Transport Phenomena Fall 2013

Stirling Engine

A Study in Fluid Flow and Heat Transfer

Morgan Bassford, Nick Eyre, Riva Kahn Hallock, Dante Santos, & Gaby Waldman-Fried12/18/2013

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Table of ContentsAbstract.......................................................................................................................................3Background & Motivation.......................................................................................................3

Available Technologies.............................................................................................................4

Problem/Challenge Statement................................................................................................5Project Goals and Objectives..................................................................................................5

Materials and Methods............................................................................................................5

Engine Prototype 1: Beta Type Aluminum Can Engine.................................................5

Engine Prototype 2: Gamma Type Machined Engine....................................................6

Analysis & Discussion...............................................................................................................7

Fluids........................................................................................................................................8

Heat..........................................................................................................................................8Dynamic Simulation................................................................................................................10

Simulink Modeling..................................................................................................................12

Conclusions...............................................................................................................................13

Reflection on Goals and Objectives.................................................................................13

Lessons Learned..................................................................................................................13

Appendix: Equations...............................................................................................................14

Equations of Heat Transfer in Transport Phenomena.....................................................14

Convection (Forced)............................................................................................................14

Convection (Natural or Free)............................................................................................14

Conduction............................................................................................................................15

Radiation................................................................................................................................15

For The Stirling Engine Project...........................................................................................15

Q1 - Forced Convection – Boiling Water.......................................................................15

Q2 – Conduction – Bottom Plate.......................................................................................16

Q3 – Convection – Working Fluid.....................................................................................16

Q4 – Conduction – Top Plate.............................................................................................17

Q5 – Natural Convection – Ice Water..............................................................................17

Total Heat Transfer.................................................................................................................18

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Table of FiguresFigure 1: Gamma type engine (left) and a beta type engine (right)..........................3Figure 2: Our first “soda can” Stirling engine prototype..........................................4Figure 3: CAD render of our second prototype. The crankshaft and flywheel are hidden........................................................................................................................ 6Figure 4: A simplified heat flow diagram of our Stirling engine design. Air is heated and cooled by the two plates as it compresses and expands in the two chambers. The crankshaft and flywheel are hidden....................................................................6Figure 5: the five stages of heat transfer within the Stirling Engine........................8Figure 6: a picture of the temperatures within our Stirling Engine under operating conditions................................................................................................................... 8Figure 7: Note that with a negative initial velocity, the system corrects and stabilizes at a positive rotational velocity..................................................................9

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AbstractStirling engines are closed-cycle heat engines that work by the cyclic convection driven expansion and contraction of a gas. Due to its high Carnot efficiency, the Stirling engine can be practically driven by a wide variety of heat sources. In this project, we examined the design and characterization of a low temperature differential Stirling engine designed to be powered by an organic energy source. We began by building a very low-cost (VLC) beta-type Stirling engine prototype which suffered from high frictional losses and provided some proof of concept but was not fully functional. Next we built a second prototype, a gamma-type design designed to have low friction and be much more efficient than the first prototype. The second prototype, while based on a known design, was substantially modified to optimize manufacturability and ease characterization while maintaining similar performance. It was functional, and we worked to simulate and characterize the engine’s heat transfer and fluid flow. Finally, we created a simplified dynamic simulation which aligned with our experimental results.

Background & MotivationStirling engines are closed-cycle heat engines that generate power

through the Stirling cycle. In a Stirling engine, a working fluid (commonly air, although noble gases are frequently used in high-performance engines to reduce explosion risks) is compressed and heated with an external heat source.1 The working fluid expands, pushing on a piston (the power stroke) and then is forced to the cool area with a displacer piston. Once the gas cools, it is compressed by the pistons as they continue to move from the momentum stored in the flywheel.i1 The compressed gas is then heated and the cycle begins again.

