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

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H E A T TRANSFER FROM A CIRCULAR CYLINDER SUBJECT TO AN OSCILLATING CROSSFLOW AS 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 October c Robert Alan BRITISH COLUMBIA 1987 Stowe, 1987 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Mechanical Engineer ing The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date October, 1 9 8 7  DE-6(3/81) 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 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. ii TABLE OF CONTENTS Abstract ii List of Figures v List of Tables viii I. Introduction 1 II. Literature Review 7 A. Unsteady Flow Past a Circular Cylinder 7 1. Analytical 7 2. Experimental 8 B. Regenerator Flow 10 1. Analytical and Numerical 10 2. Experimental 12 III. Experimental Apparatus and Procedure 16 A. Apparatus 16 1. Test Rig 16 2. Working Fluid 20 3. Instrumentation 20 a. Test-Cylinder and Anemometer 20 b. Thermocouple 22 c. Pressure Transducer 22 d. Crank Angle Measurement 23 e. Oscilloscopes 23 4. Test Section 24 B. Procedure 27 1. Calibration 27 2. Testing 30 3. Data Analysis 32 IV. Presentation of Results 34 A. Experimental Matrix 34 B. Determination of Measurement Response 38 C. Gas Velocity 39 D. Fluid Properties 47 E. Heat Transfer Results 54 V. Discussion of Results 88 A. Gas Velocity 88 B. Fluid Property Variations 91 C. Instantaneous Results 92 D. Averaged Heat Transfer Results 96 VI. Conclusions 100 VII. Recommendations for Further Work 103 iii Bibliography 104 Appendices 106 A. Properties of Freon-114 106 B. Test-Cylinder 107 C. Error Analysis and Sample Calculations 115 1. Error Analysis 115 a. Measurement Errors 115 b. Correlation Errors 117 c. Temperature Variation with Compression 118 2. Sample Calculations 120 a. Piston Kinematics 120 b. Velocity Calculations 122 c. Heat Transfer Calculations 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 31. Nusselt number versus Crank Angle and Reynolds number versus Crank Angle for Trial 3 68 Figure 32. Nusselt number versus Crank Angle and Reynolds number versus Crank Angle for Trial 6 69 Figure 33. Nusselt number versus Crank Angle and Reynolds number versus Crank Angle for Trial 9 70 Figure 34. Nusselt number versus Crank Angle and Reynolds number versus Crank Angle for Trial 13 71 Figure 35. Nusselt number versus Crank Angle and Reynolds number versus Crank Angle for Trial 16 72 Figure 36. Nusselt number versus Crank Angle and Reynolds number versus Crank Angle for Trial 17 73 Figure 37. Nusselt number versus Crank Angle and Reynolds number versus Crank Angle for Trial 21 74 Figure 38. Nusselt number versus Crank Angle and Reynolds number versus Crank Angle for Trial 25 75 Figure 39. Nusselt number versus Crank Angle and Reynolds number versus Crank Angle for Trial 28 76 Figure 40. Nusselt number versus Crank Angle and Reynolds number versus Crank Angle for Trial 30 77 vi Figure 41. Nusselt number versus Crank Angle and Reynolds number versus Crank Angle for Trial 33 78 Figure 42. Nusselt number versus Crank Angle and Reynolds number versus Crank Angle for Trial 36 79 Figure 43. Nusselt number versus Crank Angle and Reynolds number versus Crank Angle for Trial 37 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 46. Average Nusselt number versus Average Reynolds number, 80mm Stroke Trials, by Speed and Phase Angle 84 Figure 47. Average Nusselt number versus Average Reynolds number, 80mm Stroke Trials, by DSR 85 Figure 48. Average Nusselt number versus Average Reynolds number, 90 Degree Phase Angle Trials, by Stroke-to-Test-Cylinder-Diameter Ratio 86 Figure 49. Average Nusselt number versus Average Reynolds number, 180 Degree Phase Angle Trials, by Stroke-to-Test-Cylinder-Diameter Ratio 87 Figure 50. Test-Cylinder Cold Resistance Dependence on Temperature I l l 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 engines are closed-cycle, external combustion engines that show promise as power sources in a wide variety of applications because of their smooth power characteristics, quietness, efficiency, and ability to use a wide variety of heat sources. These engines use heat exchangers to transfer heat to and from the working fluid. This causes the temperature changes that drive the cycle. The heater and cooler transfer heat between the working fluid and the surroundings, but the heat exchanger within the engine, called the regenerator, is the one that has the most profound effect on engine performance. A regenerator is a special heat exchanger in which the hot fluid passes through the same flow passages 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 cold spaces. As the working fluid shuttles between these spaces, the regenerator matrix transfers heat to or from the fluid. A 1% drop in regenerator effectiveness can cause a 4% drop in overall engine efficiency; therefore regenerator flow and heat transfer characteristics should be a priority in Stirling engine research since much still remains unknown about them. Figure 1 shows a diagram of a typical Stirling engine. 1 Figure 1. Typical Stirling Engine Introduction / 2 REGENERATOR COMPRESSION SPACE PISTON CYLINDERS SHOWN IN CROSS-SECTION EXPANSION SPACE PISTON 90 DEGREES Dead Space: Volume unswept by either piston. Dead space r a t i o : (Unswept volume)/(Volume swept by one piston) Heater and Cooler: Tubes Regenerator: Wire mesh Introduction / 3 Unfortunately, the complexity of flow within Stirling engine regenerators has meant that past studies have tended to focus on overall performance data necessary for design purposes, with less emphasis on fundamental studies. Flow within a Stirling engine regenerator is highly unsteady, oscillates at such amplitude that gas at one part of the matrix never passes completely through, and has continually changing fluid properties. Instantaneous Reynolds numbers are commonly from 0 to 1000 (based on matrix wire diameter, typically 0.025 to 0.125mm); this means the flow can be laminar, transitional, or turbulent on average and over a complete cycle. Fluid temperature can vary from 300K to 1000K, and pressures can double over an entire cycle and be as high as 20 MPa. Air, hydrogen, and helium are typical working fluids, and these cycle through the engine up to 100 Hz. Different engine configurations mean that flow paths and volume variations differ from engine to engine as well, making it difficult to identify a typical Stirling engine regenerator flow. Early theoretical and experimental work on regenerators had in general been limited to simple flow conditions, such as almost-steady flow for a long blow period (long "blow" times). The development of gas turbines with regenerators required the study of shorter blow times, typically on the order of a few per second. While the residence time of a particle in the gas turbine regenerator matrix is very short, it nonetheless passes through the matrix, unlike the Stirling engine case. As such, gas turbine regenerator results are 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 assumption of a linear temperature distribution along the length of the matrix approximates the situation in an actual regenerator fairly well. However the assumption of steady temperatures throughout does not, as the entrance and exit conditions of the matrix change continually. Better models described in the next chapter use empirical correlations of flow friction and convective heat transfer rates to estimate the effects of imperfect regeneration and flow losses. These have been based on steady flow data past stacks of wire screens since unsteady flow data was unavailable. Quasi-steady flow is assumed during each of many time increments through a cycle. While this method currently gives the best estimate of Stirling engine performance, overestimation may still be as large as 50%. This is largely due to improper estimates of regenerator heat transfer rates and flow friction. The lack of knowledge of even the fundamental mechanisms of heat transfer and flow friction in Stirling engine heat exchangers have prompted recent work on incompressible oscillating flow in tubes. The general conclusion from these experiments is that shear rates at the tube wall are much higher under laminar, transitional, and turbulent conditions in oscillating flow than in steady or quasi-steady flow. In the laminar case this conclusion is supported by analytical models, and in all conditions this conclusion is supported by observations in actual engines. Analogously, if wall shear is higher, then 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 flow heat transfer correlations for convection from wire screens is based on a Reynolds number that uses wire diameter as a length scale. This is supported by extensive experimental and theoretical evidence. Convective heat transfer rate and drag coefficient correlations for single circular cylinders in crossflow are also based on cylinder diameter. 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 respect to drag and lift coefficients. On each reversal of the flow past the cylinder, vortices and other disturbances generated during the previous flow pass back over the cylinder and cause a change in shear stresses compared to the steady flow case. For Reynolds numbers up to about 50,000, drag coefficients are much higher (up to twice as high) as in the steady flow case. A parameter called the Keulegan-Carpenter number (similar to a Strouhal number and based on the velocity amplitude divided by cylinder diameter) complements the Reynolds number as an experimentally-corroborated parameter of the flow. Unfortunately, Keulegan-Carpenter numbers for flow situations similar to those in Stirling engine regenerators have Reynolds numbers at least a factor of ten higher, so empirical results are not applicable. However, results still support the conclusion that the Reynolds number as well as a type of Strouhal number is important for the determination of shear stresses in oscillating flows past transverse circular cylinders. Introduction / 6 Since the best way to solve a complex problem is often to break it into smaller parts and study each part in the absence of the others, an experimental project to examine the fundamental aspects of regenerator heat transfer under conditions present in a Stirling engine was carried out. The objective was to gain qualitative and quantitative insight into the fundamental problem of oscillating flow past a single heated transverse cylinder, with Reynolds number and volume variation conditions similar to the actual case. A test rig was designed and constructed to allow measurements of convective heat transfer rates from a single transverse cylinder in the absence of wide temperature variations and flow interferences present in Stirling engines. Under these conditions (within a range of parameters that would encompass those found in Stirling engine regenerators) other parameters of the flow geometry, such as a Strouhal number, could be evident. Just as convective heat transfer rates in steady flow past a single, transverse circular cylinder are related to steady flow past stack of wire screens, knowledge of convective heat transfer rates for various oscillating flows past a single, transverse circular cylinder may be related to the same flow in a Stirling engine regenerator. II. LITERATURE REVIEW A. UNSTEADY FLOW PAST A CIRCULAR CYLINDER 1. Analytical Analytical solutions for special cases of both oscillating flow and suddenly accelerated flow exist. In Sarpkaya and Isaacson [1] analytical solutions for any time dependent flow about a transverse circular cylinder are presented for the inviscid, ideal fluid case. Separation and the presence of boundary layers are not taken into account, but this case 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 [2] presents a solution to the suddenly-accelerated circular cylinder problem using the similarity transformation technique of Blasius [3]. A viscous term 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 separation, grow, and are Finally swept downstream. The wake behind the cylinder becomes very unsteady. Attempts to describe pressure distribution characteristics of this flow with potential flow theory reported by Schwabe [4] were not very successful, and the differences between the real flow and the calculated pressure distribution increased with the time after separation. This indicates that other 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 amplitude oscillations (amplitude very much less than the diameter of the cylinder, known as acoustical streaming flow). Pictures of streamlines from experiments agree well with the calculated streamlines. However, for oscillations of the same order as the radius of the cylinder streamlines are substantially different because of separation (which is the case in this author's work). 