Stirling-α motors

Riccardo Berti – riccardo.berti@studenti.unipd.it / riki9191@gmail.com

Introduction

Stirling motors are alternative heat engines with external heat input, equipped with regenerator and working with single-phase gas working fluid. These motors are called “hot air” even though in modern engines hydrogen and helium have replaced the air initially employed on these engines. Hot air engines run in closed loop, they haven’t cams or suction and exhaust pipes, so they are particularly silent, require little maintenance and have a regular running. The operating cycle is reversible and consists of:

ciclo stirling

Mechanically the cycle is achieved by cycling the working fluid between two heat exchangers (hot and cold) by means of two pistons which are suitably offset (90 ° to 120 ° angles are chosen). A regenerator with a thermal flywheel function is inserted in the circuit to limit the amount of cyclic heat exchange and improve its efficiency.

In the design and construction of a Stirling engine, attention should be paid to a few details, because unlike internal combustion engines, where specific powers are large enough to forgive heavy construction defects, in the case of Stirling engines, the specific powers are so low which leave no margin for defects and inefficiencies.

The coupling between plunger and cylinder is the most critical aspect for the creation of a working Stirling engine. For this reason, it has been decided to simulate three different kinematics to identify which solution has smaller contact forces between the cylinder and the plunger, resulting in less stress for the seal and therefore greater life of the same and the engine.

There are basically 3 configurations in which a Stirling engine can be made, called α, β and γ. It was chosen to analyze Stirling-α motors because it is the configuration that generally reaches higher rotation regimes, thus presenting a greater stress for the piston cylinder coupling.

Here are the three engines used in the simulation:

biella manovella ross-yoke wobble-plate

in the first part, the geometric design of the three mechanisms was performed, under the condition of having the same displacement in all engines. In the second part, the Smhidt model was applied to calculate pressures within the cylinders, and consequently such pressures were imported into adams as punctual forces acting in the center of the plunger. Finally, the contact forces between the cylinder and the plunger were created by testing different plunger geometries to obtain fluid simulations.

Objectives

The goal was to achieve the trend and the magnitude of the friction forces between plunger and cylinder, implementing them in Adams as contact forces. From the comparison of the results in the three motors we wanted to identify which kinematic layout has the lowest piston cylinder contact forces. To do this, the values ​​of the friction forces and the normal forces between the cylinder and the plunger for all three engines have been graphed.

The modelling problem

Description of the models

All parts of the three models were designed by CAD and imported into Adams as a Parasolid file. The parts were imported one at a time into Adams to assemble the engine in the initial configuration. The mass and moments of inertia of the parts were calculated by Adams, providing as input data the material and geometry of the part. The joints choice was made to avoid redundant constraint conditions.

Biella manovella

Parts:

PARTS MATERIAL MASS [kg] Ixx [kg*mm2] Iyy [kg*mm2] Izz [kg*mm2]
Albero steel 7.91 38595 38592 11548
Biella-sx aluminum 0.22 828.2 826.8 16.37
Biella-dx aluminum 0.22 828.2 826.8 16.37
Pistone-sx aluminum 0.11 51.59 50.82 46.66
Pistone-dx aluminum 0.11 51.59 50.82 46.66
Cilindro-sx copper 2.54 6374 6330 2441
Cilindro-dx copper 2.54 6374 6330 2441
Volano steel 5 75688 38007 38000

Constraints:

  • Lock_cil_sx: fixed joint between cilindro-sx and ground;
  • Lock_cil_dx: fixed joint between cilindro-dx and ground;
  • Lock_Volano: fixed joint between volano and albero;
  • Trs_pist_sx: prismatic joint between cilindro-sx and pistone-sx;
  • Trs_pist_dx: prismatic joint between cilindro-dx and pistone-dx;
  • Rvl_albero: rotational joint between albero and ground;
  • Cil_biella_sx: cylindrical joint between albero and biella_sx;
  • Cil_biella_dx: cylindrical joint between albero and biella_dx;
  • Sph_pist_sx: spherical joint between biella_sx and pistone-sx;
  • Sph_pist_dx: spherical joint between biella_dx and pistone-dx.

with these constraints the mechanism has 1 degree of freedom and no redundant constraint. In order to perform the analysis of normal forces and friction between plunger and cylinder, the trs_pist_sx and trs_pist_dx constraints have been suppressed and replaced by the contact forces:

