The Philips stirling thermal engine: Analysis of the Rhombic drive mechanism and efficiency measurements

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THE PHILIPS

STIRLING THERMAL ENGINE

ANALYSIS OF THE RHOMBIC DRIVE MECHANISM

AND

EFFICIENCY MEASUREMENTS

PROEFSCHRIFT

TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETEN-SCHAP AAN DE TECHNISCHE HOGE-SCHOOL TE DELFT OP GEZAG VAN DE RECTOR MAGNIFICUS DR. R. KRONlG, HOOGLERAAR IN DE AFDELING DER TECHNISCHE NATUURKUNDE, VOOR EEN COMMISSIE UIT DE SENAAT TE VERDE-D~GENOPWOENSDAG 16 NOVEMBER 1960 . f

r

DES NAMIDDAGS TE 2 UUR

8

~ldo

DOOR

ROELF JAN MEIJER

WERKTUIGKUNDIG INGENIEUR GEBOREN TE ALTEVEER (GEM. ONSTWEDDE)

Dit proefschrift is goedgekeurd door de promotor

Aan mijn ouders

Aan mijn vrouw

- - - -- - - -- - - .

Addendum to the thesis : "The Philips Stirling Thermal Engine. Analysis ofthe rhombic drive mechanism and efficiency measure-ments" by R. J. Meijer

Errata

page 16, line 6: for fiction read friction

(~d)2(ade

)

page 52, line 6: for

t

r2 \rw

L

mde

+

-!-

mp

read

-!-

r2 (Xd

f

(ade mde

+-!-

md) rw. L

.

X

)2

page 53, line 2: for trmo {cos 2 tp

+

(Àw }w2

read -tr2mo {cos 2 tp

+ (

ÀXw ) 2} w2 page 57, line 9: the expression for Fd3 should read

- _ 1_ (Fd _ mo Xd

+

2 Jde

X

sin

x)

2cos X L

page 57, line 9 from bottom:

for Fa-Fd6-Fp6

=

0 re ad Fa

+ Fd6 -

Fp6

=

0 page 69: equation (12t}) should read

Tr

=

-

2T

= -

2 (ao

+ n~l

Pn cos ntpm +

n~l

Qn sin n ljJm) where Pn and Qn are given in equations (110) and (lil)

page 86, line 3 from bottom: for (-4) read (-2)

page 98, fig. 57: for Pmax

=

140 kg/cm read Pmax

=

140 kg/cm2 page 102, line 3 from bottom: for Fourrier read Fourier

page 112, line 1: for

h [ dn + 2bn

~ e~f

-1

~

read h [dn

+

2bn

~ (~~)

_{2 - I}

~

_] page 112, line 4: for Tr

=

-

[kar

+ ...

]

read Tr

=

-

2T see page 111

C"'I N .... ~o 0)0

VOORWOORD

Bij het verschijnen van dit proefschrift, stel ik er prijs op ir. F. J. Philips en ir. H. Rinia dank te zeggen voor hun voortdurende belangstelling in de heet-gasmotor en het vertrouwen in mij gesteld. Hun onwankelbaar geloof in de bruikbaarheid van de stirlingmachine en hun groot doorzettingsvermogen zijn de bron geweest waaruit jarenlang de aansporingen tot het volbrengen van een moeilijke taak zijn gekomen. Moge de onlangs ontstane samenwerking tussen de N.V. Philips en de General Motors Corporation, en de reeds langer bestaande samenwerking tussen de N.V. Philips en de N.V. Werkspoor leiden tot een versnelde toepassing van de stirlingmotor naast de gebruikelijke motoren en daarmee de gehele mensheid ten goede komen.

Beste dr. Köhler. Wat heb ik veel van je geleerd! Je scherp kritisch denken en de waarheid zoeken zonder aanzien des persoons, hebben mij het ware speurwerk doen kennen. Je lijfspreuk: per aspera ad astra kan ik nu heel goed begrijpen. Voor dit alles mijn welgemeende dank.

Beste collega's en assistenten! U allen dank ik hartelijk voor de steun bij de totstandkoming van dit proefschrift ondervonden. Bijzondere dank ben ik ver-schuldigd aan dr. R. H. Bathgate voor zijn accuratesse en de prettige samen-werking bij het vertalen.

De Directie van het Natuurkundig Laboratorium der N.V. Philips' Gloei-lampenfabrieken te Eindhoven ben ik zeer erkentelijk voor de verleende vrijheid tot het publiceren van een gedeelte van mijn werk in de vorm van een proefschrift.

Samenvatting Summary

I. General introduction

CONTENTS

1.1. A short history of the höt-gas engine . . . . 1.2. The principle of the Stirling engine . . . . 1.3. Drive mechanisms used in displacer-piston engines References. . . . . . . . . . . . . . . . . . . . .

Il. Analysis of the rhombic drive mechanism

2 3 3 7 12 16 17 U.I. Symbols. . . . . . . . 17

U.1.1. Symbols for the general form 17

11.1.2. Symbols for the special form . 18

11.2. Balancing . . . 26

11.2.1. Introduction . . . . . . 26 11.2.2. The balancing conditions for the general form 27 11.2.3. The balancing conditions for the special form . . 32 Il.3. The determination of various quantities from the dimensions of the drive

mechanism. . . . . . . . . 35 11.3.1. Introduction. . . . . 35 II.3.2. S, CPdr and Xmax. . . 36 11.3.3. The volume of the expansion space, VE . 38 11.3.4. The volume of the compression space, Vc 39 11.3.5. Vo, w, cpvo and (X • • • • • • • • • • • 42

11.3.6. Summary of the equations derived above 42 UA. The pressure variation in circuit and buffer space . 44 UA.I. The pressure variation in the circuit. . . 44 1104.2. The pressure variation in the buffer space of a multi-cylinder engine 45 11.5. The torque on the crankshafts due to the gas forces and inertia forces . . . 46 II.5.1. Introduction . . . 46 11.5.2. The energy balance. . . . . . . . . . . . . 47 11.5.3. The torque on one crankshaft due to the gas forces of the circuit and

crankcase on the piston and the displacer . . . . . . . 48 11.504. The torque on one crankshaft due to the gas forces of the buffer space

acting on the fust piston of a mul ti-cylinder engine . . . 50 11.5.5. The torque on one crankshaft due to the inertia forces . . . 50 11.6. The forces on various parts of the drive mechanism due to gas forces and

inertia forces . . . 54 11.6.1. Introduction. . . . . . . . , . 54 11.6.2. The forces on the piston drive mechanism . . 54 II.6.3. The forces on the displacer drive mechanism . 57 U.6A. The forces on the crankshaft . . . 57 II.6.5. Sirnplified equations for the forces . . . 58 U.7. The power and the torque due to the friction on one crankshaft 59 U.8. Some applications of the torques calculated above 62 11.8,1. Introduction . . . 62 11.8.2. Motion of the crankshafts . . . 62 1I.8.3. The torque on the gearwheels . . . 66 U.8A. Determination of the size of the tlywheels . 67 11.8.5. Motion of the driven machine . . . . . 68 U.8.6. The reaction force and reaction torque on the base 69

II.9. Specimen calculation for a single-cylinder hot-gas engine. . . . U.9.I. Nominal data and the determination of numerical quantities .

II.9.2. The torque (Tt

+

Tr) for w = constant . . . . II.9.3. The effective moment of inertia Je . . . .

II.9.4. Calculation of the forces occurring in the system . . . II.9.5. An estimate of the torque Tr and the friction power Pr II.9.6. Motion of the crankshaft . . . .

