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Takin' the Heat Senior Design Team 8

Final Design Report

Calvin College Engineering 340

Spring 2007

Neil Bruinsma • Geoff VanLeeuwen • Ben Mead • Shalom Kundan

Team 8 Takin’ the Heat

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Abstract

The goal of this project was to design a Stirling engine that converts renewable energy into mechanical power while staying applicable to underdeveloped nations. The final design choice is a two cylinder gamma-type Stirling engine that uses atmospheric air as its working fluid. It is designed to use solar power as its heat source and ambient air as its heat sink. The design reflects our core goals of simplicity, variability, stewardship and performance.


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Table of Contents 1 Project Introduction and Objectives ..................................................................................................... 5 2 Background .......................................................................................................................................... 5 3 Final Design ......................................................................................................................................... 6

3.1 Support Structure......................................................................................................................... 6 3.2 Chamber ...................................................................................................................................... 8 3.3 Linkage ........................................................................................................................................ 8 3.4 Displacer.................................................................................................................................... 10 3.5 Troubleshooting......................................................................................................................... 11 3.6 Budget........................................................................................................................................ 11

4 Design Norms..................................................................................................................................... 13 4.1 Cultural Appropriateness........................................................................................................... 13 4.2 Stewardship ............................................................................................................................... 13

5 Results ................................................................................................................................................ 14 6 Future Improvements ......................................................................................................................... 14 7 Acknowledgments.............................................................................................................................. 14 8 Appendix ............................................................................................................................................ 15


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Table of Figures Figure 1. Final Design................................................................................................................................... 6 Figure 2. Bottom Plate .................................................................................................................................. 7 Figure 3. Top Plate........................................................................................................................................ 7 Figure 4. Support Tower ............................................................................................................................... 8 Figure 5. Crank Arm Optimization ............................................................................................................... 9 Figure 6. Offset Angle Optimization .......................................................................................................... 10 Figure 7. Displacer/Regenerator ................................................................................................................. 11 Table of Tables Table 1. Bill of Materials ............................................................................................................................ 12


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1 Project Introduction and Objectives

The project idea came from a third party. Matt Weeda, of Innotec Corporation, proposed the project

during the summer of 2006. Innotec Corporation is a manufacturing company located in Zeeland, MI and is primarily a Tier 2 supplier of automotive accessories. However, their product line extends to several other industries including office furniture and home appliance. Currently, Innotec is developing means for energy production that utilize renewable resources. One of these means is a cost-effective, mid-size Stirling engine.

There are two general types of Stirling engines available on the market. The first type are model Strirling engines that are capable of running off of the heat from a cup of hot water or a person’s hand. These models are fairly simple, inexpensive, and run off of a small temperature difference; however they do not produce any useful energy. The second type are the large industrial Stirling engines. They are extremely complex, expensive, and run off of a large temperature difference; however they do produce useable power. With its high cost the current technology has a limited customer base due to the availability of other comparable gas-powered electric generators at lower costs.

The new design is a comprise between the two currently available types. The engine runs off of a small temperature difference and has a fairly simple design. In order to still provide a useable amount of power, the decrease in temperature difference must be balanced by an increase in surface area. The advantage of using a smaller temperature difference is that a wider range of heat sources can be used such as solar arrays, wood fires, or geothermal activity. This makes the engine ideal for uses in remote locations where gasoline and electricity are unavailable.

The goal of this project is to develop a simple and inexpensive Stirling engine that converts thermal energy into mechanical energy capable of maintaining motion. In addition, it is to be easily distributed and used in third world locations.

2 Background

The Stirling engine was first invented in 1816 by Robert Stirling a Scottish minister. Stirling

intended to create a safe alternative to the explosive steam engines of his time. This dates the engine before the Diesel or the gasoline engine. In 1850 J. Ericsson a Swedish inventor produced Stirling engines between the 0.5 and 5 hp range. However, the demand for more horsepower gave way to the Otto engines that were capable of producing more power. Due to high oil prices in the 1970’s car manufactures Ford and GM spent millions of dollars developing Stirling engines, but due to the slow response time of the engine and decreasing oil prices by the 1980’s the Stirling engine was no longer considered as an alternative power source.1 Currently, Stirling engines are being substantially researched due to their potential for high efficiency, and their low to zero emissions output. 2

Stirling Engines come in all shape and sizes but they all generate power on the same principle – using a temperature difference. A Stirling engine is unlike a conventional combustion engine in that the fluid never leaves the engine. It uses an external heat source which can be anything: the sun, fire, steam, or natural gas. One side of the engine is heated using an external heat source which causes an increase in temperature and expansion of gases in the cylinder. The gases are then cooled on the other side of the engine by using a heat sink such as ambient air or water. This pressure difference between the inside cylinder and the outside air causes a piston to move creating mechanical power.

1 http://www.stirlingengine.com/ 2 http://en.wikipedia.org/wiki/Stirling_engine


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3 Final Design

Figure 1. Final Design

The final design calls for two gamma type Stirling engines to be placed side by side and connected by the same output shaft. It is a low temperature Stirling engine that uses atmospheric air as its working fluid. The advantage of having two cylinders 180 degrees out of phase is that the rotating masses of the two engines are balanced. Using two cylinders also gives the engine two separate power strokes causing the output torque of the engine to be smoother. In order for the design to stay applicable to underdeveloped nations the engine must be easily reparable. The gamma type is the simplest type of Stirling engine which uses relatively simple geometries and materials. It is therefore easier to manufacture and maintain than the other types. This design has four main components to it; the support structure, chamber, linkage, and displacer.

3.1 Support Structure

There are two main components to the support structures. First are the plates. In addition to

providing the main body for the engine the plates also act as the heat source and heat sink. In order to perform both functions the plates must be strong while at the same time have a high thermal conductivity. The material in addition to its physical characteristics, must also be readily available in the developing world at a reasonable price. As a result aluminum was chosen as the material that best fits these requirements.


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The plates were manufactured by Jamesway Tool and Die a waterjet company out of Zeeland, Michigan. Waterjet cutting was chosen to achieve rectangular holes and at a high level of accuracy. This accuracy would help with later alignment. For the top plates rectangular slots were cut to hold the support towers, and circular holes were cut for the piston and the plunger. A matching pattern of circles were cut into the top and bottom plates which would later be used to bolt the two plates together. Fins were also welded to the bottom plate to increase the surface area and therefore increase the heat transfer from the heat source.

Figure 2. Bottom Plate

Figure 3. Top Plate

The second component of the support structures is the towers. The towers were also manufactured

by Jamesway Tool and Die, and out of aluminum. They include two tabs at the base that fit into the slots in the top plate. This accurately locates each tower with respect to the plates, which in turn helps the alignment. The overall height of the towers, 29 inches, was determined by the linkage lengths. The base was wide enough to rigidly support the side moments caused by the rotating shaft. A five thousandths undersized hole was cut at the top of the towers to allow a ball bearing to be press fit in. This bearing holds the output shaft. To add rigidity to the towers, brackets were made that connect the two towers together on each chamber. An L bracket was also made that connects the tower to the sink plate. Once the alignment of the towers was set the edges of each tower were welded to the plate.


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Figure 4. Support Tower

3.2 Chamber

The chamber is the main body of the engine that is supported on either side by the two plates. It

encloses the working fluid, provides structure to the engine and insulates the heat source from the heat sink. The chamber has to be made out of material capable of withstanding 400 ˚F, and must also be a poor conductor of heat. Steel was found to be available in the size needed and could withstand the temperature rangers however, it was heavy, expensive, and provided no insulation. PVC was found to be light weight a good insulator and inexpensive but at the higher temperature it did not retain its strength. Eventually a fiberglass pipe was found that met all of the specifications. The chamber itself is 12 inches tall and has an inner diameter of 25 inches.

A Stirling engine works off of the pressure difference between the enclosed fluid and the outside air. In order to achieve this pressure, the chamber must be air tight. To create an air tight seal the top and bottom lips of the fiberglass pipe were lined with a rubber gasket. The epoxy that was used to stick the rubber to the chamber has a rated melting temperature well above 400˚F. This gasket was compressed between the plates and the chamber by tightening the bolts around the perimeter. Under compression the gasket forms an air tight seal. The bolts should only be tightened until they are snug, too much force will cause the fiberglass pipe to crack.

3.3 Linkage

The linkage system of the engine is used to convert the changing pressure inside the chamber into a

rotational shaft power. Each cylinder of the Stirling engine consists of two 3-bar linkages; one for the piston and one for the displacer. Each of these linkages consisted of a crank arm, a connecting arm, and


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an output rod. The first aspect of the linkages examined were the lengths, which ensure that the engine has the correct motion. First, a theoretical mechanical model was performed to optimize the lengths of the linkages, see Appendix C. The length of the crank arm determines the travel of the linkage. The displacer travels 7-1/2 inches in the center of the chamber and the piston travels 4 inches. To achieve this motion the arms must be half of the travel length. The theoretical analysis showed that a longer piston arm provided a greater power output; however the air cylinders only allow for 4 inches of travel.

