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Heat transfer from a circular cylinder subject to an oscillating crossflow as in a stirling engine regenerator Stowe, Robert Alan 1987

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H E A T T R A N S F E R F R O M A CIRCULAR CYLINDER SUBJECT T O A N OSCILLATING CROSSFLOW  A S IN A STIRLING  ENGINE  REGENERATOR  by ROBERT ALAN STOWE A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE  in THE FACULTY OF GRADUATE STUDIES Department of Mechanical Engineering  We accept this thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA October 1987 c  Robert Alan Stowe, 1987  In  presenting  degree  this  at the  thesis in  University of  partial  fulfilment  of  this  department  or  publication  of  thesis for by  his  or  her  representatives.  Mechanical Engineering  DE-6(3/81)  October, 1 9 8 7  for  an advanced  Library shall make it  agree that permission for extensive  It  this thesis for financial gain shall not  The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date  that the  scholarly purposes may be  permission.  Department of  requirements  British Columbia, I agree  freely available for reference and study. I further copying of  the  is  granted  by the  understood  that  head of copying  my or  be allowed without my written  ABSTRACT  An  experiment  was  designed  and carried out on the  fundamental, but poorly  understood problem of oscillating flow past a single, transverse, circular cylinder. This is an approximation of the flow about a single element in a matrix-type regenerator used in Stirling-cycle engines. The experimental rig was designed and built  to  allow  parameters  tests  to  characteristic  be of  carried various  out  for  Stirling  the  wide  engines.  range  The  of  influence  fluid flow of  these  parameters on convective heat transfer rates was measured so the approximate effects  of  these  same  parameters  on  a  Stirling  engine  regenerator  could be  determined. The main conclusion from the experiment was that average Nusselt numbers, based on test-cylinder diameter and subject to flow conditions similar to those found in Stirling engine regenerators, were 40 to 80% higher than those predicted by a steady flow correlation, for a given Reynolds number. This may be  due  to  the  high  levels  of turbulence  generated  near  the  test-cylinder.  A  secondary conclusion is that the compression and expansion of the working fluid due to a 90  degree  phase angle difference  between the motion of the pistons  raises convective heat transfer rates from the test-cylinder substantially over the 180 degree phase angle, or "sloshing" motion case.  ii  TABLE OF CONTENTS Abstract  ii  List of Figures  v  List of Tables  viii  I. Introduction  1  II. Literature Review A. Unsteady Flow Past a Circular Cylinder 1. Analytical 2. Experimental B. Regenerator Flow 1. Analytical and Numerical 2. Experimental  7 7 7 8 10 10 12  III. Experimental Apparatus and Procedure A. Apparatus 1. Test Rig 2. Working Fluid 3. Instrumentation a. Test-Cylinder and Anemometer b. Thermocouple c. Pressure Transducer d. Crank Angle Measurement e. Oscilloscopes 4. Test Section B. Procedure 1. Calibration 2. Testing 3. Data Analysis  16 16 16 20 20 20 22 22 23 23 24 27 27 30 32  IV. Presentation of Results A. Experimental Matrix B. Determination of Measurement Response C. Gas Velocity D. Fluid Properties E. Heat Transfer Results  34 34 38 39 47 54  V. Discussion of Results A. Gas Velocity B. Fluid Property Variations C. Instantaneous Results D. Averaged Heat Transfer Results VI. Conclusions  88 88 91 92 96 100  VII. Recommendations for Further Work  103 iii  Bibliography  104  Appendices A. Properties of Freon-114 B. Test-Cylinder C. Error Analysis and Sample Calculations 1. Error Analysis a. Measurement Errors b. Correlation Errors c. Temperature Variation with Compression 2. Sample Calculations a. Piston Kinematics b. Velocity Calculations c. Heat Transfer Calculations  106 106 107 115 115 115 117 118 120 120 122 123  iv  List of Figures  Figure 1. Typical Stirling Engine  2  Figure 2. Test Rig  18  Figure 3. Piston Positions for Various Phase Angles  19  Figure 4. Test-Cylinder  21  Figure 5. Test Section  25  Figure 6. Data Acquisition  26  Figure 7. Calculated Bulk Fluid Velocity at Test-Cylinder for Trial 3  41  Figure 8. Calculated Bulk Fluid Velocity at Test-Cylinder for Trial 9  42  Figure 9. Calculated Bulk Fluid Velocity at Test-Cylinder for Trial 13  43  Figure 10. Calculated Bulk Fluid Velocity at Test-Cylinder for Trial 16  44  Figure 11. Calculated Bulk Fluid Velocity at Test-Cylinder for Trial 25  45  Figure 12. Calculated Bulk Fluid Velocity at Test-Cylinder for Trial 36  46  Figure 13. Pressure versus Crank Angle for Trial 28  48  Figure 14. Volume versus Crank Angle for Trial 28  49  Figure 15. Temperature versus Crank Angle for Trial 28  50  Figure 16. Pressure versus Crank Angle for Trial 9  51  Figure 17. Volume versus Crank Angle for Trial 9  52  Figure 18. Instantaneous Nusselt number versus Reynolds number for Trial 3  55  Figure 19. Instantaneous Nusselt number versus Reynolds number for Trial 6  56  Figure 20. Instantaneous Nusselt number versus Reynolds number for Trial 9  57  Figure  21.  Instantaneous Nusselt number  versus Reynolds number  for Trial  13 58  Figure  22.  Instantaneous Nusselt number  versus Reynolds number  for Trial  16 59  Figure  23.  Instantaneous Nusselt number  versus Reynolds number  for Trial  17 60  v  Figure 24.  Instantaneous Nusselt number versus Reynolds  number for Trial  21 61  Figure 25.  Instantaneous Nusselt number versus Reynolds number for Trial  25 62  Figure 26.  Instantaneous Nusselt number versus Reynolds number for Trial  28 63  Figure 27.  Instantaneous Nusselt number versus Reynolds number for Trial  30 64  Figure 28.  Instantaneous Nusselt number versus Reynolds  number for Trial  33 65  Figure 29.  Instantaneous Nusselt number versus Reynolds number for Trial  36 66  Figure 30.  Instantaneous Nusselt number versus Reynolds number for Trial  37 67  Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure  31. Nusselt number versus Crank versus Crank Angle for Trial 3  Angle  32. Nusselt number versus Crank versus Crank Angle for Trial 6  Angle  33. Nusselt number versus Crank versus Crank Angle for Trial 9  Angle  34. Nusselt number versus Crank versus Crank Angle for Trial 13  Angle  35. Nusselt number versus Crank versus Crank Angle for Trial 16  Angle  36. Nusselt number versus Crank versus Crank Angle for Trial 17  Angle  37. Nusselt number versus Crank versus Crank Angle for Trial 21  Angle  38. Nusselt number versus Crank versus Crank Angle for Trial 25  Angle  39. Nusselt number versus Crank versus Crank Angle for Trial 28  Angle  40. Nusselt number versus Crank versus Crank Angle for Trial 30  Angle  vi  and  Reynolds  number 68  and  Reynolds  number 69  and  Reynolds  number 70  and  Reynolds  number 71  and  Reynolds  number 72  and  Reynolds  number 73  and  Reynolds  number 74  and  Reynolds  number 75  and  Reynolds  number 76  and  Reynolds  number 77  Figure Figure  Figure  41. Nusselt number versus Crank versus Crank Angle for Trial 33  Angle  42. Nusselt number versus Crank versus Crank Angle for Trial 36  Angle  43. Nusselt number versus Crank versus Crank Angle for Trial 37  Angle  and  Reynolds  number 78  and  Reynolds  number 79  and  Reynolds  number 80  Figure 44. Cycle-to-Cycle Variation of Instantaneous Nusselt number versus Reynolds number for Trial 27  81  Figure 45. Average Nusselt number versus Average Reynolds number, All Trials, by Speed and Phase Angle  83  Figure  Figure  46. Average Nusselt number versus Average 80mm Stroke Trials, by Speed and Phase Angle  Reynolds  47. Average Nusselt number 80mm Stroke Trials, by DSR  Reynolds  versus  Average  number, 84 number, 85  Figure 48. Average Nusselt number versus Average Reynolds number, 90 Degree Phase Angle Trials, by Stroke-to-Test-Cylinder-Diameter Ratio Figure 49. Average Nusselt number versus Average Reynolds number,  86  180  Degree Phase Angle Trials, by Stroke-to-Test-Cylinder-Diameter Ratio  87  Figure 50. Test-Cylinder Cold Resistance Dependence on Temperature  Ill  Figure 51. Test-Cylinder Cold Resistance Dependence on Pressure  112  Figure 52. Temperature Coefficient of Resistance for Test-Cylinder  113  Figure 53. Nodal Analysis Results  114  Figure 54. Pressure versus Volume  119  Figure 55. Test Rig Dimensions  121  Figure 56. Piston Position and Velocity Coordinates  122  vii  List of Tables  Table 1. Test Matrix and Calculated Results for Trials 1-14  36  Table 2. Test Matrix and Calculated Results for Trials 15-39  37  Table 3. Typical Fluid Property Values  53  Table 4. Cycle-to-Cycle Variation of Test Run 27  54  Table 5. Properties of Freon-114  106  Table 6. Values from Test-Cylinder Calibration  110  Table 7. Measurement Errors  116  Table 8. Correlation Errors  117  viii  I. INTRODUCTION  Stirling as  engines  power  power heat the  are closed-cycle, external  sources  in  a  wide  variety  sources. These  engines  use  ability  heat exchangers  working fluid. This causes the temperature  because to use  a  show  promise  of their  smooth  wide  heat  variety of  to and  from  changes that drive the cycle. fluid  and  The  the surroundings,  within the engine, called the regenerator, is the one  has the most profound effect on engine  that  performance.  regenerator is a special heat exchanger same flow passages  that  to transfer  cooler transfer heat between the working  but the heat exchanger  the  engines  of applications  characteristics, quietness, efficiency, and  heater and  A  combustion  in which  the hot fluid passes  through  as the cold fluid, but at a different times. It consists of  stacks of wire screens or densely-packed metal or ceramic wool. Its purpose is to maintain a temperature  gradient in the engine between the hot and  As  shuttles  the  working  fluid  transfers heat to or from cause a 4%  between  the fluid. A  these 1%  drop  spaces,  remains  Stirling  regenerator  matrix  in regenerator effectiveness can  drop in overall engine efficiency; therefore regenerator flow and  transfer characteristics should be a priority in Stirling still  the  cold spaces.  unknown  about  them.  Figure  engine.  1  1  shows  heat  engine research since much a  diagram  of  a  typical  Introduction Figure  1. Typical Stirling Engine  CYLINDERS SHOWN IN CROSS-SECTION REGENERATOR EXPANSION SPACE  COMPRESSION SPACE PISTON  PISTON  90 DEGREES  Dead Space: Volume unswept by either Dead space r a t i o :  (Unswept volume)/(Volume  Heater and C o o l e r : Tubes Regenerator:  piston.  Wire mesh  swept by one piston)  / 2  Introduction / 3 Unfortunately, meant  that  necessary within  a  past  Stirling  from  0  0.125mm); this and  and  with  to  (based  the  flow  complete can and  up  on  less emphasis on is  engine  highly  on can  over  helium  are  variations  be  fundamental  never  wire  typical  such  through,  Reynolds numbers are typically  transitional, can  or  vary  and  working  data  at  completely  diameter,  entire cycle  has  studies. Flow  oscillates  passes  temperature  an  performance  unsteady,  laminar,  cycle. Fluid  double  to 100  matrix  regenerators  overall  fluid properties. Instantaneous  1000  pressures  Stirling  focus  part of the matrix  a  volume  tended  at one  Air, hydrogen,  paths  within  regenerator  means  the engine  flow  engine  to  over  1000K, and  through  of  have  continually changing  commonly  MPa.  studies  that gas  has  average  complexity  for design purposes,  amplitude and  the  be  turbulent  from as  fluids,  0.025 to  and  on  300K  to  high as  20  these  cycle  Hz. Different engine configurations mean that flow differ  from  engine  to engine  as  well, making it  difficult to identify a typical Stirling engine regenerator flow.  Early  theoretical  limited period  to  and  simple  flow  (long "blow"  required  the  second.  While  matrix  is very  study  Stirling engine  experimental  the  work  conditions, such  times). The  on as  regenerators almost-steady  development  of gas  of shorter blow times, typically residence  short, case. As  time  of a  it nonetheless such, gas  particle passes  had flow  in  general  for a  turbines with on  in the through  the  been  long blow regenerators  order of a  few  per  gas  turbine regenerator  the  matrix,  turbine regenerator results are  unlike  the  not applicable  to Stirling engines.  The  complexity  of the flow in a  Stirling engine  regenerator has  led to the  use  Introduction / 4 of very  simple  and often unrealistic assumptions in heat transfer models. Perfect  regeneration and absence of flow friction cause the most drastic overestimates of performance compared to an actual engine. The frequently used linear  temperature  distribution  along  assumption  the length of the matrix  approximates the  situation in an actual regenerator fairly well. However the assumption temperatures matrix  throughout  change  empirical estimate  continually.  correlations  not, as the entrance  Better  of flow  models  friction  described  and  increments  through  Quasi-steady  flow  heat  chapter  transfer  use  rates to  losses. These have been  screens  is assumed  of steady  conditions of the  in the next  and flow  flow data past stacks of wire  unavailable.  and exit  convective  the effects of imperfect regeneration  based on steady was  does  of a  since unsteady flow data  during  each  of many  time  a cycle. While this method currently gives the best estimate  of Stirling engine performance, overestimation may is largely due to improper estimates  still be as large as 50%. This  of regenerator heat  transfer rates and flow  friction.  The flow  lack of knowledge of even the fundamental mechanisms of heat transfer and friction  in Stirling  incompressible  oscillating  engine flow  heat  exchangers  in tubes.  The  have  prompted  general  recent  conclusion  work on  from  these  experiments is that shear rates at the tube wall are much higher under laminar, transitional,  and  quasi-steady  flow. In the laminar  models,  turbulent  conditions  and in all conditions  case  this  in oscillating  flow  than  in steady  this conclusion is supported  conclusion  is supported  actual engines. Analogously, if wall shear is higher, then  by  or  by analytical  observations in  convective heat transfer  rates must also be higher. Limited work on "incompressible" oscillating flow past  Introduction / 5 stacks of wire  screens indicates that mesh  dimensions  other than  wire  diameter  affect convective heat transfer rates.  Steady on  flow  a  heat  Reynolds  supported transfer  by rate  transfer correlations for convection from number  that  extensive and  uses  wire  experimental  drag  coefficient  crossflow are also based  on  diameter  and  as  theoretical  correlations  cylinder diameter.  