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The Stirling Cycle


A very simple a low frequency example of the Stirling cycle is given by the drinking bird.

Where the energy for the pulsing fluid is provided by a temperature difference dT between the two glass bulbs linked by a concentric glass tube.

The beak of the bird dips into a cold volume of water cooling the bulb of the left whilst the ambient warm air heats the glass bulb on the right creating the temperature gradient required to put energy into the system. The bird will continue to oscillate about its pivot so long as the temperature difference is maintained.

Note: This  usually requires the water level to be topped up as it evaporates. Our bird has worked for over a year.


The drinking bird


Video of cycle

The principle elements of a stirling engine are:

An enclosed volume of working fluid or gas.

A hot chamber at one end and a cold at the other connected by a specific size of small bore tube or a matrix of finely spaced plates .

The gap between the plates and the diameter of the tube has to be such as to allow a certain working fluid  boundary film thickness to be established at the surface. This boundary film helps absorb and regenerate the energy of the working fluid by creating a fluid film nozzle across the system that expands and compresses the working fluid as it oscillates between the hot and cold ends.

The original Stirling engine was patented in 1816 by  Rev. Robert Stirling.  Principally a closed system. This means that a working fluid is enclosed within the system and that energy is added externally. The heat energy required to drive the cycle can therefore be supplied by any fuel or heat source available.

The Stirling cycle is created when the temperature gradient across the system is sufficient to drive an oscillating  fluid pulse between the hot and cold extremities of the system.

The speed and frequency of the cycle will vary with the density and pressure of the working fluid used and the temperature gradient  driving the system. An integral part of the Stirling cycle is the regenerator which is found in the middle of the hot and cold volumes. The effeciency of which improves the overall thermal efficiency of the system.



Modern versions of the stirling engine are produced by Whispergen and are used in both marine and domestic applications.

One way to reduce pollution would be for everybody to install one of these in there home!


There  needs to be a 900 (p/2) phase difference between hot and cold fluid displacement..


Its revolutionary yet based on an 1816 patent By Rev Robert Stirling> With a thermal efficiency greater than the modern SI and CI engines.

But then there is the government tax revenue shortfall/paradox. 


This illustration of a Stirling engine manufacture in 1842 shows a  sectioned layout of the engine.

The enclosed vessel is heated from the bottom. 

Heat input to the working fluid which in this case is air at above atmospheric pressure. Expands the air volume below the displacer piston D. Moving it upwards.

Which in turn displaces the cold air above the piston. Chilled by passing cold water through  a matrix of  cooling tubes C.

The cold and hot ends of the system are connected by a coaxial duct that has a larger diameter than the displacer cylinder

This cooler denser air then passes via an external coaxial duct that contains a large array of closely spaced steel plates. That form the regenerator R. With an air gap between plates of 0.78mm.

The oscillating fluid drives the displacer piston up and down guided by a central sliding shaft that has three pressure seals to conserve the cycle pressure.


Closely packed radial plates in the coaxial cylindrical regenerator. 1842 Dundee Stirling engine.
This experimental engine was used to run a foundry in Dundee

Engine specification:

Cylinder bore diameter  305mm (12ins )

Displacer piston stroke  610mm (24 ins.)

Speed 40 rpm (0,67c/s)

Coal consumption  22.7kgs/hr (50lbs/hr.)  1.43kg/kwhr.

Power output 15.82*103 W


Note:I have temporarily used these images whilst I am preparing some drawings.

Reverend Robert Stirling Doctor of Divinity (DD)


Some reference reading:


Stirling Engine regenerator design by Allan Organ  University of Cambridge



Stirling air engines there history and development by Allan Organ and Theordor Finkelstien.


A life history by Robert Sier is informative but fails to reach the quality of LTC Rolt's biography of Robert Stephenson.

Rev Robert Stirling by Robert Sier ISBN 0 9526417 0 4


Andy Ross Stirling engines

One of the best books on the development of  practical Stirling engines and the problems encountered with scaling designs.

"Making Stirling Engines" by Andy Ross


Stirling Engine with Ross yoke

Andy Ross, a prominent Stirling engine experimenter, invented the linkage illustrated here.3  The engine is identical in operation to the two cylinder Stirling. In this illustration, the left cylinder is the hot cylinder.

The linkage allows the engine to be more compact and reduces side loads on the pistons and connecting rods (since their travel is almost linear). 

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Copyright 2000, Matt Keveney. All rights reserved.

The D-90 Ross Yoke-drive Engine  The Ideal Adiabatic Solution

The 90 cc D-90 engine is fully described in Andy Ross' fascinating book "Making Stirling Engines" (1993). At Ohio University we have a D-90 engine which forms part of the Senior Lab course - a required course for all Mechanical Engineering students. In this section we examine the results of performing an Ideal Adiabatic simulation of the D-90 engine under specific typical operating conditions as follows

  • Mean operating pressure - pmean = 2 bar.
    (The crankcase is sealed, and the output shaft power is obtained by a magnetic coupling. Andy typically pressurizes the crankcase with a bicycle pump to about 2 bar.)
  • Cooler temperature 27 degrees Celsius (300 K), and heater temperature 650 degrees Celsius (923 K)
  • Operating frequency 50 Hz. (Note that the Ideal Adiabatic model is independent of operating speed - all results are presented per cycle)

In order to simulate the engine by means of the Ideal Adiabatic model equation set given previously, we require the equations for the Yoke-drive volume variations and derivatives Vc, Ve, dVc and dVe (all functions of crank angle q), as well as the void volumes of the heat exchangers Vk, Vr, and Vh.