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0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

Crank Arm Length (m)

Po

wer

Ou

tpu

t (W

)

Figure 5. Crank Arm Optimization

Once the lengths of the crank arms were set the lengths of the corresponding connecting arms determined the pressure angle. The pressure angle is the angle the connecting rod makes with the vertical axis. The shorter the connecting rod is the greater the pressure angle. This causes the engine to have a harder time transferring forces. Making the connecting rods longer will decrease the angle but it will also make the overall height of the engine taller. It was determined that a pressure angle of 12 degrees or less is desired for most Stirling engines. This made the displacer linkage the limiting factor. The connecting rod for the displacer is 18.25 inches, giving it a 12 degree pressure angle. For the piston the connecting rod length is 15.5 inches, giving it a 7.4 degree pressure angle. The next step in designing the linkage was to determine how they would be constructed. The crank arms needed to be mounted rigidly to the output shaft. The initial design for this was to grind a square onto the shaft end and cut a corresponding square onto the crank arms using a broach. The squares were cut slightly oversized and then press fit into the crank arms. However, after a short time the press fits began to give way and the crank arms started slipping off. A new design was developed that did away with the press fitting and instead welded the shafts to the crank arms. The connecting rods were machined out of 3/8 inch aluminum. Holes were machined at each end and undersized by 0.002 inches so that bearings could be press fit. One main characteristic of a Stirling engine is the offset angle between the piston and displacer crank arms. Changing this angle changes the timing of the entire engine. The location of the plunger in its cycle determines what pressure the chamber is at. The highest pressure happens when the piston crank is about a third of the way through its upstroke. Changing the angle between the two changes the point in


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the cycle which the maximum pressure is reached. Changing this angle is a lot like changing the ignition timing on an internal combustion engine.

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0 20 40 60 80 100 120 140 160 180 200

Offset Angle (deg)

Po

wer

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Figure 6. Offset Angle Optimization

The mechanical model was set up to accept the output pressures from the thermal model and use

them as inputs to determine an optimal offset angle. Using this information a power curve could be generated which shows the total power output at various offset angles. Using this model a 90 degree offset was found to be the optimum.

The pistons were made out of old air cylinders donated by Innotec Corporation and then cut down to size. The pistons have a 5 inch bore and a 4 inch travel. All of the seals were removed from these pistons to reduce the friction.

3.4 Displacer

The displacer’s main purpose is to cycle air from the heat source to the heat sink. It is usually a

solid disk with a gap between the outer diameter and the chamber walls that allows the air to pass. In this new design the displacer has a dual purpose. In addition to cycling the air the displacer also acts as a regenerator. A regenerator is a temporary energy storage element much like a capacitor in an electrical circuit. A regenerator is commonly placed in the flow between the hot and cold sides of a Stirling engine. When the hot air passes through it transfers some of its heat energy to the regenerator. When the cold air passes back through the regenerator it can pick that heat back up again. This cuts down on the time it takes to heat and cool the fluid, by reusing some the heat.

The displacer is made of a disk of plywood with three holes cut out of it. Five layers of fine brass wire mesh is then placed in these holes to act as the regenerator. Brass has a high thermal conductivity and the mesh configuration gives it a large surface area making it a good regenerator. There is one flaw however, as stated the main function of a displacer is to move air from one side of the chamber to the other which is hard to do with a mesh design. In order to fix this problem the gap between the chamber and the edge of the displacer was shortened. This forces the majority of the air to flow through the wire mesh while still cycling most of the air.


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Figure 7. Displacer/Regenerator

3.5 Troubleshooting

Once assembly of the engine was finished there was quite a bit of troubleshooting to do before it

would work. The first goal was to reduce the internal friction as much as possible. Low temperature Stirling engines do not produce a large amount of excess power and will not run if the internal friction is too large. One source of friction is binding in the links due to a misalignment in the shaft. To fix this problem the engines had to be adjusted until the supports on each were exactly aligned. A coupler was added on the output shaft between the two cylinders as a solution to this difficult task. This coupler compensates for both displacement and angular misalignment. After the alignment is set it needs to be periodically checked for slip or travel between the two engines.

The second goal was to optimize the displacer. Given the complex nature of the system and the geometries involved a theoretical model of the displacer was not used. Instead the dimensions of the displacer were found experimentally. If the displacer is too small it will not be able to cycle the air correctly and the system will not generate enough power. If the displacer is too large the air friction caused by trying to force too much air through too tight of a space will drag the engine to a halt. The same is true of the mesh. If the mesh is too thin the air will flow too freely through it and the system will not generate enough power. If the mesh is too thick the air will not be able to penetrate it and the engine will drag to a halt. After building a model and testing various designs a working displacer design was found.

3.6 Budget

Innotec Corporation of Zeeland, MI, supported this project financially in addition to allowing access to their machine shop for the construction of the prototype. Below is a list of materials used in the engine.


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Part # Qty Cost Description Total

1 1 $ 890.00 Water Jet Material $ 890.00

2 5 $ 19.00 Water Jet Labor ~ Towers $ 95.00

3 4 $ 12.50 Water Jet Labor ~ Plates $ 50.00

4 6 $ 8.88 3/4" Shaft Bearing $ 53.28

5 1 $ 39.09 3/4"x48" Drive Shaft $ 39.09

6 3 $ 112.27 24" Drain pipe and shipping from WI $ 336.81

7 1 $ 192.07 Brass Mesh plus shipping $ 192.07

8 1 $ 21.84 Plunger rod #2, more precise $ 21.84

9 2 $ 11.02 Plunger sleeve #2, more precise $ 22.04

10 30 $ 2.83 14" zinc plated bolts (1/2-13) $ 84.90

11 8 $ 6.08 Ball Bearings 3/8 thick, 1 1/8 OD, 3/8 hole, Double shielded $ 48.64

12 4 $ 10.14 Rod Ends 3/8" hole, 5/8-18 male connect $ 40.56

13 4 $ 20.17 3/8 thick unpolished aluminum plate (2"X36") $ 80.68

14 1 $ 6.02 HI Temp Silicon $ 6.02

15 1 $ 32.25 4'x8' Plywood delivered to Calvin $ 32.25

16 1 $ 28.63 Hi Temp rubber strip for gasket - 3/8"x30' $ 28.63

17 2 $ 19.08 Pancake Griddle -Neil $ 38.16

18 2 $ 21.19 Pancake Griddle -Ben $ 42.38

19 1 $ 5.29 High Heat Black Paint $ 5.29

Materials $ 2,107.64

Labor Rate $ 25.00

Turnover Time 12

Labor Cost $ 300.00

Total: $ 2,407.64

Table 1. Bill of Materials

This cost represents a single prototype, if the engine was put into production, costs could be lowered because materials could be ordered in bulk. It is estimated that a total production cost of around $1200 per unit would be reasonable. If the design is further developed to the point of being capable of producing 1 hp a Stirling engine would break even with a conventional gasoline water pump after only 148 hours of use. This is based on gas prices in South Africa and a conventional water pump using one gallon of fuel every 90 minutes.


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4 Design Norms

Many engineering students think that ethics and other such philosophical issues are too abstract to be relevant to their engineering work, according to Professor S. VanderLeest and Professor G. Ermer. In their report, Using Design Norms to Teach Ethics, Ermer and VanderLeest explain how to categorize moral guidelines in a decision matrix with technical aspects of engineering design. Once such a structure is in place, it is possible to strategically select an optimal design that considers previously overlooked factors such as stewardship, cultural appropriateness, trust, caring, and integrity. This section of the report defines several moral factors and describes how they were included in the final design decision.

4.1 Cultural Appropriateness

Cultural appropriateness describes a necessity to consider the customer’s desires and comfort

during design. Problems can arise if the design conflicts with existing processes or local perspectives. During the design process, it is imperative that the cultural context is analyzed at all levels to ensure that smooth integration is possible.

For the Stirling engine project, several cultural factors adjusted crucial design decisions. The first of these was the initial project scope. Most successful Stirling engines operate in high temperature conditions, using combustion exhausts as energy sources. These engines produce significant power at a relatively high efficiency. The project scope of this design was to integrate Stirling technology in a region where such combustion processes do not exist. Instead, this engine was designed to operate at temperatures below 200º F (100º C). This adjustment limited the engine’s power producing capacity.

The second major adjustment was component material. Most low operating temperature Stirling engines use exotic materials such as anodized aluminum, helium gas, and elaborate pressure-tight gaskets. Such materials are not available in the region this product is intended for, so more common materials were selected such as aluminum sheets, atmospheric air, and bronze bushings. In addition to designing the materials around the target region’s resources, the processes required to machine the engine’s components were designed around the region’s existing machine shops. Therefore, standard equipment such as lathes, end mills, and welders were used to make every engine component.