2. Experimental Sarpkaya and Isaacson [1] deal with unsteady flow about a transverse circular cylinder with respect to drag and lift coefficients. Another parameter, known as the Keulegan-Carpenter number (K) (velocity amplitude times flow period divided by cylinder diameter) is introduced to describe the amplitude of the fluid motion as compared to cylinder size. This parameter is similar to the Strouhal number (but inverted) and is shown to be important to determine drag and inertia Literature Review / 9 coefficients in oscillatory flow at high Reynolds numbers (greater than 10000). The Reynolds analogy between heat and momentum transfer shows that skin friction drag and convective heat transfer rates are related in a laminar boundary layer. In [1] drag coefficients reported for oscillatory flow are much higher (up to twice as high) as those for quasi-steady flow at Reynolds numbers between 10000 and 50000 and Keulegan-Carpenter number in the range relevant to Stirling engine regenerators. For the range of Reynolds numbers between 50 and 10000, heat transfer rates from transverse, oscillatory flow past a circular cylinder might also be expected to be substantially higher than in the quasi-steady flow case, and should be related to a parameter such as the Strouhal or Keulegan-Carpenter number. Richardson [5] presented a theoretical and experimental study of heat transfer in acoustical streaming flow. He built upon the analytical solutions for the streamlines calculated by Schlichting [2]. The experimental and theoretical results are similar when natural convection effects are ignored. Once again, however, the streaming flow case infers small amplitude oscillations with no separation, unlike the flow expected in the Stirling engine case. B. REGENERATOR FLOW Literature Review / 10 1. Analytical and Numerical The regenerator is the heat exchanger in a Stirling engine that is most important for performance, but it is also the most complex from a fluid mechanics and heat transfer point of view. As a result, there are no analytical methods that deal with the problem in sufficient detail to allow for design. The need to estimate Stirling engine and therefore regenerator performance has lead to simple treatments of the problem. The simplest analyses for the Stirling cycle assume perfect regeneration. One such example is the Schmidt analysis [6,7]. Temperatures throughout the engine are assumed to stay constant with time and the linear temperature distribution in 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 numerical methods to estimate performance of Stirling engines The ideal adiabatic model [6] still assumes perfect regeneration and a steady, linear temperature distribution in the regenerator. This contributes to the overestimation of engine performance. However, improvements to the model in areas other than the regenerator allow a more accurate estimate of performance to be made than with the Schmidt analysis. Literature Review / 11 Attempts to incorporate more accurate approximations of regeneration into a Stirling engine analysis technique use empirical data to estimate heat transfer and flow friction. Because little experimental work has been done on unsteady flow regeneration (as conceded by several Stirling engine researchers [6-12]) steady flow data is used for the calculations. The quasi-steady flow model [6] divides the regenerator flow cycle into many time increments; steady 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 assumptions still lead to overestimates of engine performance compared to an actual engine. Urieli and Berchowitz [6] report that flow friction was underestimated by a factor of four compared to the real case. A method that concentrates on regenerator design rather than engine performance prediction was reported by Miyabe et al [13]. Steady flow correlations of Nusselt number and flow friction factor versus Reynolds number (based on wire diameter or mesh size) for a stack of screens were used. The correlations presented correspond to the range of Reynolds numbers and wire parameters used in Stirling engine regenerators. The correlation for heat transfer relates Nusselt number to Reynolds number, with the geometric parameter being the diameter of the wire. Flow friction factor is correlated with Reynolds number based on the spacing of the wires in the mesh. The engine and regenerator parameters are matched to a graph of the Biot number (surface heat convection of wire compared to its internal heat conduction) and Fourier number (diameter of wire compared to thermal penetration during the regenerator blow period), so that the Literature Review / 12 temperature at the centre of the wire will be close to the temperature of the working fluid. The number of screens needed for the regenerator is determined from a theoretical analysis that assumes idealized flow conditions through the matrix. These assumptions include constant mass velocity, constant heat transfer coefficient, constant inlet temperature, constant fluid and matrix properties, and uni-directional flow (for each half-cycle). Regenerator effectiveness versus Number of Transfer Units (NTU) graphs for various ratios of matrix heat capacity and to fluid heat capacity are used to choose the number of screens required to yield an effectiveness greater than 0.95. Flow friction is then calculated and if excessive, the procedure is repeated for a different wire diameter. Results from this method compared favourably with data from tests of regenerators in one engine. These results should not be regarded as conclusive, but the validity of the method may be supported if results from unsteady flow experiments yield the same type of correlations as the steady flow data. 2. Experimental Recently have researchers have obtained experimental data that supports the assumption that heat transfer rates and flow friction in Stirling engine heat exchangers are much higher than in steady flow. Experiments by Dijkstra [14] and Taylor and Aghili [15] on oscillating flow in tubes are more relevant to heaters and coolers than regenerators but they do show that under laminar, transitional, and turbulent flow, pressure drops measured are much higher than in the steady flow case. A survey of work on oscillating flow by Seume and Literature Review / 13 Simon [16] discusses duct flow under laminar, transitional, and turbulent conditions. The fully-developed laminar case 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 not long enough to assume fully-developed flow. In transitional flow, turbulence appears to be enhanced during deceleration and is reduced during acceleration. This is the region of flow that is probably most important in Stirling engines. They state that turbulent flow pressure drops are higher than in steady flow, but means to predict them are disputed by several researchers. Seume and Simon [16] propose that the Reynolds number, the kinetic Reynolds number (or Valensi number which is the frequency times a diameter squared divided by four times the kinematic viscosity), and another geometric parameter, such as a length-to-diameter ratio describe oscillatory flow sufficiently to compare flows in different engines and experiments. However, in regenerator flow the kinetic Reynolds number is very small and is not expected to be an important parameter. One interesting note is that in any given situation, the Reynolds number is related to the kinetic Reynolds number by a simple geometric ratio, such as the stroke divided by the wire diameter. Rice et al [12] deal with oscillating flow in a stack of wire screens in a test rig that varies the flow velocity sinusoidally, but without compression or expansion. The range of Reynolds numbers is at the low end of that expected in an actual engine. Correlations of Nusselt number with Reynolds number (based on wire diameter) are presented. These correlations depend on wire diameter and Literature Review / 14 mesh size, unlike the steady flow correlations used in [13] that were based on Reynolds number. Therefore heat transfer rates in oscillating flow may depend on parameters other than the Reynolds number. In Rice's work the flow is heated and cooled at opposite ends of the regenerator; the flow in and out of the regenerator is influenced by the heat exchangers. A comparison of the absolute values of heat transfer rates with the steady flow data in [13] is difficult in this case. The main conclusion that can be drawn from the experimental work discussed is that much more work needs to be done on regenerator flow. Since little is understood about heat transfer from 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 conditions similar to the flow in a Stirling engine regenerator and compare them to steady flow values. 2. To measure the effect, if any, that phase angle difference of the piston motions had on the convective heat transfer rates in this experiment. 3. To discover the effect, if any, that dead space ratio had on the convective heat transfer rates in this experiment. (Dead space ratio is a parameter widely used to describe a Stirling engine configuration. It is the unswept or dead, volume in the engine divided by the swept volume of one piston.) 4. To discover the effect, if any, that stroke-to-test-cylinder-diameter ratio had on convective heat transfer rates in this experiment. This will be analogous to the stroke-to-wire-diameter ratio in a Stirling engine regenerator. 