Multi-Vane Expanders-Geometry and Vane Kinematics_Badr

Applied Energy 19 (1985) 159 182

Multi-vane Expanders: Geometry and Vane Kinematics

O. Badr, P. W. O'Callaghan and S. D. Probert

School of Mechanical Engineering, Cranfield Institute of Technology, Cranfield, Bedford MK43 0AL (Great Britain)

S U M M A R Y

A multi-vane expander (MVE) offers considerable promise as a prime mover for organic Rankine-cycle engines utilising solar energy or waste heat to provide the power input. 1 Therefore, mathematical models describing an expander's behaviour have been constructed; these models are applicable to a wide range of MVEs. For two existing designs, the geometrical characteristics and the vane kinematics have been evaluated using the computer sub-routines composed.

The results indicate that the neglect of the vanes' thickness when evaluating the geometrical characteristics of the expanders can lead to considerable errors in their predicted volume expansion ratios, especially when numerous vanes and small inlet angles are employed. The expected effects of the geometrical parameters of the expanders on their performances are discussed.

NOMENCLATURE

A[Oz, 0~] Area of the eccentric annulus, between the rotor and the stator-cylinder surfaces, normal to the axes of the rotor and stator in the angular-position range 01---,0 2 see eqn (A6) (m2).

Acc(O) Sliding acceleration of the vane with respect to the rotor see eqns (B5) and (B6) (m s-2).

159

Applied Energy 0306-2619/85/$03.30 :c, Elsevier Applied Science Publishers Ltd, England, 1985. Printed in Great Britain

160 O. Badr, P. W. O'Callaghan, S. D. Probert

ARC(i) Spread angle of the stator-cylinder's ith arc measured from the BASE LINE, with respect to the rotor's centre--see Figs 3, 4 and A2 (rad).

d s Diameter of the pressurising semi-cylindrical slots machined on the trailing sides of the vanes (m).

E(i) Eccentricity of the centre of the stator-cylinder's ith arc with respect to the rotor's centre--see Figs 3, 4 and A2 (m).

H v Vane height--see Fig. A3 (m). i Numbers defining the stator-cylinder's arcs; i = 1,2, . . . , k, . . . ,

n - - l , n . L Axial length of the stator cylinder (m). n Number of stator-cylinder's arcs. N S Number of pressurising slots on one vane. r v Volume expansion ratio of the MVE (-- V2/V1). R(O) Radius of the stator cylinder with respect to the centre of the

ro tor- -see Figs A l, A2 and A3 (m). RARC(i) Radius of the stator-cylinder's ith arc, with respect to the

centre of the rotor, at the end of the spread angle of arc i~ see Figs 3, 4 and A2 (m).

RR Radius of the rotor see Figs 3 and 4 (m). RS(i) Radius of the stator-cylinder's ith arc with respect to its

centre--see Figs 3, 4 and A2 (m). t Time (s). t v Thickness of the vane--see Fig. A3 (m). TI P(i) Angle subtended by the radii RA RC(i) and RS(i) at the end of

the spread angle of the ith arc--see Figs 3, 4 and A2 (rad). V(O) Volume of the expander's cell see Fig. A3 (m3). V 1, V 2 Inlet and discharge volumes of the expander--see Fig. 6 (m3). Vid(0 ) Volume of the expander's cell if the idealised (i.e. zero-

thickness) vanes are employed see Fig. A3 (m3). Vel(O) Sliding velocity of the vanes with respect to the rotor (m s - 1). X(O) Vane extension outside the rotor's slot--see Fig. A3 (m). Z Number of vanes in the MVE. 6 Angle between two successive vanes, defined by eqn (A8) and

shown in Fig. 6 (rad). 0 Angular position, measured from the BASE LINE with

respect to the rotor's centre--see Figs 3 and 4 (rad). 0~n Spread angle of the inlet port opening--see Figs 3 and 4

(tad).

Multi-vane expanders." geometry and vane kinematics 161

(0ou,) s Angular position of the beginning of the exhaust port opening--see Figs 3 and 4 (rad).

~0( i ) Angular coordinate of the stator-cylinder ith arc's centre measured from the BASE LINE with respect to the rotor's centre--see Figs 3, 4 and A2 (rad).

92 Angular speed of the rotor (rad s-1).

THE PROBLEM

Accurate predictions of the geometrical characteristics of an MVE, together with the kinematics of its vanes, are desirable for computer simulation of the behaviour of the expander and for any further attempt to optimise its design. The basic geometrical characteristics of an MVE are the variation of the cell volume as a function of the angular displacement of its rotor and the expander's volume expansion ratio. The principal vane kinematics are the variations of the protrusions of the vanes outside their rotor's slots and the vanes' sliding velocities and accelerations with respect to the rotor, as functions of the rotor's angular displacement. The accuracies of the evaluations of these parameters for an MVE depend, mainly, on the accuracy to which its stator-cylinder's shape can be described mathematically and to what extent the assumptions, made in order to simplify the calculations, accord with reality.

