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Purdue e-Pubs International Compressor Engineering Conference

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2006

Novel Geometrical Model of Scroll Compressors for the Analytical Description of the Chamber Volumes Benjamin Blunier University of Technology of Belfort-Montbéliasd

Giansalvo Cirrincione University of Picardie-Jules Verne

Abdellatif Miraoui University of Technology of Belfort-Montbéliasd

Follow this and additional works at: http://docs.lib.purdue.edu/icec Blunier, Benjamin; Cirrincione, Giansalvo; and Miraoui, Abdellatif, "Novel Geometrical Model of Scroll Compressors for the Analytical Description of the Chamber Volumes" (2006). International Compressor Engineering Conference. Paper 1745. http://docs.lib.purdue.edu/icec/1745

This document has been made available through Purdue e-Pubs, a service of the Purdue University Libraries. Please contact [email protected] for additional 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.html

C074, Page 1

NOVEL GEOMETRICAL MODEL OF SCROLL COMPRESSORS FOR THE ANALYTICAL DESCRIPTION OF THE CHAMBER VOLUMES Benjamin BLUNIER1,* , Giansalvo CIRRINCIONE2 , Abdellatif MIRAOUI1 1

Laboratory in Electrical Engineering and Systems, L2ES University of Technology of Belfort-Montbéliard, UTBM UTBM (Bât. F) – 13, rue Thierry Mieg – 90010 Belfort, France email: [email protected] 2

Department of Electrical Engineering University of Picardie-Jules Verne 33, rue Saint Leu – 80039 Amiens, France email: [email protected] *

Corresponding author ABSTRACT

Most scroll compressor models are based on a geometrical description of the scroll wraps. Since geometry is one of the main factors affecting the efficiency of the compressor, a complete description of the geometry of scroll wraps has to be known in order to establish an accurate thermodynamic model. Most authors do not yield an analytical expression of the discharge chamber, also because their assumptions are questionable. This paper gives a novel description of the geometry based on the parametric equations for the circle involutes in a novel reference system, in order to exploit the symmetry of the compressor. This approach simplifies the scroll model and yields an exact analytical expression of the compression and discharge chamber volumes, together with all the geometrical parameters and constraints, thus allowing the compressor to be optimized geometrically. The existence conditions of the conjugacy points, as well as the manufacturing constraints are also considered.

1.

INTRODUCTION

The scroll compressor is a machine used for compressing air or refrigerant, which was originally invented in 1905 by a French engineer named Léon Creux (Creux, 1905). The device consists of two nested identical scrolls constituted in the classical design by involutes of circle as shown in F IG. 1(a). The two scrolls, whose axes of rotation do not meet each other, are assembled with a relative angle of π, so that they touch themselves at different points and form a series of growing size chambers. The main advantages of the scroll compressor are the small number of moving parts, a high efficiency and a low level of noise and vibrations. However, one of the main problems encountered in developing scroll compressors is the design of the scroll profile which plays a key role in their performances. Many geometric shapes for scroll compressors have been investigated in a lot of papers and patents. Most of the works have focused only on the interpolating curves by using circle involutes (Lee and Wu, 1995; Hirano et al., 1990; Chen et al., 2002; Halm, 1997). One of the authors (Gravensen et al., 1998; Gravensen and Henriksen, 2001) redefined the entire geometry of the spiral with two planar curves where the involute is a special case. This paper proposes a new way to describe the geometry inspired by the work of Halm (1997) where the symmetries can be exploited and the optimization can be made taking into account the physical constraints.