4.2 Stewardship

Incorporating stewardship into the design process is a responsibility for those who feel called by

God to care for His creation. The first aspect of His creation comes in the form of the world around us. Stirling engines can be used to harness renewable energy sources such as solar or geothermal. Stirling engines have no emissions, little noise, and do not use hazardous materials or chemicals. In addition, Stirling engines can be used as cogeneration plants. In order to reject their excess heat many power plants run their output streams into the atmosphere or nearby lakes and rivers. In addition to changing and in many cases hurting the environment around them the plants are also wasting heat that they have paid for. Instead of wasting this resource Stirling engines could be used to recapture this heat thus increasing the overall efficiency of the plant.

The second aspect of His creation comes in the form of the people around us. Engineers have been given a gift in the form of their education and they should use this gift to help out their fellow man. A low temperature Stirling engine is capable of serving the needs of people in remote areas without access to electricity or fossil fuels. The engine has a wide variety of uses from running a grain mill or water pump to spinning an alternator to produce electric power. It is also able to accept a variety of heat sources including solar, geothermal, or an open fire. Engineers are supposed to solve problems and this project was geared towards providing a solution to these needs.


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5 Results

After many attempts the engine was finally able to run and sustain its own motion. The heat source

achieved a temperature of around 200ºF and the heat sink was at about 90ºF. At these temperatures the engine ran at around 10 rpm. When dry ice was added to the sink its temperature dropped to around 0 ºF increasing the engine speed to 18 rpm. A pressure gauge attached to the chamber read a change in pressure of about three quarters of a psi. The engine was able to sustain this motion for hours and even after the input energy stopped the engine continued to run for 5 minutes off of residual heat. The engine does not produce much excess power meaning that almost all of the power produced is used to overcome the internal friction; however the Stirling engine project is still considered a success as it proves that low temperature Stirling engines are a viable technology. With more time and further development a low temperature Stirling engine that produces useful power would be achievable.

6 Future Improvements

If this project were done over again there are some changes that should be made. A new linkage

system should be examined called a bell crank linkage. Although more complicated than the three bar design a bell crank linkage would decrease the amount of horizontal force on the piston and plunger thus reducing friction. This is achieved using pivoting triangles and provides a more vertical force on the output arm. For more information on these linkages see appendix D.

If the three bar linkage is still used instead of having a rectangular crank arm, a disk should be used in its place to allow for adjustable crank angles. This would allow for easy adjustment of the offset angle between the piston and the plunger. A greater angle would give the air more time to heat and would hopefully increase the pressure difference. Another suggestion for the linkage system is to use a system that allows the displacer to dwell at the top and bottom of the cycle. This would be most easily done using a cam surface. This would allow the displacer to move in a non-harmonic fashion and again allow the air more time to heat.

When examining the thermodynamics, including a layer of insulation around the outside of the heat source would increase the efficiency of the heat source. This would guarantee that the majority of the heat would enter the cylinder instead of convecting into the outside air. Fins should also be placed on the outside and inside of the heat sink allowing a greater surface area to reject heat.

7 Acknowledgments

We would like to acknowledge the following people for their assistance with our project. Without their help this project would have been much more difficult if even possible. Dave Ryskamp: for assisting in the fabrication processes. Professor Gayle Ermer: for helping with the mechanical model. Glenn Remelts: providing assistance in researching patents and engineering articles. Greg Bock: for giving constructive criticism. Greg Stapert: for assistance and advice in the design process. Harley Jongekrijg: for delegating a budget for this project. Matt Weeda: for advice, for the project idea, and for the invitation to assist Innotec in this development. Professor Heun: for helping with the thermal model. Professor J. Aubrey Sykes: for mentoring and giving constructive criticism. Innotec Corporation: For the availability of resources (personnel, machinery, knowledge, and materials).


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8 Appendix

Appendix A: Engine Manual Appendix B: Budget Appendix C: Mechanical Model Appendix D: Thermal Model

Appendix A: Engine Manual

CALVIN COLLEGE ENGINEERING Takin’ the Heat with Stirling Engines

Engine Manual

Team

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C A L V I N C O L L E G E E N G I N E E R I N G & I N N O T E C C O R P O R A T I O N

Stirling Engine Assembly Manual

© Calvin College Engineering 3201 Burton St SE

Grand Rapids, MI, 49546

Table of Contents

Initial Preparation 1 Engine Construction 2 Engine Frame 2 Flywheel 2 Regenerator 2 Linkage Components 3 Piston Assembly 3 Tower Assembly 4 Plate Assembly 4 Chamber Assembly 4 Plate Alignment 4 Piston Installation 5 Tower and Linkage Installation 5 Linkage and Flywheel Connection 5 Engine Troubleshooting 6

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Team

8 T A K I N ’ T H E H E A T D E S I G N M A N U A L

Product Description tirling engines capitalize on a previously wasted sources of energy. They are primarily used in situations where there is an abundant heat source, as in the exhaust flow of a combustion system or furnace. However, low energy sources are adequate if the Stirling system is designed carefully. This manual summarizes

the manufacturing and operating instructions for a low temperature difference Stirling engine.

S The engine was intentionally designed to be easily manufactured in large quantities. It utilizes the fewest possible components that are made from only standard materials. Its design allows for anyone to easily build one with access to a standard metal shop.

Initial Preparation Most engine components were purchased online from McMaster. Before beginning construction, setup an account on their website to decrease the shipping time for items as they are ordered. Once an account is activated, products can be delivered within 1-2 business days of the order. Their website is: http://www.mcmaster.com

For the few components that weren’t available through McMaster, contact local venders that can supply these parts. The towers and plates require a water-jet cutting facility that is capable of cutting aluminum plates sized 30”x50”. The piston assembly requires two large pneumatic cylinders with a 5” bore and 6” stroke. The mounting characteristics of these cylinders are arbitrary, so use what is readily available. The chamber is made from 24” fiberglass duct pipe that can be ordered from vendors over the internet. Have this pipe precut into 12” sections to reduce shipping costs. The regenerator is made from ¾” plywood available at any hardware store.

After component vendors have been selected, engine construction can begin.

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T A K I N ’ T H E H E A T D E S I G N M A N U A L

Engine Construction Begin by sending off CAD drawings of the tower and plates to the water-jet facility. These components will arrive about two weeks after placing the order. Next, order the drive shaft, drive bearings, regenerator shaft, regenerator bushings, chamber bolts, link bearings, link material, link rod-end connectors, hi-temp silicon sealer, hi-temp rubber gasket, and brass mesh from McMaster per the part numbers listed in the part list at the end of this document. Order the 24” fiberglass drain pipe cut into two 12” lengths, and insure that it is packaged well. Lastly, purchase a 4’x8’ sheet of layered plywood from a local hardware store.

E N G I N E F R A M E

Most of the components should arrive within 1-2 business days. While waiting for them to ship, begin construction of the engine frame. Using any reasonable square or channel stock available, cut out lengths to make two rectangles: 31” x 62” and 31” x 31”. After welding these rectangles, cut four 10” leg connectors from the same steel stock as before. Connect the two rectangles together by first welding the 10” legs to the corners of the smaller rectangle, creating the base assembly. Next, weld the larger rectangle to the base assembly.

The next step is to create what are called hold-er-on clamps. These are simple. First cut out eight hold-er-on plates: ¼” steel plates that are 4”x4”. Next plasma cut a 2”x3” hole offset from the center of each plate. Weld these plates to the frame where the engines will connect to the frame. Upon the construction’s completion, through bolts will connect the engines to these hold-er-on plates.

F L Y W H E E L

The flywheel should be made from a large disk about ¼” thick. If one is not available, have one laser cut by a local facility with a 1” hole in the center. The disk should weight about 80 pounds. Insert a 1”-3/4” bushing adaptor inside the center hole and weld in place, ensuring that the bushing is square in every dimension. Note that if the bushing isn’t square, the flywheel will wobble. Set flywheel aside.

R E G E N E R A T O R

Cut the plywood into two 23.75” diameter circles. This large diameter will leave close tolerances between the regenerator and the chamber walls. Trace out four wire mesh port holes onto each circle. Cut out these ports, leaving rounded corners for increased

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strength. Once the wire mesh arrives, cut 10 layers large enough to fit inside each port hole and staple them inside the ports of the disks.

The next step is to construct the regenerator mounting rod. Cut the ½” diameter steel regenerator shaft to the desired length and turn one end of the shaft down to ¼” about 1” long. Next, take a rod-end connector and ream a ¼” hole in the center to fit over the end of the regenerator shaft. After ensuring that the rod end is centered on the shaft, weld the rod end onto the shaft.

On the other end of the regenerator shaft, tap a ¼-20 hole with plenty of thread depth. Stamp out a small 5”x5” steel plate, and screw the plate to the center of one of the plywood disks. Drill a ¼” hole through the center of the plywood disk and steel plate. Repeat these steps to make the two regenerators and bolt the free end of the regenerator shaft to the center of the plate.

L I N K A G E C O M P O N E N T S

Cut the 3/8” aluminum bars into four 4”, four 8”, two 18”, and two 10” lengths. Mill two bearing holes into the ends of each pair of longer links, and take out extra side material as desired. Round the ends of these longer links so they fit inside the rod ends smoothly. Press fit the link bearings into each end of these longer links. For the eight shorter links, machine 3/8” pin holes into the correct locations.