Wolgemuth and Olson, 2 Somayajulu, 3 Marsters and Ogbuefi, 4 Ramprasad and Radha, 5 Barszcz 6 and Hussein 7 have all presented mathematical models for predicting the cell-volume variations of traditional positive-displacement vane-type machines (e.g. pumps, compressors, expanders and air motors) with a complete-circular stator cylinder forming only axial seals with the rotors. However, their calculations for cell volume were based on the assumption of having idealised vanes of zero thickness. Chlumsky 8 and Kumar Paul 9 considered the effect of the vanes being of finite thickness when calculating the cell volume.

Hussein 7 developed a mathematical model for evaluating the geometrical characteristics and the vane kinematics of an MVE with the periphery of the stator cylinder made up of several circular arcs. For the reliable and efficient operation of the MVE, the circular arcs were assumed to have common tangents at their connecting-points, whereas


the last arc was considered to form a circumferential seal with the rotor 's periphery (i.e. a sealing arc). The model could be used for the geometrical analyses of the most familiar expanders with completely-circular stator cylinders.

Two existing designs of MVE, with two geometrically-different stator cylinders, were examined in the present study:

(i) An MVE with its stator-cylinder's periphery consisting of five circular arcs four tangential and a sealing arc~( i .e , the non- circular M V E ) : it is a reversed and modified version of the

o ,2 ° , ~,o. ~ . S O . l O O m m

Fig. 1. The non-circular MVE considered with its rear end-plate removed.

Multi-vane expanders: geometry and vane kinematics" 163

commercially-available AGR 140 refrigeration rotary-vane com- pressor manufactured by Denco Air Ltd, Hereford, U K (see Fig. 1).

(ii) A prototype MVE with a stator cylinder having two circular arcs, one of them acting as a sealing arc (i.e. the circular M V E ) designed at Cranfield Institute of Technology, and built by Denco Air Ltd (see Fig. 2).

Hussein's 7 mathematical model has been extended in order to carry out the present analysis for the two considered expanders• The model, using

Fig. 2. The circular MVE mounted on the Rankine-cycle test bed at Cranfield.


relatively few variables to describe the stator-cylinder's shape, is suitable for conducting the design optimisation using a digital computer. For both of the considered expanders, a computer sub-routine was composed in order to calculate the complete set of the geometric parameters of the expander's stator cylinder. It permits the evaluation of the expander's cell volume, vane extension outside the rotor's slot, and the vane's sliding velocity and acceleration, with respect to the rotor, as functions of the angular displacement of the rotor. Hence the optimal angular position of the beginning of the exhaust port opening (i.e. the optimal discharge angle), for a given number of vanes, in order to achieve the maximum cell volume at the start of the exhaust port opening (i.e. maximum volume expansion ratio of the MVE) can be predicted. For the calculated discharge angle and inlet-port spread angle (i.e. the inlet angle), a sub- routine was developed to evaluate the volume expansion ratio.

A study concerned with the effect of the vanes' thickness on the expanders' cell-volume variations and volume expansion ratios was carried out, assuming:

(i) idealised vanes of zero thickness; (ii) 4mm-thick vanes.

The effects of changing the number of vanes and the inlet angle on the expanders' volume expansion ratios were also studied.

GEOMETRICAL DESCRIPTION OF THE STATOR- CYLINDERS' SHAPES (see Figs 3 and 4)

The number of necessary and sufficient geometrical parameters to describe fully an expander's stator-cylinder shape, can be reduced by mathematical manipulation of the geometrical relationships between the parameters. The geometry of a stator cylinder, made up with n circular arcs, i.e. ( n - 1) tangential ones and a sealing arc, neglecting the radial clearance between the rotor and the stator's sealing arc, can be described completely if values of the following parameters are known:

(i) The rotor radius RR. (ii) The eccentricity of the first circular arc of the stator cylinder E(1). (iii) The angular co-ordinates of the centres of the stator-cylinder's

circular arcs ~k(i), where i = 1, 2 . . . . . n - 1. (iv) The spread angles of the circular arcs ARC(i), where

i = 1 , 2 . . . . . n - 2 .

Multi-vane expanders: geometry and vane kinematics 165

BASE LINE

0i

EXHAUST I:~T

5

Fig. 3. Specification of the geometrical parameters for the non-circular MVE: points A, B, C, D and F indicate the start of each of the stator-cylinder's circular arcs 1,2, . . . , 5 respectively; points G, H, I and J are the geometric centres of the arcs 1, 2, 3 and 4, respectively; point O is the rotor's centre. E(1) is the eccentricity of the centre of the first

arc of the stator cylinder with respect to the centre of the rotor.