The cyclic convergence behaviour of the Ideal Adiabatic model is extremely good, and using 360 increments over the cycle, the system effectively converges within 5 cycles. The convergence criterion chosen is that after a complete cycle both variable temperatures Te and Tc must be within one degree Kelvin of their initial values. We now consider the solution of the temperature variables Tc and Te, the heat energy variables Qk, Qr, Qh, and the work energy variables Wc, We, and the net work done W. These results are presented as plots showing the variation of these parameters with the crank angle q.

In the temperature-theta diagram we observe a large cyclic temperature variation of the gas in the expansion space (> 100 K), its mean value being less than that of the heater temperature of 923 K. Similarly the mean gas temperature in the compression space is higher than the coller temperature. This suggests that the adiabatic working spaces effectively reduce the temperature limits of operation, thus reducing the thermal efficiency to less than that of the Carnot efficiency.

The energy-theta diagram shows the accumulated heat transferred and work done over the cycle. Notice that the work done W starts with the (positive slope) expansion process then the compression process, and again returning to the expansion process, Thus the total work excursion is almost 15 joules, however the net work done at the end of the cycle is only 3 joules. The most significant aspect of the energy-theta diagram is the considerable amount of heat tranferred in the regenerator over the cycle, almost ten times that of the net work done per cycle. This tends to indicate that the engine performance depends critically on the regenerator effectiveness and its ability to accomodate high heat fluxes. This aspect will be revisited in the section on the "Simple" analysis, when we examine the effect of imperfect heat exchangers on Stirling engine performance. Significantly the energy rejected by the gas to the regenerator matrix in the first half of the cycle is equal to the energy absorbed by the gas from the matrix in the second half of the cycle, thus the net heat transfer to the regenerator over a cycle is zero. It is for this reason that the importance of the regenerator was not understood for about 100 years after Stirling's original patent describing the function and importance of the regenerator. The Lehmann machine on which Schmidt did his analysis was apparently not fitted with a regenerator, and it is conceivable that Schmidt did not appreciate its importance, He refers to the textbook by Zeuner as containing a "complete, simple and clear theory" of air engines, but in the same textbook Zeuner decries the use of regenerators for air engines (Finkelstein, T., 1959, Air Engines in The Engineer part 1, 27 March.)

It is of interest to examine the two components, Wc and We, which added together gives the net work done W. These are shown as dashed lines in the following diagram.

Notice in particular that the expansion space work done (We) undergoes a vastly different process from that of heat transferred to the heater (Qh), however at the end of the cycle they have equal values (Qh = We). Similarly for the compression space work done (Wc) and the heat transferred to the cooler (Qk). In retrospect this must be so in order to retain an energy balance, however it did catch us unawares and surprised us when we first noticed this. The ideal regenerator thus behaves as the perfect isolator, isolating the energy balance of the heater and expansion space from that of the cooler and compression space. Thus for the Ideal Adiabatic model over a complete cycle

Qh = We; (Qe = 0)
Qk = Wc; (Qc = 0)
W = Wc + We

Recall that for the Ideal Isothermal model

Qe = We; (Qh = 0)
Qc = Wc; (Qk = 0)
W = Wc + We

Furthermore the Ideal Adiabatic model in itself does not give results which are significantly different from those of the Ideal Isothermal model. The pressure-volume diagram is of similar form, and the power output and efficiency are quantitatively similar (albeit the efficiency of the Ideal Adiabatic model is about 10% lower for reasons described above). However the the behaviour of the Ideal Adiabatic model is more realistic, in that the various results are consistent with the expected limiting behaviour of real machines. Thus the heat exchangers become necessary components without which the engine will not function. The required differential equation approach to solution reveals the considerable amount of heat transferred in the regenerator, indicating its importance in the cycle, and provides a natural basis for extending the analysis to include non-ideal heat exchangers (Simple analysis). Thus the solution of the Ideal Adiabatic model equations is equivalent to a simulation of the engine behaviour in all respects, from setting up the initial conditions until convergence to cyclic steady state is attained. Throughout this process all the variables of the system are available as by-products of the simulation and can be used for extending the analysis. Thus for example the mass flow rates through all the heat exchangers can be used in order to evaluate the heat transfer and flow friction effects over the cycle.



On to the Simple Analysis

Back to the computer program function set 'adiab'

Back to the Stirling Cycle Machines home page



American Stirlings MM-7 Stirling Engine


The ultimate DT engine that will run on a temperature difference of 40C




Linear Stirling engines

NASA  reasearch on the application of linear stirling engines to drive alternators and generate electric power.

The benefits of linear Stirling engines are that they remove the need for a mechanism to tranform linear diaplacement into rotary power and are simple in concept with fewer working parts.

Benefits of linear Stirling engines:

  • Simple construction
  • Easier to encapsulate. and run at a higher working fluid  pressure P


  • Difficult to maintain fluid phase difference between cold and hot chambers.
  • Integration of a thermal break between hot and cold parts of the system.

Important parameters for designing an efficient Stirling engine:

  • Efficient Regenerator design 
  • High working gas pressure.
















ŠJohn Gwynn and sons2003 

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 as long as you give us proper credit (by referring to "The Water-Rocket Explorer" http://waterocket.explorer.free.fr).