International Compressor Engineering Conference at Purdue, July 17–20, 2006

C074, Page 2 2. 2.1

FIXED SPIRAL GEOMETRY

Circle involutes

The shape of the investigated scroll can be described by an involute of circle. The scroll is therefore defined by two involutes that develop around a common basic circle with radius rb and are offset by a constant distance. The fixed spiral can be described by: ∀ ϕ ∈ If o = [ϕos , ϕmax ] xf o (ϕ) = rb (cos ϕ + ϕ sin ϕ) Sf o (1) yf o (ϕ) = rb (sin ϕ − ϕ cos ϕ) ∀ ϕ ∈ If i = [ϕis − ϕi0 , ϕmax − ϕi0 ] xf i (ϕ) = rb (cos(ϕ + ϕi0 ) + ϕ sin(ϕ + ϕi0 )) Sf i yf i (ϕ) = rb (sin(ϕ + ϕi0 ) − ϕ cos(ϕ + ϕi0 ))

(2)

where Sf o and Sf i are respectively the outer and inner involutes. The angles ϕos and ϕis are the outer and inner starting angles, ϕmax the involute ending angle and ϕi0 the initial angle of the inner involute. The thickness of the scroll is e = rb ϕi0 with ϕi0 > 0. (3)

2.2

Interpolating circle

The involutes describing the scroll start at an outer angle ϕos at Ao (Sf o (ϕos )) and an inner angle ϕis at Ai (Sf i (ϕis − ϕi0 )), respectively, and are connected by an arc belonging to the circle Cc , constrained to be tangent to the inner involute at Ai as shown in F IG. 1(b). Hence the circle equations can be written as: xC (ϕ) = rc (tc ) cos(ϕ) + xc (tc ) Cc = (4) yC (ϕ) = rc (tc ) sin(ϕ) + yc (tc ) where  xc (t)     yc (t)      rc (t)      tc      am          bm

= t = a qm t + bm 2 2 = (xc (t) − xf i (ϕis − ϕi0 )) + (yc (t) − yf i (ϕis − ϕi0 )) xf i (ϕis − ϕi0 ) + (yf i (ϕis − ϕi0 ) − bm ) tan(ϕis ) = 1 + am tan(ϕis ) xf i (ϕis − ϕi0 ) − xf o (ϕos ) = − yf i (ϕis − ϕi0 ) − yf o (ϕos ) yf i (ϕis − ϕi0 )2 − yf o (ϕos )2 + xf i (ϕis − ϕi0 )2 − xf o (ϕos )2 = 2 yf i (ϕis − ϕi0 ) − yf o (ϕos )

Depending on ϕis and ϕos , the circle, as seen from the outside of the scroll, can be either concave (F IG. 2(a)) or → convex (F IG. 2(b)). The concavity/convexity condition can be deduced from the relative positions of the vectors − τ −−→ and OAi , −−→ < 0 concavity → the circle goes from Ai to Ao (F IG. 2(a)) → − τ (ϕis − ϕi0 ) ∧ OAi (5) > 0 convexity → the circle goes from Ao to Ai (F IG. 2(b)) Moreover, −−→ − → τ ∧ OAi = rb2 (ϕis − ϕi0 ) where

  c1 c2  c3

c1 cos(ϕos − ϕis ) − c2 sin(ϕos − ϕis ) + c3 ϕos cos(ϕos − ϕis ) − sin(ϕos − ϕis ) + (ϕi0 − ϕis )

= 2 (ϕos (ϕi0 − ϕis ) − 1) = 2 (ϕos + ϕi0 − ϕis ) = ϕ2os + ϕ2i0 − 2 ϕi0 ϕis + ϕ2is + 2


(6)

C074, Page 3 The existence domain IC of the circle is deduced from the previous conditions (5) 0 0 [ϕi , ϕo ] if concave IC = [ϕ0o , ϕ0i ] otherwise

(7)

where ϕ0i ϕ0o

= =

ϕi − 2π ϕi

if concave and (ϕi > ϕo ) otherwise

(8)

ϕo − 2π ϕo

if convex and (ϕi < ϕo ) otherwise

(9)

and

ϕi ϕo

= atan2 xf i (ϕis − ϕi0 ) − xc (tc ) , yf i (ϕis − ϕi0 ) − yc (tc ) = atan2 xf o (ϕos ) − xc (tc ) , yf i (ϕos ) − yc (tc )

