be regarded as approximate .because .of the scatter of the data, and are only v a l i d i n the Reynolds number range 20,000 55,000 for an annulus of radius r a t i o  3:1.  Numerically the 50% eccentric annulus gave f r i c t i o n factors which scarcely d i f f e r e d from those i n the concentric annulus, being very s l i g h t l y 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 i s q u a l i t a t i v e l y the same type of v a r i a t i o n as predicted by Deissler  \f[  whose f r i c t i o n factors were 10% below the concentric ones  at 607. e c c e n t r i c i t y and 30% below at 100% e c c e n t r i c i t y .  It therefore  appears that e c c e n t r i c i t i e s up to 50% have l i t t l e effect on the-average f r i c t i o n f a c t o r , but greater e c c e n t r i c i t i e s cause i t to decrease considerably from i t s concentric value. .Shear Stress Measurements The shear probe c a l i b r a t i o n curve, F i g . 10, i s i t s e l f of interest.  Assuming that the dynamic pressure obtained from Bernoulli's  equation, and the pressure d i r e c t l y proportional to shear stress as predicted by Taylor  [ l i ] , are additive, the curve of shear stress  to pressure head r  should be of the form:-  P - P The factor ^/jJi  =  A  + BT"  (6>  varied by only a few percent and so the c a l i b r a t i o n  curve should also be of this form. Fig. 10.  T  This i s seen to be the case, i n  Taylor predicted that the constant. A i n equation (6) would  be independent of the probe dimensions.  This i s supported by the fact  40 that a c a l i b r a t i o n curve obtained for a d i f f e r e n t sized shear probe, which was not subsequently used, d i f f e r e d considerably from F i g , 10 at large values of shear stress where the second term of equation (6) was s i g n i f i c a n t , but coincided very closely to F i g . 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 e c c e n t r i c i t i e s of 50% and.100% are shown i n Figs. 29 and 30,  The curve for 100% e c c e n t r i c i t y i s p a r t i c u l a r l y . i n t e r e s t i n g .  It  shows that the shear stress i s .not a maximum at the point 9 =0 as would be expected, but i s : a maximum i n the region 6 =40  degrees.  i s , a l s o apparent i n F i g . 23 where the v e l o c i t y p r o f i l e s at 9=  This effect 45 degrees  and 0 = 60 degrees have steeper v e l o c i t y gradients near to the .wall than those -for other values of 0 .. shear stress tends to zero at 0=  Also i n F i g , 30 i t can be seen that the 180 degrees, where the tubes touch,  this i s as expected since the v e l o c i t i e s must also be zero at that point. In F i g , 29 for the-50% eccentric annulus the v a r i a t i o n of shear stress around the perimeter i s 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 s l i g h t dip shown i n the curve of F i g . 29 at0 = 150 degr ees  is also associated with lower v e l o c i t y gradients near the wall for 9=  120 and 150 degrees as shown i n F i g . 18. A comparison i s possible between the average measured shear  stress on the inner surface,.and the average deduced from the v e l o c i t y contours.  The lines of zero shear stress can be drawn reasonably  41  a c c u r a t e l y . o n the v e l o c i t y contours as shown i n F i g s . 20 and 25. a f o r c e b a l a n c e over the a r e a i n s i d e t h i s  l i n e t h e mean shear  By  stress  on the i n n e r s u r f a c e can be o b t a i n e d as a f r a c t i o n o f the o v e r a l l mean shear s t r e s s .  T h i s can then be compared w i t h t h e mean h e i g h t s o f t h e  curves i n F i g s . 29 and 30.  The r e s u l t s " o f t h i s procedure a r e t a b u l a t e d  i n T a b l e 2 below. T a b l e 2.  Comparison o f Measured and C a l c u l a t e d Shear S t r e s s e s  Eccentricity F r a c t i o n a l area i n s i d e l i n e of zero shear s t r e s s  AVG-/^TftVGFrom  countours  "YTAVGFrom  / ^YftVfr probe The  . 