11.9.7. Motion of the driven machine (brake) . 11.9.8. The torque on the gearwheels . 11.9.9. The reaction torque on the base References . . . . . . . .

lIl. Efficiency measurements on a single-cylinder hot-gas engine with 70 70 72 75 76 77 83 85 86 86 87

rhombic drive mechanism 88

IIJ.I. Principal engine data . 88

111.2. Constructional details . 88

IIL3. The measuring set-up 93

1II.4. Results . . . 95

Conclusions ₁₀₁

References ₁₀₃

Appendix I. List of equations ₁₀₄

Appendix II. The coefficients Bn 114

Appendix _{lIl. The coefficients Gn 118}

Samenvatting

In hoofdstuk I wordt een kort overzicht van de geschiedenis van de heetgas-motor gevolgd door een bespreking van het stirlingproces. Vervolgens wordt een nieuw drijfwerk voor de verdringermotor besproken, dat ten aanzien van de balancering en bij het beperken der gaskrachten, grote voordelen biedt.

Hoofdstuk II bevat de analyse van dit "ruitdrijfwerk" , nl. a. De afleiding van de balanceringsvoorwaarden.

b. De afleiding van reken grootheden uit de afmetingen van het drijfwerk. c. De berekening van de drukvariaties in de gemeenschappelijke bufferruimte

van een veelcilindermotor, waarvan de krukken willekeurige hoeken met elkaar maken.

d. De berekening van het koppel afkomstig van de gas- en traagheidskrachten.

e. De berekening van de krachten in het drijfwerk.

f. Een schatting van de wrijvingsenergie en van het koppel ten gevolge van de wrijving.

g. Enkele toepassingen van de formules die voor de koppels zijn afgeleid. Dit hoofdstuk eindigt met een rekenvoorbeeld dat betrekking heeft op een ééncilinder-heetgasmotor die in het Natuurkundig Laboratorium van de N.V. Philips' Gloeilampenfabrieken te Eindhoven is geconstrueerd en beproefd.

In hoofdstuk III worden rendementsmetingen beschreven, die gedaan zijn aan de proefmotor welke in hoofdstuk II als rekenvoorbeeld is gekozen. Na een korte beschrijving van de motorconstructie, volgt een overzicht van de gebruikte meetinrichtingen alsmede een samenvatting van de meetresultaten.

Onder de Conclusies wordt de heetgasmotor op enkele punten vergeleken met de zuigermotor met inwendige verbranding. Een aanhangsel geeft een over-zicht van de afgeleide formules; drie verdere aanhangsels geven tabellen en grafieken voor de coëfficiënten van reeks ontwikkelingen die voor de in hoofd-stuk II genoemde berekeningen nodig zijn.

-Summary

Chapter I contains a short history of the hot-gas engine and a discussion of the principle'of the Stirling system, followed by a description of a new drive mechanism for the displacer-piston engine which offers great advantages in connection with the balancing of the engine and the reduction of the gas forces. In chapter Il are given various calculations concerning this rhombic drive mechanism:

a. the determination of the balancing conditions ;

b. determination of various quantities needed for further calculations from the dimensions of the drive mechanism;

c. calculation of the variations in the pressure of the common buffer space of a multi-cylinder engine with arbitrarily chosen crank angles;

d. calculation of the torque due to gas forces and inertia forces;

e. calculation of the forces in the drive mechanism;

f

estimate of the friction energy and the torque due to friction; g. some applications of the equations involving the torque.

This chapter ends with a sample calculation for a single-cylinder hot-gas engine which has been built and tested in the Research Laboratory of N.V. Philips' Gloeilampenfabrieken, Eindhoven.

Chapter III describes efficiency measurements which have been made on this experimental engine: af ter a short description of the construction of the engine, the measuring equipment is described; the results of the measurements are summarized in tables and graphs.

In the conclusions, some properties of the hot-gas engine are compared with those of the internal-combustion piston engine. Appendix I gives a list of the equations which have been derived; the other three give tables and graphs containing the coefficients of the series expansions used in the calculations of chapter Il.

L GENERAL INTRODUCTION

1.1. A short history of the hot-gas engine

Hot-air engines were used in considerable numbers during the 19th century, and it was thought for a time that they might riyal the steam engine as a source of power 1.2).

Slaby 3.5) has divided the systems appearing in this period into the following types:

a. Open systems in which a fresh charge of air is taken in for each cycle and

heated directly in a furnace situated outside the cylinder of the engine.

b. Open systems in which a fresh charge of air is taken in for each cycle, but is heated indirectly (i.e. by external combustion).

c. Closed systems in which the same amount of air is used in successive cycles. The hot-air engine built by Sir George Cayley 1.3) in 1807, which was probably the fust one to work properly, belongs to group a.

Fig.!. Hot-air engine (about 1880), built according to the design of Sir George Cayley, 18076). A is the check valve through which the fresh charge of air is admitted; the outlet

valve Band the inlet valve Care operated by the engine itself. As the piston P goes from right to left, A and B are open, so new air is sucked into the cylinder via A while the air to the left of the piston is forced out through B. As the piston goes the other way, A and Bare c10sed and C is open. The fresh charge of air is thus blown through the fire D and the hot flue gases enter the cylinder to the left of P via C. The increased pressure due to this heating is the same on both sides of the piston, but the area difference due to the presence of the piston rod

causes the piston to move to the right.

Figure 1 shows an example of this group, which was used for pumping water and was still built in 1880.

The four hot-air engines built by the Swede Ericsson in 1853 for the 22oo-ton ship which bore his name (see fig. 2) are typical examples of group b. These are probably the biggest hot-air engines ever made: the power piston~ had a diameter of 4.2 mand a stroke of 1.8 m. These were meant to give an LH.P. of 600, but were found on testing only to deliver about half this. However, the fuel consumption (ab out 1 kg coal per LH.P.) was considerably lower than that ofthe marine steam engines used at that time (about 1.4 kg coal per I.H.P.) 1.4.6).

The ingenious invention of the Scottish minister Robert Stirling belongs to 3

-R

Fig. 2. Ericsson's hot-air engine, 18334_).

The power piston A and the compressor piston Bare rigidly connected by means of rods. The compressor sucks in air during its downward stroke and forces it through the regenerator R during its upward stroke. This preheated air flows past the heated wall C to the power cylinder H, where it is further heated through the wall D. About 2/3 of the way through the stroke the valve E doses the openings F and G so that the air then expands in the hot cylinder H. E is pushed rapidly downwards at the end of the stroke, so the air can pass via F and the regenerator Rl to the atmosphere during the downward stroke. The valves K and Kl are changed af ter about 50 revolutions, so that the regenerator which gave up heat now accumulates

it and vice versa.

Fig. 3. The first Stirling engine according to the patent of 18166).

A long vertical cylinder A is heated on top by the flue gases from the fumace B; the bottom is cooled either with water or by convection with the atmosphere. The cylinder contains the displacer C (of rather smaller diameter than A and centred by small rollers) and the power piston D. As the displacer moves to and fro the air in the cylinder A passes altemately from the cold space E (space between the bottom of the displacer and the top of the piston) and the hot space F (above the displacer) via a regenerator and back. The regenerator (not shown here) is in the annular space between the displacer and the cylinder, and probably consisted of thin wire wound around the displacer. The air is thus altemately in the cold and the hot space, and undergoes temperature and pressure variations which cause it to perform work

--~----~--- - - -- -- - -

--~~~~~~~~~~~~~~~~~~---group c. He introduced bis hot-gas engine, wbich contained a power piston,

a dis placer piston and a regenerator, in 1816. Tbis is shown in fig. 3. The fust

model was already in use in 1818, for pumping water out of a quarry. Together with bis brother James he introduced improvements into later modeIs, such as a compressor to increase the specific power and a double-action power piston to reduce leakage. The latter was of doubtful value, ho wever, since it entailed the provision of two displacer pistons in separate cylinders, thus increasing the dead space and decreasing the specific power.

