Ph.D Sindhu Preetham Burugupally

Cecilia D Richards

DEVELOPMENT OF A SMALL SCALE RESONANT ENGINE FOR MICRO AND MESOSCALE APPLICATIONS Abstract

the opportunity to work on the MEMS/mesoscale engine

v cycles were 38.4% and 47.6%, respectively. It has been realized that when modeling the compliant engine in a more realistic way, an open cycle would be more appropriate. The open cycle results highlight some unique features of the complaint engine. For instance, the engine volume excursions are found to be dependent on the heat input or equivalence ratio and load condition.

To ascertain the feasibility of a practical engine, super compliant structures fabricated from metals and, or composites that can sustain extreme pressures and temperatures are evaluated. In addition, a prototype engine is realized to practically demonstrate the compliant engine concept.

The engine is motored to estimate the parasitic losses at resonant operation, and the results are presented in the form of an energy flow diagram. The results showed that friction losses associated with the flexible cavity are less than that in a piston-cylinder assembly.

An example simulation based on open cycle with practical engine parameters showed that the ratio of friction work to indicated work is about 8%; less than targeted 10%. This result reinforces that a compliant resonant engine has the potential to outperform the contemporary engines at small scale. vi

Motivation

There has been an increasing demand in the past decade for delivering power from milliwatts to kilowatts in small sized packages for consumer and military applications [START_REF] Dunn-Rankin | Personal power systems[END_REF][START_REF] Annen | Linearly oscillating miniature internal combustion engine (MICE) for portable electric power[END_REF][START_REF] Jacobson | An informal survey of power MEMS[END_REF][START_REF] Walther | Advances and challenges in the development of powergeneration systems at small scales[END_REF]. Some of the day-to-day applications include power tools and portable electronics such as laptop computers, GPS receivers, and display devices. Currently, rechargeable lithium batteries are used to power these devices. With recent technical advancements in portable electronics, the low power density and power to weight ratio features of batteries have made them less suitable. Hence a need for an alternative power source arises. Some of the alternative power sources to batteries are fuel cells, thermoelectric devices, and heat engines [START_REF] Dunn-Rankin | Personal power systems[END_REF][START_REF] Walther | Advances and challenges in the development of powergeneration systems at small scales[END_REF]. The high cost and limited power density of a fuel cell, and poor thermal efficiency of a thermoelectric device make these technologies less suitable to power portable devices [START_REF] Schaevitz | A combustion-based MEMS thermoelectric power generator[END_REF][START_REF] Nielsen | A thermophotovoltaic micro-generator for portable power applications[END_REF]. On the other hand, a heat engine operated on a hydrocarbon fuel has high power density that can potentially surpass batteries. For instance, gasoline has a specific energy of about 45 MJ/kg while a lithium/thionyl chloride battery has about 2.6 MJ/kg [START_REF] Walther | Advances and challenges in the development of powergeneration systems at small scales[END_REF]. Therefore, a 3-4% efficient small scale engine run on gasoline is considered as a potential candidate to replace the existing battery technology.

A Ragone chart comparing various power sources is shown in Figure 1.1 [START_REF] Papac | Performance assessment of a centimeter scale four-stroke engine[END_REF]. In the figure, a boundary or curve of an energy source is determined by losses or leakages present [START_REF] Christen | Theory of Ragone plots[END_REF]. The parts of the curve where high energy/power densities prevail are referred as optimum working regions.

As can be seen from the figure, fuel cells have less power density compared to heat engines (model airplanes and automobiles). In terms of energy density, rechargeable lithium batteries are comparable to fuel cells, but both perform inferior to heat engines. This plot clearly manifests the potential of heat engines as a portable power source. Figure 1.1: Ragone chart of various power delivering sources [START_REF] Papac | Performance assessment of a centimeter scale four-stroke engine[END_REF].

To meet portable power requirements, small scale engines operating on different thermodynamic cycles have been proposed [START_REF] Annen | Linearly oscillating miniature internal combustion engine (MICE) for portable electric power[END_REF][START_REF] Epstein | Macro Power From Micro Machinery[END_REF][START_REF] Suzuki | Experimental study on mechanical power generation from MEMS internal combustion engine[END_REF][START_REF] Fu | Design and fabrication of a silicon-based MEMS rotary engine[END_REF][START_REF] Wan | Computational analysis of the feasibility of a micro-pulsejet[END_REF]. Commonly hydrogen or hydrocarbons are combusted in these engines to obtain mechanical work or electrical energy. Other applications of these engines include propulsion and stationary power generation in residential cogeneration systems [START_REF] Wan | Computational analysis of the feasibility of a micro-pulsejet[END_REF][START_REF] Aliabadi | Efficiency Analysis of Natural Gas Residential Micro-cogeneration Systems[END_REF].

The well-established reciprocating engine technology has motivated researchers to develop them at small scale. Most of these engines are spark ignited and are modeled as an Otto cycle whose theoretical efficiency depends on the compression ratio of the engine and the specific heat ratio of the working gas [START_REF] Annen | Linearly oscillating miniature internal combustion engine (MICE) for portable electric power[END_REF][START_REF] Suzuki | Experimental study on mechanical power generation from MEMS internal combustion engine[END_REF]. However, in practice, performance is limited by heat, friction, throttling, and combustion losses. As the scale of the engine is reduced, these losses can become significant; affecting the engine performance. A glimpse of thermal efficiencies and power densities for various internal combustion (IC) engines of different sizes are presented in Figure 1.2 [START_REF] Papac | Performance assessment of a centimeter scale four-stroke engine[END_REF]. From the figure, the power densities of small scale engines (RC model engines) are higher compared to their large scale counterparts (automobile engines). However, the thermal efficiencies of small scale engines are lower due to higher losses in regards to heat transfer, friction, and geometric tolerances. Lately, there has been a renewed interest in free-piston engines coupled with linear alternators as candidates for various small scale applications [START_REF] Clark | Fundamental Analysis of a Linear Two-Cylinder Internal Combustion Engine[END_REF][START_REF] Mikalsen | Performance simulation of a spark ignited free-piston engine generator[END_REF][START_REF] Zheng | Investigation of a 7pole/6-slot Halbach-magnetized permanent-magnet linear alternator used for free-piston stirling engines[END_REF][START_REF] Atkinson | Numerical Simulation of a Two-Stroke Linear Engine-Alternator Combination[END_REF]. The unconstrained piston motion in free-piston engines has numerous advantages over traditional reciprocating engines, such as fuel flexibility, simpler design, and reduced frictional losses [START_REF] Annen | Linearly oscillating miniature internal combustion engine (MICE) for portable electric power[END_REF][START_REF] Mikalsen | A review of free-piston engine history and applications[END_REF]. In addition, the low frictional losses and pure linear motion of the piston may allow the engine to operate with no or little lubrication thus reducing exhaust emissions. However, the issues of poor sealing and leakage losses are significant at small scale resulting in smaller engine efficiencies.

Recent advances and challenges faced in developing small scale engines are presented by Ahn et al [START_REF] Walther | Advances and challenges in the development of powergeneration systems at small scales[END_REF]. Ahn et al. reviewed various engine programs at small scale and emphasized the need for mitigating heat loss, friction loss, leakage loss, and the development of combustion systems that can address fuel-air mixing and pumping for low Reynolds number flows, and combustion in small volumes.

For the above mentioned reasons, to date, a successful miniaturized engine based on a traditional reciprocating engine or a free-piston engine has not been realized. Thus there is need for development of an efficient small scale engine which can meet the demands of portable power for both consumer and military applications.

Organization of dissertation

This dissertation is laid out in chapter wise format. The motivation for developing small scale engines, snapshots of past and ongoing engine development programs, and objectives of this study are briefed in chapter one.

Chapter two introduces the concept of a compliant engine by describing its four-stroke operating cycle. Then, design challenges, materials for fabrication, engine assembly, and design calculations are presented at fundamental level to demonstrate the compliant engine feasibility and reliability.

To determine the engine dynamics and performance, a physics-based mathematical model is developed by taking an air-standard cycle approach. The assumptions, mathematical statements, and methodology to obtain a steady state solution for open and closed cycle operations are discussed in chapter three.

A prototype engine realized at small length scale, fabricated from flexible metal bellows is presented in chapter four. The prototype engine is motored in a test stand to determine the parasitic losses at resonant operation. The engine components, apparatus used in the motoring test and the test procedure are described.

The results from the motoring test and mathematical model are presented in chapter five. The computed open and closed cycle model results are compared and their differences are highlighted. Then, a comparison of performance is made between the compliant engine and a traditional IC engine to exhibit the inherent design merits. Finally, all the positive outcomes of this study are concluded in chapter six.

The supporting material and relevant MATLAB code are available in the appendices.

Literature review

The high energy and power density characteristics of heat engines make them a preferred choice for portable power applications. To meet the portable power requirements, as a first approach, researchers have considered miniaturizing the existing large scale engines to small scales [START_REF] Fu | Design and Experimental Results of Small-scale Rotary Engines[END_REF][START_REF] Epstein | Millimeter-Scale, Micro-Electro-Mechanical Systems Gas Turbine Engines[END_REF]. Over the past two decades various studies have been pursued in developing the small scale or miniature engines [START_REF] Schaevitz | A combustion-based MEMS thermoelectric power generator[END_REF][START_REF] Fu | Design and fabrication of a silicon-based MEMS rotary engine[END_REF][START_REF] Epstein | Millimeter-Scale, Micro-Electro-Mechanical Systems Gas Turbine Engines[END_REF][START_REF] Whalen | Design, fabrication and testing of the P3 micro heat engine[END_REF][START_REF] Herrault | Selfresonant, MEMS-fabricated, air-breathing engine[END_REF].

In developing the small scale engines, it is important to know the effect of engine size on performance, as beyond a critical dimension the parasitic losses exceed the power produced. In order to identify the critical dimension, a two-stroke piston engine is chosen for scaling analysis [START_REF] Menon | Miniaturization Limits of Small IC Engines[END_REF]. In the analysis, performance tests are conducted on seven two-stroke engines weighing between 15 grams to 500 grams using a dynamometer. Upon extrapolating the experimental data, a break-even in power is observed in a 27 mm 3 sized engine. At the break-even, the engine efficiency dropped to zero. This implies that an engine must have dimensions more than 3 mm × 3 mm × 3 mm (length × breadth × height) to generate useful work. The study also shows that the engine power output y versus displacement volume x obeys a power-law of the form y=Ax b for both large and small scale engines [START_REF] Menon | Miniaturization Limits of Small IC Engines[END_REF][START_REF] Mcmahon | On size and life[END_REF]. However, the power-law for small scale engines are found to be more sensitive to size in comparison with large scale engines [START_REF] Menon | Miniaturization Limits of Small IC Engines[END_REF].

One of the first studies on micro engines was conducted by Nakajima et al. [START_REF] Nakajima | Study on micro engines-miniaturizing Stirling engines for actuators and heatpumps[END_REF]. The study comprised of performing dimensional analysis and computer simulations on a Stirling engine.

The dimensional analysis was used to investigate how design parameters changed when an engine size was shrunk. The computer simulations were performed to analyze the design and to decide upon the design parameters. The results from the dimensional analysis suggested following considerations while designing a micro-Stirling engine:

1. Heat transfer through the engine walls become more effective as the surface area to volume ratio increases.

2.

Frictional work becomes dominant compared to output work. Hence a sliding mechanism is not suitable.

3. Traditional crank-flywheel or free-piston mechanism is considered ineffective for a small scale engine. These statements hold true for any engine in general which converts heat energy into mechanical energy by motion of a piston or displacer. These considerations are extremely important while designing small scale or miniature engines.

Design aspects at small scale

Development of a combustion based small scale engine requires consideration of various design aspects such as friction, heat transfer, sealing, combustion, and gas exchange [START_REF] Sher | Theoretical limits of scaling-down internal combustion engines[END_REF][START_REF] Aichlmayr | Micro-Homogeneous Charge Compression Ignition (HCCI) Combustion: Investigations Employing Detailed Chemical Kinetic Modeling and Experiments[END_REF].

The four main aspects, namely: mode of combustion, leakage, friction, and heat transfer losses are discussed in the following subsections.

Mode of combustion

Engine efficiency depends on the quality of combustion. The quality of combustion is quantified by combustion efficiency. Typically, combustion efficiencies are between 95% -98%.

In order to achieve stable combustion, residence time must be larger than chemical reaction time; a typical condition in macro-scale engines. However, as the size of the combustion chamber reduces, the residence time decreases resulting in incomplete combustion [START_REF] Sher | Theoretical limits of scaling-down internal combustion engines[END_REF]. Therefore, the size of the combustion chamber limits combustion and engine efficiencies.

The spark ignition engines whose combustion is driven by flame front propagation are likely to suffer from flame quenching. The issue of flame quenching gets amplified at small length scales. Therefore, it is necessary to implement a mode of combustion where length scale factor becomes irrelevant. One such mode is Homogeneous Charge Compression Ignition (HCCI) where length scale effects are less relevant, and the effects of flame quenching and improper combustion are minimized. In addition, the HCCI also offers advantages such as: fuel flexibility, lean combustion, with no compression ratio limits. Thus, HCCI seems to be one possible mode of combustion at small scales.

Leakage loss

In a conventional reciprocating engine, a gap is formed between the piston and cylinder wall which allows leakage of cylinder gases. To reduce such leakage losses, pistons are fitted with rings called piston rings. A typical tolerance value of piston-cylinder gap is about 10-20 µm, which is a very small fraction of cylinder bore [START_REF] Sher | Miniaturization limitations of HCCI internal combustion engines[END_REF]. As length scales diminish, the ratio of piston-cylinder gap to cylinder bore becomes large, aggravating leakage losses. The gravity of leakage loss is evident in MEMS IC engine [START_REF] Suzuki | Experimental study on mechanical power generation from MEMS internal combustion engine[END_REF]. Suzuki et al. showed that about 18% of the total chamber mass is discharged from the engine as blow-by, resulting in significant power loss [START_REF] Suzuki | Experimental study on mechanical power generation from MEMS internal combustion engine[END_REF].

A numerical study by Sher et al. showed that an engine of 1 cc with a leakage gap of 20 µm operating at 24000 rpm can have 58% loss of chamber mass by leakage [START_REF] Sher | Scaling-Down of Miniature Internal Combustion Engines: Limitations and Challenges[END_REF]. They also showed that, a scaled up version of the engine to 10 cc volume with leakage gap of 20 µm operating at 24000 rpm can have 17% loss of chamber mass by leakage. Thus scaling down the engines aggravates leakage losses.

To circumvent such leakage losses, Sher et al. proposed to operate engines at higher speeds [START_REF] Sher | Miniaturization limitations of HCCI internal combustion engines[END_REF]. However, operating engines at higher speeds reduces the combustion gas residence time causing incomplete combustion and poor engine efficiency [START_REF] Sher | Miniaturization limitations of HCCI internal combustion engines[END_REF]. In addition, operating engines at higher speeds results in increased frictional losses [START_REF] Pulkrabek | Engineering Fundamentals of the Internal Combustion Engine[END_REF]. Therefore, the attempt to mitigate leakage losses by operating the engines at higher speeds might be counter-productive and detrimental.

Friction loss

A study on effects of friction on general class of heat engines was conducted by Gordon et al [START_REF] Gordon | On optimizing maximum-power heat engines[END_REF][START_REF] Gordon | General performance characteristics of real heat engines[END_REF]. Gordon et al. focused on understanding the effects of internal and external dissipative (or frictional) elements on engine performance. For the analysis, they assumed that the internal and external frictional losses are proportional to square of engine speed with same frictional coefficients [START_REF] Gordon | On optimizing maximum-power heat engines[END_REF]. From the study, they concluded that engines with externally dissipative friction perform better than engines with internally dissipative friction. This result was expected as the internally dissipative engines dissipate energy by heating the engine working fluid resulting in lesser output power than compared to external dissipative engines that dissipate energy directly to the surroundings [START_REF] Gordon | On optimizing maximum-power heat engines[END_REF].

Studies focused on friction in reciprocating engines have been widely pursued [START_REF] Yagi | Experimental Analysis of Total Engine Friction in Four Stroke S. I. Engines[END_REF][START_REF] Fujii | Total Engine Friction in Four Stroke S.I. Motorcycle Engine[END_REF][START_REF] Wakuri | Piston ring friction in internal combustion engines[END_REF]. In macro-scale reciprocating engines friction losses, although not insignificant, are on the order of 10% of the input fuel energy. The frictional losses arise from internal and external dissipative sources such as pumping and mechanical linkage losses. Figure 1.3 shows various internal and external dissipative elements in a reciprocating engine with their individual contribution to total friction [START_REF] Heywood | Internal combustion engine fundamentals[END_REF].

A successful attempt to reduce frictional losses by 10% of total friction was achieved at macro-scale by Hoshi et al. [START_REF] Hoshi | Reducing friction losses in automobile engines[END_REF]. This was accomplished by refining the shape of the pistons, reducing the tension and the cross-sectional area of the piston rings. The friction losses depend on contact area, engine speed, and lubricant viscosity [START_REF] Yagi | Experimental Analysis of Total Engine Friction in Four Stroke S. I. Engines[END_REF]. As the scale of the engine reduces, friction losses can become very dominant due to increased surface area to volume ratio and higher speeds [START_REF] Sher | Theoretical limits of scaling-down internal combustion engines[END_REF]. To reduce frictional losses at small scale, engines based on free-piston concept are being developed [START_REF] Annen | Linearly oscillating miniature internal combustion engine (MICE) for portable electric power[END_REF][START_REF] Suzuki | Experimental study on mechanical power generation from MEMS internal combustion engine[END_REF]. The free-piston design eliminates crankshaft friction, bearing friction, thereby hypothetically improving the overall engine efficiency. Even though the friction caused by sliding piston is significant, it received less attention. Recently, research to address sliding friction has been taken up at concept level by proposing a compliant engine design [START_REF] Richards | A Resonant Air-Standard Heat Engine[END_REF].

Heat loss

Heat transfer in an internal combustion engine is important and necessary for proper operation as the engineering materials cannot withstand temperatures of about 2700 K. In a macro-scale reciprocating engine about 30% of input fuel chemical energy is dissipated to the surroundings by conduction, convection, and radiation heat transfer [START_REF] Pulkrabek | Engineering Fundamentals of the Internal Combustion Engine[END_REF].

Heat transfer models for macro-scale engines have been studied [START_REF] Oguri | On the Coefficient of Heat Transfer between Gases and Cylinder Walls of the Spark-Ignition Engine[END_REF][START_REF] Alkidas | Heat Transfer Characteristics of a Spark-Ignition Engine[END_REF]. Figure 1.4 shows that the heat loss per cycle depends on engine speed and load conditions [START_REF] Heywood | Internal combustion engine fundamentals[END_REF]. In the figure, the heat loss is presented as a percentage of input fuel chemical energy. From the figure, for a fixed load condition (or bmep), the heat loss decreases with increase in engine speed. When the scale of the engine is reduced, heat losses can become very dominant due to increased surface area to volume ratio, and temperature gradients [START_REF] Walther | Advances and challenges in the development of powergeneration systems at small scales[END_REF]. For instance, a heat transfer study in a small scale 125 cc two-stroke spark ignition (SI) engine showed that about 50% of the chemical energy of the fuel is lost as heat transfer [START_REF] Franco | Evaluations on the Heat Transfer in the Small Two-stroke Engines[END_REF].

In compression ignition engines, the increased heat transfer at small scales can decrease the compressive heating effect thereby unfavorably affecting the engine performance [START_REF] Aichlmayr | Miniature free-piston homogeneous charge compression ignition engine-compressor concept-Part II: modeling HCCI combustion in small scales with detailed homogeneous gas phase chemical kinetics[END_REF].

Aichlmayr et al. studied the effect of stroke to bore ratio (or aspect ratio) on engines in regards to heat transfer [START_REF] Aichlmayr | Miniature free-piston homogeneous charge compression ignition engine-compressor concept-Part II: modeling HCCI combustion in small scales with detailed homogeneous gas phase chemical kinetics[END_REF]. They found out that a large aspect ratio engine has lesser heat transfer compared to a small aspect ratio engine. Unfortunately, there is a point beyond which the large aspect ratio effect becomes less effective. Aichlmayr et al. found that to be 10. Figure 1.4: Heat transfer rate in an SI engine as a function of speed and load [START_REF] Heywood | Internal combustion engine fundamentals[END_REF].

Apart from the aforementioned design aspects, mechanical stresses and simpler geometries are two other important design features for a successful small scale engine.

Classification of small scale engines

Small scale engines can be classified based on engine size and type of operation. a. Engine size: Microscale and Mesoscale b. Type of operation: Resonant and Off-resonant Engines whose dimensions are in the range of 1 μm to 1 mm are referred as microscale engines, and are commonly fabricated using semiconductor fabrication technology [START_REF] Suzuki | Experimental study on mechanical power generation from MEMS internal combustion engine[END_REF][START_REF] Whalen | Design, fabrication and testing of the P3 micro heat engine[END_REF]. On the other hand, mesoscale engines are those whose characteristic dimensions are in the range of 1 mm to 10 cm, and are fabricated by traditional and non-traditional machining techniques [START_REF] Annen | Linearly oscillating miniature internal combustion engine (MICE) for portable electric power[END_REF][START_REF] Fu | Design and Experimental Results of Small-scale Rotary Engines[END_REF][START_REF] Wang | Micro Newcomen steam engine using two-phase working fluid[END_REF][START_REF] Peirs | A microturbine for electric power generation[END_REF].

Engines which operate at their resonant frequencies are resonant engines [START_REF] Richards | A Resonant Air-Standard Heat Engine[END_REF][START_REF] Bardaweel | Cyclic operation of a MEMS-based resonant micro heat engine: Experiment and model[END_REF], while those that do not operate at resonant frequencies are non-resonant engines [START_REF] Schaevitz | A combustion-based MEMS thermoelectric power generator[END_REF]. Researchers have demonstrated that operating engines at resonant frequencies as desirable as it yields higher output work [START_REF] Richards | A Resonant Air-Standard Heat Engine[END_REF][START_REF] Bardaweel | Cyclic operation of a MEMS-based resonant micro heat engine: Experiment and model[END_REF].

A review of previous and current work on small scale engines are discussed in the following sub-sections.