for  a  wire  screens is based  length  evidence. single  scale.  This is  Convective  circular  heat  cylinders  in  This situation can be regarded  as  similar to the wire screen case, but a single element of the screen is examined in  the  absence of flow  disturbances from  the  surrounding  elements.  Much  work  has been done on harmonically-oscillating water flow past cylinders in the field of wave  mechanics, with  the flow  respect to drag  and  past the cylinder, vortices and  lift  other  previous flow pass back over the cylinder and compared drag  to the  coefficients  steady are  case. A  parameter  number  and  flow  much called  based  on  case. For  higher the  the  (up  numbers  as  up  to about 50,000,  high) as in the steady  number  (similar to a Strouhal  velocity  divided  by  amplitude  cylinder  diameter)  experimentally-corroborated parameter of numbers for flow situations similar to  regenerators have Reynolds numbers at least a factor of  higher, so empirical results are not applicable. However, results still  important  for the  flow  Keulegan-Carpenter  the conclusion that the Reynolds number as well as a is  during the  cause a change in shear stresses  to twice  the flow. Unfortunately, Keulegan-Carpenter  ten  each reversal of  disturbances generated  Reynolds  complements the Reynolds number as an  those in Stirling engine  coefficients. On  determination  transverse circular cylinders.  of  shear  support  type of Strouhal number  stresses in oscillating  flows  past  Introduction / 6 Since the best way parts and  study  to solve a complex problem is often to break it into smaller  each part in the absence of the others, an  to examine the fundamental aspects of regenerator present  in  qualitative  a and  flow past a variation  Stirling  engine  was  quantitative insight  single heated  carried into  the  heat transfer under conditions  out.  The  to  the  objective  fundamental  transverse cylinder, with  conditions similar  actual case.  experimental project  was  problem  to  of oscillating  Reynolds number and  A  test  gain  rig was  volume  designed  and  constructed to allow measurements of convective heat transfer rates from a single transverse  cylinder  interferences  present  in  the  absence  of  in Stirling engines.  wide Under  of parameters that would encompass those other  parameters  evident. Just  as  of the  flow  convective  transfer  these  found  geometry, such  heat  temperature  as  variations  conditions (within a  in Stirling engine a  and  flow range  regenerators)  Strouhal number, could  rates in steady  flow  past  a  be  single,  transverse circular cylinder are related to steady flow past stack of wire screens, knowledge of convective  heat  transfer  single, transverse circular cylinder may engine  regenerator.  rates for various oscillating  flows  past  a  be related to the same flow in a Stirling  II. LITERATURE REVIEW  A. UNSTEADY FLOW PAST A CIRCULAR CYLINDER  1. Analytical  Analytical  solutions  accelerated  flow exist. In Sarpkaya and Isaacson  time  dependent  flow  for special  about  cases  of both  a transverse circular  oscillating  into account, but this case  and  suddenly  [1] analytical solutions for any cylinder  inviscid, ideal fluid case. Separation and the presence taken  flow  are presented  for the  of boundary layers are not  is useful to show that particles far from the  cylinder are disturbed, so therefore in a real case, boundary conditions far from the cylinder can affect the flow.  Schlichting problem term  [2] presents  using  a  solution  the similarity  to the suddenly-accelerated  transformation technique  circular  cylinder  of Blasius [3]. A  viscous  allows for the no-slip condition at the fluid-cylinder boundary. The point of  separation  of the boundary  layer  from  the cylinder  can be  found  from  the  resulting equations: it may  start after the cylinder has moved a little more than  a third of its radius from  its initial point. This indicates that separation could be  a  very  important  phenomena  for the  determination  of cylinder-to-fluid  heat  transfer rates, even for flows with small accelerations.  Once  separation  occurs,  the flow  pattern  outside  the boundary  layer  changes  greatly. Pictures taken by L. Prandtl in [2] show that two similar vortices form  7  Literature Review / 8 behind  the  cylinder  soon  after  downstream. The wake behind describe pressure reported the  separation,  flow  and  the cylinder becomes very  are  Finally  theory  successful, and the differences between  and the calculated pressure  after separation. This indicates that other  swept  unsteady. Attempts to  distribution characteristics of this flow with potential flow  by Schwabe [4] were not very  real  grow,  distribution  increased  with  the time  characteristics of the flow, particularly  heat transfer rates, could not be calculated by analytical means after separation occurs.  The  case of a harmonically-oscillating cylinder in fluid at rest is also reported in  [2]. Schlichting calculated an analytical solution to the case for small oscillations  (amplitude  very  as acoustical streaming with the  much  less than the diameter of the cylinder, known  flow). Pictures of streamlines from experiments agree well  the calculated streamlines. However, for oscillations of the same radius  of the cylinder  streamlines  are substantially different  separation (which is the case in this author's  2.  amplitude  order as  because of  work).  Experimental  Sarpkaya  and Isaacson  cylinder with the  [1] deal with  respect to drag  Keulegan-Carpenter  unsteady  flow  about a transverse  circular  and lift coefficients. Another parameter, known as  number (K) (velocity  amplitude  times flow  period divided  by  cylinder diameter) is introduced  as  compared to cylinder size. This parameter is similar to the Strouhal number  (but  inverted)  and is shown  to describe the amplitude of the fluid  to be  important  to determine  drag  and  motion  inertia  Literature Review / 9 coefficients in oscillatory flow at high Reynolds numbers (greater than  The  Reynolds  friction  drag  boundary  analogy and  between  convective  layer. In [1] drag  heat heat  and  momentum  transfer  coefficients  rates  reported  transfer are  10000).  shows  related  for oscillatory  in  that a  flow  10000 and 50000 and Keulegan-Carpenter  number  regenerators. For the range of Reynolds numbers  and  transfer  cylinder  might  quasi-steady  flow  also  rates from  be  case,  expected and  should  Strouhal or Keulegan-Carpenter  Richardson  [5] presented  acoustical  streaming  transverse, oscillatory to be  be  substantially  related  to a  numbers  in the range relevant  to Stirling engine 10000, heat  laminar  are much  higher (up to twice as high) as those for quasi-steady flow at Reynolds between  skin  between  50  flow past a circular higher  parameter  than such  in the as the  number.  a theoretical and experimental study of heat transfer in  flow.  He  built  streamlines calculated by Schlichting  upon  the  analytical  [2]. The experimental  solutions  for the  and theoretical results  are similar when natural convection effects are ignored. Once again, however, the streaming  flow case  the flow expected  infers small amplitude  oscillations with  in the Stirling engine case.  no separation, unlike  Literature Review / 10  B. R E G E N E R A T O R  1. Analytical and  The  regenerator  important  for  mechanics and  FLOW  Numerical  is  the  heat  performance,  but  exchanger it is  to  to estimate  Stirling  simple treatments  simplest  analyses  such  example  is the  in  assumed  Stirling  engine  that  is  most  the  most  complex  from  a  fluid  engine  a result, there are no  analytical  in sufficient detail to allow for design.  and  therefore regenerator performance  The  has  lead  regeneration.  One  of the problem.  The  are  also  a  heat transfer point of view. As  methods that deal with the problem need  in  for the  Stirling  Schmidt analysis  to stay constant with  cycle  assume  perfect  [6,7]. Temperatures throughout the engine  time  and  the  linear  temperature  distribution  the regenerator remains constant as well. Flow losses in the regenerator are  also neglected. This analysis estimates engine  performance at about twice that of  a similar, but real, engine.  More complex analysis techniques use of Stirling engines The and to in  numerical  methods to estimate performance  ideal adiabatic model [6] still assumes perfect regeneration  a steady, linear temperature  distribution in the regenerator. This contributes  the overestimation of engine performance. However, improvements to the model areas  other  than  the  regenerator  allow  a  more  performance to be made than with the Schmidt analysis.  accurate  estimate  of  Literature Review / 11 Attempts Stirling and  to  incorporate  engine  analysis  flow  flow  friction.  regeneration  more  accurate  technique  use  approximations  empirical data  Because little  experimental  (as  by  conceded  steady  flow  data  is used  divides  the  regenerator  for the  flow  work  several  into  to  has  Stirling  calculations. The  cycle  many  of  time  regeneration  estimate been  into  heat  transfer  done on  engine  unsteady  researchers  quasi-steady  flow  increments;  a  [6-12])  model  steady  [6]  flow is  assumed during each increment. Temperature is allowed to vary with time within the  regenerator  matrix.  This  method  gives an  improved  result  for performance  characteristics over the methods that assume perfect regeneration, but the steady flow an  assumptions actual  engine.  underestimated  A  by  lead  Urieli  to overestimates  of engine  and  [6]  Berchowitz  performance  report  that  flow  compared friction  to was  a factor of four compared to the real case.  method that concentrates on regenerator design rather than engine performance  prediction was number and or  still  mesh  to  for a the  engine  stack  range  of  of  regenerators.  screens  Reynolds The  were  used.  numbers  correlation  the wire. Flow  friction  spacing of the  wires  matched  graph  compared  a  factor is correlated in the of  the  to its internal heat  mesh. The Biot  with  engine  number  conduction) and  (based on  The  and  for  number to Reynolds number, with the geometric  to  flow correlations of Nusselt  flow friction factor versus Reynolds number  size)  correspond Stirling  reported by Miyabe et al [13]. Steady  wire  heat  wire  correlations  diameter presented  parameters  transfer  relates  used  in  Nusselt  parameter being the diameter Reynolds number and  (surface  regenerator heat  based  on  the  parameters  are  convection  Fourier number  of  (diameter  of  wire  of wire  compared to thermal penetration during the regenerator blow period), so that the  Literature Review / 12 temperature  at the centre of the wire  working  fluid. The  number  from  theoretical  analysis  a  matrix. These  of screens that  uni-directional flow  inlet  (for each  close to the for the  idealized  temperature  regenerator  flow  is  of the  determined  conditions through  from  this  regenerators in one  fluid  and  matrix  the  heat  Number  capacity  and  to choose the number of screens required to yield  than  0.95.  Flow  friction  is  is repeated for a different wire  method  properties, and  effectiveness versus  graphs for various ratios of matrix  effectiveness greater  excessive, the procedure  constant  half-cycle). Regenerator  to fluid heat capacity are used  compared  favourably  engine. These results should  but the validity of the method may experiments  needed  assumes  temperature,  of Transfer Units (NTU)  Results  be  assumptions include constant mass velocity, constant heat transfer  coefficient, constant  an  will  be  then  and  if  tests  of  diameter.  with  data  not be  supported  calculated  from  regarded  as conclusive,  if results from  unsteady  flow  yield the same type of correlations as the steady flow data.  2. Experimental  Recently  have  researchers  assumption  that  exchangers  are  and  and  the  steady  transfer  obtained rates  higher than  Aghili  [15] on  coolers than  transitional, and in  much  Taylor and  heaters  heat  have  and  experimental  data  flow  in  in steady  oscillating  regenerators  flow  but  case. A  survey  of work  Stirling  flow. Experiments in tubes  they  turbulent flow, pressure drops flow  friction  that  do  show  measured on  are  more  that  are  oscillating  by  the  engine  heat  Dijkstra  by  [14]  relevant to  under  much  flow  supports  laminar,  higher Seume  than and  Literature Review / 13 Simon  [16]  discusses  conditions. The  duct  flow  under  fully-developed laminar  laminar,  case  transitional,  and  turbulent  shows that the wall shear  stress is  eight times higher in oscillatory flow than in unidirectional flow. However, in the Stirling engine  case  duct lengths are  flow. In transitional and  is reduced  higher  flow, turbulence  long enough  appears  to be  during acceleration. This is the  most important are  not  to assume fully-developed  enhanced during deceleration  region of flow that is probably  in Stirling engines. They state that turbulent flow pressure  than  in steady  flow, but  means  to predict  them  are  drops  disputed  by  several researchers.  Seume and number  [16] propose that the  (or Valensi number  divided by such  Simon  four times  which  the kinematic  is the  frequency  viscosity), and  times  a  another  Reynolds  diameter  geometric  squared  parameter,  as a length-to-diameter ratio describe oscillatory flow sufficiently to compare  flows  in different  kinetic  number  engines  Reynolds number  parameter.  One  and  is very  interesting  is related  to the  note  Rice et al [12] that  deal with  varies  expansion. The  the  experiments. small and is that  in  However,  flow  the wire  any  regenerator  given by  to be  situation,  a  simple  flow  the  an  important  the  Reynolds  geometric  ratio,  diameter.  oscillating flow in a velocity  in  is not expected  kinetic Reynolds number  such as the stroke divided by  rig  Reynolds number, the kinetic  stack of wire  sinusoidally,  but  range of Reynolds numbers is at the low Correlations of Nusselt number  with  screens in a  test  compression  or  without end  an  actual engine.  