Fig. 4. Example of a Stirling engine from about 1880 4).

The displacer A and the piston Bare both in the same cylinder. The displacer fits closely into the cylinder at C, so that the air is forced through the cooler D and along the heater wall E. There is no separate regenerator. The hot flue gases from the furnace F flow round the heater wall E and pass out of the chirnney H via the damper G. If the ~ngine is used for pumping

water, a pump rod can be fitted directly on to the projecting end L of the rocker K.

The fust, and in principle better, model of 1816 was forgotten until 1868, when W. Lehmann realized the advantage of a small dead space, and again placed the displacer piston and the power piston in the same cylinder. Mter all sorts of mechanical improvements, tbis led to an engine as shown in fig. 4 (usually used without the regenerator), wbich was sold in large numbers in the second half of the 19th century. According to the price-list of a German factory from about 1885, the largest Stirling engine wbich they made gave 2 H.P.,

weighed 4100 kg (with packing and masonry) and had a volume of 21 m3•

5

It is no wonder that the internal combustion engine with its much greater specific power and better efficiency displaced this hot-air engine, despite a certain charm which the latter possessed.

Mter the internal combustion engine had been brought to perfection, so that all its advantages and disadvantages could be seen, intensive research on !he Stirling hot-air engine was carried out in the Research Laboratories of N.V. Philips' Gloeilampenfabrieken, Eindhoven 7-11). This was prompted by the fact that before the second world war Philips felt the need of a heat-driven source of power for radios and similar equipment for use in parts of the world where the fuel needed for such a device was easier to obtain than batteries. The only power sources which came into consideration, owing to the special demands made for this purpose, were the thermoelement, the closed-circuit steam engine and the hot-air engine. The choice feIl on the latter after preliminary studies showed that the hot-air engines on the market at the time (see fig. 5) had been completely neglected by the development of modern materials and new knowledge about fluid flow and heat transfer, and that in fact Stirling's system was no longer fully used, since the regenerator has graduaIly fallen out ofuse.

Fig. 5. Hot-air engine with generator for a radio (1938).

The improvements in radio tubes and particularly the introduction of transistors, which reduced the power consumption ofbattery models enormously, made the hot-air engine less attractive for the purpose for which it had originaIly been intended, so the research then turned to higher-power models. Some small generators working on this principle (see fig. 6) were however built on a special occasion in 1953, iri memory of the original intention. The hot-air engines of these generators were the fust to be fitted with the rhombic drive mechanism 12.13). The analysis of this drive mechanism and the test results of an experimental high-power engine are the subjects of this thesis.

1

Fig. 6. Generator (with radio) built in 1953 and incorporating the fust rhombic drive for the Stirling system.

1.2. Tbe principle of the Stirling eogioe

An internal-combustion engine provides a surplus of work in virtue of the compression at low temperature of a certain quantity of air, to which atomized fuel is added either before or after the compression, the subsequent heating of the mixture by rapid combustion, and its expansion at high temperature.

The hot-gas engine is based on the same principle, i.e. the compression at low temperature and expansion at high temperature of a given quantity of gas. The heating takes place, however, in an entirely different manner, the heat being supplied to the gas from outside, through a wal!. For this reason the description "external-combustion engine" is appropriate. Owing to the high heat capacity of the wall, it is not of course possible to heat and cool the gas simply by rapid heating and cooling of the wal!. We have already seen that Stirling made the gas temperature change periodically by causing a "displacer piston" (hereafter simply cal1ed "displacer") to transfer the gas back and forth between two spaces, one at a fixed high temperature and the other at a fixed low temperature - see fig. 7. If we raise the displacer in fig. 7, the gas will flow from the hot space via the heater and cooler ducts into the cold space. If now the displacer is moved downwards the gas will return to the hot space along the same path. During the first transfer stroke the gas has to yield up a large quantity of heat to th tcwolei; an equal quantity of heat has to be taken up during the second stroke from' the heater. The regenerator shown in fig. 7 is inserted between the heater duct and cooler duct in order to prevent unnecessary wastage of this heat. lt is a space filled with porous material to which the hot gas yields heat before entering the cooler; when the gas streams back, it takes up the stored heat again prior to its entry into the heater.

---+----hot spoox

2689

Fig. 7. Principle of the displacer system.

As the displacer moves up and down, the gas fiows from the hot space to the cold space via the heater, regenerator and cooler, and back.

The displacer system, which serves to heat and cool the gas periodically, is combined with a power piston (hereafter simply called the "piston") which compresses the gas while it is in the cold space and allows it to expand while

in the hot space (all dead spaces in cooler, heater etc. being disregarded). Since compression takes place at a lower temperature than expansion, a surplus of work results. Fig. 8 shows foUT stages of the cycle through which the whole

IJ JII IJl 2690

Fig. 8. PriIiciple of the hot-gas process.

We will assume for the sake of simplicity that the piston and the displacer move discontinuously.

The cycle can then be divided into four stages:

I. Piston' in its lowest position, displacer in its highest. All the gas is in the cold space. IT. The displacer is still in its higbest position; tbe piston has compressed tbe gas at low

temperature.

IIT. The piston is still in its highest position; the displacer has transferred the gas from the cold space to the hot space.

IV. The hot gas has expanded, and both the piston and the displacer are in their lowest positions. The displacer will now return the gas to the cold space while the piston remains where it is, to give stage I again.

- - t 2691

Fig. 9. The discontinuous displacements of the piston (P) and the displacer (D) plotted as a function of time. Band Erepresents the volume variations of the hot space (VE), band C those of the cold space (V c). These variations are plotted separately at the bottom of the

figure. III p

I

II I

- v

2692

Fig. 10. The p,V diagram of the hot-gas cyc1e represented in fig. 9.

system passes if a discontinuous movement of piston and displacer is presupposed. The displacements they are assumed to undergo are plotted as functions of time in fig. 9; the ordinates in band Erepresent the variation in the volume of the hot space, and those in band C the variation in the volume ofthe cold space. The volume variations are plotted separately in the lower part of the diagram.

Fig. 10 is the p,V diagram of the cycle

CV

is the total volume of the gas).

-In a practical version of the engine the movements of piston and displacer must of course be continuous, not discontinuous, as they have been assumed to be in these figures; the continuo us movements will be obtained with the aid

of some kind of crank and connecting rod mechanism. It will not then be

possible to distinguish any sharp transitions between the four stages but this will not alter the principle ofthe cycle (or detract from its efficiency - see below).

The movements of piston and displacer might now be as indicated in fig. 11,

Fig. 11. As fig. 9, except that the motion of the piston and displacer is now continuous, and the displacements are plotted as a function of the crank angle IX.