Microscale heat engines

Some of the sincere efforts to develop MEMS-based engines began in mid-1990s with the development of Micro-Gas Turbine Engine [START_REF] Epstein | Macro Power From Micro Machinery[END_REF][START_REF] Epstein | Millimeter-Scale, Micro-Electro-Mechanical Systems Gas Turbine Engines[END_REF]. Soon after, the PowerMEMS research gained recognition and other researchers began developing such small scale engines operating on various thermodynamic principles to provide either an alternative portable power, or a propulsion device. MEMS Rotary Engine [START_REF] Fu | Design and fabrication of a silicon-based MEMS rotary engine[END_REF], MEMS Air-breathing Engine [START_REF] Herrault | Selfresonant, MEMS-fabricated, air-breathing engine[END_REF], and MEMS IC Engine [START_REF] Suzuki | Experimental study on mechanical power generation from MEMS internal combustion engine[END_REF] are some of the examples. These engines are combustion driven and are mere scaled down versions of existing macro-scale engines. The approach of mere scaling down of engines proved to be challenging and is evident in Micro-Gas Turbine Engine [START_REF] Epstein | Millimeter-Scale, Micro-Electro-Mechanical Systems Gas Turbine Engines[END_REF], MEMS Rotary Engine [START_REF] Fu | Design and fabrication of a silicon-based MEMS rotary engine[END_REF],

and MEMS IC Engine [START_REF] Suzuki | Experimental study on mechanical power generation from MEMS internal combustion engine[END_REF].

Micro-Gas Turbine Engine

The development of Micro-Gas Turbine Engine initiated at Massachusetts Institute of Technology was based on Brayton cycle due to the superior features of Brayton cycle such as: power density, fabrication simplicity, and ultimate efficiency. The project focused on developing a propulsion and power production device at 10-100 W scale [START_REF] London | Microfabrication of a high pressure bipropellant rocket engine[END_REF]. The engine comprised of a centrifugal compressor and a radial turbine rotor with diameters measuring 8 mm and 6 mm respectively. The engine had an overall dimension of 21 mm with rotor designed to tolerate high stresses when operated at 1.2 Mrpm. The engine was designed to produce a shaft power of 17 W with 400 micron tall airfoils by pumping 0.36 g/s of air. The feasibility of the engine was demonstrated at component level (turbine, bearings, and combustor), but a complete device has yet to be tested. Epstein et al. identified that stringent process control involving complex fabrication sequence, shorter residence time, geometric tolerances, thermal isolation, and increased surface heat losses as some of the challenges in developing the engine [START_REF] Epstein | Millimeter-Scale, Micro-Electro-Mechanical Systems Gas Turbine Engines[END_REF].

MEMS Rotary Engine

A MEMS Rotary Internal Combustion Engine capable of delivering power on the order of few milliwatts was systematically investigated at University of California at Berkeley. To understand the engine behavior and design issues, a larger version was built from steel using wireEDM. The engine had a displacement volume of 348 mm 3 and when ran on hydrogen fuel at 9300 rpm produced a positive power of 2.7 W [START_REF] Fu | Design and Experimental Results of Small-scale Rotary Engines[END_REF]. Later the group designed and fabricated a micro-Wankel engine [START_REF] Fu | Design and fabrication of a silicon-based MEMS rotary engine[END_REF]. Though flame quenching has been addressed by means of thermal management; heat losses and sealing are the key issues that are to be addressed for successful running of the engine.

MEMS Internal Combustion Engine

One of the first investigations pertaining to the development of a micro-scale reciprocating IC engine was undertaken by Suzuki et al [START_REF] Suzuki | Experimental study on mechanical power generation from MEMS internal combustion engine[END_REF]. This MEMS-based engine was built from silicon wafer stacks and Pyrex glass by adopting microfabrication technology as shown in Figure 1.5.

The engine mainly consisted of a silicon cylinder case with combustion chamber, silicon piston connected to an elastic spring, and platinum microprobes for spark plug. The dimensions and surface-to-volume ratio of combustion chamber were 5 mm × 3 mm × 1 mm and 3.07 mm -1 respectively. The engine operated on a two-stroke principle with premixed hydrogen and oxygen as its fuel. Hydrogen was chosen instead of hydrocarbons due to its wide flammability-limit range, high specific heating value, low minimum ignition energy, and smaller quenching distance.

Based on the two-stroke Otto cycle analysis, the maximum cycle pressure was estimated to be 349 kPa, while the experimentally recorded peak combustion pressure was found to be 142.6 kPa. It was assumed that the disparity in pressure values was a result of blow-by and heat transfer to the cylinder walls. To justify the assumption, a combustion model based on first law of thermodynamics for an open system that includes losses due to blow-by and heat transfer was developed. A back of the envelope calculation for the MEMS IC engine showed that about 18% of the total chamber mass was discharged from the engine as blow-by, resulting in significant power loss. The engine when operated at 3 Hz with a displacement volume of 0.9 mm 3 produced a mechanical power output of 29.1 mW. Suzuki et al. remarked that with the increase in engine speed, the effect of heat transfer, fluid leakage, and friction on the engine power can become very dominant. Significant fluid leakage loss seemed to be one of the hurdles in materializing the engine. Thus to develop a successful MEMS-based engine the following challenges such as flame quenching, poor thermal isolation, significant heat loss, larger viscous and friction losses, large mechanical stresses, poor geometric tolerances, and intricate designs have to be mitigated [START_REF] Sher | Theoretical limits of scaling-down internal combustion engines[END_REF][START_REF] Aichlmayr | Micro-Homogeneous Charge Compression Ignition (HCCI) Combustion: Investigations Employing Detailed Chemical Kinetic Modeling and Experiments[END_REF].

MEMS P 3 Heat Engine

Contrary to the concept of scaling of existing engines to small scale, a unique design was conceived in developing a MEMS-based external heat engine to harvest waste heat (P 3 heat engine) [START_REF] Whalen | Design, fabrication and testing of the P3 micro heat engine[END_REF]. Its unique design addressed most of the challenges predominant at microscale. For instance, fabrication was made easy by adopting a simple two dimensional structure. The simple structure facilitated stacking such engines in a cascade form (arranged in series fashion) to obtain higher output power. The design incorporated flexing components thereby reducing the high stresses. The engine was an external heat engine, thus the question of flame quenching doesn't arise. The engine consisted of a cavity filled with a saturated two-phase working liquid (3M™ PF-5060DL), bounded by two membranes: top and bottom membranes. The engine was built from silicon by adopting semiconductor fabrication technology. Figure 1.6 shows a crosssectional view of a prototype engine. The top membrane of the engine was a thin composite layer of silicon and PZT, while the bottom layer was solely a silicon membrane. The PZT was a part of piezoelectric membrane generator whose function was to convert mechanical work into electrical energy. The engine operation could be idealized as: compression, isothermal and isobaric heat addition, expansion, and isothermal and isobaric heat rejection processes. A 2 mm × 2 mm bottom membrane and a 3 mm × 3 mm top membrane were separated by a 60 µm thick semiconductor tape that formed a 0.6 mm 3 total chamber volume. The engine when operated at its resonant frequency of 240 Hz, produced a peak output power of 0.8 µW. Whalen et al. claimed that the novel engine operating principle could possibly approach ideal Carnot vapor cycle [START_REF] Whalen | Design, fabrication and testing of the P3 micro heat engine[END_REF]. A scaled up version of the engine was operated at resonance and off-resonance; and the results demonstrated resonant operation as beneficial [START_REF] Bardaweel | Cyclic operation of a MEMS-based resonant micro heat engine: Experiment and model[END_REF].

With its simple design, it had addressed flame quenching, intricate designs, large mechanical stresses, and poor geometric tolerances. The engine demonstrated successful conversion of heat energy to electric energy. However, the engine is best suited for energy harvesting applications from external heat sources.

A scaling analysis of the MEMS P 3 heat engine was carried out to further improve its existing design. As a part of this study, P 3 heat engines of various sizes fabricated from different materials were tested. The results indicated that scaling up of engine volume, more compliant structure, and slower engine speeds as desirable [START_REF] Bardaweel | MEMSbased resonant heat engine: Scaling analysis[END_REF]. The outcome of this study highlighted the importance of scaling up of such micro resonant engines for improved performance. Details of the scaling analysis are presented in Appendix A1: SCALING ANALYSIS OF MEMS P3 HEAT ENGINE.

Mesoscale heat engines

Recently attempts have been made to develop pulsejet and free-piston engines at mesoscale [START_REF] Annen | Linearly oscillating miniature internal combustion engine (MICE) for portable electric power[END_REF][START_REF] Tian | Miniature HCCI free-piston engine compressor for orthosis application[END_REF][START_REF] Geng | Experimental and numerical investigation of an 8-cm valveless pulsejet[END_REF]. For obtaining increased output work, these engines are operated at resonant frequencies.

A pulsejet engine is considered to be one the simplest propulsion device as it does not require turbomachinery with no or fewer moving parts [START_REF] Geng | Experimental and numerical investigation of an 8-cm valveless pulsejet[END_REF]. This makes it light in weight, and is regarded as an ideal device for thrust generation. Pulsejet engines are based on the Humphrey thermodynamic cycle with no mechanical compression. Hence, their thermodynamic efficiencies are lower compared to gas turbine engines that operate on Brayton cycle [START_REF] Geng | Combined Numerical and Experimental Investigation of a 15-cm Valveless Pulsejet[END_REF]. The low thermodynamic efficiency is somewhat offset by the fundamental simplicity of the engine. Being less efficient, these engines are not competitive for large scale applications; however at small scale there exists a tradeoff between simple engine operation and ultimate efficiency. This propelled research into various feasibility studies on valveless pulsejet engines at the small scale domain for propulsion applications [START_REF] Geng | Experimental and numerical investigation of an 8-cm valveless pulsejet[END_REF][START_REF] Geng | Combined Numerical and Experimental Investigation of a 15-cm Valveless Pulsejet[END_REF].

The concept of free-piston dates back to 1928 [START_REF] Pescara | Motor compressor apparatus[END_REF]. In a free-piston engine, the piston is "free" to move in the cylinder and its motion is not constrained by crankshaft assembly. This unconstrained motion has numerous advantages like fuel flexibility, simpler design, and reduced frictional losses [START_REF] Annen | Linearly oscillating miniature internal combustion engine (MICE) for portable electric power[END_REF][START_REF] Mikalsen | A review of free-piston engine history and applications[END_REF]. The low frictional losses and pure linear motion of the piston may allow the engine to operate with no or little lubrication thus reducing the emissions [START_REF] Annen | Linearly oscillating miniature internal combustion engine (MICE) for portable electric power[END_REF]. It is possible to optimize the combustion process and facilitate fuel-flexibility by adjusting the compression ratio of the engine [START_REF] Xiao | Motion characteristic of a free piston linear engine[END_REF]. These features encouraged researchers to develop them at small scale [START_REF] Suzuki | Experimental study on mechanical power generation from MEMS internal combustion engine[END_REF][START_REF] Tian | Miniature HCCI free-piston engine compressor for orthosis application[END_REF]. A group at West Virginia University built a prototype of a spark ignited free-piston linear alternator (FPLA) that generated a peak electrical power of 316 W at an operating frequency of 23.1 Hz [START_REF] Atkinson | Numerical Simulation of a Two-Stroke Linear Engine-Alternator Combination[END_REF]. A hydrogen based free-piston linear alternator is being studied to provide a high efficiency and low emission electrical generator [START_REF] Van Blarigan | Advanced Internal Combustion Electrical Generator[END_REF]. A variant in a free-piston engine is a rotationally oscillating Micro Internal Combustion Swing Engine (MICSE) is being investigated at University of Michigan [START_REF] Dahm | Micro internal combustion swing engine (MICSE) for portable power generation systems[END_REF].

MEMS-Fabricated Valveless Pulsejet Engine

An air-breathing, self-resonant, MEMS-fabricated, and ceramic-based mesoscale valveless pulsejet engine was fabricated and characterized by Herrault et al. [START_REF] Herrault | Selfresonant, MEMS-fabricated, air-breathing engine[END_REF]. The primary elements of the engine were fabricated from ceramic sheets. The engine components included a hydrogen feed line, air inlet, fuel inlet, combustion chamber, spark plug, and resonant tailpipe. Figure 1.7 shows a 3-D CAD model of the engine whose combustion chamber was approximately 0.8 cm 3 .

The engine was operated on hydrogen fuel in the kHz range with pressure excursions of up to 10 kPa. For a fuel flow rate of 2.5 L/min, a pressure differential of 9 kPa and a resonant frequency of 1.38 kHz were measured. An output power of 2.5 µWrms was generated at an operating frequency of 1.5 kHz by coupling the engine with a Lorentz-force type generator [START_REF] Herrault | Selfresonant, MEMS-fabricated, air-breathing engine[END_REF]. Herrault et al. anticipates that the use of hydrocarbon fuel and a vibration based magnetic power generator may improve the performance of the engine. However, the issue of viscous and heat losses in the engine has not been highlighted. The scaling laws suggest these losses may be dominant due to the nature of the engine size. A computational study by Qun Wan et al. revealed that the viscosity plays an important role in a micro-pulsejet engine [START_REF] Wan | Computational analysis of the feasibility of a micro-pulsejet[END_REF]. Geng et al. emphasized that for a self-sustained combustion process, the excessive heat loss must be prevented as it lowers the temperatures of residual gas and engine walls [START_REF] Geng | Experimental and numerical investigation of an 8-cm valveless pulsejet[END_REF].

Miniature Internal Combustion Engine

An innovative motor-generator consisting of a miniature linear engine (abbreviated as MICE) coupled with a linear electric alternator is being developed at Aerodyne Research, Inc. [START_REF] Annen | Linearly oscillating miniature internal combustion engine (MICE) for portable electric power[END_REF]. To achieve high efficiency, most of the movable parts are eliminated such as crankshaft, bearing surfaces etc. that are found in a standard IC engine-generator set. The main components of the engine include a two-stroke free-piston engine, a double helix spring, and a linear alternator as depicted in Figure 1.8. The double helix spring is intended to ensure pure linear motion of the moving parts. The free-piston design facilitates variable compression ratio and easy use of HCCI combustion. The HCCI mode of combustion favors low emissions and high efficiency characteristics, and fuel flexible capability [START_REF] Annen | Glow Plug-Assisted HCCI Combustion in a Miniature Internal Combustion Engine (MICE) Generator[END_REF].

The MICE generator is being developed for three power ranges of 5-10 W, 100 W, and 300-500 W for micro air vehicles and powered prosthesis applications. A 300 W engine-generator with an overall diameter of 8 cm and a height of 28.5 cm (includes the generator) is fabricated and tested over a wide range of conditions on propane and Jet-A fuel. The equivalence ratio is varied between 0.5 and 1.35. The scavenging ratios ranged from 0.25 to 0.85. The tests showed that a thermal efficiency of 16% is achievable when run on Jet-A fuel with scavenging ratio of 0.67 and an equivalence ratio of 0.67 at 110 Hz. The obtained mechanical output power from experiment and model were 197 W and 244 W respectively. The reasons for the difference in power output are believed to be caused by leakage losses and lower combustion efficiency. By refining the existing design, Annen et al. expects the efficiency to go up to 25% [START_REF] Annen | Linearly oscillating miniature internal combustion engine (MICE) for portable electric power[END_REF]. To meet 100 W power requirements, the 300 W engine-generator is scaled down such that its overall diameter is about 3.6 cm and height of 13 cm. This scaled down version is reported to be currently undergoing combustion tests.

The group mentioned a demonstration of a net power output from a 10 W size engine, but its efficiency has not been reported. It is predicted that the friction, heat, and leakage losses can become dominant due to the nature of the design and engine size [START_REF] Sher | Miniaturization limitations of HCCI internal combustion engines[END_REF]. A numerical study by Sher et al. on leakage losses shows that engines operate poorly at smaller length scales [START_REF] Sher | Miniaturization limitations of HCCI internal combustion engines[END_REF]. This indicates that a 5-10 W engine shall perform poorly compared to a 300 W engine. Therefore, the efficiency of a 5-10 W engine may be far less than 16%.

Free Elastic-Piston Engine Compressor

A free elastic-piston engine is being developed to provide a human-scale power capability in the form of pneumatic potential energy [START_REF] Riofrio | Experimental Assessment of a Free Elastic-Piston Engine Compressor with Separated Combustion Chamber[END_REF]. Figure 1.9 depicts a sketch of a prototype engine.

The unique feature of the engine is its elastic free-piston made from a silicone rubber membrane. This concept prevents the blow-by and provides near zero frictional loss. The engine comprises of an on/off valve that controls the flow of the pressurized mixture of propane and air in to the combustion chamber. To maintain the proper air-fuel mixture in the mixing line, a metering valve is used. A computer controlled miniature sparkplug initiates fuel ignition. The combustibles are expelled in to an expansion chamber by the use of a passive combustion valve that marks the beginning of the power stroke. During the power stroke, the free elastic-piston performs work by compressing air (working fluid) into a high-pressure supply tank. This high-pressure tank functions as a pneumatic energy storing device. Figure 1.9: A Pro/ENGINEER drawing of a free elastic-piston compressor [START_REF] Riofrio | Experimental Assessment of a Free Elastic-Piston Engine Compressor with Separated Combustion Chamber[END_REF].

Lack of scavenging issues, ability to stop and start on demand, elimination of intake and compression strokes are few benefits this design has to offer. Due to the absence of kinematic linkages the engine is simpler and suitable for portable power. It is possible to operate the engine at any desired frequency by tuning the piston dynamics. The use of completely passive combustion valve has made over-expansion cycle possible.

Few tests have been conducted on a prototype engine. A full device operation was conducted and successful pumping was achieved at low pressures to up to 260 kPa. Results showed that the slow response of valves resulted in backflow of air from the reservoir to the pump which in turn resulted in energy loss. This indicated that a tradeoff between the flow capacity and response time has to be reached. A separate test on the silicone piston under no load conditions showed that short term thermal effects did not affect the piston as no signs of stress were found.

Riofrio et al. identifies that a careful tuning of mass and stiffness of the piston, response time of valves can result in optimized performance in terms of desired power and efficiency. Modelbased simulations showed that very low mass of piston can make the engine operation inefficient. The authors highlight the need for further research to operate the engine more efficiently.

This unique engine design is a novel idea and requires further study. Firstly, the long term thermal effects on the silicone rubber membrane and its durability have to be studied to materialize the concept. A setback in the design is the engine's inability to breath, and therefore relies on injection of pressurized air and propane for continuous operation.

Four Stroke Free Piston Engine

Until now most of the free-piston engines fabricated and tested were based on two-stroke principle. Xu et al. fabricated a prototype of a four-stroke, free-piston SI engine with a linear electric alternator [START_REF] Xu | Prototype testing and analysis of a novel internal combustion linear generator integrated power system[END_REF]. Figure 1.10 shows the prototype engine. The engine is single cylinder with bore of 62 mm and a maximum stroke of 70 mm. The free-piston linear alternator comprises of a spark plug, combustion chamber, electromagnetic valves, kick back device, reversible energy storage device, linear alternator, displacement sensors, and an electronic control unit. A spring connected to the piston functions as a kickback device. The reversible energy storage device has the ability to store and supply the generated electrical energy as per the situation. The reversible energy storage device supplies necessary energy for ingesting fresh air-fuel mixture during the intake stroke. The spark ignition initiates fuel combustion and results in power stroke.

The energy generated in the power stroke is partly stored in kickback device, and partly converted to electrical energy. It has been observed that the stroke lengths are different in each process which provides a possibility for optimized performance [START_REF] Xu | Prototype testing and analysis of a novel internal combustion linear generator integrated power system[END_REF]. The prototype engine when operated had a compression ratio (CR) of 6.5 and an expansion ratio (ER) of 10.5, resulting in 32% engine efficiency. The engine when operated at 25 Hz generated an output power of 2.2 kW.

Xu et al. studied the effects of intake stroke length, intake air pressure, and expansion stroke on the power output. The possibility of the variable intake stroke in the engine is found to be advantageous, as it can control the amount of air-fuel mixture ingested in to the cylinder. This eliminates the need for an intake throttle. The effect of increasing the intake air pressure showed an increase in power output. This is similar to supercharging/turbocharging. The unrestricted expansion stroke can result in an Atkinson type cycle where the cylinder pressure at the end of power stroke is close to atmospheric. The outcome is obtaining additional work which was not possible in an Otto cycle based traditional reciprocating engine. Figure 1.10: A prototype of four-stroke free-piston engine [START_REF] Xu | Prototype testing and analysis of a novel internal combustion linear generator integrated power system[END_REF].

Comparison of results from experiments and numerical simulations showed a difference of 10%. Even then, the engine efficiency obtained from the experiments showed better performance compared to traditional reciprocating engines with an identical geometric configuration.

Though the engine proved to be more efficient than traditional engines of an identical size, the presence of sliding friction is a bottle neck for increasing the efficiency of the engine. Therefore, development of engines at small scale with compliant chambers can mitigate losses and increase efficiency.

Research objective

A small scale internal combustion engine operating on a four-stroke principle will be developed by acknowledging the outcomes from scaling analysis of the MEMS-based P 3 heat engine. Important design aspects at small scale such as friction and leakage losses will be addressed by adopting a compliant chamber design. To ascertain the feasibility of a practical compliant engine, super compliant structures fabricated from metals and, or composites that can sustain extreme pressures and temperatures will be evaluated. A mathematical model shall be developed to assess the performance of the engine for a range of equivalence ratios at various load conditions in both open and closed cycle operations. The outcome of the study is aimed at proposing a small scale compliant resonant engine for micro and mesoscale applications with a prime focus on improved performance.

The study towards the development of the engine will be conducted in the order presented below.

Firstly, a combustion-based, small scale compliant engine operating on a four-stroke Otto cycle principle at its resonance will be proposed. To ascertain the feasibility of a 1 cc practical engine, super compliant structures fabricated from composites and, or metals that can sustain extreme pressures and temperatures will be evaluated. As a part of this evaluation, design challenges, materials for fabrication, engine assembly, and design calculations will be presented at fundamental level.

Secondly, a physics-based mathematical model of the engine will be developed by taking an air-standard cycle approach. With the mathematical model it becomes possible to predict the engine dynamics and performance. Since a real engine operates in an open cycle, the model will be solved primarily to simulate an open cycle operation. However, for completeness, closed cycle operation results will also be presented. In addition, both the open and closed cycle results will be compared to highlight the differences.

Thirdly, a prototype of the engine will be realized using readily available off-the-shelf flexible metal bellows to practically demonstrate the compliant engine concept. The prototype engine will be tested using an in-house developed four-stroke motoring technique for estimating the parasitic losses.

Finally, the obtained motoring test results will be used in the mathematical model to demonstrate the potentiality of the compliant engine. This will be accomplished by comparing the performance of the complaint engine with a typical IC engine. It is anticipated that in a compliant engine, for equivalence ratios of octane-air mixture of 0.6 or higher, the frictional dissipative losses are less than 10% of indicated work. If such is the case, the compliant engine concept is consolidated and could be seen as a promising alternative to the traditional IC engines.

