The Theory of a Scroll Profile

Transcription

1 Purdue Universit Purdue e-pus International Compressor Engineering Conference School of Mechanical Engineering 006 The Theor of a Scroll Profile Huert Bukac Little Dnamics Follow this and additional works at: Bukac, Huert, "The Theor of a Scroll Profile" (006). International Compressor Engineering Conference. Paper This document has een made availale through Purdue e-pus, a service of the Purdue Universit Liraries. Please contact epus@purdue.edu for additional information. Complete proceedings ma e acquired in print and on CD-ROM directl from the Ra W. Herrick Laoratories at Herrick/Events/orderlit.html

2 C080, Page 1 THE THEORY OF SCROLL PROFILE Huert Bukac Little Dnamics, Inc., 1 Count Road 138, Vinemont, AL Tel.: (56) , hukac@littlednamics.com ABSTRACT Onl a pair of parallel curves can create a pocket that compresses gas. A review of theor of parallel curves that is presented here shows that there is more general wa how to design an aritrar scroll profile. Several selected tpes of plane spirals are discussed here. A common wa of creating a curve that is parallel with a given involute of a circle is to rotate the given involute of a circle plus or minus one hundred eight degrees, ut this is not possile for all tpes of spirals. The same parallel curve can e generated application of equations of parallel curves. 1. INTRODUCTION Although, the involute of a circle is the most frequentl used tpe of scroll profile, there exist other tpes of plane spirals that can e used to generate alternative scroll profiles. The necessar condition for a pair of spirals to create a pocket that can compress gas is to form a pair of parallel curves. These curves are also known as Bertrand curves (Bertrand Joseph, , French mathematician). An infinite numer of parallel curves can e generated from a given plane curve, ut not ever spatial curve ma have a parallel curve. In most papers on the theor of scroll profile, the authors start with vector form of intrinsic equation, where the coordinate is the trajector, the path. Such a concise notation is not ver useful for a practicing engineer. This paper skips this tpe of generalization and rather than that it starts with parametric equations of a spiral in the x- sstem of coordinates. It is shown that an involute of a circle that is rotated ± π is actuall a parallel curve that is parallel with the given involute of a circle. Some spirals can achieve higher compression ratio in the same space than the other ones. Each tpe of spiral has specific properties that can also affect noise and viration of a scroll compressor. These properties are more important for radiall compliant scroll compressors rather than for the fixed-oriting-radius scroll compressors. Two curves = ( s ), and = ( s) r s is original curve ( ) r ( s) is parallel curve. PARALLEL CURVES r r r r form a pair of parallel curves if n ( s) is a normal of the curve s c ( s) = ( s) + c ( s) is the natural coordinate of the curve, the length of the arc of curve [m] is an aritrar constant, positive or negative [m] r r n (1) The aove concise vector notation (1) is of a little practical use. More practical is the notation in the Cartesian sstem of coordinates. A plane-curve that has parametric equations x = ( t ), = ( t) are ψ has a corresponding parallel curve, which equations ( ) x = t + c ψ ( t) ( t) + ψ ( t) () International Compressor Engineering Conference at Purdue, Jul 17-0, 006

3 ( ) = ψ t c ( t) ( t) + ψ ( t) C080, Page In the equations () and (3) the dash indicates derivative the parameter. Fig. 1 shows two parallel curves k and k (3) k k t P c. ρ n O S C x Fig. 1: Formation of two parallel curves If we express location of point P in Fig. 1 means of radius vector ρ and angle, equations () and (3) will take the form sin cos x cos c ρ + ρ = ρ + (a) ρ + ρ cos sin sin c ρ ρ ρ = ρ ρ ρ ρ + ρ In some cases, it ma e more convenient to use equations (a) and (3a) rather than equations () and (3). In the case of a spiral, the maximum theoretical distance etween parallel spirals cannot e aritraril long. As we can e see in Fig. 1, the maximum possile oriting radius R OR is equal to c. If the innermost tip of the vane has thickness s o [m] then the oriting radius is R OR = c-s o. 1 c = x ( 0 + π ) x ( 0 ) + ( 0 + π ) ( 0 ) (4) 0 is eginning angle [rad] is oriting radius [m] R OR As a case stud, we will show geometr of a scroll profile generated a power spiral. The length ρ of radius vector of a power spiral increases with the power of the parameter φ. Thus, the radius vector of a power spiral is a is an aritrar constant, a>0, [m] is exponent, it is a constant (3a) ρ = a (5) Three special values of in equation (5) define three tpes of commonl known spirals. Thus, if = 0 the spiral degenerates into a circle; if = 1, equation (5) defines the spiral of Archimedes; if =-1, equation (5) defines a hperolic spiral. The parametric equations of a power spiral are x = a cos (6) = a sin (7) International Compressor Engineering Conference at Purdue, Jul 17-0, 006