(10) (11)

mobile spiral

fixed spiral

Sf i (Cc )

point of conjugacy

Sf o (dm )

rc Ao

O xc (tc ), yc (tc ) Ai

discharge chamber

−−→ OAi

− → τ (ϕis − ϕi0 )

compression chamber suction chamber (a) Schematic view of a scroll compressor

(b) Overview of a spiral

Figure 1: Schematic of a scroll compressor and the spiral

3. 3.1

SCROLL COMPRESSOR GEOMETRY

Orbiting scroll

The geometry of the orbiting scroll is the same as the fixed one but is offset by π and the two scrolls are in conjugacy. Defining θ as the orbiting angle, it follows: ∀ ϕ ∈ Imo = [ϕos , ϕmax ] xmo (ϕ, θ) = rb (cos(ϕ + π) + ϕ sin(ϕ + π)) + ro cos(θ + 3π/2) Smo (12) ymo (ϕ, θ) = rb (sin(ϕ + π) − ϕ cos(ϕ + π)) + ro sin(θ + 3π/2) ∀ ϕ ∈ Imi = [ϕis − ϕi0 , ϕmax − ϕi0 ] xmi (ϕ, θ) = rb (cos(ϕ + ϕi0 + π) + ϕ sin(ϕ + ϕi0 + π)) + ro cos(θ + 3π/2) Smi ymi (ϕ, θ) = rb (sin(ϕ + ϕi0 + π) − ϕ cos(ϕ + ϕi0 + π)) + ro sin(θ + 3π/2) where ro is the orbiting radius and ro = rb (π − ϕi0 ) with 0 < ϕi0 < π and rb > 0.


(13)

C074, Page 4

−−→ OAi ϕo

ϕi

Ai

− → τ

− → τ ϕi

Ao

ϕo

O(xc , yc ) Ao

−−→ OAi

O(xc , yc )

Ai

(a) Concavity

(b) Convexity

Figure 2: Convexity and concavity of the interpolating circle

3.2

Novel reference frame

In literature the reference frame shown in F IG. 3(a) is employed. Here a novel reference frame is presented (see F IG. 3(b)) in order to exploit the symmetry of the two scrolls. In this frame, the parametric equations of the fixed scroll are given by: 0 xf o (ϕ) = rb (cos ϕ + ϕ sin ϕ) − 1/2 ro cos (θ + 3 π/2) 0 Sf o (14) yf0 o (ϕ) = rb (sin ϕ − ϕ cos ϕ)) − 1/2 ro cos (θ + 3 π/2) 0 xf i (ϕ) = rb (cos(ϕ + ϕi0 ) + ϕ sin(ϕ + ϕi0 ))) − 1/2 ro cos (θ + 3 π/2) Sf0 i (15) yf0 i (ϕ) = rb (sin(ϕ + ϕi0 ) − ϕ cos(ϕ + ϕi0 ))) − 1/2 ro cos (θ + 3 π/2) and of the orbiting scroll are given by: 0 Smo

0 Smi

3.3

x0mo (ϕ, θ) = −x0f o (ϕ, θ) 0 (ϕ, θ) = −yf0 o (ϕ, θ) ymo

(16)

x0mi (ϕ, θ) = −x0f i (ϕ, θ) 0 ymi (ϕ, θ) = −yf0 i (ϕ, θ)

(17)

Points of conjugacy

The kth point of conjugacy ϕCf imo−f i,k between the fixed inner involute (f i) and the orbiting outer involute (mo) on the fixed inner involute (f i) is given by: ϕCf imo−f i,k = θ − ϕi0 + 2kπ

k ∈ Z.

(18)

The kth point of conjugacy ϕCf imo−mo,k between the fixed inner involute (f i) and the orbiting outer involute (mo) on the orbiting outer involute (mo) is given by: ϕCf imo−mo,k = θ − π + 2kπ

k ∈ Z.