SMALL SCALE RESONANT ENGINE Chapter 2

Introduction

A reciprocating engine with a variation in basic design is proposed; the traditional pistoncylinder assembly is replaced by a flexible cavity (or compliant chamber) with a mass or piston at one end. Hence the volume change occurs through deformation of the flexible cavity or compliant chamber. The engine operates in a linear motion therefore the need for crankshaft assembly does not arise. Due to the elimination of a traditional sliding seal and crankshaft assembly the frictional losses are substantially reduced. There are several advantages to this design. Blow-by does not occur since the cavity is sealed. The compression ratio is not fixed by engine geometry but is variable and may be changed by operating conditions. Hydrocarbon emissions are reduced since the need for lubricant is removed and crevice volumes are, if not eliminated, substantially reduced. Finally, the geometric complexity of the chamber is reduced significantly which leads to a reduction in cost. This concept shares some features with freepiston engines; such as, a variable compression ratio which allows for fuel flexibility [START_REF] Flynn | Observations on 25,000 Hours of Free-Piston-Engine Operation[END_REF].

However, unlike a single piston free-piston engine [START_REF] Mikalsen | A review of free-piston engine history and applications[END_REF] which uses a rebound device to store energy, the flexible cavity itself performs the task of a rebound device thus enabling a reduction in engine size. A study focused on analyzing the dynamics and performance of the engine for small compression ratios was conducted by Richards et al. [START_REF] Richards | A Resonant Air-Standard Heat Engine[END_REF]. Since the compression ratios were small, they assumed the system to be linear in nature and developed a linearized lumpedparameter model to predict the engine performance. The model was also used to demonstrate that operating the engine at resonance was beneficial. Insight into dynamic behavior of the engine obtained by the linearized lumped-parameter model is presented in Appendix A2: ANALYSIS OF LINEARIZED MODEL.

Engine operation

The resonant engine operates on a four-stroke Otto cycle principle comprising of intake 6→7→1, compression 1→2, expansion 3→4, and exhaust processes 4→5→6. During the intake stroke 7→1, the piston moves towards bottom-dead-center-I (BDC I ) by utilizing the potential energy stored in the flexible cavity structure. As the chamber expands, induction of the fresh fuel-air mixture takes place. Throughout the intake process 7→1, the intake valve remains open (IO) while the exhaust valve remains closed (EC). When the piston reaches state 1, its velocity becomes zero and the intake valve closes (IC).

The compression process 1→2 begins when the piston starts to moves from BDC I towards top-dead-center-I (TDC I ). During the process, a part of the stored potential energy in the flexible cavity structure is used in compressing the fuel-air mixture. Ideally, the compression process 1→2 is assumed isentropic as the heat and friction losses are neglected.

At the end of compression process 1→2, the fuel-air mixture is ignited (IGO, ignition on) at near constant volume, thereby increasing the pressure and temperature in the cavity or chamber. This process (combustion or heat addition 2→3) is assumed instantaneous and ends at state 3 (IGC, ignition off). The increased pressure in the chamber pushes the piston towards bottomdead-center-II (BDC II ) and the engine undergoes expansion process 3→4. Just like the compression process 2→3, the expansion process 3→4 is also treated to be isentropic. The expansion process 3→4 is the power stroke of the cycle.

As the piston reaches BDC II its velocity becomes zero and the gas attains thermodynamic state 4. Now the exhaust valve opens (EO) and the blowdown process 4→5 (initial phase of exhaust process) begins. The hot gases are expelled out of the chamber due to the pressure gradient across the exhaust valve.

As the pressure in the chamber drops to near atmospheric, the piston begins to move from BDC II to TDC II marking the beginning of exhaust stroke 5→6. During this process, the piston physically pushes the left over hot gases out of the chamber by utilizing the potential energy stored in the flexible cavity structure (gained from the previous process 3→4). Processes 4→5 and 5→6 are together called the exhaust process 4→5→6. When the piston reaches TDC II , the exhaust valve closes (EC) and intake valve opens (IO).

During this time the thermodynamic process 6→7 takes place. In real situations, both the valves remain open for some duration of time identified as valve overlap. The process 6→7 is a part of intake process 6→7→1. After process 6→7, the piston travels from TDC II to BDC I , and the next operating cycle begins.

The work done by the hot gases is spent in performing mechanical work and overcoming frictional work. The flexible cavity behaves as a mechanical spring and is capable of storing potential energy.

Realization of a prototype engine

In the following sub-sections, design challenges, materials for fabrication, engine assembly, and design calculations will be presented at fundamental level to demonstrate the compliant engine feasibility and reliability.

Design challenges

Consider a situation where the engine size is about a cubic centimeter (say 0.785 cc, 1 cm in diameter and 1 cm in height) with an allowable compression ratio CR=8. This engine when fueled with a rich mixture of air and octane (equivalence ratio Φ≈1.3) will experience a peak pressure and peak temperature of about 3.5 MPa and 1350 K respectively. Therefore, there arises a need for a mechanically robust and a compliant structure that can withstand extreme temperature and pressure conditions. For small volume excursions metal bellows may be used.

However, for better efficiency (or output work) the volume excursions must be higher, which requires a super compliant structure. For a reliable operation, there is a necessity for good heat transfer in order to prevent overheating of engine components. Usually in a traditional reciprocating IC engine, the temperature of the cylinder wall on the gas-side is below 180 °C to avoid deterioration of the lubricant [START_REF] Heywood | Internal combustion engine fundamentals[END_REF]. Therefore, in the proposed engine, compliance, mechanical robustness, and thermal management are the main design challenges.

Materials to use

As the pressure inside the chamber could be on the order of few mega Pascal, the compliant chamber structure ought to be robust. For this purpose, Kevlar ® is chosen for its superior tensile strength σ t =3000 MPa. To ensure good thermal management, copper is selected for its good thermal conductivity K c =400 W/m-K. In order to overcome damages due to warping, a flexible material has to be used. For this purpose, Silastic ® HCR silicone rubber is chosen. To prevent Kevlar ® from excessive heat, it has to be insulated. Pyrogel ® XT is a good candidate for insulation due to its low thermal conductivity K p =0.1 W/m-K. The mechanical and thermal properties of these materials are listed in Table 2.1.

Engine assembly

A square-shaped diaphragm bellows fabricated from composite laminates makes the flexible engine cavity. The flexible cavity consists of two laminates: top and bottom. Each composite laminate is made by strategically stacking sheets of Kevlar ® , Pyrogel ® XT, Silastic ® HCR silicone rubber, and copper to form a sandwich structure as shown in Figure 2.2. The contact areas between Kevlar ® and copper are thermally insulated by Pyrogel ® XT, as mechanical properties of Kevlar ® degrade beyond 150 °C. Also Pyrogel ® XT is used where Kevlar ® is prone to direct exposure of hot chamber gases. Since the corners of the composite laminate is susceptible to warping, Silastic ® HCR silicone rubber is used to ease it.

Design calculations

A back of the envelope calculation to estimate thicknesses of various sheets in the composite laminate are presented in this section.

Kevlar ® fibers design thickness

A single ply of Kevlar ® fibers is used in designing the walls of the compliant engine cavity.

So as to determine the thickness of the Kevlar ® fibers, a force balance equation is written for a sheet of Kevlar ® as shown here.

Referring to the above figure, let's say the pressure force and tension force in the Kevlar ® sheet are equal in magnitude and opposite in direction then, Pressure × area of the sheet = Tension force in the sheet

𝑃 × 𝑎 2 = 𝑡 × 4𝑎 × 𝜎 𝑡 thickness, 𝑡 = 𝑃 × 𝑎 4𝜎 𝑡 (2.1)
For a 1 cc engine whose side length a=1 cm, design pressure P=3.5 MPa, and tensile strength σ t = 2600 MPa (at 150 °C), the thickness of Kevlar ® fibers ought to be about t=3.25 µm (for single ply).

Pyrogel ® design thickness A temperature of 1350 K (~ 1077 °C) in the engine cavity can damage the Kevlar ® fibers as its properties deteriorate at temperatures beyond 150 °C. Hence, Pyrogel ® sheets are used. A maximum design temperature of 50 °C is set at the interface of Kevlar ® and Pyrogel ® sheets. It is assumed that the heat lost from the engine walls is about a third of 60 W (=20 W) of generated output power. If fifty percent of the cavity surface 1 cm × 1 cm is covered by Kevlar ® and 10 W of heat is lost through it. Then the required thickness of Pyrogel ® to prevent temperatures exceeding 50 °C at the interface of Kevlar ® and Pyrogel ® ought to be 0.51 mm. Below is the relevant calculation.

Using Fourier's law of heat conduction to the Pyrogel ® sheet we have,

Heat rate = 𝐾 × 𝐴 × ∆𝑇 𝛥𝑡
where K is thermal conductivity of Pyrogel ® , A is surface area of Pyrogel ® sheet, ΔT is temperature difference, and Δt is thickness of Pyrogel ® sheet.

For the aforementioned design values we have,

0.1 W m -K × 5 × 10 -5 m 2 × (1350 -323)K 𝛥𝑡 m = 10 W
Upon simplifying we get, thickness 𝛥𝑡 =0.513 mm.

So, therefore a thickness 𝛥𝑡 =0.513 mm will ensure the temperature at the interface of Kevlar ® and Pyrogel ® sheets to be 323 K (50 °C). However, the 10 W of heat lost from the engine cavity occurs through the least thermal resistant copper pathway instead of Pyrogel ® as shown below. Let's assume the dimensions (thickness) of the various sheets or components to be as labeled in the below figure.

mm Copper

Kevlar ® Pyrogel ® 1350 K 300 K R c R k R p 300 K 1350 K 323 K 1 mm 0.5 mm ≡ (Thermal resistance circuit) (Simplified physical sketch)
The thermal resistances of copper sheet, Pyrogel ® , and Kevlar ® are calculated as shown below.

Copper, 𝑅 𝑐 = 𝑡 𝑐 𝐾 𝑐 𝐴 𝑐 = 5 mm 400 W m -K ⁄ × 5 × 10 -5 m 2 = 0.25 K W (2.2 a-c) Pyrogel ® , 𝑅 𝑝 = 𝑡 𝑝 𝐾 𝑝 𝐴 𝑝 = 1 mm 0.1 W m -K ⁄ × 5 × 10 -5 m 2 = 200 K/W Kevlar ® , 𝑅 𝑘 = 𝑡 𝑘 𝐾 𝑘 𝐴 𝑘 = 0.5 mm 0.04 W m -K ⁄ × 5 × 10 -5 m 2 = 250 K/W
The ratio of heat flow from copper to composite layer (Pyrogel ® and Kevlar ® ) is given by,

𝑞 𝑐 𝑞 𝑘+𝑝 = 𝑅 𝑝 + 𝑅 𝑘 𝑅 𝑐 = 1800 (2.3)
From above expression (2.3) it is clear that heat prefers the low resistant copper pathway.

Therefore, out of 10 W of heat lost, 99.94 % is lost through the 5 mm copper block. The remaining 0.006 W of heat is lost from Pyrogel ® , which is negligible. Thus a thickness of 0.52 mm Pyrogel ® sheet will ensure proper insulation of Kevlar ® from extreme temperatures. The sandwich structure made from 0.5 mm Pyrogel ® and 13 µm Kevlar ® fibers shall together tolerate extreme temperatures and pressures (or mechanical loads). In this heat transfer analysis, heat transfer by convection is neglected.

Copper design thickness

The thickness of copper is estimated based on the maximum pressure load of 3.5 MPa. As the copper block is subjected to high temperatures, its yield strength is accordingly chosen (at elevated temperatures of about 760 °C) for design calculations.

Referring to the figure used in Kevlar ® fiber calculations, let's say the pressure force and tension force in the copper sheet or block are equal in magnitude and opposite in direction then, Pressure × area of the sheet = Tensile force in copper block Upon simplifying we get, thickness, 𝑡 = 𝑃 × 𝑎 4𝜎 𝑡 (2.4)

For a 1 cc engine whose side length, a=1 cm, design pressure, P=3.5 MPa, and tensile strength, σ t =70 MPa (at 760 °C), the thickness of copper block can be estimated using expression (2.4). It has been found that thickness should be at least t=0.125 mm. However, for additional mechanical strength and higher piston inertia a thickness of 5 mm is selected.

Silicon rubber design calculation

During the expansion process, the corners of the cavity might undergo warping. To ease warping, an additional compliance is necessary, this is achieved by using flexible silicon rubber Silastic ® FL 40-9201. Below is its thickness calculation.

Referring to the figure used in Kevlar ® fiber calculations, let's say the pressure force and tension force in the silicon rubber sheet are equal in magnitude and opposite in direction then, Pressure × area of the sheet = Tensile force in silicon rubber Upon simplifying we get, thickness, 𝑡 = 𝑃 × 𝑎 4𝜎 𝑡 (2.5)

For a 1 cc engine whose side length, a=1 cm, design pressure, P=3.5 MPa, and tensile strength, σ t =8.2 MPa (at 200 °C), the thickness of silicone rubber can be estimated using expression (2.5). It has been found that thickness should be at least t=1 mm.

MATHEMATICAL MODEL

Four-stroke air-standard Otto cycle

For a specific compression ratio, Otto cycle based engines operate more efficiently than diesel or dual cycle based engines. Therefore, the complaint engine will be designed and modeled on an Otto cycle.

Assumptions

Due to the complexity in the analysis of a real engine operation, the engine is modeled by taking an air-standard cycle approach with few assumptions in place. The assumptions are listed below:

1. The working fluid is considered to be air with constant properties, and obeys ideal gas.

2. The specific heat ratio of air is taken to be 1.4.

3. The combustion process is modeled as constant volume heat addition process for a short duration t q →0. [START_REF] Walther | Advances and challenges in the development of powergeneration systems at small scales[END_REF]. For open cycle simulation, the blowdown process is assumed to be an instantaneous and isentropic; occurring at the end of the expansion stroke (i.e. at zero piston velocity).

5. For closed cycle simulation, the exhaust process is replaced with a heat rejection process by applying a cooling pulse at the end of the expansion stroke (i.e. at zero piston velocity) for a short duration t q →0. In practice, as in real engines, fuel-air mixture is burnt to produce heat. In this analysis, as aforementioned, constant rate periodic heat addition is used instead. The engine is modeled to have a flexible cavity with a mass or piston at one end. The flexible cavity of the engine deforms elastically to allow compression and expansion without the necessity of a sliding seal. Like analyses of free-piston engines [START_REF] Annen | Linearly oscillating miniature internal combustion engine (MICE) for portable electric power[END_REF][START_REF] Suzuki | Experimental study on mechanical power generation from MEMS internal combustion engine[END_REF][START_REF] Mikalsen | Performance simulation of a spark ignited free-piston engine generator[END_REF][START_REF] Xiao | Motion characteristic of a free piston linear engine[END_REF], and Stirling engines [START_REF] Benvenuto | Analysis of Free-Piston Stirling Engine/Linear Alternator Systems Part 2 -Results[END_REF][START_REF] De Monte | Reflections on Free-Piston Stirling Engines, Part 2: Stable Operation[END_REF][START_REF] Formosa | Nonlinear dynamics analysis of a membrane Stirling engine: Starting and stable operation[END_REF], the analysis leads to a nonlinear lumped-parameter model based upon Newton's law for the dynamics of the piston, a thermodynamic model for the operating cycle of the working fluid, a model for friction, and a mass-flow rate equation for the working fluid flowing through the valve (for open cycle simulation). The analysis also provides for the possibility of an external spring. Unlike other approaches, which approximate nonlinearity using an equivalent stiffness [START_REF] Xiao | Motion characteristic of a free piston linear engine[END_REF], the four-stroke thermodynamic model makes no approximation. The four-stroke thermodynamic model shall predict the engine's performance at higher heat inputs (or fuel rich combustion) and practical compression ratios, where the predictions of a linearized model [START_REF] Richards | A Resonant Air-Standard Heat Engine[END_REF] can become invalid.

Open cycle

A physics-based nonlinear lumped-parameter model will be developed on an air-standard open cycle approach to predict the performance of the engine for a four-stroke resonant operation.

Governing equations

A schematic diagram of the engine modeled on Otto cycle is shown in Figure 3 As the heat added per cycle is periodic, it is assumed that the dependent variables T(t), P(t), (t), and V(t) will also be periodic at steady state. Figure 3.2 shows the heat-rate function q(t) used for modeling the open cycle. The heat addition pulse associated with combustion phenomenon is assumed impulsive for a duration t q 0. The heating pulse q(t) at a constant rate of q H is applied at the end of the compression stroke at t=t 1 . When t q 0, lim 𝑡 𝑞 →0 𝑞 𝐻 𝑡 𝑞 = 𝐸, where E is the finite energy supplied to the engine per cycle during the heating pulse. The heat addition process is followed by expansion, exhaust, and intake processes. At the time t=t 1 +t 2 +t 3 +t 4 =t p , the cycle repeats, hence t p is the cycle period of the engine. The time lapsed in each process need not be equal. In the analysis, it is assumed that the heating pulse q H and the blowdown process occur at zero piston velocity for an infinitesimal duration, therefore 𝛥𝑉 3 ̇= 𝛥𝑉 2 ̇= 0 and 𝛥𝑉 5 ̇= 𝛥𝑉 4 ̇= 0.

𝜌 𝑜 + ∆𝜌 = 𝑃 𝑜 + ∆𝑃 𝑅(𝑇 𝑜 + ∆𝑇) , (3.2)
𝑚 𝑆 2 ∆𝑉 ̈+ (𝑏 + 𝑏 𝑓 + 𝑏 𝑝1 ) 𝑆 2 ∆𝑉 ̇+ 𝑠 𝑆 2 ∆𝑉 = ∆𝑃 , (3.3)
The assumption that the piston velocity is zero when the impulsive heating and blowdown are applied is a condition of resonance at steady state.

Steps to solve open cycle

To characterize the pressure-volume (P-V) behavior of the engine, the model equations [(3. ] ,

(3.15)

∆𝑃 ̇= 1 (1 + ∆𝑉 ̅ ) [ 𝑀𝑅𝑇 𝑜 ∆𝑇 Ṗ𝑜 𝑉 𝑜 -(1 + ∆𝑃 ̅ ) ∆𝑉 ̅ ̇] , (3.16)
where of blowdown process (exhaust valve opens EO), end of blowdown process and beginning of displacement process, end of displacement process and exhaust valve closes (EC), and intake process begins with intake valve opening (IO) respectively. In the figure, compression ratio CR is defined as V 1 /V 2 and the expansion ratio ER as V 4 /V 3 . In the heat addition process 2→3, the temperature inside the flexible cavity before and after the heat addition can be computed from equations (3.14) and (3.15) and are given by

𝛺 = 𝐶(1 + ∆𝜌̅ )(1 + ∆𝑉 ̅ ), (3.17
𝛥𝑉 3 = 𝛥𝑉 2 , (3.18
)

𝛥𝑇 3 = 𝛥𝑇 2 + 𝛹 (1 + ∆𝜌 1 )(1 + ∆𝑉 1 ) , (3.19)
where

𝛹 = 𝑞 𝐻 𝑡 𝑞 𝜌 𝑜 𝑉 𝑜 𝑐 𝑣 𝑇 𝑜 . (3.20)
The temperature inside the cavity at the end of blowdown process 4→5 can be computed by assuming an isentropic expansion from pressure 𝑃 4 ̅ to ambient pressure 𝑃 5 ̅̅̅ = 1 given by equation (3.21),

∆𝑇 5 = (1 + ∆𝑇 4 ) ( 1 + ∆𝑃 4 1 + ∆𝑃 5 ) 1-𝛾 𝛾 -1 , (3.21
)

∆𝑉 5 = ∆𝑉 4 , (3.22)
where ∆𝑃 5 = 0.

Closed cycle

Though it is a common practice to model reciprocating engines as closed cycle (without intake and exhaust strokes), this approach is questionable for internal combustion engines with varying stroke lengths. Though, the open cycle simulation allows for better understanding of engine dynamics and performance, for completeness a mathematical model for the closed cycle is presented. Here, the governing equations and the steps involved to obtain the solution for closed cycle are discussed.

Governing equations

The The imposed heat-rate q(t) is periodic with periodicity t p as shown diagrammatically in The heat-rate waveform q(t) in equation (3.28) consists of impulsive heating and cooling.

The heating and cooling pulses are of duration t q . The heating pulse at a constant rate of q H is applied at the end of the compression stroke at t=0. At the time t=t 1 , a cooling pulse at a constant rate q H -q o is applied. At the time t=t 1 +t 2 =t p , the cycle is repeated, so that t p is the period of the cyclic heat-rate. This specification for the heat-rate q(t) allows for asymmetry in the cycle, in that the times t 1 and t 2 , and the magnitude of the heating and cooling pulses, are not necessarily equal.

As it is assumed that the impulsive heating and cooling pulses occur at zero piston velocity, 𝛥𝑉 3 ̇= 𝛥𝑉 2 ̇= 0, and 𝛥𝑉 4 ̇= 𝛥𝑉 1 ̇= 0. The assumption that the piston velocity is zero when the impulsive heating and cooling are applied is a condition of resonance at steady state.

Figure 3.5: Heat-rate function q(t) for modeling the closed cycle.

Steps to solve closed cycle

For the subsequent analysis, the nondimensional scales as defined in equation

)

𝑑∆𝑉 ḋ𝑡 = -𝛬∆𝑉 -2𝜁∆𝑉 ̇+ 1 𝛾 [ ∆𝑇 ̅ -∆𝑉 ̅ 1 + ∆𝑉 ̅ ] , (3.34)
𝑑∆𝑇 𝑑𝑡 = - 1 𝐶 [ 1 + ∆𝑇 ̅ 1 + ∆𝑉 ̅ ] ∆𝑉 ̇-𝐻 𝐶 ∆𝑇 + 𝛤 𝐶 𝑞 , (3.35)
where ∆𝑉, ∆𝑉 ̇, and ∆𝑇 are the state variables.

To characterize the P-V behavior of the engine, the system equations [(3.29) -(3.32)] are integrated numerically. It is assumed that the duration of the heating and cooling pulses t q 0

while lim 𝑡 𝑞 →0 𝑞 𝐻 𝑡 𝑞 = 𝐸, where E is the finite energy supplied to the engine per cycle during the heating pulse. In this circumstance the P-V behavior of the engine for the closed cycle at steady state takes the form shown in

𝛥𝑉 3 = 𝛥𝑉 2 , (3.36
)

𝛥𝑉 1 = 𝛥𝑉 4 , (3.37
)

𝛥𝑇 3 = 𝛥𝑇 2 + 𝛹 , (3.38
)

𝛥𝑇 1 = 𝛥𝑇 4 -𝛹 (1 - 𝑞 𝑜 𝑞 𝐻 ) , (3.39)
where Ψ is defined by equation (3.20).