0%  507»  100%  0.330  0.328  0.320  1 .320  1 ,312  1 .288  1.320  1.190  1.003  exact agreement o f the two v a l u e s o f ^T[AV6./T^V6-  A  T  zero e c c e n t r i c i t y i s assumed i n t h e c a l i b r a t i o n o f t h e probe and i s not an e x p e r i m e n t a l r e s u l t .  The v a l u e s ofAVC-/TAX*obtained  i n the  e c c e n t r i c a n n u l i by t h e two methods a r e n o t i n s a t i s f a c t o r y agreement. No e x p l a n a t i o n o f t h i s c o u l d be found, as both the shear  stresses  measured by.the probe and t h e l i n e s o f zero shear s t r e s s were thought to be r e a s o n a b l y a c c u r a t e . D e s p i t e t h i s disagreement to measured l o c a l  shear s t r e s s e s  i t i s thought  t h a t t h e method used  i s v a l i d , and g i v e s a t l e a s t a  q u a l i t a t i v e p i c t u r e o f the shear s t r e s s v a r i a t i o n around  the perimeter,  which c o u l d n o t have been o b t a i n e d e i t h e r by i n t u i t i o n o r from the velocity  profiles.  Ol  o  :  L. 10  3.0  DISTANCE  FIG-. II  STATIC  1+0  30  PFtoM  PRESSURE  FLOW  SCREEN  &RRDI E N T S  So  6o  1o  (.INCHES)  IN  THE  ANNULI  •ooS  o)S  •OIO  WALL^ DISTANCE  (.INCHES)  POSITI ON OF M A * IMOH 1N  Ft r O W ~ ~  I. H H I N H R  -QiS  •OIO  •oo5  VELO CITY -  (  V> /  LU  A-—-  P F  ,OFI L E NE( R WALL  01  INNER  _J  PROF I L E  NEA1I WALL  1/  H o  OUTER-  a  U-  o  •hFOR  4 Al N STREAM PROFILE  FOR  BOO NDARN  L-U.  IKE  AND  SCALE'S  Lov, '£R  r et V-  P« S.OFIL&?;  LRVEPf  USE  R-H- A N D  UPPER  SCALES  cc  Lftk_CjULATl i D S L O P E ' AT O U T E R WfiLL-  z  //  1  .ALCULPITEC >  INNER  SLOPE  AT  cr 0  z o o  OD  V*J B L U  1  >  >  1 u«  FIG-TO.  VELOCITY  \ i  i n  u v- c  r  PROFILE.  «N VJ i i  IN  i  ctv,  THE  w  n i-1_  ( . I N L H C ^  CONCELMTRtC  ANNULUS.  Re = 53,poo  DISTANCE  FIG. 18  PROFILES  FROM INNER  PERPENDICULAR TO  INNER  io  W A L L (iNCHEi)  WALL  AT  S0°/  o  ECCENTRICITY  ^ o  •/  FIG-.  -7. PROFILES  -B DISTANCE  '6  -kFROM  PERPENDICULAR  OUTER TO  "6 VftLL  OUTER  -8  "1  -9  l-o  (iNCHes) WALL  AT  So'/*  ECCENTRIC  IT7  FIG. Z O  VELOCIT 7  CONTOURS  IN  THE  50 %  ECCENTRIC  ANNULUS  FIG-. 2Ll  THEORETICAL IN  THE  LAMINAR 5 0%  FLoW  ECCENTRIC  VELOCITY ANNULUS  CONTOURS  >  6  DISTANCE  FI&23  PROFILES  PERPENDICULAR  TO  FROM  INNER  INNER  W R L L I. INCHES]  WALL  AT l O O ^  ECCENTRICITY  4>-  I-O DISTANCE  FIG.  %l+  PROFILES  F-RoM  PERPENDICULAR  OUTER  To  VfiLL  ^INCHES)  OUTER WALL  AT  lOO°/o  ECCENTRICITY  R e =1 5 6 , 0 0 0  F\G.  15  VELOCITY  CONTOURS  IN  T H E  100°/  ECCENTRIC  ANNULUS  57  REYNOLDS  FIG.  X b  FRICTION CONCENTRIC  NUMBER  FACTORS  IN  ANNULUS  THE  58  lo  o  ^=o.lS5 R e 6w6  o H u  a.  z 0  p  z*/tr  i-sW-  id"  1  REYNOLDS  FIG-. I ! 1  FRICTION  NUMBER  FACTORS  ECCENTRIC  IN T H E  ANNULUS  S0%  59  cc o \o  r  <t a  r  o  Pi u  ff  u.  Sx40 V  REYNOLDS  PI&.  18  FRICTION ECCENTRIC  NUMBER  FACTORS  IN  ANNULUS  THE  IOO %  IS  10  io  \xo  So Q  FIG-- X I  SHEAR S T R E S S E S ON THE ECCENTRIC ANNULUS  \8D  ISO  ( DEG-REEs)  INNER  WALL  OF  THE  5 0 % o  FIG.  30  SHEAR  STRESSES ECCENTRIC  ON  INNER  ANNULUS  WALL  OF  THE  IOO°/>  CHAPTER  IV. 1.  IV  SUMMARY OF RESULTS The results obtained for flow i n a concentric annulus show  good agreement with the results of previous investigations. a useful check on the accuracy of the whole experiment  This i s  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 i n concentric annuli, but serve to substantiate the few previous investigations. .No previous experimental results i n eccentric annuli are available for comparison. Taylor  \j\  in  The p r o f i l e s obtained by Deissler and  semi t h e o r e t i c a l analysis do not agree well with the  experimental p r o f i l e s .  This i s thought to show that the Deissler-  Taylor method i s not applicable to annuli. Complete v e l o c i t y p r o f i l e s were obtained i n annuli of 507. and 1007. e c c e n t r i c i t y and are plotted as contours of equal v e l o c i t y . Themost s t r i k i n g feature of these contours i s the r e l a t i v e l y , s m a l l variation, of mainstream v e l o c i t y around the annular gap.  