P I S T O N A S S E M B L Y

The machining procedure will depend on the configuration of the available pneumatic cylinders. Completely disassemble the cylinders and remove the primary o-rings from around the bore and from inside the top seal. Drill several medium-sized holes inside the top seal to minimize flow restriction in the top portion of the cylinder. Machine the cylinder shafts down to 10” lengths, and tap a 5/8-18 threaded hole in the end. Next, screw the rod ends into these holes. Finally, spray 3-36 lubrication oil inside and around all seals and test the cylinder for friction resistance, which should be minimal.

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T O W E R A S S E M B L Y

After the towers arrive, validate all dimensions. Press fit the drive shaft bearings into each tower and set these towers aside.

P L A T E A S S E M B L Y

Starting from a large 1/8” aluminum sheet, stamp out the ½” strips needed to make the fin array. Weld the fin array onto one face of each of the four plates, ensuring that the funs to not block any bolt or locator holes on the plates. NOTE: Welding may cause thermal deformation of the plates, so ensure that these fins are only spot welded.

Cut eight 3”x1” aluminum strips from the large 1/8” sheet. These strips are covers for the sink-tower slots, and will be welded from the bottom side of the sink. Weld these plates in place, ensuring that no air bubbles are present so to create an air-tight seal.

Turn the 1” OD bronze regenerator bushings down to 0.751” and press fit these bushings into the sink plate by heating the sink plate with any torch available. Ensure that the heat is applied uniformly to both sides of the sink plate, so to not thermally deform the plate. Next, slide the regenerator shaft through the bronze bushing with the welded rod end on the outside of the sink plate. Attach the plywood regenerator assembly to the shaft using a ¼-20 bolt.

C H A M B E R A S S E M B L Y

Remove the protective backing from the high temperature rubber strip, exposing the adhesive on the rubber. Place the rubber strip around the perimeter of the chamber on both sides.

Center the fiberglass drain pipe onto the source plate. Note that the source plate is the plate that only has holes for the chamber bolts; it doesn’t contain the tower slots. Place the heat sink on top of the fiberglass pipe, lowering the regenerator assembly neatly inside the chamber as the sink approaches the chamber. This will enclose the regenerator inside the chamber. Secure the source and sink plates together by loosely bolting them with the long 14” chamber bolts. Repeat this process with the other chamber so that the two engines can be placed next to each other on the frame.

P L A T E A L I G N M E N T

Begin this process by placing the source plates side by side so that the edges are flush in all dimensions. Next, center a sink plate of one of the engines using a tape measure. Do this by measuring the distance between the chamber and the sink edge along all four sides and adjusting until the measurements are the same. Tighten this aligned engine together by carefully putting 10in-lb on each chamber bolt. Next, align the final

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sink plate against the first plate in all dimensions and tighten this sink down in the same way as the first. Ensure that both chambers are rigid and centered.

P I S T O N I N S T A L L A T I O N

Drill and tap four holes into the tops of the sink plates around the piston holes. These tapped holes will secure the air cylinders to the top plate. Apply a later of silicon caulk to the edge of the piston cylinder, center the piston cylinder over the hole in the sink, and tighten the four piston bolts. Ensure that the piston can slide freely, adjusting the cylinder alignment as necessary.

T O W E R A N D L I N K A G E I N S T A L L A T I O N

Cut the ¾” steel drive shaft into the desired lengths and weld the shaft sections to the crank arms, ensuring that the 90 degree phase offset is achieved on certain sections. Place the tower and drive shaft assemblies on top of the sink plate into the tower slots in the sink. Caulk around the outside of the slots ensuring an air-tight seal between the sink and the towers. Align the drive shafts between the two engines by adjusting the position and heights of the individual engines and towers. Rigidly mount the towers in place by adding supports as necessary. Using the hold-er-on plates, clamp the engines to the frame so that the shaft alignment is conserved.

Finally, spot weld the corners of each tower to the sink plate to insure the system’s rigidity.

L I N K A G E A N D F L Y W H E E L C O N N E C T I O N

Connect each link to the corresponding crank arm using 3/8” shoulder bolts. Lift the flywheel into position, and slide onto the driveshaft between the two engines. Tighten the set screw on the flywheel, securing it in place. Connect the two drive shafts together with a love-joy coupling, ensuring that the alignment between the shafts is mostly centered. Tighten the coupling so that the drive shafts offset the two engine phases by 180 degrees. This offset will keep the system masses balanced.

Manually rotate the drive shaft and adjust the engine position, tower rigidity, shaft position, and bearing lubrication as necessary to minimize the internal friction of the system. If possible, attach a large pulley to the drive shaft and run the engine for 10,000 cycles to wear in the engine components. This wear will reduce friction and increase air-tightness of the bushing connections.

5


Engine Troubleshooting Now that the engine is built, the source plate can be heated to 250 F and the engine should work well. If there are still problems, the following section may provide some insight. One of the first things to consider is the alignment. Make sure that the two chambers are aligned accurately in both the x and y directions. Also, make sure that the shafts are off set by 90 degrees as intended in the original design. Then, check that there is no air leak in the system. This will ensure that all the pressure difference created by the heated air is being used most efficiently. Next, check and make sure that the two different heat sources are at the same temperature. If the problem still persists try using ice. Placing ice on the top plates of the two chambers will help in increasing the temperature difference, thus increasing the pressure difference and this should help in running the engine. Using dry ice is also a good idea as it evaporates faster, thus increasing the rate at which the engine runs. However, be careful about placing the dry ice on the top plate. Make sure the dry ice isn’t too close to the brass sleeves and any other crucial connections because it freezes the lubrication and contracts the brass.

6

Appendix B: Budget

Bill of Materials Prototype





4 6 $ 8.88 3/4" Shaft Bearing $ 53.28

5 1 $ 39.09 3/4"x48" Drive Shaft $ 39.09









14 1 $ 6.02 HI Temp Silicon $ 6.02






Materials $ 2,107.64

Labor Rate $ 25.00

Turnover Time 12

Labor Cost $ 300.00

Total: $ 2,407.64

Bill of Materials Production





4 6 $ 8.88 3/4" Shaft Bearing $ 53.28

5 1 $ 39.09 3/4"x48" Drive Shaft $ 39.09









14 1 $ 6.02 HI Temp Silicon $ 6.02






Materials $ 694.68

Labor Rate $ 25.00

Turnover Time 12

Labor Cost $ 300.00

Total: $ 994.68

Appendix C: Mechanical Model The purpose of the mechanical model is to calculate the amount of shaft work that the engine is capable of producing. The pressures determined from the thermal model are used as inputs for the mechanical model. There are two reasons for creating a mechanical model. First, it calculates the output power for a specific pressure curve. Secondly, the mechanical model can be used to optimize the power output by changing the link lengths, masses, or timing.

Slider Crank Analysis

For the mechanical model two main approaches were used. The first is a Slider Crank Analysis. This model is used to find the position, velocity, and acceleration of each link throughout the engine cycle. The engine has two linkages that can be modeled as slider cranks, the piston and the displacer.

Figure 1. Slider Crank Design

The base case Stirling engine calls for the piston and the displacer linkages to be connected to the same crank shaft, and as a result both crank arms move at the same speed. Since the two links are running at the same speed the only way to change the timing of the engine is to change the offset angle between them.

Linkage Force Analysis

The second part of the mechanical model is a Linkage Force Analysis on the piston linkage. Now that the location of each link is determined, the next step is to see how the forces are transferred through the linkage. This analysis is done on the piston and not the displacer because the pressure difference is only seen on either side of the piston. The displacer is enclosed in the chamber of our engine and therefore does not have a pressure difference. The Linkage Force Analysis examines each link individually by summing the forces and moments acting on that link. To use this model some assumptions had to be made, however, these assumptions can be changed to more accurately reflect the actual parts. A piston diameter is chosen and the pressure is used as an input force on the bottom of the piston. This force travels through the piston linkage and creates a torque on the crank shaft.

Model Flaws

There are a few problems with the mechanical model. First, it does not take friction into account. One of the biggest problems when building a Stirling engine is overcoming the internal friction. The power output of the model is an extremely ideal case. The actual value will be quite a bit less. Secondly this model assumes the engine is moving at a constant rotational speed. In actuality an engine accelerates during its power stroke and decelerates the rest of the time. Using this ideal condition of a constant speed means that the model does not take into account the forces needed to sustain this motion. This approximation again overestimates the total power output. It is important to realize that the model is a completely idealized version of the engine and that the actual output will be considerably less than what is calculated.