Stirling engines are still in development for many industries, but are most often used in hybrid technologies or to power pumps. Stirling engines are heat engines and can run on much smaller temperature differentials than the their internal combustion cousins.1 Furthermore, their external heat source need not necessarily be a product of combustion--they may be run on sustainable alternatives such as solar, compost, bio-gas and steam. This makes them a very attractive technology to developing nations, and follows with the UN Millenium Development Goal, “Ensure environmental sustainability”. Our aim is to possibly also address the National Academy of Engineering’s Grand Challenge entitled “Make solar economical”. However,

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the precision and materials typically required in the manufacture of Stirling engines have hampered their widespread adoption in countries that could benefit from them most.

No industrial-scale solar Stirling plants currently operate in the United States despite attempts to implement them in heavily sunlit states. In 2010, Stirling Energy Systems commissioned and built the 1.5 MW Maricopa plant (a single unit proof of concept) in Arizona with future plans to build two additional full scale plants (709 MW and 850 MW). Unfortunately they did not have sufficient equity to finish the additional plants and the Maricopa Plant closed in 2011 after the company went bankrupt trying to fund the additional projects.ii However, full scale solar Stirling plants exist in Germany and other European countries.

In terms of power generated per unit area of space, solar Stirling plants are superior to photovoltaics. The average efficiency of the Carnot cycle with an 100 degree temperature differential is 17%, which is higher than the average efficiency of a solar cell at 12%. Furthermore, industrial scale solar Stirling engines operate at a 700 degree temperature differential which gives a 30% Carnot efficiency.iii However, due to incredibly high manufacturing and maintenance costs (largely driven by the tight tolerances required by the engines to function), solar Stirling plants are not a viable option for large scale energy generation, especially in developing nations.

Available Technologies

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Figure 1: Gamma type engine (left)iv and a beta type engine (right)v.

We have chosen to build one beta and one gamma type Stirling engine. A beta Stirling engine has a single power piston and a displacer mechanism. Gas is exchanged between expansion and compression spaces through a temperature gradient. In gamma Stirling engines, the power piston and displacer are in different cylinders. While gamma Stirling engines require less of a temperature differential to function, they also produce less specific power. Stirling engines of a variety of types, sizes, and qualities can be purchased in kit form as well as fully assembled. We borrowed a low temperature differential Stirling engine kit, which cost $98 for a small engine. The gamma type that we built cost us $150 for materials and was designed to produce less than 10W of power.

Problem/Challenge StatementIn general, Stirling engines are not known for their overall general

efficiency nor cost-effectiveness. Our biggest challenge was to create a more efficient, working Stirling engine out of readily available materials. It is particularly challenging to create a low-cost Stirling engine that runs on very low temperature differentials. We designed a Stirling engine to run on the heat generated by hot compost. Hot compost can hit maximum temperatures of around 160oFvi, giving us a temperature differential of

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around 130-140 degrees F to work with. According to our research, an engine operating with a temperature differential of ~130oF should yield a maximum thermal (Carnot) efficiency of about 17.4%vii. Stirling engines have a high theoretical efficiency but characteristically low efficiency in practice because it is difficult to maintain a large enough temperature differential for the engine to function: either heat is lost to the environment or the cool side of the engine heats up via conduction through the device. It has been a challenge to reach or exceed even this percent efficiency using the naturally occurring heat from organic matter decomposing.

Project Goals and ObjectivesWe wanted to pursue two consecutive physical prototypes, a

mathematical model and simulations for both fluid flow and heat transfer in the system. We completed a sketch model of a beta type engine (with power output of less than 10W) using found materialsviii, and used what we learned from that to make a larger Stirling. Then, using the specifications and measurements from the gamma engine, we wanted to create mathematical models of the system. Eventually we used simulations to derive what changes could be made to the engine in order to increase its efficiency to cost ratio.

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Materials and MethodsEngine Prototype 1: Beta Type Aluminum Can Engine

Figure 2: Our first “soda can” Stirling engine prototype.