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 peculiarities in the flow parameters that may have affected the convective heat transfer results over those expected from a purely sinusoidal oscillating flow case. The flow field was established by the use of a test rig with a similar configuration to the engine shown in Figure 1. Heat transfer was measured by the use of a heated-film test-cylinder, similar to a hot film probe. A pressure transducer and thermocouple measured the pressure and temperature of the gas, and bulk gas velocity at the test-cylinder was calculated from crank position and rotational speed. Nusselt numbers and Reynolds numbers for the test-cylinder were calculated for the various test runs. III. EXPERIMENTAL APPARATUS AND PROCEDURE A. APPARATUS 1. Test Rig The motivation behind the design and construction of the test rig was to produce a flow situation similar to a Stirling engine with simplicity and versatility. The maximum peak Reynolds number that had to be accommodated was about 1000, and by using a heavy gas as the working fluid and a large diameter hot film "probe", or test-cylinder, pressures and speeds could be kept low enough to keep the rig design simple, safe, inexpensive, and easily 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 Stirling engines use an opposed-piston arrangement, as shown in Figure 1. Since the goal of the experiment was to gain insight into heat transfer rates from a circular cylinder in a "simple" crossflow (absent from influences such as twists and constrictions in the flow path), a horizontally-opposed piston setup was chosen with a constant-diameter flowpath from one piston to the other. The test section in the middle contained the test cylinder, gas temperature thermocouple, pressure transducer, and two observation windows. 16 Experimental Apparatus and Procedure / 17 The piston diameter chosen was 32mm, since this is the smallest diameter piston commercially available for common internal combustion engines (50cc). Each piston travelled in its own cylinder, honed for a small piston-cylinder clearance. Solid rectangular teflon rings were machined to allow a good seal and easy sliding under dry lubrication conditions. (Dry lubrication was necessary to prevent contamination of the test section by oil.) A piston pin fitted 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 was placed directly below the test section. The crank plates were drilled to accommodate strokes 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 phase angles. Two connecting rods and a pinned link transferred the rotary 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 complete adjustability. To drive the pistons at the desired speed under a pressure of 1 MPa, a 1/2 hp source was needed. A 1/2 hp DC electric motor with an SCR controller was chosen to allow speed adjustability, but little variation in speed once the controller was set. A 20:1 speed reducer and a pair of high-torque drive pulleys (3:2 ratio) and toothed belt stepped the maximum rotational speed of the motor down to about one Hertz. All drive components were mounted on a welded steel frame made of 2-inch-square tubes that provided more than adequate strength and rigidity. Figure 2. Test Rig Experimental Apparatus and Procedure / 18 Experimental Apparatus and Procedure / 19 Figure 3. Piston Positions for Various Phase Angles. PHASE ANGLE 0 DEGREES 90 DEGREES 180 DEGREES COMPRESSION AND EXPANSION COMBINATION AS IN A S T I R L I N G ENGINE SLOSHING 2. Working Fluid Experimental Apparatus and Procedure / 20 To achieve a peak Reynolds number of 1000 a heavy gas that could be used at room temperature and pressure was required. Freon-114 (dichlorotetrafluoroethane) was chosen because of its availability, low toxicity, high molecular weight (and therefore low kinematic viscosity), and low vapour pressure of about twice atmospheric pressure at room temperature. By heating the cylinders and test section of the rig with a heating tape to about 90° C, the pressure of gaseous Freon-114 could be raised to 1 MPa. A 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 required Reynolds numbers was about 2mm. This size 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 equipment. The test-cylinder chosen was a glass capilliary tube of 1.8mm diameter, cut to 28mm length. It would then fit inside the test section Experimental Apparatus and Procedure / 21 but was of sufficient length to minimize 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 was mounted between acrylic rods and glued with epoxy. A thin wire (used for winding transformer coils) ran along the top of the acrylic rod to the end of the film, where it was connected electrically with one loop and some silver paint. This proved to provide a rigid, thermally insulating mounting system for the test-cylinder and allowed an easy electrical hookup to the anemometer. Figure 4. Test-Cylinder Test-cylinder as seen through end of test rig Experimental Apparatus and Procedure / 22 The test-cylinder and mount were placed in the test section between two Conax fittings. These fittings sealed the wire leads completely and prevented Freon-114 leaking from the test section. The leads were hooked 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 1/32 inch diameter exposed junction copper-constantan thermocouple probe was chosen. This probe was hooked up to a cold junction compensator, then to a small battery-operated 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 Transducer A 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 fast response to pressure variations of less than one millisecond. 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 are mounted so the outside holes pass by one of them, and the reset hole passes by the other. Passage of a hole causes the photodetector to emit a pulse to an electronic circuit 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 of the output is very good, so calculations involving crank angle (such as piston velocity) are simplified. e. Oscilloscopes Two 2-channel Nicolet digital oscilloscopes were used to sample the output voltages from each of the four measurement setups (test-cylinder voltage, temperature, pressure, and crank angle). The sweeps were triggered by the reset signal from the crank angle circuit and the oscilloscopes were set to sample at 1000 Hz. Each channel could store 4000 points, so sweep time was four seconds. This data could then be transferred in binary format to an IBM-compatible personal computer for storage on a 5.25 inch diameter floppy disk, where it could be analyzed later. Experimental Apparatus and Procedure / 24 4. Test Section The test section had to be designed to accommodate the test-cylinder, thermocouple, pressure transducer, a port for Freon-114 supply, and two observation windows. It was made of aluminum. Holes drilled into the test section were kept as small as possible to minimize extra dead (unswept) volume and secondary flows. The Freon-114 supply was fitted with a check valve in the test section so the supply tube between the test section and the first shut-off valve would not become part of the dead volume. The windows were filled with acrylic plugs since they consisted of an inch long hollow plug with 1/4 inch of glass at the outside end. A thin, flexible, clear plastic film was glued to the inside of the test section bore to allow gas "communication" across itself, but to reduce secondary flow effects and streamline the flow along the bore discontinuities. A photograph of the test section is in Figure 5. Figure 6 shows a schematic diagram of the data acquisition system. Figure 5. Test Section Experimental Apparatus and Procedure / 25 i THERMOCOUPLE AMPLIFIER PRESSURE TRANSDUCER CRANK ANGLE DETECTOR AMPLIFIER AMPLIFIER AND STAIRCASE GENERATOR DIGITAL OSCILLOSCOPE TEST-CYLINDER HOT FILM ANEMOMETER DIGITAL OSCILLOSCOPE PERSONAL COMPUTER Figure 6. Data Acquisition B. PROCEDURE Experimental Apparatus and Procedure / 27 1. Calibration Measurement setups for temperature and pressure were supposed to provide linear outputs over the range of values expected during the tests. This would allow linear algebraic relations to be used to correlate the output voltages with temperatures and pressures. This linearity was checked by exposing the thermocouple to different temperatures and the pressure transducer to different pressures. Results indicated that a linear correlation could be used in both cases (see Appendix C). Some interesting properties of the test-cylinder were discovered during its calibration, and were taken into account during the data analysis. Film resistance was found to increase almost linearly with pressure, so a linear correlation was incorporated into the data analysis to allow for changing pressures when heat transfer rates were calculated. This dependence on pressure can be attributed to the fact that the test-cylinder is hollow, and the pressure difference across the wall causes a slight distortion of the test-cylinder. The resistance of the nickel film therefore changes, much like a strain gauge. Film resistance varied with temperature in a linear fashion, as expected, but the temperature coefficient of resistance that resulted depended on the way the test-cylinder was heated. A pure nickel film should have a temperature coefficient of resistance of 0.006 ohms/ohms/°C. A film of 98% nickel and 2% manganese Experimental Apparatus and Procedure / 28 has a value of 0.0045 ohms/ohms/°C, and this indicates impurities in the nickel film will decrease the value. The exact composition of the nickel film on the test-cylinder is not known, so the test-cylinder was heated, in air, by setting the resistance on the constant temperature anemometer to different values. Surface temperatures were measured directly with a thermocouple. The thermocouple readings indicated that the temperature distribution along the length of the test-cylinder was approximately parabolic and that there was little circumferential variance in temperature, so the test-cylinder could be assumed to have circumferentially invariant properties. Because the temperature distribution along the test-cylinder length was approximately parabolic, the average temperature coefficient of resistance was found from the graph of thermocouple readings taken at one point. The value was found to be 0.005297 ohms/ohms/°C (to within 2.5%). The test-cylinder was also subjected to different temperatures by heating the cylinders and test section with the heating tape. This meant that the entire test-cylinder and its mounting were subjected to the same temperature. The temperature coefficient of resistance was found to be 0.001885 ohms/ohms/°C for this type of heating, less than 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 coefficient for uniform heating was used to adjust the cold resistance of the test-cylinder when the gas temperature varied during a test run. The coefficient for electrical heating (parabolic distribution) was used to calculate the surface temperature of the test-cylinder. Experimental Apparatus and Procedure / 29 Free convection losses from the test-cylinder were thought to be small. The main power loss with the pistons motionless was conduction out the ends. The free convection loss for these conditions was also taken into account, and was approximately 10-15% of the total power loss. Changes in the amount of free convection were not included in the analysis. Even at small Reynolds numbers however, free convection from cylinders in steady crossflow accounts for less than 10% of the total heat transfer, so any change in this would affect the overall Nusselt number minimally. Three-dimensional effects were reduced by the length-to-diameter ratio of the test-cylinder being about 15:1. "Trailing" vortices from the edges of the acrylic rod would not reach in far enough to disturb the flow over the effective part of the test-cylinder. The parabolic temperature distribution means that the temperature