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Scrutinizing the sources of inefficiencies in the piston-cylinder clearance of an oil-free linear compressor
International Journal of Refrigeration 104 (2019) 513–520
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International Journal of Refrigeration journal homepage: www.elsevier.com/locate/ijrefrig
Scrutinizing the sources of inefficiencies in the piston-cylinder clearance of an oil-free linear compressor Claudio J. Santos, Thiago Dutra, Cesar J. Deschamps∗ POLO Research Laboratories for Emerging Technologies in Cooling and Thermophysics, Federal University of Santa Catarina, CEP 88048-300, Florianopolis, SC, Brazil
a r t i c l e
i n f o
Article history: Available online 17 May 2019 Keywords: Linear compressor Piston-cylinder clearance Aerostatic bearing Leakage
a b s t r a c t The piston of linear compressors is subjected to low side loads and metal contact can be avoided by feeding high-pressure refrigerant gas from the discharge system into the piston-cylinder clearance. This paper reports a thorough study on the inefficiencies associated with leakage and viscous friction brought about by the transient compressible fluid flow in the piston-cylinder clearance of an oil-free linear compressor. The main objective is to understand the effect of design parameters, such as compressor speed, piston oscillation amplitude, pressure ratio and radial clearance, on the volumetric and isentropic efficiencies. The compressor efficiency was found to be mainly affected by the clearance and pressure ratio. A reduction of 2.5% in the volumetric efficiency and 4.2% in the isentropic efficiency, for instance, were predicted when the radial clearance was varied from 2.5 to 10.0 μm. © 2019 Elsevier Ltd and IIR. All rights reserved.
Examiner les sources d’inefficacités dans le jeu piston-cylindre d’un compresseur linéaire sans huile Mots-clés: Compresseur linéaire; Jeu piston-cylindre; Palier aérostatique; Fuite
1. Introduction The drive mechanism of a linear compressor consists of a linear actuator coupled to a resonant spring and an electric motor. Fig. 1 shows the main components of an oil-free linear compressor driven by a magnet motor. The linear drive mechanism with a resonant spring allows the use of part of the kinetic energy of the system, reducing the current required in the motor and, therefore, the power input. The piston of a linear compressor is subjected to low side loads and metal contact can be avoided without the use of lubricating oil, by feeding high-pressure refrigerant gas from the discharge system into the piston-cylinder clearance. An oil-free linear compressor can bring some benefits, such as an increase in the heat transfer in the evaporator and condenser, reduction in the friction between moving parts and flexibility in the mounting orientation of the compressor next to the refrigeration system.
∗
Corresponding author. E-mail address: [email protected] (C.J. Deschamps).
https://doi.org/10.1016/j.ijrefrig.2019.05.015 0140-7007/© 2019 Elsevier Ltd and IIR. All rights reserved.
Most of the models developed to simulate linear compressors are comprised of submodels for the piston dynamics, compression cycle and leakage in the piston-cylinder clearance (Davies et al., 2010; Bradshaw et al., 2011; Jomde et al., 2018). In these models, the leakage in the piston-cylinder clearance is analytically solved for a combined Couette-Poiseuille incompressible flow. However, this type of approach is not suitable for predicting the leakage of gas through the clearance due to presence of compressibility effects (Braga and Deschamps, 2017). Nishikawa et al. (1998) developed a finite volume model to solve the compressible fluid flow in the piston-cylinder clearance of a four-stage compressor operating without lubricating oil. In addition to accuracy, the results should be presented in a form that is convenient for analysis. Pérez-Segarra et al. (2002) and Pérez-Segarra et al. (2005) proposed a convenient method to identify the thermodynamic inefficiencies of each process of the compression cycle (suction, compression, discharge and expansion). The authors also isolated the main inefficiencies affecting the volumetric efficiency, such as losses due to the suction and discharge processes, re-expansion of the cylinder clearance volume,
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Nomenclature cv Dcap Dpist f h Lamp Lcap Lcfo Lcyl Linstop Lpist m m˙ p T Tcond Tevap Tsuc Tw v
∀
W˙ W˙ f W˙ ind W˙ s
specific heat at constant volume (m2 s−2 k−1 ) restrictor channel diameter (m) piston diameter (m) operating frequency (s−1 ) specific enthalpy (j kg−1 ) piston oscillation amplitude (m) length of the restrictor channel (m) length of the feeding hole (m) cylinder length (m) distance from datum position to valve plate (m) piston length (m) mass (kg) mass flow rate (kg s−1 ) pressure (Pa) temperature (K) condensing temperature (K) evaporating temperature (K) suction line temperature (K) wall temperature (K) specific volume (m3 kg−1 ) volume (m3 ) compressor power consumption (W) piston friction power consumption (W) indicated power (W) isentropic power (W)
Greek letters γ specific heat ratio (-) δ piston-cylinder clearance (m) ɛv volumetric inefficiency (-) ɛs isentropic inefficiency (-) ηc overall efficiency (-) ηele electrical efficiency (-) ηmec mechanical efficiency (-) ηs isentropic efficiency (-) ηv volumetric efficiency (-) ρ density (kg m−3 ) ϕ rate of viscous dissipation in the piston-cylinder clearance (W) Subscripts cyl cylinder dis discharge exp expansion fo feeding orifice gap piston cylinder clearance gsk piston skirt gtp piston top i compression chamber in entering the control volume ini initial manc aerostatic bearing out exiting the control volume pist piston s isentropic sh superheating suc suction vol volume wall wall backflow in the valves and leakage in the piston-cylinder gap. Schreiner et al. (2010) suggested that the suction gas superheating could be split into two contributions: the first in the suction system and the second during the suction process inside the cylinder.
