Thermo- and CFD modelling of Stirling machines
DEVELOPMENT OF A COMPUTER MODEL TO SIMULATE A LOW
TEMPERATURE DIFFERENTIAL RINGBOM STIRLING ENGINE
A Robson, T Grassie
School of Engineering, Napier University Edinburgh, EH10 5DT, UK
ABSTRACT
The Stirling cycle engine of 1816 and subsequent closed cycle regenerative heat engines
have been the subject of many analyses. The complex nature of the mechanical, fluid, and
thermal processes within the engine makes an accurate analysis challenging. In describing
engine performance and operation, methods thus far employed have considered idealised
cycles, combined cycles, empirical data and finite element techniques to predict engine
performance. This paper presents a third order analysis of a low temperature differential
Ringbom Stirling engine [1] undertaken at Napier University Edinburgh. The laws of
conservation of mass, momentum and energy, as well as the ideal gas law, are applied to each
space and element in the engine, and a numerical model is developed. The numerical model
describes the variation of the kinematic, fluid and thermal processes within the engine with
time, and will be used to optimise the engine design. A physical model will be used to obtain
operational data. The preliminary analysis suggests that in respect of optimising a physical
engine several approaches may be adopted. The optimised physical engine, coupled with an
electric generator or pump will to exploit energy sources such as process waste heat, solar hot
water and geothermal low grade heat.
INTRODUCTION
The analyses of the Stirling cycle engine have evolved over the years [2]. These range,
in increasing rigour, from the trivial, zeroth order analysis reproduced in undergraduate level
text books to the third order analysis. The third order analysis considers the engine as
comprising discrete nodes or elements. Laws of conservation of mass, momentum and energy
are applied to each of the nodes and gas state equation is applied to the working fluid at each
relevant node. This allows the development of a set of differential equations, describing the
processes occurring at all the nodes at any given time. This analysis may also be categorised
as a Cl-1+CL-2+CL-3+ideal gas analysis [3]. For this paper the term third order analysis will
be used. To enable coding for computer simulation of engine operation, the above mentioned
equations are manipulated to form discrete terms within the time domain.
Presently, there are several third order analyses that use numerical methods to return
nodal solutions, namely the Martini-Weiss code [4], Ureili-Berchowitz code [5], and the
SNAP code [6]. These analyses concentrate upon sealed engines with positive drive to the
displacer. As yet there is no third order analysis of the Ringbom variant of the Stirling
engine. The Ringbom Stirling engine has a unique feature of operation, in that the engine
uses a pressure differential to drive the displacer. The more common method attaches the
displacer to the drive shaft or flywheel utilising a discontinuous motion link. This pressure
differential is between the atmosphere and the working fluid, acting across the displacer rod,
thus removing the need for mechanical connections between the displacer and the drive shaft
[7].
The objective of this work is to produce and develop a computer model that may be
used to predict the operation of a Low Temperature Differential Ringbom Stirling Engine
(LTDRSE). By changing the design parameters and property variables within the computer
program the effect upon performance may be considered, and the design of the physical
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Thermo- and CFD modelling of Stirling machines
engine, optimised. Verification of the computer model will be through comparison with a
physical engine of identical dimensions and properties.
The main aim of the present work is therefore to produce a design optimisation tool for
low temperature heat engines, working below the boiling point of water (under 373K). Such
an engine would use low grade energy sources such as process waste heat, geothermal springs
or enhanced solar hot water systems. When coupled with a suitable generator, they could
produce several hundreds of watts of electricity, depending upon the size of the engine, and
the temperatures of the hot and cold reservoirs.
ANALYTICAL APPROACH
In the analysis developed in this work the engine is divided into discrete spaces and
elements. The relevant thermal and physical properties are shown in Figure 1, and defined in
the text.
r
ℓ
hP
Piston
AP
pA,
ADR
xP
Cold end heat
exchanger
TC, AC
hSK
xD
Compression Space
AR
Regen
hSE
pK, TK, EK, mK
Regen
Displacer
Expansion Space
AD
pE, TE, EE, mE
Hot end heat exchanger
TH, AH
hD
hC
Figure 1: Schematic of a Low Temperature Differential Ringbom Stirling Engine
Conservation of mass for the expansion, mE and compression space, mK, gives
respectively:
dm E
  m R
(1)
dt
dmK
 m R  m A
(2)
dt
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Thermo- and CFD modelling of Stirling machines
 R is the gas transfer rate through the regenerator from the expansion to the
where t is time, m
 A is the leakage rate from the atmosphere to the compression space.
compression space, and m
It is assumed that the transfer and leakage rate are proportional to the pressure difference.
It is next assumed that the regenerator operates in its ideal state, in that the fluid
temperature leaving the regenerator is equal to the bulk temperature of the respective space.
Application of the unsteady flow energy equation for the expansion and compression space
gives the following equation for the variation of the expansion space temperature TE with
time:
dTE
dx
1 