The various stages of the cyc1e can no longer he c1eady distinguished.

in which the volume variations of the cold and hot spaces have again been plotted separately. The only essential condition for obtaining a surplus ofwork is that the volume variation of the hot space should have a phase lead with respect to that of the cold space; this is equivalent to requiring that the ap-propriate p,V diagram, shown in fig. 12, should be traced out in the clockwise direction. The variation of the pressure and the power may readily be calculated

if certain simplifying assumptions are made. We wiIl assume that V E, the volume

of the hot (expansion) space, and Vc, the volume of the cold (compression)

space, vary with the crank angle (X in a purely sinusoidal fashion:

VE

=

! Vo(1

+

cos (X)

The crank angle IX (for constant angular velocity, proportional to the time: IX = rot) is measured from the position at which VE has its maximum value Vo; qJ is the phase angle between the volume variations of the hot and cold spaces,

p

r

_ v

2694

Fig. 12. The p-V diagram for the cyc1e represented in fig. 11.

and w is the ratio between their maximum volumes. Another important quantity introduced is the gas teinperature ratio,

Tc

.

=

- - '

TE '

(.

<

1)

The condition that the mass of the working fl.uid remains constant throughout the cycle now leads to the formula for the pressure p as a function of the crank angle:

1-15

p

=

pmax 1

+

15 COS(IX-O)

in which pmax is the maximum pressure occurring during the cycle and

15

=

V·

2

+

w2

+

2 .w cos qJ . + w + 2s tan 0

=

w sin qJ

• +

w cos qJ

(s

=

L Vs' Tc the relative dead space reduced to temperature Tc, where V s and VO'Ts

Ta are the volume and absolute temperature of the dead spaces). 1 1

-From the above we obtain the mean pressure,

and the power output

1/

1 - <5

P

= pmax

YT+b

P =twVop(1-r) <5 sin 8

1

+

V(1-

<52₎

We are concerned here with a reversible cyclic process in which, in accordance with the "idealized" conditions assumed (isothermal behaviour in cold and hot spaces and 100% efficient regenerator action), the supply of heat takes place at only one temperature TE and the removal of beat at only one temperature Tc. Tbere is a tbeorem in thermodynamics whicb states tbat under these conditions the efficiency with which heat is converted into work (the thermal efficiency) is that of the Carnot cycle:

From this we may obtain the quantity of heat supplied per second, P 1 < 5 . 8

qE

=

-

=

"2-wVo

p

sm 17 1

+

V(1-<52

)

The equation for the power given above shows that the power is proportional to the mean pressure in the engine. In order to obtain a large specific power (power output per unit volume swept out by the piston), a high mean pressure must therefore be used, among other things.

1.3. Drive mechanisms used in displacer-piston engines

There are various ways of making the piston and displacer perform the desired movements. Fig. 3 shows the earliest method used, and fig. 4 shows one of the mechanisms in use at the end of the last century. These engines had the working gas at a low pressure (minimum prèssure generally 1 atm. abs.). Stirling introduced a double-action piston with two separate displacers in 1827, in order to reduce the gas leaks at higher pressures and also to balance the forces exerted on the piston to a certain extent; this ho wever also gave a large dead space.

This disadvantag~ bas been removed in the smaller hot-air engines and the gas refrigeration machines made by Philips by placing the drive mechanism in a gastight crankcase and filling this with the working gas at a pressure (the

"2695

Fig. 13. The drive mechanism used in small hot-air engines.

D = forked piston connecting rod, K = crankshaft, T = rocker, connected by hinged rods Sl and S2 to the displacer rod V and to a point on D.

2696

Fig. 14. The drive mechanism used in the gas refrigerating machine.

Two parallel connecting rods Dl and D2 on cranks Kl and K2 are connected to the piston, and one connecting rod V' on a crank K' between the other two is connected to the displacer

rod V.

buffer pressure) equal to the minimum or mean circuit pressure (figs. 13 and 14). It is dear, however, that such a pressurized crankcase leads to rather

-heavy constructions, especially with large engines; and if it is desired to operate at very high pressures in order to increase the specific power, this becomes a great disadvantage.

We shall describe below the principle of operation of a drive mechanism for a displacer-piston engine which allows the buffer pressure to be applied, while keeping the crankcase at atmospheric pressure if desired. This drive mechanism also has the advantage of allowing even a single-cylinder engine to

be completely balanced. It consists of twin cranks and connecting-rod

mecha-nisms offset from the central axis of the engine; the cranks rotate in opposite senses and are coupled by two gearwheels.

Fig. 15. The rhombk drive mechanism.

1 = piston. 6 = displacer. 5,5' = cranks on two shafts which rotate in opposite directions and are coupled by the gear wheels 10 and 10'. 4,4' = connecting rods pivoted on the ends of a yoke 3 fixed to the hollow piston rod 2. 9,9' = connecting rods pivoted on the ends of a yoke 8 fixed to the displacer rod 7, which passes through the hollow piston rod. 11, 12 =

stuffing boxes. 13 = buffer space filled with gas under high pressure.

Fig. 15 is a schematic diagram of the system. Fixed to the piston 1 by way

of piston rod 2 is a yoke 3. One end of the yoke is linked by connecting rod .

4 to crank 5, the other end by connecting rod 4' to crank 5'. The displacer is actuated by a precisely similar arrangement: the displacer rod 7, which passes through the hollow rod 2, has fixed to it a yoke 8, which is linked to cranks 5

and 5' by connecting rods 9 and 9' respectively. if 9 and 9' are given the same

length as 4 and 4', the two pairs of connecting róds will form a rhombus, of which only the angles vary when the system is in motion; it is for that reason that we have adopted the name "rhombic drive". Gearwheels 10-10' en,sure

exact symmetry of the system at all times. The two crankshafts being geared together, the entire shaft output can be taken off either.

The symmetry of the system and the coaxial arrangement of piston and displacer rods make it an easy matter to avoid putting the crankcase under high pressure. The stuffing-box 11 for the displacer rod is inside the hollow piston rod. One more seal, namely the stuffing-box round the piston rod (12 in fig. 15), is all that is necessary to form a comparatively small cylindrical chamber 13 under the piston, separate from the crankcase ; this "buffer space" ean be filled with gas at the desired buffer pressure. The minimum permissible volume of the buffer space is determined only by the range within which it is desired to limit the pressure variations inside the chamber. In a multi-cylinder engine the buffer chambers can be interconnected; this allows the volume of the individu al spaces to be made even smaller.

_ O l 2b98

Fig. 16. The volume variations V E and V c of the hot and cold spaces respective1y of a hot-gas engine with rhombic drive mechanism, plotted as in fig. 11.

The piston and displacer movements resulting from the new drive are

dis-played graphically in fig. 16, in the same way as in figs. 9 and 11. It will be seen

that if the direction of rotation is as indicated in fig. 15, the volume variation

of the hot space will have a phase lead with respect to that of the cold space, as is required. The piston and displacer do not by any means move in simple harmonie motion, but it is found that a very good version of the Stirling cyc1e is obtainable. This might seem surprising, for on the face of it the use of the rhombic drive may appear to have restricted the freedom of choice with regard

to parameters wand cp (the amplitude ratio and relative phase of the variations

in volumes V E and V c - see above), which play a large part in the design of a

hot-gas engine. However, we will see in Il.3 that it is in fact possible, by altering the offset of the crankshafts, the proportions of cranks and connecting rods and the ratio of piston and displacer diameters, to vary these parameters over quite a wide range.