File:X:\ENGINEERING\TEAMS\TEAM 8\MECHANICAL CALCULATIONS\MECHMODEL.EES 12/7/2006 8:18:06 PM Page 1

EES Ver. 7.699: #1896: For use only by students and faculty in the Calvin College Engineering Grand Rapids, MI

l+s <= other two

Constants

Piston

r3 = 0.2032 [m] 3 bar linkage connecting the Piston to the crank shaft

r2 = 0.0508 [m]

theta1 = 270 [deg]

ω rpm = 853

OMEGA2 = ω rpm · 2 · π

60

alpha2 = 0 [1/s2]

Plunger

r3pl = 0.5032 [m] 3 bar linkage connecting the Plunger to the crank shaft

2 · r2pl = 0.25 [m]

theta1pl = 270 [deg]

offset = 90 [deg]

Force Analysis

r32 = 0.0508 [m]

r34 = r3 – r32

m3 = 0.042260294 [kg]

m4 = 0.028173529 [kg]

I3 = 0.0075 · 1.35582 · ·N–s2–m

lb f–s2–ft

g = 9.807 [m/s2]

Rpiston = 0.0254 [m]

Area = π · Rpiston2

Lookup Table begins

N rows = 20

theta2 i = Lookup 'Lookup 1', i , 'THETA2' for i = 1 to N rows Crank Shaft Piston

r1 i · cos theta1 = r2 · cos theta2 i + r3 · cos theta3 i for i = 1 to N rows Position Piston

r1 i · sin theta1 = r2 · sin theta2 i + r3 · sin theta3 i for i = 1 to N rows



v i · cos theta1 = – r2 · OMEGA2 · sin theta2 i – r3 · OMEGA3 i · sin theta3 i for i = 1 to N rows

Velocity Piston

v i · sin theta1 = r2 · OMEGA2 · cos theta2 i + r3 · OMEGA3 i · cos theta3 i for i = 1 to N rows

a i · cos theta1 = – r2 · alpha2 · sin theta2 i – r2 · OMEGA22 · cos theta2 i – r3 · alpha3 i · sin theta3 i

– r3 · OMEGA3 i2 · cos theta3 i for i = 1 to N rows

Acceleration Piston

a i · sin theta1 = r2 · alpha2 · cos theta2 i – r2 · OMEGA22 · sin theta2 i + r3 · alpha3 i · cos theta3 i – r3

· OMEGA3 i2 · sin theta3 i for i = 1 to N rows

theta2pl,i = theta2 i + offset for i = 1 to N rows Crank Shaft Plunger

r1pl,i · cos theta1pl = r2pl · cos theta2pl,i + r3pl · cos theta3pl,i for i = 1 to N rows Position Plunger

r1pl,i · sin theta1pl = r2pl · sin theta2pl,i + r3pl · sin theta3pl,i for i = 1 to N rows

vpl,i · cos theta1pl = – r2pl · OMEGA2 · sin theta2pl,i – r3pl · omega3pl,i · sin theta3pl,i for i = 1 to N rows

Velocity Plunger

vpl,i · sin theta1pl = r2pl · OMEGA2 · cos theta2pl,i + r3pl · omega3pl,i · cos theta3pl,i for i = 1 to N rows

apl,i · cos theta1pl = – r2pl · alpha2 · sin theta2pl,i – r2pl · OMEGA22 · cos theta2pl,i – r3pl · alpha3pl,i

· sin theta3pl,i – r3pl · omega3pl,i2 · cos theta3pl,i for i = 1 to N rows

apl,i · sin theta1pl = r2pl · alpha2 · cos theta2pl,i – r2pl · OMEGA22 · sin theta2pl,i + r3pl · alpha3pl,i

· cos theta3pl,i – r3pl · omega3pl,i2 · sin theta3pl,i for i = 1 to N rows

Force Analysis

r23,x,i = r2 · cos theta2 i for i = 1 to N rows

r23,y,i = r2 · sin theta2 i for i = 1 to N rows

r32,x,i = – r32 · cos theta3 i for i = 1 to N rows

r32,y,i = – r32 · sin theta3 i for i = 1 to N rows

r34,x,i = r34 · cos theta3 i for i = 1 to N rows

r34,y,i = r34 · sin theta3 i for i = 1 to N rows

ag3x,i = – r2 · OMEGA22 · cos theta2 i – r32 · OMEGA3 i

2 · cos theta3 i – r32 · alpha3 i · sin theta3 i

for i = 1 to N rows

ag3y,i = – r2 · OMEGA22 · sin theta2 i – r32 · OMEGA3 i

2 · sin theta3 i + r32 · alpha3 i · cos theta3 i

for i = 1 to N rows

ag4y,i = a i for i = 1 to N rows

Link 2

F12,x,i + F32,x,i = 0 for i = 1 to N rows



F12,y,i + F32,y,i = 0 for i = 1 to N rows

r23,x,i · F32,y,i – r23,y,i · F32,x,i + Tshaft,i = 0 for i = 1 to N rows

Link 3

– F32,x,i + F43,x,i = m3 · ag3x,i for i = 1 to N rows

– F32,y,i + F43,y,i – m3 · g = m3 · ag3y,i for i = 1 to N rows

– r32,x,i · F32,y,i + r32,y,i · F32,x,i + r34,x,i · F43,y,i – r34,y,i · F43,x,i = I3 · alpha3 i for i = 1 to N rows

Link 4

– F43,y,i + Fpress,i – m4 · g = m4 · ag4y,i for i = 1 to N rows

Pressure Analysis

height i = r2pl + r3pl – r1pl,i for i = 1 to N rows

pressure i = 502.05 [kPa/m2] · height i

2 + 25.767 [kPa/m] · height i + 125.48 [kPa] – 101.325 [kPa]

for i = 1 to N rows

pressure i = Fpress,i

Area · 0.001 ·

kPa

Pa for i = 1 to N rows

Tavg = Average Tshaft,1..N,rows

Power = – Tavg · OMEGA2

SOLUTION

Unit Settings: [kJ]/[K]/[kPa]/[kg]/[degrees]

alpha2 = 0 [1/s2] Area = 0.002027 [m2] g = 9.807 [m/s2]

I3 = 0.01017 [N-s2-m] m3 = 0.04226 [kg] m4 = 0.02817 [kg]

Nrows = 20 offset = 90 [deg] OMEGA2 = 89.33 [1/s]

ωrpm = 853 [-] Power = 89.22 [W] r2 = 0.0508 [m]

r2pl = 0.125 [m] r3 = 0.2032 [m] r3pl = 0.5032 [m]

r32 = 0.0508 [m] r34 = 0.1524 [m] Rpiston = 0.0254 [m]

theta1 = 270 [deg] theta1pl = 270 [deg] Tavg = -0.9988 [N-m]

No unit problems were detected.

Arrays Table

theta2i Tshaft,i r1i theta3i r1pl,i theta2pl,i theta3pl,i omega3i

[deg] [N-m] [m] [deg] [m] [deg] [deg] [1/s]

1 0 -6.252 0.1967 255.5 0.3782 90 270 -0.000002989

2 18 -3.757 0.1817 256.2 0.3828 108 274.4 7.105

3 36 -1.468 0.1691 258.3 0.3967 126 278.4 13.4

4 54 -0.04788 0.1599 261.5 0.4195 144 281.6 18.26

5 72 0.3605 0.1543 265.6 0.4503 162 283.7 21.3

6 90 1.640E-07 0.1524 270 0.4874 180 284.4 22.33

7 108 -0.6704 0.1543 274.4 0.5276 198 283.7 21.3

8 126 -1.095 0.1599 278.5 0.5664 216 281.6 18.26



Arrays Table

theta2i Tshaft,i r1i theta3i r1pl,i theta2pl,i theta3pl,i omega3i

[deg] [N-m] [m] [deg] [m] [deg] [deg] [1/s]

9 144 -0.7759 0.1691 281.7 0.5989 234 278.4 13.4

10 162 0.4827 0.1817 283.8 0.6206 252 274.4 7.105

11 180 2.358 0.1967 284.5 0.6282 270 270 9.965E-07

12 198 4.1 0.2131 283.8 0.6206 288 265.6 -7.105

13 216 4.948 0.2289 281.7 0.5989 306 261.6 -13.4

14 234 4.508 0.2421 278.5 0.5664 324 258.4 -18.26

15 252 2.771 0.2509 274.4 0.5276 342 256.3 -21.3

16 270 2.047E-09 0.254 270 0.4874 360 255.6 -22.33

17 288 -3.275 0.2509 265.6 0.4503 378 256.3 -21.3

18 306 -6.239 0.2421 261.5 0.4195 396 258.4 -18.26

19 324 -7.984 0.2289 258.3 0.3967 414 261.6 -13.4

20 342 -7.939 0.2131 256.2 0.3828 432 265.6 -7.105

Arrays Table

vi omega3pl,i vpl,i ai alpha3i apl,i alpha3pl,i r23,x,i

[m/s] [1/s] [m/s] [m/s2] [1/s2] [m/s2] [1/s2] [m]