The ma themat i ca l re la t ionships fo r ca lcula t ing the o ther d ep en d en t geomet r ica l pa ramete r s o f the expander ' s s ta tor cyl inder are given in A p p e n d i x A. These include:

(i) The spread angle A R C ( n - 1) o f the (n - 1)th c i rcular arc o f the s ta tor cylinder.

(ii) The radii R S ( i ) of the s ta tor -cyl inder ' s c i rcular arcs, measu red f r om their respective centres , where i = 1, 2, . . . , n.

(iii) The eccentrici t ies E( i ) of the centres o f the s ta tor-cyl inder ' s arcs with respect to the ro tor ' s centre , where i = 2, 3 , . . . , n.

(iv) The radii R A R C ( i ) o f the c i rcular arcs at the ends o f their spread angles, measu red f rom the ro tor ' s centre , where i = 1, 2, . . . , n.

The geometr ica l shape o f the expander ' s s ta tor -cy l inder can be descr ibed comple te ly in po la r co -o rd ina tes by R(O) with respect to the


BASE LINE

Fig. 4. Circular MVEgeometrical parameters: points A, B, G and O indicate the start of the circular arc 1 and 2, the geometric centre of arc 1 and the rotor's centre, respectively.

rotor's centre and the BASE El N E (i.e. point O and line OA respectively in Figs 3 and 4). The mathematical formulation of R(O) is given in Appendix A.

CELL-VOLU ME VARIATIONS

At any instant of time, the considered cell of an expander, as shown in Fig. A3 of Appendix A, is the space bounding it between the two neighbouring vanes, the rotor, the stator cylinder and the two end plates of the expander. In calculating the volume of a particular cell the following assumptions apply:

(i) The vanes' tips maintain continuous contact with the stator- cylinder's surface.

(ii) The clearances between the relatively-moving parts of the expander, namely the rotor-end plate and the vane-end plate axial clearances, vane-rotor's slot tangential clearance, and the rotor- sealing arc radial clearance, are ignored.


In order to determine the effect of the thickness of the vanes on the volume of an MVE's cell, two approaches have been used see Fig. A3:

(i) The first approach considers idealised zero-thickness vanes. (ii) The second approach takes the thickness of the vanes into

account. In this case, the ideal cell's volume, calculated assuming zero-thickness vanes, should be reduced by half the volumes of the extended parts of the leading and the trailing vanes of the cell outside their slots in the rotor. However, in the expanders under consideration, semi-cylindrical slots were machined into the trailing faces of the vanes in order to allow the high-pressure working fluid to flow into the empty chambers of the rotor slots and so to act underneath the vanes thereby helping to maintain the required continuous contact between the tips of the vanes and the stator cylinder. Under these circumstances, the volume of the chamber underneath the leading vane together with the volumes of the pressurising slots should be considered as part of the volume of the cell under consideration.

To enable a comparison of the geometrical characteristics of the two expanders considered, their cell volumes have been calculated, per unit axial lengths of their respective stator cylinders, for the same number of vanes (i.e. Z = 8). The predictions, calculated via the analysis presented in Appendix A, as functions of the angular displacement of the leading vane of the considered cell are presented in Fig. 5. The following conclusions can be drawn:

(i) For the two expanders under consideration, the cell volumes calculated taking into account the thickness of the vanes are higher

than those for the idealised zero-thickness vanes over most of the 360 ° range of the rotor angular displacement. However, in the angular displacement ranges 278 ~ 340 ° and 236 ~ 302 °, for the non-circular and the circular MVE respectively, the calculated cell volumes, considering the thickness of the vanes, are very slightly smaller than those for the zero-thickness vanes.

(ii) The ratios [V(O)-Vid(O)]/Vid(O), for the non-circular and the circular MVEs, are greatest at small and large angular displacements 0 of the rotor, i.e. during the charging and final stages of the exhaust processes respectively. Therefore, the vane thickness should be taken into account when studying the expanders' breathing.


t Z=8 600 - - - VANE THICKNESS tv= 0 mm - - t v = Z, mm

,oo / / .~

300 .~//,,

200 'S / 100 / / /

/ / / 71 I

0 20 ~ 60 ~ ,6o ~0 i~0~;0 i~02602~02',02~02~03~3~03~0360 : ANGULAR DISPLACEMENT, e,[ deg]

Fig. 5. Variations of the cell's volume for the expanders with the angular position of the leading vane of the considered cell.