on  wire diameter) are presented. These correlations depend on  of that expected in  Reynolds  number  (based  wire diameter  and  Literature Review / 14 mesh  size, unlike the  steady  flow  correlations used in [13] that were based  Reynolds number. Therefore heat transfer rates in oscillating flow may parameters other than and  cooled  regenerator values  at  the  opposite  on  Reynolds number. In Rice's work the flow is heated  ends  is influenced by  of heat  depend  on  of the the  regenerator;  heat  transfer rates with  the  exchangers. A  the  steady  flow  flow  in and  comparison data  out  of the  of  the  absolute  in [13] is difficult in  this case.  The that  main conclusion that can much  understood  more  work  about heat  needs  be drawn from to  be  transfer from  done  the experimental  on  regenerator  work discussed is  flow.  Since  little is  a cylinder subject to similar flow conditions  as well, this experiment focussed on this more fundamental problem. The  specific  objectives of this experiment were: 1.  To  measure the convective heat transfer coefficients from a circular cylinder  in  oscillating  flow  regenerator and 2.  To  measure  motions had 3.  To  the  conditions  similar  to  flow  in  a  transfer  Stirling  engine  compare them to steady flow values. effect, if any,  that phase  angle  difference of the piston  on the convective heat transfer rates in this experiment.  discover the effect, if any, that dead space ratio had  heat  rates in this  experiment.  widely used to describe a Stirling engine dead, volume in the engine divided by 4.  the  (Dead  space  ratio  on  the convective  is a  parameter  configuration. It is the unswept or  the swept volume of one piston.)  To  discover the effect, if any,  on  convective heat transfer rates in this experiment. This will be analogous  to the stroke-to-wire-diameter  that stroke-to-test-cylinder-diameter ratio  ratio in a Stirling engine  regenerator.  had  Literature Review / 15 5.  To  discover  any  other parameters  that  may  have  affected  convective heat  transfer rates in this experiment apart from Reynolds number, phase angle, dead space ratio, and stroke-to-test-cylinder-diameter ratio. 6.  To discover any the  convective  peculiarities in the flow parameters that may heat  transfer  results  over  those  expected  have affected  from  a  purely  sinusoidal oscillating flow case.  The  flow  field  configuration the use  was  established  the  use  to the engine shown in Figure  of a heated-film  transducer  by  and  test-cylinder, similar  thermocouple measured  speed.  Nusselt  numbers  a  1. Heat to a  test  and  were calculated for the various test runs.  rig with  transfer was hot  the pressure and  and bulk gas velocity at the test-cylinder was rotational  of  film  a  similar  measured  probe. A  pressure  temperature of the gas,  calculated from crank position  Reynolds  by  numbers  for the  and  test-cylinder  III. EXPERIMENTAL APPARATUS AND PROCEDURE  A. APPARATUS  1. Test Rig  The a  motivation behind  flow situation  maximum and  the design and construction of the test rig was to produce  similar to a Stirling engine  with  simplicity  peak Reynolds number that had to be accommodated  by using  was about 1000,  a heavy gas as the working fluid and a large diameter  "probe", or test-cylinder, pressures and speeds could be kept the  and versatility. The  rig design  simple,  safe, inexpensive, and easily  hot film  low enough to keep  achievable. Adjustability of  phase angle, piston strokes, and connecting rod lengths was necessary  so a wide  range of flow parameters could be examined.  To  achieve the necessary  peak Reynolds number with Freon-114 as the working  fluid, the parameters chosen were a maximum pressure of 1 Mpa (150 psia) and piston  speeds  of 0.5m/sec. The simplest  arrangement, as shown  in Figure  gain  transfer  insight  into  heat  1. Since  rates  (absent  path),  horizontally-opposed piston setup  flowpath the  test  from  influences such  from  crossflow a  from  Stirling  engines  use an  opposed-piston  the goal of the experiment was to a  circular  cylinder  in a  as twists and constrictions was  chosen  with  a  in the flow  constant-diameter  one piston to the other. The test section in the middle  cylinder,  gas temperature  thermocouple,  observation windows.  16  pressure  "simple"  transducer,  contained and two  Experimental Apparatus and Procedure / 17 The  piston diameter chosen was 32mm, since this is the smallest diameter piston  commercially travelled  in its own  rectangular under  available for common internal combustion engines (50cc). Each piston cylinder, honed  teflon rings  dry lubrication  contamination  were  machined  conditions.  of the test  for a small  section  to allow  piston-cylinder a good  (Dry lubrication by oil.) A  seal  was  piston  clearance. and easy  necessary  pin fitted  Solid sliding  to prevent  to each  piston  allowed a self-aligning spherical rod end to be used. Some misalignment could be tolerated without having to resort to increasing the size of the linkages and pins.  As  can be seen  from  Figure  2, the crank  section. The crank plates were drilled  was placed  to accommodate  directly strokes  below  the test  of 50, 100, and  150mm, and phase angles of 0, 90, and 180 degrees between the pistons. Figure 3  shows piston positions for the various  a  pinned  link  transferred  the rotary  phase angles. Two connecting rods and  motion of each  crank  plate  to the nearly  sinusoidal reciprocating motion of the piston. The connecting rod adjacent to each piston was threaded to allow  To  drive the pistons at the desired  source chosen  was needed. A to allow  1/2 hp D C  speed  controller was set. A (3:2  complete adjustability.  speed under a pressure of 1 MPa, a 1/2 hp electric  motor  adjustability, but little  with  an SCR  variation  controller was  in speed  once the  20:1 speed reducer and a pair of high-torque drive pulleys  ratio) and toothed belt stepped  the maximum  rotational speed of the motor  down to about one Hertz. A l l drive components were mounted on a welded steel frame and  made  rigidity.  of 2-inch-square  tubes  that  provided  more  than  adequate  strength  Experimental Apparatus and Procedure / 18 Figure 2. Test Rig  Experimental Apparatus and Procedure / 19 Figure 3. Piston Positions for Various Phase Angles.  PHASE  0  DEGREES  90  EXPANSION  DEGREES  COMBINATION  COMPRESSION AND  ANGLE  AS  IN  A STIRLING  ENGINE  180  DEGREES  SLOSHING  Experimental Apparatus and Procedure / 20  2. Working Fluid  To achieve a peak Reynolds number of 1000 a heavy  gas that could be used at  room temperature and pressure was required. Freon-114 was  chosen  therefore  because  low  atmospheric  of its availability, low toxicity, high  kinematic  viscosity),  pressure at room  and  low  vapour  temperature. By  section of the rig with a heating tape to about Freon-114  could  (dichlorotetrafluoroethane)  be raised  to 1 MPa. A  molecular weight (and  pressure  heating  of about  the cylinders  twice  and test  90° C, the pressure of gaseous  table of properties of Freon-114  is  in  Appendix A.  3. Instrumentation  o. Test-Cylinder and Anemometer  Because  measurements with hot film probes usually require minimum  dependence  on the Reynolds number of the surrounding flow as well as fast time response, the  diameter  of  these  commercially-available  probes  is  small,  usually  0.025-0.050mm. The diameter of the test-cylinder needed  for the test rig to yield  the  size  required  Reynolds  numbers  was  about  2mm.  This  is not commercially  available, so one had to be designed and built. Nickel was chosen  as the film  since it can easily be deposited on a glass substrate in a vacuum chamber, and its high temperature coefficient of resistance made it ideal for use with hot-film anemometer 1.8mm  equipment.  The test-cylinder  diameter, cut to 28mm  chosen  length. It would  was  a glass  capilliary  then fit inside the test  tube of section  Experimental Apparatus but  was  of  sufficient  length  to  minimize  and Procedure / 21  three-dimensional  effects  at the  test-cylinder ends. The glass substrate provided the desired thermal properties for a hot film probe by insulating the nickel film. During deposition of nickel vapour in the vacuum  chamber on the glass, the test-cylinder was rotated to ensure an  even coating. Enough nickel was deposited to yield a film resistance of about five ohms.  The  test-cylinder  thin wire rod and  mounted  (used for winding  between  some  silver system  paint.  acrylic  rods  and glued  with  epoxy. A  transformer coils) ran along the top of the acrylic  to the end of the film, where  mounting the  was  This  proved  for the test-cylinder  it was  connected  electrically  to provide  a  rigid,  and allowed  an easy  with  thermally  one loop insulating  electrical hookup to  anemometer.  Figure 4. Test-Cylinder  Test-cylinder as seen through end of test r i g  Experimental Apparatus The  and Procedure / 22  test-cylinder and mount were placed in the test section between two Conax  fittings. These fittings leaking from  sealed the wire leads completely and prevented  the test section. The leads were hooked  Freon-114  via a coaxial cable to a  Thermo-System Incorporated Model 1010A Constant Temperature Anemometer. The anemometer allowed any hot resistance setting up to 30 ohms to be selected in increments  of 0.01 ohms. The output gave a voltage present in the bridge; this  could be used  to find the power dissipated from  the test-cylinder. (See Appendix  B.)  b.  Thermocouple  The  thermocouple  probe  chosen  had to be sturdy enough to withstand insertion  into the test rig, but small enough to ensure fast thermal response. A diameter probe  exposed  was  junction  hooked  battery-operated  up  copper-constantan to a  cold  thermocouple  junction  probe  compensator,  1/32 inch  was chosen. This then  to a  small  100:1 gain amplifier, and finally to an amplifier that boosted the  output by another factor of ten. This allowed one millivolt of thermocouple  output  to be the equivalent of one volt of amplifier output.  c. Pressure  A  Transducer  strain-gauge  type  absolute  pressure  transducer  was  used  to measure gas  pressure in the rig. It was hooked up to an amplifier calibrated so an output in millivolts corresponded  to the pressure in psia. This setup yielded extremely  response to pressure variations of less than one millisecond.  fast  Experimental  Apparatus and Procedure  / 23  d. Crank Angle Measurement  One  crank  plate was fitted with  a ten-inch diameter  plate drilled with  180 holes  around its circumference, plus one "reset" hole closer to the centre. Two infra-red photo-emitter/detectors and  the reset  photodetector  are mounted  hole  to emit  passes  by  a pulse  so the outside holes the other.  Passage  to an electronic  pass  by one of them,  of a  circuit  hole  causes the  that contains  a staircase  generator. Passage on an outside hole boosts the output voltage of the circuit by about  20mV, while  the reset hole causes the output  voltage to be set back to  the minimum value. The setup allows the position of the crankplate to be known at  any point in time within 2 degrees of rotation. The linearity  very  good, so calculations  involving  crank  angle  (such  of the output is  as piston  velocity) are  simplified.  e. Oscilloscopes  Two  2-channel  voltages  Nicolet  from  each  digital  of the four  temperature,  pressure, and crank  signal from  the crank  1000  Hz. Each  seconds.  This  IBM-compatible  angle  channel data  oscilloscopes were  personal  setups  to sample  (test-cylinder  circuit and the oscilloscopes were store  then  be  4000  later.  points,  transferred  computer for storage  disk, where it could be analyzed  the output voltage,  angle). The sweeps were triggered by the reset  could  could  measurement  used  on a  set to sample at  so sweep in binary 5.25 inch  time  was four  format diameter  to an floppy  Experimental Apparatus  and  Procedure  / 24  4. Test Section  The  test  section  thermocouple, observation  had  pressure windows.  to  be  designed  transducer, It  was  a  made  to  port  of  accommodate  for  Freon-114  aluminum.  section were kept as small as possible to minimize and  secondary  test section  flows. The  so  the  Freon-114 supply was  supply  tube  between  the  Holes  glass  at the  consisted of an  outside end.  A  reduce  secondary  discontinuities. A a schematic  flow  test section  thin, flexible, clear  effects  and  and  into  two  the  test  (unswept) volume  fitted with a check valve in the and  the  first shut-off  windows were filled with  inch long hollow  inside of the test section bore to allow gas  test-cylinder,  supply,  drilled  extra dead  valve would not become part of the dead volume. The acrylic plugs since they  the  plastic  plug with film  was  1/4  inch of  glued  to the  "communication" across itself, but to  streamline  the  flow  along  the  bore  photograph of the test section is in Figure 5. Figure 6 shows  diagram of the data acquisition system.  Experimental Figure 5. Test Section  i  Apparatus and Procedure / 25  THERMOCOUPLE PRESSURE TRANSDUCER  AMPLIFIER AMPLIFIER  CRANK ANGLE DETECTOR  AMPLIFIER AND STAIRCASE GENERATOR  TESTCYLINDER  HOT FILM ANEMOMETER  DIGITAL OSCILLOSCOPE DIGITAL OSCILLOSCOPE  Figure 6. Data Acquisition  PERSONAL COMPUTER  Experimental Apparatus  and  Procedure / 27  B. PROCEDURE  1. Calibration  Measurement  setups  linear  outputs  allow  linear  over  for the  temperature range  and  of values expected  algebraic relations to be  temperatures  and  thermocouple  to  pressures.  different  pressure  used  This  and  supposed  during the  to  provide  tests. This would  to correlate the output voltages with  linearity  temperatures  were  was  the  checked  by  exposing  pressure transducer  pressures. Results indicated that a linear correlation could be  used  the  to different  in both cases  (see Appendix C).  Some  interesting  calibration, and was  properties  of  the  test-cylinder  discovered  into  the  data  analysis  to allow  for changing  transfer rates were calculated. This dependence on the fact that causes  during its  were taken into account during the data analysis. Film resistance  found to increase almost linearly with pressure, so a  incorporated  wall  were  the test-cylinder is hollow, and  a  slight distortion  linear correlation pressures when  pressure can be  heat  attributed to  the pressure difference  of the test-cylinder. The  was  across the  resistance of the nickel  film therefore changes, much like a strain gauge.  Film resistance varied with temperature temperature  coefficient  test-cylinder was  of  heated. A  resistance  in a linear fashion, as expected, but the that  resulted  depended  on  the  pure nickel film should have a temperature  of resistance of 0.006 ohms/ohms/°C. A  film  of 9 8 %  nickel and  2%  way  the  coefficient manganese  Experimental Apparatus and Procedure / 28 has  a value of 0.0045 ohms/ohms/°C,  film  will  decrease  the value. The  and  this indicates impurities in the nickel  exact composition of the  test-cylinder is not known, so the test-cylinder was resistance  on  temperatures readings  the were  indicated  test-cylinder was variance  in  measured that  test-cylinder  one  temperature  so  invariant length  point. The  the  value  thermocouple.  