difference is smaller at the ends, effectively reducing the importance of the end losses further. For each test, temperature, pressure, and bridge voltage were recorded while the test-cylinder was heated electrically, but before the rig was operated. This gave reference values so that during the test run, the change in cold film resistance could be calculated. From this the surface temperature of the test-cylinder could be found because the effective overheat ratio would change (the hot resistance setting remains fixed during a run). The initial bridge voltage reading yields power dissipated in the test-cylinder due to conduction and 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 test runs were made over a varietj' of strokes, phase angles, working fluid pressures, and dead-space ratios. Runs at a phase 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 (complete sloshing, no compression or expansion) were done 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 settings and pressures were varied so a wide range of Reynolds numbers and dead-space ratios that encompass those found in actual engines were examined. Each set of tests (usually four to six) was preceded by setting the desired stroke, phase 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 apart (minimum volume). For 180 degrees the crank plate was positioned at the Experimental Apparatus and Procedure / 31 arbitrary 0 degree position. The working space was then filled with Freon-114 vapour to a desired pressure and the rig cycled to make sure that no condensation 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 pressure was obtained, the cold resistance of the film was determined, and the corresponding hot resistance (an overheat ratio of 1.2 was always used) was set on the constant temperature anemometer. The unit was then set to run, supplying the test-cylinder with power. At this point, temperature, pressure, and bridge voltage were recorded to provide the initial reference settings for the later analysis. The test rig was then operated at the desired speed until the oscilloscope traces indicated that cycle-to-cycle variation was sufficiently small and a "quasi-steadiness" had been reached. A four second sweep of the measurements was then taken, stored, and transferred to floppy disks in the computer. The cold resistance was then checked to ensure that it had not changed from before the test, and after the data had been stored, the next test was done at a different speed and/or pressure. 3. Data Analysis Experimental Apparatus and Procedure / 32 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 voltages had to be converted to gas velocities so the Reynolds number of the flow could be calculated. A third-degree polynomial was fitted by regression analysis to a set of voltages that represented one crank angle rotation, and the voltages were scaled so the lowest could be 0 degrees and the highest 360 degrees. This gave a plot of crank angle position versus time. Approximately 180 points (spaced at 5msec or 10msec intervals, depending on the speed of the test run) were stored as a file in ASCII format so they could be read into a SYMPHONYt spreadsheet. Once they were read into the spreadsheet, the crank angle values were manipulated to yield the positions of each piston at each point in time. The difference in piston position between each point gave piston velocity, and gas velocity was calculated by a "weighted" average of both piston velocities. This gas velocity was used to calculate the Reynolds number at each point. For each point, ' the corresponding pressure, temperature, and bridge voltage reading was read into GRAFPAK and stored as part of an ASCII file. These files, along with gas velocity, were read into another spreadsheet. There, the temperature and pressure readings were manipulated to give values from linear correlations of density, kinematic viscosity, and thermal conductivity. Nusselt t® Alan Jones, University of British Columbia. t° Lotus Development Corporation. Experimental Apparatus and Procedure / 33 number and Reynolds number at each point was calculated from these results. Sample calculations and property correlations are presented in Appendix C. Only 180 points were used for the data analysis for each cycle since the spreadsheet memory quickly filled because of the complexity of the calculations. As it was, four spreadsheets had to be used to get the four inputs to Nusselt number and Reynolds number outputs. Because the measurement instruments had been well grounded, the error due to noise for the outputs was small. Averaging the readings over five or ten points around each of the approximately 180 points may have "damped out" the noise mathematically, but the additional accuracy would not have been worth the additional work. 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 the 0 to 600 range. At the 40mm stroke, runs were done at dead space ratios of 1.5 and 2.0. The rig could not be adjusted for the dead space ratio of 1.0. The majority of the tests were done at the stroke of 80mm because dead space ratios of 1.0, 1.5, and 2.0 could be attained. This would show the dependence of heat transfer rates upon dead space ratio. At the 120mm stroke, only the dead space ratio of 1.0 was examined. At each stroke and dead space ratio, runs were done at either full or half speed, and at different gas pressures so results over a wide range of Reynolds numbers could be obtained. Runs for identical geometric parameters and similar Reynolds numbers, but different gas pressures and speeds, were made to expose the possible dependence of the heat transfer results on other parameters besides Reynolds number, dead space ratio, and stroke. Because other investigators have examined the 180 degree phase angle case exclusively, runs at this phase angle were made 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 300. 34 Presentation of Results / 35 The lower range of Reynolds numbers in the 180 degree phase angle case result because the peak pressure throughout the test is almost constant, while in the 90 degree phase angle case there are wide pressure variations. Average Reynolds numbers are therefore used as a comparison between the different phase angle cases. Tables 1 and 2 show the test matrix and calculated results for Reynolds and Nusselt numbers. T r i a l Phase Angle S t r o k e Speed Reynolds No. N u s s e l t No. P r e s s u r e (Degrees) (mm) (Average) (Average) (Average,kPa) 1 180 40 50% 56. 10 3.57 152.2 2 180 40 50% 81 .04 4 . 36 224.4 3 180 40 100% 89.85 5.87 110. 1 4 180 40 100% 97.08 6 . 19 133.4 5 180 40 100% 163.15 8.56 240.0 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 Trials 1-14 oo T r i a l Phase Angle S t r o k e Dead Space Speed Reynolds No. N u s s e l t No. P r e s s u r e (Degrees) (mm) R a t i o (Average) (Average) (Avg.kPa) 15 90 120 1 . 105 50% 158 . 19 11 . 23 164. . 4 16 90 120 1 . 105 100% 314 .66 15.55 156. .4 17 90 120 1 . 105 50% 281 .25 19. 18 410 .5 18 90 120 1. . 105 100% 520 .52 27.41 364. . 1 19 90 80 2 .082 50% 99. 74 6 . 18 170. .4 20 90 80 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 33 90 80 0. .998 100% 246 . 29 16.93 220. 3 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 38 90 40 1 . 500 100% 86.! 99 8.48 148. 8 39 90 40 1 . 500 100% 139 .84 1 1 .79 268. 6 Table 2. Test Matrix 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. DETERMINATION OF MEASUREMENT RESPONSE Before testing at the 90 degree and 180 degree phase angles, the test rig was set for 0 degree phase angle operation. This mode allowed no bulk gas velocity at the test-cylinder, but compression and expansion of the gas occurred. This gave the time response of the gas temperature thermocouple and the test-cylinder system The comparatively fast response of the pressure transducer meant that the time lag between peaks of the pressure transducer voltage and the anemometer bridge and thermocouple voltages would give an accurate estimate of the response. At a moderate system pressure 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 variations in temperature over 50msec during the test runs meant that improvements in accuracy of the results by the inclusion of the thermocouple response lag in the calculations would 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 100msec response lag meant that rapid variations in heat transfer rates could not be measured. At full speed the variation would have to be over 40 Presentation of Results / 39 degrees of crankshaft rotation to be measured with a high degree of accuracy. However, this resolution is acceptable since the application of the results is meant to be Stirling engine regenerators. Changes in heat transfer rates over less than 40 degrees of crankshaft rotation have little relevance in this case, where the average heat transfer rate over a half-cycle (180 degrees) is of much greater interest. This lag was not considered in the calculations, but is taken into account in the discussion of the overall heat transfer results. C. GAS VELOCITY Experimental investigations have concentrated upon the 180 degree phase angle case, so assumed bulk fluid velocities have been taken to be identical to the piston velocity. This is a good assumption. However, in the 90 degree case, the gas velocity at the test-cylinder is related to an average of the velocity of both pistons. Since the distance between the pistons is never more than 8 or 9 piston diameters, and the gas velocity adjacent to each piston is identical to that piston's velocity, the calculated gas velocity is a good estimate of the actual bulk gas velocity at the test-cylinder. This calculated velocity is an average of the piston velocities, weighted according to the distance of the test-cylinder from each piston. The closer piston has more influence on the velocity at the test-cylinder. (See Appendix C.) Graphs of calculated gas velocity at the test-cylinder are presented for each stroke and phase angle in Figures 7-12. Velocity 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 1 0 90 180 270 360 Crank Angle (degrees) Figure 7. Calculated Bulk F l u i d V e l o c i t y at Tes t -Cy l inder for T r i a l 3 -0.3 H 1 1 1 1 0 90 180 270 360 Crank Angle (degrees) Figure 8. Calculated Bulk Fluid Velocity at Test-Cylinder ^ for Trial 9 0.6 -0.6 H 1 1 1 1 0 90 180 270 360 Crank Angle (degrees) Figure 9. Calculated Bulk Fluid Velocity at Test-Cylinder for Trial 13 Figure 10. Calculated Bulk Fluid Velocity at Test-Cylinder for Trial 16 Figure 11. Calculated Bulk Fluid Velocity at Test-Cylinder for Trial 25 D. FLUID PROPERTIES Presentation of Results / 47 The fluid properties for Freon-114 are calculated from the measurements of temperature and pressure. The initial temperature of each test was raised high enough to ensure that no condensation of the Freon-114 took place; this is how the higher gas pressures and therefore higher Reynolds numbers were achieved. The cylinder walls provided neither adiabatic nor isothermal boundary conditions for the flow and as a result the bulk temperature variations in the gas were only a few degrees Kelvin in the 90 degree phase angle case. Pressure varied smoothly as well. For the 180 degree phase angle case small pressure and temperature variations result because of volume changes due to the geometry of the test rig. The magnitudes of these variations increase with stroke. Graphs of temperature, pressure, and volume