Fig. 1. Schematic showing the main moving parts of a linear compressor.
The concept of an oil-free linear compressor has many advantages, but the effect of the aerostatic bearing formed by the pistoncylinder clearance on the compressor efficiency is not yet fully understood. This paper reports an assessment of the thermodynamic inefficiencies brought about by the transient compressible fluid flow in the piston-cylinder clearance of an oil-free linear compressor. First, a simulation model is developed by combining three models: (i) a three-dimensional finite-volume model for the fluid flow in the clearance; (ii) a lumped-parameter approach for the compression cycle; and (iii) analytical determination of the mass flow rate through the restrictor channels. The different sources of volumetric and isentropic inefficiencies in the piston-cylinder clearance are then identified and quantified for the baseline operating conditions. Finally, a sensitivity analysis is performed to evaluate the effect of design parameters on these inefficiencies. 2. Simulation model Three submodels were developed to form the simulation model adopted to analyze the inefficiencies in the piston-cylinder clearance of an oil-free linear compressor. These models take into account different phenomena: i) compression cycle in the cylinder; ii) transient compressible fluid flow in the piston-cylinder clearance; and iii) compressible fluid flow through the restrictor channels. These submodels are detailed in the following sections. 2.1. Compression chamber model The compression chamber model (CC) is used to solve the compression cycle. The indicated power and the mass flow rate predicted by this model are required to evaluate the thermodynamic inefficiencies. The CC model was developed via a lumped formulation for the mass and energy conservation equations. Applying the ideal gas assumption, the energy conservation equation applied to the compression chamber assumes the following form:
dTi 1 = dt mi cv,i
−hi
d mi − m˙ j h j dt
− Ti
R d ∀i d mi −R vi dt dt
, (1)
C.J. Santos, T. Dutra and C.J. Deschamps / International Journal of Refrigeration 104 (2019) 513–520
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Fig. 2. Piston-cylinder clearance solution domain.
where T is temperature and the subscript i refers to the gas in the compression chamber, m is mass, cv is specific heat at constant volume, h is specific enthalpy, v is specific volume and R is the gas constant. Eq. (2) neglects the convective heat transfer at the cylinder walls. The specific volume is determined from vi = ∀i /mi and the instantaneous pressure is calculated from the ideal gas equation, pi = R Ti /vi . The term dmi /dt in Eq. (1) represents the rate of change of the mass inside the compression chamber, which is obtained from the mass conservation equation:
d mi = m˙ in,i − m˙ out,i + m˙ out ,gt p − m˙ in,gt p dt
(2)
where m˙ in,i , m˙ out,i , m˙ out ,gt p and m˙ in,gt p represent the mass flow rate entering and exiting the cylinder through the valves and cylinder clearance, respectively. Finally, the term m˙ j h j corresponds to the net balance of energy entering and leaving the control volume:
m˙ j h j = − m˙ in,i hin,i − m˙ out,i hi + m˙ out ,gt p hgt p − m˙ in,gt p hi .