(19)

C074, Page 5

(x0f i (ϕ) , yf0 i (ϕ))

θ Symetry center

θ

ro 1 2

ro

ro

(−x0f i (ϕ) , −yf0 i (ϕ))

(a) Old reference frame

(b) Novel reference frame

Figure 3: Novel reference frame Using the same notations, the kth point of conjugacy between the fixed outer involute and the orbiting inner involute, repectively, on the fixed outer involute and the orbiting inner involute is given by: ϕCf omi−f o,k ϕCf omi−mi,k

= θ − π + 2kπ k ∈ Z, = θ − ϕi0 + 2kπ k ∈ Z.

(20) (21)

Then, finally, the coordinates of the points of conjugacy are: fixed inner involute :

x0f i (ϕCf ime−f i,k , θ) , yf0 i (ϕCf ime−f i,k , θ)

fixed outer involute :

−x0f i (ϕCf ime−f i,k

, θ) ,

(22)

−yf0 i (ϕCf ime−f i,k , θ)

(23)

Assuming that for k > 2 being k ∈ {1..α}, where α (α ∈ N) is the number of involute rotations (i.e., ϕmax = α 2π) all the points of conjugacy exist for all θ (θ ∈ ]0, 2π]), then it can be deduced: ϕCf imo−f i,k = θ − ϕi0 + 2(k − 1)π k ∈ {2..α}, ∀θ ∈]0, 2π] (24) ϕCf imo−f i,1 = θ − ϕi0 exists if max(ϕis , ϕos + π) ≤ θ where 0 ≤ ϕos ≤ π and ϕi0 ≤ ϕis ≤ 2π.

4. 4.1

ESTIMATION OF THE CHAMBER VOLUMES

Compression chamber

The surface of the kth compression chamber between the fixed inner involute and the orbiting outer involute can be computed straightforward by the involute between two consecutive points of conjugacy: Sf i,mo,k (θ)

= =

1 2

Z

ϕCf ime−f i,k+1 ϕCf ime−f i,k

x0f i

dyf0 i dx0f i − yf0 i dϕ dϕ

dϕ −

1 2

Z

ϕCf ime−me,k+1 ϕCf ime−me,k

x0mo

0 dymo dx0mo 0 − ymo dϕ dϕ

rb2 π (π − ϕi0 )(4kπ − 3π + 2θ − ϕi0 )

Taking into account the existence conditions of the points of conjugacy: Sf i,mo,k (θ) = rb2 π (π − ϕi0 )(4kπ − 3π + 2θ − ϕi0 ) ∀k ∈ [2..α − 1] Sf i,mo,1 (θ) = rb2 π (π − ϕi0 )(4π − 3π + 2θ − ϕi0 ) exists if max(ϕis , ϕos + π) ≤ θ The volume is immediately deduced by multiplying it with the constant scroll height.


dϕ (25)

(26)

C074, Page 6 4.2

Discharge chamber

The computation of the surface of the discharge chamber takes into account the concavity and convexity cases (F IG. 4). The whole discharge surface Sd is computed in accordance with what is shown in F IG. 4: 2 (S1 + S2 ) = 2 (S1 + S21 − S22 + S23 ) if concave Sd = (27) 2 (S1 + S2 ) = 2 (S1 + S21 + S22 − S23 ) otherwise where: Z S1

ϕCf omi−mi,k

=

x0mi

ϕis −ϕi0

S22 S23

Z

ϕCf omi−f o,k

dϕ −

x0f o

ϕos

dx0f o dyf0 o − yf0 o dϕ dϕ

1 if max(ϕis , ϕos + π) ≤ θ 2 otherwise

with k = S21

0 0 dymi 0 dxmi − ymi dϕ dϕ

dϕ (28)

= At (0, 0, xAi , yAi , xAo , yAo ) 1 = rc (tc )2 |ϕ0i − ϕ0o | 2 = At (xc (tc ), yc (tc ), xAi , yAi , xAo , yAo )

(29) (30) (31)

being At (xA , yA , xB , yB , xC , yC ) the area of triangle whose vertices are ABC, computed as: At (xA , yA , xB , yB , xC , yC ) =

1 |yC · xA + xC · yB − xC · yA − yC · xB + yA · xB | 2

(32)