-The balancing conditions for this rhombic drive mechanism will be deter-mined in section 1I.2, from those for the general form (fig. 18), while after caIculating the pressure variation in the buffer space in 1I.4 the torque due to gas forces and inertia forces wiU be treated in II.S. The forces occurring in the drive mechanism will be calculated in 11.6, and with the aid of the equations thus derived the fiction losses will be estimated in 1I.7. Section 11.8 gives some applications of the torques calculated in 11.5 and 1I.7. The caIculations in 1I.8 are concluded with a specimen caIculation for an engine which was constructed under my direction in Philips' Research Laboratories, Eindhoven, while chapter III will give the results of efficiency measurements carried out on this engine.

REFERENCES

1yW. J. M. RANKINE, Mechanics Magazine, London, 1854.

2) J. N. O. B. KITCHING, Ericsson's Calorie engine, New York, 1860.

3) J. O. KNOKE, Die Kraftmaschinen des Kleingewerbes, Berlin, 1887.

4) SIR J. ALFRED EWING K. C. B., The Steam-Engine and other Heat-Engines, Cambridge, 1926.

5) T. FlNKELSTEIN, The Engineer, 492-497, 522-527, 568-571, 720-723, 1959.

6) CR. DELAUNAY, Mechanica (translated by F. A. T. DELPRET), A. W. SYTHOFF, Leyden, 916-922, 1854.

7) H. RINIA and F. K. DU PRÉ, Air engines, Philips Tech. Rev., 8, 129-136, 1946.

8) H. DE BREY et al., Fundamentals for the development of the Philips' air engine, Philips Tech. Rev., 9, 97-104, 1947.

9) F. L. VAN WEENEN, The construction of the Philips' air engine, Philips Tech. Rev.,9, 125-134, 1947.

- 10) J. W. L. KÖHLER and C. O. JONKERS, Fundamentals of the gas refrigerating machine,

Philips Tech. Rev., 16,69-78, 1954.

, 11) idem., Construction of a gas refrigerating machine, Philips Tech. Rev., 16, 105-115, 1954.

12) R. J. MEDER, The Philips hot-gas engine with rhombic drive mechanism, Philips Tech. Rev.,

20, 245-262, 1958.

Il. ANAL YSIS OF THE RHOMBIC DRIVE MECHANISM

n.l.

Symbols

11.1.1. Symbols for the general form (see figs. 18 and 19) r L cL ST e e

+

Lle rer reo (j Yo d'fJ' W

=

( f t . d2_'fJ' w = dt2 1 L

T

=

r

c À cL r ST v = -r = crank radius

=

length of piston connecting rod

=

length of displacer connecting rod

=

distance from crank pin to swivel point

=

eccentricity of piston drive mechanism

=

eccentricity of displacer drive mechanism

=

di stance from centre of gravity of crankshaft to centre of crankshaft

= distance from centre of gravity of counterweight to centre

of crankshaft

. crank angle ('fJ' = 0 when the crank is at OT') = (constant) angle STP

=

angle which the piston connecting rod makes with the x-axis

=

angle which the dis placer connecting rod makes with the x-axis

= angle between radius of centre of gravity of counterweight

and crank radius

=

angular velocity of crankshaft

=

angular acceleration of crankshaft

=

relative length of piston connecting rod

=

relative length of displacer connecting rod

=

relative di stance fr om crank pin to swivel point e

e

=

-

=

relative eccentricity of piston drive mechanism

r

A e

+

Lle l ' . . f d' 1 d . h '

e

+

LJe = - - -

=

re atIve eccentncIty 0 ISp acer nve mec am sm

rer eer = -r rco eco = -r r

=

relative di stance from centre of gravity of cranks haft to centre of cranks haft

=

relative distance from centre of gravity of counterweight to centre of cranks haft

-ma

=

mass concentrated in the point P: half of the mass of the

piston, piston rod and piston yoke and part of the mass of the piston connecting rod

=

mass concentrated in the point T: the rest of the mass of

the piston connecting rod and part of the mass of the arm ST

= mass concentrated in the point S: the rest of the mass of

the arm ST and part of the mass of the displacer connect-ing rod

=

mass concentrated in the point Q: the rest of the mass of

the displacer connecting rod and half of the mass of the displacer, displacer rod and displacer yoke

= mass of the crankshaft concentrated in Cer

= mass of the counterweight concentrated in Ceo

=

abscissa of mI = abscissa of m2

=

abscissa of ma

=

abscissa of ID4

=

abscis sa of Ill5

=

abscissa of m6

11.1.2. Symbolsfor the specialform (see figs. 20 and 21)

Angles tp tpm =

rot

Lttp tpI IX X Xmax qJvo Yo qJdr

=

crank angle (tp = 0 when the crank is at OB')

=

mean angular velocity times time

=

tp-tpm

=

crank angle for the first cylinder of a multi-cylinder engine

= crank angle corresponding to the fust harmonic of the

volume of the expansion space (IX

=

0 when this volume

is maximum)

=

angle which the connecting rods make with the x-axis

= maximum value of X

=

phase angle by which the fust harmonic of the volume of

the expansion space leads the fust harmonic of the volume of the compression space

= phase angle by which the cranks of the ith _{piston lead}

those of the first piston

= angle between radius of centre of gravity of counterweight

and crank radius (balancing conditions lead to 'Yo = n)

= phase angle by which the fust harmonic of the motion of

Points

A

pe Ade Bpe Bde epe ede

= shaft angle of the driven machine

=

wt-.xe

= angle of twist

=

ffJ-ffJm

=

middle of the small end of the piston connecting rad

=

middle of the small end of the displacer connecting rad

=

middle of the big end of the piston connecting rad

=

middle of the big end of the displacer connecting rad

=

centre of gravity of piston connecting rad

=

centre of gravity of displacer connecting rad

Lengths, distances, reduced lengths

r Lde Lpe L pe

=

Lde

=

L e Sp = Sd = S rer reo ape ade 1 L

T

=

r

e 8 = -r rer eer = -r reo eeo = -r Xp Xd Xc, Ye Diameters

D

p Dd

=

crank radius

=

length of displacer connecting rad

=

length of piston connecting rad (balancing conditions)

=

eccentricity of drive mechanism

=

stroke of piston = stroke of displacer

=

distance from centre of gravity of crankshaft to centre of crankshaft

=

distance from centre of gravity of counterweight to centre of crankshaft

=

distance from epe to Bpe

=

distance from ede to Bde

=

relative length of connecting rad

=

relative eccentricity of drive mechanism

=

relative distance fr om centre of gravity of crankshaft to centre of crankshaft

=

relative distance from centre of gravity of counterweight to centre of crankshaft

= abscissa of piston = abscis sa of displacer

=

co-ordinates of crank pin

=

diameter of piston = diameter of displacer

-dCl Masses mp mpc mpc mdc m" _dc mco mcr mc mo Radii of gyration ipc ipc idc ." lpc ." ldc ic

=

diameter of piston rod

=

diameter of displacer rod

=

diameter of piston yoke pin

=

diameter of displacer yoke pin

=

diameter of crank pin

=

diameter of crank-pin bearing shell around which the .

connecting rod of the displacer or the piston moves

=

diameter of crankshaft journal

=

mass of piston, piston rod and piston yoke

=

mass of piston connecting rod

=

mass dynamically equivalent to part of mpc, concentrated in Apc

=

mass dynamically equivalent to part of mpc, concentrated in Bpc

=

mass of displacer, displacer rod and displacer yoke

=

mass of displacer connecting rod

=

mass dynamically equivalent to part of mdc, concentrated in Adc

= mass dynarnically equivalent to part of mdc, concentrated

in BdC

=

mass of counterweight

=

mass of crankshaft

=

mass of crankshaft and counterweight

2apc 2adc

= mp

+

- - L -mpc = md

+

--L-mdc

=

radius of gyration of the piston connecting rod ab out its centre of gravity, Cpc