This i s i n  contrast to the laminar flow solution i n eccentric annuli which has been obtained for the 507. eccentric annulus. Average f r i c t i o n factors i n the annuli decrease with increasing e c c e n t r i c i t y , but the change i s :small for e c c e n t r i c i t i e s of less than 507..  F r i c t i o n factors i n the concentric annulus  agree  well with Davis' equation. The l o c a l shear stress varies much less around the inner wall of an eccentric annulus than i n t u i t i o n would suggest.  This i s  63 an important result as the l o c a l heat transfer c o e f f i c i e n t would vary in a similar manner.  The fact that the maximum shear stress i n an  eccentric annulus need not occur at the position of greatest wall spacing i s an interesting result which could not have been predicted on simple theoretical IV. 11.  grounds.  CONCLUSION As was stated i n the introduction the i n i t i a l purpose of  this work was to study the relationship between flow and heat transfer i n 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 v e l o c i t y contours obtained i n eccentric annuli may be considered reasonably accurate, yet they do not permit the accurate drawing of v e l o c i t y gradient lines i n the annuli.  From such lines i t  would be possible to calculate the l o c a l shear stress at any point i n the f l u i d , dividing this by the l o c a l v e l o c i t y gradient and density, would give the eddy d i f 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 d i f f u s i v i t y of heat, a l l the information needed to obtain the heat transfer from the v e l o c i t y d i s t r i b u t i o n and the thermal boundary conditions would be available.  In practice extremely accurate and  detailed p r o f i l e s 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 v e l o c i t y gradients are very small over most of the area of flow.  In p a r t i c u l a r on the l i n e of zero  shear stress the eddy d i f f u s i v i t y of momentum i s not defined. \) If. the eddy d i f 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 c o e f f i c i e n t s . -As an example consider the case of heat transfer from the inner tube of an eccentric annulus to a f l 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 v i s u a l i z e d 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 d i f f u s i v i t y , and which i n addition, acts as a non-uniformly distributed heat sink, with strength proportional to the l o c a l v e l o c i t y i n the annulus.  I f subjected to the same  thermal boundary conditions:the above model would have the same heat transfer c h a r a c t e r i s t i c s 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 v e l o c i t y p r o f i l e s seems to;be impracticable. A great s i m p l i f i c a t i o n would be possible i f a mathematical  description of the p r o f i l e could  be found, similar to the 'universal v e l o c i t y -profile' i n c i r c u l a r tubes.  Considering the complexity of the solution i n 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 p r o f i l e .  .Another p o s s i b i l i t y  would be the construction of an analogue system on the lines of the model described above. The most p r a c t i c a l method of obtaining heat transfer data from a v e l o c i t y p r o f i l e would be to use a large d i g i t a l computor. This would be woEthwhile only i f an extremely detailed p r o f i l e were available.  The solution thereby obtained would,however^not be a  general solution but would apply to one p a r t i c u l a r annulus at one  flow rate. For p r a c t i c a l purposes.it would seem to be better to rely.on direct experiment to obtain values of the heat transfer c o e f f i c i e n t s in non-symmetrical ducts.  