The volume is immediately deduced by multiplying it with the constant scroll height. Two examples (concave and convex) are given in F IG. 5. Ai x0f i (ϕis − ϕi0 , θ), yf0 i (ϕis − ϕi0 , θ)

S2

S1

Ai S2 S1

xc (tc ), yc (tc )

Ao Ao

Ai

Ao

S2

=

=

x0f o (ϕos , θ), yf0 o (ϕos , θ)

S21

+

=

−

+

S22

−

Ai S23

−

+

Ao S2

=

S21

−

S22

Sd

Sd

(a) Convex case

(b) Concave case

+

Figure 4: Discharge chamber


S23

C074, Page 7 0.06

0.06 Sf i,mo,1 (θ)

0.05

0.04 S (cm2 )

S (cm2 )

0.04 0.03

0.03

0.02

0.02

0.01

0.01

0

Sf i,mo,1 (θ) Sf i,mo,2 (θ) Sd (θ)

0.05

Sf i,mo,2 (θ) Sd (θ)

0

1

2

3 θ (rad)

4

5

6

0

0

1

2

3 θ (rad)

4

5

6

(a) Concave: rb = 1.91 · 10−3 m, ϕis = 2.571 rad, ϕi0 = (b) Convex: rb = 1.91 · 10−3 m, ϕis = 3.471 rad, ϕi0 = 1.571 rad, ϕos = 2 rad 1.471 rad, ϕos = 1 rad

Figure 5: Surfaces of the chambers vs. orbiting angle (examples)

5.

CONCLUSION

The efficiency of the scroll compressor is dictated by its geometry. Hence, a complete description of the model is required, also to determine an accurate thermodynamic model. For this purpose, a novel geometry description is proposed here. It exploits the symmetry of the scroll compressor by using a novel reference frame. This choice allows the scroll model to be simplified and yields exact analytical expressions for the compressor and discharge chamber volumes, together with all the geometrical parameters and constraints for the optimization of the scroll geometry.

REFERENCES Chen, Y., Halm, N. P., Groll, E. A., Braun, J. E., 2002. Mathematical modeling of scroll compressor – part I: compression process modeling. International Journal of Refrigeration 25, 731–750. Creux, L., 1905. Rotary engine. U.S Patent 801182. Etemad, S., Nieter, J., 1989. Design optimization of the scroll compressor. International Journal of refrigeration 12, 146–150. Gravensen, J., Henriksen, C., 2001. The geometry of the scroll compressor. Society for Industrial and Applied Mathematics 43 (1), 113–126. Gravensen, J., Henriksen, C., Howell, P., 1998. Danfoss: Scroll optimization. Final report, Department of Mathematics, Technical University of Denmark, Lyngby, 32nd European Study Group with Industry. Halm, N. P., 1997. Mathematical modeling of scroll compressor. Master’s thesis, Herrick kab., School of Mechanical Engineering, Purdue University. Hirano, T., Hagimoto, K., Maada, M., 1990. Scroll profiles for scroll fluid machines. MHI Tech Rev 27 (1), 35–41. Lee, Y.-R., Wu, W.-F., 1995. On the profile design of a scroll compressor. International journal of refrigeration 18 (5), 308–317. Morishita, E., Sugihara, M., Inaba, T., Nakamura, T., Works, W., 1984. Scroll compressor analytical model. In: Purdue International Compressor Engineering Conference Proceedings. pp. 487–495. Tseng, C.-H., Chang, Y.-C., 2006. Family design of scroll compressors with optimization. Applied Thermal Engineering 26, 1074–1086.


C074, Page 8 ACKNOWLEDGEMENT Giansalvo Cirrincione is with the Department of Electrical Engineering University of Picardie-Jules Verne and funded with a grant of ISSIA-CNR, Palermo, Italy in the framework of the project “Automazione della gestione intelligente della generazione distribuita di energia elettrica da fonti rinnovabili e non inquinanti e della domanda di energia elettrica, anche con riferimento alle compatibilità interne e ambientali, all’affidabilità e alla sicurezza”.
