Gnutek, Z.

Kalinowski, E.

Pietrowicz, S.

English

Purdue E-Pubs

Purdue UniversityPurdue e-PubsInternational Compressor Engineering Conference School of Mechanical Engineering2000Analysis of Thermodynamic Process In the WorkChamber of a Spiral Machine in the Function of theRotation AngleZ. GnutekWroclaw University of TechnologyE. KalinowskiWroclaw University of TechnologyS. PietrowiczWroclaw University of TechnologyFollow this and additional works at: https://docs.lib.purdue.edu/icecThis document has been made available through Purdue e-Pubs, a service of the Purdue University Libraries. Please contact epubs@purdue.edu foradditional information.Complete proceedings may be acquired in print and on CD-ROM directly from the Ray W. Herrick Laboratories at https://engineering.purdue.edu/Herrick/Events/orderlit.htmlGnutek, Z.; Kalinowski, E.; and Pietrowicz, S., "Analysis of Thermodynamic Process In the Work Chamber of a Spiral Machine in theFunction of the Rotation Angle" (2000). International Compressor Engineering Conference. Paper 1467.https://docs.lib.purdue.edu/icec/1467ANALYSIS OF THERMODYNAMIC PROCESSES IN THE WORK CHAMBER OF A SPIRAL MACHINE IN THE FUNCTION OF THE ROTATION ANGLE Zbigniew GNUTEK, Eugeniusz KALINOWSKI, Slawomir PIETROWICZ Institute of Heat Engineering and Fluid Mechanics Wroclaw University of Technology 27 Wybrzeze Wyspianskiego Str. 50- 370 Wroclaw, POLAND ABSTRACT The paper describes the dependences of thermodynamic parameters in the work chamber upon the position of the rotor. It discusses the process of compression in individual working spaces. An analysis is made of the influence of friction onto the process of the adiabatic expansion (the influence of using different oils). It considers the possibility of applying a spiral expander in the ORC power stations. Further research is being made for considering the influence of friction on the process of the adiabatic expansion (compression) and the expander's efficiency. rb- parameter of the spiral; re - eccentricity; ¢ - forming angle; Ps - suction pressure, NOMENCLATURE Vs = B· Ss- volume of the chamber during the suction; V -volume of the chamber in the function of the rotation angle 8; Ss - cross section area of the chamber being filled, K - adiabatic exponent, B- height of the spiral. 1. INTRODUCTION One of the basic constructional elements of the ORC power station is the motor. It seems, the spiral expander can fulfil this role. The main properties of the spiral engines are: - a simple mechanism - only two elements, an active and a passive spiral are required to compress gas. (In piston compressors around fifteen elements cooperate with one another in order to perform the same work.) Fifteenth International Compressor Engineering Conference at Purdue University, West Lafayette, IN, USA- July 25-28, 2000 815 - low level of noise, - simplicity and self-balance of power, which also means a reduction of the machine's free vibration to minimum. When compared to the piston compressors, the spiral compressors have three basic features which make them different: 1. The processes of suction and discharge in the spiral compressors are physically separate, which reduces the heat exchange between the input and output gases. In the piston compressors this process takes place in one working chamber-the heat flow between the input and output gas is significant and so the compressor's efficiency decreases. 2. The process of compression and discharge is more balanced. The processes of suction and discharge recur approximately every 30 - 60°. 3. In the spiral compressors there are no valves. Thanks to this leakages are eliminated in the set of valve- valve seat. The further part of the article analyses the spiral machine, treating it as an expansive machine. The idea of applying the spiral machine as an expander appeared when in the Institute of Heat Engineering and Fluid Mechanics of the Technical University ofWroclaw a C-R cycle station was designed and constructed, in which the working medium was a low-boiling liquid. The greatest problem was to select a suitable machine for expanding the low-boiling medium [3], [4]. 