for test run 54 are shown in Figures 13-15. Graphs of pressure and volume for test run 23 are in Figures 16 and 17; there was no measureable change in temperature during the 180 degree phase angle runs. 500 400 300 200 100 0 Crank Angle (degrees) Figure 13. Pressure versus Crank Angle for T r i a l 28 250 90 180 270 Crank Angle (degrees) 360 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 Crank Angle (degrees) 360 Figure 17. Volume versus Crank Angle for T r i a l 9 O l Presentation of Results / 53 Relations for density, dynamic viscosity, thermal conductivity, and constant pressure specific heat were derived from tables. Over the range of temperatures and pressures examined, linear relations for dynamic viscosity, thermal conductivity, and constant pressure specific heat versus temperature provided sufficient accuracy (see Appendix C). A relation for density dependent on pressure and temperature was also used. Table 3 shows typical values for each of the properties. Table 3. Typical Fluid Property Values Pressure (MPa) Temperature (K) Density (kg/m3) 0.05-0.69 295-360 3.4-45.3 11.4-13.6 x 10"6 0.011-0.014 0.698 Dynamic Viscosity (Ns/m2) Thermal Conductivity (W/m°C) Prandtl Number Prandtl number variation is less than 1% over this temperature range. E. HEAT TRANSFER RESULTS Presentation of Results / 54 Figures 18-43 present calculated and measured data during several selected test runs that give a cross-section of all the data taken. The "Instantaneous Nusselt number versus Reynolds number" graphs in Figures 18-30 show the values at each of the approximately 180 data points spaced throughout one cycle of the test rig. These graphs are plotted on a logarithmic scale so that all the test runs can be plotted on the same scale for comparison, but good resolution occurs at the lower values, where most 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 variation in the instantaneous points on the graph, Table 4 shows the values for Nusselt number and Reynolds number, both peak and average, and the average pressure. Table 4. Cycle-to-Cycle Variation of Test Run 27 Trial Reynolds No. (Max) 27A 626.11 27B 633.93 27C 637.91 Reynolds Nusselt No.(Avg) No. (Max) 249.59 22.38 250.35 21.64 249.73 21.58 Nusselt Pressure No.(Avg) (Avg,kPa) 15.60 225.8 15.60 225.3 15.79 225.1 Man a n 1 •sg fi % ° o • > 8 c • 0-90 Degrees • 90-180 Degrees o 180-270 Degrees o 27C I-36C > De gre es 10 Reyno 100 ds number 1000 Figure 18. Instantaneous Nusselt number versus Reynolds number for Trial 3 * f I - - .A E9 sr f • 0-90 Degrees ° *> % O O " • • • • • • 90-180 Degrees o 180-270 Degrees o 27< 3-36I 3D< >gr< »et 10 100 1000 Reynolds number Figure 19. Instantaneous Nusselt number versus Reynolds number for Tr i a l 6 3? > < \ P 9 m • 9 o • • 0 • • • 0-90 Degrees • 90-180 Degrees o 180-270 Degrees • 27C )-36( ) De gre 0 8 10 Reyno 100 ds number 1000 Figure 20. Instantaneous Nusselt number versus Reynolds number for T r i a l 9 E C CD CO CO 3 10 • e- • o — f 0 —C > I • 0-90 Degrees • 90-180 Degrees o 180-270 Degrees o 27C >-36C > De gre es 0 1 00 10 00 Reynolds number Figure 21. Instantaneous Nusselt number versus Reynolds number for Tr i a l 13 ^ 0 0 • • « • • • • • a • i *-->• • • • • 0-90 Degrees • 90-180 Degrees o 180-270 Degrees o 27C )-36C >De gre es 10 100 Reynolds number 1000 Figure 22. Instantaneous Nusselt number versus Reynolds number for Tria l 16 7 _ i \ • • .-. • • 1 * ,* / • • I • 1 • D 1 [ • • i 3 1 • • * Q • • 0-90 Degrees • 90-180 Degrees o 180-270 Degrees o 27C )-36C ) De gre es ) 1 0 0 1 01 OO Reynolds number Figure 23. Instantaneous Nusselt number versus Reynolds number for Tria l 17 E c (D W W 10 , " l i i l . '% I —• • •—m- I 1 • • D n E E _ < V 1 o • 0-90 Degrees • 90-180 Degrees o 180-270 Degrees D • • • • o 27C )-36( ) De gre tes 10 100 Reynolds number 1000 Figure 24. Instantaneous Nusselt number versus Reynolds number for Tria l 21 ^ • i " i , 1 • mi — > m 1 ial>V •< n o • • 0 ti at J w _ * • D • • « • i • 0-90 Degrees • 90-180 Degrees o 180-270 Degrees o 27C )-36C ) De gre es 10 100 1000 Reynolds number Figure 25. Instantaneous Nusselt number versus Reynolds number for Tria l 25 • • . 1 • • • mi II , , c V ' o •o—O • J LI H i - #§*..j • • « • » • 0-90 Degrees • 90-180 Degrees 0 180-270 Degrees o 27C )-36( ) De gre tea 10 100 Reynolds number 1000 Figure 26. Instantaneous Nusselt number versus Reynolds number for Tria l 28 • • •• 1 w • • I" i • • d D • » « • i • 0-90 Degrees • 90-180 Degrees o 180-270 Degrees o 27C )-36C ) De gre es 10 100 Reynolds number 1000 Figure 27. Instantaneous Nusselt number versus Reynolds number for Tria l 30 1 1 • • * • • • • • a < r a « • • a i • • • E t j O n C l^ 3 • • # , • • .* f • 0-90 Degrees • 90-180 Degrees o 180-270 Degrees a 27C )-36( ) De gre tea 1 10 0 1 01 DO Reynolds number Figure 28. Instantaneous Nusselt number versus Reynolds number for T r i a l 33 • . • •• • %" < * 1 n * I n n • • • 1 C EJ t. tr • 9 • 0-90 Degrees • 90-180 Degrees © 180-270 Degrees o 27C )-36( ) De grc ies 10 100 1000 Reynolds number Figure 29. Instantaneous Nusselt number versus Reynolds number for Tr i a l 36 • 0-90 Degrees • 90-180 Degrees o 180-270 Degrees a 270-360 Degrees 10 100 Reynolds number 1000 Figure 30. Instantaneous Nusselt number versus Reynolds number for Tria l 37 / 68 10 9 0 180 270 Crank Angle (degrees) 3 6 0 150 .Q £ ZJ c CO TD O c >-<D OC 100 5 0 <9 ^ \ § 8 8 o » o° O n \ / / \ % — v o o o o o o o o o ° < k $ 9 0 270 180 Crank Angle (degrees) 3 6 0 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 Tr i a l 6 / 70 Figure 33. Nusselt number versus Crank Angle and Reynolds number versus Crank Angle for Tr i a l 9 / 71 6 0 0 9 0 180 270 3 6 0 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 3 6 0 Crank Angle (degrees) Figure 35. Nusselt number versus Crank Angle and Reynolds number versus Crank Angle for T r i a l 16 / 73 6 0 1000 0 9 0 180 270 3 6 0 Crank Angle (degrees) Figure 36. Nusselt number versus Crank Angle and Reynolds number versus Crank Angle for T r i a l 17 / 74 Figure 37. Nusselt number versus Crank Angle and Reynolds number versus Crank Angle for T r i a l 21 / 75 4 0 0 0 9 0 180 270 3 6 0 Crank Angle (degrees) Figure 38. Nusselt number versus Crank Angle and Reynolds number versus Crank Angle for Tr i a l 25 / 76 0 9 0 180 270 3 6 0 Crank Angle (degrees) Figure 39. Nusselt number versus Crank Angle and Reynolds number versus Crank Angle for Tr i a l 28 / 77 15 OH 1 1 1 1 O 9 0 180 270 3 6 0 Crank Angle (degrees) 3 0 0 0 9 0 180 270 3 6 0 Crank Angle (degrees) Figure 40. Nusselt number versus number versus Crank Angle Crank Angle and for T r i a l 30 Reynolds / 78 8 0 0 0 9 0 180 270 3 6 0 Crank Angle (degrees) Figure 41. Nusselt number versus Crank Angle and Reynolds number versus Crank Angle for Tr i a l 33 4 0 0 0 9 0 180 270 3 6 0 Crank Angle (degrees) Figure 42. Nusselt number versus Crank Angle and Reynolds number versus Crank Angle for Tr 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 OJ -O E c w CO Z 10 • o .->«• o • • o / i • • [ Uf" • Trial 27A • Trial 27B o Tr al 2 7C 10 100 Reynolds number 1000 Figure 44. Cycle-to-Cycle Variation of Instantaneous Nusselt number versus Reynolds number for T r i a l 27 00 Presentation of Results / 82 Figure 45 presents the results for average Nusselt number versus Reynolds number for each test run. A line that represents a steady flow correlation for heat transfer from a circular cylinder in cross-flow is also on the graph to illustrate the difference between the oscillating and steady flow results. The results are plotted by test rig piston speed and phase angle to show the dependence of heat transfer rates upon these. No distinction for dead space ratio or stroke-to-test-cylinder-diameter ratio is made. The lowest Reynolds number points for the 180 degree phase angle case are very close to the steady flow correlation, but all of the other points are higher, some substantially. The 90 degree phase angle results are generally higher than those for the 180 degree phase angle case. There is a slight dependence of the 180 degree phase angle results upon test rig piston speed, but no dependence is evident for the 90 degree case. Figure 46 presents the same results with only the data points for the 80mm stroke (47.6 stroke-to-test-cylinder-diameter ratio) shown. This illustrates the dependence of the results on test rig piston speed and phase angle better than the previous graph. Figure 47 shows the heat 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 with stroke-to-test-cylinder-diameter ratio in the 180 degree phase angle case. " • 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 Trials, 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 ^ w m ^ m 30 100 1000 Reynolds number (average) Figure 46. Average Nusselt number versus Average Reynolds number, 80mm Stroke Trials, 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 1000 Reynolds number (average) gure 47. Average Nusselt number versus Reynolds number, 80mm Stroke Trials, by DSR 1000 Reynolds number (average) Figure 48. Average Nusselt number versus Average Reynolds number, 90 Degree Phase Angle Trials, by Stroke-to-Test-Cylinder-Diameter Ratio oo 03 1000 Reynolds number (average) Figure 49. Average Nusselt number versus Average Reynolds number, 180 Degree Phase Angle Trials, by Stroke-to-Test-Cylinder-Diameter Ratio 00 V. DISCUSSION OF RESULTS A. GAS VELOCITY The velocity graphs show the calculated bulk gas velocity at the test-cylinder versus test rig crank angle. The sinusoidal piston motion curves illustrate the difference between the velocity that results from the actual piston motions and piston motions assumed in most theoretical work. Interestingly, very few Stirling cycle machines use truly sinusoidal piston motion, but it seems to have become an accepted standard for most of the test rigs used in experimental work as well. (The use of a perfectly-sinusoidal drive for this test rig was not feasible.) However, since the flow field present in this test rig should be considered turbulent rather than laminar, heat transfer results should not be much different than if sinusoidal piston motions were used, and they are therefore applicable to a wide range of Stirling engine configurations. Figure 7 shows gas velocity at the test-cylinder for the 180 degree phase angle case and the 40mm stroke. In all cases, the rotational speed is assumed to be constant, when in fact, especially at higher system pressures, it varies slightly as the motor drive for the test rig responds to the changes in load. Full test rig 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 angle but 80mm stroke. The piston motion curves are still very close to each other, so for both the 40mm and 80mm 88 Discussion of Results / 89 stroke cases, test result variation due to the different rig geometries would not be significant. The nature of the recirculating wake is the main parameter in convective rates and localized velocities about the test-cylinder. The amount of vortex shedding from the test-cylinder depends on a "Strouhal number type" parameter such 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 order as the test-cylinder diameter, and the length of the wake is always of the order of the stroke length, so the exact nature of the velocity should not have too much effect on the heat transfer 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 case is presented in Figure 9. The two curves are close until 270 degrees