  • Contact_dx
  • Contact_sx

In order to guarantee the individual degree of freedom, the joints Sph_pist_sx and Sph_pist_dx have been disabled and replaced by:

  • Rvl_pis_dx
  • Rvl_pis_sx

Ross Yoke

Parts:

PARTS MATERIAL MASS [kg] Ixx [kg*mm2] Iyy [kg*mm2] Izz [kg*mm2]
Albero steel 2 2913 2803 1382
Biella-sx aluminum 0.22 828.2 826.8 16.37
Biella-dx aluminum 0.22 828.2 826.8 16.37
Pistone-sx aluminum 0.11 51.59 50.82 46.66
Pistone-dx aluminum 0.11 51.59 50.82 46.66
Cilindro-sx copper 2.54 6374 6330 2441
Cilindro-dx copper 2.54 6374 6330 2441
Volano steel 5 75688 38007 38000
Perno steel 0.122 28.59 28.59 6.12
Triangolo steel 1.2 3873 2314 1639

Constraints:

  • Lock_cil_sx: fixed joint between cilindro-sx and ground;
  • Lock_cil_dx: fixed joint between cilindro-dx and ground;
  • Lock_vol: fixed joint between volano and albero;
  • Lock_perno: fixed joint between perno and ground;
  • Rvl_alb: rotational joint between albero and ground;
  • Trs_pist_sx: prismatic joint between cilindro-sx and pistone-sx;
  • Trs_pist_dx: prismatic joint between cilindro-dx and pistone-dx;
  • Cilindrico: cylindrical joint between albero and triangolo;
  • Cil_biella_sx: cylindrical joint between triangolo and biella_sx;
  • Cil_biella_dx: cylindrical joint between triangolo and biella_dx;
  • Sph_pist_sx: spherical joint between biella_sx and pistone-sx;
  • Sph_pist_dx: spherical joint between biella_dx and pistone-dx.
  • Prim_perno_tri: linear primitive between triangolo and perno;

with these constraints the mechanism has 1 degree of freedom and no redundant constraint. In order to perform the analysis of normal forces and friction between plunger and cylinder, the trs_pist_sx and trs_pist_dx constraints have been suppressed and replaced by the contact forces:

  • Contact_sx
  • Contact_dx

In order to guarantee the individual degree of freedom, the joints Sph_pist_sx and Sph_pist_dx have been disabled and replaced by:

  • Rvl_pist_dx
  • Rvl_pist_sx

Wobble plate

Parts:

PARTS MATERIAL MASS [kg] Ixx [kg*mm2] Iyy [kg*mm2] Izz [kg*mm2]
Albero-1 steel 1.11 850 768 552
Albero-2 steel 1.11 850 768 552
Biella-1 aluminum 0.29 1066 1056 28
Biella-2 aluminum 8.55 354 353 3
Biella-3 aluminum 8.55 354 353 3
Biella-4 aluminum 8.55 354 353 3
Pistone-1 aluminum 0.12 56 55 54
Pistone-2 alluminum 0.16 63 63 63
Pistone-3 aluminum 0.16 63 63 63
Pistone-4 aluminum 0.16 63 63 63
Cilindro-1 copper 2.23 3964 3964 2208
Cilindro-2 copper 2.23 3964 3964 2208
Cilindro-3 copper 2.23 3964 3964 2208
Cilindro-4 copper 2.23 3964 3964 2208
Volano steel 5 75688 38007 38000
Wobble steel 3.2 3237 3237 2793
Flangia aluminum 1 5604 2847 2847

Constraints:

  • Fix_2: fixed joint between tree-2 and tree-1;
  • Fix_volano: fixed joint between flywheel and tree-1;
  • Fix_cil_1: Fixed between cylinder-1 and flange;
  • Fix_cil_2: fixed joint between cylinder-2 and flange;
  • Fix_cil_3: Fixed between cylinder-3 and flange;
  • Fix_cil_4: fixed joint between cylinder-4 and flange;
  • Fix_flangia: fixed joint between flange and ground;
  • Rvl_albero_1: Rotational joint between tree-1 and ground;
  • Rvl_albero_wobble: rotational joint between tree-1 and wobble;
  • Rvl_1: rotational joint between connecting rod 1 and piston-1;
  • Uni_2: universal joint between connecting rod 2 and piston-2;
  • Uni_3: universal joint between connecting rod 3 and piston-3;
  • Uni_4: universal joint between connecting rod 4 and piston-4;
  • Trs_1: prismatic joint between cylinder-1 and piston-1;
  • Trs_2: prismatic joint between cylinder-2 and piston-2;
  • Trs_3: prismatic joint between cylinder-3 and piston-3;
  • Trs_4: prismatic joint between cylinder-4 and piston-4;
  • Sph_1: spherical joint between connecting rod 1 and piston-1;
  • Sph_2: spherical joint between connecting rod 2 and piston-2;
  • Sph_3: spherical joint between connecting rod 3 and piston-3;
  • Sph_4: spherical joint between connecting rod 4 and piston-4.

with these constraints the mechanism has 1 degree of free and no redundant constraint. In order to perform the analysis of normal forces and friction between piston and cylinder, the constraints trs_2, trs_3 (since it was decided to examine contact forces only on two pistons for not to weigh the simulation) were suppressed and replaced by forces contact:

  • Contact_2
  • Contact_3

Geometric considerations for assembly and simulation

A Stirling-α engine requires two cylinders to operate, one hot and one cold, according to what has been described above, however while the crankshaft and ross-yoke crankshaft mechanisms are designed to work with two cylinders, the Wobble-plate would be too much unbalanced by operating with only two cylinders offset by 90 °. For this reason, the Wobble-plate was made with 4 cylinders, as if they had two Stirling motors hinged on the same shaft and offset by 180 ° in operation.

To be able to compare the mechanisms, we chose to fix the total displacement, which must be the same for the three engines. to achieve the same displacement, the borehole and race of the piston rods were fixed, equal to all three motors, while the crank radius value was calculated in order to obtain the fixed displacement.

Therefore the calculations were developed by fixing the same stroke between crank handle and Ross-yoke, and half stroke for the Wobble-plate.

Biella manovella

Fixed: rm=50[mm]; c= 2*50=100[mm]; d=50[mm];

formula 1

Ross yoke

ross yoke 1

The Ross-yoke mechanism was invented by Andy Ross, whose idea was to design a mechanism that would limit the lateral movement of the connecting rods and thus limit the lateral forces on the plunger. The key component of the mechanism is the yoke. For the following calculations, consider the angle θ positive clockwise from the horizontal position; yc and ye represent the displacements of the two pistons, while x indicates the displacement of the center point of the base of the yoke, all referring to the horizontal axis passing through the shaft. For the analysis it is assumed that the lateral displacement of the two connecting rods connected to the yoke end and the center point at the base of the yoke are negligible.

ross yoke 2

ross yoke 3

 

 

 

 

 

 

Now consider the conditions of maximum and minimum displacement of one of the two plungers, conditions that are obtained when the crankshaft arm is aligned with the yoke side, as can be seen from the above figures.

The assumption of no lateral displacement implies:formula 2

And then by solving the two rectangular triangles you get the measurements of ymax and yminformula 3

For the same stroke of the previous case, it sets c = (y_max)-(y_min) = 100 [mm] Fixed the values ​​of          yoke = 110 [mm] and b2 = 55 [mm], resolves in r by attempts. For rm = 42 [mm] you get a c = 101 [mm] value for which you are satisfied, the crank radius value is fixed in this way.

Wobble plate

Wobble-plate as well as Swash-plate are mechanisms used to convert the translational motion of the pistons into the rotating shaft motion.

wobble plate 1

Fixed: c=50[mm]; Int=110[mm]; Hoff=50[mm]

you have it: tg(α)=c/int; α=atg(c/int)=24.44[°]

formula 4

In this way, from the last three reports the geometry of the crankshaft is known.

Forces

To move the three mechanisms, punctual forces are applied to the center of the plunger, they are directed orthogonally to the upper face of the plunger, a module equal to the product between a crank angle function, which represents the pressure of a Stirling engine , and the plunger area. For the calculation of pressures, the Smhidt model, one of the most used methods, has been used, which proposes an approximate solution.