=

radius of gyration of the displacer connecting rod about its centre of gravity, Cdc

=

radius of gyration of the reduced mass m'pc about Apc

=

radius of gyration of the reduced mass mdc about AdC

=

radius of gyration of the reduced mass m;;c about Bpc

= radius of gyration of the reduced mass md'c about Bdc

=

radius of gyration of the crankshaft with counterweight about its centre

Areas :n; 2

Ad = -Dd

4 = cross-sectional area of displacer

Ad = :

(D~-d~)

=

cross-sectional area ofdisplacer minus that ofdisplacer rod

:n; 2

Ap = -Dp

4 = cross-sectional area of piston

A~

= :

(D~-d~)

=

cross-sectional area of piston minus that of displacer rod

A;{ = :

(D~

-

d~)

=

cross-sectional area of piston minus that of piston rod

f Ad A~ Ad f' -Ad

X'

f" = p -Xp Volumes VE Vc Vo wVo Vb Vbo

V

bl Vb! Vbc Vbt Pressures

= volume of expansion space = volume of compression space

= maximum value of fint harmonic of VE

= maximum value of first harmonic of V c = varying volume of buffer space

= maximum value of first harmonic of Vb

= fust and second harmonics of Vb (piston of fust cylinder)

= fust and second harmonics of Vb (piston of ith cylinder) = constant volume of buffer space

= total buffer volume of multi-cylinder engine (constant part

+

fust and second harmonies of varying part)

= fictive dead space at the expansion end

= fictive dead space at the compression end

=

VbO

= pressure in expansion space

= mean pressure in expansion space

= pressure in compression space

-21-Pc

Pb Pb Pee g

=

PE .

pc '

Forces

F;;

Frp Frd Fp6

= mean pressure in compression space

=

pressure in buffer space

=

mean pressure in buffer space

=

pressure in cranJecase

g '

=

Pb. " pee

P

c

'

g

=

p

c

=

gas force on displacer

=

gas force on piston

=

gas force on piston due to circuit pressure

+

crankcase pres su re

=

gas force on piston due to buffer pressure

=

friction force on piston and piston rod

=

friction force of the displacer rod on the stuffing box in the piston

=

vertical component of force on piston yoke pin

=

horizontal component of force on piston yoke pin

= component ofthe force on the massless rod in the direction

of the piston connecting rod

=

component of the force on the massless rod perpendicular

to the piston connecting rod

=

component of the force of the big end Bpe of the piston

connecting rod on the cranJe. pin (or crank-pin bearing shell) in the direction of the cranJe

=

component of the force of the big end Bpc of the piston connecting rod on the crank pin (or cranJe-pin bearing shell) perpendicular to the cranJe

= vertical component of the force on the displacer yoke pin

= horizontal component of the force on the displacer yoke

pin

=

component ofthe force on the massless rod in the direction

of the displacer connecting rod

=

component of the force on the massless rod perpendicular

to the displacer connecting rod

=

component of the force of the big end Bde of the displacer connecting rod on the crank pin (or cranJe-pin bearing shell) in the direction of the cranJe

=

component ofthe force ofthe big end Bde ofthe displacer

connecting rod on the cranJe pin (or cranJe-pin bearing shell) perpendicular to the cranJe

Ps

=

component of the force of the main bearing on the

crankshaft journal in the direction of the crank

=

component of the force of the main bearing on the

crankshaft journal perpendicular to the crank

Torques on one crankshaft

T gc Tgb Tg Tm T~ Tmw Tf ff T",m Tt Tfpl TfP2 TtP3 Tf~3 Tfdl Tfd2 Ttd3

Tfd3

Tf4 Tf5

= torque due to gas forces from circuit and crankcase

=

torque due to gas forces from buffer space

=

Tgc

+

Tgb

=

torque due to inertia forces

=

Tm-lew

= torque due to inertia forces at constant angular velocity

=

torque due to friction forces

=

mean value of Tf

=

Tg -

T

f - Tmöi

= resultant torque

= torque due to PfPI

=

torque due to Pfp2

=

torque due to Pip3 = torque due to Pfp3

=

torque due to Pfdl

=

torque due to Pfd2

=

torque due to· Ptd3

=

torque due to

Pfd3

= torque due to Pf4

=

torque due to Pf5

Other torques and moments

T

=

torque transmitted by the gearwheels of the drive

mecha-nism

=

reaction torque on the base

=

torque of the driven machine

= moment of the massless rod acting on m'pc

=

moment of the massless rod acting on m~c

=

moment of the massless rod acting on mdc

=

moment of the massless rod acting on mdC

Moments of inertia

Jpc

=

sum of the moments of inertia of m~c about Apc and m';c

about Bpc

-Energies ~E ~U Ee Ep Powers Pt PtP1 PtP2 PtP3 Ptd1 Ptd2

=

sum of the moments of inertia of mde about Ade and mde

about Bde

=

effective moment of inertia of half of the drive mechanism

ab out the corresponding crankshaft

=

moment of inertia of the flywheel which is not connected

with the coupling

=

moment of inertia of the flywheel which is connected with

the coupling

=

mean value of Je

=

Je

+

JFl

=

Je

+

h2

=

moment of inertia of the driven machine

= Jo

+

J1

+

J2

= total kinetic energy of half of the drive mechanism

=

total potential energy of half of the drive mechanism

=

kinetic energy of crankshaft and counterweight

=

half of the kinetic energy of the piston, piston rod and piston yoke

-.:... kinetic energy of the ·piston connecting rod

= half of the kinetic energy of the displacer, displacer rod and displacer yoke

=

kinetic energy of displacer connecting rod

=

total power consumed by friction of half of the drive mechanism

=

half of the power consumed by friction of the piston and

piston rod

= power consumed by friction of the piston yoke pin

=

power consumed by friction of the crank-pin bearing shell

on the crank pin (crank-pin bearing shell being fixed to piston connecting rod)

=

power consumed by friction of the big end of the piston

connecting rod on the crank-pin bearing shell (the latter being fixed to the displacer connecting rod)

= half of the power consumed by the friction of the displacer

rod on the stuffing box in the piston

=

the power consumed by the friction of the displacer yoke

Ped3

P

e

d3

Pf4

PfS

=

power consumed by the friction of the crank-pin bearing shelJ on the crank pin (the crank-pin bearing shell being fixed to the displacer connecting rod)

=

power consumed by friction of the big end of the displacer connecting rod on the crank-pin bearing shell (the latter being fixed to the piston connecting rod)

=

power consumed by the friction ofthe crankshaftjournals in the main bearings

=

power consumed in gearwheel transrnÏssion Coefficients of friction flp2 flP'3 fl3 fld2 fld3 fl4 Further symbols d'!jJ w = w = -'I' dt . d2_'!jJ w= - -dt2 l(; N Xp Xd Xe )ie Bn en, Dn

G

n Hn c5 wrp k

=

coefficient of friction corresponding to PtP2

= coefficient of friction corresponding to Pf~3

=

coefficient of friction corresponding to Pip3 or Pfd3 = coefficient of friction corresponding to Ptd2

= coefficient of friction corresponding to Pfd3

=

coefficient of friction corresponding to Pf4

=

angular velocity of crankshaft

=

angular acceleration of crankshaft

=

ratio of principal specific heats of working gas

=

number of cylinders

=

piston velocity

=

displacer velocity

=

acceleration of crank pin in the x direction

=

acceleration of crank pin in the y direction

=

coefficients (appendix II)

=

coefficients (appendix I)

=

coefficients (appendix lIl)

=

coefficients (appendix IV)

=

coefficient of non-uniformity

=

angular velocity of the driven machine

=

torsional stiffness of the coupling

-25-Wo

11.2. Balancing

=

V

k(Jo

+

Jl

+

J2) JoCh

+

J2)

=

fraction of power lost in gearwheel transmission, per crankshaft

11.2.1. Introduction

As has already been mentioned in chapter I, each cylinder of a hot-gas engine contains two reciprocating bodies with a phase difference between them, the piston and the displacer. Since these two bodies have different functions, they also have different shapes.

