A general geometrical model of scroll compressors ...

International Journal of Refrigeration 28 (2005) 958–966 www.elsevier.com/locate/ijrefrig

A general geometrical model of scroll compressors based on discretional initial angles of involute Baolong Wang, Xianting Li*, Wenxing Shi Department of Building Science, Tsinghua University, Beijing 100084, People’s Republic of China Received 11 May 2004; received in revised form 1 December 2004; accepted 11 January 2005 Available online 23 May 2005

Abstract Based on discretional initial angles of involute of scroll compressors, a general mathematical representation of scroll wraps, working chamber volume and leakage areas is presented. The scroll wraps’ geometrical expressions, including interaction arc, of the involute angle with discretional initial angles of involute are developed. By using it, a new calculation formula of working chamber volume without restriction to special involute initial angles is set up and the expression of the volume during all the suction, compression and discharge processes in a general subsection function style is given. A geometrical model of leakage areas, including flank in, flank out, radial in and radial out, is also developed based on the discretional initial angles condition. Finally, this geometrical model is applied in a thermodynamic model and the simulation results are compared with some former experimental results. It is found that this model has a satisfactory accuracy and is easy to be used in thermodynamic simulation. q 2005 Elsevier Ltd and IIR. All rights reserved. Keywords: Scroll compressor; Modelling; Geometry; Volume; Compression

Compresseurs a` spirale : mode`le ge´ome´trique general fonde´ sur les angles initiaux de la volute Mots cle´s : Compresseur a` spirale ; Mode´lisation ; Ge´ome´trie ; Volume ; Compression

1. Introduction The scroll theory was brought forward by Creux at the beginning of previous century [1]. But it was impossible to manufacture a working pair of scrolls until the mid 1970’s due to a very small tolerance required. Since then, the scroll compressor became more and more popular because of its

* Corresponding author. Tel.: C86 1 62785860; fax: C86 10 62773461. E-mail address: [email protected] (X. Li).

0140-7007/$35.00 q 2005 Elsevier Ltd and IIR. All rights reserved. doi:10.1016/j.ijrefrig.2005.01.015

unique advantages, such as low level of noise, high efficiency and high reliability. The scroll compressor’s geometry is one of the main factors affecting the efficiency of the compressor. In order to establish a thermodynamic model for a scroll compressor, the geometry of the scroll has to be known and completely understood. The shapes of scroll wraps include the involute of a circle, involute of a square [2], hybrid wraps [3,4] and so on. The most popular shape for scroll wraps is the first and it will be studied in this paper. The wraps of scrolls directly affect the built-in compression ratio and the performance of the compressor,

B. Wang et al. / International Journal of Refrigeration 28 (2005) 958–966

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Nomenclature a h L L(4) N r R S V x y

radius of the basic circle scroll height length of the involute; length of the leakage line length of the point on the involute to its tangent point on the base circle loops of the scroll orbiting radius of the rotating scroll radius of the interaction arc, equal to (PKt)/2 area work volume X coordinate Y coordinate

Greek letters d leakage gap q orbiting angle

so many researchers have done a lot of work in this field. For a scroll compressor with the warps of the involute of a circle, the research emphasis is always put on the inner portion because the outer portion of the scroll wraps is the involute and simple. Liu [5,6] presented a graphic and mathematical method for the modified design of the scroll wraps. The perfect meshing profiles (PMPs) of a scroll compressor, a design with zero clearance volume, are researched by Hirano [7] and Lee [8]. It can be found that those foregoing researches always put the scroll in a special angle and make aiZKa0. So a and Ka can be used in the simulation to stand for ai and a0, respectively, which reduce a parameter in the geometrical expressions. This method reduces the computation complexity of scroll wraps. But when the loop of the scroll is not an integer, this method make the starting orbiting angle not equal to 0, which raises a lot of difficulties to comprehend the mesh process of the scroll pair. Those previous models can not be used when the scroll can not or is not easy to be put to the required special position and angle, such as the reproduction of the scroll in numerical control machine. Therefore, these coordinates’ functions of scroll wraps, including interaction arc, with discretional involute initial angles and discretional loop will be presented in this paper. Other researches done on geometrical model of scroll compressor is to develop the geometry relationship between the compressor chamber volume, leakage areas and the orbiting angle. Morishita [9] first advanced a chamber volume expression of the compression process and discharge process. They gave the functions of compression chamber volume for every 2p orbiting angle and based on special involute initial angle (aiZKa0): q (1) Vpi Z pPðP K 2tÞh ð2i K 1Þ K p Here, i means the i-th chamber. His formulation used to

q* 4 4out,start, 4e a

discharge angle involute angle 4os starting angle of the outer involute involute ending angle, equal to 2pNC(p/2) initial angle of the involute