The assumptions of the Smhidt model are:

  • Mass and engine fluid temperature are fixed;
  • Pressure from the law of the ideal gas;
  • No pressure loss during heat exchanges and no internal pressure difference;
  • Isothermal expansion and compression;
  • Fluid with ideal gas behavior;
  • Perfect regeneration;
  • Expansion volume keeps Te constant temperature;
  • Compression volume keeps constant Tc temperature;
  • The regeneration temperature is the average between Te and Tc;
  • Expansion and compression volumes vary sinusoidally.

The model was implemented in an excel sheet from which the values ​​of the crank angle pressures were obtained by inserting data on the geometry of the motor and the operating conditions of the thermal cycle.

Below are the p-v diagrams and the trend of the pressures depending on the crank angle obtained from the model.

p-v aggiornato f angolo aggiornato

 

 

In adams the forces were introduced by inserting the following formula, function of the crank angle:

Immagine

adapting it from time to time to the engine geometry.

Other forces

to make the simulation more adherent to reality and to limit the engine rotation regime, friction forces on constraints have been introduced with the following features:

Biella manovella
force constraints materials μ s μ d
Attrito_cil_sx Cil_biella_sx steel-aluminum 0.3 0.2
Attrito_cil_dx Cil_biella_dx steel-aluminum 0.3 0.2
Attrito_trs_sx Trs_pist_sx copper-aluminum 0.28 0.23
Attrito_trs_dx Trs_pist_dx copper-aluminum 0.28 0.23

 

Ross-yoke
force constraints materials μ s μ d
Attrito_cil_sx Cil_biella_sx steel-aluminum 0.3 0.2
Attrito_cil_dx Cil_biella_dx steel-aluminum 0.3 0.2
Attrito_trs_sx Trs_pist_sx copper-aluminum 0.28 0.23
Attrito_trs_dx Trs_pist_dx copper-aluminum 0.28 0.23

 

Wobble-plate
force constraints materials μ s μ d
Attrito_trs_1 Trs_1 copper-aluminum 0.28 0.23
Attrito_trs_2 Trs_2 copper-aluminum 0.28 0.23
Attrito_trs_3 Trs_3 copper-aluminum 0.28 0.23
Attrito_trs_4 Trs_4 copper-aluminum 0.28 0.23

For the calculation of normal forces and the friction between plunger and cylinder, contact forces have been inserted with the following characteristics in an attempt to obtain a sufficiently fluid simulation and force movements with a small number of peaks. The internal diameter of the cylinders is 50.4 [mm] while the outer diameter of the piston is 50 [mm]. In order to simplify the modeling of the contact, two elastic bands of 50.35 [mm] diameter integrated in the plungers have been hypothesized. Between the cylinder and the plunger, a small game was specifically left to observe how normal forces are influenced by the movement of the connecting rods. The contact model used for normal forces is the continuous impact model, while the coulomb model has been used for frictional forces. The engine used for simulation of contacts is RAPID with 800 mesh mesh.

Biella-manovella
model features CONTACT_sx CONTACT_dx
continuous impact stiffness 4000 4000
  Force exponent 2 2
damping 10 10
Penetration depth 0.05 0.05
coulombiano Static coeff. 0.28 0.28
  Dynamic coeff 0.23 0.23
Stiction transition vel. 100 100
Stiction transition vel. 1000 1000

 

Ross-yoke
model features CONTACT_sx CONTACT_dx
continuous impact stiffness 4000 4000
  Force exponent 2 2
damping 10 10
Penetration depth 0.05 0.05
coulombiano Static coeff. 0.28 0.28
  Dynamic coeff 0.23 0.23
Stiction transition vel. 100 100
Stiction transition vel. 1000 1000

 

Wobble_plate
model features CONTACT_2 CONTACT_3
continuous impact stiffness 4000 4000
  Force exponent 2 2
damping 10 10
Penetration depth 0.05 0.05
coulombiano Static coeff. 0.28 0.28
  Dynamic coeff 0.23 0.23
Stiction transition vel. 100 100
Stiction transition vel. 1000 1000

Simulation

To perform dynamic simulations, the GSTIFF I3 integrator has been set to a tolerance of 0.0001, which is the solution that has made it possible to get more fluid simulations. The simulation was launched using a script_solver with the following characteristics, common to all three simulations:

  • Deactivating contact forces;
  • Deactivate constraints rvl_sx and rvl_dx;
  • Dynamic simulation with pitch = 0.001 and duration = 0.2 [sec];
  • Activating contact forces; • Activating constraints rvl_sx and rvl_dx;
  • Deactivate constraints trs_sx, trs_dx, sph_sx, and sph_dx;
  • Dynamic simulation with pitch = 0.0001 and duration = 0.2 [sec]

The initial conditions for simulation are the imposition of an initial angular velocity of 105 [rad / s], corresponding to approximately 1000 rpm, applied to the rotational joint of each engine shaft.