The Improper choice of q o /q H and initial conditions ∆𝑉 2 and ∆𝑇 2 would lead to monotonically drifting state variables ∆𝑉 and ∆𝑇 precluding periodic behavior. To determine the steady-state behavior of the engine q H is specified, the ratio q o /q H is chosen, and integration is started from the quiescent initial state of ∆𝑉 2 = ∆𝑉 2 ̇= ∆𝑇 2 = 0. An example integration illustrating the first cycle in a transient dwell-up from this initial state is shown in Figure 3 and is determined by the conditional integration. Since the system undergoes a transient phase, the cycle is not closed and ∆𝑉 2 ≠ ∆𝑉 2 ′ and ∆𝑇 2 ≠ ∆𝑇 2 ′ . Therefore, to obtain a steady state solution, an objective function is defined that computes square error of state variables at 2 and 2', and a function minimization computational procedure is adopted to determine proper choice of ∆𝑉 2 , ∆𝑇 2 and q o /q H for a specified heat input q H that would result in a steady-state periodic cycle.

Once the objective function satisfies a specified tolerance the computation stops, the cycle closes In the circumstance that the heat input q H t q per cycle is small, volume ∆𝑉 ̅ and temperature ∆𝑇 ̅ excursions will be small. where

𝐸 ̅ (𝑡̅ ) = { 1, (𝑘 -1)(𝑡 ̅ 1 + 𝑡 ̅ 2 ) < 𝑡 ̅ < 𝑘 𝑡 ̅ 1 0, 𝑘𝑡 ̅ 1 < 𝑡 ̅ < 𝑘 (𝑡 ̅ 1 + 𝑡 ̅ 2 ) . (A2 .4)
In Appendix A2: ANALYSIS OF LINEARIZED MODEL, it is shown that the natural period for resonant operation is 𝑡 ̅ 𝑝 = 2𝜋/𝜔 𝑑 , where 𝜔 𝑑 = 𝜔 𝑛 √1 -𝜁 2 and 𝜔 𝑛 = √1 + 𝛬. The heat is added at 𝑡 ̅ =0, heat is rejected at 𝑡 ̅ = 𝑡 ̅ 𝑝 /2, and that the efficiency of the cycle at resonance is given by

𝜂 = (𝛾 -1) 𝛾 𝛹 1 + 𝛬 𝑓(𝜁) , (A2.13) 𝑓(𝜁) = 1 + 𝑒 -𝜋𝜁 √1-𝜁 2 1 -𝑒 -𝜋𝜁 √1-𝜁 2 .
(A2.11) EXPERIMENT Chapter 4

Introduction

Realization of the engine described in chapter three requires super compliant structures that can sustain the pressures, temperatures, and elastic deformations that are encountered in practical applications. A prototype of such an engine at small scale is realized using flexible metal bellows to demonstrate the compliant engine concept. To characterize the advantages of this engine over traditional engine designs, parasitic losses are measured and compared with traditional reciprocating engines. In this regard, an experiment station is built and a novel four-stroke motoring technique is developed to estimate parasitic losses in the compliant engine at resonant operation. The nonlinear lumped-parameter model discussed in chapter three is used in conjunction with motoring experiment data to evaluate the engine performance.

The first part of this chapter briefs the components and apparatus used, while the later part is devoted to the experimental setup and procedure.

Experimental apparatus

The prototype engine is fabricated and motored. The measurements from motoring experiment are recorded electronically using various instruments. In the following subsections, an overview of all the instruments and devices with their specifications is presented. For experimentation, some of the instruments and components are calibrated and thus obtained calibration curves are included. Instrument uncertainties are available in Appendix D3: UNCERTAINTY ESTIMATION.

Edge welded metal bellows

To experimentally realize the compliant engine concept, edge welded flexible metal bellows are used that meet AMS 5666F standards. The bellows are made by stamping and welding 0.0762 mm thick INCONEL ® 625 sheets to form a long and flexible assembly.

Electromagnetic vibration shaker

To actuate the engine piston, an electromagnetic vibration shaker (make: MB Electronics) is used instead of an electric motor. The MB Electronics vibration shaker unit comprises of a Vibramate vibration exciter (MB Model EA 1500) and an electronic power amplifier (2120 MB Amplifier). The vibration exciter is capable of delivering strokes up to 1.25 cm with a capacity to generate force in the range 0-50 lbs. vector (or 220 N). The force generated and strokes delivered are dependent on the excitation frequency. The vibration shaker and amplifier operate in a frequency range of 5 Hz-10 kHz and 5 Hz-20 kHz respectively. The 2120 MB Amplifier has an input impedance of 20,000 ohms and allows a maximum input voltage of 2 Vrms. To prevent damage to the equipment, the maximum current is restricted to 0.5 A. In order to monitor the total output stage current, the selector switch on the front panel of amplifier is set to T position.

Pressure transducer

The instantaneous engine chamber (or bellows cavity) pressure is measured by flush mounting a compression-preloaded, Quartz based sensor (Kistler 211B5) to the interior of engine chamber. The sensor is acceleration compensated. The internal micro signal conditioning unit converts charge developed into readable voltage. The pressure transducer has an operating range of 0-690 kPa, and can be used for frequencies between 0.025 Hz to 50 kHz. The pressure signal is amplified by ten times using an electronic amplifier (make: PCB Piezotronics). The pressure transducer is dynamically calibrated with respect to a standard pressure sensor.

Force sensor

The dynamic force acting on the piston during the motoring experiment is measured by sandwiching an ICP ® Force sensor (PCB 208A03) between the piston and the shaker table. The force sensor has an inbuilt MOFSET microelectronic amplifier and does not require a charge amplifier. However, an electronic amplifier (make: PCB Piezotronics) with a gain set to 100 is used to amplify the force signal. The measurement ranges for compressive and tensile forces are ±2200 N. A compressive force results in a positive voltage signal, while a tensile force results in a negative voltage signal. The operating frequency range is 0.003 Hz-70 kHz.

A static calibration test using standard weights is performed to determine the sensitivity of the force sensor. The result of calibration test is plotted as force versus output voltage in

Solenoid valve

Instead of using a cam driven intake and exhaust valves, a direct acting solenoid valve with tight shutoff is used to allow gas exchanges. The solenoid valve is normally closed in deenergized state. The valve is rated at 120 V and 60 Hz. During normal operation it consumes an average power of 17.1 W. Its 7.14 mm orifice is rated for 620 kPa (inert air medium). The response time of the solenoid valve is very critical for normal operation. Therefore accurate response time measurements are obtained by measuring the lag in rise (or open) and fall (or close) times of the poppet valve (inside the solenoid valve). The experiment is conducted by supplying the solenoid valve with a square pulse of 19 Hz with 50% duty cycle, and the response times are measured using Laser Doppler Vibrometer (Polytec OFV-5000) and Digital Phosphor Oscilloscope (Tektronix, TDS5034B). Thus obtained open and close times are found to be 5.32 ms and 10.28 ms respectively.

Laser doppler vibrometer

A Polytec OFV-5000 laser vibrometer is used to measure the displacement and velocity of the piston or mass in non-contact mode. Its electronics include a displacement decoder DD-200 and a velocity decoder VD-02.

The helium neon laser beam generated in the fiber interferometer unit (Polytec OFV 511 Fiber Interferometer) is focused on the piston to measure its vibration characteristics such as displacement and velocity. To ensure accurate displacement measurements at mesoscale, the resolution is set to 10240 nm on the front panel of the Polytec OFV 5000 vibrometer controller.

The maximum measureable displacement is relative to the depth of focus of the laser beam, and measurements of up to several centimeters can be obtained. The tracking filter is set to "fast" and the maximum velocity is set to 10 m/s. The proper focusing of the laser beam and use of retroreflective tape on the piston can greatly improve the signal strength. To reduce the signal drift, the equipment is allowed to warm up for at least six hours before the measurements are recorded. This allows the optics in the sensor head to stabilize. The displacement and velocity readings acquired at the BNC outlets on the front panel of the vibrometer controller are fed to the computer (National Instruments interface BNC-2110, PCI-6052E) for recording.

Solenoid valve amplification circuit

The square pulse generated in the computer to actuate the solenoid valve needs amplification to meet valve electrical specifications.

Experimental procedure

A sequence of experiments is performed to estimate parasitic losses in the prototype engine fabricated from bellows. At first, a frequency response plot of the engine is generated by treating the engine as a spring mass damper system. This plot is used to identify the natural frequencies of the engine. Then, the engine is motored at its resonant frequencies in the experiment station.

As a part of motoring experiment, to characterize the losses due to heat transfer and fluid flow friction, (loss)mep-speed-frequency relations (loss mean effective pressure) are obtained. These relations are obtained by varying the piston speeds and operating frequencies on a one at a time basis. Finally, the nonlinear lumped-parameter model discussed in chapter three is used in conjunction with motoring experiment data to evaluate the engine performance. Thus obtained parasitic losses are illustrated in the form of an energy flow diagram.

Engine configuration

To experimentally realize the concept illustrated in Figure 3.1, a flexible metal bellows is used. Figure 4.7 shows a photograph of the engine fabricated from bellows in the test set-up. The engine assembly consists of a piston or mass attached at one end of the enclosed bellows, thus replacing a traditional piston-cylinder assembly. An electronically actuated solenoid valve takes the place of cam driven intake and exhaust valves to allow airflow during the intake and exhaust processes. The mass of the piston and the stiffness of the metal bellows are chosen such that the resonant frequencies of the engine are below the solenoid valve's frequency of operation.

The metal bellows shown in Figure 4.7 have s=3050 N/m, V o =21.88 cc with 2.08 cm in radius and 1.6 cm of nominal length (all dimensions in mesoscale range), and m=0.18 kg. Figure 4.8 shows a schematic sketch of assembled prototype engine for four-stroke motoring test. To obtain a frequency response plot, the engine is excited by motion of a support point (an acrylic plate) [START_REF] Thomson | Theory of vibration with applications[END_REF] using a vibration shaker table as shown in Figure 4.9. In this experiment, the mass or piston is free to move, while the acrylic plate (also called support point) is connected to the vibration shaker. The sine function on a function generator is fed into the vibration shaker to displace the support point harmonically over a frequency ranging from 0 to 50 Hz. The harmonic displacement of support point is the input, while the piston displacement is the output of the engine. The magnitude of transfer function for the engine is defined as ratio of piston to support

Motoring test

To quantitatively determine all the parasitic losses in the engine, the engine is motored in a four-stroke mode (Figure 4.7). The motoring experiment is conducted at ambient conditions for experimental simplicity. Figure 4.10 shows a schematic sketch of the experimental apparatus.

The top end of the engine is rigidly clamped to a fixed support. Instead of using an electric motor to actuate the piston, an electromagnetic vibration shaker capable of delivering strokes up to 1.25 cm is used to drive the engine at the piston. The piston is rigidly linked to the vibration shaker through a Quartz based force sensor (PCB 208A03) to measure the instantaneous force applied on the piston. A Kistler 211B5 pressure transducer is flush mounted to the interior of the flexible cavity (bellows cavity) to measure instantaneous cavity pressure. The displacement and the velocity of the piston are measured optically using the Polytec laser vibrometer. In the motoring experiment, the air (or working fluid) is considered to be the system under study. The experimentally obtained pressure and displacement information is used to compute the work done on the system (or air) by the vibration shaker. In this context, the work done by the system is boundary work or PdVwork. The force and displacement information is used to compute the work done by the vibration shaker on the engine. The difference between the work done by the vibration shaker and work done on the system is attributed to the inherent structural damping of the flexible metal bellows. The work done on the system is dissipated by three mechanisms. First, there is friction due to fluid flow. Second, because there is compressive heating of the air there is heat transfer. Finally, there is net enthalpy out of the system due to the intake and exhaust processes. To characterize the losses due to friction and heat transfer a mean effective pressure of losses is formulated. This parameter is then obtained over a range of engine speeds. The (loss)mep-speed-frequency relations are obtained for a closed condition as it excludes the enthalpy lost to the surroundings. The enthalpy lost to the surroundings can be estimated using the nonlinear mathematical model in conjunction with the experimental data and this is described in the subsequent section.

To obtain (loss)mep-speed-frequency relationships for the engine, the motoring experiment is performed in the closed condition for varying piston speeds and operating frequencies. In the engine, the mean piston speed is proportional to stroke length and operating frequency.

Therefore, for a fixed operating frequency, an increase in piston speed is achieved by increasing the stroke length. Similarly, for a fixed speed, an increase in operating frequency is achieved by decreasing the stroke length. Operating the engine at a fixed frequency holds the heat transfer by compressive heating constant, while operating at fixed speed holds the friction constant.

Therefore, the heat transfer and friction effects are separated by varying the operating frequency and engine speed, on a one at a time basis. It is worth mentioning that the heat transfer is inversely related to the operating frequency, while the fluid flow friction is proportional to mean piston speed. For laminar flows the pressure drop required for fluid flow is linearly related to fluid velocity, while for turbulent flows, the pressure drop is proportional to square of fluid velocity. Thus the fluid flow friction component of (loss)mep contains both linear and quadratic velocity terms [START_REF] Pulkrabek | Engineering Fundamentals of the Internal Combustion Engine[END_REF].

Estimation of performance parameters

The mathematical model developed in chapter three is used in conjunction with motoring experiment data to estimate various performance parameters. A brief description on implementing the mathematical model to determine the parameters such as b+b f and h is presented. In this context, the dissipative element b+b f shown in Figure 3.1, models the PdVwork or work done by the system on surroundings (or vibration shaker). In this analysis, it is worth mentioning that the damper b p1 =0 N-s/m. A negative value of b+b f models the work done on the system by the vibration shaker. The work done on the system need not be same in each process;

RESULTS

Chapter 5

Introduction

In this chapter, the results from the motoring experiment and mathematical model are presented in order. First, as a part of motoring experiment, the generated frequency response plots, (loss)mep-speed-frequency plots, and energy flow diagrams are presented. Next, the results from the mathematical model simulating a fueled operation for both open and closed cycle operations are discussed. Then, the results from both the cycles are compared and the differences are highlighted. Finally, the obtained motoring test results will be used in the mathematical model to demonstrate the potentiality of the compliant engine. This will be accomplished by comparing the performance of the complaint engine with a typical IC engine.

Estimation of parasitic losses

The prototype engine shown in Figure 4.7 is motored in the ambient temperature and pressure conditions of T o =295 K and P o =93.4 kPa respectively. The properties of air used are specific gas constant R=286.9 J/kg-K, density 𝜌 𝑜 =1.10 kg/m 3 , heat capacity at constant volume c v =717.25 J/kg-K, and speed of sound c=344 m/s.

Determination of resonant frequencies

In order to determine the operating frequencies for the four-stroke motoring experiment, a frequency response plot is generated in open and closed conditions as shown in Figure 5

Four-stroke motoring

The prototype engine is motored at its resonant frequencies using the vibration shaker at ambient temperature as illustrated in later falling to 78.4±1.97 kPa at the end of expansion stroke 3→4. Note that the expansion stroke is bigger than the compression stroke. The thermodynamic states 2 and 3 overlap on each other as there is no change in thermodynamic properties of the engine during 2→3 since there is no heat addition or combustion. At the end of expansion stroke 3→4, the pressure inside the flexible cavity is below atmospheric. When the solenoid valve opens, sudden inflow of ambient air into the flexible cavity takes place. As a result, the pressure inside the flexible cavity rises during the process 4→5. Next in the displacement process 5→6, the piston is pushed to top-dead-center (point 6) to expel the air from the flexible cavity. During the process 5→6, the pressure in the flexible cavity rises slightly and then falls at the end of the displacement process (point 6) due to the flow constriction in the solenoid valve. Finally, the motored engine undergoes an intake process 7→1, thereby marking the completion of an operating cycle 1→2→3→4→5→6→7→1.

Each operating cycle lasts for about 76.3 ms. The longest stroke recorded is 9.5 mm (corresponding to the displacement process 5→6).

In an ideal P-V diagram for a motored engine, the expansion process 3→4 should retrace the compression process 1→2, and extend beyond point 1, up to point 4. Similarly, the intake process 7→1 should retrace the displacement process 5→6. However, the experimental data shows a P-V loop enclosing an area, due to fluid flow friction, heat due to compressive heating, and enthalpy transfer from the system. The enthalpy transfer during the exhaust and intake processes includes the pumping work. The direction of the P-V loop is found to be anticlockwise indicating negative work by the system. The work done by the vibration shaker is expended in overcoming structural friction in flexible metal bellows and work done on the system. Experimentally, the work done by the vibration shaker is about 101.8mJ/cycle±17.7% while the work done on the system is about 72.8 mJ/cycle ±15.7%. Thus, the structural friction work by the flexible metal bellows is 29 mJ/cycle ±23.66%. This structural friction work corresponds to a damping coefficient of 1.5±0.35 N-s/m for a mean piston speed of 0.4 m/s. The structural friction work in the flexible metal bellows accounts for 28% of work done by the vibration shaker, while the work on the system accounts for 72%.

(Loss)mep-speed-frequency plot

Experimentally determined (loss)mep-speed-frequency plots are used to capture the effects of heat loss (or heat transfer) due to compressive heating and fluid flow friction on the engine performance. implies the presence of fluid flow friction in the crevices formed by bellows corrugations and becomes very significant at higher velocities. Thus, it is recommended to operate the engines at higher resonant frequencies to limit losses due to heat transfer.

81

In the four-stroke motoring experiment, the average frequency of operation is 13.1 Hz and mean piston speed is 0.4 m/s, thus the (loss)mep corresponding to heat loss due to compressive heating and fluid flow friction are 1240 Pa and 677 Pa respectively. This implies that heat transfer losses are 29% higher than the fluid flow friction.

Estimation of performance parameters

As previously mentioned in chapter four, the mathematical model is used in conjunction with motoring experiment data to evaluate the engine performance. In the model, the damping The values of damping coefficient b+b f for each thermodynamic process are tabulated in Table 5.1. The damping coefficient varies from process to process. From Table 5.1, the product of velocity, piston displacement, and damping coefficient gives the work done on the system.

Based on the sign, the compression process 1→2 and the intake process 7→1 are the processes where the work is done on the system, while the expansion process 3→4 and displacement process 5→6 correspond to the work done by the system. Summing the work done on the system from each process adds up to 110 mJ/cycle. b Numerically "Net enthalpy out" is negative of "Net enthalpy in".

c The PdVwork error of 15.7% is equivalent to the maximum error in Heat lost, Net enthalpy out, and Internal energy change. The end result of the motoring experiment is used in the subsequent sections to evaluate the engine performance for a fueled operation. The motoring technique presented here could also be extended to free-piston engines to estimate losses quantitatively.

Mathematical model

The comparison is made with a traditional IC engine.

Open cycle

The engine, shown schematically in Subsequently, the engine undergoes displacement process 5→6 to the minimum volume V 6 =0.22 cc, followed by intake process 6→7→1 until volume V 1 =87.8 cc. Finally, an adiabatic compression 1→2 takes place resulting in completion of one operating cycle 1→2→3→4→5→6→7→1. It can be seen from the figure that the stroke lengths are unequal 1 The minimum volume of the cylinder when the piston is at state 6 86 which makes the engine operation unique in comparison with traditional reciprocating engines.

The nondimensional terms for this reference engine at Φ=0.65 are C=2.5, H=0, Г=3488, Λ=0.274, and ζ=0.24.

To determine the work done per cycle for Φ=0.65, a numerical integration of the P-V diagram (Figure 5.6) for one operating cycle is performed and found to be 72.47 J. The corresponding indicated thermal efficiency is calculated to be 41.2%. In addition, corresponding compression and expansion ratios are found to be about 1.756 and 4.646, respectively.

Equivalence ratio and heat input relationship

For a specific fuel, an equivalence ratio is dependent on mass of fuel injected (or associated heat input) and mass of air inducted in to the engine. bringing it to abrupt stop. In a free-piston engine, such impulsive forces occur when the piston hits the cylinder head at the top-dead-center. The sudden piston stoppage phenomenon could be treated as an elastic or an inelastic collision. For a conservative estimation of engine efficiency and performance, the collision at state 6 is assumed to be inelastic in nature. The fact that beyond the inflection point, the mass of air intake during process 6→7→1 remains constant can be understood by analyzing the engine dynamics as discussed below.

In an inelastic collision, the piston or mass is assumed to lose all the kinetic energy irrevocably to the surroundings. As a result, the energy available with the spring mass damper system (bellows-piston combination) after the collision is only the stored spring energy at state 6.

Since there exists a lower bound on the engine volume (i.e. clearance volume V c ), the negative spring excursions are bounded ∆𝑉 6 = ∆𝑉 𝑐 . Hence, the stored spring energy at state 6 is a fixed value beyond the inflection point. Beyond the inflection point, the intake stroke length remains constant; and so the mass of inducted air is also constant. Therefore beyond the inflection point, the equivalence ratio depends solely on the heat input in a linear relationship.

The change in nonlinear to a linear relation is observed for loads b=20 and 25 N-s/m.

However, for load b=30 N-s/m such a trend is not observed in the range Φ<1.3. It is hypothesized that it occurs in the domain Φ>1.3.

A brief comparative study of elastic and inelastic collision modeling on the engine performance showed some interesting facts. The difference in indicated thermal efficiencies for Φ=0.6 is about 5% between the elastic and inelastic cases, and it widens with equivalence ratio Φ. For a fixed load (say b=20 N-s/m), the resonant frequency in elastic collision is higher than the inelastic. A similar trend is observed for b=25 N-s/m. The effect of an inelastic collision at top-dead-center did not affect the engine performance significantly. Supporting plots and detailed discussions are included in the Appendix A3: ELASTIC COLLISION.

Effects of equivalence ratio and load on engine performance

Using the nonlinear model equations [(3.7) -(3.11)], the effect of equivalence ratio Φ and load b on the engine is determined in terms of engine efficiency, volume ratios (CR and ER), operating frequency, and output power.

Efficiency and volume ratios

For a fixed load b=20 N-s/m, the effect of equivalence ratio on indicated thermal efficiency and volume ratios (CR and ER) for the reference engine are presented in Figure 5.8. From the figure, it can be seen that efficiency increases with equivalence ratio. The efficiency rises from 40.29% to 47.53% for an increase in equivalence ratio from Φ=0.6 to Φ=1.2. This indicates that it is desirable to operate the engine at higher equivalence ratios. Of course, in practice flammability limits impose the upper and lower bounds on equivalence ratio. From Figure 5.8, the efficiency holds a linear relation with equivalence ratio for smaller Φ (Φ<0.2), and the relation starts to asymptote for higher Φ (Φ>0.2). Similar findings have been reported for a closed cycle modeling of a resonant heat engine [START_REF] Preetham | Modeling of a resonant heat engine[END_REF].