(3)
Leakage at the top clearance, m˙ gt p , can occur in two directions, from the clearance toward the compression chamber and in the opposite direction. When the leakage occurs toward the clearance, m˙ in,gt p , the enthalpy of the gas in the cylinder is considered in the energy balance, Eq. (3). When the leakage occurs toward the compression chamber, m˙ out ,gt p , the enthalpy of the gas coming from the top clearance is considered. The leakage at the top clearance is calculated from the finite-volume model adopted for the clearance, as will be detailed in Section 2.2. The conservation Eqs. (1) and (2) are solved by the Euler method. The suction and discharge processes are modeled as isobaric, i.e., pressure drops in the valves and mufflers are not taken into account. Therefore, the mass flow rates m˙ in,i and m˙ out,i are calculated instantaneously for a fixed value of pi and satisfying the mass and energy conservation in the compression chamber. 2.2. Piston-cylinder clearance model The solution domain of the piston-cylinder clearance model (PCC) consists of the clearance region shown in Fig. 1. Perfect alignment between the piston and the cylinder was considered. The dimension of the piston-cylinder radial clearance is 5 μm, much smaller than the piston diameter (by a factor of < 1/10 0 0). Therefore, curvature effects can be disregarded, and the solution domain can be simplified to the geometry of two parallel plates. The solution domain is further simplified due to planes of symmetry in the circumferential direction (Fig. 2). The mass, momentum and energy conservation equations are required to solve the transient compressible fluid flow. The dynamic viscosity is obtained using data from the library REFPROP® v8.0 (Lemmon et al., 2002). The hypothesis of ideal gas is adopted and the viscous dissipation rate, , is calculated at each time step from the results of velocity field. The governing equations were
solved via the finite-volume method available in the computational fluid dynamics software Fluent v14.0.0. An implicit formulation was used for time discretization and a second-order upwind scheme was employed to evaluate advective terms in the transport equations. The SIMPLE algorithm was adopted for the coupling between the pressure and velocity fields during the iterative solution procedure. The boundary conditions for the flow through the piston-cylinder clearance are those indicated in Fig. 2: (a) Compression chamber interface: pressure and temperature given by the CC model at each piston position. Gas flow is allowed in both directions across the interface. (b) Cylinder surface: no-slip condition and prescribed temperature (T = 328.15 K). (c) Restrictor channels interface: mass flow rate analytically determined in the restrictor channel. The gas temperature is assumed to be equal to that in the discharge system. (d) Symmetry boundary condition: normal velocity components and normal gradient components of any other property are set to zero. (e) Piston surface: no-slip condition and prescribed temperature (T = 328.15 K). (f) Internal environment interface: Evaporating pressure and suction gas temperature are prescribed. It should be mentioned that the temperature of 328.15 K prescribed to the cylinder wall was obtained from measurements for the evaporating and condensing temperatures of 249.85 and 313.65 K, respectively. Naturally, the wall temperature varies with the pressure ratio, but a sensitivity analysis revealed that the volumetric and isentropic inefficiencies (ɛv and ɛs) for this compressor design virtually does not change when this temperature is varied from 328.15 to 373.15 K. Simulations are initialized with the piston positioned at the top dead center (TDC). The velocity field is set to zero and the temperature field is set to 328.15 K. At the TDC, interfaces a) and f) are submitted to condensing and evaporation pressure conditions, respectively. A linear variation in the pressure between these two surfaces is set as the initial condition. 2.3. Restrictor channels model The restrictor channels are grooves positioned on the outer wall of the cylinder. The mass flow rate leaving the restrictor channels is estimated by assuming the hypothesis of laminar, isothermal compressible fluid flow, using the following expression:
m˙ f o = ρ f o U¯
fo
Acap ,
(4)
where ρ fo and U¯ f o are the density and mean velocity at the channel inlet conditions, and Acap is the cross-section area of the channel. The mean velocity U¯ f o is calculated from the Mach number at
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C.J. Santos, T. Dutra and C.J. Deschamps / International Journal of Refrigeration 104 (2019) 513–520 Table 2 Output quantities provided by the simulation. Model
Quantity
Compression chamber (CC) Piston-cylinder clearance (PCC)
m˙ in,i , m˙ out,i , hout,i , Ti , hi , pi , W˙ ind m˙ gsk , m˙ in,gt p , m˙ out ,gt p , hgsk , W˙ f , Q˙ gap , hgt p , Tgap , pgap
Restrictor channel Suction system (energy balance) Discharge system (energy balance)
m˙ f o Q˙ suc Q˙ dis