1 -4.538 22.19 -7.252E-07 104.7 2060 749.6 0.0001607 0.0508

2 -3.972 21.17 2.633 209.1 1941 746.2 -579.8 0.04831

3 -3.12 18.15 5.23 268.7 1611 725.9 -1129 0.0411

4 -2.122 13.31 7.687 294.8 1136 660.7 -1601 0.02986

5 -1.068 7.057 9.78 302.7 583.1 513.1 -1928 0.0157

6 -2.941E-07 9.897E-07 11.17 304 0.0001616 255.8 -2046 -2.195E-09

7 1.068 -7.057 11.46 302.7 -583.1 -103.4 -1928 -0.0157

8 2.122 -13.31 10.38 294.8 -1136 -511.8 -1601 -0.02986

9 3.12 -18.15 7.896 268.7 -1611 -887.9 -1129 -0.0411

10 3.972 -21.17 4.268 209.1 -1941 -1151 -579.8 -0.04831

11 4.538 -22.19 -1.150E-12 104.7 -2060 -1245 -5.073E-11 -0.0508

12 4.659 -21.17 -4.268 -41.45 -1941 -1151 579.8 -0.04831

13 4.222 -18.15 -7.896 -207.8 -1611 -887.9 1129 -0.0411

14 3.213 -13.31 -10.38 -361.1 -1136 -511.8 1601 -0.02986

15 1.737 -7.057 -11.46 -468.3 -583.1 -103.4 1928 -0.0157

16 -9.242E-24 9.897E-07 -11.17 -506.7 3.082E-20 255.8 2046 6.584E-09

17 -1.737 7.057 -9.78 -468.3 583.1 513.1 1928 0.0157

18 -3.213 13.31 -7.687 -361.1 1136 660.7 1601 0.02986

19 -4.222 18.15 -5.23 -207.8 1611 725.9 1129 0.0411

20 -4.659 21.17 -2.633 -41.45 1941 746.2 579.8 0.04831

Arrays Table

r23,y,i r32,x,i r32,y,i r34,x,i r34,y,i ag3x,i ag3y,i ag4y,i Fpress,i

[m] [m] [m] [m] [m] [m/s2] [m/s2] [m/s2] [N]

1 0 0.0127 0.04919 -0.0381 -0.1476 -304 -26.16 104.7 125.6

2 0.0157 0.01208 0.04934 -0.03624 -0.148 -289.1 -146.2 209.1 123

3 0.02986 0.01027 0.04975 -0.03082 -0.1493 -245.9 -245.9 268.7 115.6

4 0.0411 0.007465 0.05025 -0.02239 -0.1507 -178.7 -319.6 294.8 104.2

5 0.04831 0.003925 0.05065 -0.01177 -0.1519 -93.94 -364.8 302.7 90.44

6 0.0508 -5.487E-09 0.0508 1.646E-08 -0.1524 0.00002299 -380 304 76.48

7 0.04831 -0.003925 0.05065 0.01177 -0.1519 93.94 -364.8 302.7 64.51

8 0.0411 -0.007465 0.05025 0.02239 -0.1507 178.7 -319.6 294.8 56.07

9 0.02986 -0.01027 0.04975 0.03082 -0.1493 245.9 -245.9 268.7 51.36

10 0.0157 -0.01208 0.04934 0.03624 -0.148 289.1 -146.2 209.1 49.41



Arrays Table

r23,y,i r32,x,i r32,y,i r34,x,i r34,y,i ag3x,i ag3y,i ag4y,i Fpress,i

[m] [m] [m] [m] [m] [m/s2] [m/s2] [m/s2] [N]

11 -4.390E-09 -0.0127 0.04919 0.0381 -0.1476 304 -26.16 104.7 48.96

12 -0.0157 -0.01208 0.04934 0.03624 -0.148 289.1 104.3 -41.45 49.41

13 -0.02986 -0.01027 0.04975 0.03082 -0.1493 245.9 230.6 -207.8 51.36

14 -0.0411 -0.007465 0.05025 0.02239 -0.1507 178.7 336.2 -361.1 56.07

15 -0.04831 -0.003925 0.05065 0.01177 -0.1519 93.94 406.2 -468.3 64.51

16 -0.0508 -6.584E-09 0.0508 1.975E-08 -0.1524 -0.00005582 430.7 -506.7 76.48

17 -0.04831 0.003925 0.05065 -0.01177 -0.1519 -93.94 406.2 -468.3 90.44

18 -0.0411 0.007465 0.05025 -0.02239 -0.1507 -178.7 336.2 -361.1 104.2

19 -0.02986 0.01027 0.04975 -0.03082 -0.1493 -245.9 230.6 -207.8 115.6

20 -0.0157 0.01208 0.04934 -0.03624 -0.148 -289.1 104.3 -41.45 123

Arrays Table

pressurei heighti F12,x,i F12,y,i F32,x,i F32,y,i F43,x,i F43,y,i

[kPa] [m] [N] [N] [N] [N] [N] [N]

1 61.97 0.25 -147.8 -123.1 147.8 123.1 134.9 122.4

2 60.7 0.2454 -138.1 -122.6 138.1 122.6 125.9 116.9

3 57.03 0.2315 -112.9 -117.7 112.9 117.7 102.5 107.7

4 51.41 0.2087 -77.82 -108.7 77.82 108.7 70.27 95.62

5 44.62 0.1779 -38.86 -96.64 38.86 96.64 34.89 81.64

6 37.73 0.1408 3.696E-07 -83.28 -3.696E-07 83.28 6.018E-07 67.63

7 31.83 0.1006 36.85 -70.71 -36.85 70.71 -32.88 55.71

8 27.66 0.06179 70.67 -60.58 -70.67 60.58 -63.12 47.49

9 25.34 0.02927 99.6 -53.49 -99.6 53.49 -89.21 43.51

10 24.38 0.007603 120.1 -49.01 -120.1 49.01 -107.9 43.25

11 24.15 -1.111E-09 128 -46.42 -128 46.42 -115.1 45.73

12 24.38 0.007603 121.2 -45.48 -121.2 45.48 -109 50.31

13 25.34 0.02927 101.3 -46.77 -101.3 46.77 -90.94 56.94

14 27.66 0.06179 72.38 -51.34 -72.38 51.34 -64.83 65.97

15 31.83 0.1006 37.9 -59.85 -37.9 59.85 -33.93 77.43

16 37.73 0.1408 0.000009354 -71.86 -0.000009354 71.86 -0.00001171 90.47

17 44.62 0.1779 -39.91 -85.78 39.91 85.78 35.94 103.4

18 51.41 0.2087 -79.53 -99.47 79.53 99.47 71.98 114.1

19 57.03 0.2315 -114.6 -111 114.6 111 104.2 121.2

20 60.7 0.2454 -139.2 -119.1 139.2 119.1 127 123.9

Appendix D: Thermal Model The thermal model was constructed because there was a need to calculate the pressure drop in the Stirling engine cycle. This information is necessary to determine the power of the engine itself. Throughout the engine’s cycle, the fluid pressure is constantly fluctuating between low and high pressures. During the low pressure, the piston is driven down by atmospheric pressure. During the high pressure, the piston is driven upwards by the fluid pressure inside the chamber. Thus, as the gap between these pressures increase, the capability of the engine increase. The inputs to this model are component dimensions and materials, inlet radiation energy, and surrounding temperature, pressure, and wind conditions. The outputs are system temperatures and pressures at different points during the Stirling cycle. The thermal model consists of three main components: conduction, convection, and radiation. A thermal model for this system is vital for two reasons. The first is that this system is completely driven by transferring energy via a heat exchange from some abundant source. This model confirms the feasibility of this system by calculating the effects of the surroundings on the system. These effects can be analyzed to predict if a certain energy input can power the system. The second reason why a thermal model is vital for this system is because it allowed for a comparison of design alternatives without expensive prototypes. Component material, dimensions, and properties can be varied thus allowing for quick adjustments and graphical comparisons of any desired calculation. Because the model is segmented into individual components, optimization can target specific parts of the design.

Conduction

The conduction model considers only heat flow through the material medium. This is the most basic form of the conduction equation used in this model.

In this equation, ‘k’ is the conductivity of the component material, ‘A’ is the cross sectional area of the energy flow, ‘∆T’ is the temperature difference between sides of the material, and ‘∆x’ is the thickness of the material. In the gamma type Stirling engine, there are four main conduction flows. They are: conduction through the heat source, vertically through the fluid chamber walls, horizontally through the fluid chamber walls, and through the heat sink. The pairs of red arrows in Figure 7 represent these heat flows.

x

T -kA Q Conduction

∆

∆ =&

Negligible conduction heat transfers are from the heat sink to the mechanical components above the sink and from the heat source to the hardware that mounts the engine. These were ignored in this model. Also, the displacer is modeled as a lumped solid, wiping away the need to consider conduction in it.

