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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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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Title | Heat transfer from a circular cylinder subject to an oscillating crossflow as in a stirling engine regenerator |
Creator |
Stowe, Robert Alan |
Publisher | University of British Columbia |
Date Issued | 1987 |
Description | 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 tests to be carried out for the wide range of fluid flow parameters characteristic of various Stirling engines. The influence 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. |
Subject |
Stirling engines Heat exchangers Heat -- Transmission |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2010-07-21 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0097234 |
URI | http://hdl.handle.net/2429/26741 |
Degree |
Master of Applied Science - MASc |
Program |
Mechanical Engineering |
Affiliation |
Applied Science, Faculty of Mechanical Engineering, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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