4 C080, Page 3 When we sustitute equations (6) and (7) and their derivatives into equations () and (3), we get governing equations of a parallel power spiral. Fig. shows an example of parallel power spirals and creation of a pocket. sin + cos x = x + c cos sin = c (8) (9) Fig. : Parallel power spirals and creation of a pocket.1 Creating Vanes We can see, in the Fig., the original power spiral makes a wall of one vane while the parallel one makes wall of the other vane. Each spiral represents actuall a surface of fixed and oriting scroll. In order to make further calculations simpler, we start with the spiral that is the centerline of the gap etween fixed and oriting scrolls. Fig. shows that a single spiral can generate onl pocket on one side. However, we need another pocket that is smmetrical with the first one. We can achieve that creating a mirror image of the first centerline. The first centerline has the equations x 1 1 is x-coordinate of the first centerline [m] is -coordinate of the first centerline [m] The second centerline that is a mirror image of the first one has the equations x 1 is x-coordinate of the first centerline [m] is -coordinate of the first centerline [m] x1 1 x = ρ cos (10) = ρ sin (11) = ρ cos (1) = ρ sin (13) The parallel curve that forms the outer wall of the first gap has the equation International Compressor Engineering Conference at Purdue, Jul 17-0, 006

5 C080, Page 4 x sin cos + c = x (14) x cos sin c = 11 1 is x-coordinate of the outer wall of the first gap [m] is -coordinate of the outer wall of the first gap [m] (15) The parallel curve that forms the inner wall of the first gap has the equations sin cos + c x1 = x1 x 1 1 cos sin c = is x-coordinate of the inner wall of the first gap [m] is x-coordinate of the inner wall of the first gap [m] The outer wall of the second (mirror image) gap has the equations sin cos + c x1 = x + (16) (17) 18) x 1 1 cos sin c = 1 is x-coordinate of the outer wall of the second gap [m] is x-coordinate of the outer wall of the second gap [m] (19) International Compressor Engineering Conference at Purdue, Jul 17-0, 006

6 C080, Page 5 The inner wall of the second (mirror image) gap has the equations sin cos + c x = x x 1 1 cos sin c = + is x-coordinate of the inner wall of the first gap [m] is x-coordinate of the inner wall of the first gap [m] (0) (1) Fig. 3: Centerline of the gap spiral and its mirror image (left), gap and its mirror image (right) Fig. 4: Centered fixed and oriting vanes (left), engaged vanes (right) International Compressor Engineering Conference at Purdue, Jul 17-0, 006