(iii) The finite vane thickness results in a relatively small increase of the maximum cell volumes of the expanders. It also leads to a slight decrease of the values of the angular displacements at which these maxima are achieved (i.e. the optimal discharge angles of the expanders). Practically, the effect of the vane thickness can be neglected when predicting the optimal discharge angles for the considered expanders.

V O L U M E EXPANSION RATIOS OF THE MVEs

The volume expansion ratio r~ of an MVE can be defined as the ratio of the discharge volume V 2 (i.e. the cell volume, when the leading vane is at the angular position (0out)~) to the inlet volume V 1 (i.e. the cell volume with the leading vane at the angular position (0in + 6), which is sometimes referred to as the 'cut-off ' angle)--see Fig. 6.

For a given geometry of the stator cylinder and a given number of vanes, the developed computer sub-routine permits r V to be evaluated for any specified value of the inlet angle 0i,, using the calculated cell volume


BASE LINE BASE LINE

TRAILING V A N E ~ , ~ . ~ ~ ,

~IEAOING VANE -- ~ -

Fig. 6. Terminology of an MVE: inlet volume V~ and discharge volume V 2.

9 OPTIMAL NUMBER OF DISCHARGE ANGLE

VANES [ (0o~) s Iop t [ degrees I \ z 8 \ ~ \ 4 266 6

\ ~ 6 251.6 ? . ~ \ 8 244.1

10 239.6 ~ i ~ 12 236.6

i 6 tv = O mm uJ • ~x x t v = 4 mm xxx' operoting point of the ~x x 10 existing design

~x x 12

.." .-...

1

0 INLET ANGLE, Bin[degrees]

Fig. 7. Predicted values for the volume expansion ratio of the non-circular MVE.


variation V(O). The discharge angle (0ou0~ is determined as the optimal angle for the considered expander employing the given number of vanes.

Figures 7 and 8 indicate the effects of" the inlet angle, number of vanes and the vanes' thickness on the expanders' volume expansion ratios. It can be concluded that:

(a) The value of r v increases with the number of vanes but decreases with the inlet angle of the expander. The rate of increase of the expansion ratio, with the number of vanes, and its rate of decrease, as a function of the inlet angle, fall with both the number of vanes and the inlet angle. The maximum number of vanes of specified thickness and height is limited by size and strength considerations of the rotor. At the same time, with a large number of vanes, one should expect high friction losses in the expander. On the other hand, the minimum inlet port spread angle (i.e. the inlet angle 0i. ) is limited, mainly, by the required breathing characteristics of the expander.

(b) The greater the inlet angle, the better the filling of the expander's cells, during the charging process. On the other hand, the larger the inlet

I NUMBER OPTIMAL / OF V A N E S DISCHARGE ANGLE

7 F Z [(%ul)s]op ~ [deg ] i

> | 4. 205.0 I~ 6 1 9 0 0

6P, 8 182.5 / , lO 178. o k \ 12 1 7 5 0

S ~ ~ - - - - tv = 0 mm L ~ ' \ \ ' - - tv = 4 mm

x r~'~ \ ' k ' , @ operoting point o~ the ] %, ~ ' . \ ",, exisbng design

& ~x \x 3 , , x

' xxx x

3 2" "" " -

L

0 I~O 3n 35 I ~ J I I I L 20 30 40 50 60 70 60 90 INLET ANGLE, Bin [degrees]

Fig. 8. Predicted values for the volume expansion ratio of the circular MVE.


angle the greater the internal leakage flow through the clearances in the expander due to:

(i) increasing the spread angle of the high pressure zone around the rotor periphery;

(ii) increasing the pressure differential between the adjacent cells as a result of the shortening of the expansion zone for given inlet and exhaust manifold pressures.

(c) The predicted values of the volume expansion ratios, assuming a finite thickness for the vanes, are, generally, less than those with the idealised zero-thickness vanes. The effect of the vanes' thickness increases as the number of vanes increases and the inlet angle decreases. For the expanders under consideration, with eight vanes and inlet angles of 36 ° and 35 ° for the non-circular and the circular MVE respectively, assuming zero thicknesses for the vanes will result in a considerable error in evaluating their volume expansion ratios.

THE VANE KINEMATICS

In this analysis, the extension of a vane outside its rotor's slot (i.e. the vane protuberance) X(O) and the vane's sliding-velocity I/el(O) and acceleration Acc(O) with respect to the rotor, were formulated mathematically, as functions of the angular displacement 0 of the rotor. The derivations of the mathematical formulae are given in Appendix B: the following assumptions were made:

(i) The vanes are perfectly rigid: i.e. they do not distort. (ii) The vanes are in contact with the stator cylinder at all times.