Convection

The convection model considers only heat flow between a solid medium and a fluid. This is the most basic form of the convection equation used in this model. In this equation, ‘h’ is the convection coefficient, ‘A’ is the surface area of exposure, and ‘∆T’ is the temperature difference between the fluid and the surface. There are several convection flows, each modeled as forced convection. The heat source has convection flows to both the ambient air and to the system fluid. Convection to the ambient air from the heat source is to be minimized, while convection to the system fluid is to be maximized to achieve maximum performance. The chamber walls also have convection flows to both ambient air and the system fluid. Optimally, these are both minimized so the fluid keeps a maximal amount of energy. The heat sink has both convection flows; however these flows are both maximized to achieve maximum performance. The logic behind these initial assumptions is: as the fluid cycles up and down in the chamber, it caries energy from the heat source to the heat sink. Thus the energy flow is vertical. Any flow that is not upward is minimized. Figure 8 presents this idea: the large red arrows represent that most of the convective heat flow is vertical. Additional convection flows occur inside the chamber because the displacer constantly transfers heat in and out of the working fluid.

T A h Q conv Convection ∆ = &

Negligible convection heat transfers are from the displacer connecting rod to the fluid and from the piston (not pictured, but on top of the heat sink) to the fluid. These were ignored in this model.

1.1.1 Radiation

The radiation model considers only heat flow between a heat source and sink that are separated by a certain medium. This is the most basic form of the radiation equation used in this model.

)( 4TAQRadiation ∆⋅= σε&

In this equation, ‘ε’ is the surface emissivity, ‘A’ is the surface area of exposure, ‘σ’ is the Stefan-Boltzmann constant, and ‘∆(T4)’ is the difference between two surface temperatures, each raised to the fourth power. Note that for incident radiation, the higher surface temperature is that of the ambient air. There are several radiation flows. The driver of the system is the sun’s radiation heat flow to the system heat source. However, this heat source could also be an open flame or a geothermal source as described above. In addition, there is a radiation flow from the source to the chamber wall and to the displacer. The chamber walls have radiation flows from the sun on the outside, and radiation flows from the source and to the sink as well as to the displacer on the inside. The sink also has a radiation flow from the sun. The sink’s radiation flows are minimized in order to maximize its effectiveness as a heat sink. The sink also has radiation flows from the displacer and walls inside the chamber.

Negligible radiation heat transfers are from the ambient air to the sides of the source and sink and from the displacer to the displacer connecting rod. These were ignored in this model.

Model Flaws

This first thermal model not only utilized these three heat transfer types, it was completely constructed around them as well. (The model had three sections: conduction, convection, and radiation.) For instance, under the conduction section, all energy flows accounted for were listed at once, one right after another. This made the model impossible to compile. It couldn’t be broken down to calculate guess values because every variable depended on something else. A new method was necessary that utilize the same heat transfer categories, but allowed the model to be compiled in pieces.

Thermal Model #2: Control Volume

After re-examining the system, a different approach was used. Instead of breaking the system down strictly by heat transfer (i.e.: conduction, convection, and radiation), a control volume process was used. First, all control volumes were identified. Then, the first law of thermodynamics was applied by establishing energy accounting equations on the control volumes one by one. To find these energy terms, certain variables were given reasonable guess values until another control volume solved for that variable. In this way, the thermal model assuredly solved for every energy term, leaving no thermal guesses. The second model has only dimension, material, and mechanical assumptions. Each of these can be varied to produce estimates for an optimal solution. Please refer to the Analysis section below for these results. Note that this second approach still utilized the same heat transfer means as the first. However rather than compiling the entire system at once, one control volume was compiled at a time. Below, figure 10 depicts these control volumes, each separated by blue borders. Table 3 breaks down the heat transfer terms that are included in each control volume.

CV Energy Accounting Equation – (In’s = Out’s)

1 SourceCondAirSourceConvSourceIncidentRad QQQ ,,,&&& +=

→→

2 DisplacerSourceRadFluidSourceConvVerticalWallCondSourceeCond QQQQ→→

++= ,,,,&&&&

3 AirWallConvHorizontalWallCondWallIncidentRad QQQ→→

=+ ,,,&&&

4 SinkWallRadWallCondFluidWallConvWallSourceRad QQQQHorizontal →→→

+=+ ,,,,&&&&

5 SinkDisplacerRadFluidDisplacerConvDisplacerSourcetRad QQQ→→→

+⋅= ,,, 2 &&&

6 SinkCondSinkDisplacerRadSinkFluidConvVerticalWallCond QQQQ ,,,,&&&& =++

→→

7 AirSinkConvSourceIncidentRadSinkCond QQQ→→

=+ ,,,&&

8 SinkFluidConvFluidWallConvFluidDisplacerConvFluidSourceConv QQQQ→→→→

=+⋅+ ,,,, 2 &&&&

In Table

• RadQ& refers to radiation heat transfer. BA→refers to a transfer of energy from component ‘A’ to ‘B’.

• ConvQ& refers to convection heat transfer. BA→refers to a transfer of energy from component ‘A’ to ‘B’.

• CondQ& refers to conduction heat transfer for a particular component.

• Each term represents a red arrow from one of the above figures.

File:THERMAL INTERIM NEW1G.EES 5/7/2007 4:38:50 PM Page 1


Function Nu (Re, Pr)

If Re < 500000 Then Dummy := 0.664 · Re0.5

· Pr1 / 3 Else

Dummy := 0.037 · Re0.8

· Pr1 / 3 EndIf

ν := Dummy

End Nu

FUNCTION KEY

For NUSSELT & REYNOLDS NUMBERS

1 Sourcebottom

2 Sourcetop & Sinkbottom

3 Displacer

4 Walloutside

5 Wallinside

6 Sinktop

For PRANDTL numbers

1 air

2 fluid

Adjust these variables:

SINK$ = 'Aluminum'

SOURCE$ = 'StainlessAISI347'

WALL$ = 'Glass fiber batt rho=40'

FLUID$ = 'Air'

radiussource = 0.75 [m] Assume reasonable value here - ADJUST THIS LATER

Lsource = 0.01 thickness of source

Lsink = 0.01 [m] Thickness of sink

Lwall = 0.25 [m] Height of wall

Thicknesswall = 0.01 [m]

radiusdisp = radiussource · 0.9 Assume reasonable value here - disp is slightly smaller

thickdisp = 0.02 [m]

ANGVELOCITY = 800 [rev/min] BEN MEAD: SET THIS

Velair = 3.03 [m/s] Wind velocity at source

Velair,wall = 6 [m/s] Side wind velocity

Velair,sink = 10 [m/s] Wind velocity at sink



factor2 = 0.2 Fractional temperature difference b/w plate and wall surfaces

VARYTHIS = 0.5 Adjusts height of displacer - move from 0-1

Source - Bottom

Qconv,source,air = hconv,source,1 · Asource · Tsource,bot – T∞

Qcond,source = ksource · Asource · Tsource,bot – Tsource,top

Lsource

ksource = k SOURCE$ , Tsource

Asource = π · radiussource

2

Nusselt1 = hconv,source,1 · Lc,air

kair

kair = k 'Air' , T =Tair

Nusselt1 = Nu RE1 , Pr1

RE1 = Velair · Lc,air

νair

Lc,air = radiussource · 2 Assumes flow covers only a certain fraction of whole plate

νair = Visc 'Air' , T =Tair

ρair

ρair = ρ 'Air' , T =Tair , P =Pair

Pr1 = Pr 'Air' , T =Tair

Tair = T∞

T∞

= 303.15 [K]

Pair = 101.3 [kPa]

Tsource = Tsource,top + Tsource,bot

2

BIOTsource = hconv,source,1 · Lc,air

kair

Guesses

Tsource,bot = 373 [K]

Energy Accounting

0 = Qconv,source,air – Qcond,source

Source - Top



Qcond,wall = kwall · Awall · Twall,bot – Twall,top

Lwall

Qconv,source,fluid = hconv,source,2 · Asource · Tsource,top – T fluid

Q rad,source,disp = F1 · εsource · σ · Asource · Tsource,top

4 – Tdisp

4

kwall = k WALL$ , Twall

Awall = π · radiuswall

2 – radiuswall – Thicknesswall

2

Twall = Twall,top + Twall,bot

2

Twall,top = Tsink,bot · 1 + factor2

Twall,bot = Tsource,top · 1 – factor2

radiuswall = radiussource

Nusselt2 = hconv,source,2 · Lc,fluid

k fluid

k fluid = k FLUID$ , T =T fluid

Nusselt2 = 0.037 · Re2

0.8 · Pr2

1 / 3

Re2 = Vel fluid · Lc,fluid

ν fluid

Pr2 = Pr FLUID$ , T =T fluid

Lc,fluid = radiussource · 1.5 Assumes flow covers only a certain fraction of whole plate

ν fluid = Visc FLUID$ , T =T fluid

ρ fluid

ρ fluid = ρ FLUID$ , T =T fluid , P =P fluid

Vel fluid = VELplate Fluid velocity inside chamber

F1 = 0.5 · S1 – S1

2 – 4 ·

radiusdisp

radiussource

2 0.5

S1 = 1 +

1 + radiusdisp

heightdisp

2

radiussource

heightdisp

2

heightdisp = Lwall · VARYTHIS VARY THIS IN A PARAMETRIC TABLE

σ = 5.67 · 10– 8

· 1 [W/m2-K

4]