of crank angle, when the actual piston motion curve flattens and deviates further from the sinusoidal piston motion curve. The 120mm stroke case is close to the limit for this particular test rig geometry if close approximation of sinusoidal piston motion is required. However, since very few Stirling engines employ sinusoidal piston motion, results from "other" velocity conditions are certainly applicable. The next three graphs illustrate the gas velocity at the test-cylinder for the 90 degree phase angle case. An important observation from these graphs is that the velocity curves for the sinusoidal piston motion cases are far from sinusoidal. The compression and expansion of the gas causes the velocity in the positive Discussion of Results / 90 direction (compression stroke) to be of shorter duration than the velocity in the negative direction (expansion stroke). Therefore, results from test rigs for the 180 degree phase angle case (the only published results up to now) should not be applied directly to the 90 degree phase angle case because of the differences in velocity conditions. This is supported by the fact that the heat transfer results from this test rig depend significantly on phase angle. Figure 10 is for the 40mm stroke case. The compression peak velocity for the actual piston motion case is 0.02m/sec higher and comes 10-15 degrees crank angle sooner than for the sinusoidal motion case. However, for points at which there is high acceleration of the gas, the curves are of the same slope. The actual piston motion curve flattens during the expansion stroke, and the peak is about 0.2m/sec less than the sinusoidal piston motion case. For one cycle, the effects on heat transfer of the difference in the velocity peaks should average out, so the differences in the velocity curves should not affect the average heat transfer rates significantly. Peak heat transfer rates, presented on the "Instantaneous Nusselt number versus Reynolds number" graphs should be affected to a greater extent, but this was not obvious. This is discussed later. Similar differences in the velocity peaks are illustrated in Figure 11 for the 80mm stroke case, except they are twice as high. The velocity curve for the actual piston motion case during the expansion stroke is flatter than in the 40mm case. The slopes along the high acceleration periods are still similar. Figure 12 presents the 120mm stroke case. Velocity peaks are about 0.8m/sec Discussion of Results / 91 different, but the slopes during the high acceleration periods are similar. The actual piston motion curve shows a "dip" of 0.2m/sec in speed during the expansion stroke. This did not appear to affect the average heat transfer results, and its effect on the instantaneous results is discussed later. In all cases, deviations in the velocity curves between sinusoidal piston motions and actual piston motions were as great as those between the different stroke cases. Since the average heat transfer results indicated no significant dependences on stroke-to-test-cylinder-diameter ratio, the differences in velocities present in this test rig and a rig with assumed sinusoidal piston motion would not be expected to cause significant differences in average heat transfer rates. These results should be valid for a wide range of Stirling engine configurations. B. FLUID PROPERTY VARIATIONS Figure 15 shows that the temperature varies smoothly, but only a few degrees Kelvin, with crank angle, for test run 54. It is approximately in phase with the pressure variations shown in Figure 13, the difference being due to the thermal response lag of the thermocouple. Pressure varies smoothly as well in this 90 degree phase angle case. The volume curve in Figure 14 follows the first two, but inversely, and appears to be very close to being sinusoidal, despite the fact that the piston motions are not exactly sinusoidal. Figure 16 shows the pressure variation with crank angle for test run 23, a typical 180 degree phase angle case. If the piston motions were truly sinusoidal, Discussion of Results / 92 the curve in this case would be a straight horizontal line. This is also true for the volume variation curve shown in Figure 17. However, the finite, connecting rod lengths of the test rig mean that while the rotational motion of the cranks are 180 degrees out of phase with each other, the pistons are not. Also, the forward and reverse motions are not identical because on the forward pass, the cranks are moving downward, while on the reverse pass, they are going upwards. This accounts for the variations present in both the pressure and temperature curves that occur at twice the rate of the variations in the 90 degree phase angle case, and the different magnitudes of the variations on the forward and 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 and are presented as "Instantaneous Nusselt versus Reynolds number" graphs, "Nusselt number versus Crank Angle" graphs, and "Reynolds number versus Crank Angle" graphs. The instantaneous plots are all scaled identically so direct comparisons between the test runs can be made easily. To obtain good resolution at the higher Reynolds numbers, the data below a Reynolds number of 10 has been eliminated; this means only one or two points from each run are lost, however. The other graphs are scaled linearly to fit the data. The abscissa runs from 0 to 360 degrees, or one crank revolution. Zero degrees corresponds to the beginning of the compression stroke on the 90 degree phase angle case and to near zero piston velocity on one of the half-strokes in the 180 degree phase angle case. Discussion of Results / 93 All of the instantaneous graphs reveal an important fact: at very small Reynolds numbers, the Nusselt number is always substantially higher than the steady flow correlation values shown in Figures 45-49, usually 100% or greater. This means that there must be substantial local fluid motion near the test-cylinder, even though the calculated bulk fluid motion is near zero. The vorticity in the wake does not diffuse totally 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 show the instantaneous Nusselt number plotted against instantaneous Reynolds number for the 180 degree phase angle case. The first graph, Figure 18, shows two loops, one almost superimposed on the other. This is to be expected if each stroke has close to the same motion as the reverse one. The heat transfer is expectedly highest at the highest Reynolds number. There are approximately four crank angle degrees between every two adjacent points. Figures 19 and 20 are for the 80mm stroke case and show more of a difference between the stroke "loops". The geometry of the test rig means that the forward and reverse strokes will not be identical, and there will also be a slight compression and expansion of the gas. This effect is even more pronounced for the 120mm 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 0 to 130 crank angle degrees. Figures 22-38 for the test runs at 120mm and 80mm strokes show that the highest values for Nusselt number and Reynolds number are near 90 degrees. The last two graphs, Figures 29 and 30 have the highest Reynolds number points at 90 degrees as well, but the highest Nusselt number values occur just as the flow reverses, from 130 to 180 degrees. In all cases, the expansion stroke from 180 to 270 degrees has values of Nusselt number and Reynolds number that do not change appreciably, so the points are clustered in a small area. The cycle is completed from 270 to 360 degrees as the Nusselt numbers decrease with Reynolds number to the initial cycle values. The differences between the 40mm stroke curves and the curves for the other strokes come about from a higher intensity of local motion about the test-cylinder in the 40mm case just as the flow reverses. The reverse must occur after the expansion 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 stroke curves, however. Differences in the curves due to different dead space ratios are not apparent. This means that this parameter affects all of the instantaneous values to the same extent, which is significant only for an averaged 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 a graph of Nusselt number versus crank angle and Reynolds number versus crank angle for each selected test run. These graphs show how the Nusselt number follows the rises and falls of the Reynolds number curve. Any rapid change in Reynolds number is often accompanied by fluctuations in the Nusselt number. For the 180 degree phase angle case, this is illustrated by Figures 33 and 34. Figures 31 and 32 show only smooth variations in Nusselt and Reynolds numbers, and they are in phase (within the response lag error). Figures 35-43 deal with the 90 degree phase angle case. The Nusselt number varies almost proportionately with the Reynolds number in all cases during the compression stroke (about 0 to 130 degrees). There, is a rise in Nusselt number after the reversal of the flow, when the bulk gas velocity is low but the local velocities are high. The Nusselt number peaks before or at 180 degrees for all cases, and then tails 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 indicate that there are some regions in the reversed wake that have higher local velocities than the rest of the wake. Figure 44 shows three successive cycles during test run 27. This shows that r there is little cycle-to-cycle variation in heat transfer rates. Table 4 shows the average values for Nusselt number and Reynolds number, and the differences Discussion of Results / 96 here are negligible. This means the flow conditions from cycle-to-cycle are consistent, and the results obtained from these tests are repeatable. D. AVERAGED HEAT TRANSFER RESULTS Figure 45 presents all of the test runs on a graph of average Nusselt number versus average Reynolds number. The data is plotted according to test rig speed and phase angle to demonstrate the dependence of the results on these parameters. A logarithmic scale is used since any power-law dependence of heat transfer rates upon Reynolds number will put similar data near a straight line. The 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 on phase angle. The 90 degree phase angle points generally yield Nusselt numbers 30 to 40% higher than values for the 180 degree case for a given average Reynolds number. A steady flow correlation N u = 0 . 