Figures 9 and 10 show the vane kinematics, evaluated using the developed computer sub-routine, for the non-circular and the circular MVEs considered, respectively. In order to make the data of more general interest, the vane sliding velocity is presented as Vel(O)/~ and the vane sliding acceleration as Acc(O)/~ 2 where ~ is the angular speed of the rotor.

There are discontinuities in the sliding acceleration and the rate of change of the sliding velocity with respect to the angular displacement of the vane at the points of contact of the successive arcs of the stator cylinder as a result of the variation of the arcs' curvatures. These can effect the reliable operation of the vanes in the expanders considered.


141

%o~ ;o

i _ ~

03:,o, u ' ~ 2 ~ .... ..j ..... ~o.,~ 4o ~o~ ~o ,~o

I ANGULAR -21 "~ .... "~'e"e-(.o.,..~. ~ !DISPLACEMENT, - 4 ................. . \ ! e { a ~ ]

• I -6 \ j - 8 \. /

\ / -10 ~ . J "

-12 /

-14 I .-" /

- 16

-18

Fig. 9. Non-circular MVE: vane kinematics.

14

E~ ~7 \ ,%~, 8 / \

... 2 i . . . . . . . . / / " ' \~ - . \ ......... ]

i320 , , ol ~ . . o 14o ",. ~ 2oo ~2o ..... z6o 28o 30o i 34o 36o :

L ....... ~ X /" J ANGULAR - L r ".~o~ "~ .."" ./DISPLACEMENT _~ .... ~ \ ... / .

"...... ....-'>'~ . /

Fig. 10. Circular MVE: vane kinematics.


CONCLUSIONS

The present geometrical analyses of the MVEs represent some of the computer simulation necessary for the complete behaviour prediction and performance evaluation. Only then will the necessary design optimisation for a proposed specific application be possible. Nevertheless the following guide lines emerged from the present study:

(i) Due to the smaller eccentricity E(1) of the first arc of the stator cylinder of the non-circular expander compared with that for the circular machine, the rate of increase of the cell volume of the former, during the charging process, is lower. On the other hand, the inlet port width-to- stator length ratio for the non-circular MVE is 0.737 compared with 0.638 for the circular device. With a low rate of cell volume increase and high port width-to-stator length ratio, less throttling losses at the intake port, less pre-expansion of the incoming working fluid, higher cell pressures at the cut-offangle and, therefore, better filling of the expander's cells occur.

(ii) For the same number of vanes and inlet angle, the volume expansion ratio of the MVE with the non-circular stator is higher than that with the circular stator. This is due to the higher maximum cell volume and the lower inlet volume of the former compared with the latter. The maximum radius of the stator cylinder (RS)max and the maximum vane protuberance [X(0)]ma x for the non-circular expander are 53.39 mm and 12"93 mm respectively, compared with 51.01 mm and 12.36 mm for the circular expander. The two expanders have the same rotor radius RR =45 mm and the same vane height H v =25 .5mm. The following practical limitations 4'6'7 apply:

RR/(RS)m,x = 0"75-0"95

[X(O)]m,x/H v = 0"5-0"6

Hv/RR = 0"5-0"6

Accordingly, the non-circular expander can perform more efficiently than the circular device at relatively high working pressure ratios.

(iii) The angular spread of the expansion zone of the non-circular expander is greater and the rate of the cell volume increase is slower than those for the circular device. Accordingly, the pressure differential between the adjacent cells of the former is lower and, therefore, less cell- to-cell internal leakage can be expected.


(iv) For the same number of vanes, the spread angle of the exhaust port opening of the non-circular MVE (i.e. { A R C ( 4 ) - (0oot)~}) is less than that of the circular expander (i.e. {ARC(I) - (0ou,)s}). A long exhaust port opening causes an extended low-pressure region around the rotor's circumference and, therefore, high leakage flows of the working fluid from the high-pressure cells across the existing clearances.

(v) The spread angle of the sealing arc of the circular stator is bigger than that of the non-circular one. A small sealing arc will yield a high pressure gradient along the rotor's circumference and, therefore, a high internal leakage flow across the existing axial clearance between the rotor and the two end plates. Moreover, for the same number of vanes, assuming the same probability for loss of contact to occur between the tips of the vanes and the stator cylinder, a shorter sealing arc will lead to a higher leakage of the working fluid from the high-pressure cell facing the inlet port, to its low-pressure successor, facing the exhaust port, through the radial clearance between the rotor and the sealing arc of the stator cylinder.