Because  approximately  along  on  the  could  the  The  be  the  thermocouple length  of  the  little circumferential assumed  temperature  parabolic,  values. Surface  the  that there was  test-cylinder  properties. was  a  to different  distribution  approximately parabolic and  coefficient of resistance was at  with  film  heated, in air, by setting the  anemometer  directly  the  temperature,  circumferentially the  constant temperature  nickel  to  have  distribution  along  average  temperature  found from the graph of thermocouple readings taken was  found  to be  0.005297  ohms/ohms/°C  (to within  2.5%).  The  test-cylinder  cylinders  and  test-cylinder  was  test and  also  section  subjected with  its mounting  the were  to  type of heating, less than  temperatures  heating tape. This subjected  temperature coefficient of resistance was this  different  to  the  meant same  by  heating the  that  the  entire  temperature.  The  found to be 0.001885 ohms/ohms/°C for  half that  indicated  for heating the test-cylinder  electrically. This can be explained by the fact that in each case, the test-cylinder and  mounting is subjected to a different set of thermal stresses, so each type of  heating will have a different temperature coefficient of resistance. The for  uniform heating was  when the gas temperature heating  used  to adjust the cold  resistance  varied during a test run. The  (parabolic distribution) was  used  coefficient  of the test-cylinder  coefficient for electrical  to calculate the surface temperature  of  Experimental Apparatus  and  Procedure / 29  the test-cylinder.  Free convection losses from power  loss with  convection  loss  approximately convection  the for  the test-cylinder were thought to be small. The  pistons motionless these  conditions  10-15% of the  were  not  of the total heat  was  conduction also  taken  out  analysis. Even  the  into  power loss. Changes  included in the  however, free convection from 10%  total  was  ends. The  account,  in the  main free  and  was  amount of free  at small Reynolds  numbers  cylinders in steady crossflow accounts for less than  transfer, so any  change in this would  affect the overall  Nusselt number minimally.  Three-dimensional  effects  test-cylinder being about  were  reduced  15:1.  "Trailing"  by  the  length-to-diameter  vortices from  the  ratio  edges of the  of  the  acrylic  rod would not reach in far enough to disturb the flow over the effective part of the  test-cylinder.  temperature of the end  For each  The  parabolic  temperature  distribution  means  difference is smaller at the ends, effectively reducing the  test, temperature, heated  pressure, and  electrically, but before the rig was  calculated. From  found  setting  importance  bridge voltage were recorded while the operated. This gave  reference values so that during the test run, the change in cold  be  the  losses further.  test-cylinder was  could be  that  because  remains  the  fixed  this the surface temperature  effective during  a  overheat  ratio  run). The  power dissipated in the test-cylinder due  would  initial  film  resistance  of the test-cylinder could change  bridge  to conduction and  (the hot  voltage  resistance  reading  yields  free convection at the  Experimental Apparatus and Procedure / 30 initial  conditions,  so the power  dissipated  during  the test  run due to forced  convection and additional free convection could be calculated.  Calibration results are presented in Appendix  B.  2. Testing  Many fluid  test  runs  were  made  over  pressures, and dead-space  a varietj' of strokes, phase  ratios.  Runs  at a  phase  angles, working  angle  of 0  degrees  ("pure" compression or expansion with no "sloshing" motion) were done to get an idea of the time response of the instrumentation. Runs at a phase  angle of 180  degrees  done  (complete  sloshing,  no  compression  or expansion)  were  as all  experimental work on oscillatory flow to this point in time has been done in this mode  and a  direct  comparison  of results  would  be possible. These  runs  were  done at different speeds and pressures to yield results over the desired range of Reynolds numbers. The phase  angle setting of 90 degrees comprised the majority  of the runs since this is the case most relevant to conditions in a real Stirling engine.  Equipment  Reynolds  numbers  settings  and  and dead-space  pressures ratios  were  that  varied  encompass  so  a  wide  those  found  range  of  in actual  engines were examined.  Each  set of tests  stroke, phase  (usually  four  to six) was  preceded  by  setting  the desired  angle, and connecting rod length on the test rig. For phase angles  of 0 or 90 degrees the crank was set so the pistons would  be furthest  (minimum  positioned  volume).  For 180  degrees  the crank  plate  was  apart  at the  Experimental Apparatus arbitrary vapour  0 degree to  a  condensation  position. The working  desired  pressure  and  space  was  and Procedure / 31  then  filled  to  make  the rig cycled  with  Freon-114  sure  that  no  of the Freon-114 took place. If it did, some fluid was bled off (or  the cylinders and test section were heated).  After  the proper  determined, always then  pressure  was  obtained,  the cold  and the corresponding hot resistance  used) was set to  temperature,  resistance  (an overheat  set on the constant temperature  run, supplying  the  pressure, and bridge  test-cylinder voltage  were  of the ratio  film  of 1.2 was  anemometer. The unit  with  power.  recorded  At  this  to provide  speed  until  the oscilloscope  traces indicated  that  cycle-to-cycle  was sufficiently small and a "quasi-steadiness" had been reached. A sweep  of the measurements  was  disks in the computer. The cold had  not changed from  then  taken,  resistance  checked  point,  at the  variation  four second  stored, and transferred  was then  was  the initial  reference settings for the later analysis. The test rig was then operated desired  was  to floppy  to ensure  that it  before the test, and after the data had been stored, the  next test was done at a different speed and/or pressure.  Experimental  Apparatus  and Procedure  / 32  3. Data Analysis  Voltage values stored in the digital oscilloscopes were transferred  to binary files  on floppy disks in a personal computer by the use of a data acquisition program called  GRAFPAK.t  The  crank  angle  velocities so the Reynolds number  voltages  had  to  be  converted  of the flow could be calculated. A  to gas  third-degree  polynomial was fitted by regression analysis to a set of voltages that represented one  crank  degrees versus  angle rotation, and the voltages were scaled so the lowest could be 0  and the highest 360 time.  Approximately  depending on the speed  degrees. 180  This gave a plot of crank  points  (spaced  they  were  manipulated difference velocity  read  to yield  in piston  was  position  gas velocity was used  For  each  reading files,  point, ' the  was  along  read with  into  the  into  10msec  intervals,  by  of each  between each a  "weighted"  the  piston  crank  point gave  average  GRAFPAK  gas velocity,  pressure,  and pressure  readings  correlations  of density, kinematic  read  velocity,  piston  another  were manipulated viscosity,  t® Alan Jones, University of British Columbia. t° Lotus Development Corporation.  and  The  and gas  velocities. This  at each point.  temperature,  into  were  point in time.  piston  of both  values  and  bridge  and stored as part of an ASCII  were  temperature  angle  at each  to calculate the Reynolds number  corresponding  format  spreadsheet.  spreadsheet,  the positions  calculated  or  position  of the test run) were stored as a file in ASCII  so they could be read into a S Y M P H O N Y t  Once  at 5msec  angle  spreadsheet.  file.  These  There, the  to give values from  thermal  voltage  linear  conductivity. Nusselt  Experimental Apparatus number  and  Reynolds  number  Sample calculations and  Only  180  points  were  used  memory  As  four spreadsheets  number and  Reynolds  point was  quickly  for the  data  filled because had  to be  "damped  analysis  of the used  for each  complexity  these  out"  the  noise  / 33  results.  C.  cycle  of the  since  the  calculations.  to get the four inputs to Nusselt  to noise for the outputs was  the readings over five or ten points around have  from  number outputs. Because the measurement instruments  been well grounded, the error due  may  calculated  Procedure  property correlations are presented in Appendix  spreadsheet it was,  at each  and  small. Averaging  each of the approximately  mathematically,  would not have been worth the additional work.  but  the  had  180 points  additional  accuracy  IV. PRESENTATION  OF RESULTS  A. EXPERIMENTAL MATRIX  Adjustability was one of the design criteria of the test rig so that heat transfer rates over a range of parameters relevant to the Stirling engine regenerator case could  be measured.  For the 90 degree  phase  angle  case, maximum  Reynolds  numbers in the range 0-1500 were achieved during the tests, with the bulk of the  tests from  dead  the 0 to 600 range. A t the 40mm  space ratios of 1.5 and 2.0. The rig could  stroke, runs  were done at  not be adjusted  for  the dead  space ratio of 1.0. The majority of the tests were done at the stroke of 80mm because show  space  ratios  the dependence  120mm and  dead  stroke, only  dead  space  of 1.0, 1.5, and 2.0 could  of heat  the dead  ratio,  runs  transfer  rates  upon  be attained. This  dead  space  space ratio of 1.0 was examined. were  done  at either  full  or half  ratio.  would A t the  A t each stroke speed,  and at  different gas pressures so results over a wide range of Reynolds numbers could be  obtained.  numbers, possible  Runs  for identical  but different dependence  geometric  parameters  gas pressures and speeds,  of the heat  transfer  results  were  and similar made  Reynolds  to expose the  on other parameters besides  Reynolds number, dead space ratio, and stroke.  Because  other  investigators  have  exclusively, runs at this phase  examined  the 180 degree  angle were made  phase  angle  case  as well. Several runs at each  stroke were made at full and half speed and at various gas pressures. Maximum peak Reynolds numbers close to 900 were achieved, with most runs up to  34  300.  Presentation of Results / 35 The  lower range of Reynolds numbers in the 180  because  angle case result  the peak pressure throughout the test is almost constant, while in the  90 degree phase numbers  degree phase  angle case there are wide pressure variations. Average  are therefore used  as  a comparison  between  Reynolds  the different phase  angle  cases.  Tables  1 and  2  Nusselt numbers.  show  the test matrix and  calculated  results  for Reynolds  and  Trial  Phase  Angle  Stroke  (Degrees)  (mm)  1  180  40  2  180  40  Speed  Reynolds  No.  Nusselt  No.  Pressure  (Average)  (Average)  50%  56. 10  3.57  152.2  50%  81 .04  4 . 36  224.4  (Average,kPa)  3  180  40  100%  89.85  5.87  110. 1  4  180  40  100%  97.08  6 . 19  133.4 240.0  5  180  40  100%  163.15  8.56  6  180  80  50%  60.23  5.02  75.2  7  180  80  100%  125. 14  9. 36  78.0  8  180  80  50%  130.02  8.42  175.6  9  180  80  100%  178.43  10.28  119.8  10  180  80  50%  217.89  9.60  350.7  11  180  80  100%  339.17  14 . 14  275.6  12  180  120  50%  201.73  9.57  172.0  13  180  120  100%  364.55  15.58  153.5  14  180  120  100%  609.32  19.57  317.7  Table 1. Test Matrix and Calculated Results for T r i a l s 1-14  oo  Trial  Phase  Stroke  Dead  (Degrees)  Angle  (mm)  Ratio  15  90  120  1 . 105  16  90  120  17  90  120  18  90  120  19  90  80  20  90  80  Space  Speed  Reynolds  No.  Nusselt  No.  Pressure  (Average)  (Average)  (Avg.kPa)  50%  158 . 19  11 . 23  164.. 4  1 . 105  100%  314 .66  15.55  156..4  1 . 105  50%  281 .25  19. 18  410 .5  1.. 105  100%  520 .52  27.41  364.. 1  2 .082  50%  99. 74  6 . 18  170..4  2 .082  100%  218 . 12  12.49  176..3  21  90  80  2,.082  50%  124 .58  8.30  222,.0  22  90  80  2 .082  100%  288 .76  15.90  275,.9  23  90  80  2 .082  50%  183 .25  12.22  385..8  24  90  80  2 .082  100%  335 .59  19.01  348.. 8  25  90  80  1 .527 ,  100%  144 .02  10. 12  118.. 7  26  90  80  1 .527  50%  115 .03  8.97  228.,7  27  90  80  1 .527 .  100%  249 .59  15.60  225. 8  28  90  80  1 .527 ,  50%  156 . 10  1 1 .55  320. 9  29  90  80  1 .527 .  100%  287 .94  17. 16  279. 9  30  90  80  0..998  50%  89. 77  7.35  154 ..0  31  90  80  0..998  100%  181 .52  12.40  153..7  32  90  80  0..998  50%  121 .59  12.76  225. 2 220. 3  33  90  80  0..998  100%  246 . 29  16.93  34  90  40  2..005  50%  38. 78  3.54  135. 5  35  90  40  2.,005  100%  79 .!91  7. 17  135. 3  36  90  40  2..005  100%  134 .64  1 1 .04  255. 9  37  90  40  1 .500 .  50%  47..41  6 . 19  165 .4  40  1 .500  100%  86.!99  8.48  148. 8  40  1 .500  100%  139 .84  1 1 .79  268. 6  38 39  90 90  T a b l e 2. T e s t M a t r i x and C a l c u l a t e d R e s u l t s f o r T r i a l s  15-39 oo -0  Presentation of Results / 38 B. D E T E R M I N A T I O N  OF MEASUREMENT  RESPONSE  Before testing at the 90 degree and 180 degree phase set for 0 degree phase at  the test-cylinder,  angles, the test rig was  angle operation. This mode allowed no bulk gas velocity  but compression  and expansion  of the gas occurred.  This  gave the time response of the gas temperature thermocouple and the test-cylinder system the  The comparatively fast  time  lag between  peaks  response  of the pressure transducer meant  of the pressure  transducer  anemometer bridge and thermocouple voltages would the  response.  At  a  moderate  system  pressure  voltage  that  and the  give an accurate estimate of that  could  be  considered  an  average of the pressures during 90 degree and 180 degree phase angle tests, the lag of the thermocouple was about 50msec.  The  small  that  variations  improvements  thermocouple  in temperature in  accuracy  over  of  50msec  the  results  response lag in the calculations  would  during by  the test the  runs  meant  inclusion  of the  be small, so this  lag was  neglected.  The  lag of the anemometer bridge voltage was about 100 msec, which will cause  a damping effect on the heat transfer  measurements. The peak Nusselt numbers  would  not be as high as with a faster response probe, but the average values  would  not be affected significantly.  