m˙ = m˙ th − 1 − ηv,idc m˙ th − m˙ gt p(exp) − m˙ gt p(suc ) − (ρini,comp − ρsuc )
Fig. 3. Interaction between the models. Table 1 Input quantities required in the simulation. Model
Quantity
Compression chamber (CC) Piston-cylinder clearance (PCC) Restrictor channel Suction system (energy balance) Discharge system (energy balance)
hin,i , m˙ out ,gt p , hgt p , pevap , pcond hi , pi , pevap , m˙ f o , h f o pcond , pgap m˙ , hsuc , m˙ gsk , hgsk , m˙ in,i , hin, i m˙ , hdis , m˙ out,i , hout,i , m˙ f o
the inlet, as suggested by Hülse and Prata (2016):
⎡ 1−
p pfo
2
⎢1 γ ln p f o 2 + p
Mfo = ⎣
4 fD L Dh
⎤ 12 ⎥ ⎦ ,
(5)
where γ is the ratio of specific heats (= c p /cv ), fD is the Darcy friction factor ( fD = 64/Re), L is the length of the restrictor channel, pfo is the pressure at the inlet of the channel, which is equal to the pressure in the discharge system (pcond ), and p is the pressure at the outlet of the channel, calculated by the PCC model in the feeding hole (surface c of Fig. 2). The mass flow rate in each restrictor channel is provided as a boundary condition for the PCC model. 3. Analysis of thermodynamic inefficiencies The parameters commonly adopted to evaluate the performance of compressors are the volumetric efficiency and isentropic efficiency. In the following sections, a method is presented to identify the main volumetric and isentropic inefficiencies associated with the flow in the piston-cylinder clearance of a small oil-free linear compressor. In order to support the next sections, Fig. 3 shows a diagram of the mass and energy fluxes between different regions of the compressor and Tables 1 and 2 indicate the input and output quantities associated with each model during the simulation. 3.1. Volumetric inefficiency The mass flow rate provided by the compressor (m˙ ) can be written as a function of the ideal mass flow rate, m˙ th = ∀sw ρsuc f , where ∀sw is the swept volume and ρ suc is the gas density in the suction line at the compressor entrance, and reductions associated with different effects:
∀BDC f − m˙ gsk
(6)
The second term on the right-hand side of Eq. (6) corresponds to the reduction in the flow caused by the dead volume (ηv, idc ). The terms m˙ gt p(exp) and m˙ gt p(suc ) represent the mean flow rates through the piston top clearance during the expansion and suction processes, respectively. A positive value for m˙ gt p indicates mass input into the compression chamber, i.e., leakage occurs from the gap to the compression chamber. This would represent a reduction in the mass flow rate, since a smaller amount of mass would enter the compression chamber during the suction process due to the mass intake from the piston-cylinder clearance. The fourth term expresses the loss due to the difference between the gas densities in the suction line and inside the cylinder at the beginning of the compression process (ρsuc − ρini,comp ). The variable ∀BDC is the volume of the compression chamber when the piston is positioned at the bottom dead center (BDC). The last term states for the mass flow loss concerning the leakage through the piston skirt. Eq. (6) disregards losses in the valves, since ideal suction and discharge processes were assumed in the present analysis, which focused on inefficiencies of the fluid flow through the piston-cylinder clearance. Dividing Eq. (6) by m˙ th and introducing the parameter ɛvk to represent the volumetric inefficiencies, i.e., the fraction of the total mass flow rate reduction due to a specific effect k (ε vk = m˙ k /m˙ th ), one obtains:
ηv = ηv,idc − ε vgt p,exp − ε vgt p,suc − ε vsh − ε vgsk ,
(7)
where ηv (= m˙ /m˙ th ) is the volumetric efficiency. The total volumetric inefficiency is
ε v = ε vgt p,exp + ε vgt p,suc + ε vsh + ε vgsk ,
(8)
3.2. Isentropic inefficiency The isentropic inefficiencies can be determined by calculating the amount of heat that should be withdrawn from the control volume (dashed line in Fig. 3) to make the entropy of the gas at the outlet equal to the entropy of the gas at the inlet. This means that the heat transfer rates indicated in Fig. 3 can be regarded as energy transfer rates that make the actual process different from the isentropic process. It should be noted that our study was restricted to evaluate the inefficiencies brought about by the fluid flow in the pistoncylinder clearance. Therefore, the electric motor losses were not taken into account in the analysis. As a result, the compressor power input is defined herein as W˙ = W˙ ind + W˙ f , where W˙ ind is the indicated power and W˙ f is the bearing loss due to the viscous friction on the piston, and the energy balance in the control volume of Fig. 3 yields:
W˙ ind + W˙ f + m˙ (hsuc − hdis ) = Q˙ gap + Q˙ suc + Q˙ dis ,
(9)
where hsuc and hdis represent the specific enthalpy at the inlet and outlet of the control volume, respectively. The term Q˙ gap is the heat rejection rate in the piston-cylinder clearance calculated
C.J. Santos, T. Dutra and C.J. Deschamps / International Journal of Refrigeration 104 (2019) 513–520
Fig. 4. Linear compressor geometric parameters – piston at the TDC.