Subscripts i inner involute o outer involute circle, o center of the intervene arc inter intersection point of the intervene arc and the outer involute s suction c compression d discharge r radial f flank

calculate the volume of the discharge chamber during 0! q%q* is different from the one during q*!q%2p. Hayano [10] developed a geometrical model of the chamber volume during compression process, which also set up a different expression for every 2p. Tojo [11] and Etemad [12] applied Morishita’s model of the compression pocket volume in their scroll compressor simulation. Nieter [13] presented a geometry model of the suction chamber, which was used to simulate the dynamics of the scroll suction process. Zhu [14] and Liu [15] gave a working pocket volume model including suction, compression and discharge process. Their modeling for compression and discharge process had a similar style with Morishita’s model. It’s difficult to apply all of the above models in thermodynamic simulation of scroll compressors because of their complexity. Yanagisawa [16] and Hirano [17] promoted the geometrical model of the scroll compressor into a new stage. They developed three general expressions for suction, compression and discharge processes respectively. The main difference between Morishita’s model and Yanagisawa’s model is that the later can use a general formulation to calculate the volume during whole compression process and a general formulation to calculate the volume during whole discharge process. The expression of compression process is: Vc Z 2pharð24e K 2q K 3pÞ

(2)

Here, q starts from zero. Their researches simplified the geometrical model greatly and made its application in scroll compressor simulation easier than before. However, this model still has its weakness, i.e. all the result was based on the special involute initial angle assumption (aiZKa0). This made the application of the model limited. Halm [18, 19] had woken up to this. He set up a geometrical model of scroll compressors, including suction, compression and

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discharge, which was based on another special involute initial angle (aiZp and a0Z0). In foregoing investigations on leakage areas, the leakage areas are always divided into two types, flank and radial, and the areas of leak in and leak out are supposed to be same. In actual, the areas of leak in are quite different from the areas of leak out for whole working process. In the same time, these previous models just deals with the leakage areas in compression process and the leakage in suction process and discharge process must be calculated using other methods or be treated as an ideal process (there is no suction/discharge pressure loss in suction/discharge process), which leads to the increase of the complexity in leakage calculation. To sum up, the current models of scroll are mostly restricted to special initial angles of the involutes and respectively set up the suction, compression and discharge processes models (all q of three processes start from zero), which leads to difficulties in practical application of this model. The present paper intends to develop a general geometrical model of scroll compressors. The scroll wraps’ geometrical expressions, including interaction arc, of the involute angle with discretional initial angle of involute is brought forward. Based on it, a new calculation formula for working chamber volume is set up, which is not restricted to special involute initial angles and gives the expression of the volume during all the suction, compression and discharge processes in a general subsection function style. A leakage area model, including flank in, flank out, radial in and radial out, is also developed based on the discretional initial angles condition.

2. Geometrical model of the scroll profiles In order to avoid the restriction of the initial angle, the model of the scroll profiles is built on discretional initial angle. The shape of scrolls is an involute of a circle. From the definition of the involute, the distance of the point on the involute to its tangent point on the base circle satisfied this differential relation: vLð4Þ Za v4

(3)

Therefore, the coordinates of the point on the involute with a zero involute initial angle can be: x Z aðcosð4Þ C 4 sinð4ÞÞ

(4)

yZ aðsinð4ÞK 4 cosð4ÞÞ If the involute initial angle is considered (Fig. 1), the coordinates of the points on the inner involute and outer involute are: xi Z aðcosð4i C ai Þ C 4i sinð4i C ai ÞÞ

(5)

yi Z aðsinð4i C ai Þ K 4i cosð4i C ai ÞÞ x0 Z aðcosð40 C a0 Þ C 40 sinð40 C a0 ÞÞ y0 Z aðsinð40 C a0 Þ K 40 cosð40 C a0 ÞÞ For a scroll compressor with discretional initial angles, the expression of the discharge angle is different from the one with a symmetric initial angle [20]: ai K a0 * * f*2 0 C 2f0 sin f0 K 2 a K a i 0 C 2 cos f*0 K 2 (6) a K a0 2 K2 Z pK i 2 3 q* Z p K f*0 K a0 2 Therefore, the profiles of the scroll with discretional initial angles are produced as following: For outer involute, x0 Z aðcosð40 C a0 Þ C 40 sinð40 C a0 ÞÞ y0 Z aðsinð40 C a0 Þ K 40 cosð40 C a0 ÞÞ