Further research on flow i n such ducts i s  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 A n a l y s i s o f t h e S h e a r Probe  P  Fig. An above. flow, be  31  I d e a l i z e d Cross  S e c t i o n o f Shear Probe  i d e a l i z e d c r o s s s e c t i o n o f t h e shear probe i s shown  I t i s assumed to be c o m p l e t e l y the f l u i d  i n contact with  surrounded by f l u i d  the w a l l s b e i n g a t r e s t .  the case i f t h e probe i s w i t h i n a laminar  i n laminar This  will  sub l a y e r and i t s  t h i c k n e s s , b , i s s m a l l compared t o the t h i c k n e s s o f the l a y e r . The  —yU.C. The  shear s t r e s s i n the f l u i d  and on the w a l l . i s g i v e n by,  p r e s s u r e head p i c k e d up by,the probe i s  around the probe i s completely v a r i a b l e s ,0 jx t  , b  and t h e fow l  s p e c i f i e d by,the v a l u e s o f t h e  andC...  Thus i t f o l l o w s that.,  (p*_p)  -  Choosingp>|CA and t the methods o f d i m e n s i o n a l  R e p l a c i n g C by  (p,jUL,t,c) as independent v a r i a b l e s and a p p l y i n g  a n a l y s i s i t can be p r e d i c t e d t h a t ,  ^ J  }  I f a s i n g l e probe i s c o n s i d e r e d dropped from t h e r e l a t i o n s h i p l e a v i n g ,  b i s constant  and mdy be  67  Thus there exists a unique relationship between the f l u i d properties, the shear stress on the wall and the pressure head picked up by,the probe.  If a means-of c a l c u l a t i n g 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 i n the concentric case and then used to measure shear stresses with the core tube eccentric. It should be pointed out that the v a l i d i t y . o f the method does !  not depend on the v e l o c i t y p r o f i l e being linear over the probe mouth. It i s s u f f i c i e n t that the p r o f i l e should be determined e n t i r e l y by the shear stress at the wall and by the f l 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 S o l u t i o n i n an E c c e n t r i c Annulus  The symbols used i n t h i s d i s c u s s i o n a r e d e f i n e d i n F i g . 32. 1  ^  yV/  0  c  ^  ^  -<3^  \  1  c  >  \  c  J > s.  •  >  Fife. 32 Geometry.of an E c c e n t r i c Annulus  For e s t a b l i s h e d laminar flow o f an i n c o m p r e s s i b l e newtonian fluid  i n a duct the a x i a l v e l o c i t y , V , obeys the e q u a t i o n (Ref. [_22]  /x. V V X  =  - P  w i t h boundary c o n d i t i o n s V = 0 on a l l s o l i d Let  V  Equation  *  =  (i)  boundaries.  T^(X*+L£)  (1) reduces t o ,  V^U^  F o r a more d e t a i l e d d i s c u s s i o n see R e f .  (2) —  O  .  )  and the boundary conditions to,  ^-/^ i * ^ * * ^ )  l|i -  °  n  a l l s o l i d boundaries. Apply the conformal transformation (Ref. \22\ )  "Z  C o t (§/>0 - -  i c  -  "Z. -  where,  §  =  (3)  X +• L  1 +• CY}  Under this transformation an annular region i n the 2. plane transforms into a rectangle i n the ? plane bounded by the l i n e s ; =  YJ  = ^  , | - O  and  |  = XTT.  Geometrically the value of Y^ at any point i n theZplane i s equal to the logarithm of the r a t i o of i t s distances from the points (+C, o) and ( - C  . The value of f at a point i s the value of the  ;  angle subtended at the point.by the l i n e C C . It may e a s i l y be shown that, -y  .^rJl  r  —  U - c ^ L ° .,  and  i  ^  ^(ccr^yli n ^ Y) j  _^c e?o |^ )  (4)  In the new coordinate system the f u n c t i o n s a t i s f i e s  a.  equationV^-0 j with boundary conditions,  ^  —  V  (\ Co^vjn Ccrvh  »_ cceeoo | /  o  n  a  1  1  boundaries.  Assume a solution of the form  Substituting this i n the equation V  y — O  and solving gives.  the  70  where  ^5  ^  Since ^  , Cw, and  O  w  are a r b i t r a r y constants.  i s an even function of \  solution i s therefore ^ =  ,  Aw,  ^ (Cm61- D £  CoOrr\  ~  O  . A  ^).The most general  w  solution i s therefore, CO  f (|,*)) - ^  D ^ e ^ ) Co-^ro^  + CoY) + IX - -(5)  where vn i s now a p o s i t i v e integer, andCr^and CCare functions of w  .  This may be rewritten as, . OO  =  *Cvj f  Y L CU(vj) C o ^ w ^  U  (6)  0  where C U A * ) ) i s an unknown function oft(\ a n d .  