Fig. 5. Comparison between the results obtained with the PCC model and experimental data reported by Suefuji et al. (1992).
Table 3 Geometric parameters. Parameter
Value
Parameter
Value
Parameter
Value
Lpist /Lcyl Lcfo1 /Lcyl Lcfo2 /Lcyl
0.82 0.42 0.78
Linstop /Lcyl Lcap1 /Dcap Lcap2 /Dcap
0.12 5.15 × 103 6.53 × 103
Dfo /Lcyl Dcap /Lcyl Dpist /Lcyl
1.60 × 10−2 8.40 × 10−4 0.31
4.1. Model validation
with the PCC model. The heat transfer rate Q˙ suc is evaluated by assuming that the specific enthalpy at the compression chamber entrance (hin, i ) is equal to that at the compressor inlet (hsuc ), which is known from the evaporating pressure, pevap , and suction line temperature, Tsuc , both prescribed. Finally, Q˙ dis is the heat rejection rate required to reach the thermodynamic state at the compressor outlet represented by the condensing pressure (pcond ) and specific entropy equal to that at the compressor inlet. Q˙ suc and Q˙ dis are determined from energy balances:
Q˙ suc = m˙ hsuc + m˙ gsk hgsk − m˙ in,i hin,i ,
(10)
Q˙ dis = m˙ out,i hout,i − m˙ f o hdis − m˙ hdis ,
(11)
where m˙ gsk and hgsk are predicted with the PCC model, m˙ in,i , m˙ out,i and hout, i are calculated with the CC model and m˙ f o is obtained from the restrictor channels model. Replacing the definitions of isentropic power, W˙ s = m˙ (hdis − hsuc ), and isentropic efficiency, ηs = W˙ s /W˙ , in Eq. (9) and introducing the parameter ɛsk to represent the increase in power input, i.e., isentropic inefficiency due to the specific effect k (ε sk = Q˙ k /W˙ s ), gives:
εs =
1
ηs
− 1 = ε sgap + ε ssuc + ε sdis
517
(12)
where ɛs is the total isentropic inefficiency. 4. Results Fig. 4 shows a schematic of the linear compressor, highlighting the most important geometric parameters. The line located at a distance Linstop from the valve plate represents the position around which the piston oscillates with an amplitude Lamp . Table 3 shows the values of these geometric parameters. The nominal dimension of the piston-cylinder clearance is δ = 5 μm. All simulations carried out in the present study considered the evaporating pressure of 114.8 kPa. Considering R134a as the refrigerant fluid, the evaporating temperature is 249.85 K. Moreover, based on experimental data, the temperature of the gas at the compressor inlet was prescribed as 305.15 K and the temperatures of the piston and cylinder walls were set at 328.15 K.