To better understand the difficulties involved in building Stirling engines, we began by constructing a beta type engine out of found materials according to the instructions on Scraptopower.comix. This proof of concept allowed us to experiment with the amount of heat needed to overcome the shaft resistance. Unfortunately, imprecise components and assembly lead to major friction losses (primarily around the crankshaft) that ultimately did not allow our beta type to generate enough power to have the flywheel spin freely. However, we did notice that the flywheel spun more easily with a temperature differential--i.e., when the hot end of the engine was placed in boiling water--so it probably would have worked if our assembly had been more precise. Experimentation with the beta type model highlighted the importance of assembling and manufacturing our components carefully and within tolerances, especially the crankshaft.

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Engine Prototype 2: Gamma Type Machined EngineFor our second prototype, we built a gamma type Stirling engine

roughly based on plans from ridders.nu. We decided to use a known design to reduce the uncertainties in our design process and allow us to focus on the analysis and characterization rather than design of the system. Gamma type engines are similar to beta engines but have an offset power piston, which simplifies them mechanically and reduces the weight of the power piston.

This engine is constructed from two aluminum plates which form the top and bottom edges of a large diameter cylinder. The plates are round because it is easier to machine round parts to a higher tolerance. The two plates are also insulated from each other by a plastic cylinder and plastic standoffs. Aluminum was chosen for its ease of machining and high thermal conductivity. The displacement piston is constructed of a round disk of balsa wood, chosen for its light weight and ease of machining. The displacement piston is machined from graphite--selected for its low coefficient of thermal expansion and self-lubricating properties--and slides in a glass tube. A pair of aluminum plates support a crankshaft in ball bearings above the top plate of the system. The crankshaft is manufactured from steel shaft brazed to connecting offset links. Connecting rods attach the crankshaft to the power and displacement pistons. A CAD render of our design (without flywheel and crankshaft) is shown below.

Figure 3: CAD render of our second prototype. The crankshaft and flywheel are hidden.

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Analysis & Discussion

Figure 4: A simplified heat flow diagram of our Stirling engine design. Air is heated and cooled by the two plates as it compresses and expands in the two chambers. The crankshaft and flywheel are hidden.

FluidsA Stirling engine is a closed system, using the relationship between

temperature and pressure at a constant volume to derive work from a temperature difference. As such, there is no bulk flow of fluid. The displacer piston shuttles fluid back and forth between the hot and cold plates and fluid expands into the power piston, but there is no net flow in and out.

To characterize the flow that we do have, we assumed that the engine is working between 1 and 10 Hz. Assuming this frequency and taking into account the geometries of our engine, the volume of fluid displaced is so small that the flow everywhere can be taken as laminar. There are no doubt vortices caused by the geometry of the engine, but the flow itself is not turbulent.

This is somewhat unfortunate because the boundary layer in laminar flow is much thicker than in turbulent flow, reducing the heat transfer from a fluid to a wall. We want to maximize the heat transfer between the plates and the working fluid, and therefore make the flow along the plates turbulent. There are methods of roughing the surface just enough to make

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the flow turbulent without creating pockets of stagnant fluid that could be used to increase this heat flow.

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HeatThe fluid flow in our system is not particularly interesting, but the meat

of the project lies in the ways that heat is transferred. The Stirling engine works off of a temperature difference and five stages of heat transfer. As shown in Figure 5, the heat is transferred from the source (in this case, a pot of boiling water) to the bottom plate of the engine, to the working fluid, which is then forced to transfer heat to the top plate of the engine, which transfers heat to the ice water bath. The total heat transfer through the system is 307 Watts. This is not the amount of power that our engine outputs. That would be determined by a series of ideal gas law and torque equations. Our calculations to acquire these numbers can be found in the Appendix.

Figure 5: the six stages of heat transfer within the Stirling Engine.

Figure 6: a picture of the temperatures within our Stirling Engine under operating conditions.

