Analyzing Failure in an Alpha Type Stirling Engine

Abstract: This analysis using ABACUS software will help to understand the failure of an air cylinder undergoing the normal stresses and strains of a Stirling engine cycle. This cycle includes changing pressures and temperatures of the working fluid inside the cylinder several times per second.

Analysis done by

An Son Leong

and

Matt Palmer

Stirling Engine Background:

The Stirling engine is a closed cycle heat engine with a gaseous working fluid. Essentially,

it is powered by the temperature difference between the hot and cold side of the engine.

Thermodynamically, the Stirling cycle consists of four parts:

1) Compression- The working fluid inside the system is compressed

2) Heat Addition- Heat is added to the system and the gas expands

3) Expansion- The gas expands against the piston, doing work

4) Heat Removal – Heat is removed from the system and the cycle starts all over again

The Stirling Engine and Stirling Cycle was patented by Reverend Robert Stirling in 1816.

It is very efficient, approaching Carnot efficiency; however the engines are plagued by several

issues. They are not very versatile and have trouble adapting to temperature differences that

they were not designed for. It is also hard to vary the speed of the engine. The engines fell out

of favor because of advances in steam engine technology. Since the 60’s various companies

have researched Stirling engines for automobiles, power generation, etc but the engines have

never been developed commercially until recently. Due to their high efficiency and their ability

to run off of any heat source, several initiatives have been made to develop Stirling technology.

NASA developed a Stirling engine to generate power for a deep space probe using plutonium as

a source of heat. A California company has developed a 25 kW generator that runs off of

sunlight focused by a parabolic mirror.

Motivation:

This project is an extension of a project for a class in which the authors are enrolled. The

objective was to design a Stirling engine that could run off of a low temperature difference.

Typical commercial Stirlings run off of a temperature difference that can exceed 1000 degrees

Fahrenheit and pressures of up to 100 atm. The Stirling engine designed by the authors was

designed using off the shelf air cylinders for a first generation demonstration model. The

temperature difference is 85 degrees Fahrenheit. That is, the cold side will be kept a constant

60 degrees and the hot side a constant 145 degrees. Originally, the design called for the

working fluid to be pressurized with helium at 10 atmospheres with the system at maximum

volume, as cycle efficiency increases as pressure increases. This means that the system will see

pressures of 30 atm when the gas is compressed.

The air cylinders ordered are rated to withstand 250 psi, or approximately 17 atm

before the seals on the pistons burst. The question was then posed that if the seals were not

the limiting factor, how much pressure could the cylinder withstand without failure? What

effect does temperature have on the stainless steel cylinder? These questions are relevant

because they will allow the authors to optimize the design of the first generation engine

without compromising the structural integrity of the engine. The picture below is the engine

designed by the authors.

The two air cylinders are connected by brass tubing that allows transfer of the gas

between hot and cold sides. Notice the copper piping that surrounds the hot and cold cylinders.

This will circulate water at a given temperature to keep each cylinder at the designated hot or

cold temperature. The two steel flywheels will store the kinetic energy required to compress

the gas and begin the cycle over again.

Objectives:

The objective of this structural analysis is to find the criteria for failure in our air

cylinders. We will know the project is successful when the Stirling designed by the authors is

safely operated as a result of the structural analysis done by ABACUS.

Analysis:

Two simple models were developed to analyze the air cylinder. The picture on the left

shows the top down profile of the cylinder with x and y symmetry. The picture on the right

shows the sides view of the cylinder and will show how the cylinder will deform lengthwise.

The radial model was subject to four boundary conditions: It is assumed to have x and y

symmetry, the temperature of the inside of the cylinder is forty degrees Celsius warmer than

room temperature (assumed to be zero) and the outside of the cylinder is assumed to be 50

degrees warmer than the ambient temperature. The difference between inner and outer wall

temperature is because of the assumption that not all the gas will be the maximum hot or cold

temperature. This is because the temperature exchange is happening at a very high frequency.

The model for the length was also subject to four boundary conditions: X symmetry, the

ends of the cylinder are constrained against movement, and the same inner and outer wall

temperature of 40 and 50 degrees Celsius above ambient.

Results:

For the given model, it was found that pressure had a much greater effect on the failure

of the cylinder than temperature. Given the cylinder dimensions, it was found that failure will

occur along the top edge of the model in the area where the threads connect brass tubing to

the air cylinder. This failure occurs at an internal gauge pressure of 50 kPa, much less than the

rated pressure of the cylinder. This could be because the dimension of the top of the cylinder

was not given in the part schematic drawings and was inaccurately represented in our model.

An assumption was made that the top and bottom wall were the same thickness as the side

wall of the cylinder. On the next page are the analysis for the radial and length model at failure.

It is assumed the model fails when any node reaches the yield stress for stainless steel of

approx 250 MPa. Notice the red at the top and bottom of the length model for a stress of 272

MPa. Notice also that the radial stress at the inside of the cylinder is a mere 200 MPa.

ABACUS is also capable of producing graphical results that relate to temperature. For

the purposes of this analysis, the two plots shown are heat flux and nodal temperature. The

temperature plot of the radial section is shown here. Notice the outer and inner walls are 50

and 40 degrees above the ambient temperature, with a fairly linear distribution in between.

This is a result that one would expect. When the heat flow vectors are plotted, the vectors all

point radially inward as heat is flowing from hot to cold. It is not shown here because the vector

plot is very messy and difficult to read.

The plots of the nodal temperature and heat flux for the length model are shown above. Notice

the upper half of the cylinder has become the hot temperature and the lower half has become

the ambient temperature. The heat flux plot matches up nicely with this result, as one can see

that a lot of heat is flowing out of the lower half of the cylinder.

Conclusions:

This analysis shows that the air cylinder will fail at an internal gauge pressure of .5

atmospheres or 50 kPa. This is much less than the 17 atm promised by the manufacturer. As

shown above, the failure occurs at the ends of the cylinder where it is constrained against

deflection. This is a realistic constraint, as the design of the engine does not allow it to translate

in any direction. However, the analysis could be somewhat flawed because of the lack of

accurate dimensions provided in the schematic drawings. Needless to say, during test runs of

the engine it will not be run at gauge pressures higher than 50 kPa. This same analysis can be

done for the second generation engine to help determine how thick the cylinder walls need to

be to withstand a higher working pressure that will increase efficiency.