6 8 3 * R e ° - 4 6 6 * P r 0 - 3 3 3 from Hilpert [17] is also plotted on the graph to show the relative heat transfer values between steady and oscillating flow. The values for 180 degree phase angle are from 10 to 30% higher than the steady flow values, with the difference increasing with Reynolds number. The 90 degree phase angle values are 40 to 80% higher than steady flow, with the difference increasing with Reynolds number as well. Discussion of Results / 97 Average values for Nusselt number and Reynolds number are used as a means of comparing the oscillating flow values with the steady flow correlation because these are the most useful values for design of a regenerator. Until now, regenerator design has relied on steady flow correlation or "trial 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 stroke. This eliminates dependence of the results on 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 is obvious. A slight dependence is indicated for the 180 degree phase angle case, with the 100% speed values being 5 to 10% higher for a given Reynolds number. A good part of this difference may be 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 heat transfer rate dependence on phase angle as Figure 45. The reasons for this can be explained by the fact that the flow for the 90 degree case is more unsteady and turbulent than the 180 degree case due to the compression and expansion of the gas. When a vortex is compressed, its rotational speed increases to conserve angular momentum. This causes higher shear rates in the flow and therefore a greater turbulence in the flow results. Some turbulence spreads throughout the flow; therefore the wake spreads out more 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 ratio (DSR) for the 90 degree phase angle case (dead space ratio has no relevance in the 180 degree phase angle case). The average Nusselt number for the 1.0 DSR case is 10% higher than in the 1.5 DSR case for a given average Reynolds number, which in turn is 10% higher than in the 2.0 DSR case. This difference may be partly due to experimental error, but there are enough points for each dead space ratio to support the trends. These trends can be explained by the higher shear rates present in the lower DSR cases. There is more compression and expansion of the gas in these 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 cases. The next two graphs that show average Nusselt number versus Reynolds number present the data for different stroke-to-test-cylinder-diameter ratios. Figure 48 shows the results for the 90 degree phase angle case and Figure 49 shows the results for the 180 degree phase angle case. In both cases no dependence of average Nusselt number on stroke-to-test-cylinder-diameter ratio for a given average Reynolds number is indicated. This shows that the wakes for each case are similar in their influence on the cylinder when they pass over it. In an actual regenerator, the stroke-to-test-cylinder-diameter ratios are an order of Discussion of Results / 99 magnitude higher, so a direct comparison between these results and a regenerator case cannot 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 case would be expected. The regenerator case also consists of a series of adjacent cylinders rather than just one, so the wake would be different as well. The size of the vortices in the wake however, are close to the cylinder diameter, so the effects of dead space ratio and phase angle should have similar effects, though the overall average rates could be expected to be different. VI. CONCLUSIONS Average Nusselt numbers from a circular cylinder in an oscillating crossflow are higher than those for steady flow at the same Reynolds number. Instantaneous Nusselt number values were also higher than steady flow at most points in the cycle, even at the points with almost no calculated bulk flow velocity; therefore the recirculating wake from the cylinder must raise convective heat transfer rates over steady flow values. There is fluid motion even when there is no indicated bulk fluid velocity. Phase angle differences accounted for the greatest variation in the average Nusselt numbers for the oscillating flow. The 180 degree phase angle Nusselt numbers were 10 to 30% higher than the steady flow correlation values for a given average Reynolds number. The 90 degree phase angle Nusselt numbers were 30 to 40% higher than the 180 degree phase angle case, or about 40 to 80% higher than the steady flow values for a given average Reynolds number. Dead space ratio (DSR) had a significant effect on average Nusselt numbers. Heat transfer increased with a decrease in dead space ratio. The 1.5 DSR case had values for average Nusselt numbers about 10% higher than the 2.0 DSR case; this is true for the 1.0 DSR case as compared to the 1.5 DSR case as well. The fluid motion was obviously more intense in the smaller dead space ratio case, and this results from the wake from the cylinder being more significant in terms of total volume than in the case of greater dead volume. There was also more compression and expansion in the lower DSR cases, the 100 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 ratio had no significant effect on average Nusselt numbers for either the 90 degree or the 180 degree phase angle cases. This means that the wake characteristics are about the same for the range of cases examined. However, the results might be extrapolated to higher stroke-to-wire-diameter ratios, such as those in a Stirling engine 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 had a very slight effect on average Nusselt numbers for the 180 degree phase angle case. No effect was noticed for the 90 degree phase angle case. The "Instantaneous Nusselt number versus Reynolds number" graphs showed that heat transfer from the test cylinder during the compression stroke is generally higher than during the expansion stroke. During the middle of the expansion stroke, the Nusselt number variation is small due to the fact that Reynolds number varies little at this point. 'i Graphs for Nusselt number and Reynolds number versus crank angle revealed 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 angle case show that even for sinusoidal piston motion, calculated bulk fluid velocity at the test cylinder cannot be considered sinusoidal. This means that Nusselt numbers for flow conditions similar to the 180 degree phase angle case cannot be compared directly to the Stirling engine case. VII. RECOMMENDATIONS FOR FURTHER WORK Further experimental work should begin with a flow visualization study of the fluid motion within the test rig, both with and without the test-cylinder in place. This will provide information as to the nature of the flow within the test rig and give an idea of the importance of the recirculating wake and the turbulence in the rest of the flow on the heat transfer rates. Accurate measurements of local and bulk velocities with hot-wire or laser-doppler techniques would complement the flow visualization and better estimates of the Reynolds numbers within the flow field could be made. Tests similar to those in this experiment could be carried out with various arrangements of cylinders about the test-cylinder. This would approximate the flow situation in a regenerator more closely than the single cylinder case. The test-cylinder should also be improved so its temperature coefficient of resistance can be determined more accurately and its response to changes in convective rates quickened. 103 BIBLIOGRAPHY 1. Sarpkaya, T. and Isaacson, M. Mechanics of Wave Forces on Offshore  Structures. Van Nostrand Reinhold Publishing Company, New York, New York, 1981. 2. Schlichting, H. Boundary Layer Theory. McGraw-Hill Book Company, New York, New York, 1968. 3. Blasius, H. Grenzshicten in Fliissigkeiten mit Kleiner Reiburg. Z. Math. u. Phys. 56,1. (1908). 4. Schwabe, M. Uber Druckermittlung in der instationaren ebener Stromung. Ing.-Arch. 6, 34-50 (1935); NACA TM 1039 (1943). 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. Stirling Cycle Engine Analysis. Adam Hilger Ltd., Bristol, England. 1984. 7. Walker, G. Stirling Engines. Oxford University Press, Oxford, England, 1980. 8. West, CD. Principles and Applications of Stirling Engines. Van Nostrand Reinhold Company, New York, New York, 1986. 9. Reader, G.T., Hooper, C. Stirling Engines. E.&F.N. Spon, London, England, 1983. 10. Roach, P.D. Measurements With The Reversing Flow Test Facility. Proc. 21st IECEC, Paper 869119. 11. Krazinski, J.C, Holtz, R.E., Vherka, K.L., Lottes, P.A. An Analysis of  Pressure Drops Under Reversing Flow Conditions. Proc. 21st IECEC, Paper 869116. 12. Rice, G., Thonger, J.C.T., and Dadd, M.W. Regenerator Effectiveness  Measurements. Proc. 20th IECEC, Paper 859144. 13. Miyabe, H., Takahashi, S., Hamaguchi, K. An Approach To The Design of  Stirling Engine Regenerator Matrices Using Pack of Wire Gauzes. Proc. 17th IECEC, Paper 829306. 104 / 105 14. Dijkstra, K. Non-Stationary Heat Transfer In Heat Exchangers. Proc. 19th IECEC, paper 849092. 15. Taylor, D.R. and Aghili, H. An Investigation of Oscillating Flow in Tubes. Proc. 19th IECEC, Paper No. 849176. 16. Seume, J.R. and Simon, T.W. Oscillating Flow in Stirling Engine Heat  Exchangers. Proc. 21st IECEC, Paper 869118. 17. Hilpert, R. Warmeabgabe von geheizen Drahten. und Rohren., Forsch. Geg. Ingenieurwes., vol 4, p 220, 1933. APPENDICES A. PROPERTIES OF FREON-114 Table 5. Properties of Freon-114 Temperature Dynamic V i s c o s i t y Thermal C o n d u c t i v i t y ( K e l v i n ) ( N s / m 2 x l 0 - 6 ) (W/mK) 290 11.27 0.0106 300 11.59 0.0110 310 11.92 0.0115 320 12.25 0.0120 330 12.59 0.0125 340 12.92 0.0131 350 13.26 0.0137 360 13.60 0.0144 370 13.94 0.0151 Temperature (K) Constant P r e s s u r e S p e c i f i c Heat (kJ/kg/K) 276.9 0.641 298 0.667 400 0.760 Temp(K) D e n s i t y (kg/m 3) ( P r e s s u r e ( P r e s s u r e ( P r e s s u r e ( P r e s s u r e 137.9 kPa) 275.8 kPa) 413.7 kPa) 551.6 kPa) 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 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 D e n s i t y from ASHRAB Thermodynamic P r o p e r t i e s of  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 York, 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 Research F o u n d a t i o n , I,F.I./Plencum Data C o r p o r a t i o n , New York, 1970. 106 / 107 B. TEST-CYLINDER The design, construction, and calibration of a device to measure heat transfer rates was the most challenging aspect of outfitting the equipment for instrumentation. Because average heat transfer rates around the entire circular cylinder were of interest rather than localized rates about the cylinder, a design similar to that of a hot film probe was chosen. The film chosen had to be highly thermally conductive, easily deposited on the chosen substrate to a thickness that would yield a film of sufficient resistance to be compatible with the anemometer equipment, and have a high temperature coefficient of resistance. Nickel was chosen as the film as it produced the best combination of properties. The substrate material had to be electrically insulative and have a low thermal conductivity so that the response of the test-cylinder would be suitably fast. Borosilicate glass capillary tubes of ideal size (about 2mm diameter) were readily available and were found to be a good choice for the test-cylinder substrate. Rods and tubes of various plastics were considered because of their much lower conductivities than glass, but their lower melting temperatures prevented the deposition of the nickel film in a vacuum chamber. Attempts to deposit a nickel film on these plastics electrolessly were not successful because this technique produces a nickel film with impurities. These impurities alter the electrical properties significantly and a film thick enough to be of suitably low resistance could not be produced. The first test-cylinders were constructed with 20 gauge solid copper wires pushed about 5mm into a 30mm long piece of capillary tube and glued with silicone / 108 sealer. These were rotated about their axis in the vacuum chamber so that the rising nickel vapour would deposit evenly on the glass. A film less than one micron thick provided the necessary electrical conductivity. Nickel paint provided electrical contact from the film 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 as the glue caused the nickel paint to crack and the contact to be broken. High thermal