The displacer consists of a displacer body, which forms a moving seal between the hot and cold gas, and a thermally insulating displacer dome. The pressure differences in tbe gas over the displacer are caused by the flow losses in the engine, which are usually smalI, so it is not generally necessary to use piston rings for this seal as long as the displacer body fits closely into the cylinder. The displacer therefore has the form shown in fig. 17a.

The piston has a more usual function, which may be compared with that of the piston of a compressor, and it forms the moving seal between the cold gas of varying pressure in the circuit and the gas in the buffer space, which has a

a b

Fig. 17a. The form of displacer u~d.

Fig. 17b. The form of piston used.

more or less constant pressure. The piston is shown in fig. 17b.

The masses of the displacer and the piston would in general be different if

they were both designed exclusively for the above-mentioned purposes; but they can also be used for another purpose, i.e. balancing the motion of tbe

moving parts of the engine, which does not interfere with their basic functions since they are differently shaped in any case. The ratio of the masses of the displacer and the piston must therefore satisfy certain conditions which are derived below.

11.2.2. The balancing conditions for the general form

The new drive mechanism described in section 1.3 is only one form of a more general system. The general form of the drive mechanism is shown in fig. 18.

Fig. 18. The genera] form of the rhombic drive mechanism.

This differs from the special form described above in that one of the connect-ing rods (considerconnect-ing one ofthe two symmetrical halves ofthe drive mechanism) is not connected to the crank pin, but to a point on the other connecting rod. Figure 18 shows the piston connecting rod equipped with such a swivel point S. The eccentricity of the piston yoke pin Pand of the displacer yoke pin Q also differ, being denoted by e and e

+

4e respectively. The swivèl point S on the piston connecting rod is at a di stance ST from the crank pin, and makes an angIe cpa with the main direction TP of the connecting rod. The centre of

gravity ofthe crankshaft Cer is at a distance rer from the centre 0 ofthe crank-shaft in the direction of the crank pin, while the position of the centre of

-gravity of the counterweight with respect to 0 is given by the angle Yo and tbe

distance r co.

We will now determine the conditions which must be imposed on the various quantities in order to give complete balancing.

This calculation is based on fig. 19, which only gives one half of the drive mechanism, since the other half is its mirror-image. The lengths in this figure are expressed in terms of the crank radius.

Fig. 19. The general form of the rhombic drive mechanism, with dimensions expressed relative to the crank radius.

It follows directly from the symmetry ofthe system that the sum ofthe inertia forces in the y direction is zero: :E Y = 0, and the sum of the torques due to the inertia forces about the z-axis is also zero: :E T z

=

O.

In order to obtain balancing, therefore, it is only necessary that the sum of the forces in the x direction, :E X, should be zero.

Since we are only concerned with forces, we may replace the mass of each moving part by two equivalent masses, which are chosen so that:

1. the sum of the two masses is equal to the mass which they replace,

2. the centre of gravity of the two masses coincides with the centre of gravity of the part which they replace.

The drive mechanism may thus be represented by point masses mI, m2, ma, m4, ms and fi6 with abscissae XI-X6 respectively.

It may be seen from fig. 19 that:

and

. 1

Xl

=

SlO tp -

T

cos X X2

=

sin tp

Xa

=

sin tp- 'JI cos(X

+

gJo)

C

X4

=

Xa

+

T

cos f}

Xs

=

(!er sin tp X6

=

(!eo sin(tp

+

yo)

sin X

=

À (e - cos tp)

sin () =

~

[sin X - 'JIÀ sin(x

+

gJo)

+

ÀLle)

c

~

cos X and

~

cos () can be expanded in series in terms of tp:

1 00

T

cos X

=

n~o Bn cos ntp

c 00

T

cos ()

=

n~o An cos ntp

(1) (2)

(3) (4) where Bn is a function of À and e, and An is a function of À, c, e, Lle, 'JI and gJo.

If we substitute these expressions for sin X, cos X and cos () into the equations for Xl,

xa

and X4, we obtain:

00

Xl

=

sin tp- L Bn cos ntp n-o

X2

=

sin tp

00

Xa

=

'JIÀe sin gJo-'JIÀ sin gJo cos tp

+

sin tp- 'JIÀ cos gJo L Bn cos ntp n-o

00 00

X4 = 'JIÀe singJo-'JIÀ singJocos tp

+

sin tp- 'JIÀ cOSgJo L Bncos ntp

+

~ An cos ntp

n-o n-o

Xs = eer sin tp X6 = (!eo sin (tp

+

yo)

Differentiating twice with respect to time and multiplying by the correspond-ing masses, we obtain the forces in the X direction.

The sum of these forces must be zero: 6

LX = ~Xmmm=O

m=l

Xlml = ml[w 2{-sin tp

+

~n2Bncos ntp} +w{costp

+

LnBnsin ntp}]

x2m2 = m2[-w2sin tp +wcostp]

2 9

X3ms =ms[W2{'JIÀ sinlPocos1p-sin 1p

+

'JIÀCOSlPo ~n2BnCosn1p}

+

+w{'JIÀ sinIPo sin 1p

+

cOS1p

+

'JIÀ cos 9'10 ~nBnsinn1p}]

X41ll4=1ll4[w2{'JIÀsinlPocos1p-sin 1p

+

'JIÀCOSlPo ~n2BnCosn1p-~n2AnCosn1p}

+

+w{'JIÀ sin 9'10 sin 1p

+

cos1p

+

'JIÀ cos 9'10 ~nBnSinn1p- ~nAnSinn1p}]

X5tns = ecrm5[-W2 sin 1p

+

W cos 1p]

x6m6= eCOm6[-w2sin (1p

+

"0)

+

wcos (1p

+

"0)]

<Xl

where ~ means ~.

n-l

The counterweight m6 can only give a harmonically varying force, whose magnitude and phase can be chosen so as to make the fust harmonic of the forces zero. The other parameters of the system must now be chosen so that

the higher harmonics add up to zero at any time:

<Xl <Xl

w2 [{mI

+

vÀ cos 1P0(m3

+

m4)} ~ n 2Bn cos n1p- m4 ~ n 2An cos n1p]

+

n~2 n=2 (6)

<Xl <Xl

+

dJ [{mI

+

'JIÀ cos 9'10 (ma

+

m4)} ~ nBn sin n1p- m4 ~ nAn sin n1p] = 0

n=2 n=2

For n ~ 2, therefore, the following condition must hold: Bn

An (7)

It may be seen from equation (7) that Bn/An must have a value which is

independent ofn. We will now discover what this means in terms ofthe geome-try of the system.

Put Bn/An = D.