A preliminary study was carried out to verify the time and space discretizations required for the PCC model and the number of cycles necessary to reach the solution for the fully periodic operating condition. The Richardson extrapolation method was adopted (Cadafalch et al., 2002), with three levels of mesh refinement and a growth rate of 2. The truncation errors were analyzed based on the results for the viscous dissipation () and the total volumetric and isentropic inefficiencies (ε v and ε s). For the simulations, the following parameters were adopted: pcond = 1030.3 kPa; Tcond = 313.65 K; δ = 5 μm; f = 120 Hz; and Lamp /Lcyl = 0.215. It was found that the time step size should be set at 5.79 × 10−7 s and that 15 cycles should be simulated to reach the fully periodic operating condition. The best compromise between computational cost and solution accuracy was found with a mesh of 20 elements in the radial direction. We found that mesh refinements in the other directions are much less important for solution accuracy. The results of the PCC model were validated through comparison with the experimental data of Suefuji et al. (1992) for mass flow rate of R22 through a channel with a clearance (h). The length and width of the channel were 4.5 and 126 mm, respectively, whereas the clearance value and the inlet pressure were varied in the tests. The temperature was set at 300 K and the outlet pressure was kept at 100 kPa. The PCC model was adapted to match the fluid flow condition of the experiments, i.e. the restrictor channels and piston motion were not considered in the simulations. Fig. 5 presents results of mass flow rate for three clearances (h = 3.6 μm, 9.0 μm and 21.0 μm) as a function of the pressure ratio. As can be seen, the leakage mass flow rate predicted by the PCC model and the measurements of Suefuji et al. (1992) are in reasonable agreement, with a mean difference of 14%, and a maximum deviation of 35% occurring for the most challenging case for measurements (h = 3.6 μm). 4.2. Baseline operating condition The baseline operating condition is defined by the following parameters: pressure ratio = 8.97; pcond = 1030.3 kPa; Tcond = 313.65 K; δ = 5 μm; f = 120 Hz; and Lamp /Lcyl = 0.215. Data presented in Table 4 indicate a reduction of 54.0% in the mass flow rate due to the dead volume brought about by the small amplitude of Lamp /Lcyl for the baseline operating condition. On the other hand, the volumetric inefficiency resulting from the flow in the pistoncylinder clearance (ɛv) is only 0.79%, mostly attributed to the inef-
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Table 4 Mass flow rates and different sources of the volumetric inefficiency. Parameter
Value
Parameter
Value (%)
Parameter
Value (%)
m˙ th [kg/s] m˙ [kg/s] ηv [−]
2.20E-03 9.96E-04 45.25%
ɛvidc [-] ɛvgsk [-] ɛvgtp, suc [-]
53.96 0.21 0.18
ɛvgtp, exp [-] ɛvsh [-] ɛv [-]
0.39 0.01 0.79
Fig. 7. Rates of energy flow through the top clearance and heat transfer through the piston and cylinder walls.
Fig. 6. Instantaneous mass flow rates through the piston-cylinder clearance. Table 5 Power input and different sources of the isentropic inefficiency. Parameter
Value
Parameter
Value
Parameter
Value
W˙ s [W] W˙ ind [W] W˙ f [W]
60.65 61.37 0.13 98.61%
ηc [-] ηmec [-]
98.81% 99.79% 0.14% 0.12%
ɛsgap [-] ɛs [-] [W]
1.15% 1.41% 0.69
ηs [−]
ɛssuc [-] ɛsdis [-]
ficiencies ɛvgtp, suc and ɛvgtp, exp associated with leakage through the top gap (m˙ gt p ) during the expansion and suction processes. Fig. 6 shows the results for the mass flow rate through the restrictors and leakage through the top and skirt gaps throughout the compression cycle. Negative values indicate that the gas leaves the piston-cylinder clearance domain. The instantaneous leakage through the top gap reaches magnitudes much greater than those in the skirt gap or feeding orifices. In fact, the piston-cylinder clearance acts as a reservoir, receiving, accumulating and delivering mass to the compression chamber periodically. This transient behavior is relevant with regard to the volumetric efficiency. For instance, leakage from the top gap to the compression chamber during the expansion and suction processes shown in Fig. 6 reduce the amount of mass of refrigerant that can enter the compression chamber through the suction valve. On the other hand, a positive net value of m˙ gt p during the cycle contributes to an increase in m˙ gsk and hence a decrease in the volumetric efficiency. The results for the compressor power, isentropic efficiency and inefficiencies are reported in Table 5. The total inefficiency, ɛs, increased the power consumption (W˙ ind + W˙ f ) by 1.41% with reference to the isentropic power W˙ s . Only 0.21% of this value is due to the piston friction power W˙ f , also referred to as mechanical loss.
On the other hand, the compression efficiency, ηc = W˙ s /W˙ ind , undergoes a more significant change, greater than 1%. Table 5 also shows that the main factor responsible for the increase in the energy consumption is the viscous dissipation released as heat in the piston-cylinder clearance (ɛsgap ). Fig. 7 shows energy flow rates through the boundaries of the PCC indicated in Fig. 2. According to Fig. 7, the instantaneous values for the energy entering the top gap and the heat transfer at the walls are similar in magnitude but opposite in sign. This means that a great part of
Fig. 8. Instantaneous viscous dissipation rate integrated throughout the domain.