εsource = 0.17 for polished stainless: see p724 in 328 book

Guesses

Energy Accounting

Qcond,source = Qcond,wall + Qconv,source,fluid + Q rad,source,disp

Displacer

Q rad,disp,sink = F2 · εdisp · σ · Adisp · Tdisp

4 – Tsink,bot

4

Qconv,disp,fluid = hconv,disp,1 · Adisp · Tdisp – T fluid

Givens

Nusselt3 = hconv,disp,1 · Lc,disp

k fluid


RE3 = Vel fluid · Lc,disp

ν fluid

Lc,disp = radiusdisp · 2

F2 = 0.5 · S2 – S2

2 – 4 ·

radiussink

radiusdisp

2 0.5

S2 = 1 +

1 + radiussink

Lwall – thickdisp – heightdisp

2

radiusdisp

Lwall – thickdisp – heightdisp

2

εdisp = 0.07 for alluminum foil: see p724 in 328 book

radiussink = radiussource assumes source and sink are same size

Adisp = radiusdisp

2 · π

Guesses

Energy Accounting

Q rad,source,disp = Q rad,disp,sink + 2 · Qconv,disp,fluid

Wall - Outside

Q rad,incident,wall = εwall · σ · Awall · Twall

4 – T

∞

4

Qcond,wall,horizontal = kwall · Awall,horizontal · Twall,outside – Twall,inside

Thicknesswall

Qconv,wall,air = hconv,wall,2 · Awall · Twall – T∞



Givens

Nusselt4 = hconv,wall,2 · Dwall

kair

Nusselt4 = 0.027 · RE4

0.805 · Pr1

1 / 3For Re > 40000....... see p867 in Cengel #1 for other values

Dwall = radiuswall · 2

RE4 = Velair,wall · Lc,wall

νair

Lc,wall = Lwall

εwall = 0.07

Awall,horizontal = radiuswall · 2 · π · Lwall

Guesses

Energy Accounting

Q rad,incident,wall + Qcond,wall,horizontal = Qconv,wall,air

Wall - Inside

Qconv,wall,fluid = hconv,wall,1 · Awall,horizontal · T fluid – Twall,inside

Q rad,source,wall = 0

Q rad,wall,sink = 0

Givens

Nusselt5 = hconv,wall,1 · Lwall

k fluid

Nusselt5 = 0.023 · RE5

0.8 · Pr2

0.4Assumes TURBULANT & that wall heats air

RE5 = Vel fluid,wall · Lwall

ν fluid

Vel fluid,wall = VELwall Fluid velocity inside chamber

Guesses

Energy Accounting

Q rad,source,wall + Qconv,wall,fluid = Qcond,wall,horizontal + Q rad,wall,sink

Sink - Bottom

Qcond,sink = ksink · Asink · Tsink,bot – Tsink,top

Lsink



Qconv,fluid,sink = hconv,sink,2 · Asink · T fluid – Tsink,bot

Givens

Nusselt2 = hconv,sink,2 · Lc,fluid

k fluid

Asink = radiussink

2 · π

ksink = k SINK$ , Tsink

Cp,sink = c SINK$ , Tsink

ρsink = rho SINK$ , Tsink

Tsink = Tsink,top + Tsink,bot

2

Guesses

Energy Accounting

Qconv,fluid,sink + Q rad,disp,sink + Qcond,wall = Qcond,sink

Sink - Top

Q rad,incident,sink = εsink · σ · Asink · Tsink,top

4 – T

∞

4

Qconv,sink,air = hconv,sink,1 · Asink · Tsink,top – T∞

Givens

Nusselt6 = hconv,sink,1 · Lc,air,sink

kair


RE6 = Velair,sink · Lc,air

νair

Lc,air,sink = radiussource · 1.9 Assumes flow is restricted b/c of flywheel mount

εsink = 0.07 for polished stainless: see p724 in 328 book

Energy Accounting

Qcond,sink + Q rad,incident,sink = Qconv,sink,air

Fluid - To Find Tfluid

Qconv,source,fluid + 2 · Qconv,disp,fluid – Qconv,fluid,sink + Qconv,wall,fluid = 0

NOW FIND VELOCITIES

VOLbelow,disp = Adisp · ANGdisp,velocity,avg



VELwall = VOLbelow,disp

AREAaround,disp

VELplate = VOLbelow,disp

Asource

ANGcycles,per,sec = ANGVELOCITY · 0.016667 · min

sec

ANGdisp,velocity,avg = 2 · Lwall · 0.99

1 [rev] · ANGcycles,per,sec

AREAaround,disp = Asource – Adisp

Radiation Flux Terms

qwall = Q rad,incident,wall

Awall

qsink = Q rad,incident,sink

Asink

qsink2 = Q rad,disp,sink

Asink

SOLUTION

Unit Settings: [kJ]/[K]/[kPa]/[kg]/[degrees]

ANGcycles,per,sec = 13.33 [rev/sec] ANGdisp,velocity,avg = 6.6 [m/s]

ANGVELOCITY = 800 [rev/min] AREAaround,disp = 0.3358 [m2]

Adisp = 1.431 [m2] Asink = 1.767 [m

2]

Asource = 1.767 [m2] Awall = 0.04681 [m

2]

Awall,horizontal = 1.178 [m2] BIOTsource = 317.4 [-]

Cp,sink = 0.9094 [kJ/kg-K] Dwall = 1.5 [m]

εdisp = 0.07 εsink = 0.07

εsource = 0.17 εwall = 0.07

factor2 = 0.2 [-] FLUID$ = 'Air'

heightdisp = 0.125 [m] kair = 0.02588 [W/m-K]

kfluid = 0.02889 [W/m-K] ksink = 236.6 [W/m-K]

ksource = 15.39 [W/m-K] kwall = 0.035 [W/m-K]

Lc,air = 1.5 [m] Lc,air,sink = 1.425 [m]

Lc,disp = 1.35 [m] Lc,fluid = 1.125 [m]

Lc,wall = 0.25 [m] Lsink = 0.01 [m]

Lsource = 0.01 [m] Lwall = 0.25 [m]

νair = 0.00001608 [m2/s] νfluid = 0.00003807 [m

2/s]

Pair = 101.3 [kPa] Pfluid = 53.42 [kPa]

Qcond,sink = 681.9 [W] Qcond,source = 676 [W]

Qcond,wall = -0.5151 [W] Qcond,wall,horizontal = 5.905 [W]

Qconv,disp,fluid = 18.02 [W] Qconv,fluid,sink = 657.6 [W]

Qconv,sink,air = 690.8 [W] Qconv,source,air = 676 [W]

Qconv,source,fluid = 615.6 [W] Qconv,wall,air = 6.747 [W]

Qconv,wall,fluid = 5.905 [W] Qrad,disp,sink = 24.82 [W]

Qrad,incident,sink = 8.939 [W] Qrad,incident,wall = 0.8416 [W]

Qrad,source,disp = 60.86 [W] Qrad,source,wall = 0 [W]

Qrad,wall,sink = 0 [W] qsink = 5.058 [W/m2]



qsink2 = 14.05 [W/m2] qwall = 17.98 [W/m

2]

radiusdisp = 0.675 [m] radiussink = 0.75 [m]

radiussource = 0.75 [m] radiuswall = 0.75 [m]

ρair = 1.164 [kg/m3] ρfluid = 0.5405 [kg/m

3]

ρsink = 2699 [kg/m3] σ = 5.670E-08 [W/m

2-K

4]

SINK$ = 'Aluminum' SOURCE$ = 'Stainless_AISI347'

Thicknesswall = 0.01 [m] thickdisp = 0.02 [m]

Tair = 303.2 [K] Tdisp = 346.6 [K]

Tfluid = 344.3 [K] T∞ = 303.2 [K]

Tsink = 314 [K] Tsink,bot = 314 [K]

Tsink,top = 314 [K] Tsource = 372.9 [K]

Tsource,bot = 373 [K] Tsource,top = 372.8 [K]

Twall = 337.5 [K] Twall,bot = 298.2 [K]

Twall,inside = 344.2 [K] Twall,outside = 345.6 [K]

Twall,top = 376.8 [K] VARYTHIS = 0.5

Velair = 3.03 [m/s] Velair,sink = 10 [m/s]

Velair,wall = 6 [m/s] Velfluid = 5.346 [m/s]

Velfluid,wall = 28.14 [m/s] VELplate = 5.346 [m/s]

VELwall = 28.14 [m/s] VOLbelow,disp = 9.447 [m3/sec]

WALL$ = 'Glass fiber batt rho=40'

No unit problems were detected.

Arrays Table

Fi hconv,disp,i hconv,sink,i hconv,source,i hconv,wall,i Nusselti Pri REi Si

[W/m2-K] [W/m2-K] [W/m2-K] [W/m2-K] [-] [-]

1 0.7335 5.539 36.06 5.476 38.04 317.4 0.7268 282646 1.838

2 0.9271 12.27 12.27 4.195 477.6 0.7175 157982 2.259

3 258.8 189578

4 243.1 93283

5 329.1 184774

6 1985 932826

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