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Figure 6 is a picture taken by an IR camera displaying the temperatures of our engine in Fahrenheit. The white oval near the bottom is the flame, the red area above it the iron ring from the ring stand on which we were resting the engine, above that is the green and blue bottom plate, then the spectacular gradient of the working fluid of the engine, followed by the very cool upper plate. We hypothesize that the bottom plate is not the same red as the ring stand and the bottom most air because the portion that the camera can see is exposed to the ambient air and cools quickly. We assume that the portion of the bottom plate closer to the flame is much warmer.

Dynamic SimulationA dynamic simulation was made in MATLAB to help understand the behavior of our engine. The simulation simplifies our system into a hot and cold chamber with a displacer piston that moves the air betweenx. The cold chamber contains the region of the power piston cylinder below the piston. The simulation has two state variables: the rotational position and velocity of the crankshaft. These in turn are used to calculate the positions of the pistons and the volumes of the two chambers. The system applies the simplifying assumption that the gas in each chamber is of uniform temperature and uses the ideal gas law to calculate the pressure of the two-volume system. This pressure creates a force on the power piston which creates a moment on the crankshaft, causing the system to continue spinning. Viscous damping is applied on the crankshaft to counteract the force from the engine.

An animation was used to visualize the system. Due to the viscous damping, the system asymptotically stabilizes at a rotational velocity and power output. The power output of the simulated system is very low (< ¼ W). Note that this is an idealized version of the system. As verified by thermal imaging, the actual engine has a temperature gradient across the displacement cylinder, resulting in lower effective temperatures and less pressure in the system.

The simulation was compared to the second prototype engine and matched up very well. The geometry was equivalent to our prototype and when tested with a 100 K temperature differential and a viscous damping coefficient of 0.1 Newton-seconds, the system stabilized at a rotational speed of about 1 rotation per second, just as our prototype did (Figure 7).

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Using the simulation, we performed some system and geometry optimization studies. We found that the larger the diameter of the power piston, the higher the power produced. This makes sense as there is a larger area for the pressure to act on. When taken to the limit of the diameter of the displacer piston, a beta type Stirling engine results. However, this is a trade-off with the increased rotational mass of the crankshaft caused by a larger power piston. Next, we looked into the effect of the contained gas on performance. We found that when alternative gases are used, atmospheric pressure overpowers the pressure within the engine and causes decreased performance. We also found that the taller the displacement cylinder, the higher the power output of the system. This aligns with the expected behavior as there is more gas present to expand and contract. Furthermore, we found that the diameter of the displacer piston is not especially critical to the performance of the system.

Figure 7: Stirling Engine Dynamic Simulation. Note that with a negative initial velocity, the system corrects and stabilizes at a positive rotational velocity.

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Simulink ModelingWe attempted to model heat flow, temperature and pressure using Simulink in MATLAB. However, due to difficulty modeling thermo-fluid systems using the Simulink software we did not achieve this goal. We did successfully complete a model showing temperature changes as the fluid is heated and cooled via conduction and convection.

Figure 8: Simulink Diagram for Heat Flow in the Stirling Engine

ConclusionsReflection on Goals and ObjectivesIn the end, we did accomplish our goal of creating two physical prototypes, a mathematical model, and simulations for fluid flow and heat transfer. Though the first prototype was not fully functional, we learned a lot for the next iteration. The gamma engine helped us collect data regarding the heat transfer in our model; we were able to show why the theoretical efficiency of a Stirling engine and the actual efficiency are so different.

Lessons LearnedThe power to cost ratio of Stirling engines is far too low for them to be a viable energy source. Our Stirling engine produced a sub-watt power output, rendering it nearly useless for any energy applications. On top of that, Stirling engines are precision machines that rely on near-perfect machined parts to work. Because of this, a Stirling engine would probably not be the right choice for power generation in the developing world. Solar

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panels are impossible to repair on site, but they are becoming more and more mass-produced, meaning their price is being driven down daily. A Stirling engine would be near-impossible to service, and implausible to make locally with any desirable success rate. The only way this could become a viable solution is if significant startup capital was put into creating millions of small temperature difference Stirlings that could be sold at low cost in developing countries.Recommendations and Future WorkIf anyone wanted to pursue this in the future, we would recommend optimizing the geometries of the pistons, the insulative nature of the walls, and the thickness of the top and bottom plates. It would be valuable to have a machine shop manufacture parts that could reach the necessary tolerances. For us, air leakage around the power piston and chamber was the difference between the engine working or not so it would be necessary to have the Stirling engine precision machined.