On the secondary axis of the figure, corresponding compression and expansion ratios are plotted. For any given Φ, the expansion ratio is greater than the compression ratio, and the difference between them grows as Φ increases. It is evident from the figure that the efficiency is a strong function of expansion ratio compared to the compression ratio; in fact the compression ratio remains unchanged for Φ>0. [START_REF] Menon | Miniaturization Limits of Small IC Engines[END_REF]. The point where Φ=0.23 is the inflection point for the reference engine. The unaltered value of CR for Φ>0.23 can be understood by analyzing the engine dynamics during the processes 7→1 and 1→2. Since beyond the inflection point the intake stroke length for process 7→1 remains constant, the engine volume V 1 remains fixed. As the engine volume V 2 is purely dictated by the stored spring energy at state 1, its volume at state 2 is predetermined by volume V 1 . So beyond the engine's inflection point, since the volumes V 1 and V 2 are fixed, the compression ratio of the engine doesn't change. However, the expansion ratio increases with equivalence ratio (or heat input), as higher heat input gives rise to higher chamber pressure that pushes the piston further outward. The effect of load on the indicated thermal efficiency and expansion ratio is shown in Figure 5.9. Three loading conditions are shown b=20, 25, and 30 N-s/m. It can be observed that for 91 fixed Φ (say Φ=1.0), as the load applied increases from 20 N-s/m (ζ=0.240) to 25 N-s/m (ζ=0.296) the efficiency drops monotonically from 45.75% to 43.45%. The efficiency drops further down to 28.34% as load increases to 30 N-s/m (ζ=0.352). This reiterates that it is desirable to design an engine with as small damping ζ as possible to operate efficiently [START_REF] Bardaweel | MEMSbased resonant heat engine: Scaling analysis[END_REF][START_REF] Preetham | Modeling of a resonant heat engine[END_REF].

However, in practice the minimum damping is limited by the physically achievable expansion ratios. On the secondary axis of the figure, corresponding expansion ratios for the three cases of 20, 25, and 30 N-s/m are plotted. For any given Φ, the expansion ratio drops as the load applied increases. This is attributed to a decrease in the amplitude of piston oscillation. Similar such findings have been reported for a closed cycle modeling of a resonant heat engine [START_REF] Preetham | Modeling of a resonant heat engine[END_REF]. A practical volume ratio (ER) of 6 is obtained from the model for Φ=1.0 and b=20 N-s/m. Figure 5.9: Effect of load on indicated thermal efficiency and expansion ratio.

Thermodynamic states

In order to explore the effects of equivalence ratio Φ (or heat input) and load b on the thermodynamic states of an operating cycle, a P-V diagram is plotted. To capture the effect of equivalence ratio on the reference engine, P-V diagrams for various equivalence ratios are plotted in Figure 5.10 and Figure 5.11. Figure 5.10 is plotted for equivalence ratios Φ<0.23, i.e.

for Φ=0.12 and Φ=0.15, while Figure 5.11 is plotted for equivalence ratios Φ>0.23 i.e. for Φ=0.65 and Φ=1.0. These two figures together capture the effect of equivalence ratios on the thermodynamic states of the engine operating cycle 1→2→3→4→5→6→7→1. From the figures, increasing the equivalence ratio Φ results in higher cavity pressure and thereby an increase in amplitude of oscillation yielding higher expansion ratio. It can be seen from Figure 5.10 that the intake and compression stroke lengths increase with Φ for all Φ <0.23. However, in 94

The effect of load on the engine is studied for a fixed heat input of 39.25 J/cycle as shown in Figure 5.12. Upon increasing the load b from 20 N-s/m to 30 N-s/m the volume excursions decrease while the area enclosed by the P-V loop increases from 9.31 J/cycle to 12.1 J/cycle. The decreased volume excursions, especially the intake stroke length 7→1 results in reduced engine breathing capacity. Hence, less air gets ingested during the intake stroke 7→1. This results in an increase in the equivalence ratio from Φ=0.19 to Φ=1.09 since the heat input is fixed and the ingested air is decreased. To maintain a fixed equivalence ratio Φ for increasing load b, the heat input should be accordingly reduced to tally with the reduced air intake. The increased work output for a fixed heat input indicates the engine begins to operate more efficiently with higher load b; contrary to fixed equivalence ratio operating conditions. there is no inflection point in the equivalence ratio range plotted the behavior is different. Figure 5.13 suggests that the inflection point shifts towards the higher Φ for increasing load.

Output power

The engine output power is computed by only accounting for the energy dissipated from the useful damper b (also called load). Figure 5.14 shows the output power and the operating frequency of the engine for loads from 12 N-s/m to 30 N-s/m. The plots in the figure are generated for fixed equivalence ratios of Φ=0.6 and Φ=1.0. The plots show that a richer fuel air mixture generates higher output power from the engine. The difference in output power from rich and lean combustion (Φ) is larger at smaller loads compared to higher loads. It is worth mentioning that both the work output and the operating frequency determine the output power from the engine. From Figure 5.14, it is clear that the difference in frequencies for Φ=0.6 and Φ=1.0 are marginal. Therefore, the large difference in output power for Φ=0.6 and Φ=1.0 is due to useful work from the engine. The numerical values of useful work, operating frequency for each loading condition for Φ=1.0 are presented in Table 5.3. The decreasing output power with load is a result of large drop in work output (or useful work) from the engine. As both the heat input and engine efficiency decrease for a fixed Φ and increasing load, their product (i.e. work output) also follows the decreasing trend. The figure suggests that if Φ is fixed, smaller loads yield higher output power and vice-versa. In other words, to maintain a fixed output power (say 320 W) for increasing load (from b=25 N-s/m to 28 N-s/m), the engine must be supplied with richer air-fuel mixture (Φ=1.0 instead of Φ=0.6).

Fixed speed and fixed frequency operation

An engine's mean piston speed is proportional to stroke length and operating frequency. In a traditional reciprocating engine, the piston speed and the frequency of crankshaft revolution (RPM) are related by a proportionality constant (i.e. fixed stroke length) [START_REF] Heywood | Internal combustion engine fundamentals[END_REF]. Therefore, fixed speed conditions are same as fixed frequency (RPM) conditions. However, in the resonant engine studied here, the stroke length varies with load and equivalence ratio. Therefore, in the engine a fixed speed operating condition may be different from a fixed frequency condition.

The effect of load on equivalence ratio for fixed speed and fixed frequency operations are plotted in Figure 5.15. From the figure, for a fixed speed operation of 6 m/s, the engine demands a richer fuel-air mixture as the load increases. Similar behavior is observed for a fixed frequency operation of 13.6 Hz. Such trends are expected and show resemblance to traditional reciprocating engines. In a traditional engine, the fixed speed and fixed frequency conditions are identical as described above. However, in the proposed engine the fixed speed and fixed frequency conditions are different; a feature unique to the engine. This is very evident from the figure for higher loads (b>20 N-s/m). Interestingly for loads b<20 N-s/m, the fixed frequency and fixed speed conditions are found to be nearly identical. [START_REF] Pescara | Motor compressor apparatus[END_REF]. This is because the heat input is fixed. Increasing equivalence ratio implies decreasing mass of intake air in to the engine. As a consequence, the temperature at the end of the heat addition process 2→3 is higher for higher equivalence ratios. The higher temperature T 3 translates to higher pressure P 3 , thereby resulting in larger expansion ratios as evident in Figure 5.16. The intake stroke length for b p1 =0 N-s/m is found to be shorter than rest of the cases (b p1 <0), as a negative damping coefficient b p1 indicates an additional force that can push the piston further outward.

Effect of heat loss

Though heat loss from an engine has negative effects on engine efficiency, an optimal heat transfer is desired for proper operation [START_REF] Pulkrabek | Engineering Fundamentals of the Internal Combustion Engine[END_REF]. The effect of heat transfer 2 on the reference engine performance for fixed heat input E=138.55 J/cycle is shown in Figure 5.17. The steady state results show that for h=0.01, 0.1, and 3 W/K the corresponding equivalence ratios are 0.505, 0.504, and 0.504 respectively. For a fixed heat input E=138.55 J/cycle and an equivalence ratio 2 Heat transfer from the engine to the surroundings during the heat addition process 1→2 is neglected as it is assumed instantaneous.

102

Φ=0.50, the temperature T 3 at the end of heat addition process in each of the cases are equal. The heat transfer during the expansion process 3→4 is largest for h=3 W/K and is manifested as pressure drop in Figure 5.17. Since lower cavity pressure results in smaller strokes, the engine efficiencies are customary to be low. Additional information on the effect of heat loss on engine performance is presented in Table 5.5.

Figure 5.17: Effect of heat loss.

Closed cycle

The

Thermodynamic cycle

A relationship between the sealed cavity pressure and volume is shown in To determine the work done per cycle for Ψ=18.2, a numerical integration of the P-V diagram (Figure 5.18) for one thermodynamic cycle is performed and found to be 2.106 J. In addition, the CR and engine thermal efficiency for this condition are found to be 9.01 and 58.5% respectively.

Thus found thermal efficiency agrees with the Otto cycle efficiency for same given CR=9.01.

Effects of heat input and load

The effects of heat input and loading on the engine performance are predicted using the mathematical model. Predictions of efficiency and CR using both the nonlinear and linearized mathematical models are discussed in the following sub-section.

Efficiency and compression ratio

The effects of load and heat input on efficiency and compression ratio (CR) are presented in [START_REF] Jacobson | An informal survey of power MEMS[END_REF]. This indicates that it is desirable to design an engine with as small damping ζ as possible to operate efficiently [START_REF] Richards | A Resonant Air-Standard Heat Engine[END_REF]. However, in practice the minimum damping is limited by the physically achievable compression ratios.

For small heat inputs Ψ the nonlinear model shows behavior which agrees with the linear model (A2.13) and (A2.10). But as heat input Ψ increases the nonlinear behavior starts to become apparent. For higher heat inputs, the linear model over-predicts the efficiency which can be seen in the figure .   106 On the secondary axis of the figure, corresponding compression ratios for the loads b=0.7 Ns/m and 4.9 N-s/m are plotted. For a given Ψ the compression ratio drops as applied load increases. This is attributed to a decrease in the amplitude of piston oscillation which occurs with an increase in load. For a given load, increasing the heat input Ψ results in higher cavity pressure and thus an increase in the amplitude of piston oscillation yielding increased volume excursions or compression ratios. The compression ratio of a traditional IC engine is constrained by the crank radius; [START_REF] Heywood | Internal combustion engine fundamentals[END_REF] however, the compression ratio of this engine varies and depends on operating conditions like Ψ (or 𝛷) and b which is an interesting feature of the engine design. The compression ratios obtained from the model are CR=8 to 15 (for Ψ=18.2) and are practical for IC engines. For the closed cycle, it is noteworthy to mention that the CR and ER are numerically equal. The efficiencies and compression ratios for various loads with a fixed heat input of Ψ=18.2 are calculated and listed in Table 5.6. a Reference load on the engine.

Thermodynamic states

In order to explore the effects of load on the thermodynamic states of the cycle a P-V diagram for three different loads is shown in Figure 5.20. A log-log scale is chosen for presenting the thermodynamic cycle in a compact form. The plot with the solid line (for b=4.9 N-s/m) when drawn on linear scale resembles the plot in Figure 5.18. As the applied load b on the engine is increased the swept volume decreases, as pointed out above. For instance, the swept volume for b=4.9 N-s/m is 6.28 cc while for b=0.7 N-s/m it is 23.39 cc. The figure shows that the peak pressure increases with a decrease in the load for same heat input. However, the rise in pressure 108 due to heat input for all the three loads b=0.7, 1.75, and 4.9 N-s/m is about 1.84 MPa which is as expected, since the heat input is held constant for all the three loads.

Operating Frequency

The engine operating frequency depends on both the load and the heat input. that the operating frequency decreases for increasing Ψ. This means that the nonlinearity caused by the thermo-physics of the working fluid is softening in nature unlike other engines which contain an external hardening spring [START_REF] Formosa | Nonlinear dynamics analysis of a membrane Stirling engine: Starting and stable operation[END_REF]. The natural frequency of a harmonically forced Duffing oscillator is known to increase or decrease depending on whether a hardening or softening spring is present [START_REF] Nayfeh | Nonlinear oscillations[END_REF]. However it is acknowledged that the model equations [(3.29) -(3.32)] is 3 rd order and the engine is impulsively forced.

Calculations performed for load variations from 0.7 N-s/m to 7 N-s/m with fixed Ψ=18.2,

showed that the operating frequency initially increased from 8.29 Hz to 10.12 Hz, and then dropped to 4.86 Hz. The results are given in Table 5.6. This behavior suggests that there is a particular load b for optimal output power. A literature review on engine operating frequencies in existing free-piston engines showed that some engines operate at low frequencies of 8.5 Hz [START_REF] Hu | Mathematical modelling of a hydraulic free-piston engine considering hydraulic valve dynamics[END_REF] to 115 Hz [START_REF] Annen | Linearly oscillating miniature internal combustion engine (MICE) for portable electric power[END_REF]. The proposed engine when modeled in closed cycle fits within the range of these frequencies.

The engine operating frequency could be tuned by varying the mass and or stiffness of flexible sealed cavity. The engine operating frequency can also be controlled by stopping the piston movement at bottom-dead-center as implemented in the CHIRON Free Piston Engine [START_REF] Achten | Horsepower with Brains: The Design of the CHIRON Free Piston Engine[END_REF]. Exploiting this idea, it is possible to stop the piston at top-dead-center to increase the dwell time, in case of inadequate dwell time at top-dead-center.

Time for compression and expansion stroke

As mentioned earlier, the specification for the heat-rate q(t) allows for asymmetry in the time of expansion and compression. A plot of instantaneous piston speed against time for one cycle, for various loads is shown in Figure 5.22. From the figure for load b=0.7 N-s/m, upon heat addition at t=0 s the piston accelerates. A maximum speed of about 10.66 m/s is reached before the piston comes to rest at the end of expansion stroke which occurs at t=0.052 s. After the end of heat rejection process the piston accelerates towards the top-dead-center and finally reaches zero velocity. This particular process takes approximately 0.0686 s which is longer than the expansion time. It is noteworthy to mention that the time of expansion and compression become equal for small heat inputs [START_REF] Richards | A Resonant Air-Standard Heat Engine[END_REF].

Maximizing output power and power density

Output power from the engine for the above presented cases is computed assuming that the only dissipation mechanism present is the energy conversion damping b (also called load). From Figure 5.23, it is clear that the output power steadily rises with an increase in amount of fuel added per cycle Ψ (or Φ). In addition, the output power depends on the load b for a given heat input Ψ. To find the maximum output power, the load b is varied and the heat input Ψ is held constant at Ψ=18.2. The results presented in Figure 5.24 show that the maximum output power occurs at load b=1.75 N-s/m while the maximum power density occurs at load b=4.9 N-s/m.

Comparison of open and closed cycle analyses

In the previous sections, the open and closed cycles are simulated for two different engine sizes. Here a common size is chosen to compare the results from both the cycles. The engine configuration and ambient conditions are same as prescribed in section 5.3.1 Open cycle.

Overall engine performance in presence of friction and heat loss

The open cycle models a more realistic engine behavior compared to closed cycle, and highlights the advantages in the complaint engine design. For instance, if the reference engine operates in a lean condition Φ=0.5 with real engine parameters (h=0.1 W/K, b p1 =-40 N-s/m, and b f =1.5 N-s/m), then the corresponding ratio of friction work to indicated work is about 8% (Table 5.5). This value is found to be less than (<10%) the friction losses in traditional reciprocating engines with a piston-cylinder assembly [START_REF] Heywood | Internal combustion engine fundamentals[END_REF]. To ascertain the feasibility of a practical compliant engine, super compliant structures fabricated from metals and, or composites that can sustain extreme pressures and temperatures are evaluated by performing back of the envelope design calculations. The design calculations showed that it is possible to fabricate a 1 cc compliant engine from metals and composites.

In addition, a prototype of the compliant engine is realized using readily available off-theshelf flexible metal bellows to practically demonstrate the compliant engine concept. The engine is tested in a motoring test station to estimate the parasitic losses at resonant operation. In the motoring test, instead of using an electric motor to actuate the piston, a vibration shaker is used to drive the engine. A (loss)mep-speed-frequency surface plot is obtained and a nonlinear regression fit is performed to understand the energy distribution among various parasitic elements. The motoring results are presented in the form of an energy flow diagram. The diagram revealed that friction losses associated with flexible cavity is 28% of total motoring work; 22% less than the sliding friction in piston-cylinder assembly. Therefore, the frictional losses can be drastically reduced by adopting a compliant design.

Coupling the motoring test results with the mathematical model showed that the ratio of friction work to indicated work is about 8%, which is smaller than friction loss in traditional reciprocating engines. This result reinforces that a compliant resonant engine has the potential to outperform the contemporary engines at small scale. It also encourages us to pursue further developmental studies to materialize the concept.

The results of this study reinforce the idea that a compliant engine design can be a plausible solution to mitigate frictional and leakage losses at small length scales. This small scale engine bridges the existing gap between the micro and macroscale engine domains. To the best of our knowledge, this sort of comprehensive work on a small scale compliant engine is first of its kind.

Though HCCI is one preferred mode of combustion at small scale, the proposed engine is not modeled on HCCI. As a next step towards demonstrating the engine feasibility, an HCCI mode modeling is suggested. A linear-system transfer function approach is then used to interpret the experimental data. In this approach, the input to the system is taken to be the heat rate input into the engine, q(t), computed from the measured heater voltage using q(t)=V h 2 (t)/R h , where R h is the resistance across the heater. The output from the system is taken to be the velocity of the expander membrane u(t), computed from the measured displacement of the membrane, 𝑢(𝑡) = 𝑥(𝑡) ̇. .

Mathematical modeling of the micro heat engine: review

A lumped-parameter model of the micro heat engine is developed and validated against measured data. A brief description of the model is presented. Details on model development and validation can be found in [START_REF] Bardaweel | Cyclic operation of a MEMS-based resonant micro heat engine: Experiment and model[END_REF]. The lumped-parameter model of the engine is obtained by applying conservation laws to the vapor bubble and liquid film above the evaporator, and Newton's law of the motion for the expander and evaporator membranes.

                                                                         l g g o o o o o h i o o o p o o fg o T o o f T T p V V T V RT V s r T V T c V h C r m m r b b 0 0 1 0 0 0 0 0 0 0 0 0 0 ) ( 0 0 0 0 1 2 2 2 2 2 2         ) ( 0 0 1 0 0 2 2 2 1 2 0 0 0 0 0 0 2 2 2 1 2 0 0 0 0 1 0 0 0 0 1 0 2 2 t q h T T p V V R M BS T RS R M T R M S T C U R M BS h U T RS R M T R M S r s fg o l g g u o o u o u o o P o B u o fg o c o u o u o o                                                                                                    (A1.3)
where V is the volume change of the vapor bubble, M is the molecular weight of the vapor, and m is the mechanical inertia caused by fluid movements. The universal gas constant is given by R u =8.31 J/mol-K, R=R u /M is the mass-specific gas constant, and h fg is the latent heat of the liquid. The area of the bottom surface of the vapor bubble is given by S=r i 2 . U B is the overall heat transfer coefficient intended to account for heat transfer from the bubble to its surroundings and c U is the coefficient intended to account for heat transfer from the evaporator to its immediate surroundings. The coefficient C T models thermal inertia of the engine. Here, the timeaveraged power  generated by the damper b can be computed with

2 ) ( 2 1     k k k f Q b (A1.4)
The time-averaged power  delivered to the load b is proportional to the square of the velocity amplitude, i.e. U k =Q k (f k ), summed over the frequency spectrum. Thus, for given load b, higher velocity amplitudes are desirable.

Scaling model

In scaling analysis, the following nondimensional scales are used.

Results

To investigate the scaling issues related to the micro heat engine, a set of experiments are performed. In these experiments the expander membrane material and thickness, and the thickness of engine cavity are varied. In the experiments, the side lengths of the expander and evaporator membranes are fixed at 10 mm and 8 mm respectively. The engine cavity is filled with a two phase-mixture of 3M  PF5060DL working fluid. The size of the vapor bubble is approximately 9 mm. The wicks are 5 μm thick, 10 μm high, and spaced 90 μm apart. An "added mass" of approximately, 4.00 and 0.56 grams are placed on top of the expander membranes.

To operate the micro heat engine, periodic unipolar heat pulses of durations of T=100, 10 and Table A1.1 lists the specifications of the micro heat engines used in scaling analysis. In Table A1.1, engine I is used as the nominal case for engine performance comparison. For this engine, the engine cavity was 450 µm thick and the expander membrane was a 2.2 µm thick silicon membrane. The parameters obtained from model fitting for engines I, II, and III are given in Table A1.3.

Table A1.3 suggests that engine I suffered from large thermal storage C T . In addition, the obtained parasitic heat loss coefficients from the bubble, U B , are negligible. Figure A1-6 reveals that velocity amplitude of the engine at the resonance frequency V max increased from 0.6 mm/s/W to 1.75 mm/s/W when the thickness of the engine cavity is reduced from 450 µm (engine I) to 150 µm (engine II) while maintaining the same expander membrane, i.e. 2.2 µm thick silicon membrane. Also, the velocity amplitude increased from 0.6 mm/s/W to 11 mm/s/W when the expander membrane is changed from 2.2 µm thick silicon membrane (engine I) to 300 nm silicon nitride expander membrane (engine III). In addition, the resonant frequency of the micro heat engine shifted from 180 Hz to 110 Hz for engine I and engine III respectively. It becomes clear that changing one specific engine component, like expander membrane or cavity thickness, results in a complex combination of engine parameters. That is the performance of the engine is a consequence of coupling between dynamics, thermodynamics, and heat transfer within the engine. Thus, it becomes crucial to use scaling analysis to understand the performance of the engine.

A nondimensional representation of the collected data is shown in Figure A1-7. In Figure A1-7, the performance of the engines in Table A1.1 is compared based on the maximum velocity amplitudes V max appearing in the transfer functions of the engines. In Figure A1-7 the ratio of maximum velocities, V max / V I is plotted on the vertical axis. The nominal velocity, V I is defined as the maximum velocity amplitude V max of engine I in Table A1. V I =18.33. The effect of time-geometry factor  and nondimensional thermal inertia C is evident.