(vi) Because the non-circular MVE stator-cylinder's surface consists of five circular arcs compared with two for the circular MVE, the number of the discontinuities in the sliding acceleration and the rate of change of sliding velocity with respect to 0 of the vanes of the former is higher and, therefore, less reliable vane operation can be expected. Moreover, because of the greater eccentricities, the values of the sliding velocities and accelerations along the spread angles of the third and fourth arcs of the stator cylinder of the non-circular MVE are higher than those for the circular device, during the corresponding range of angular displacements. As a result, greater friction losses in the former are expected there whereas higher friction losses are expected with the circular stator throughout the low displacement-angle range.

A C K N O W L E D G E M E N T

The authors wish to thank the Egyptian government and the British ORS scheme for supporting this research programme. They also wish to express their gratitude to Dr M. Hussein and Denco Air Limited, Hereford, UK, for their suggestions and encouragement. The computer sub-routines developed in this investigation are available to subscribers to Applied Energy, subject to the approval of the project's sponsors.


R E F E R E N C E S

1. O. Badr, P. W. O'Callaghan, M. Hussein and S. D. Probert, Multi-vane expanders as prime movers for low-grade energy organic Rankine-cycle engines, Applied Energy, 16(2) (1984), pp. 129-46.

2. C. H. Wolgemuth and D. R. Olson, A study of breathing in vane-type expanders, Proceedings of the 1971 IECEC, Aug. 1971, No. 719163, pp. 1249-58.

3. K. D. S. R. Somayajulu, An analysis of vane-in-rotor pumps, Trans. ASME, J. Basic Eng., Dec. 1971, pp. 505-17.

4. G. F. Marsters and E. Ogbuefi, Rotary vane expander development: some design considerations, Proceedings of the 1973 IECEC, Sept. 1972, No. 729048, pp. 249 54.

5. B. S. Ramprasad and T. S. Radha, On some design aspects of rotary vane pumps, Vacuum, 23(7) (1973), pp. 245-9.

6. Z. Barszcz, Dynamic analysis of pneumatic vane motors, A rchiwum Budowy Maszyn, XXVII(I) (1980), pp. 25-39.

7. M. Hussein, Low grade energy engines, PhD Thesis, School of Mechanical Engineering, Cranfield Institute of Technology, Bedford, UK, 1981.

8. V. Chlumsky, Reciprocating and rotary compressors, SNTL--Publishers of Technical Literature, Prague, Czechoslovakia, 1965, pp. 281-3.

9. P. Kumar Paul, Rotor cell volumes in sliding-vane compressors, Fluid Power Int. (June 1971), pp. 40 1.

A P P E N D I X A: G E O M E T R I C A L ANALYSIS OF THE MVE

The stator cylinder

Referring to Fig. A1, the following mathematical relations can be developed for the geometrical parameters of the stator-cylinder's first arc:

RS(1) = (RR 2 + [E(1)] 2 - 2 E ( I ) R R c o s ~(1)) 1!2

R(O) = E(1)cos [~(1) - 0]

+ ([RS(1)]2 _ [E(I)]2 sin 2 [~p(1) - 0]) ~:2

0 < 0 < ARC(l ) (A1)

RARC(1) = E(1) cos [~b(1) - ARC(l)]

+ ([RS(1)] 2 - [E(1)] 2 sin 2 [~(1) - ARC(l)]) 1'2

TIP(l) =s in -~ ( E ( I ) sin [•(1) - ~ ( i j ARC(1)])


BASE LINE

A e

Fig. AI. Geometrical details of the first circular arc of the stator cylinder (with centre G): O represents the centre of the rotor.

For the second arc of the s ta tor cylinder, similar expressions can be derived together with an expression for the eccentricity, E(2), taking into cons idera t ion the condi t ion that successive arcs should have a c o m m o n tangent at their connect ing points. Figure A2 shows the connect ing point o f the second arc with the first arc (point B). The centre o f the second arc (point H) lies on the line connect ing the contac t point B with the centre o f the first arc (point G) as a necessary condi t ion for bo th arcs to have a c o m m o n tangent at B. For the second arc the following relations can be developed:

( sin[tfi(2)-ARC(1)] ) RS(2) = RARC(I) sin [5(2) ~T-Ifi-(1)~A-R-C(1)]

E(2) = E(1) RS(2) sin [~,(1) - ARC(1 )] RS( I ) sin [tp(2) - ARC(l)]

R(O) = E(2) c o s [ ~ , ( 2 ) - 0]

+ ([RS(2)] 2 - [E(2)] 2 sin 2 [~9(2) -- 01) ~/2 (A2)

ARC(l) <_ O <_ ARC(2)

RARC(2) = E(2)cos [~,(2) - ARC(2)]

+ ([RS(2)] 2 - [E(2)] 2 sin 2 [~(2) - ARC(2)]) 1/2

TIP(2) = s i n ~ ( E ( 2 ) s i n [~9(2))~_2)- ARC(2)])