the energy entering the piston-cylinder top gap (positive value) is transferred to the walls and does not return to the compression chamber when the flow through the top gap changes direction. The results for the viscous dissipation rate () integrated throughout the solution domain of the PCC model during the compression cycle are shown in Fig. 8. The peak for occurs close to the TDC when the pressure is highest and hence the same occurs with the velocity magnitudes and gradients in the pistoncylinder clearance. The mean value for calculated over the cycle is 0.69 W, which corresponds to 75% of the energy that enters the gap (= 0.92 W) during the compression cycle. 4.3. Parametric analysis The compressor performance and thermodynamic inefficiencies were analyzed by varying four parameters: pressure ratio ( ), piston-cylinder radial clearance (δ ), operating frequency (f), and amplitude of oscillation (Lamp ). Each of these parameters was varied individually, while for the others the values adopted for the baseline operating conditions were maintained. It should be mentioned that experiments to validate the results of our parametric analysis is quite an intricate task because several phenomena affect the compressor efficiency and take place in a coupled manner, such as valve dynamics, suction gas superheating and motor losses. In fact, in order to isolate the inefficiencies brought about by the fluid flow in the piston-cylinder clearance from the inefficiencies associated with other phenomena, the simulations
C.J. Santos, T. Dutra and C.J. Deschamps / International Journal of Refrigeration 104 (2019) 513–520
519
Fig. 9. (a) Volumetric and (b) isentropic inefficiencies as a function of pressure ratio.
Table 6 Numerical results as a function of pressure ratio.
Table 7 Numerical results as a function of radial clearance.
[-]
3.61
5.15
8.97
12.8
δ [μm]
2.5
3.0
4.0
5.0
7.5
10.0
m˙ [kg/s] W˙ ind [W] W˙ f [W]
1.78E-03 60.74 0.13 80.81% 81.10% 99.65% 99.86%
1.55E-03 68.77 0.13 70.26% 70.70% 99.46% 99.66%
9.96E-04 61.50 0.13 45.25% 46.04% 98.61% 98.81%
4.68E-04 35.20 0.12 21.22% 22.39% 95.69% 96.00%
m˙ [kg/s] W˙ ind [W] W˙ f [W]
1.01E-03 61.84 0.21 45.73% 46.04% 99.13% 99.47%
1.01E-03 61.76 0.18 45.64% 46.04% 99.07% 99.36%
1.00E-03 61.63 0.15 45.45% 46.04% 98.87% 99.11%
9.96E-04 61.50 0.13 45.25% 46.04% 98.61% 98.81%
9.80E-04 61.36 0.11 44.47% 46.04% 97.19% 97.37%
9.54E-04 61.20 0.11 43.28% 46.04% 94.91% 95.08%
ηv [-] ηv, idc [-] ηs [-] ηc [-]
were carried for ideal valves and without considering convective heat transfer in the compression chamber. 4.3.1. Pressure ratio The effect of the pressure ratio was analyzed by fixing the evaporating pressure and varying the condensing pressure. Simulations were carried out for the following pressure ratios ( ): 3.61, 5.15, 8.97 and 12.8. The results for the mass flow rate, indicated power, friction power and volumetric, compression and isentropic efficiencies are reported in Table 6. As expected, the increase in the pressure ratio gives rise to a decrease in ηv , because the portion of the swept volume required to expand the vapor left in the dead volume after the discharge process also increases. Therefore, vapor from the suction line is only able to fill the portion of the swept volume that has not been filled with the expanded vapor. The leakages in the piston-cylinder clearance also increase with increasing pressure ratio, contributing to the reduction in ηv . The indicated power (W˙ ind ) reaches a maximum when = 5.15, since the mass flow rate increases and the specific work of compression decreases with the pressure ratio. The compression and isentropic efficiencies, ηc and ηs , decrease in the same proportion (approximately 4% in the range analyzed), since the friction power is very small when compared to the indicated power. According to Fig. 9(a), the total volumetric inefficiency caused by the aerostatic bearing formed by the piston-cylinder clearance, ɛv, increases by almost 1% when varies from 3.61 to 12.8. The effect of leakage through the skirt gap, ɛvgsk , is significant. The volumetric efficiency is also reduced by the leakage into the cylinder during the expansion process, ɛvgtp, exp , which occurs because the pressure difference in the gap increases with the pressure ratio. The results in Fig. 9(b) show that the heat rejection rate in the piston-cylinder clearance, ɛsgap , is the most important source of isentropic inefficiency. For instance, a reduction of 4.3% in the isentropic efficiency is observed when the pressure ratio is
ηv [-] ηv, idc [-] ηs [-] ηc [-]