4out;start ! 40 ! 4e K a0

(7) For inner involute, xi ¼ aðcosð4i þ ai Þ þ 4i sinð4i þ ai ÞÞ yi ¼ aðsinð4i þ ai Þ K 4i cosð4i þ ai ÞÞ Fig. 1. Expressions of the inner involute and outer involute.

pK

ai K a0 ! 4i ! 4e K ai 2

(8)


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For interaction arc, xcircle ¼ R cosð4circle Þ þ xcircle;0 ycircle ¼ R sinð4circle Þ þ ycircle;0

yinter K ycircle;0 arc tan xinter K xcircle;0 !4circle !

p ai þ a0 þ 2 2 (9)

Here, xcircle,0, ycircle,0 are the coordinates of the center of the intervene arc, xinter, yinter are the coordinates of the intersection point of the interaction arc and the outer involute (Fig. 2). They could be gained from: a C a0 xcircle;0 Z a cos p C i (10) 2 a C a0 ycircle;0 Z a sin p C i 2 ðxinter K xcircle;0 Þ2 C ðyinter K ycircle;0 Þ2 Z R2

(11) Fig. 3. Schematic of area calculation.

xinter Z a cosð4out;start C a0 Þ C 4out;start sinð4out;start C a0 Þ yinter Z a sinð4out;start C a0 Þ K 4out;start cosð4out;start C a0 Þ

ð4 SZ

0

1 1 ða4Þ2 d4 Z a2 43 2 6

(13)

By using this general calculation formula, the volume of the compression chamber can be calculated. 3. Geometrical model of the volume of the work chamber 3.1. Suction chamber Because the suction, compression and discharge are continuous processes, the change of the volume is also continuous and the geometrical model of the volume of the compression chamber should be advanced in a continuous manner. The tiny area dS in the Fig. 3 can be calculated by [20]: 1 1 dS Z Lð4ÞLð4Þd4 Z ða4Þ2 d4 2 2 So the whole area S is:

(12)

The main area of the suction chamber is S1 (Fig. 4). ð4 ð 1 e 1 4eKp S1 Z h L2i d4 K L20 d4 (14) 2 4eKq 2 4eKqKp Here, LiZa(4Kai) and L0Za(4Ka0). Therefore, the main area S1 is: 1 S1 Z arð2q4e K q2 K qðai C a0 C pÞÞ 2

(15)

Actually, the area S1 has a difference with the practical suction pocket. The correction area can be calculated as [19]:

Fig. 2. Schematic of the interaction arc.

Fig. 4. Schematic of volume calculation of suction chamber.

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i 1 h p S2 Z ar 2ð1 K cos qÞ K 2ð4e K pÞsin q K sin 2q 2 4 (16)

model [18]) are applied in function (20), the Yanagisawa’s model (function (2)) and Halm’s model can be easily gotten (Pay attention to the different starting angle of q).

The whole suction volume and its derivative are then: 3.3. Discharge chamber

Vs Z 2hðS1 C S2 Þ h Z har ð2q4e K q2 K qðai C a0 C pÞÞ C 2ð1 K cos qÞ K 2ð4e K pÞsin q K

i p sin 2q 4

h dVs Z har ð24e K 2q K ðai C a0 C pÞÞ dq i p C 2 sin q K 2ð4e K pÞcos q K cos 2q 2

(17)

(18)

0!q%2p. 3.2. Compression chamber Using the same calculation method like suction process, the compression chamber volume can be obtained. ð 4 C2pKq ð 1 e 1 4eCpKq 2 Vc Z 2h L2i d4 K L0 d4 (19) 2 4eKq 2 4eK2pKq Vc Z 2pharð24e K 2q K ðai C a0 K pÞÞ

(20)

dVc Z K4phar dq

(21)

5 2p! q% 4e K p C q* 2 It should be mentioned that the q in this formulation starts from 2p, which makes the whole volume model be a subsection function style and easier to use in the scroll simulation. If the special initial angles (aiCa0Z0) for Yanagisawa’s [16] model and aiCa0Zp/2 for Halm’s