An engine with higher time-geometry factor  and lower nondimensional thermal inertia C is desirable. respectively. Increasing the effective mass of the engine or reducing the stiffness of the expander membrane will reduce the reference frequency ω, and thereby increase the time-geometry factor . Reducing the reference frequency of the engine may be achieved by adapting more compliant structures or larger engines. For example, although engine II and engine III, both had the same added mass, i.e. m added = 4.0 grams, their resonant frequencies shifted from 170 Hz for engine II to 110 Hz for engine III. This is, presumably, due to reduction in stiffness of the top expander membrane. That is, the 300 nm thick silicon nitride membrane used in engine III is more compliant than the 2.2 µm thick silicon membrane used in engine II. Interestingly, even though very small amount of mass of 0.5 g is added to the engine IV, the resonant frequency is lesser than engines II and III, which is attributed to the low stiffness of PDMS membranes. The timegeometry factor and the velocity amplitude of engine IV are found to be 1.77 and 28.47 respectively. The high velocity amplitude of engine IV suggests lesser damping compared to engine I.

Another important scaling issue to consider here is the ratio of evaporation-area to engine volume S/V o . Increasing the ratio S/V o increases time-geometry factor  which unfavorably increases heat transfer from the engine. Therefore it is desirable to keep the ratio of evaporationarea to engine volume S/V o constant or even decrease it. It is possible to scale up the engine while keeping the ratio S/V o constant by increasing the side length of the membrane and maintaining the thickness of the engine cavity, h.

Laser Doppler Vibrometer

The displacement error information (Displacement Decoder DD-200) is obtained from the manufacturer's manual for the measurement range 5120 μm/V.

Resolution: 5120 nm Total error: ±5120 nm

Repeatability data from motoring experiment

The following data sets are obtained by repeating the motoring experiment at same conditions.

8 :Figure 1 . 1 :

 811 Figure 1.1: Ragone chart of various power delivering sources [7]. ................................................

Figure 1 . 2 :

 12 Figure 1.2: Thermal efficiencies of various IC engines [7]. ...........................................................

Figure 1 . 3 :

 13 Figure 1.3: Dissipative pathways in a four cylinder spark-ignition engine from motoring

Figure 1 . 4 : 11 Figure 1 . 5 :

 141115 Figure 1.4: Heat transfer rate in an SI engine as a function of speed and load [36]. .....................11

Figure 1 . 6 :

 16 Figure 1.6: Cross-sectional view of a prototype engine [21]. .......................................................

Figure 1 .

 1 Figure 1.7: 3-D CAD model of MEMS valveless pulsejet engine [22]. .......................................

Figure 1 . 8 :

 18 Figure 1.8: The design of a MICE Generator [2]. .........................................................................

Figure 1 . 9 :

 19 Figure 1.9: A Pro/ENGINEER drawing of a free elastic-piston compressor [56]. .......................

Figure 1 .

 1 Figure 1.10: A prototype of four-stroke free-piston engine [57]. .................................................

Figure 2 . 1 :

 21 Figure 2.1: P-V diagram for an ideal cycle. ..................................................................................

Figure 2 . 2 :

 22 Figure 2.2: Assembled composite laminate (top view). ................................................................

Figure 2 . 3 :

 23 Figure 2.3: Assembled compliant engine in compressed and expanded states. ............................

Figure 3 . 1 :

 31 Figure 3.1: Schematic sketch of the compliant resonant engine. ..................................................

Figure 3 . 2 :

 32 Figure 3.2: Heat-rate function q(t) for modeling the open cycle. .................................................

Figure 3 . 3 :

 33 Figure 3.3: Steady state open cycle P-V diagram. ........................................................................

Figure 3 . 4 :

 34 Figure 3.4: Transient state open cycle P-V diagram. ....................................................................

Figure 3 . 5 :

 35 Figure 3.5: Heat-rate function q(t) for modeling the closed cycle. ...............................................

Figure 3 . 6 :

 36 Figure 3.6: Steady state closed cycle P-V diagram. ......................................................................

Figure 3 . 7 :Figure 4 . 1 :

 3741 Figure 3.7: Transient state closed cycle P-V diagram. ..................................................................

Figure 4 . 2 :

 42 Figure 4.2: Schematic sketch of the bellows with dimensions. ....................................................

Figure 4 . 3 :

 43 Figure 4.3: Stiffness estimation of edge welded metal bellows. ...................................................

Figure 4 . 4 :

 44 Figure 4.4: Pressure transducer calibration curve. ........................................................................

Figure 4 . 5 :

 45 Figure 4.5: Force sensor calibration curve. ...................................................................................

Figure 4 . 6 :

 46 Figure 4.6: Solenoid valve amplification circuit...........................................................................

Figure 4 . 7 :

 47 Figure 4.7: Prototype engine in the test set-up for four-stroke motoring. ....................................

Figure 4 . 8 :

 48 Figure 4.8: Schematic sketch of assembled prototype engine for four-stroke motoring test. ......

Figure 4 . 9 :

 49 Figure 4.9: Experimental setup for generating frequency response plots. ....................................

Figure 4 . 10 :

 410 Figure 4.10: Sketch of the four-stroke motoring test set-up. ........................................................

Figure 4 . 11 :

 411 Figure 4.11: Signals generated for four-stroke motoring experiment. ..........................................

Figure 5 . 1 :

 51 Figure 5.1: Generated frequency response plots. ..........................................................................

Figure 5 . 2 :

 52 Figure 5.2: Four-stroke P-V diagram obtained from motoring experiment and model. ...............

Figure 5 . 3 :

 53 Figure 5.3: (loss)mep-speed plots for various frequencies of operation. ......................................

Figure 5 . 4 :

 54 Figure 5.4: Surface fit of (loss)mep-speed-frequency data. ..........................................................

Figure 5 . 5 :

 55 Figure 5.5: Energy flow diagram for the motored engine. ...........................................................

Figure 5 . 6 :

 56 Figure 5.6: Open cycle P-V diagram for heat input E=176 J/cycle (Φ=0.65). .............................

Figure 5 . 7 :

 57 Figure 5.7: Relation between heat input and equivalence ratio for various loads. .......................

Figure 5 . 8 :

 58 Figure 5.8: Effect of equivalence ratio on indicated thermal efficiency and volume ratios for the

Figure 5 . 9 :

 59 Figure 5.9: Effect of load on indicated thermal efficiency and expansion ratio. ..........................

Figure 5 . 10 :Figure 5 . 11 :Figure 5 . 12 :Figure 5 . 13 :Figure 5 . 14 :Figure 5 . 15 :Figure 5 . 16 :Figure 5 . 17 :Figure 5 .Figure 5 . 19 :Figure 5 .Figure 5 . 21 :Figure 5 . 22 :Figure 5 . 23 :Figure 5 . 24 :Figure 5 . 25 :Figure 5 . 26 :Figure 5 . 27 :Figure 5 . 28 :

 51051151251351451551651755195521522523524525526527528 Figure 5.10: Open cycle P-V diagrams for fixed load b=20 N-s/m and equivalence ratios Φ= 0.12

Figure 1 . 2 :

 12 Figure 1.2: Thermal efficiencies of various IC engines [7].

Figure 1 . 3 :

 13 Figure 1.3: Dissipative pathways in a four cylinder spark-ignition engine from motoring

Figure 1 . 5 :

 15 Figure 1.5: The MEMS IC engine [10].

Figure 1 . 6 :

 16 Figure 1.6: Cross-sectional view of a prototype engine [21].

Figure 1 .

 1 Figure 1.7: 3-D CAD model of MEMS valveless pulsejet engine [22].

Figure 1 . 8 :

 18 Figure 1.8: The design of a MICE Generator [2].

  Figure 2.1 shows a pressure-volume (P-V) diagram of an ideal, part throttled operating cycle 1→2→3→4→5→6→7→1.

Figure 2 . 1 :

 21 Figure 2.1: P-V diagram for an ideal cycle.

Figure 2 . 2 :

 22 Figure 2.2: Assembled composite laminate (top view).

Figure 2 . 3 :

 23 Figure 2.3: Assembled compliant engine in compressed and expanded states.

Chapter 3 3. 1 Introduction

 31 Reciprocating engines are modeled as both open and closed cycles. In the open cycle, exchange of heat and mass occurs with the surroundings, while in the closed cycle only exchange of heat takes place. In reciprocating engines with fixed stroke length (i.e. traditional IC engines), the piston dynamics during the intake and exhaust strokes are not very different, thus modeling as a closed cycle (without intake and exhaust strokes) appears reasonable. But for an engine with varying stroke lengths this approach is questionable. Usually as a first step, for simplicity reasons, IC engines are modeled as a closed cycle. However, in this study the engine is modeled as an open cycle. A real engine operates in an open cycle therefore, modeling the engine in an open cycle is appropriate. For completeness, a mathematical model for the closed cycle is also presented towards the end of the chapter.

  .1. The heat addition to the engine normally associated with combustion is modeled by the heat-rate q(t) into the flexible cavity of nominal volume V o at the left end. The heat rejection from the engine takes place through the valve by mass transfer during the exhaust process. The working fluid of the engine is considered to be air trapped in the cavity at the beginning of compression stroke. The pressure inside the cavity represented by bellows corrugations oscillates about standard atmospheric pressure P o . The deformable corrugations of total stiffness s allow horizontal displacement x(t) of the right end of the volume, modeled to have a lump-parameter inertia m. When the displacement of the mass m is zero, the volume V o is V o =LS, where L is the nominal length and S is the cross-sectional area of the engine cavity or chamber. The movement of the mass m is impeded by the dampers b f and b. These dissipative elements model energy loss to friction and energy converted to useful work respectively. In addition, a damper with negative damping coefficient b p1 (b p1 <0) is incorporated to model the work done on the engine during the intake stroke. For a steady state operation, the air temperature T o +T(t), pressure P o +P(t), density  o +(t), and engine volume V o +V(t) undergo cyclic variation, where the ambient and time-varying cyclic components are denoted with the "o" subscript and , respectively.

Figure 3 . 1 :

 31 Figure 3.1: Schematic sketch of the compliant resonant engine.

)

  𝑀 ̇𝑖,𝑒 = -𝑘𝛥𝑃 . (3.5) In the equation set [(3.1) -(3.5)], c v is the constant volume heat capacity of air and is assumed constant over the temperature range, h is a coefficient that models conduction/convection heat losses from the enclosed air volume to the outside environment, k is the proportionality constant that relates the mass-flow rate of working fluid and pressure drop across the valve, R is the mass-specific gas constant of air, e i and e e are the specific enthalpies of the air entering and exiting the engine, and 𝑀 ̇𝑖and 𝑀 ̇𝑒are the rates of mass of air entering and exiting the engine respectively. The damping coefficient b p1 is zero for cases where no external work is done on the engine during the intake process, and/or for other processes in the engine's operating cycle. If the heat added per cycle and initial conditions T(0), P(0), and V(0) are specified, equations [(3.1) -(3.5)] constitutes a nonlinear model for the determination of the intermediate thermodynamic state variables T(t), P(t), (t), and V(t).

Figure 3 . 2 :

 32 Figure 3.2: Heat-rate function q(t) for modeling the open cycle.

  ) and M, 𝑀 ̇𝑖,𝑒 are given by equations (3.9) and(3.11) in terms of state variables ∆𝑉, ∆𝑉 ̇, ∆𝑇, and ∆𝑃.To characterize the pressure-volume (P-V) behavior of the engine, the model equations [(3.7) -(3.11)] are integrated numerically. A steady state P-V behavior of the engine takes the form shown in Figure3.3. The thermodynamic states 1→2→3→4→5→6→7 on the diagram correspond to the intake valve closing (IC) and beginning of the compression stroke, end of the compression stroke and beginning of impulsive heat addition (IGO), end of impulsive heat addition (IGC) and beginning of expansion stroke, the end of the expansion stroke and beginning

Figure 3 . 3 :

 33 Figure 3.3: Steady state open cycle P-V diagram.

  q H that would result in a steady-state periodic cycle. Once the objective function satisfies a specified tolerance the computation stops, the cycle closes (i.e. ∆𝑉 1 = ∆𝑉 1 ′ , ∆𝑇 1 = ∆𝑇 1 ′, and ∆𝑃 1 = ∆𝑃 1 ′ ), and the engine begins to operate in a steady state.

Figure 3 . 4 :

 34 Figure 3.4: Transient state open cycle P-V diagram.

  four-stroke mathematical model developed for the open cycle in the previous section is simplified to model a closed cycle operation. The lumped parameters defined for open cycle are applied to model the closed cycle. Since no mass transfer can occur in the closed cycle operation, the valve shown in Figure 3.1 is sealed. Therefore, the flexible cavity is modeled as sealed flexible cavity. The heat transfer to and from the engine normally associated with combustion and exhaust is modeled by the heatrate q(t) in and out of the sealed flexible cavity of volume V o at the left end. Just like the open cycle, the pressure inside the cavity oscillates about standard atmospheric pressure P o . Contrary to open cycle, the negative damping coefficient b p1 (b p1 <0) that models the work done on the engine during the intake stroke is set to zero, as intake and exhaust processes does not exists (in true sense "ignored") here. The mathematical statements [(3.1) -(3.5)] for the closed cycle

Figure 3 . 5 𝑞

 35 Figure 3.5

  (3.6 a-f) is adopted. After application of these scales, the nonlinear model [(3.23) -(3.26)] becomes terms C, H, Γ, 𝜁, and Λ are defined by equation (3.12 a-f). For analysis, the equations [(3.29) -(3.32)] are written in state space format, the system equations [(3.29) -(3.32)] then become 𝑑∆𝑉 𝑑𝑡 = ∆𝑉 , ̇

Figure 3 . 6 .

 36 Thermodynamic states 1→2→3→4 on the diagram correspond to the end of impulsive cooling and beginning of the compression stroke, end of the compression stroke and beginning of the impulsive heating, end of impulsive heating and beginning of expansion stroke, and the end of the expansion stroke respectively.

Figure 3 . 6 :

 36 Figure 3.6: Steady state closed cycle P-V diagram.

  (i.e. ∆𝑉 2 = ∆𝑉 2 ′ and ∆𝑇 2 = ∆𝑇 2 ′ ), and the engine begins to operate in a steady state. A flow chart diagram highlighting major computational steps involved in solving the nonlinear lumped-parameter model for the closed cycle operation is presented Appendix B2: CLOSED CYCLE COMPUTATION ALGORITHM. A MATLAB code is written by defining an objective function to solve the nonlinear model to arrive at a steady state solution.

Figure 3 . 7 :

 37 Figure 3.7: Transient state closed cycle P-V diagram.

  If, in addition, parasitic heat losses are neglected by setting H=0, it is shown in the Appendix A2: ANALYSIS OF LINEARIZED MODEL that the model equations [(3.33) -(3.35)] can be approximated by ∆𝑉 ̈+ 2𝜁𝛥𝑉 ̇+ (1 + 𝛬)𝛥𝑉 = 𝛹 𝛾 𝐸 , (A2.5)

  Figure 4.1 shows the bellows used in the motoring experiment. The inner and outer diameters of the bellows are 35.2 mm and 48.0 mm respectively. Its free length is 20.3 ± 3.0 mm. The operating extended and compressed lengths of the bellows are 26.4 mm and 4.8 mm respectively. Therefore, its operating stroke length is 21.6 mm. These bellows are designed to tolerate an internal pressure of 0.31 MPa. A schematic sketch of the metal bellows with dimensions is presented in Figure 4.2.

Figure 4 . 1 :

 41 Figure 4.1: Edge welded metal bellows.

Figure 4 . 2 :

 42 Figure 4.2: Schematic sketch of the bellows with dimensions.

Figure 4 . 3 :

 43 Figure 4.3: Stiffness estimation of edge welded metal bellows.

  Figure 4.4 shows the calibration equation of the pressure transducer.

Figure 4 . 4 :

 44 Figure 4.4: Pressure transducer calibration curve.

Figure

4. 5 .

 5

Figure 4 . 5 :

 45 Figure 4.5: Force sensor calibration curve.

  Figure 4.6 shows an amplifier circuit designed using diodes, transistors, and resistors. Diode 1N4007, Transistors NPN 2N3773 and PNP 2N6609 are used. In the figure, the collector terminal of the PNP 2N6609 transistor is connected to a 120 V DC power supply. The unamplified square pulse or signal is supplied to the Diode 1N4007. The amplified square pulse or signal is obtained at the emitter terminal of the PNP 2N6609 transistor where the solenoid valve is connected.

Figure 4 . 6 :

 46 Figure 4.6: Solenoid valve amplification circuit.

Figure 4 . 7 :

 47 Figure 4.7: Prototype engine in the test set-up for four-stroke motoring.

Figure 4 . 8 :

 48 Figure 4.8: Schematic sketch of assembled prototype engine for four-stroke motoring test.

4. 3 . 2

 32 Frequency response plot As the engine is motored for four-stroke operation, the engine encounters open and closed condition operations. During the intake and exhaust processes, the solenoid valve remains open and the piston shuttles back and forth thus forcing the fluid flow in and out of the engine, hence an open condition operation. During compression and expansion strokes, the solenoid valve remains closed as the piston moves back and forth, hence the closed condition operation. Due to the presence of Helmholtz stiffness of the air (or working fluid) in the closed condition, the effective stiffness in the closed condition is higher than in the open condition. As a result the resonant frequencies of the closed (f c ) and open (f o ) conditions are different. Hence, frequency response plots for the open and closed conditions are generated separately to obtain their natural frequencies f o and f c by treating the engine as spring mass damper system.

Figure 4 . 9 :

 49 Figure 4.9: Experimental setup for generating frequency response plots.

  point displacement amplitudes. A Polytec laser vibrometer is used to measure the displacement amplitudes optically. Frequency response plots made for open and closed conditions are used to identify the natural frequencies f o and f c. These plots can then be used to calculate quality factors for both open and closed conditions to estimate the parasitic losses. The engine resonant frequencies can be tuned by varying the mass and or stiffness of the closed flexible cavity formed by metal bellows.

Figure 4 . 10 :

 410 Figure 4.10: Sketch of the four-stroke motoring test set-up.

Figure 4 .

 4 Figure 4.11 shows the generated waveforms supplied to the vibration shaker and solenoid valve from the computer. A graphical program in LabVIEW ® is written to generate the signals shown

Figure 4 . 11 :

 411 Figure 4.11: Signals generated for four-stroke motoring experiment.

  hence the numerical values of the damping coefficient b+b f may be different. The proportionality constant k in equation(3.5) is experimentally determined by measuring the mass-flow rate and pressure drop across the solenoid valve. Its value is estimated to be about k= 6.8 × 10 -7 kg/s-Pa ± 20%. This mass flow-rate model is found to be accurate for flow rates up to 0.0005 kg/s encountered in the motoring experiment. The compressibility effects are neglected as the fluid flow is found to be incompressible, since Mach number=0.1. The relevant calculation is presented in Appendix D1: CALCULATION OF MACH NUMBER FOR THE FLUID FLOWING IN THE ENGINE. The experimentally obtained volume and pressure excursions at zero piston velocity points (𝑉 ̇=0), and initial conditions T(0), P(0), ρ(0), and V(0) are specified to solve the nonlinear model equation set [(3.1) -(3.5)] for determining the intermediate thermodynamic state variables T(t), P(t), and V(t). In this regard, the coefficients b+b f and h are chosen such that a steady state operation is predicted by equations [(3.1) -(3.6 a-f)] during the motoring test that matches most closely with the experimental measurements. Improper choice of the floating parameters (b+b f and h) would result in unsteady state motoring simulation. Hence, a function minimization computational procedure is adopted by defining an objective function which is the difference between the values obtained from experiment and nonlinear model. Upon minimizing the objective function the model reaches a steady state.

  .1. From the figure, the resonant frequencies in the open and closed conditions are found to be f o =19.2±0.1 Hz and f c =41.4±0.1 Hz with quality factors of 19.68 and 29.67 respectively. As pointed out earlier, the closed condition resonant frequency is higher than the open condition due to additional Helmholtz stiffness s h =11.1 kN/m of encapsulated air in the flexible metal bellows. The quality factor in the open condition is lower than the closed condition indicating higher parasitic losses during open condition operation. These parasitic losses are due to fluid flow friction, heat lost due to compressive heating, pumping losses etc.

Figure 5 . 1 :

 51 Figure 5.1: Generated frequency response plots.

Figure 4 . 7 and

 47 Figure 4.10. The engine is motored in the closed condition for duration of 24.1 ms and in the open condition for 52.2 ms. Thus, the duration of one cycle is about 76.3 ms.

Figure 5 . 2 :

 52 Figure 5.2: Four-stroke P-V diagram obtained from motoring experiment and model.

Figure 5 .

 5 3 shows (loss)mep-speed plots generated for various frequencies ranging from 10 Hz to 60 Hz. From Figure 5.3 for any fixed frequency, the (loss)mep increases with mean piston speed. This rise in (loss)mep is due solely to the increase in fluid flow friction as the heat transfer rate remains constant. For a fixed mean piston speed, the (loss)mep value drops as the frequency of operation increases. This drop captures the heat transfer effect as the fluid flow friction remains constant.

Figure 5 . 3 :

 53 Figure 5.3: (loss)mep-speed plots for various frequencies of operation.

Figure 5 . 4 :

 54 Figure 5.4: Surface fit of (loss)mep-speed-frequency data.

  Figure 5.2.

Figure 5 . 5 :

 55 Figure 5.5: Energy flow diagram for the motored engine.

  nonlinear lumped-parameter model presented in chapter three is used to simulate a fueled operation for both open and closed cycles. The open and closed cycles are simulated for two different engine sizes. First, the open cycle results are presented, followed by the closed cycle. Later, a common engine size is chosen to compare the results from both the open and closed cycles. At last, results for an open cycle with heat and friction losses are presented and a

Figure 3 . 1 , 5 . 3 . 1 . 1

 315311 with h=0 W/K, b f =1.5 N-s/m, k=6.8×10-7 kg/s-Pa ± 20%, V o =21.8 cc (2.08 cm in diameter and 1.6 cm of nominal length), s=3050 N/m, b=20 N-s/m, b p1 =-40 N-s/m, and m=0.18 kg is treated as the reference engine. The numerical values of frictional damping coefficient b f , mass-flow rate coefficient k, nominal volume V o , stiffness s, and mass m belong to the prototype engine that were experimentally determined in the section 5.2 Estimation of parasitic losses. The clearance volume 1 V c is chosen to be 0.01 percent of nominal volume V o . Standard temperature and pressure conditions of T o =295 K and P o =93.4 kPa are chosen for the ambient state and for the thermodynamic properties of air. The properties of air used are R=286.9 J/kg-K, ρ=1.09 kg/m 3 , c v =717.25 J/kg-K, and c=346 m/s. The nonlinear model equations [(3.7) -(3.11)] are used to predict the engine performance and behavior. Operating cycle A relationship between the flexible cavity pressure and volume for four-stroke operation is shown in Figure 5.6. It is obtained using the nonlinear model equations [(3.7) -(3.11)] at steady state operation by impulsively supplying a heat E=176 J/cycle at the end of the compression stroke 1→2 to the reference engine. The heat of E=176 J/cycle supplied corresponds to lean combustion (equivalence ratio Φ=0.65) of octane in the engine. The figure shows that the pressure inside the flexible cavity at the end of heat addition process (i.e. at V 2 =50.1 cc) rises to ~1.6 MPa. Following the heat addition process 2→3, the engine expands adiabatically 3→4 to about ten times of its nominal volume, which is 232.7 cc. Next, the blowdown process 4→5 takes place resulting in instantaneous drop in pressure to ambient state P 5 =93.4 kPa.