The could  100msec  response  lag meant  not be measured. A t full  that  speed  rapid  variations  the variation  in heat transfer  would  have  rates  to be over 40  Presentation of Results / degrees of crankshaft rotation However,  this  meant  to  be  Stirling  engine  than  40  degrees  of crankshaft  less  where the greater  resolution  to be  measured  is acceptable  since  regenerators.  with the  high  in  have  heat  little  lag  was  not  considered  in  the  of  of accuracy.  the  results  transfer  relevance  average heat transfer rate over a half-cycle (180  interest. This  degree  application  Changes  rotation  a  39  is  rates  over  this  case,  in  degrees) is of much  calculations, but  is taken  into account in the discussion of the overall heat transfer results.  C. GAS  VELOCITY  Experimental case, so piston gas  investigations  assumed  velocity. This  velocity at the  pistons. Since the diameters, piston's gas  This  bulk  and  have  fluid  concentrated  velocities have  phase  angle  is a good assumption. However, in the  90  degree case, the  to an  taken  degree  identical to  gas  velocity, the  180  be  distance  been  the  to  test-cylinder is related  the  upon  average of the  velocity of both  between the pistons is never more than 8 or 9 velocity  calculated gas  adjacent  to  each  piston  is  the  identical  velocity is a good estimate of the  to  piston that  actual bulk  velocity at the test-cylinder.  calculated  to the  distance  influence  on  Graphs  of  stroke  and  velocity is an of the  average of the  piston  test-cylinder from each piston. The  the velocity at the test-cylinder. (See  calculated phase  velocities, weighted  gas  angle  velocity in  at  Figures  the 7-12.  closer piston  Appendix  test-cylinder Velocity  according has  more  C.)  are  presented  for  each  curves  assuming  sinusoidal  Presentation of Results / 40 motion  of each  piston  are also  presented  for each  graph.  In all cases, the  test-cylinder is situated in the centre of the test rig, at the centre of the mean of each  piston position. This  means that the adjustable  connecting  rod for each  piston was set to the same length. This was true for all tests as well.  0.15  -0.15  H  1  1  1  0  90  180  270  1 360  Crank Angle (degrees)  F i g u r e 7.  C a l c u l a t e d Bulk F l u i d V e l o c i t y for T r i a l 3  at  Test-Cylinder  -0.3  H  0  1  90  1  180  1  270  1  360  Crank Angle (degrees)  Figure 8. Calculated Bulk F l u i d Velocity at Test-Cylinder for T r i a l 9  ^  0.6  -0.6  H  0  1  1  1  90  180  270  1 360  Crank Angle (degrees)  Figure 9. Calculated Bulk F l u i d Velocity at Test-Cylinder for T r i a l 13  Figure 10. Calculated Bulk Fluid Velocity at Test-Cylinder for T r i a l 16  Figure 11. Calculated Bulk F l u i d Velocity at Test-Cylinder for T r i a l 25  Presentation of Results / 47  D. FLUID PROPERTIES  The  fluid  properties  temperature  and  for  pressure. The  enough to ensure that no the  higher  The  gas  cylinder  for the flow only  a  smoothly  few as  Freon-114  pressures  and  as  a  For  Reynolds  neither adiabatic nor  result the  the  from  the  180  bulk 90  test  temperature,  runs.  no  raised  variations  degree phase phase  high how  achieved.  isothermal boundary conditions  temperature  degree  of  in the  gas  were  angle case. Pressure varied  angle  case  small  pressure  and  to the geometry of  magnitudes of these variations increase with stroke. Graphs of  pressure, and  Graphs of pressure and was  was  numbers were  temperature variations result because of volume changes due the test rig. The  measurements  of the Freon-114 took place; this is  therefore higher  degrees Kelvin in the well.  calculated  initial temperature of each  condensation  walls provided and  are  measureable  volume for test run  volume for test run  change in temperature  23  54  are shown in Figures 13-15.  are in Figures 16  during the  180  and  degree  17; there  phase  angle  500  400  300  200  100  0  Crank Angle (degrees)  Figure 13. Pressure versus Crank Angle for T r i a l 28  250  90  180  360  270  Crank Angle (degrees)  Figure 14. Volume versus Crank Angle for T r i a l  28 CD  Figure 15. Temperature versus Crank Angle for T r i a l 28 ©  120  100  80  60  40  20  90  180  270  360  Crank Angle (degrees)  Figure 17. Volume versus Crank Angle for T r i a l 9 Ol  Presentation of Results / 53 Relations  for density,  pressure specific heat and  pressures  conductivity,  and  sufficient accuracy and  temperature  dynamic  viscosity,  were derived from  examined, constant  relations  for  pressure  specific  heat  also used.  Table  conductivity, and  constant  tables. Over the range of temperatures  linear  (see Appendix C). A was  thermal  dynamic versus  viscosity,  thermal  temperature  provided  relation for density dependent on pressure 3 shows typical  values for each  properties.  Table 3. Typical Fluid Property  Values  Pressure (MPa) Temperature (K) Density (kg/m ) Dynamic V i s c o s i t y (Ns/m ) Thermal Conductivity (W/m°C) Prandtl Number 3  2  0.05-0.69 295-360 3.4-45.3 11.4-13.6 x 10" 0.011-0.014 0.698  6  Prandtl number variation i s less than 1% over t h i s temperature range.  of the  Presentation of Results / 54  E. HEAT TRANSFER RESULTS  Figures runs  and measured  data  during several selected  that give a cross-section of all the data taken. The "Instantaneous  number each  18-43 present calculated  versus  Reynolds  number"  of the approximately  test rig. These  graphs  graphs in Figures  180 data  are plotted  points spaced  18-30 show throughout  on a logarithmic scale  runs can be plotted on the same scale for comparison, at  the lower  values, where  most  test  Nusselt  the values at  one cycle  of the  so that all the test  but good resolution occurs  of the points reside.  Figures  31-43 include  linear plots of Nusselt number and Reynolds number versus crank angle for each of these test runs.  Figure  44 is an "Instantaneous  Nusselt number  versus Reynolds number"  graph  that presents three sucessive cycles of test run 27. It shows that while there is some values  variation  in the instantaneous  for Nusselt number  points on  and Reynolds  the graph,  number, both  Table  4  shows the  peak and average,  and  the average pressure.  Table 4. Cycle-to-Cycle Variation of Test Run  Trial 27A 27B 27C  Reynolds No. (Max) 626.11 633.93 637.91  Reynolds No.(Avg) 249.59 250.35 249.73  27  Nusselt No. (Max) 22.38 21.64 21.58  Nusselt No.(Avg) 15.60 15.60 15.79  Pressure (Avg,kPa) 225.8 225.3 225.1  o  •  %° > 8  a  n  1  •sg  fi  Man  c  •  0-90 Degrees  •  90-180 Degrees  o 180-270 Degrees o 27CI-36C> Degre es  10  100 Reyno ds number  1000  Figure 18. Instantaneous Nusselt number versus Reynolds number for T r i a l 3  *f -  °  *> %  .A sr f  E9  O O " • • •  I  • •  •  0-90 Degrees  •  90-180 Degrees  o 180-270 Degrees o  10  100  27<3-36I3D< >gr<»et  Reynolds number  1000  Figure 19. Instantaneous Nusselt number versus Reynolds number for T r i a l 6  o 0  10  F i g u r e 20.  \  •9 • m • 9 ><  3? P  • •  •  0 - 9 0 Degrees  •  90-180 Degrees  o  180-270 Degrees  •  27C)-36( ) De gre 0 8  100  1000  Reyno ds number  Instantaneous N u s s e l t number versus number for T r i a l 9  Reynolds  •  E 10  e-  •o—  f  0 —C  > I  C CD CO CO  3  •  0 - 9 0 Degrees  •  90-180 Degrees  o 180-270 Degrees o 27C>-36C> Degre es  0  100  10 0 0  Reynolds number  Figure 21. Instantaneous Nusselt number versus Reynolds number for T r i a l 13  ^ 00  • • • •  • •  •  a  • • «  • i  >• • *--  •  0-90 Degrees  •  90-180 Degrees  o 180-270 Degrees o  10  100  27C)-36C >De gre es  1000  Reynolds number  Figure 22. Instantaneous Nusselt number versus Reynolds number for T r i a l 16  •  •  •  •  .-.  •  _ i  1  •  •  •  *  Q  I1  • D  •  1  [  •  *  ,*  \ 7 /  • 3 i1  •  • 0-90 Degrees • 90-180 Degrees o 180-270 Degrees o 27C)-36C)  )  10  0  Reynolds number  Degre es  101OO  Figure 23. Instantaneous Nusselt number versus Reynolds number for T r i a l 17  E c (D W W  i i l .  10  —• • • D  10  , " 1l  •—m-  I  • n  I  '%  EE _  D  V  1  • • • •  o  <  100  •  0-90 Degrees  •  90-180 Degrees  o  180-270 Degrees  o  27C)-36( ) De gretes  1000  Reynolds number  Figure 24. Instantaneous Nusselt number versus Reynolds number for T r i a l 21  ^  •  m D  •  n  mi  • i, 1  i"  o  •  •  «  —  •  1ia  l>V  0 tiat J w  •  >  •<  _* •  i  •  0-90 Degrees  •  90-180 Degrees  o 180-270 Degrees o 27C)-36C ) De gre es  10  100  Reynolds number  1000  Figure 25. Instantaneous Nusselt number versus Reynolds number for T r i a l 25  .  1  •  • o  •  •o—O  •  •  •  »  •  •  J  LI  mi II  ,, c Hi  • «  - #§*..  V ' j  • 0-90 Degrees • 90-180 Degrees 0 180-270 Degrees o 27C)-36() Degretea  10  100 Reynolds number  1000  Figure 26. Instantaneous Nusselt number versus Reynolds number for T r i a l 28  •  •  I"  •  •  •• 1  w  i  •d  • • D  »  «  •  i • 0-90 Degrees • 90-180 Degrees o 180-270 Degrees o 27C)-36C) Degre es  10  100  1000  Reynolds number  Figure 27. Instantaneous Nusselt number versus Reynolds number for T r i a l 30  •  • •  r  11  •  •*  •  •  i  •  •  E t j O C^l n  ••  a <  a  «•  a  3  •  •  #  .*  f  ,••  •  0-90 Degrees  •  90-180 Degrees  o 180-270 Degrees a 27C)-36( ) De gretea  1  10 0  Reynolds number  101DO  Figure 28. Instantaneous Nusselt number versus Reynolds number for T r i a l 33  •  . • •• n  tr  n  •  • •  %"  < n  EJ • •1 C  *  1  *I  t.  9  • 0-90 Degrees • 90-180 Degrees © 180-270 Degrees o 27C)-36() Degrcies  10  100  1000  Reynolds number  Figure 29. Instantaneous Nusselt number versus Reynolds number for T r i a l 36  • 0-90 Degrees • 90-180 Degrees o 180-270 Degrees a 270-360 Degrees  10  100  1000  Reynolds number  Figure 30. Instantaneous Nusselt number versus Reynolds number for T r i a l 37  / 68  10  90  180  270  360  Crank Angle (degrees)  150  .Q £  <9  100  8 8  c  CO TD  O c ><D  OC  \  §  ZJ  50  »  — O  o o°  v  o o o o o o o o o ° n  90  <  / / k$  180  ^\  270  \ %  360  Crank Angle (degrees) Figure 31. Nusselt number versus Crank Angle and Reynolds number versus Crank Angle for T r i a l 3  / 69  Figure 32. Nusselt number versus Crank Angle and Reynolds number versus Crank Angle for T r i a l 6  / 70  Figure 3 3 . Nusselt number versus Crank Angle and Reynolds number versus Crank Angle for T r i a l 9  / 71  600  90  180  270  360  Crank Angle (degrees) Figure 34. Nusselt number versus Crank Angle and Reynolds number versus Crank Angle for T r i a l 13  / 72  1000  0  90  180  270  360  Crank Angle (degrees) Figure 35. Nusselt number versus Crank Angle and Reynolds number versus Crank Angle for T r i a l 16  / 73  60  1000  0  90  180  270  360  Crank Angle (degrees) Figure 36. Nusselt number versus Crank Angle and Reynolds number versus Crank Angle for T r i a l 17  / 74  F i g u r e 37. N u s s e l t number versus Crank Angle and Reynolds number versus Crank Angle f o r T r i a l 21  / 75  400  0  90  180  270  360  Crank Angle (degrees) Figure 38. Nusselt number versus Crank Angle and Reynolds number versus Crank Angle for T r i a l 25  / 76  0  90  180  270  360  Crank Angle (degrees) Figure 39. Nusselt number versus Crank Angle and Reynolds number versus Crank Angle for T r i a l 28  / 77  15  OH O  1  90  1  180  1  270  1  360  Crank Angle (degrees)  300  0  90  180  270  360  Crank Angle (degrees) Figure 40. Nusselt number versus Crank Angle and Reynolds number versus Crank Angle for T r i a l 30  /  800  0  90  180  270  360  Crank Angle (degrees) Figure 41. Nusselt number versus Crank Angle and Reynolds number versus Crank Angle for T r i a l 33  78  400  0  90  180  270  360  Crank Angle (degrees) Figure 42. Nusselt number versus Crank Angle and Reynolds number versus Crank Angle for T r i a l 36  / 80  150  Crank Angle (degrees) Figure 43. Nusselt number versus Crank Angle and Reynolds number versus Crank Angle for T r i a l 37  o  OJ  -O  E c  10  • o  .->«•  • o  •  i •  •  [ Uf"  /  w CO  Z  10  •  Trial 27A  •  Trial 2 7 B  o  Tr al 2 7C  100  1000  Reynolds number  Figure 44. Cycle-to-Cycle Variation of Instantaneous number versus Reynolds number for T r i a l 27  Nusselt 00  Presentation of Results / 82 Figure  45 presents  number heat  for each  transfer  illustrate results  the results  test  from  run. A  are plotted  cylinder  between  by test  Nusselt  number  line that represents a steady  a circular  the difference  for average  versus  flow correlation for  in cross-flow is also  the oscillating  rig piston  speed  and steady and phase  Reynolds  on the graph to flow  results. The  angle  to show the  dependence of heat transfer rates upon these. No distinction for dead space or  stroke-to-test-cylinder-diameter ratio  points for  the 180 degree phase  correlation, but all of the other degree  phase  phase angle results  upon  angle  results  case. There test  is made.  angle  case  are very  is a slight  than  Reynolds  number  close to the steady  points are higher, some  are generally higher  rig piston  The lowest  ratio  substantially. The 90  those  for  the 180 degree  dependence of the 180 degree phase  speed, but no dependence  flow  is evident  angle  for the 90  degree case.  Figure  46 presents  the same  results  with  only  the data  stroke  (47.6 stroke-to-test-cylinder-diameter ratio)  dependence of the results on test rig piston speed the  previous  graph.  Figure  47 shows the heat  shown.  points for This  the 80mm  illustrates the  and phase angle better than  transfer  results  for the 80mm  stroke case (and 90 degree phase angle); the increase of Nusselt number with a decrease in dead space ratio (DSR) can be seen. Nusselt number does not change with  stroke-to-test-cylinder-diameter ratio,  as shown  in Figure  48 for the 90  degree phase angle case. Figure 49 also shows no significant variation of Nusselt number case.  with  stroke-to-test-cylinder-diameter ratio  in the 180 degree  phase  angle  "•  60% Speed, 90 Deg Phase  O 50% Speed. 180 Deg Phase •  100% Speed. 90 Deg Phase  •  100% Speed. 