j^i}  a n  d j ^ ( ^ 5 ^ ^  A  R  known from the boundary  E  conditions, therefore for ^=^iandYj=T| equation (6) may be treated z  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  using the theory of residues to be equal to P . C>  -  y o /  )'  CUl<) ) x  WT  - /X €  II  V  CO^Gh <~j  From equations (5) and (6)  and  c u (^)x)  = Cw, e  £.  C^rOn Y"^ (Ref.jji})  L|  P . C * -*»\*)x Similarly,  may be shown by  '  +- D m  e  _  ( ) 8  71  - JJT  Hence  \  ^  _  )  ]  and  For the special case of vr\ = O , tp(Y)  7  |)  Co/j^-Do  —  (9)  PuttingTjr-fj | and integrating over the range^ = O to ^ — TT  Co  Similarly  Hence  r  C  0  V], +  Co ^ ^  fco  + Tj„ P.C X  ^  X  =  ^ ^ ( x a r t h  - fyj  ^1  C<rtk  (io)  fy- |)  '(10)  ( Go-tin V). - c c ^ q n \  ^  x  j  Substituting the values obtained for Cw\, D»v^Co> L\>,back into equation (5) the complete solution for ^ i s obtained. Converting back fromty* to V using equations (2) solution for the a x i a l v e l o c i t y i s obtained as,  and (4) the  72 For any eccentric annulus the values of f| Y^and C may (>  be  calculated from the geometry, and the v e l o c i t y d i s t r i b u t i o n computed from the above equation. This was done for the annulus used in the experimental work at an e c c e n t r i c i t y of 50%.  The calculations were performed on the IBM  1620 computer at this University and the results are plotted as v e l o c i t y contours i n F i g . 21.  BIBLIOGRAPHY  Rothfus, R. R, "Velocity D i s t r i b u t i o n and F l u i d F r i c t i o n i n Concentric -Annuli" D. Sc. Thesis, Carnegie Institute of Technology. 1948 Rothfus, R. R. Monrad, C. C , Sehecal, V. E. "Velocity Distribution and F l u i d Flow i n Smooth Concentric Annuli" Ind. Eng. Chem. V o l . 42,.1950 pp 2511 Rothfus, R. R. Walker, J . E., Whan, G.A. "Correlation of Local V e l o c i t i e s i n Tubes, Annuli and between P a r a l l e l 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 F r i c t i o n i n Flow Through Annular Sections." Ind. Eng. Chem. V o l . 47, 1955, pp 913 Knudsen, J.G., Katz, D.L. "Velocity P r o f i l e s i n Annuli.V Proc. Midest Conf. F l u i d Dynamics. 1st Qonf., .1950, pp 175 Knudsen, J.G., Katz, D.L. " F l u i d 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 i n an Annulus with Various Eccentricities.." N.A.C.A. T.N. 3451, 1955 Barrow, H. " F l u i d Flow and Heat Transfer i n an Annulus with a Heated Core Tube." Proc I.M.E.. V o l . 169, 1955, pp 1113 Barrow, H. "A Semi-theoretrical Solution of Asymmetric Heat Transfer i n Annular Flow" Journal Mech. Eng. Sc. Vol 2, 1960, pp 331  Ower, E. "The Measurement of Airflow" Chapman and H a l l , Ltd.!1927 Davis, E.S. N "Heat Transfer and Pressure Drop i n 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 P i t o t Tube." Proc. Roy. Soc. London. Sec. A. Vol 166, 1938 Barker, M. "On the Use of Very Small P i t o t Tubes for Measuring Wind V e l o c i t y . " 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 F l u i d in Turbulent Motion". Proc. Roy. Soc. London, Sec. A. Vol. 97, 1920 Fage, A., Falkner, V.M. "An Experimental Determination of the Intensity of F r i c t i o n on the Surface of an A e r o f o i l " . Proc. Roy Soc, London. Sec. A. Vol 129, 1930 M i l l e r , B. "The Laminar Film Hypothesis" Trans. A.S.M.E. Vol. 71, 1949, pp 357 Mizushina, T. "Analogy Between F l u i d F r i c t i o n and Heat Transfer in A n n u l i " 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 F r i c t i o n Factors for Turbulent Flow Through Rectangular Ducts .' A.S.M.E. Jour. Heat Transfer Feb. 1962, pp 82. 1  Stein, R.P., Hooper, J.W., Markets, M., Selke, W.A., Bemdler, A.J., B o n i l l a , C.F. "Pressure Drop and Heat Transfer to Non B o i l i n g and B o i l i n g Water i n Turbulent. Flow i n an Internally Heated Annulus". .Chem. Eng. Prog. Symposium Series. No. 11, 1954, pp 115 Lamb, H. "Hydrodynamics", 5th E d i t i o n pp 555 Cambridge University,Press Milne - Thomson, L.M. "Theoretical Hydrodynamics" 4.th E d i t i o n 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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