(3)

k HH TH  TE   p E AD D  R m RTE 

dt
cV mE 
dt

where cV is the specific heat at constant volume. The first term in the bracket takes into
account the heat transferred from the hot reservoir at temperature at temperature TH to the
expansion space, with the constant of proportionality kHH. The second term considers the
work done against the displacer, where pE is the pressure in the expansion space, AD is the
cross sectional area of the displacer, and xD is its position. The final term takes into account
the gas flow between the expansion and compression spaces, with R being the gas constant.
Similarly, for the compression space temperature TK, the following equation applies:
dx D







k
T

T

p
A

A
 cV m AT3  R m ATK 
HC
K
C
K
D
DR

dt
dTK
1 

(4)

2




dt
cV m K
 r sin  cos 
 d
*
 r sin  
 R m AT3  p K AP 

0.5
  2  r sin  2
 dt




where
for
p A  pK
T3  TA  TK
T3*  TA
(5)
for
p A  pK
T3  0
T  TK
(6)
*
3
where pA is the ambient pressure, TA is the ambient temperature, ADR is the cross sectional
area of the displacer rod and r is the length of the crank arm, l is the length of the connecting
rod, θ is the angular position of the flywheel, and all other symbols have similar meaning to
those defined above.
The pressure in the expansion and the compression space, pE and pK, are given
respectively as
pE 
m E RTE
AD  x D  0.5hD 
(7)
pK 
m K RTK
 AD  ADR hC  0.5hD  x D   AP x P  0.5hP 
(8)
where hD is the height of the displacer, hC is the height of the chamber, hP the height of the
power piston and xP is its position.
The equation of motion of the flywheel-power piston assembly is given by:
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Thermo- and CFD modelling of Stirling machines

sin  cos
 3
r mP cos  2
2

  r sin  




0.5


 d 2
 r mP cos  sin   I F cos  2 

 dt
2



2
2
2
2


r 2 cos   sin  
 r 2 sin  cos
 d 
 rmP cos  

 r cos 

2 1.5
2 0.5
2
2
dt










r
sin



r
sin



d
 rmP g cos   r  pK  p A AP cos   k DF cos
dt



(9)
where mP is the mass of the power piston, IF is the second moment of the flywheel, α and γ are
auxiliary angles defined respectively as
r
sin   sin 

cos  sin    
(10)
(11)
and where the last term on the right-hand side of equation (9) takes into account the effect of
the assumed viscous damping on the flywheel, with constant kDF.
Figure 1 indicates that, as the displacer approaches either the top or the bottom surface
of the chamber, its motion is controlled by stub springs with length hSE and hSK, of stiffness
kSE and kSK respectively. The equation of motion for the displacer is then given by:
mD
d 2 xD
 mD g   p E  p K AD   p K  p A ADR  FDA
dt 2
(12)
where mD is the mass of the displacer, g is the gravitational constant, and the force FDA is
given as:
for 0.5hD  xD  hSE  0.5hD
for hSE  0.5hD  xD  hC  hSK  0.5hD
for hC  hSK  0.5hD  xD  hC  0.5hD
FDA  kSE hSE  0.5hD  xD 
FDA  0
FDA  kSK hC  hST  0.5hD  xD 
(13)
(14)
(15)
The above set of equations can be solved numerically, and the results obtained
compared with experimentally determined values.
The present analysis employs several assumptions to simplify the initial numerical
solution. These simplifications allow a basic analytical tool to be created quickly. To reduce
the error inherent in this approach, each assumption will be addressed, incorporated in the
analytical solution and then encoded to form part of the numerical solution program. The
assumptions used are:
 The engine alignment is vertical
 Pressures in all spaces are uniform
 Temperatures in all spaces are uniform
 The working fluid acts as a perfect gas
 All Flow is through the regenerator
 No loss of fluid, so mass is constant
 The mass of working fluid in the regenerator is constant
 Heat transfer within the matrix is solely between the matrix material and the working
fluid
 Heat addition at start-up is instantaneous
 Work done on the system is negative
 Work done by the system is positive
353
Thermo- and CFD modelling of Stirling machines