There are two areas in discharge process: Vd (Vd1 and Vd2) and Vdd (Fig. 5). For a real discharge process (Fig. 5), the opening position of the discharge port always makes the discharge processes dissimilar for those two pockets. For pocket 1 (Vd1), the main throttling phenomena happens at Adl, which is decided by the shape of the discharge port and the orbiting angle. For pocket 2 (Vd2), the throttling phenomena will appear at Ad2, which is decided by the shape of the scroll and the orbiting angle. But for an ideal scroll, the discharge port always located at central part of these two scrolls, for example, the region of Vdd, to make a balance discharge. This makes discharge processes in the ideal process similar to the pocket 2. So the authors considered that the Ad2 is the critical area for discharge and the Vd1 and Vd2 are the volume of the discharge. In the same time, because the area of Vdd connected to discharge port is far bigger than Ad2, the Vdd is same treated as outer of discharge port. This comprehension is different from Chen’s [19]. ð 4 C2pKq ð 1 e 1 4eCpKq 2 Vd Z 2h L2i d4 K L0 d4 2 52 pKq* 2 32 pKq* ð 4 C2pKq ð 1 e 1 4eCpKq 2 L2i d4 K L0 d4 (22) Z 2h 2 4osCpCa0 2 4osCa0 Here, 4osCpCa0Z(5/2)pKq* p 7p Vd Z har 4e K q C q* K 4e K q K q* K ai K a0 C 2 2

(23) dVd Z harðK24e C 2q C ai C a0 K 3pÞ dq

(24)

5 p 4e K p C q* ! q% 4e K C q* 2 2

Fig. 5. Schematic of discharge chamber.

Fig. 6. Change of the volume during suction, compression and discharge.


The whole change process of the volume during suction, compression and discharge can be plotted in Matlab (Fig. 6).

1 3p w4e K p C q : 2 ð 4eC2pKq a C a0 a fK i df LZ 2 4eCpKq

963

(29)

4. Geometrical model of the leakage area In this paper, the suction and discharge processes are also treated as leakage. The main difference between suction/discharge process and leakage process in compression is the value of leakage areas, so these processes can be treated similarly. According to the difference of the external pressure, the leakage areas of radial in, radial out, flank in and flank out are calculated respectively. Due to existence of two or more different external pressures in a single leakage process, such as radial out, the leakage area in a period of orbiting angle could be two parts. This phenomenon has never been reported in other papers. According to difference of leakage mode and leakage direction, the leakage could be divided into four kinds (Fig. 7): radial in, radial out, flank in and flank out. 4.1. Radial out The leakage line length of radial out can be calculated as: ð 4e a C a0 0 wp: L Z df (25) a fK i 2 4eKq

p w2p:

2p w3p:

a ai C a0 2 Kp L Z 2p 4e K 2 2 ð 4eC2pKq ai C a0 df L1 Z a fK 2 4eKp

(26)

As a result, the leakage area of the radial out leakage is shown in Table 1. It can be found that there are two expressions during 2p!q!3p, that’s because the refrigerant leaks into two pockets, i.e. suction chamber and outer pocket, with different pressures at that moment. 4.2. Radial in The leakage line length of radial in can be calculated as: a C a0 df a fK i 2 4eKpKq

ð 4eKp 0 wp: L Z

(30)

5 p w4e K p C q* : 2 ð 4eKq a C a0 df LZ a fK i 2 4eKpKq

(31)

5 1 4e K p C q* w4e K p C q* : 2 2 ap ½4p K 2q* K ðai C a0 Þ LZ 2

(32)

As a result, the leakage area of the radial in leakage is shown in Table 2. (27) 4.3. Flank out

a C a0 df L2 Z a fK i 2 4eCpKq ð 4eKp

(28)

The length of the leakage line can be calculated as: 0 w2p: L Z rð1 K cos qÞ C df

(33)

1 2p w4e K p C q* : 2

(34)

L Z df

As a result, the leakage area of the flank out leakage is shown in Table 3. The leakage area of flank out during 0! q!2p is the suction area. 4.4. Flank in The length of the leakage line can be calculated as:

Fig. 7. Schematic of four kinds of leakage.

5 0 w4e K p C q* : L Z df 2

(35)

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Table 1 Leakage area of the radial out leakage Orbiting angle (q)

Leakage area

0–p p–2p 2p–3p

dr a½2qð4e K ððai C a0 Þ=2ÞÞK q2 dr a½2pð4e K ððai C a0 Þ=2ÞÞK p2 dr a½2ð3pK qÞð4e K pK ððai C a0 Þ=2ÞÞC ð3pK qÞ2 dr a½K2ð2pK qÞð4e K pK ððai C a0 Þ=2ÞÞK ð2pK qÞ2 dr a½2pð4e C pK qK ððai C a0 Þ=2ÞÞC p2

3p–4eK(1/2)pCq*

5 3 4e K p C q e4e K p C q : 2 2 L Z R K r2 C ða K rÞ2 C 2rðR K rÞ

ðkC1Þ=ðkK1Þ 1=2 dm k 2 Z Cd Apup dt RTup k C 1 (36)

1=2 5 !cos q K 4e K p C q Cdf 2

3 1 4e K p C q* w4e K p C q* : 2 2

L Z 2ðR K rÞ C df (37)

As a result, the leakage area of the flank in leakage is shown in Table 4. The leakage area of flank in during qO4eK5p/2Cq* is the discharge area.