Figure 5 . 6 :

 56 Figure 5.6: Open cycle P-V diagram for heat input E=176 J/cycle (Φ=0.65).

Figure 5 .

 5 7 graphically shows the relation between heat input and corresponding equivalence ratio for various loads b=20, 25, and 30 Ns/m. Each plot in the figure is obtained by supplying the engine with varying heat and corresponding equivalence ratios are calculated based on the mass of air inducted during the intake process 6→7→1. It becomes clear from the figure that the relationship between the equivalence ratio and heat input is dependent on the load b on the engine.

Figure 5 . 7 :

 57 Figure 5.7: Relation between heat input and equivalence ratio for various loads.

For a fixed

  load (say b=20 N-s/m), the equivalence ratio holds a nonlinear relationship for smaller heat inputs and a linear relation for higher heat inputs. The point where the nonlinear relation transmutes to linear is defined as the inflection point (represented by an open circle symbol). Beyond the inflection point, the mass of air intake during intake process 6→7→1 remains constant; as a result, the equivalence ratio is linearly proportional to heat input. The inflection point is indicative of the first occurrence of engine's volume V 6 becoming equal to the clearance volume V c . Such an occurrence results in an impulsive force on the piston at state 6

Figure 5 . 8 :

 58 Figure 5.8: Effect of equivalence ratio on indicated thermal efficiency and volume ratios for

Figure 5 .

 5 Figure 5.11 i.e. for Φ>0.23, the intake and compression stroke lengths remain constant for

Figure 5 . 10 :

 510 Figure 5.10: Open cycle P-V diagrams for fixed load b=20 N-s/m and equivalence ratios Φ=

Figure 5 . 11 :

 511 Figure 5.11: Open cycle P-V diagrams for fixed load b=20 N-s/m and two equivalence ratios

Figure 5 . 12 :Figure 5 .

 5125 Figure 5.12: Open cycle P-V diagrams for fixed heat input of 39.25 J/cycle for loads b=20

Figure 5 . 13 :

 513 Figure 5.13: Effect of load and equivalence ratio on engine operating frequency.

Figure 5 . 14 :

 514 Figure 5.14: Effect of load on output power for fixed equivalence ratios Φ =0.6 and 1.0.

Figure 5 . 15 : 5 . 3 . 1 . 4

 5155314 Figure 5.15: Effect of load on equivalence ratio to regulate operating speed and frequency.

Figure 5 . 16 :

 516 Figure 5.16: Effect of intake work.

  mathematical model developed for the open cycle in chapter three is simplified to model the closed cycle. Here a centimeter size engine is considered for analysis. The engine, shown schematically in Figure 3.1, with h=0 W/K, b f =0 N-s/m, V o =0.785 cc (1 cm in diameter and 1 cm of nominal length), s=2.435 N/m, b=4.9 N-s/m, and m=0.0367 kg is treated as the reference engine. Standard temperature and pressure conditions of T o =298 K and P o =100 kPa are chosen for the ambient state and for the thermodynamic properties of air. The properties of air used are R=286.9 J/kg-K, ρ=1.19 kg/m 3 , c v =717.25 J/kg-K, and c=343.7 m/s. The nonlinear model equations [(3.29) -(3.32)] are used to predict the engine behavior and performance.

Figure 5 .

 5 18. It is obtained using the nonlinear model equations [(3.29) -(3.32)] at steady state operation by impulsively supplying a nondimensional heat of Ψ=18.2 at the end of the compression stroke to the reference engine. The ratio 𝑞 𝑜 𝑞 𝐻 = 0.5848 is required for steady state operation of the nonlinear

Figure 5 . 19 .

 519 Figure 5.19. Two loads b=0.7 and 4.9 N-s/m are shown. It can be observed that when the load

Figure 5 . 19 :

 519 Figure 5.19: Efficiency and compression ratio (CR) plot for loads b=0.7 and 4.9 N-s/m.

Figure 5 .

 5 Figure 5.20: P-V diagram for various loads b=0.7, 1.75, and 4.9 N-s/m (at Ψ=18.2).

  Figure 5.21 shows a plot of frequency versus heat input for two loads. The figure shows that the engine operating frequency changes as the applied load on the engine and/or the heat added per cycle change. This figure is plotted for two loads b=0.7 and 4.9 N-s/m (ζ=0.0548 and 0.3838 respectively). For a fixed load b=0.7 N-s/m, the resonant frequency (or operating frequency) obtained from the nonlinear model for smaller heat inputs converges to the resonant frequency calculated using linear model represented with an open circle symbol. It is clear from the plot

Figure 5 . 21 :

 521 Figure 5.21: Operating frequency and heat input Ψ for loads b=0.7 and 4.9 N-s/m.

Figure 5 . 22 :

 522 Figure 5.22: Variation in expansion and compression times for loads b=0.7, 1.75, and 4.9 N-

Figure 5 . 23 :

 523 Figure 5.23: Output power against heat input Ψ for loads b=0.7, 1.75, and 4.9 N-s/m.

Figure 5 . 24 :

 524 Figure 5.24: Output power and power density for various loads (Ψ =18.2).

Figure 5 .

 5 25 shows a comparison of open and closed cycle P-V diagrams generated for the reference engine (defined in section 5.3.1 Open cycle) for an equivalence ratio Φ=0.51. In the figure, the thermodynamic states of open and closed cycles are denoted by subscripts "o" and "c", respectively. It is evident from the figure that the P-V diagram for the closed cycle (1,c→2,c→3,c→4,c→1,c) is very different from the open cycle (1,o→2,o→3,o→4,o→5,o→6,o→7,o→1,o) in terms of stroke length, peak pressure, and indicated work.

Figure 5 . 25 :

 525 Figure 5.25: Open and closed cycle P-V diagrams for b=20 N-s/m, h=0 W/K, and Φ=0.51.

Figure 5 .

 5 Figure 5.26 graphically shows the relation between heat input and corresponding equivalence

Figure 5 . 26 : 5 . 3 . 3 . 1 5 . 3 . 3 . 2 Efficiency

 52653315332 Figure 5.26: Heat input and equivalence ratio relation for the reference engine.

Figure 5 . 27 : 5 . 3 . 3 . 3

 5275333 Figure 5.27: Comparison of open and closed cycle in terms of efficiency and volume ratios

  In a closed cycle, the cavity is sealed hence its stiffness and frequency remain constant though out the thermodynamic cycle. But in the open cycle, the stiffness varies and depends on the engine process. In addition, for a closed cycle only two strokes are involved in calculating the frequency, while in an open cycle all the four strokes are counted.

Figure 5 . 28 :

 528 Figure 5.28: Open and closed cycle operating frequencies.

Figure 5 .

 5 Figure 5.28 shows the trend in engine operating frequency for open and closed cycles with

Figure A1- 1 :

 1 Figure A1-1: Schematic sketch of MEMS P 3 resonant heat engine [47].

Figure A1- 2 :

 2 Figure A1-2: Working cycle of the MEMS P 3 heat engine [47].

(

  KOH) etching, and DRIE are performed in order to form the 950 nm thick PDMS membranes. A schematic sketch of the fabrication process is shown in Figure A1-3.

Figure A1- 3 :

 3 Figure A1-3: Fabrication of PDMS membranes.

Figure A1- 4 :

 4 Figure A1-4: Experimental set up.

  Figure A1-5 shows model schematic of the P 3 heat engine. The cavity of the engine is assumed to have a cylindrical shape with diameter 2r o and nominal thickness h. In the center of the cavity is saturated-vapor bubble. The bubble is surrounded by liquid in the radial annulus. The liquid also occupies a film of small thickness at the bottom of the vapor bubble. The vapor bubble is 2r i in diameter. The instantaneous density, temperature, and pressure of the bubble are given by ρ o +Δρ, T o +ΔT, and P o +ΔP, respectively. The instantaneous temperature of the liquid layer is T o +ΔT l , and the saturation pressure of the liquid is given by P l . The subscript "o" indicates an ambient and/or static component, while  indicates a departure from the ambient value. The top of the cavity is bounded by a movable expander of mass m, constrained to move by a spring of stiffness s and two dampers with coefficients b f and b respectively. The damper b f is included to account for frictional losses in the engine. The damper b is used to model the useful work harvested by an energy converter that converts the mechanical power into electrical power. A movable rigid evaporator provides the lower boundary. Its motion is constrained by the spring of stiffness s h . The motions of the expander and evaporator are measured by the displacements x and x h , respectively. A linearized version of the model takes the form

Figure A1- 5 :

 5 Figure A1-5: Schematic sketch of the P 3 heat engine.



  and the * superscript denotes a non-dimensional dependent or independent variable. Applying Equation (A1.5a-e) to the lumped-parameter model of the engine Equation (A1.3), and neglecting evaporator membrane flexibility and parasitic heat losses from the vapor bubble, the system model in nondimensional form becomes 𝑜 ℎ 𝑓𝑔 𝑉 𝑜 (A1.7a-f) The scaling model described by (A1.6) reveals the nondimensional numbers Q, C, U and products FΦ and NΦ. The nondimensional numbers C and U account for thermal inertia and parasitic conductive heat losses respectively. The nondimensional number, Q is the quality factor describing energy losses due to damping and viscous effects. The quantities F, N, and Φ account for mechanical properties associated with engine flexing, thermal properties of the working fluid, and time-geometry properties of the engine respectively.

  1.0 ms, at fundamental frequencies of f o =0.1, 1, and 10.0 Hz respectively are applied. The displacement of the expander membrane x(t) is measured with the laser vibrometer. Both the displacement of the expander membrane x(t) and the heater voltage V h (t) are sampled with a digital oscilloscope and stored on a personal computer. During the experiment, the input energy E per cycle, delivered to the engine is approximately E=9 mJ. Under these conditions, the input heat rate q(t) and the velocity of the expander membrane 𝑢(𝑡) = 𝑥(𝑡) ̇ are given by Equations (A1.1) and (A1.2), respectively. Experimental determination of Q k and U k is performed by taking the FFT of the sampled heat rate and expander membrane velocity respectively. The choice of frequencies f o and heat pulse durations T, allow the collection of heat rate Q k , and velocity amplitudes U k , in the frequency bandwidth of 0.1-1000 Hz.

Figure A1- 6

 6 Figure A1-6 shows, both, measured and modeled transfer functions of the tested engines.

Figure A1- 6 :

 6 Figure A1-6: Velocity transfer functions of the tested engines.

1 .

 1 The ratios of thermal inertial C/ C I , and time-geometry factor / I , are plotted on the horizontal axes. The nominal values of both time-geometry factor  I , and nondimensional thermal inertial C I , are computed, for engine I, using Equations (A1.7a-f). Next, consider engine I and engine III in TableA1.1. For engine I, nominal engine case, the ratios of time-geometry factor and nondimensional thermal inertia are / I =1 and C/C I =0.1, respectively. However, for engine III, the time-geometry factor is raised by a factor of 4.7, i.e. / I =4.7 and the nondimensional thermal inertia is reduced to one third, i.e. C/C I =0.335. The velocity amplitude for engine III is increased by a factor of 18.33, i.e. V max /

  the working fluid evaporation property N are the parameters that dominate the performance of the engine. The composition of these quantities implies that the performance of the engine is defined by three factors: geometry of the engine, speed of operation, and thermal physical properties of the engine components. performance. The reference frequency of the engine ) both the effective mass of the engine and stiffness of the expander membrane, i.e.

Figure A1- 7 : 1 ) 2 (Figure 3 . 6 , 6 ,Figure 3 . 7 ) 6 ,Figure 3 . 7 )

 71236637637 Figure A1-7: Performance comparison of the micro heat engines.

TABLE OF CONTENTSTable 2 .

 OF2 1: Materials and their properties used in engine realization ............................................ 32

Table 5 .

 5 1: Damping coefficients obtained from the mathematical model. ................................... 82

Table 5 .

 5 2: Energy distribution in a motored engine cycle. ........................................................... 83

Table 5 .

 5 3: Effect of load on engine performance for fixed equivalence ratio Φ=1.0. .................. 97

Table 5 .

 5 4: Effect of intake work on the reference engine performance. ..................................... 101

Table 5 .

 5 5: Effect of heat loss on reference engine performance. ................................................ 102

Table 5 .

 5 6: Engine performance for various loads at Ψ=18.2. ..................................................... 107

Table 5 .

 5 7: Output power, power density, and work output for various loads at Ψ=18.2. ............113

Table 5 .

 5

Table 2 .

 2 1: Materials and their properties used in engine realization

	Material	Property exploited	Yield strength (MPa)	Thermal stability or Melting point	Thermal conductivity W/(m-K)
	Kevlar ® 49	High tensile strength	3000	150 -170 °C	0.04
	Silicone rubber
	Silastic ®	Mechanical flexibility	8.2	200 °C	0.18-0.28
	FL 40-9201
	Pyrogel ®	Thermal insulator	Not available	up to 650 °C	0.1
	Copper	Thermal conductivity	70	1000 °C	401

  The resulting volumes and temperatures ∆𝑉 4 , ∆𝑇 4 and ∆𝑉 2 , ∆𝑇 ∆𝑃 5, ∆𝑉 5 ̇= 0 and ∆𝑉 7 , ∆𝑇 7 , ∆𝑃 7 , ∆𝑉 7 ̇= 0. Since the state variables at 6 and 7 are assumed to be same, therefore, ∆𝑉 7 = ∆𝑉 6 , ∆𝑇 7 ≈ ∆𝑇 6 , and ∆𝑃 7 ≈ ∆𝑃 6 .The system of equations [(3.9),(3.11), and (3.13) -(3.16)] can only reach steady state when the proper choice of load b and initial conditions ∆𝑉 1 , ∆𝑃 1 , and ∆𝑇 1 are chosen for cycle integration. Improper choice of load b and initial conditions ∆𝑉 1 , ∆𝑃 1 , and ∆𝑇 1 would lead to monotonically drifting state variables ∆𝑉, ∆𝑃, and ∆𝑇 precluding periodic behavior.To determine the steady-state behavior of the engine q H is specified, initial load b is chosen, and integration is started from the quiescent initial state of ∆𝑉 1 = ∆𝑉 1 ̇= ∆𝑇 1 = ∆𝑃 1 = 0. An example integration illustrating the first cycle in a transient dwell-up from this initial state is shown in Figure3.4. The volume ∆𝑉 2 and temperature ∆𝑇 2 are obtained by integrating the equations [(3.13) -(3.15)] from the initial conditions ∆𝑉 1 , ∆𝑇 1 , ∆𝑉 1 ̇= 0 for a time 𝑡 1 such that ∆𝑉 2 ̇= 0 with 𝑀 ̇𝑖,𝑒 = 0. Note that the time 𝑡 1 is not known 'a priori', and is determined during the integration. The volume ∆𝑉 3 and temperature ∆𝑇 3 are determined using equations ∆𝑉 7 , ∆𝑇 7 , ∆𝑃 7 , ∆𝑉 7 ̇= 0 until a time 𝑡 4 such that 𝛥𝑉 1′ ̇=0. Like the time of compression 𝑡 1 and time of expansion 𝑡 2 , the time of exhaust 𝑡 3 , the time of intake 𝑡 4 are not known 'a priori', and are determined by the conditional integration. Since the system undergoes a transient phase, the engine operating cycle is not closed i.e. ∆𝑉 1 ≠ ∆𝑉 1 ′ , ∆𝑇 1 ≠ ∆𝑇 1 ′ , and ∆𝑃 1 ≠ ∆𝑃 1 ′ . Therefore, to obtain a steady state solution an objective function is defined that 50 computes square error of state variables at 1 and 1', and a function minimization computational procedure is adopted to determine proper choice of ∆𝑉 1 , ∆𝑇 1 , and ∆𝑃 1 for a specified heat input

2 from expansion and compression processes 3→4 and 1→2 are computed by numerically integrating equations [(3.13) -(3.15)] from the initial conditions ∆𝑉 3 , ∆𝑇 3 , ∆𝑉 3 ̇= 0 and ∆𝑉 1 , ∆𝑇 1 , ∆𝑉 1 ̇= 0 with 𝑀 ̇𝑖,𝑒 = 0. On the other hand, the resulting volumes, temperatures, and pressures ∆𝑉 6 , ∆𝑇 6 , ∆𝑃 6 and ∆𝑉 1 , ∆𝑇 1 , ∆𝑃 1 from exhaust and intake strokes 5→6 and 7→1 are computed by numerically integrating equations [(3.9), (3.11), and (3.13) -(3.16)] from the initial conditions ∆𝑉 5 , ∆𝑇 5 , [(3.18), (3.19)]. The volume ∆𝑉 4 and temperature ∆𝑇 4 are obtained by integrating the equations [(3.13) -(3.15)] from the initial conditions ∆𝑉 3 , ∆𝑇 3 , ∆𝑉 3 ̇= 0 for a time 𝑡 2 such that ∆𝑉 4 ̇= 0 with 𝑀 ̇𝑖,𝑒 = 0. Here, the time 𝑡 2 is not known 'a priori', and is determined during the integration. Then, the temperature ∆𝑇 5 and volume ∆𝑉 5 are determined from equations [(3.21), (3.22)]. Next, the volume ∆𝑉 6 , temperature ∆𝑇 6 , and pressure ∆𝑃 6 , are obtained by integrating the equations [(3.9), (3.11), and (3.13) -(3.16)] from the initial conditions ∆𝑉 5 , ∆𝑇 5 , ∆𝑃 5 , ∆𝑉 5 ̇ = 0 for a time 𝑡 3 such that ∆𝑉 6 ̇= 0. Now, the state variables at 7 are obtained from ∆𝑉 7 = ∆𝑉 6 , ∆𝑇 7 = ∆𝑇 6 , and ∆𝑃 7 = ∆𝑃 6 . Finally, a volume ∆𝑉 1 ′ , temperature ∆𝑇 1 ′ , and pressure ∆𝑃 1 ′ are obtained by integrating the equations [(3.9), (3.11), and (3.13) -(3.16)] from the initial conditions

  resulting volumes and temperatures ∆𝑉 4 , ∆𝑇 4 and ∆𝑉 2 , ∆𝑇 2 from expansion and compression processes 3→4 and 1→2 are computed by numerically integrating equations [(3.33) -(3.35)] from the initial conditions ∆𝑉 3 , ∆𝑇 3 , ∆𝑉 3 ̇= 0 and ∆𝑉 1 , ∆𝑇 1 , ∆𝑉 1 ̇= 0.The system equations [(3.33) -(3.35)] can only reach steady state when the proper cooling pulse magnitude q o /q H and initial conditions ∆𝑉 2 and ∆𝑇 2 are chosen for cycle integration.

  The volume ∆𝑉 4 and temperature ∆𝑇 4 are obtained by integrating the equations [(3.33) -(3.35)] from the initial conditions ∆𝑉 3 , ∆𝑇 3 , ∆𝑉 3 ̇= 0 until a time 𝑡 1 such that ∆𝑉 4 ̇= 0. Note that the time 𝑡 1 is not known 'a priori', and is determined during the integration. Then, the volume ∆𝑉 1 and temperature ∆𝑇 1 are determined from equations (3.37) and (3.39). Finally, a volume ∆𝑉 2 ′ and temperature ∆𝑇 2 ′ are obtained by integrating the equations [(3.33) -(3.35)] from the initial conditions ∆𝑉 1 , ∆𝑇 1 , ∆𝑉 1 ̇= 0, until a time 𝑡 2 such

.7. First, temperature and volume are computed for the thermodynamic cycle 1→2→3→4→1 as explained below and later equations (3.29) and (3.31) are used to calculate corresponding pressure. To determine ∆𝑉 3 and ∆𝑇 3 , equations (3.36) and (3.38) are used.

that 𝛥𝑉 2′ ̇=0. Like the time of expansion 𝑡 1 , the time of compression 𝑡 2 is not known 'a priori',

Table 5 .

 5 1: Damping coefficients obtained from the mathematical model. When a row wise summation is performed, a zero value is obtained, indicating satisfaction of the first law of thermodynamics. When a column wise summation is performed, it becomes clear that the PdVwork is lost by heat transfer and net enthalpy lost from the system. The net enthalpy lost (or Net enthalpy out) during the air exchange processes 4→5, 5→6, and 7→1 is 20 mJ/cycle while the energy lost due to compressive heating and fluid flow friction is 83 90 mJ/cycle. Using the (loss)mep-speed curve fit equation, out of 90 mJ/cycle of heat generated, 32 mJ/cycle is from fluid flow friction, while 58 mJ/cycle is due to compressive heating.

	Event	Damping coefficient (N-s/m)	Velocity × displacement (m 2 /s)	Work done on system (mJ)
	Process 1→2	-32.617	0.0049	160
	Process 3→4	7.911	0.0066	-50
	Process 5→6	19.288	0.0042	-80
	Process 7→1	-23.031	0.0033	80
	One cycle	-	-	110

Ideally, the work done on the system for a cycle obtained from experiment should match with the predictions from the model. Instead, the work done on the system calculated from the experiment and model are 72.8 mJ/cycle and 110 mJ/cycle, respectively. This suggests that the model over predicts the experimental observation. This results in a conservative estimate of the losses present in the system. Table

5

.2 lists the energy distribution in a motored engine cycle obtained from the model. As can be seen from the table (in the "Net enthalpy out" column), processes 1→2 and 3→4 take place in the closed condition hence no energy is lost in mass transfer. While, processes 4→5, 5→6, and 7→1 take place in the open condition and mass transfer occurs.

Table 5 .

 5 2: Energy distribution in a motored engine cycle.

	Event	PdVwork (mJ)	Heat lost a (mJ)	Internal energy change (mJ)	Net enthalpy out b (mJ)
	Process 1→2	-1010	740	270	0
	Process 3→4	1060	-640	-420	0
	Process 4→5	0	0	480	-480
	Process 5→6	-1100	0	-2700	3800
	Process 7→1	940	-10	2370	-3300
	One cycle c	-110 ± 15.7%	90 ± 15.7%	0 ± 15.7%	20 ± 15.7%

a Heat generated is eventually dissipated as Heat lost.