Multi-vane expanders. geometry and vane kinematics 177

General ly , for the ith s tator-cyl inder 's arc (i = 1, 2 . . . . . n - 1 ) , the required geometr ical pa ramete r s can be evaluated using the following set of equat ions:

RS(1) = (RR 2 + [E(1)] 2 - 2E(1)RRcos ~(1)) ~:2

( s in[O( i ) -ARC(i -1)] ) RS(i) = RARC( i - 1) sin [ ~ ( t ) + ~ 0 : ~ i ) Z - A ) ~ C - ( i - 1)]

i = 2 , . . . , n - 1

RS(i) [sin O ( i - l) - ARC( i - l)] E(i) = E(i - 1)

R S ( i - 1) [sin 0( i ) - ARC( i - 1)]

i = 2 . . . . . n - 1

R(O) = E(i) cos [O(i) - 01

+ ([RS(i)] 2 - [E(i)] 2 sin 2 [0(i) - 0]) 1'2

O<_ O<ARC(1) f o r i = l

A R C ( i - 1 ) < O < A R C ( i ) for i = 2 . . . . . n - 1

RARC(i) = E(i) cos [~(i) - ARC(i)] + ([RS(i)] 2 - [E(i)] 2 sin 2 [0(i) - ARC(i)]) 1;2

i = l , . . . , n - 1

TIP(i) = sin-1 (E(i) sin 0( i ) - ARC(i)) \- ~ f i - i = l , . . . , n - - 1

(A3)

Fo r the (n - 1)th arc, the spread angle ARC(n - 1) can be de te rmined f rom the condi t ion that

RARC(n- 1) = RR

The geometr ic pa ramete rs for the sealing arc (i.e. the nth arc) are given by:

ARC(n)RS(n) = R(O) = RARC(n) = == 02r~ ARC(n - 1)_< 0__ 2re} (A4)

0 ( n ) = 0 o r rc


BASE LINE

C ~ ATOR

Fig. A2. Geometrical details of the second circular arc of the stator cylinder (with centre H): O represents the centre of the rotor.

BASE LINE

SEALING ARC

MVE*s ceil for ideollzed m ..... thick . . . . . . . . . ~ ~ ~ ~ ~

MVE's celt taking the vanes' ~ ~

{ ) ) ° ]

dA

- - " . ROTOR / I~, ?1

" ~ J ~ S T A T O R

\-LEAD,NG VANE

Fig. A3. Terminology used when analysing the volume of the MVE's cell.


T h e c e l l - v o l u m e v a r i a t i o n

Assuming zero-thickness vanes In this case, the MVE's cell projection normal to the axis of the rotor is indicated by the cross-hatched area in Fig. A3. As shown, the infinitesimal incremental annulus area dA, swept by a vane, initially at the angular location 0, through the infinitesimal angular displacement, dO, is given by

dA = ½([R(0) ] 2 - R R 2) dO (A5)

Substituting the expressions for R(O) from the sets of eqns (A3) and (A4) into eqn (A5) and integrating between the limits 01 and 02 yields

[RS(i)] 2 - RR 2 [E(i)] 2 A[02' 01] • 2 ( 0 2 - - 01) -~ 4

× {sin (2[~(i) - 01] ) - sin (2[~(i) - 02])}

[RS(i)]2{sin , ( . E ( i ) s i n [ ~ ( i ) - 0 1 ] ) + 2 RS(i)

(E(i) sin [~9(i) - \ -R~ O2]jj ~ s i n

1

E(i) + 2-- {sin [~9(i)- 0x]

× ([RS(i)] 2 - [E(i)] 2 sin 2 [~9(i) - 01]) x/2

- sin [~,(i) - 02]([RS(i)] 2

- [E(i)] 2 sin 2 [~9(i) - 02]) 1'2 }

i = 1 , 2 . . . . . n - 1

0 < 0 ~ and 0 2 <ARC ( 1 ) for i = 1

A R C ( i - 1 ) < 0 1 and O2<ARC( i ) for i = 2 , . . . , n - 1

A[O2, 01] = 0

i = n

A R C ( n - 1 ) < 0 1 and 0 2 < 2 g

(A6)

The accumulated annulus area swept by a vane, initially at the angular


position 0 = 0 (measured from the BASE LINE), as a function of the angular position of the rotor, can be calculated as

A[O, 01 0 <_ 0<<_ ARC(l)

A[ARC(I), 0] +A[O, ARC(l)] ARC(l) < 0 <_ ARC(2)

A[ARC(I ), 0] + A[ARC(2), ARC(l)] + A[O, ARC(2)]