Simulation and analisys of results

During the simulation of each engine, in the first part (duration 0.2 [sec]), the reactions of the prismatic joints along the orthogonal directions to the piston translational axis were measured in the second part (when the prismatic joints were replaced by forces of contact) the normal contact forces were measured in the same directions.  Throughout the entire simulation, the displacement of the connecting rod head was also measured in the same direction of force measurement. The measurement was made to evaluate the impact of the twisting of the connecting rod on the piston and cylinder forces. In the charts to follow, for normal forces you have:

  • red, contact forces;
  • blue, the reactions of the prismatic joints;
  • pink, the displacement of the connecting rod head.

Normal forces biella-manovella

analisi_biella_manovella_dx

Normal force cylinder_dx / pistone_dx and sliding biella_dx in z direction

analisi_biella_manovella_sx

normal force cylinder_sx / pistone_sx and sliding biella_sx in z direction

Normal forces ross-yoke

analisi_ross_yoke_dx

normal force cylinder_dx / pistone_dx and sliding biella_dx in z direction

analisi_ross_yoke_sx

normal force cylinder_sx / pistone_sx and sliding biella_sx in z direction

Normal forces wobble plate

analisi_normale__2_wobble_plate

normal forces cylinder_2 / pistone_2 and slamming biella_2 in y and z directions

analisi_normale_3_wobble_plate

normal force cylinder_3 / pistone_3 and slamming biella_3 in y and z directions

Friction forces

In the second part of the simulation, frictional forces between plunger and cylinder were also measured and in the same graph the speed of the connecting rod head was reported in the direction of the plunger sliding axis. In graphs of frictional forces there is:

  • red, frictional forces;
  • blue, the speed of the plunger.

Friction biella manovella

analisi biella manovella 1

cylinder_dx / pistone_dx friction force and axial displacement speed of the piston

att_biella_manovella_sx

friction force cylinder_sx / pistone_sx and axial displacement speed of the piston

Friction ross yoke

att_ross_yoke_dx

cylinder_dx / pistone_dx friction force and axial displacement speed of the piston

att_ross_yoke_sx

cylinder_sx / pistone_sx friction force and axial displacement speed of the piston

Friction wobble plate

analisi_attrito_2_3_wobble_plate

cilindro_2/pistone_2, cilindro_3/pistone_3 friction and relative axial displacement speed of the piston

Collected data

biella manovella
pistone_sx
  trs_sx_z contact_sx attrito_sx
max min max min statico dinamico a max vel
forza [N] 22,8 -69,5 44,6 -45,1 132,7 106,2
testa pos [mm] -42,97 49,2 -48,7 49,4 \ \
testa vel [mm/s] \ \ \ \ 186,5 5765
pistone dx
trs_dx_z contact_dx attrito_dx
max min max min statico dinamico a max vel
forza [N] 37,7 -70,2 56,5 -47,4 132,6 107,9
testa pos [mm] -39 44 -42 47,7 \ \
testa vel [mm/s] \ \ \ \ 179,9 5765

 

ross yoke
pistone_sx
trs_sx_z contact_sx attrito_sx
max min max min statico dinamico a max vel
forza [N] 41,9 -34,2 47 -38,5 133,4 109,6
testa pos [mm] 10 -6,5 10,12 -5,5 \ \
testa vel [mm/s] \ \ \ \ 71,4 5378
pistone dx
trs_dx_z contact_dx attrito_dx
max min max min statico dinamico a max vel
forza [N] 35,1 -33,8 33,3 -43,1 144 109,7
testa pos [mm] 5,45 -11,45 6,09 -12,5 \ \
testa vel [mm/s] \ \ \ \ 231,2 7127