180 Deg Phase Steady Flow  30  100  1000  Reynolds number (average)  Figure 45. Average Nusselt number versus Average Reynolds number, A l l T r i a l s , by Speed and Phase Angle  ^ oo CO  "•  60% Speed, 90 Deg Phase  O 50% Speed, 180 Deg Phase •  100% Speed, 90 Deg Phase  •  100% Speed, 180 Deg Phase S t e j B d v J f J o y ^  30  100 Reynolds number (average)  w  m  ^  m  1000  Figure 46. Average Nusselt number versus Average Reynolds number, 80mm Stroke T r i a l s , by Speed and Phase Angle  ^ oo  •  1.0 Dead Space Ratio  O 1.5 Dead Space Ratio "•  2.0 Dead Spece Ratio Steady Flow  30  100  Reynolds number (average)  1000  gure 47. Average Nusselt number versus Reynolds number, 80mm Stroke T r i a l s , by DSR  Reynolds number (average)  1000  Figure 48. Average Nusselt number versus Average Reynolds number, 90 Degree Phase Angle T r i a l s , by Stroke-to-Test-Cylinder-Diameter Ratio oo 03  Reynolds number (average)  1000  Figure 49. Average Nusselt number versus Average Reynolds number, 180 Degree Phase Angle T r i a l s , by Stroke-to-Test-Cylinder-Diameter Ratio 00  V. DISCUSSION OF RESULTS  A. GAS  The  VELOCITY  velocity  versus  show  test rig crank  difference piston  graphs  the calculated  angle. The sinusoidal  between the velocity that  gas velocity  piston  results from  motion  at the test-cylinder curves  the actual  illustrate the  piston  motions and  motions assumed in most theoretical work. Interestingly, very  cycle machines use truly sinusoidal piston an  bulk  accepted  standard  motion, but it seems to have become  for most of the test rigs used  well. (The use of a perfectly-sinusoidal drive However,  since  turbulent  rather  the flow  field  present  in experimental work as  for this test rig was not feasible.)  in this  test  rig should  be considered  than laminar, heat transfer results should not be much different  than if sinusoidal piston motions were used, and they are therefore a wide range of Stirling engine  Figure  constant, when  rig  applicable to  configurations.  7 shows gas velocity at the test-cylinder for the 180 degree phase angle  case and the 40mm  as  few Stirling  stroke. In all cases, the rotational speed is assumed to be  in fact, especially at higher  system  pressures, it varies  slightly  the motor drive for the test rig responds to the changes in load. Full test speed  is also  used  in all graphs. The actual  piston  motion  curve  deviates  very slightly from the sinusoidal piston motion curve.  Figure  8 is for the same  phase  curves  are still  to each  very  close  angle  but 80mm  stroke.  other, so for both  88  The piston  the 40mm  motion  and 80mm  Discussion of Results / 89 stroke be  cases, test result variation due to the different rig geometries  significant. The nature  convective vortex  rates  and localized  shedding  parameter  of the recirculating  from  such  velocities  about  the test-cylinder  wake  is the main  the test-cylinder.  depends  on a  would not  parameter in  The amount of  "Strouhal number  type"  as the stroke-to-test-cylinder-diameter ratio. (The Strouhal number  is equal to the frequency times the test-cylinder diameter divided by the velocity, which is closely related to the stroke-to-test-cylinder-diameter ratio at high enough Reynolds  numbers.)  The vortex  size  is always  of the same  test-cylinder diameter, and the length of the wake is always stroke  length, so the exact nature  effect  on the heat  transfer  of the velocity  should  order  as the  of the order of the not have  too much  rates if only small differences are present. This is  reflected in the heat transfer results for this test rig.  The  gas velocity at the test-cylinder for the 120mm  9. The two curves are close until 270 degrees piston  motion  curve  motion  curve. The 120mm  test rig geometry  flattens  and deviates  stroke case  if close approximation  The  from  the sinusoidal  to the limit  for this motion  piston  particular  is required.  sinusoidal piston motion, results  "other" velocity conditions are certainly applicable.  next three graphs  degree velocity The  angle, when the actual  of sinusoidal piston  However, since very few Stirling engines employ from  of crank  further  is close  case is presented in Figure  illustrate the gas velocity at the test-cylinder for the 90  phase angle case. A n important observation from curves  compression  for the sinusoidal and expansion  piston  motion  cases  of the gas causes  these graphs are far from  the velocity  is that the sinusoidal.  in the positive  Discussion of Results / 90 direction  (compression  stroke) to be of shorter duration than  negative direction (expansion stroke). Therefore, results from degree phase  angle  case  (the only  from  test rigs for the  published results up to now) should  applied directly to the 90 degree phase angle case velocity  the velocity in the  conditions. This is supported  180  not be  because of the differences in  by the fact that the heat  transfer results  this test rig depend significantly on phase angle.  Figure  10 is for  actual  piston  angle  sooner  there  is high  the 40mm  motion than  case  stroke case. The compression  is 0.02m/sec  higher  for the sinusoidal motion  acceleration  and comes  0.2m/sec less  of the gas, the curves  effects  on heat  than  transfer  the sinusoidal  piston  of the difference  10-15 degrees  case. However, for  motion  crank  points at which  are of the same  actual piston motion curve flattens during the expansion about  peak velocity for the  slope. The  stroke, and the peak is case. For one cycle, the  in the velocity  peaks  should  average  out, so the differences in the velocity curves should not affect the average transfer  rates  "Instantaneous  significantly. Nusselt  Peak  number  heat  versus  transfer Reynolds  rates, number"  presented graphs  heat  on the should  be  affected to a greater extent, but this was not obvious. This is discussed later.  Similar  differences  in the velocity  peaks  in Figure  stroke case, except  they  actual  piston  during  40mm  case. The slopes along the high acceleration periods are still similar.  Figure  12 presents  case  the 120mm  as high. The velocity  11 for  80mm  motion  are twice  are illustrated  the expansion  stroke  stroke case. Velocity  is flatter  curve  for  the  in  the  than  peaks are about  the  0.8m/sec  Discussion different, actual  but  piston  the  slopes  motion  during  curve  the  shows  high a  acceleration  "dip"  of  0.2m/sec  expansion stroke. This did not appear to affect the and  In and  its effect on  piston  cases. Since the on  motions were  cause  great  as  ratio, the  The  during  the  speed  sinusoidal piston motions  those between the  average heat transfer results indicated no  different stroke  significant dependences  differences in velocities present in this  a rig with assumed sinusoidal piston motion would not be  significant differences  should be  as  similar.  later.  velocity curves between  stroke-to-test-cylinder-diameter  test rig and to  in the  in  are  average heat transfer results,  the instantaneous results is discussed  all cases, deviations actual  periods  of Results / 91  in  average  heat  transfer  rates.  expected  These  results  valid for a wide range of Stirling engine configurations.  B. FLUID PROPERTY VARIATIONS  Figure  15  shows that the  temperature varies smoothly, but  Kelvin, with crank angle, for test run pressure variations shown in Figure response  lag of the  but  inversely, and  13,  the  volume curve  appears to be  a  few  degrees  54. It is approximately in phase with  thermocouple. Pressure  degree phase angle case. The  only  difference being due varies  smoothly  in Figure  14  as  to the well  follows  the  the  thermal  in this first  90 two,  very close to being sinusoidal, despite the fact  that the piston motions are not exactly sinusoidal.  Figure  16  typical  180  shows  the  pressure  variation with  degree phase angle case. If the  crank  angle  for test run  23,  a  piston motions were truly sinusoidal,  Discussion of Results / 92 the curve in this case would be the  volume  variation  curve  a straight horizontal line. This is also true for  shown  in Figure  17. However, the finite, connecting  rod lengths of the test rig mean that while the rotational motion  of the  are  not. Also, the  180  degrees  out of phase  with  each  other, the  pistons are  forward  and  reverse motions are not identical because on  cranks  are  moving  upwards.  This  temperature degree  accounts  curves  phase angle  forward and  downward,  that  for  while  the  occur  case, and  on  the  variations  at  twice  the forward  reverse  pass,  present  in  rate  of the  the  both  are  going  pressure  variations  the different magnitudes of the  pass, the  they the  cranks  in the  variations  on  and 90 the  reverse strokes.  C . I N S T A N T A N E O U S  R E S U L T S  Several test runs were selected as being representative of the entire experimental matrix graphs,  and  are  "Nusselt  presented  as  number  "Instantaneous  versus  versus Crank Angle" graphs. The direct  comparisons  between  the  resolution at the higher Reynolds 10  has  from  0  other graphs  to 360  to near  test runs  phase angle case.  velocity  graphs,  and  Reynolds  number"  "Reynolds  can  be  made  easily. To  number  obtain good  numbers, the data below a Reynolds or two  points from  number of  each  run  are  are scaled linearly to fit the data. The abscissa  degrees, or one  zero piston  versus  instantaneous plots are all scaled identically so  to the beginning of the compression and  Angle"  been eliminated; this means only one  lost, however. The runs  Crank  Nusselt  on  crank  revolution. Zero  stroke on one  the  90  degree  degrees phase  of the half-strokes in the  corresponds angle 180  case  degree  Discussion of Results / 93 All of the instantaneous  graphs reveal an important  fact: at very  numbers, the Nusselt number is always substantially higher than  small Reynolds the steady  correlation values  shown in Figures 45-49, usually 1 0 0 % or greater. This  that  be substantial local  there  must  though the calculated bulk does not diffuse totally  fluid  fluid  motion  near  means  the test-cylinder,  motion is near zero. The vorticity  flow  even  in the wake  between stroke reversals. The motion in the wake  after  each stroke affects the heat transfer and fluid motion on each subsequent stroke. Turbulence  in the main  flow  also  persists  between  flow  reversals  and will  produce the same effect.  Figures  18-21  instantaneous  show  Nusselt  number  for the 180 degree phase  The heat  if each  stroke has close to the same  transfer is expectedly  are approximately  four  crank  highest angle  against  case. The first  on the other. This  motion  at the highest  degrees  plotted  angle  18, shows two loops, one almost superimposed  is to be expected  There  instantaneous  Reynolds number  graph, Figure  one.  the  as the reverse  Reynolds  between every  number.  two adjacent  points.  Figures  19  and 20 are for the 80mm  stroke  case  difference between the stroke "loops". The geometry the  forward  and show  of the test rig means  and reverse strokes will not be identical, and there  slight compression and expansion for the 120mm  more  will  of a that  also be a  of the gas. This effect is even more pronounced  stroke, shown in Figure 21.  Figures 22-30 are for the 90 degree phase angle case. As was the case for the  Discussion of Results / 94 average  value  graphs,  the Nusselt  numbers  are higher  for a given  Reynolds  number than in the 180 degree phase angle case. The highest rates occur during the compression  portion of the cycle from  22-38 for the test  runs  at 120mm  values for Nusselt number  0 to 130 crank  and 80mm  and Reynolds number  angle degrees. Figures  strokes show are near  that the highest  90 degrees.  two graphs, Figures 29 and 30 have the highest Reynolds number degrees  as well, but the highest Nusselt number  reverses, from to 270 degrees change  from  270 to 360 degrees  stroke from 180  and Reynolds number that do not  appreciably, so the points are clustered  completed  points at 90  values occur just as the flow  130 to 180 degrees. In all cases, the expansion has values of Nusselt number  The last  in a small  as the Nusselt  area. The cycle is  numbers  decrease  with  Reynolds number to the initial cycle values.  The  differences between  the 40mm  strokes come about from in the 40mm expansion  stroke curves  and the curves  for  the other  a higher intensity of local motion about the test-cylinder  case just as the flow reverses. The reverse must occur after the  stroke because  the average  heat  transfer  values  did not appear to  depend on the stroke-to-test-cylinder-diameter ratio. The high stroke-to-wire-diameter ratios in actual Stirling engine regenerators mean that the curves for the 120mm and  80mm  strokes are more applicable than the 40mm  Differences in the curves This  means  that this  due to different  parameter  affects  dead  space  stroke curves, however.  ratios  are not apparent.  all of the instantaneous  same extent, which is significant only for an averaged  values  to  the  value for the entire curve  if different curves are compared. Dependence on different test rig speeds is also  Discussion of Results / 95 not apparent, but the averape values indicate that this has negligible influence on heat transfer rates for the 90 degree phase angle case.  Figures  31-43 each present  Reynolds show  number  how  number  versus  crank  the Nusselt  curve.  Any  a graph of Nusselt number angle  number  rapid change  for each  follows  selected test  the rises  in Reynolds  versus  and  number  crank angle and  run. These  falls  graphs  of the Reynolds  is often  accompanied  by  fluctuations in the Nusselt number. For the 180 degree phase angle case, this is illustrated  by  Figures  variations in Nusselt  33  and  34.  Figures  31  and  and Reynolds numbers, and they  32  show  only  smooth  are in phase (within the  response lag error).  Figures  35-43 deal with  the 90 degree  varies almost proportionately with compression stroke (about  phase  in all cases  number  during the  0 to 130 degrees). There, is a rise in Nusselt  velocities are high. The Nusselt number tails  case. The Nusselt  the Reynolds number  after the reversal of the flow, when the bulk  cases, and then  angle  number  gas velocity is low but the local  peaks before or at 180 degrees for all  off to the initial value  at 360 degrees. On  some  of the  runs, "sudden" changes during the latter part of the expansion  stroke in Nusselt  number  wake  indicate that there  are some  regions  in the reversed  that  have  higher local velocities than the rest of the wake.  Figure  44  shows  there is little average  values  three  successive  cycles during  r  cycle-to-cycle variation for Nusselt  number  in heat and  test run 27.  This  transfer rates. Table  Reynolds  number,  and  shows that 4  shows the  the differences  Discussion of Results / 96 here  are  negligible.  This  means  the  flow  consistent, and the results obtained from  D. AVERAGED  Figure  parameters.  to  demonstrate  on a graph  the  logarithmic scale is used  transfer rates upon Reynolds number The  are  of average  Nusselt number  Reynolds number. The data is plotted according to test rig speed  angle A  cycle-to-cycle  these tests are repeatable.  