5. Application This general geometrical model of scroll compressors was applied in the thermodynamic model of a scroll compressor to calculate the refrigerant properties during the suction, compression and discharge processes. The refrigerant is R22. The geometrical parameters of the scroll are same as Winandy’s scroll [21]. The mass flow process is thought as a one-dimensional compressible flow in a nozzle with an isentropic assumption: ðkC1Þ=k 1=2 dm 2k pdown 2=k p K down Z Cd Apup pup pup dt Rðk K 1ÞTup

k=ðkK1Þ pdown 2 R pup k C1

(38)

k=ðkK1Þ pdown 2 ! k C1 pup

(39)

Here, Cd is flow coefficient and is set based on the experiments. When the suction state of the refrigerant is at 630 kPa and 35.2 8C and discharged pressure is 2150 kPa, the pressure and temperature of the refrigerant as a function of the orbiting angle were plotted in Figs. 8 and 9. Both the pre-compression phenomenon at the end of suction process and the over-compression phenomenon at the start of discharge process have been demonstrated in these figures. Something must be mentioned here. The temperature and pressure in Figs. 8 and 9 refer to the properties of refrigerant in the pocket. If the variation of the discharge volume (in actual, the variation is very little) and the inlet leakage of the refrigerant can be ignored, this properties of the refrigerant in the discharge pocket can be considered as an isentropic expansion process. This process can be described by: ðkK1Þ=k T2 p Z 2 T1 p1

(40)

For refrigerant R22, k is bigger than 1. So the refrigerant temperature in the pocket at latter time T2 is smaller than the refrigerant temperature in the pocket at former time T1 and the decease of the temperature follows the decease of the pressure in the pocket. In the same time, other researchers [16,23,24] also got the same conclusions by experiments or simulations as the authors. Actually the decrease of the refrigerant temperature at the outside of the discharge pocket is very little due to a constant enthalpy process. A number of simulations were conducted and the results were compared with the experimental results of Winandy [21]. The prediction of the refrigerant flow rate and motor

Table 2 Leakage area of the radial in leakage Orbiting angle (q)

Leakage area

0–p p–4eK(5/2)pCq* 4eK(5/2)pCq*–4eK(1/2)pCq*

dr a½2qð4e K pK ððai C a0 Þ=2ÞÞK q2 dr a½2pð4e K qK ððai C a0 Þ=2ÞÞK p2 dr ap½4pK 2q* K ðai C a0 Þ


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Table 3 Leakage area of the flank out leakage Orbiting angle (q)

Leakage area

0–2p 2p–4eK(1/2)pCq*

2hrð1K cosðqÞÞC 2hdf 2hdf

Fig. 8. Pressure of the refrigerant in the pocket as a function of the orbiting angle. Fig. 10. Prediction of the refrigerant mass flow rate.

Fig. 9. Temperature of the refrigerant in the pocket as a function of the orbiting angle.

Fig. 11. Prediction of the motor power.

power were illuminated in Figs. 10 and 11, respectively. In the calculation of motor input power the motor efficiency and the mechanical efficiency are considered and these two efficiencies can be calculated by the polynomials provided by Park etc [22]. hmec Z 0:8680 C 0:0048f K 4:4444 !10K5 f 2

hmot Z 0:6980 C 0:0013 C 4:1235 !10K5 f 2 K 4:8781 !10K7 f 3 C 1:4206 !10K9 f 4 There, f is the frequency of the compressor. The relative error of the mass flow rate varied from K2.5 to C2.5%. The relative error of the motor power is between

Table 4 Leakage area of the flank in leakage Orbiting angle (q) *

0–4eK(5/2)pCq 4eK(5/2)pCq*–4eK(3/2)pCq* 4eK(3/2)pCq*–4eK(1/2)pCq*

Leakage area 2hdf 2hfRK ½r2 C ðRK rÞ2 C 2rðRK rÞcosðqK ð4e K ð5=2ÞpC q* ÞÞ1=2 gC 2hdf 4h(RKr)C2hdf

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K2.0 and C5.0%. These results show that a scroll compressor thermodynamic model using this geometrical model can simulate the compression process satisfactorily.

[11]

6. Conclusions A general geometrical model of scroll compressors is brought out in this paper. This model is not restricted by involute initial angle and includes all the suction, compression and discharge periods in a subsection function manner, which provides a base for simulation of the scroll compressor and dynamic visualization of the work process of the scroll compressor.

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