2. THE SPIRAL OF ARCHIMEDES AS A BASIS OF CONSTRUCTION OF A SPIRAL EXPANDER This is and analysis of work of the spiral expander on the basis of the spiral of Archimedes. ~ In this spiral the course radius r = OM is proportional to the angle of inclination e against the polar semi-axis: r=a· B where: a - is called the parameter of the spiral of Archimedes. Equation 1 describes the spiral of Archimedes in polar coordinates. While equation 2 provides a ,general" equation of the spiral of Archimedes in coordinates x-y. Fifteenth International Compressor Engineering Conference at Purdue University, West Lafayette, IN, USA- July 25-28, 2000 (1) 816 ~x2 +y2 =a·arctg(y/x) (2) y In the further part of the work we shall be using parametric equations of the sloping line developing a circle. X / x = OMx + MxMx =a( cost+ tsint) (2) y = MMx- MB = a(sint- tcost) (3) Fig. 1 Coordinates set used to define the parametric equation of the spiral Figure 2. shows the active spiral with all markings. The passive spiral is defined by the set of equations: - inner curve xfw =rb(cos(q))+q}sin(¢)) y fw = rh(sin(¢)- ¢cos(¢)) (4a) (4b) - outer curve x fo = rh{cos(¢) +¢sin(¢))+ bsin(r/J) (Sa) YJo = rb(sin(¢)- r/Jcos(¢))- bcos(rjJ) (Sb) Fig. 2 Active spiral Fifteenth International Compressor Engineering Conference at Purdue University, West Lafayette, IN, USA- July 25-28, 2000 Fig. 3 A spiral expander system 817 The active spiral is defined by the following relationship: - inner curve - outer curve -- xow = rb( -cos(¢)- ¢sin(¢))+ re cos( B) (6a) Yow= rb( -sin(¢)+ ¢cos(¢))+ resin( B) (6b) xoz = rb( -cos(¢)- ¢sin(¢))- bsin(¢) + re cos( B) (7a) Yoz = rb( -sin(¢)+ ¢cos(¢))+ bcos(¢) +resin( B) (7b) Width of spiral b can be calculated by: 3. A DUAL SPIRAL AS THE BASIS OF EXPANDER CONSTRUCTION (8) In the spiral expander the gas decompression is realised between two identical spirals (fig.3). One spiral - the passive one remains standstill to the still set of coordinates x - y with the middle point 0. The second spiral- the active one moves eccentrically to point 0. The coordinates of point Om (Xe, Ye) can be calculated from: xe = re cos( B) Ye =resin( B) Radius of rotation re can be calculated from the equation: (9c) (9b) re = rbJr- t (10) The two spirals have points of contact. These points lie on the inner Bik (x0w ,Yow) and outer Bok(x0 z, Yoz) side of the spiral. These points Bik and Bok can be found by defining the angle of the contact points. In this case such an angle can be evaluated as: ¢;(k) = 21r(k1 + 8) + fJr + B kt = -1..1 ¢0 (k) = 2Jr(k2 + o) + fJr + B; k2 = 0 .. 2 where: o = 0 fior 0 < 0 <l.7l-"" 2 'l'os 0 =1 for t7l-¢os <0 <27l The coordinates of the contact points can be calculated: - on the inner curve Bik -xow = rb( -cos(¢0 (k))- ¢o(k)sin(r/J0 (k))) + re cos( B) Yow = rb (- sin(¢ o ( k)) + ¢ o ( k) cos(¢ o ( k))) + re sin( B) Fifteenth International Compressor Engineering Conference at Purdue University, West Lafayette, IN, USA- July 25-28, 2000 (11) (12) (13a) (13b) 818 - on the outer curve T oi -xoz = rh( -cos(¢i(k))- ¢i(k)sin(¢i(k)))- bsin(¢i(k)) + re cos( B) Yoz = rh( -sin(¢i(k)) + ¢i(k)cos(¢i(k))) + bcos(r/Ji(k)) +resin( B) (14a) (14b) The figures below present several reciprocal positions of the active and passive spirals: Fig. 4 A system of spirals in an expander 4. PRESSURE DEPENDING ON THE ANGLE OF ROTATION <pIN AN EXPANDER The cross-section of a spiral machine is specified by the formula (15): t S = J~x' 2 + y' 2 dt 0 In the considered case the area between the spirals can be calculated as: Fifteenth International Compressor Engineering Conference at Purdue University, West Lafayette, IN, USA- July 25-28, 2000 (15) 819 0.5 L5 2.5 3.5 4.5 Rotation angle Fig. 5 Area of the cross-cut in the function of the rotation angle 5.5 Specification rb = 0.3 em re= 0.419 em b= 0.524 em (16) The gas pressure present in individual chambers of the considered unit was calculated based on the assumption that the decompression would be adiabatic. Such pressure can be calculated from the formula: p=p,(;r 16 Pres sure 14 1·---··············'·--··· _ __:_ _____________ --+-·····-·-· '----------'---------c,, ............ ;------- \-----------:---------'--·-------· I [MP a] 0_4 1_5 2 2.5 Rotation angle [RAD] 3.5 4.5 5.5 Fig. 6 Pressure in the work chamber depending on the rotation angle Fifteenth International Compressor Engineering Conference at Purdue University, West Lafayette, IN, USA- July 25-28, 2000 Specification B = 1.5 em p5 =2 Mpa ic = 5.5 rad (17) 820 5. COMPONENTS OF FORCE ACTING ON EXPANDER'S ROTOR Forces acting on the active spiral can be distributed on two Ft and Fr and moment M0 (Fig. 3) - tangent force F t, - radial force Fr, The tangent force can be deducted from: ( ro(K,) r,(Kz)+,.- ( b ) J ( ) (J8) F, =Brb p/{nos(t{J-O)dt/J- pi ¢+- cos(t/J-O)d¢ +bB Ps + pd .<oJ-n- ,(-1) rb and the radial force can be deducted from: ( !o(K1) !'(K2)+1T [ b J J Fr = - Brb p /P sin(¢- B)d¢- p 1 rjJ +- sin(¢ - B)d¢ (19) oCO)-tc ,(-1) rb The Moment is defined by: M = Br, 2(- jvlo(K,) _Avl..t. + r,(K,)+1< ·("'+!!...)a"']+ o b #.<o>_,.PJ'f'-''1' #,(-1) PJ r rb r bB(- (rh (¢0 (0) -b/2)ps + (rh (t/J;(K) + b/2)pd) (20) In these formulas K 1 and K2 is the highest number of the values k1 or k2. The force of the radial friction can be calculated from: Fft =Ft. Jl where: p- friction factor- in this case mixed friction, JL = 0.1 For the accepted data of the forces F1 and FJr are presented in Fig. 7 1.5 ..-----,----,---,---,------.----.---.--,------.---. 1.35 l----t--t---l---+---1f-----l---f---+---1f----l 0.3 l-----+----+----1---+---lf.----rt~ -~f-0.15 1-. ~ •. ~ .• f--.. ~ ..~.lf--.. ~ .. -1 .. ~ •. ~ ... -~. ~ ..~ ..-+. ~ ..... T .... 1. .. .. .. .. . .. .... 0 0 0.2 0.4 0.6 0.8 I 1.2 1.4 1.6 1.8 Rotation angle (rad) (21) 2 Fig. 7 Tangent force Ft and friction force Ffl depending on the rotation angle (assumed 11 = 0.1 ) Fifteenth International Compressor Engineering Conference at Purdue University, West Lafayette, IN, USA- July 25-28, 2000 821 Radial force is reduced to the formula: (22) In our concrete case this force equals F r = I72 .465 N and the friction force F tr = I7 .24 N Pressure pj is the pressure existing along the curve of the spiral. The example of distribution of pressure along the spiral is illustrated by Fig. 8. 2.4 2.2 2 1.8 ~ 1_6 ~ 1.4 1.2 ~ I Ul ~ 0.8 t:l.. 0.6 I· . 0.4 -I . l ---1 ···········<··············· ............ . •···-·-·:····· --: ,. ····························-· : : .!. . '-·-·-- \ I 2 Angle along spiral [rad] 4 Fig. 8 Distribution of pressure along spiral (according to Fig. 2) for angle 8 = 0.3 1t. 6. CONCLUSION Presented in this paper, the analysis of a spiral machine shows that it is possible to use it for decompression of a medium in an ORC micro power station. Such decompression takes place more evenly than in piston machines, which makes one expect that a spiral machine will better perform the function of an expansion machine in the ORC systems. The influence of individual constructional parameters on the expander's efficiency and on its power will be the subject of further research. REFERENCES [I] N. Ishii, M. Fukushina, K. Sawai, K. Sano, K. Imaichi,: Dynamic behavior of a scroll compressor, JSME International Journal, Series Ill. Vol. 3I. No. I, 1988 [2] E. Niczyporowicz: Plane curves - selected topics of the analytic and the differential geometry, PWN, Warszawa I991 [3] Z. Gnutek, S. Biernacki: Conditions of effective use of waste heat in C-R micro power stations C-R with low boiling medium, Gospodarka Pa.J.iwami i Energi~ [Energy and Fuel Management], 12/96 Fifteenth International Compressor Engineering Conference at Purdue University, West Lafayette, IN, USA- July 25-28, 2000 822 

Analysis of Thermodynamic Process In the Work Chamber of a Spiral Machine in the Function of the Rotation Angle

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Analysis of Thermodynamic Process In the Work Chamber of a Spiral Machine in the Function of the Rotation Angle

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