Table 5 . 3

 53

	Load b	Equivalence	Heat input	Frequency	Useful work	Output power
	(N-s/m)	Ratio Φ	(J/cycle)	(Hz)	(J/cycle)	(W)
		1.01	483	13.94	183	2548
		1.03	389	13.95	155	2166
		1.02	334	13.89	135	1882
		1.04	283	13.70	117	1601
		1.02	236	13.32	97	1293
		1.01	212	12.76	87	1108
		1.01	114	12.33	43	529
		1.00	75	12.33	26	320

: Effect of load on engine performance for fixed equivalence ratio Φ=1.0.

Table 5 .

 5 4: Effect of intake work on the reference engine performance.

	Intake work load, b p1 (N-s/m)	0	-20	-40
	Heat input (J/cycle)	138.55	138.55	138.55
	Equivalence ratio	1.74	1.15	0.51
	Expansion ratio	8.37	6.43	4.13
	Compression ratio	1.05	1.22	1.76
	Indicated thermal efficiency (%)	47.59	44.25	38.41
	Brake thermal efficiency (%)	43.55	40.43	34.64
	Useful work (J/cycle)	60.34	56.02	47.99
	Friction work (J/cycle)	4.53	4.26	4.21
	Cycle frequency (Hz)	13.68	14.16	13.37
	Power output (W)	825.45	793.2	641.6

Table 5 . 5 :

 55 Effect of heat loss on reference engine performance.

	Heat loss coefficient h (W/K)	0.01	0.1	3
	Heat input (J/cycle)	138.55	138.55	138.55
	Equivalence ratio	0.505	0.504	0.504
	Heat lost (J/cycle)	0.377	3.625	63.55

Table 5 .

 5 6: Engine performance for various loads at Ψ=18.2.

	Load	Efficiency	Compression	Operating frequency
	(N-s/m)	(%)	ratio	(Hz)
	0.70	74.6	30.8	8.29
	1.40	69.6	19.6	9.50
	1.75	67.8	17.0	9.81
	2.10	66.2	15.1	10.00
	2.80	63.7	12.6	10.12
	4.20	59.9	9.8	9.73
	4.90 a	58.5	9.0	9.27
	5.60	57.4	8.4	8.49
	7.00	56.6	8.1	4.86

Table 5 .

 5 7: Output power, power density, and work output for various loads at Ψ=18.2.

	Load	Output power	Power density	Work output
	(N-s/m)	(W)	(W/Liter)	(J/Cycle)
	0.70	22.34	956.6	2.69
	1.40	23.86	1635.0	2.51

Table 5 .

 5 8: Comparison of performance parameters of reference engine obtained by closed and open cycles with equivalence ratio Φ=0.51. From the Table 5.8, the operating frequency of the engine in a closed cycle operation is 37.03 Hz while for an open cycle is 13.37 Hz. The difference in frequencies is a result of varied stiffness of the bellows-cavity, and the number of strokes involved in the open and closed cycles.

	Performance parameter	Closed cycle	Open cycle
	Indicated thermal efficiency (%)	47.66	38.41
	Brake thermal efficiency (%)	44.33	34.64
	Heat added per cycle (J/cycle)	35.6	138.5
	Cycle frequency (Hz)	37.03	13.37
	Compression ratio, CR	5.05	1.76
	Expansion ratio, ER	5.05	4.13
	Average piston speed (m/s)	4.75	4.77
	Power output (W)	584.7	641.6

  steady state solution is obtained. Since, modeling the compliant engine in closed cycle is questionable; it is modeled in both open and closed cycles. The open and closed cycle results showed similar trends, but their predictions differed numerically. For example, both cycles showed that the indicated thermal efficiencies increase with heat input, but their numerical predictions differed by 10%. The high efficiency prediction by the closed cycle is a result of the absence of or "neglecting" pumping work. Since a real compliant engine operates in an open cycle, it has been realized that modeling the engine in the open cycle is appropriate. The open cycle modeling results showed that the engine performance depends on inputs parameters such as heat input, heat loss, and load on the engine. The engine performance is determined in terms of thermal efficiency, operating frequency, and output power.

  are 5 μm thick, 10μm high, and spaced 90 μm apart. Potassium hydroxide (KOH) anisotropic etching is then used to form the evaporator membranes.Fabrication of these expander membranes is different. Fabrication of the 2.2 µm thick silicon membranes starts with a 3 inch diameter double-sided polished silicon wafer. The wafer is prepared in the furnace as described above. That is, a 500 nm thick oxide layer is grown, followed by boron diffusion to provide a 2.2 µm thick etch stop. A 5 nm adhesion layer of titanium/tungsten followed by a 500 nm gold layer are sputtered on the back side of the wafer. vacuum chamber for 15 minutes. A spin rate of 3000 RPM for 60 s is used to fabricate 950 nm thick PDMS membranes. Then the wafer is baked for 15 minutes at 90 °C. The back side of the wafer is then patterned into 10 mm square window membranes using a standard contact photolithographic process. Finally, Deep Reactive Ion Etching (DRIE), Potassium hydroxide

	In this work, three types of expander membranes are used: 2.2 µm thick silicon, 300 nm thick
	silicon nitride, and 950 nm thick Polydimethylsiloxane (PDMS) expander membranes.
	Fabrication of the silicon nitride expander membranes starts with a 3 inch diameter double-
	sided polished silicon wafer. A Low Pressure Chemical Vapor Deposition (LPCVD) process is
	used to deposit a 300 nm thick layer of silicon nitride on both sides of the wafer. The back side
	of the wafer is then patterned into 10 mm square window membranes using a standard contact
	photolithographic process. Deep Reactive Ion Etching (DRIE) followed by Potassium hydroxide
	(KOH) etching is then used to form the 300 nm thick silicon nitride membranes.
	Fabrication of PDMS expander membranes starts with a 3 inch diameter double-sided
	polished silicon nitride wafer. A 10:1 (base to curing agent ratio) PDMS solution is deposited on

The gold layer on the back side of the wafer acts as an etch mask when the wafer is etched in Potassium hydroxide (KOH) later. Next, the back side of the wafer is patterned, using a standard photolithographic process, into 10 mm square window membranes. The wafer is then etched in Potassium hydroxide (KOH) etchant to form the silicon membranes.

the top side of the wafer by spin coating. To free bubbles from PDMS solution, the solution is 137 placed in

  The experimentally determined periodic heat input is represented by the Fourier series 𝑞(𝑡) = 𝑄 𝑜 + ∑ 𝑄 𝑘 cos(2𝜋𝑓 𝑘 𝑡 + 𝜃 𝑘 ) are the harmonic frequencies f k = k/T P , T P is the period of the engine cycle operation, Q k and  k are the amplitudes and phases. The k amplitudes Q k and phases  k are extracted from the heat input q(t) using an FFT algorithm. The number of k components is determined by the number of sampled points in a measured waveform.

		𝑘	(A1.1)
	where f k Likewise the experimentally determined periodic velocity of the expander membrane is
	represented by the Fourier series
	𝑢(𝑡) = 𝑈 𝑜 + ∑ 𝑈 𝑘 cos(2𝜋𝑓 𝑘 𝑡 + 𝛾 𝑘 )
	𝛤(𝑓 𝑘 ) =	𝑈 𝑘 𝑄 𝑘

𝑘 (A1.2)

As before, amplitudes, U k , and phases,  k , are extracted from the velocity of the expander membrane, u(t), using an FFT algorithm.

Interpretation of the performance of the engine is then reduced to predicting the system transfer function 𝛤(𝑓 𝑘 ) between the coefficients Q k and U k at each of the k frequencies f k , i.e.,

Table A1

 A1

			.1 Engine specifications
	Engine specification	Engine I	Engine II	Engine III	Engine IV
	Expander material	Silicon	Silicon	Silicon nitride	PDMS
	Expander thickness (µm)	2.20	2.20	0.30	0.95
	Cavity thickness (µm)	450	75	150	450
	Mass added (g)	4.00	4.00	4.00	0.56

Table A1 .

 A1 2 Environmental conditions and thermal properties of 3M  PF5060DL.

	Physical property	Value
	Liquid density ρ (kg/m 3 )	1775
	Vapor density ρ o (kg/m 3 )	3.53
	Molecular weight M (kg/mol)	0.338
	Saturated pressure P o (kPa)	25.10
	Vapor specific heat C p (J/kg.K)	755
	Vapor constant R (J /kg.K)	24.18
	Enthalpy h fg (kJ/kg)	94
	Ambient temperature T o (K)	294

Table A1

 A1 It is hypothesized that the nondimensional thermal inertia C, the time-geometry factor Φ, and

		.3 Model fitting parameters
	Parameter	Engine I	Engine II	Engine III
	C T (J/k)	0.2323	0.113	0.026
	b f (N.s/m)	0.76	0.46	0.8
	U B (W/K)	7.00x10 -4	0.10x10 -4	1.87x10 -4
	m m  (g)	2.42	2.8	4.8
	U c (W/K)	1.24	2	0.16

Table D3

 D3

			. 1: Repeatability data from motoring experiment
	Data set Work done	Work done	Structural	Velocity ×	Damping
		on piston	on system	friction work	displacement	coefficient b
		(mJ/cycle)	(mJ/cycle)	(mJ/cycle)	(m 2 /s)	(N-s/m)
	1	101.8	72.8	29	0.019	1.53
	2	99.2	78.1	21.2	0.019	1.67
	3	102.7	71.0	31	0.019	1.66
	4	102.0	78.0	24	0.0189	1.28
	5	98.0	69.0	29	0.0186	1.55
	Mean	100.8	73.8	26.8	0.0189	1.53
	Standard	2.0	4.1	4.1	0.0002	0.157
	deviation

T. R. Padmanabhan, Industrial Instrumentation: Principles and Design, SpringerLondon, 2000

x LIST OF TABLES Thus for higher efficiency, high heat-rate, low external stiffness , and low damping  are beneficial. However, it should be noted that operation in the linear regime precludes high compression ratios, so that these trends are shown to be modified when the P-V behavior of the engine modeled by equations [(3.33) - (3.35)] at practical compression ratios is explored in section 5.3.2 Closed cycle. model for Ψ=18.2. The figure shows that the pressure inside the sealed flexible cavity at the end of heat addition process rises to ~1.97 MPa. Following the heat addition process 2→3, the engine expands adiabatically 3→4 to about nine times of its minimum volume which is 7.077 cc. Next, the heat rejection process 4→1 takes place instantaneously by interacting with the surroundings.

Finally, the engine undergoes adiabatic compression 1→2 to the minimum volume V 2 =0.785 cc resulting in a closed thermodynamic cycle 1→2→3→4→1. It is observed that the minimum volume of the engine corresponds to the quiescent volume 𝑉 𝑜 . The nondimensional terms for this reference engine at Ψ=18.2 were C=2.5, H=0, Г=227.53, Λ=0.0022, and ζ=0.3838.

CONCLUSIONS Chapter 6

In this study, work on developing an internal combustion engine operating on a four-stroke Otto cycle principle at small length scales has been conducted. To mitigate friction and leakage losses at small scale, a complaint engine design has been implemented in which the pistoncylinder assembly is replaced by a flexible cavity. The compliant design offers various advantages, such as fuel-flexibility and simpler design. The aim of the study has been to demonstrate the feasibility and potentiality of the compliant engine at small scale. This has been accomplished by fabricating and quantifying the parasitic losses using mathematical modeling and experimental tools.

The study towards the development of the engine involved the following tasks:

 Developing a mathematical model that can predict the engine's performance  Fabricating and testing a prototype engine to demonstrate its feasibility and to extract real engine parameters  Coupling the mathematical model with the prototype test results to demonstrate the compliant engine potentiality To predict the performance of the engine, a physics-based lumped-parameter model is developed by taking an air-standard cycle approach. The model is derived by applying conservation of mass, conservation of energy, an ideal gas model to a moving control volume containing the air as working fluid, Newton's second law for the piston or mass, and a mass-flow rate equation for working fluid flowing through the valve. The model is solved repeatedly until a Appendix A1: SCALING ANALYSIS OF MEMS P 3

HEAT ENGINE

A scaling analysis of the MEMS engine presented in section 1.3.2.1.4 MEMS P3 Heat Engine is discussed here. For scaling analysis, a scaling model of the engine is developed and the parameters and factors which affect the overall performance of the engine are investigated. Both experiment and model are used to investigate the issues related to scaling of the micro heat engine [START_REF] Bardaweel | MEMSbased resonant heat engine: Scaling analysis[END_REF].

Overview of the MEMS-based resonant micro heat engine

Figure A1-1 shows a schematic of the resonant micro heat engine. The engine consists of four components: a thermal switch, an evaporator, an expander, and an engine cavity. The thermal switch is used to control the timing and duration of the heat addition and heat rejection.

The cavity is filled with a two-phase working fluid, bounded by the evaporator membrane from the bottom, the expander membrane from the top, and the semiconductor tape walls from the sides. Mechanical power is produced as the working fluid deforms the expander. A capillary wick fabricated on top of the evaporator controls the thickness of the liquid-phase layer inside the cavity.

The engine is an external combustion engine that converts thermal power into mechanical power through the use of a novel thermodynamic cycle [START_REF] Bardaweel | Cyclic operation of a MEMS-based resonant micro heat engine: Experiment and model[END_REF]. The output power may be used either, directly, as mechanical work done by the engine or it may be converted into electrical power through the use of a thin-film of piezoelectric material deposited on the expander. It was shown that the engine produced a net mechanical power when operated at low frequency from a constant heat source at 60 °C [START_REF] Cho | Power production by a dynamic micro heat engine with an integrated thermal switch[END_REF].

Cyclic operation of the micro heat engine results in pressure and volume changes inside the cavity and, thereby, mechanical power is produced. Due to the resonant nature of operation, the period of cyclic heat addition and removal is equal to the inverse of mechanical natural frequency of the engine, i.e. natural period of the engine. As a result, the displacement amplitude of the expander membrane is maximized and for a fixed amount of energy input to the engine, the work done by the engine increases. For a resonant operation pressure and volume in the engine cavity are decoupled, i.e. cyclically rise and fall 90 ° out of phase.

Fabrication

Fabrication of the evaporator membrane starts with a 3 inch diameter double-sided polished silicon wafer. First, the wafer is placed in a furnace at 1050 °C for 120 minutes to grow 500 nm thick oxide layers on both front and backside of the wafer. To prepare the wafer for boron diffusion, the backside of the wafer is masked with semiconductor tape, and the wafer is immersed in Buffered Oxide Etch (BOE) for 15 minutes. The boron diffusion process is then carried out for 110 minutes at 1125 °C in the furnace, to provide a 2.2 µm thick etch stop. Minor steps, including both BOE and Low Temperature Oxide (LTO) processes are then performed to remove the borosilicate glass layer formed as a result of boron diffusion. Next, a 5 nm adhesion layer of titanium/tungsten is sputtered on both sides of the wafer. Next, 300 nm and 500 nm gold layers are sputtered on the front and back side of the wafer, respectively. A standard photolithographic process is then used to pattern the front side with resistance heaters. Likewise, the back side is patterned into 8 mm square window membranes. Next, micro channel wicks are fabricated on top of the resistance heater using an SU-8 polymer. The wicks take the shape of rectangular walls, radially patterned toward the center of the evaporator membrane. The wicks with a 5 nm adhesion layer of titanium/tungsten followed by a 300 nm layer of gold. The gold is then patterned via photolithography to produce a pattern of 30 μm diameter circular gold targets.

Once the arrays of gold targets are fabricated, the wafer is diced and the rest of the fabrication process is carried out at the die level. Micro-droplet deposition occurs when the die is exposed to mercury vapor at 184 °C. As mercury preferentially condenses on the gold targets, microdroplets form and grow on the gold targets. Further details on the fabrication of engine components can be found in References [START_REF] Bardaweel | Cyclic operation of a MEMS-based resonant micro heat engine: Experiment and model[END_REF][START_REF] Cho | Power production by a dynamic micro heat engine with an integrated thermal switch[END_REF][START_REF] Cho | Fabrication and characterization of a thermal switch[END_REF].

Experimental facility

The experimental setup shown in Figure A1-4 is used to validate the scaling model. Precision control of heat addition and heat rejection is required to experimentally measure the dynamic properties of the micro heat engine using the transfer function approach. As shown in the figure, a function generator is used to deliver a periodic voltage V h (t) to the resistance heater. The use of a resistance heater enables precise control over the magnitude and duration of the heat addition process. Experimental measurement of the heat rejection temporal profile through the thermal switch is unattainable, and thus it is chosen to allow heat rejection from the engine to occur by passive conduction to the surroundings. Nonetheless, in previous work a thermal switch was used to control heat rejection from/or heat addition to the engine [START_REF] Bardaweel | Cyclic operation of a MEMS-based resonant micro heat engine: Experiment and model[END_REF][START_REF] Cho | Fabrication and characterization of a thermal switch[END_REF].

In the transfer function approach, a periodic voltage V h (t) consisting of a unipolar square wave is applied to the resistance heater. Each time the voltage is applied through the resistance heater; electrical power is dissipated as heat and transferred into the rigid evaporator membrane.

The heat transferred into the cavity evaporates liquid working fluid, causing a pressure increase inside the cavity. The expander membrane deflects up due to the increase in the pressure. Heat Thermal physical properties of the engine components play an important role in determining the performance of the engine. In the nondimensional thermal inertia C, the dimensional lumped thermal inertia parameter C T is scaled against the engine vapor volume V o , and the latent evaporation property  o h fg of the working fluid N. If the working fluid N is fixed, then the nondimensional thermal inertia C scales against the inverse of engine vapor volume V o .

Consequently, the nondimensional thermal inertia C will decrease for larger engine volume.

And, the nondimensional thermal inertia C will decrease for working fluids with higher latent evaporation property  o h fg . Moreover, the thermal inertia parameter C T which appears in the nondimensional thermal inertia C is determined by the properties of the engine components. For example, thermal inertia of the top expander membrane is proportional to its mass density ρ m , specific heat C P, m , and material volume V m . For most available materials the product

fixed. As a result, a smaller expander membrane V m will lower the thermal storage of the expander membrane. For example, thermal inertial C T decreased from 0.113 J/K to 0.026 J/K when the expander membrane was changed from 2.2 µm thick silicon membrane to 300 nm thick silicon nitride membrane, i.e. engine II and engine III, respectively. This is mainly due to reduction in the material volume of the top expander membrane and thus lower thermal storage in the 300 nm thick silicon nitride expander membrane.

The time-geometry factor  is composed of the reference frequency ω, and the ratio of evaporation-area to engine volume S /V o . For a fixed evaporation-area to volume ratio S/ V o , the time-geometry factor  is inversely proportional to the speed of the operation, which is determined by the resonant frequency of this resonant-type heat engine. As a result, lower operating speed will result in higher time-geometry factor  and, thus better engine

where =c p /c v is the specific heat ratio of the working air medium. If the parasitic heat loss is neglected, then H=0 and equation (A2.1) may be integrated to obtain

where the quantity 𝐸 ̅ is 𝐸 ̅ = ∫ 𝑞 ̅𝑑𝑡 ̅ 𝑡 ̅ 0

is the accumulated nondimensional energy added to the engine. To the level of approximation of the linearized model, the accumulated added energy

(A2.9)

From (A2.6), (A2.7) and (A2.8), it can be deduced that for resonant operation zero volume velocity at thermodynamic state 2→3 and 4→1 (i.e. that the matrix [] is diagonal), the nondimensional elapsed time must be 𝜏̅ =

At a resonant condition, with 𝜏̅ = 𝑡 ̅ 𝑛 , the volume excursion from (A2.9) simplifies to

(A2.11)

High volume excursions are associated with high heat load, and low external stiffness  and damping .

The volume excursion may then be used to compute an approximation for the efficiency of the engine. The instantaneous work  i being performed by the air volume is

The efficiency  is computed by integrating the instantaneous power during the expansion process 3→4 and compression process 1→2 and dividing by the heat added. After performing this calculation, an expression for the efficiency  is obtained as

Appendix A3: ELASTIC COLLISION

To have an optimistic estimation of engine efficiency and performance, the collision at TDC II (state 6, Figure 3.3) is assumed to be elastic in nature. In the elastic collision, the piston or mass is assumed to restore all the energy, and the energy available with the spring-mass-damper system (bellows-piston combination) is higher than in the inelastic collision. The elastic collision is modeled by assuming that the piston engages with a stiff spring s stiff (=600 × s) as it approaches TDC II state 6 (Figure 3.3). The stiff spring models the kinetic energy that would have been lost if the collision was inelastic. The recovered kinetic energy is utilized in subsequent process i.e. the intake stroke 7→1.

In this section, a brief comparative study of elastic and inelastic collision modeling on the engine performance is presented. ,'Aineq',[],'bineq',[],'Aeq',[],'beq',[],'lb',[ 0,(P.Tim+0),-0.3,0, 30],'ub',[ 5, (P.Tim+10),0,30, 30], 'x0', [ 0, 0, 0, 0, 30], 'options',options ); gs = GlobalSearch('Display','iter','TolFun',0,'TolX',0,'NumTrialPoints',800, ' % Intake manifold del_Pr_bar P.P7=P7; Pratio_67= (1+P6)/(1+P7); T7= ((1+T6)*(Pratio_67^((1-P.g)/P.g)))-1; P.T7=T7; mass_7=(P.Po*P.Vo*(1+P7)*(1+V7))/(P.R*P.To*(1+T7)); P.rho7= (mass_7/(P.Vo*P.rho*(1+V7)))-1; end %============================================================================ % PROCESS 7-1.Intake stroke %============================================================================ m_in=m_tot-mass_7; % mass ingested during intake stroke t=linspace(0,1*tau,P.N); % time for integration (s/s) yo(1)=V7; yo(2)=V7dot; yo(3)=T7; yo(4)=P7;

% Initial conditions at 7 yo=yo'; options = odeset('Events',@event71); [t,y]=ode45(@(t,y) process71(y,P),t,yo,options); % Runga-Kutta solution t71=t; N4=numel(y)/4; % #data points 7-1 V71= y(:,1); % del_Vol_bar 7-1 V71dot=y(:,2); % Velocity_bar 7-1 T71=y(:,3); % del_Temp_bar 7-1 P71=y(:,4); % del_Pr_bar 7-1 t71=t71(1:N4); % time_bar 5-6 The heat added per cycle, q h t q = mass of fuel × heating value of fuel Similarly, for an equivalence ratio Φ=1.3, the heat added/cycle, q h t q =3.6 J/cycle, and 𝛹=18.2.

Appendix D3: UNCERTAINTY ESTIMATION

The measurements from motoring experiment are recorded electronically using various instruments. Each instrument to a certain degree introduces error in the measurement. Usually, the errors or uncertainties from each instrument propagate into the final results. To determine the overall error, the root-sum-square (rss) formula is used. Here the uncertainties pertaining to individual instruments and overall experiment are presented.

Pressure transducer

The following pressure error information is obtained from the manufacturer's manual.

Linearity