Another future pursuit working off this model would be to collect experimental data (pressure, temperature, displacer piston velocity, flywheel rpm, etc.) off of the engine to compare with and validate simulation values.

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Appendix: Equations

Equations of Heat Transfer in Transport PhenomenaConvection (Forced)

ℜ→Pr→Nu→h→Q̇

ℜ= ρVDμ

where ρ=density ,V=velocity of fluid ,D=charictaristic length(diameter∈this case) ,∧μ=dynamic viscosity

Pr ¿

νKρc

= μcK

where c=specific heat ,∧K=thermal conductivity Nu=.664 Re .5Pr.33 If LaminarNu=.037 Re .8Pr .33 If Turbulent

h=NuKL

Q̇conv=hA ∆Twhere Q̇cond=theheat flow of convection , A=the crossectionalrea ,∧∆T=thetemperature difference

Convection (Natural or Free)Q̇=h A s∆T

where As=surface area

h= KLc

Nu

where K=thermal conductivity , Lc=charictaristic lencth (largest possible vortex ) ,∧Nu=the Nusselt number

Nu=.54Ra .25

where Ra=the RayleighNumber

Ra= gβ∆TLc3

ν2Pr

where g=gravitational acceleration ,β=expansion∨gascoefficient , ν=the kienematic viscosity ,∧Pr=thePrandtl number

Pr=μCK

where C=the specific heat ,∧K=thermal conductivity

Lc ( for acylendar )=D4

β= 1T f

T f=∆T2

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ConductionRconv=

1hA

where Rconv=the resistancedue ¿convection ,∧A=the crossectional area

Rcond=LKA

where Rcond=theresistacne due¿conductuction ,K=thermal conductivity ,∧A=thecrossectional area ∑R=Rconv1+Rcond+Rconv 2

where Rconv 1=the resistancedue¿convection ononeside of the thematerial ,R cond=the resistance dueconduction through thematerial ,∧Rconv 2=the resistancedue ¿convectionon the other side of thematerial

Q̇cond=T 1−T 2∑Ri

RadiationQ̇=εσ T 4 A

where ε=emmisivity (0¿1 ) , σ=the Stefan−Boltzmannconstant

For The Stirling Engine ProjectNote: All rates of heat transfer are reduced to %17 to reflect the maximum Carnot efficiency of a Stirling Engine.

Q1 - Forced Convection – Boiling Water Assumptions:

Temperature of boiling water ¿99℃ Velocity of boiling water ¿1m /s Depth of water ¿ Lc=1cm ∆T across water ¿1℃ Diameter of pot ¿25cm

Properties:

ρ=959.1 kgm3

μ=2.85×10−4 Pa∗s

C=3773 Jkg K

K= .679 WmK

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Values: ℜ=33653 (Turbulent ) Pr=1.584 Nu=180.47

h=12263 Wm2K

Q̇=102.25W

Q2 – Conduction – Bottom PlateAssumptions:

Temperature of plate ¿98℃ The depth/thickness of the bottom plate = .012 m

Properties:

hw=12253Wm2K

radius=.07m Charictaristic Length=Lc=.012m

K=240 WmK

∆T=50℃

Values:

RconvWater=.005301KW

Rcond=.003248KW

RconvAir= .5567KW

Q̇=15W

Q3 – Forced Convection – Hot Plate and Cold Working FluidAssumptions:

Frequency of motor=2Hz Air is either the temperature of the top plate or the bottom plate, but not in

between. The velocity along the plate is the same as the velocity of the air past the

displacement piston Inner Radiusof displacement cylendar=.054m Radiusof displacement piston=.053m

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We can perform lumped analysis despite Bi=84.876@2℃ ∆T=96K Lc=Diameter=.14m

Properties:

V= volumedisplacedarea∗time

=π r2

2hπ∗(r 22−r12 )∗period

=.485 ms

K air=.0245wmK

c=1006 jkg K

ρ=1.283 kgm3

μ=1.733×10−5 kgms

Values:

Q̇= .037c.33k .66 ρV ∆T

μ .466 (Lc ρV ).2=85.62W

Q4 – Forced Convection – Cold Plate and Hot Working Fluid

Assumptions: Frequency of motor=2Hz Air is either the temperature of the top plate or the bottom plate, but not in

between. The velocity along the plate is the same as the velocity of the air past the

displacement piston Inner Radiusof displacement cylendar=.054m Radiusof displacement piston=.053m We can perform lumped analysis despite Bi=65.94@98℃ ∆T=96K Lc=Diameter=.14m

Properties:

V= volumedisplacedarea∗time

=π r2

2hπ∗(r 22−r12 )∗period

=.485 ms

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K air=.0315wmK

c=1006 jkg K

ρ=.9507 kgm3

μ=2.172×10−5 kgms

Values:

Q̇= .037c.33k .66 ρV ∆T

μ .466 (Lc ρV ).2=71.69W

Q5 – Conduction – Top PlateAssumptions:

Same exposure area to water as bottom plate RconvWater∧RconvAir are the sameas∈Q̇2

Properties:

K=240 WmK

∆T=96℃ Lc=.005m

Values:

RconvWater=.005301KW

Rcond=.00135KW

RconvAir= .5567KW

Q̇=28.9W

Q6 – Natural Convection – Ice WaterAssumptions:

Water is at 1˚C

LC=D4

=.035m

Pr is the sameas∈Q̇1=1.584

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∆T=1℃ Radius=.07m

Properties:

K= .679 WmK

g=9.8ms2

ν=1.787×10−6m2

S

Values: Ra=4.168×108(Turbulent ) Nu=77.15

h=1496 Wm2K

Q̇=3.82W

Total Heat TransferQ̇Total=307W=3kW

i Kongtragool, Bancha, and Somchai Wongwises. "A Review of Solar-powered Stirling engines and Low Temperature Differential Stirling engines." Science Direct (n.d.): n. pag. Web. <http://www.inference.phy.cam.ac.uk/sustainable/refs/solar/Stirling.pdf>.

ii Stancich, Rikki. "What happened to Tessera Solar's projects?." CSP Today. FC Business Intelligence Limited, 4 Mar 2011. Web. 18 Dec 2013. <http://social.csptoday.com/markets/what-happened-tessera-solars-projects>.

iii "Solar Dish Engine." Solar Power And Chemical Energy Systems. The International Energy Agency. Web. 18 Dec 2013. <http://www.solarpaces.org/CSP_Technology/docs/solar_dish.pdf>.

iv Gamma Engine: http://www.geo.unizh.ch/~iwoodh/gamma.gif

v Beta Engine: http://scholar.lib.vt.edu/ejournals/JOTS/v37/v37n2/images/foster2.jpeg

vi "Compost Fundamentals" Washington State University. Web. 22 Sep. 2013. <http://whatcom.wsu.edu/ag/compost/fundamentals/needs_temperature.htm>

vii [i]

viii “Quick And Easy Stirling engine On Coffee Warmer” https://www.youtube.com/watch?v=MTFfZk7Bg90

ix (www.scraptopower.co.uk/can-stirling/make-a-coke-can-stirling-engine)

x Inspiration for simulation taken from Ohio University Stirling Cycle Machine Analysis class webpage <www.ohio.edu/mechanical/stirling/isothermal/isothermal.html>.