Thermal energy provided to the system is positive
Thermal energy removed from the system is negative
No pressure drop due to the regenerator (pumping losses)
Ideal regenerator (TRE = TE, TRK = TK)
No parasitic losses
Fluid expansion is instantaneous (close to the speed of sound) hence happens much
faster that the motion it causes on the displacer. Depending upon the rate of expansion
the direction of flow through the displacer may be discerned.
NUMERICAL RESULTS
The initial conditions for solving the set of equations (1) to (13) were defined. The
internal pressures and temperatures for all nodes throughout the engine were set equal to the
ambient conditions. The flywheel was started from the angular position equal to the top dead
position of the power piston, with a set angular velocity. The displacer was assumed to sit on
the bottom stub springs. The temperatures of the hot and cold reservoir were set at an equal
interval from the ambient condition. This particular assumption was used to achieve
symmetry and thus allow a faster convergence to the steady-state solution. Due to the
complex and fast initial transients, the time interval required to achieve convergence had to be
less than 10-8 seconds, resulting in very long simulation run times.
Typical preliminary results are shown in Figures 6 to 9. The constants required for the
numerical solution have been determined experimentally. A small demonstration engine built
at Napier University was used for this.
70
60
50
40
30
20
10
0
2.00
time
2.10
Figure 6: Variation of the position of the power piston and the displacer with time.
354
Thermo- and CFD modelling of Stirling machines
1.2
20
1.1
10
1.0
0
0.9
-10
0.8
2.00
time
-20
2.10
Figure 7: Variation of the pressure pE and the pressure difference pE-pK with time.
1.2
70
65
1.1
60
55
1.0
50
45
0.9
40
35
0.8
2.00
time
30
2.10
Figure 8: Variation of the pressure pE and the position of the power piston with time.
355
Thermo- and CFD modelling of Stirling machines
80
20
70
15
60
10
50
5
40
0
30
-5
20
-10
10
-15
0
2.00
time
-20
2.10
Figure 9: Variation of the positions of the power piston and the displacer, and the pressure
difference pE - pK with time.
DISCUSSION
The present analysis uses many simplifying assumptions. The above defined
simulated performance will, clearly contain a significant error. To improve accuracy these
potential errors are addressed in and the improved equations adopted into the numerical
solution.
Even in its initial form the numerical solution has highlighted several interesting
phenomena. Such as the phase difference between the displacer and piston being a function of
mass transfer through the regenerator. When running without a load on the flywheel the
amplitude of the displacer stroke reduces as engine speed increases. At the overdriven limit
the displacer can ‘jump’ cycles.
Initial modelling was performed using an excel spreadsheet, and upon the regenerator
only. It soon became evident that excel was not appropriate to manipulate the large amounts
of data being generated. This lead to the use of Microsoft VBA, which also proved
inappropriate for the large data sets. A more suitable programming language was required.
For this work FORTRAN has been chosen. These data were then exported to an excel
spreadsheet for ease of interpretation.
The inter-dependency of the mechanical and thermal cycles dictates this order of
calculation. The following progression of calculations is used.
1.
2.
3.
4.
5.
6.
7.
8.
356
Flywheel angle theta
Piston location
Displacer location
Expansion space temperature
Compression space temperature
Mass flow
Expansion space pressure
Compression space pressure
Thermo- and CFD modelling of Stirling machines
The verification strategy requires a physical model to be built so that the results may
be compared. This physical model is based upon drawings published by Senft [8], and has
been manufactured at Napier University.
CONCLUSIONS
Conservation laws and the ideal gas law have been applied to a Low Temperature
Differential Ringbom Stirling Engine. Employing simplifying assumptions a third order
numerical analysis has been developed. These equations describe the processes happening in
throughout the engine simultaneously.
The equations are manipulated and transformed into discrete terms within the time domain.
These terms are then coded to form the numerical solution. By changing variables within the
computer program the engine will be optimised. This process will be verified by comparison
of experimental data with numerical results.
Further development of the present model will produce an accurate modelling tool for a
Low Temperature Differential Ringbom Stirling Engine, and allow definition of optimum
design and operating parameters.
REFERENCES
[1] SENFT, J.R. 1993. Ringbom Stirling Engines. New York: Oxford University Press.
[2] READER, G.T and HOOPER, C. 1983. Stirling Engines. Cambridge: E & F. N. Spon.
[3] ORGAN, A.J. 1992. Thermodynamics and gas Dynamics of the Stirling Cycle Machine.
(s.l.), Cambridge university press.
[4] WALKER, G. et al. 1994. The Stirling Alternative Power Systems, Refrigerants and Heat
Pumps. OPA. Yverdon.
[5] URIELI, I. and BERCHOWITZ, D.M. 1984. Stirling Cycle Engine Analysis. Bristol:
Adam Hilger.
[6] Altman, A. 2003. Proceedings of the 11th ISEC. Paper. Rome
[7] Ringbom, O. 1907. US patent 856,107
[8] SENFT, J.R. 2000b. Miniature Ringbom Engines. Fifth Reprint. New York: Moriya
Press.
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