45 presents all of the test runs  phase  from  HEAT TRANSFER RESULTS  versus average and  conditions  dependence  of  the  results  on  these  since any power-law dependence of heat  will put similar data  near  a straight line.  Nusselt number increases with Reynolds number for each set of data points,  within  allowable  experimental  error.  Sample  calculations  and  a  discussion of  experimental errors is presented in Appendix C.  Figure  45 also displays the dependence of Nusselt number  90 degree phase angle points generally yield than  values for the 180 degree case  on phase angle. The  Nusselt numbers 30 to 4 0 % higher  for a given  average  Reynolds number.  A  steady flow correlation Nu=0.683*Re°-  from  Hilpert [17] is also plotted on the graph  values angle  between  steady  are from  10  difference are  40  increasing to 8 0 %  and  to with  higher  Reynolds number as well.  oscillating  30%  higher  Reynolds than  flow. than  4 6 6  flow,  0  3 3 3  to show the relative heat transfer The  values  the steady  number. The  steady  *Pr -  with  90  for 180 flow  degree  degree  values, phase  the difference  phase  with  angle  increasing  the  values with  Discussion of Results / 97 Average values for Nusselt number  and Reynolds number  of comparing the oscillating flow values with these  are the  regenerator  most  design  useful  values  has relied  on  the steady  for design  steady  flow  of a  are used  as a means  flow correlation because regenerator.  correlation  or  Until  "trial  now,  and  error"  experimentation. If the difference over a cycle between oscillating and steady flow for  a  single  cylinder  is known,  the heat  transfer  rates  for wire  screens in  steady flow could be scaled up by a similar amount as a first approximation to a design.  Figure 46 shows the data for only the 80mm of the results  on  stroke. This eliminates dependence  stroke-to-test-cylinder-diameter ratio.  The  data  is once  again  plotted for different test rig speeds and phase angles. For the 90 degree phase angle case, no dependence of heat transfer rates for a given Reynolds number on the test rig speed phase  angle  given  Reynolds  is obvious. A  case, with number.  slight dependence is indicated for the 180 degree  the 100% speed A  good  values  part  of  being  this  5 to 10%  difference  higher  may  be  for a due  to  experimental error, but a slight trend is indicated because the three points shown for  each  case  are consistent. This graph  displays the same  dependence on phase angle as Figure 45. The reasons by  the fact  that  the flow  for the 90  degree  heat  transfer rate  for this can be explained  case  is more  unsteady  turbulent than the 180 degree case due to the compression  and expansion  gas.  increases to  When  a  vortex  is compressed,  angular momentum. This causes greater  turbulence  in the flow  its rotational  higher shear results.  flow; therefore the wake spreads  Some  out more  speed  and of the  conserve  rates in the flow and therefore a turbulence  spreads  throughout the  in the 90 degree case  than  in the  Discussion of Results / 98 180 degree case  and the flow  is more  "homogeneous". There  are greater local  velocities and therefore higher convective rates in the 90 degree phase angle case than in the 180 degree phase angle case.  Figure 47 shows the dependence of the heat transfer rates on dead space (DSR) for the 90 degree phase angle case the case  (dead space  180 degree phase angle case). The average is 1 0 % higher  than  in the 1.5 DSR  ratio has no relevance in  Nusselt number for the 1.0  case  for a given  number, which in turn is 1 0 % higher than in the 2.0 DSR may  be partly due to experimental  dead  space  ratio  higher shear and  to support  the trends. These  of the gas in these  average  DSR  Reynolds  case. This difference  error, but there are enough points for each  rates present in the lower  expansion  ratio  DSR  trends can be explained by the cases. There is more  compression  cases, which in turn result in higher  wake  turbulence and therefore higher local velocities and convective rates. The wake is also more significant in terms of total volume between the pistons in the lower DSR  The  cases.  next two graphs that show average  present  the data  for different  Nusselt number versus Reynolds number  stroke-to-test-cylinder-diameter ratios.  Figure  48  shows the results for the 90 degree phase angle case and Figure 49 shows the results average average are  for the 180 degree Nusselt  on  angle  case. In both  cases  no dependence of  stroke-to-test-cylinder-diameter ratio  for a  given  Reynolds number is indicated. This shows that the wakes for each case  similar  actual  number  phase  in their  regenerator,  influence on the cylinder the  when  they  pass  stroke-to-test-cylinder-diameter ratios  over  are  an  it. In an order  of  Discussion of Results / 99 magnitude higher, so a direct comparison case cannot  between these results and a regenerator  be made. However, the lack  of change over the range  examined in  these results indicates that little change between these results and a much higher stroke-to-test-cylinder-diameter consists  of a  series  case  would  be expected. The regenerator case  of adjacent cylinders  rather  than  would be different as well. The size of the vortices close angle  to the cylinder should  have  diameter,  similar  expected to be different.  so the effects  effects,  though  just  one, so the wake  in the wake  of dead  the overall  also  space average  however, are  ratio rates  and phase could  be  VI. CONCLUSIONS  Average Nusselt numbers from higher  than  those  a circular cylinder in an  for steady  flow  at the  same  Nusselt number values were also higher than cycle, even  at the points with  the recirculating wake from over  steady  almost  no  oscillating crossflow are  Reynolds number.  steady  Instantaneous  flow at most points in the  calculated bulk  flow velocity; therefore  the cylinder must raise convective heat transfer rates  flow values. There is fluid motion even when there is no  indicated  bulk fluid velocity.  Phase  angle  Nusselt  differences  numbers  for the  numbers were  10  given  Reynolds  average  were 30 80%  accounted  to 4 0 %  oscillating  to 3 0 %  for  the  flow. The  higher than  number. The  higher than  the  greatest  the 90  180  180  steady degree  variation  degree  space  ratio  (DSR)  had  a  phase  values  for average  significant  Nusselt numbers  case; this is true for the well. The ratio  case,  significant There  was  fluid and  1.0  motion  was  this  results  in terms of total  DSR  effect  case  about  on  angle  the  and  Nusselt  Nusselt  numbers  higher  from  in the  expansion  100  10%  Nusselt  ratio. The  case  the  1.5  than  the  1.5  DSR  to the  intense in the  wake  volume than  also more compression  angle  Reynolds number.  average  as compared  obviously more from  average  degree phase angle case, or about 40 to  Heat transfer increased with a decrease in dead space had  phase  the  flow correlation values for a  higher than the steady flow values for a given average  Dead  in  numbers. DSR  case  2.0  DSR  case  smaller dead cylinder  being  of greater dead  in the  lower  DSR  as  space more  volume.  cases, the  Conclusions / 101 same  reason why  the Nusselt numbers for the 90 degree  phase angle case are  higher than for the 180 degree phase angle case.  Stroke-to-test-cylinder-diameter numbers means  for either  the 90  ratio degree  had no  significant  effect  or the 180 degree  phase  that the wake characteristics are about the same  examined.  However,  stroke-to-wire-diameter  the ratios,  results such  might  as  those  be  on  average  angle  cases. This  for the range of cases  extrapolated  in a  Nusselt  Stirling  to  engine  higher  regenerator.  "Strouhal number type" effects would occur at only very low Reynolds numbers, which  are expected  to be  significant  at strokes much  less  than  40mm  in the  experiment.  Test rig speed degree  had a very slight effect on average Nusselt numbers for the 180  phase angle case. No  effect was  noticed  for the 90 degree  phase angle  "Instantaneous Nusselt number versus Reynolds number" graphs  showed that  case.  The  heat transfer higher  than  from  the test cylinder  during the expansion  stroke, the Nusselt number  during the compression  stroke.  variation  During  the middle  is small due  stroke is generally of the expansion  to the fact  that  Reynolds  angle  revealed  number varies little at this point.  'i Graphs  for Nusselt number  and Reynolds  number  versus  crank  that Nusselt number varies proportionately with Reynolds number if the Reynolds number  is changing  smoothly.  Flow  reversals  and  rapid  changes  in Reynolds  Conclusions / 102 number cause fluctuations in the Nusselt number.  The  velocity  graphs  for the 90  degree  phase  sinusoidal piston motion, calculated bulk fluid be  considered  sinusoidal.  This  means  similar to the 180 degree phase Stirling engine case.  that  angle case  velocity Nusselt  show  that  even for  at the test cylinder cannot numbers  for flow conditions  angle case cannot be compared  directly to the  VII. RECOMMENDATIONS FOR FURTHER WORK  Further  experimental  work  should  begin  with  a  fluid motion within the test rig, both with and This and in  will  provide  give an the  local  information as  idea of the importance  rest of the flow and  to the  bulk  on  velocities  the  complement the flow visualization  visualization  transfer  hot-wire and  study  of the  without the test-cylinder in place. of the  flow  within the  of the recirculating wake and  heat  with  nature  flow  rates. Accurate  or  test rig  the turbulence  measurements of  laser-doppler  techniques  better estimates of the  Reynolds  would numbers  within the flow field could be made.  Tests  similar  to  those  in this  arrangements  of cylinders  flow  in a  situation  test-cylinder can rates  be  should  determined  about  regenerator also be more  experiment the  test-cylinder.  more closely  improved  could  so  accurately and  quickened.  103  than  be  carried  This the  its response  would  single  its temperature to  out  with  various  approximate  cylinder  case.  the The  coefficient of resistance changes  in convective  BIBLIOGRAPHY  1.  Sarpkaya, T. and Isaacson, M. Mechanics of Wave Forces Structures. Van Nostrand Reinhold Publishing Company, New York, 1981.  2.  Schlichting, H. Boundary York, New York, 1968.  Theory.  McGraw-Hill  Book  3.  Blasius, H. Grenzshicten in Fliissigkeiten Phys. 56,1. (1908).  mit Kleiner  Reiburg. Z. Math. u.  4.  Schwabe, M. Uber Druckermittlung in der Ing.-Arch. 6, 34-50 (1935); N A C A T M 1039  5.  Richardson, P.D. Heat Transfer from a Circular Cylinder by Acoustic Streaming. Journal of Fluid Mechanics, vol. 30, part 2, pp 337-355, 1967.  6.  Urieli, I. and Berchowitz, D.M. Ltd., Bristol, England. 1984.  Stirling Cycle Engine Analysis. Adam  7.  Walker, 1980.  Oxford  8.  West, C D . Principles and Applications of Stirling Reinhold Company, New York, New York, 1986.  9.  Reader, 1983.  10.  Roach, P.D. Measurements With 21st IECEC, Paper 869119.  11.  Krazinski, J.C, Holtz, R.E., Vherka, Pressure Drops Under Reversing Flow 869116.  12.  Rice, G., Thonger, J.C.T., and Dadd, M.W. Measurements. Proc. 20th IECEC, Paper 859144.  13.  Miyabe, H., Takahashi, S., Hamaguchi, K. A n Approach To Stirling Engine Regenerator Matrices Using Pack of Wire 17th IECEC, Paper 829306.  G.  G.T.,  Stirling  Layer  Engines.  Hooper, C. Stirling  instationaren (1943).  University  Press,  on Offshore York, New  Company,  ebener  Oxford,  Engines. Van  New  Stromung.  Hilger  England,  Nostrand  Engines. E.&F.N. Spon, London, England,  The  104  Reversing Flow  Test  Facility.  Proc.  K.L., Lottes, P.A. A n Analysis of Conditions. Proc. 21st IECEC, Paper  Regenerator  Effectiveness  The Design of Gauzes. Proc.  / 105 14.  Dijkstra, K. Non-Stationary Heat IECEC, paper 849092.  Transfer  In Heat Exchangers. Proc. 19th  15.  Taylor, D.R. and Aghili, H. A n Investigation of Oscillating Flow Proc. 19th IECEC, Paper No. 849176.  16.  Seume, J.R. and Simon, T.W. Oscillating Flow Exchangers. Proc. 21st IECEC, Paper 869118.  17.  Hilpert, R. Warmeabgabe von geheizen Drahten. und Rohren., Forsch. Geg. Ingenieurwes., vol 4, p 220, 1933.  in Stirling  in Tubes.  Engine  Heat  APPENDICES  A. PROPERTIES OF FREON-114 Table 5. Properties  of Freon-114  Temperature  Dynamic V i s c o s i t y  (Kelvin)  (Ns/m xl0- )  Thermal Conductivity (W/mK)  290 300 310 320 330 340 350 360 370  11.27 11.59 11.92 12.25 12.59 12.92 13.26 13.60 13.94  0.0106 0.0110 0.0115 0.0120 0.0125 0.0131 0.0137 0.0144 0.0151  Temperature  2  6  (K)  Constant Pressure S p e c i f i c Heat (kJ/kg/K)  276.9 298 400  0.641 0.667 0.760  Temp(K) (Pressure 137.9 k P a ) 299.81 310.93 322.04 333. 1 5 344.26 355.73 366.48  9.8633 9.4669 9.1063 8.7770 8.4707 8.1850  Density  (kg/m )  (Pressure 275.8 k P a )  (Pressure 413.7 k P a )  (Pressure 551.6 k P a )  19.7558 18.9182 18.1528 17.4564 16.8186 16.2290  29.6027 28.2704 27.0666 25.9779 24.9891  39.2983 37.4340 35.7704 34.2704  3  D e n s i t y f r o m ASHRAB Thermodynamic P r o p e r t i e s o f R e f r i g e r a n t s . American S o c i e t y of H e a t i n g , R e f r i g e r a t i n g , and A i r C o n d i t i o n i n g E n g i n e e r s , I n c . New Y o r k , 1979. Other p r o p e r t i e s from T h e r m o p h y s i c a l P r o p e r t i e s of M a t t e r . © Purdue R e s e a r c h F o u n d a t i o n , I , F . I . / P l e n c u m Data C o r p o r a t i o n , New Y o r k , 1970.  106  /  107  B. TEST-CYLINDER  The  design,  rates  construction,  was  the  instrumentation.  and  most  calibration of a  challenging  Because  average  aspect  heat  of  to  highly  thermally  thickness the  that  that  a  hot  film  conductive,  would  yield  probe  easily a  film  anemometer equipment, and  Nickel was The  of  chosen as the  substrate  conductivity  material  so  that  the  rates  measure the  around  deposited  on  equipment  chosen  chosen  for  entire circular  cylinder, a  film  the  heat transfer  the  rates about the  chosen. The  design  had  to  be  substrate  to  a  of sufficient resistance to be  have a high  film  had  was  to  outfitting  transfer  cylinder were of interest rather than localized similar  device  compatible  with  temperature coefficient of resistance.  as it produced the best combination of properties.  to be  electrically insulative and  response  of  the  have a low  test-cylinder would  be  thermal  suitably  fast.  Borosilicate glass capillary tubes of ideal size (about 2mm  diameter) were readily  available  test-cylinder substrate.  