conductivity through the sealer to the copper wire provided a hot-spot in the middle of the test-cylinder that burned out even at very low overheat ratios. The design that was used for the tests employed the same plated 30mm long glass tubes fitted to 1/16 inch diameter 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 25mm length of exposed film to the gas flow. This design was much stronger and more rigid than the original one, and no problems with electrical contact were encountered. The use of acrylic and thin copper wire kept thermal conduction from the test-cylinder to a minimum, though this conduction was still significant. Calibration of the test-cylinder encompassed many different factors. The cold resistance of the test-cylinder was found to change with ambient pressure, so this was taken into account to determine the actual overheat ratio as the gas temperature and pressure changed. This can be attributed to mechanical straining / 109 of the test-cylinder and film. When the temperature coefficient of the film was checked, the method by which the test-cylinder was heated affected the value. When the entire rig was heated (isothermal test-cylinder) the temperature coefficient was much lower than when the test-cylinder was heated electrically (parabolic-type temperature distribution). This can be attributed to different stress and strain fields in the test-cylinder and film in each 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 bead itself had a temperature distribution in it, it was calibrated against a surface of known temperature. Corrected average temperatures were calculated for each hot resistance dialed into the anemometer. The value was found to be 0.005297 ohms/ohm/K, similar to published values for pure nickel (about 0.006 ohms/ohm/K). Table 6 shows the values obtained from the test-cylinder calibration. Figures 50-52 show the results used in the calibration of the test-cylinder. A nodal thermal analysis of the test-cylinder confirmed that the temperature around the circumference of the test-cylinder could be considered to be constant even if the local convective heat transfer coefficients had a distribution expected in steady crossflow. Thermocouple measurements supported this as well. The longitudinal temperature variation was also analyzed and was similar to the measured values. The result for the longitudinal analysis is in Figure 53. Overall / 110 power dissipation between the nodal analysis and actual situation could not be matched because of uncertainties in the thermal parameters. After all the tests were run, the actual test-cylinder film resistance was estimated, since the ohmmeter leads had to be placed directly on the film. The wire-to-film resistance at the end of the test-cylinder was measured to be 0.57 ohms (this is the resistance through the nickel paint and the thin copper wire). This value was assumed to remain constant with varying ambient 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 from the calculated convective (forced) heat transfer power loss. Table 6. Values from Test-Cylinder Calibration C o l d R e s i s t a n c e Rc= measured 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 0) P= p r e s s u r e ( p s i ) P 0= p r e s s u r e ( p s i ) a t i n i t i a l c o n d i t i o n s (when Rc was measured) Temperature Dependency =0.0089912(T-T 0) T= temperature (K) T 0= temperature (K) a t i n i t i a l c o n d i t i o n s 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 w i t h h e a t i n g t a p e . Temperature C o e f f i c i e n t of R e s i s t a n c e : a=0.005297 R/O/K Taken as average v a l u e over l e n g t h of t e s t - c y l i n d e r , measured d i r e c t l y by c a l i b r a t e d thermocouple probe. 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 A n a l y s i s Measured 3 1 8 . 1 328.8 T h i n Wire •A J--25mm H - N i c k e l P a i n t T e s t - C y l i n d e r f o r Nodal A n a l y s i s A c t u a l T e s t - C y l i n d e r 300 Ambient C. ERROR ANALYSIS AND SAMPLE CALCULATIONS / 115 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 by the equipment. The thermocouple has an accuracy of 1.7°C from 0°C to 900°C, but this will affect the fluid property correlations rather than the temperature difference calculations; the 1.7°C tolerance is relevant over wide temperature swings only, since an examination 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 angle measurement was only two degrees, but a third-degree polynomial fitted to the data. This, coupled with the fact that the test rig's crankshaft had high rotational inertia, made the effective resolution much higher. Therefore, the error for the velocity measurements resulted mostly from the differentiation of discrete rotational position variations. The resistance of the test-cylinder film and leads was measured at high accuracy from the bridge in the anemometer. The resistance can be measured to within 0.01 ohms. The actual resistance of the • nickel film was calculated from an estimation of the resistance of the leads (negligible) and the lead-to-film 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 ( D i f f e r e n c e ) ±0.5% Pressure ±0.5% V e l o c i t y ±2.6% Bridge V o l t a g e ±1.9% Bridge R e s i s t a n c e (Rc and Rh) ±0.15% The above are random e r r o r s that a f f e c t the accuracy of the c a l c u l a t i o n s . Below are the estimated u n c e r t a i n t i e s of the f i l m v a l u e s that are not random, meaning that they 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 the same way. Temperature C o e f f i c i e n t of R e s i s t a n c e F i l m R e s i s t a n c e ±2.5% ±3.6% / 117 b. Correlation Errors Correlations for the fluid properties of the Freon-114 were used to simplify the calculations. Uncertainties in the values for dynamic viscosity and thermal conductivity were due to errors in the tabulated values. Density was calculated from a correlation for the gas constant that was temperature and pressure dependent. This correlation produced density values less than 1% from those tabulated. A constant pressure specific heat correlation was used to calculate the Prandtl number. Table 8. Correlation Errors Dynamic V i s c o s i t y T a b u l a t e d V a l u e s (from a c t u a l ) +1.0% C o r r e l a t i o n ( e r r o r w i t h i n ±0.1% t a b u l a t e d v a l u e s ) Thermal C o n d u c t i v i t y T a b u l a t e d V a l u e s ±0.2% C o r r e l a t i o n ±1.0% Constant P r e s s u r e S p e c i f i c Heat C o r r e l a t i o n ±0.6% D e n s i t y C o r r e l a t i o n ±1.0% / 118 c. Temperature Variation with Compression The thermocouple's response lag means that the peak temperatures will not be sensed accurately. However, temperature rise during the 90 degree phase angle tests of the Freon-114 gas is low so the error introduced by the thermal lag on the average heat transfer rates will not be significant. The values of the heat transfer rates at the temperature extremes of the cycle will be affected to a greater extent, but a measurement error of IK will only overestimate the Nusselt number by about 2%. The ratio of specific heats (Cp/Cv) for Freon-114 is 1.09, so if entropic compression is assumed, the temperature rise is about 2OK if Freon-114 can be assumed to behave like a perfect gas. A logarithmic plot of pressure versus volume (Figure 54) reveals that the polytropic coefficient is about 1.05 due to heat transfer between the gas and the test rig. This produces a temperature rise of about 10K, which is only a couple of degrees above the temperature indicated by the thermocouple. The sensitivity of the calculated temperature rise to errors in the polytropic coefficient calculation discourage its use in the data analysis. A smaller uncertainty results if the thermocouple measurements are used. 100 CD ZJ co CO CD 10-0.0001 Volume 0.001 Figure 54. Pressure versus Volume / 120 2. Sample Calculations a. Piston Kinematics A=310 B=313 C=connecting rod l e n g t h D=15 F= 1 4 1 G=305 H=157 I=crank angle J=16 K=127 L=-90-I S=stroke R=L+380 T=(R/180)*TT E=(G2+H2)'5 0=tan- 1 (G/H-(T-TT/2) ) M=(S2+E2-2SE*cos(0))'5 N = ( ( B - F ) 2 + J 2 ) - 5 P=-(cos"1(M2+N2-A2)/2MN) -sin- 1((S*sin(0))/M+tan"'* (H/G)+tan- ' ( ( B - F / J ) - i r ) Q = s i n " 1 ( ( - K - J * s i n ( P ) + F * c o s ( P ) ) / C ) 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 the equations for one piston only. The equations for the other piston are similar, with the appropriate changes in crank angle and geometry. Figure 55. Test Rig Dimensions / 6. Velocity Calculations I 122 XL=Left Piston Position(mm) XR=Right Piston Position(mm) VL=Velocity of Left Piston VR=Velocity of Right Piston VTC=Calculated Bulk Gas Velocity at Test-Cylinder VL=AXL/AT VR=AXR/AT VTC=(VL*[XR])/([XL]+[XR])+(VR*[XL])/([XR]+[XL])/1000(m/sec) VOL=( [XL] + [XR] )*TT*16.092 + 1 .0)cc DSR=[XLmin] + [XRmin]*1 6.092*7r+1 .0)/ (-[XLmin] + [XRmin]*16. 09 2*7r) [ ] = Absolute Value Test-Cylinder f \ I w VL I VR p Origin Figure 56. Piston Position and Velocity Coordinates c. Heat Transfer Calculations T r i a l 27, F i r s t P o i n t I n i t i a l C o n d i t i o n s (V) V e l o c i t y = 0.012988 m/sec (CA) Crank Angle = 2.37592 degrees (BV) B r i d g e V o l t a g e = 3.085 V (P) Pressure/1000 = 0.03705 mV (T) Thermocouple V = 3.05 mV ( B V 0 ) B r i d g e V o l t a g e = 2.165 V Rc = 5.12 ohms Rh = 6.14 ohms P 0 = Pressure/1000 = 0.0246 mV T 0 = Thermocouple V = 2.940 mV P r e s s u r e P=37.05 p s i P o=24.60 p s i ± 0.5% Temperature T=274.693+15.8222319X (x i n mV) T=322.95K T 0=321.21K ±0.6% Power P o w e r = ( ( B V ) 2 - ( B V 0 ) 2 ) / ( 2 + R h ) 2 *Rh*(Rh-1.14/Rh) =0.364475 Watts ±3.0% R c ( a c t u a l ) 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 ±6.8% F i l m Temperature Tf=(Th-T)/2+T=339.44 ±0.5% Dynamic V i s c o s i t y K =(162857+0.03321429Tf)/10 s =12.903 Ns/m 2 ±1.0% / 124 T h e r m a l ^ C o n d u c t i v i t y k=5.1786*10-**Tf-0.0451429 =0.013064 W/mK ±1.0% R (gas c o n s t a n t ) R=7.1019792-0.0442057P... ..,+(-6.679*10- 5 +9.6693*10' 5P)T = 6.65752 ( m 3 A g ) * ( p s i / K ) D e n s i t y (p) p=P/RT=(37.05*1000)/(6.65752*339.44) =16.395 ±1.0% N u s s e l t Number Nu=Power/(K*w*l*(Th*T)) (1=0.025m) =0.364475/(0.013064***0.025*32.975) =10.77 ±7.5% Reynolds Number Re=pVD/ji (D=0.00168m) =(16.395*0.012988*0.00168)/12.903*10- 6 =27.73 ±3.0% P r e s s u r e i s c a l c u l a t e d i n p s i because the t r a n s d u c e r c o u l d 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 those u n i t s . / 125 The sample calculation was for one point only; the uncertainty in the result is also for this single point. The uncertainty was estimated by the use of a method that weighs the relative uncertainties of each variable in an equation. This method involves differentiating the equation with respect to each variable, multiplying each "differentiation" by the uncertainty in the variable to which it was differentiated, and taking the root mean square of the "differentiations". The averaged results have a smaller uncertainty because the random uncertainties for each point will be averaged out. This is why the conclusion that the average Nusselt number is affected by dead space ratio should be valid despite the fact that the variations between the various dead space ratios were not much higher than the uncertainty for the calculation of Nusselt number at a single point. 

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