7 C080, Page 6 On the left in Fig. 3, we can see oth centerlines, and on the right in Fig 3, we can see two smmetrical gaps that are centered at the common origin of the x- sstem of coordinates. In the Fig 3, we can see that the vanes (scrolls) are created the space etween gaps. The outer wall of the first gap and the inner wall of the second gap will form one vane, and the inner wall of the first gap and the outer wall of the second gap will create second vane. This ma e seen in the Fig. 4. B comparing Fig. 3 and Fig 4, we can also see that certain parts of each vane, at the center and on the circumference, are cut off, ecause the are not needed.. Built-in Compression Ratio The scroll compressor is a constant pressure ration machine. Assuming poltropic compression, the gas that is taken into the suction pocket is discharged out from the discharge pocket at the pressure that is equal to p S p D n is suction pressure [Pa] is discharge pressure [Pa] is poltropic coefficient n D S R p = p C () The uilt-in compression ratio is the ratio of the volume of suction pocket to the volume of the discharge pocket. Because the volume of a pocket is equal to the area of the pocket multiplied the constant height of vane, we can consider onl the area of each pocket. VD AD h LD c LD CR = = = = (3) VS AS h LS c LS V D is volume of discharge pocket [m 3 ] V S is volume of suction pocket [m 3 A D is area of suction pocket [m ] A S is area of discharge pocket [m ] L D is length of the centerline of discharge pocket [m] L S is length of the centerline of suction pocket [m] h is height of vane [m] Equation (3) shows that the uilt-in compression ratio is the ratio of the length of centerline of the discharge gas to the length of the centerline of the suction gap. dφ φ O ρ c x Fig. 5: Area of a pocket The length L of the centerline of the gap etween scrolls is alwas equal to the length of an arc that has angle from φ to φ + π. International Compressor Engineering Conference at Purdue, Jul 17-0, 006

8 C080, Page 7 In the case of a power spiral we have ρ ( ) d (5) L = (6) 1 1 L = a d = a The result otained from equation (5) has sufficient engineering accurac and the integration is not difficult. If we wish, we can use formulae that are more precise ( ) ρ ( ) ρ ( ) (7) L = x + d = + d 1 1 In the case of a power spiral the result of integration of an expression in (7) depends on the value of exponent, and the integration ma not e eas. The advantage of power spiral over the involute of a circle is in that that the power spiral can achieve higher uilt-in compression ratio. For example, the compressor with scrolls that are involutes of a circle has uilt-in compression ratio 1.66 on the working angle π, while the compressor with power scrolls has uilt-in compression ratio.71 on the same working angle. While we can change the uilt-in compressor ratio of an involute of a circle ased compressor increasing working angle, we can change the uilt-in compression ratio of a power spiral ased compressor changing exponent while keeping working angle constant..3 Contact Between Mating Scrolls Contact etween mating scrolls is an important propert of the scroll compressor. The reactions etween mating scrolls of a scroll compressor ased on the involute of a circle have direction that is tangent to oth generating radii and thus identical with the direction of the oriting radius. In the contrar, the reactions etween mating scrolls of a scroll compressor ased on the power spiral do not have an reaction etween mating scrolls parallel with the oriting radius. As a matter of fact, the reaction at each contact point passes through that point and the center of curvature that is on the locus of centers of curvature. The center of curvature of a plane curve has the coordinates dρ dρ ρ + ρ cos + sin d d xc = ρ cos (8) dρ d ρ ρ + ρ d d dρ dρ ρ + ρ sin cos d d C = ρ sin dρ d ρ ρ + ρ d d Thus, for the value of =, and after we find all derivatives of equation (5), locus of centers of curvature has coordinates ( ) xc = a 1 sin cos ( 4) + + C = a 1 cos sin (9) (30) (31) International Compressor Engineering Conference at Purdue, Jul 17-0, 006

9 C080, Page 8 Equations (30) and (31) represent another spiral. This indicates that the normal reactions change their directions. The change in the direction of normal reaction ma generate more harmonic components and thus will contriute to generation of viration and noise. Fig. 6 is an example of the locus of centers of curvature of the spiral in Fig. 4. Fig. 6: Locus of centers of curvature 3. CONCLUSIONS The theor of parallel curves enales design of scroll compressors with vanes that are other tpes of plane spirals. It can e seen that spirals, which radius vector increases linearl, such as the involute of a circle and Archimedes spiral, can form pairs of smmetrical pockets, and that the thickness of vanes is constant. Other tpes of spirals such as power spirals, which radius vector increases nonlinearl also form pairs of smmetrical pocket, ut the thickness of vanes increases with increasing radius vector. Power spirals can achieve higher uilt-in compression ratio in the same working angle than the involute of a circle. REFERENCES Gravesen, J., Henriksen, C.,001, The Geometr of the Scroll Compressor, SIAM Review, Vol. 43, No. 1, pp Rektors, K., 1969, Surve of Applicale Mathematics, MIT Press. International Compressor Engineering Conference at Purdue, Jul 17-0, 006