Rods and  and  were  than  deposition of the on  produces  to  tubes of various  conductivities  film  found  these a  glass,  be  a  good  nickel  but  their  lower  electrolessly were  film  properties significantly  with  and  for the  plastics were considered  nickel film in a vacuum plastics  choice  a  impurities. film  because of their much lower  melting  temperatures  prevented  the  chamber. Attempts to deposit a nickel not  successful  These  because  impurities  thick enough to be  alter  this  technique  the  electrical  of suitably low  resistance  could not be produced.  The  first  about  5mm  test-cylinders were constructed into a  30mm  long  piece  with 20  gauge solid copper wires pushed  of capillary  tube  and  glued  with  silicone  / 108 sealer. These were rotated about rising  nickel  micron  thick  electrical  vapour  would  their axis in the vacuum  deposit evenly  provided the necessary  contact  from  the  film  chamber  on the glass. A  electrical  so that the  film  less than one  conductivity. Nickel  paint provided  to the copper  wire.  However,  when  these  test-cylinders were tested in a wind tunnel, the lack of rigidity between the wire and  the glass  due to the use of silicone  sealer  paint to crack and the contact to be broken. the  sealer  to the copper  test-cylinder that burned  The  design that  wire  provided  a  as the glue caused  High  the nickel  thermal conductivity  hot-spot  in the middle  through of  the  30mm  long  out even at very low overheat ratios.  was used  for  the tests employed  glass tubes fitted to 1/16 inch diameter  the same  plated  holes in the ends of 1/8 inch  diameter  acrylic rod. Epoxy was used to glue the acrylic to the tubes. A thin copper wire as then wrapped at each  end of the exposed  tube to connect the film with the  anemometer leads and electrical contact was enhanced by the application of nickel paint. This left an approximately This  design was much  problems with  25mm  length of exposed  stronger and more  electrical contact were  rigid  than  encountered.  copper wire kept thermal conduction from  film  to the gas flow.  the original one, and no  The use of acrylic  and thin  the test-cylinder to a minimum, though  this conduction was still significant.  Calibration resistance  of the test-cylinder of the test-cylinder  this was taken temperature  into account  encompassed  was found to determine  many  to change  different with  factors.  ambient  The cold  pressure, so  the actual overheat ratio as the  and pressure changed. This can be attributed to mechanical  gas  straining  / 109 of the test-cylinder  and film. When  checked, the method When  the entire  coefficient  was  by which  rig was  much  the temperature  the test-cylinder  heated  lower  than  was heated  (isothermal  when  coefficient of the film affected  test-cylinder)  the test-cylinder  strain  fields  in the test-cylinder  and  film  was heated  in each  the value.  the temperature  (parabolic-type temperature distribution). This can be attributed and  was  electrically  to different stress  case. The "isothermal"  coefficient was used to adjust the cold resistance value as the temperature of the gas changed.  The  temperature  coefficient  of the test-cylinder  was  found  by  measuring the  surface temperature of the test-cylinder by a small exposed thermocouple. Because the  thermocouple  calibrated  bead  against  a  itself surface  temperatures were calculated The  a  of  temperature known  distribution  temperature.  for each hot resistance dialed  in it, it was  Corrected  average  into the anemometer.  value was found to be 0.005297 ohms/ohm/K, similar to published values for  pure nickel the  had  (about 0.006 ohms/ohm/K). Table 6 shows the values obtained from  test-cylinder  calibration.  Figures  50-52  show  the results  used  in the  calibration of the test-cylinder.  A  nodal  around  thermal  analysis  of the test-cylinder  the circumference of the test-cylinder  even if the local convective heat transfer in  steady  longitudinal  crossflow.  Thermocouple  temperature  variation  confirmed  the temperature  could be considered  to be constant  coefficients had a distribution expected  measurements was  that  also  supported  analyzed  measured values. The result for the longitudinal analysis  this  and was  as well. The similar  to the  is in Figure 53. Overall  / 110 power  dissipation  between the nodal  matched because of uncertainties  After  all the tests  estimated, since wire-to-film  were  and actual  test-cylinder  at the end of the test-cylinder  (this is the resistance  through the nickel paint  This  value  to  assumed  not be  film  resistance  was  leads had to be placed directly on the film. The  ohms  was  situation could  in the thermal parameters.  run, the actual  the ohmmeter  resistance  analysis  remain  constant  was measured  to be 0.57  and the thin copper  with  varying  ambient  wire). and  test-cylinder film temperature. For the calculation of power dissipated in the film, the  power dissipated  in 1.14 ohms resistance  at the ends was subtracted  the calculated convective (forced) heat transfer power loss. Table 6. Values from Test-Cylinder  Calibration  Cold Resistance  Rc= m e a s u r e d c o l d r e s i s t a n c e (ohms) Rca= a c t u a l c o l d r e s i s t a n c e (ohms)  P r e s s u r e Dependency  =0.007348609(P-P ) P= p r e s s u r e ( p s i ) P = pressure (psi) at i n i t i a l c o n d i t i o n s (when Rc was m e a s u r e d ) 0  0  Temperature  Dependency  =0.0089912(T-T ) T= t e m p e r a t u r e ( K ) T = t e m p e r a t u r e (K) a t i n i t i a l conditions 0  0  P r e s s u r e dependency measured w i t h T f i x e d . Temperature dependency measured w i t h P f i x e d , t e s t r i g c y l i n d e r s heated with heating tape.  Temperature  C o e f f i c i e n t of Resistance:  a=0.005297 R/O/K Taken a s average v a l u e over l e n g t h o f t e s t - c y l i n d e r , m e a s u r e d d i r e c t l y by c a l i b r a t e d t h e r m o c o u p l e p r o b e .  from  350  Figure 50. Test-Cylinder Cold Resistance Dependence on Temperature  Figure 51. Test-Cylinder Cold Resistance Dependence on Pressure  Figure 52. Temperature Coefficient of Resistance for Test-Cylinder  Figure 53. Nodal Analysis Results  Temperatures i n K e l v i n  Nodal  Measured  Analysis  318.1  328.8  Thin Wire  •A 25mm  J-H  -  N  i  c  k  e  l  P  a  i  n  t  Test-Cylinder for Nodal A n a l y s i s 300  Ambient  Actual  Test-Cylinder  / 115  C. ERROR ANALYSIS AND SAMPLE CALCULATIONS  1. Error Analysis  a. Measurement Errors  The  measurement errors include the uncertainty in the readings from signal noise  and  limitations in accuracy. The temperature, pressure, and bridge voltage errors  were due mostly  to signal noise  generated  has  of 1.7°C from  0°C to 900°C, but this  an accuracy  property 1.7°C  correlations rather tolerance  examination  than  is relevant  over  by the equipment. The thermocouple  the temperature wide  will  difference  temperature  affect  the fluid  calculations; the  swings  only,  since  an  of the data reveals random fluctuations of only 0.15°C at 60 Hz or  greater (much higher frequencies than the temperatures present during test runs).  The  resolution of the crank  third-degree test  rig's  polynomial crankshaft  fitted had  angle  measurement  was  only  to the data. This, coupled  high  rotational  inertia,  made  two  with  degrees, but a  the fact that the  the effective resolution  much higher. Therefore, the error for the velocity measurements resulted mostly from the differentiation of discrete rotational position variations.  The from  resistance of the test-cylinder film and leads was measured at high the bridge in the anemometer. The resistance can be measured  0.01 ohms. The  actual resistance of the • nickel  estimation of the resistance of the leads  film  was  accuracy to within  calculated from  (negligible) and the lead-to-film  an  contact  / 116 that consisted of the thin copper wire  and nickel paint. The uncertainty on this  measurement is +/- 0.1 ohms.  For all cases, the uncertainty has been converted to a percentage for an average measurement value.  Table 7. Measurement Errors  Temperature  (Absolute)  +0.6%  Temperature  (Difference)  ±0.5%  Pressure  ±0.5%  Velocity  ±2.6%  Bridge Voltage  ±1.9%  Bridge Resistance  (Rc and Rh)  ±0.15%  The above a r e random e r r o r s t h a t a f f e c t t h e a c c u r a c y o f t h e c a l c u l a t i o n s . Below a r e t h e e s t i m a t e d u n c e r t a i n t i e s of t h e f i l m v a l u e s t h a t a r e n o t random, m e a n i n g t h a t t h e y a f f e c t c a l c u l a t e d v a l u e s c o n s i s t e n t l y i n t h e same way. Temperature C o e f f i c i e n t Film  Resistance  of R e s i s t a n c e  ±2.5% ±3.6%  / 117 b. Correlation  Errors  Correlations for the fluid calculations. conductivity from  a  were  This  tabulated. A Prandtl  Uncertainties in  the  due to errors  correlation  dependent.  properties of the Freon-114 were  for the  correlation  gas  values  for dynamic  used  viscosity  in the tabulated values. Density constant  produced  that  density  was  values  to simplify the and was  temperature less  constant pressure specific heat correlation was  than used  thermal calculated  and 1%  pressure  from  to calculate the  number.  Table 8. Correlation Errors  Dynamic V i s c o s i t y  Tabulated Values (from a c t u a l )  +1.0%  Correlation (error within tabulated values)  ±0.1%  Tabulated Values  ±0.2%  Correlation  ±1.0%  Constant Pressure S p e c i f i c Heat  Correlation  ±0.6%  Density  Correlation  ±1.0%  Thermal  Conductivity  those  / 118 c. Temperature  The  Variation  accurately. However, temperature  tests of the Freon-114  rates  greater  extent, but  Nusselt  number by  1.09,  2OK  so  will not be  at the temperature a  extremes  measurement  about 2%.  if entropic  if Freon-114  about  produces above  1.05  due  during  the 90  The  assumed  to heat  temperature  rise  the  temperature  indicated  calculated  temperature  discourage  its use  rise  to behave like  in the  data  will  only  be  affected  to a  overestimate the  by  between 10K, the  errors  a perfect gas. A  the  which  gas  in  the  and  the  is only  thermocouple.  analysis. A  thermocouple measurements are used.  rise  is about logarithmic  54) reveals that the polytropic coefficient  of about  to  values of the heat  cycle will  IK  angle  by the thermal lag on  is assumed, the temperature  transfer  a  of  phase  ratio of specific heats (Cp/Cv) for Freon-114  compression  can be  degree  significant. The of the  error  plot of pressure versus volume (Figure is  rise  temperatures will not be  gas is low so the error introduced  average heat transfer rates  transfer  is  Compression  thermocouple's response lag means that the peak  sensed  the  with  The  polytropic  smaller  a  test  rig. This  couple of degrees sensitivity  coefficient  uncertainty  of the  calculation  results if the  100  CD ZJ  co  CO CD  100.0001  0.001  Volume Figure 54.  Pressure versus Volume  / 120 2. Sample  Calculations  a. Piston Kinematics  A=310 B=313  C=connecting D=15  rod length  F= 1 4 1 G=305 H=157 I=crank a n g l e J=16 K=127 L=-90-I S=stroke R=L+380 T=(R/180)*TT E=(G +H )' 2  2  0=tan-  1  5  (G/H-(T-TT/2) )  M=(S +E -2SE*cos(0))' 2  2  5  N=((B-F) +J ) 2  2  - 5  P=-(cos" (M +N -A )/2MN) 1  2  2  2  -sin- ((S*sin(0))/M+tan"'* (H/G)+tan- ' ( ( B - F / J ) - i r ) 1  Q=sin" ((-K-J*sin(P)+F*cos(P))/C) 1  U=-G-J*cos(P)-F*sin(P) V=H-J*sin(P)+F*cos(P) X=U+C*cos(Q)+D Y=V-C*sin(Q)  These are  are the equations for one piston only. The equations for the other piston  similar, with the appropriate changes in crank angle and geometry.  / Figure 55. Test Rig Dimensions  I  122  6. Velocity Calculations  XL=Left P i s t o n Position(mm) XR=Right P i s t o n Position(mm) VL=Velocity of L e f t P i s t o n VR=Velocity of Right P i s t o n VTC=Calculated Bulk Gas V e l o c i t y at T e s t - C y l i n d e r VL=AXL/AT VR=AXR/AT VTC=(VL*[XR])/([XL]+[XR])+(VR*[XL])/([XR]+[XL])/1000(m/sec) VOL=( [XL] + [XR] )*TT*16.09 + 1 .0)cc 2  DSR=[XLmin] + [XRmin]*1 6.09 *7r+1 . 0 ) / (-[XLmin] + [XRmin]*16. 09 *7r) 2  2  [ ] = A b s o l u t e Value  Test-Cylinder  f\  w  VL  I I  VR  p Origin  F i g u r e 56. P i s t o n P o s i t i o n and V e l o c i t y C o o r d i n a t e s  c. Heat Transfer Calculations  Trial  27, F i r s t  Point  Initial  Conditions  (V) V e l o c i t y = 0.012988 m/sec  (BV )Bridge = 2.165 V  (CA) C r a n k A n g l e = 2.37592 d e g r e e s  Rc = 5.12 ohms  (BV) B r i d g e = 3.085 V  Rh = 6.14 ohms  0  Voltage  Voltage  (P) P r e s s u r e / 1 0 0 0 = 0.03705 mV  P = Pressure/1000 = 0.0246 mV  (T) T h e r m o c o u p l e V = 3.05 mV  T = Thermocouple V = 2.940 mV  0  0  Pressure P=37.05 p s i P =24.60 p s i  ± 0.5%  o  Temperature T=274.693+15.8222319X T=322.95K  (x i n mV)  T =321.21K  ±0.6%  0  Power Power=((BV) -(BV ) )/(2+Rh) =0.364475 W a t t s 2  2  0  2  *Rh*(Rh-1.14/Rh) ±3.0%  Rc(actual) Rca=Rc+0.008991214(T-321.21)... ...+0.007348609(P-24.60)=5.227  ohms  ±1.0%  Th=Surface Temperature of Test C y l i n d e r Th-T=(Rh/Rca-1)/a  (a=0.005297±2.5%)  =32.975K Film  ±6.8%  Temperature  Tf=(Th-T)/2+T=339.44 Dynamic V i s c o s i t y K=(162857+0.03321429Tf)/10 =12.903 Ns/m 2  ±0.5% s  ±1.0%  / 124  Thermal^Conductivity k=5.1786*10-**Tf-0.0451429 =0.013064 W/mK R (gas  ±1.0%  constant)  R=7.1019792-0.0442057P... ..,+(-6.679*10- +9.6693*10' P)T = 6.65752 ( m A g ) * ( p s i / K ) 5  5  3  Density  (p)  p=P/RT=(37.05*1000)/(6.65752*339.44) =16.395 Nusselt  ±1.0%  Number  N u = P o w e r / ( K * w * l * ( T h * T ) ) (1=0.025m) =0.364475/(0.013064***0.025*32.975) =10.77  ±7.5%  R e y n o l d s Number Re=pVD/ji (D=0.00168m) =(16.395*0.012988*0.00168)/12.903*10=27.73  Pressure  6  ±3.0%  i s c a l c u l a t e d i n p s i because t h e transducer be c o n v e n i e n t l y c a l i b r a t e d i n t h o s e u n i t s .  could  / 125 The  sample  calculation was  for one  point only; the uncertainty  also for this single point. The uncertainty was that  weighs  method  involves  multiplying was  each  relative  uncertainties  differentiating  the  "differentiation" by  estimated by the use of a method  of each  equation  variable  with  the uncertainty  in  respect  an  equation.  to  each  This  variable,  in the variable to which it  differentiated, and taking the root mean square of the "differentiations". The  averaged each  the  in the result is  results have a smaller uncertainty because  point will  Nusselt  be  averaged  number is affected by  out. This dead  is why  the random uncertainties for  the conclusion  space ratio should  that the variations between the various  dead  that  the  average  be valid despite the fact  space ratios were not much  higher  than the uncertainty for the calculation of Nusselt number at a single point.  


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