If this is substituted into equations (3) and (4), we have:

where and

c 1

D

T

cos () =

T

cos X

+

Cl cos 1p

+

Co Cl = -BI

+

DAl

Co = -Bo

+

DAo Equation (8) may be rewritten as follows:

cos () = E cos (X-(3)

+

c~

where

1

E

=

: = ; ;

-cD cos {3

tan {3 = -Cl cos {3 =

---:=====

1 SIn . {3

= -

----;:=====_ Cl

Vc~

+

1

Vc;

+

1

We have already shown (2) that:

sin () =

~

{sin X - 'JIÀ sin(x

+

9'10)

+

ÀL1e}

c

(8)

(9)

This may be written as:

sin 0

=

~

sin(x-IX)

+

~

Lle (11)

where p

=

VI - 2vA. cos cpo

+

V2A.2 and

1 vA. sin cpo

tan IX

=

;

1 - vA. cos cpo cos IX

=

-(1p - vA. cos CPo);

. 1 , .

SIn IX

= -

'1111. sm cpo p

We now have the following equations for 0:

cos 0

=

E eos(x-(3)

+

e~

sin 0

= R.

sin(x-IX)

+

~

Lle

e ' e

(9) (11)

The balaneing conditions ean now be found by eombining these two equations, making use of the relationship:

eos2₀

+

_sin2_{0 = 1}

Setting the sum of the eoeffieients of the fust harmonie and of the seeond harmonie of X equal to 0, and the sum of the constant terms equal to 1, we obtain the following conditions :

E

=

R.

=

I·

e ' IX

=

f3; Co

=

0; A.Lle

=

0

It now follows, from (10), that:

D

=

lf(eEeosf3),andsubstitutingtheabovevalues, wehaveD

=

lf(l-vA.eos CPo).

If this value of BnfAn is substituted in equation (7), it follows that:

.ml - 1114

+

vÄ. cos cpo (ma

+

21ll4)

=

0

We thus have the following conditions whieh must be imposed on the drive meehanism in order that the sum of the higher harmonies of the forces should be zero (assuming that Ä. =1= 0):

Lle =

°

e

=

Vl - 2vÄ. cos cpo

+

v2Ä.2

mI

=

1114 - VÄ. cos cpo (ma

+

21ll4)

We still have the terms for the fust harmonies to balance (see (6» :

w2_[{ml

+ vÄ.

_{cos cpo(ma}

+

_{1ll4)} BI- Ill4AI] cos 1jJ}

+

+

w[ {mI

+ V.A.

cos cpo (ma

+

1ll4)} BI - Ill4AI] sin 1jJ Now

(12) (13)

(14)

[ {mI

+

vA. cos cpo (ma

+

Il14)} Bl -1ll4AI] = m4 (

~

- Al)

=

m4VA. sin cpo

-so the reIDaining fust harmonies are:

ID4(WZ_{VÀ. sin fPo cos 1p + wvÀ. sin fPo sin 1p)}

Equation (5) thus beeoIDes:

- WZ[(ID1 + IDz + IDa + ID4 + (lerID5) sin 1p + (leoID6 sin (1p + yo)

+

-{vÀ.(IDa + 2ID4) sinfPo} cos 1p] +W[(ID1 +IDz+ IDa + ID4 + (lerID5) cos 1p+ + (leoID6 cos (1p + yo) + {vÀ.(IDa + 2ID4) sin fPo} sin 1p] = 0

So

where

Yo = n-{}; m6=--M (leo {} = _tan-1 vÀ.(ma + 2m4) sin fPo

mI + mz + ma + m4 + (lerID5

and M = V(m1 + m2 + ma + m4 + (lerm5)2 + {vÀ.(ma + 2m4)sinfPo}2

In brief, then, the drive meehanism ean be completely balaneed ifthe following conditions are satisfied 1):

Lts

=

0

e =

Vl-

2vÀ. cos fPo + v2À.z

m1- m4 + vÀ.(ma + 2m4) cos fPo = 0

while the position and mass of the eounterweight are given by:

(12)

(13)

(14)

Yo =n-tan -1 vÀ. (ma + 2m4) sin cpo (15)

ml + mz + ma + m4 + (lerm5

m6 = _1_ V(m1 + mz + ma + m4 + (lerm5)2 + {vÀ.(ma + 2m4) sin fPo}2 (16)

(Jeo

The following remarks may be made about these conditions :

a. The condition e =

Vl-

2vÀ. cos fP + vZ_{À.2 implies that (if Lts = 0) PS}

must be equal to SQ (see fig. 19).

1/

1 v

PS = SQ =

r

~

-

2

T

cos fPo + '1'2

b. If cpo = 0° or 180°, S lies on the piston eonnecting rod or an extension thereof. Then Yo = n, i.e. the eentre of gravity of the counterweight now lies diametrieally opposite the crankpin.

11.2.3. The balancing conditions Jor the special Jorm

The rhombie drive meehanism described in seetion 1.3 (fig. 15) is a special form of the general form discussed above. It differs from the general form in

that ST

=

0 (see fig. 19). All the calculations given bel ow refer to this form, which will be called simply the rhombic drive from now on (fig. 21).

Since ST

=

0 (i.e. 'V

=

0), equation (13) becomes c

=

1, which means that the piston connecting rod is the same length as the displacer connecting rod. The piston connecting rod of length Land mass mpe can be replaced by the two masses:

, ape

ffipe

=

-r-mpe (17)

" (1

ape )

ffipe

=

-

- r - ffipe (18)

while the corresponding masses for the dispJacer connecting rod are:

(19) (20) (see fig. 20).

a b

Fig. 20a. The piston connecting rod with the equivalent mass system. Fig. 20b. The displacer connecting rod with its equivalent mass system.

Equation (14) then becomes:

apempe - ademde =

t

L (md - mp) (21)

where md and mp are the masses of the moving parts of the displacer and

'accessories and the piston and accessories respectively.

The position of the counterweight is found by substituting 'V

=

0 in equation (15):

Î'o = n (22)

-i.e. the centre of gravity of the counterweight must be diametrically opposite the crank pin.

The mass of the counterweight follows from equation (16): reomeo

=

r{mpe

+

mde

+

l(mp

+

md)}

+

rermer or if the crankshaft is considered together with the counterweight:

reme

=

r{mpe

+

mde

+

l (mp

+

md)} where me

=

meo

+

mer reomeo- rermer re = meo

+

mer (23) (24)

reo, rer and re are the distances from the centre of the crankshaft to the centres of gravity of the counterweight, crankshaft and counterweight

+

crankshaft; the corresponding masses are denoted by meo, mer and me.

1r-- - --pÎslon rod

---di'ploc~r rod

, - - -- -piston )'06c~

- ----dispbccryokc 2703

Fig. 21. The special form of the rhombic drive, which is obtained from the general form

II.3. The determination of various quantities from the dimensions of the drive mechanism

IJ. 3.1. Introduction

The design of the heat exchangers (heater, regenerator and cooler) of a hot-gas engine depends to a very large extent on the purpose for which the engine is intended. In one case, for example, the stress may be laid on a large specific power or low weight, while the efficiency is of relatively little importance; while in another case it is just the specific fuel consumption which is important, while the weight does not matter so much.

Often other factors also play a part, e.g. the price, so a compromise must be made between maximum specific power and minimum specific fuel con-sumption to give the best engine for the purpose in question.

The dimensions of the heat exchangers which are determined by the above-mentioned factors can be calculated to a good approximation from the first harmonic of the volume variations and the corresponding phase angle.

p

v

+x 2104

Fig. 22. Part of the rhombic drive mechanism shown in several positions; the dimensions are expressed relative to the crank radius.