ARC(2) < 0 < ARC(3)

k - 1 i

0] + ~ ' A[ARC(i + 1), ARC(i)] + A[O, ARC(k)] A[ARC(1), ( d

i - I ARC(k) < 0 <_ A RC(k + 1) (A7)

n 3

A[ARC(1), 0] + ~ ' A[ARC(i + 1), ARC(i)] + A[O, ARC(n ~ 2 ) ~ ¢ d

i = 1

A R C ( n - 2 ) < O <_ARC(n- 1)

n - 2

A[ARC(1), 0] + ~ ' A[ARC(i + 1), ARC(i)] ( d

i = 1 A R C ( n - 1) < 0 < 2rt

If 0 in eqn (A7) is replaced by 0 - 6, where 6 (the angle between two successive vanes) is given by

2rt 6 - (AS)

Z

the calculated area will represent the accumulated area swept by the trailing vane. For 0 - 6 < 0, the accumulated area swept by the trailing vane should be considered to be zero.

Thus the MVE's cell volume, Via(0), can be calculated at any angular position 0, of the rotor (corresponding to the angular position of the leading vane with respect to the BASE LINE) as

Via(0) = L [accumulated annular area swept by the leading vane at angular position 0 (A9)

- accumulated annular area swept by the trailing vane at angular position ( 0 - 6)]


Considering the vane's thickness The single-hatched area in Fig. A3 represents the projection of the MVE's cell. The vane protrusion X(O) outside the rotor's slot at any angular position of the rotor 0 can be determined as

X(O) = R(O) - RR

Assuming that the vane height is equal to the depth of the rotor's slots, the MVE's cell volume V(O), can be evaluated, using the corresponding predicted value of Vid(O ), as

V(O) = Via(0 ) - ½Ltv[X(O ) + X(O - 6)] + LtvX(O ) + N s ~ - Hv (A10)

The second, third and fourth terms of the right-hand side of eqn (A10) represent the reduction in the calculated idealised cell-volume caused by half the volumes of the extended portions of the leading and trailing vanes outside the rotor's slots, the volume of the chamber underneath the leading vane of the cell (which is attached to the cell through the pressurising slots) and the pressurising slots' volumes, respectively.

A P P E N D I X B: THE VANE K I N E M A T I C S

The protrusion of a vane outside its rotor 's slot (i.e. the displacement of the vane with respect to the rotor), X(O), can be expressed as a function of the angular displacement 0 of the rotor, i.e.

X(O) = R(O) - RR (B1)

Substituting the expression for R(O) from eqns (A3) and (A4) into eqn (B1) yields

X(O) = E(i) cos [~(i) - 0] ) + ([RS(i)] 2 - [E(i)] 2 sin 2 [~9(i) - 0]) 1/2 - RR t (B2)

where i = 1, 2, . . . , n - 1

X(O) = 0 where i = n

The vane sliding velocity with respect to the rotor, Vel(O), as a function of 0 can be determined as

dX(O) dX(O) dO dX(O) Vel(O)- d ~ - d0 d t - d ~ f~ (B3)


Subst i tut ing f rom eqn (B2) into eqn (B3), and carrying out the differentiat ion, yields

Vel(O) = f2 (E(i) sin [~(i) - 8] \

½[E(i)] 2 sin 2 [ 0 ( i ) - 8] "~

([RS( i )~ [E-(~]2-s~n2 [-~) -_ 81)1/2 ) (g4)

where i = 1, 2, . . . , n - 1

Vel(O) = 0 where i = n

The vane sliding acceleration with respect to the ro tor Acc(O), as a funct ion of the angular displacement , 0, is

d[Vel(O)] d[Vel(O)] Acc(O) - dt - f~ dO (B5)

Subst i tut ing f rom eqn (B4) into eqn (B5) and differentiating, gives

t - E ( i ) cos [t)(i) - 8] Acc(O) f~2

[ g ( i ) ] 2 c o s 2 [~p(i) - 81 ( [ R S ( i ) ] 2 - [E ( i ) ] 2 s in 2 [ ~ ( i ) - 0]) 1,2

1[E(i)]2 sin 2 2 [~(i) - 0] ~ (B6)

([RS( i)] ~ ~ [ E ~ s(~ ~([) = 01) 3/2

where i = 1, 2 . . . . , n - - 1

Acc(O) = 0 where i = n

In eqns (B2), (B4) and (B6), the following limiting values for the angular displacement , 8, for any s ta tor sector i should be considered

0 < 0 < A R C ( l ) f o r i = l

ARC(1)<O<ARC(2) for i = 2

ARC(k-1)<O<_ARC(k) for i=k

ARC(n-2)<O<ARC(n-1) f o r i = n - 1

ARC(n-1)<O_<2rc f o r i = n

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Multi-Vane Expanders-Geometry and Vane Kinematics_Badr