 

wobble plate
pistone_2
trs_2_y contact_2_y trs_2_z contact_2_z attrito_2
max min max min max min max min statico dinamico a max vel
forza [N] -1,2 -4,7 0 -7,4 -0,08 -0,19 -0,012 -0,22 133,4 109,5
testa pos [mm] 2,5 3,7 5,3 5,5 0,13 0,1 0,12 0,13 \ \
testa vel [mm/s] \ \ \ \ \ \ \ \ 159,2 3098
pistone 3
trs_3_y contact_3_y trs_3_z contact_3_z attrito_3
max min max min max min max min statico dinamico a max vel
forza [N] 7,17 -3,9 4,17 -6,38 8,9 -0,6 12,2 -4,22 144 109,5
testa pos [mm] -5,3 5,4 -5,3 5,2 -5,4 -5,4 -5,4 -5,4 \ \
testa vel [mm/s] \ \ \ \ \ \ \ \ 132,6 2567

Conclusion

  • Observing the values ​​of the reaction forces of the prismatic joints, it can be observed that the crankshaft mechanisms and Ross-yoke have very similar values ​​and can be considered equivalent solutions.
  • The reaction forces of the prismatic joints on the wobble-plate are, however, significantly smaller (a order of magnitude), showing them as a winning solution.
  • Normal contact forces observation also leads to the same conclusion: crankshaft mechanisms and Ross-yoke are equivalent solutions, while confirming woble-plate as a preferred mechanism to reduce the stresses between plunger and cylinder, having also this case involves smaller contact normal forces of at least one order of magnitude compared to competing solutions.
  • It is noted that there are different forces in the Wobble-plate mechanism depending on the piston / cylinder pair observed; This is due to the particular implementation of the mechanism: piston_1 is coupled with the connecting rod through a rotational joint, unlike the other 3 pistons, whose junction occurs with universal joints. This is necessary to prevent the Wobble from rotating on itself, a condition that would occur if all piston / rod coupling were spherical. In some real solutions, all piston / rod couplings are made by spherical joints, but in such cases the wobble is tied to the frame by means of a prismatic joint to prevent the rotation.
  • Finally, it can be seen that the normal forces exchanged between the cylinder and the piston are closely related to the displacements suffered by the connecting rod head. In fact, what allows the Wobble-plate to reduce the effort is just the limited movement of the connecting rod head.
  • Regarding friction, almost coincident values ​​are observed in all solutions, both for static friction and dynamic friction. However, it is believed that the values ​​collected are not significant and that it is necessary to model the piston / cylinder contact area differently and implement a different friction model in adams. It is believed that for more accurate data on friction behavior, it may be useful to use FEM simulation programs to concentrate the analysis on the contact surface between the cylinder and the elastic band, starting from data collected in adams on normal forces.

Bibliography

  • W. S. Badr, M. Fanni, Ali K. Abdel-Rahman, S. Abdel-Rasoul (2015). Dynamic simulation and optimization of Rombic drive stirling engine using MSC ADAMS software. Procedia Technology 22 (2016) 754 – 761
  • R. W. Redlich, D. M. Berchowitz, (1985). Linear dynamics of free-piston Stirling engine. Proc. Instn. Mech. Engrs., Vol 199 No A3.
  • Berchowitz D., Urieli I. (1980). Stirling Cycle Engine Analysis. Washington Paper 809336 pp 1701-5
  • Hirata K.. Schmidt theory for stirling engines. National Maritime Research Institute.
  • Huo J., Wu H., Chen J. (2014). Dynamics Simulation of Stirling Engine for Solar Energy Based on the Time-Varying Support Stiffness. Open Mechanical Engineering Journal8, 710-715.
  • Immobili F., Ascari G., Molinari F. (2011). Guida pratica al motore stirling. Sandit libri.
  • https://it.wikipedia.org/wiki/Motore_Stirling
  • https://www.ohio.edu/mechanical/stirling/engines/yoke_vol.html
  • https://it.mathworks.com/help/physmod/hydro/ref/swashplate.html?requestedDomain=www.mathworks.com
  • http://www.fairdiesel.co.uk/products.htm
  • http://newenergydirection.com/blog/2008/11/ross-yoke-design-for-alpha-engines/

 

Comments are closed.