increased from 3.61 to 12.8. This effect is significant and has to be taken into account if oil-free linear compressors are designed for operating conditions with a high-pressure ratio. 4.3.2. Piston-cylinder radial clearance The effect of the piston-cylinder radial clearance of the linear compressor was evaluated for six values: δ = 2.5, 3.0, 4.0, 5.0, 7.5 and 10.0 μm. According to Table 7, the volumetric efficiency undergoes a reduction of 2.5% (ηv − ηv,idc ) and the compression and isentropic efficiencies decrease by more than 4% when the clearance is varied from 2.5 to 10.0 μm. Fig. 10(a) shows that all volumetric inefficiencies increase with the piston-cylinder clearance. In particular, the increase in the leakage through the skirt gap (ɛvgsk ) is a direct result of the greater leakage at the top gap (ɛvgtp, exp and ɛvgtp, suc ), since the mass flow rate through the feeding orifices does not vary with δ . The results in Fig. 10(b) indicate that the effect of the heat dissipated inside the piston-cylinder clearance, ɛsgap , is the most significant isentropic inefficiency. The leakage through the skirt gap is considerably increased with the clearance and therefore the same occurs with the heat rejection rate in the suction system (Q˙ suc ) and the inefficiency ɛssuc . Small clearances are associated with greater frictional power, W˙ f , due to the resistive viscous force on the piston wall. However, the effect of greater leakage as the radial clearance δ is increased results in a negative trade-off and the compressor efficiency is decreased. 4.3.3. Operating frequency and piston oscillation amplitude The inefficiencies were found to be less affected by the variation of the operating frequency and piston oscillation amplitude. For instance, when the operating frequency is varied from 60 to 240 Hz, the total volumetric inefficiency (ɛv) and isentropic inefficiency (ɛs) are decreased by only 0.5% and 0.9%, respectively. It is worth mentioning that the operating frequency is a fundamen-
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Fig. 10. (a) Volumetric and (b) isentropic inefficiencies as a function of radial clearance.
tal parameter regarding the motor efficiency of linear compressors. However, in the present study its effect was only analyzed regarding the inefficiencies associated with the fluid flow in the pistoncylinder clearance. As far as the oscillation amplitude is concerned, the total volumetric inefficiency remains approximately constant when the piston oscillation amplitude (Lamp /Lcyl ) is varied between 0.21 and 0.24. In turn, the total isentropic inefficiency is found to decrease by almost 1%, mainly due to the reduction of pistoncylinder clearance losses (ɛsgap ). 5. Conclusions A study was successfully carried out on the thermodynamic inefficiencies associated with the transient compressible fluid flow in the piston-cylinder clearance of an oil-free linear compressor. A convenient method was adopted to identify different sources of the thermodynamic inefficiency in the piston-cylinder clearance, such as viscous friction force on the piston surface, viscous dissipation associated with the flow in the clearance, and leakage at the piston skirt. The leakage through the skirt gap causes a portion of the refrigerant to circulate continuously inside the compressor, reducing the flow of gas supplied to the refrigeration system. Another significant leakage occurs from the top gap to the compression chamber during the expansion and suction processes, which reduces the amount of mass of refrigerant that can enter the compression chamber through the suction valve. This phenomenon was found to be the most significant source of volumetric inefficiency of the flow in the clearance. Regarding the isentropic efficiency, the viscous friction force on the piston had an almost negligible effect on the isentropic efficiency. On the other hand, the mechanical energy dissipated by viscous friction in the clearance is the most influential source of isentropic inefficiency. The compressor performance and thermodynamic inefficiencies were analyzed by varying the pressure ratio, piston-cylinder radial clearance, operating frequency and piston oscillation amplitude. The compressor efficiency was found to be mainly affected by the clearance and pressure ratio. A reduction of 2.5% in the volumetric efficiency and 4.2% in the isentropic efficiency, for instance, were predicted when the clearance was varied from 2.5 to 10.0 μm.
Acknowledgements This study forms part of a technical-scientific cooperation program between the Federal University of Santa Catarina and EMBRACO. Support from CNPq (grant 465448/2014–3) and EMBRAPII Unit POLO/UFSC is also acknowledged. References Bradshaw, C.R., Groll, E.A., Garimella, S.V., 2011. A comprehensive model of miniature-scale linear compressor for electronics cooling. Int. J. Refrig. 34, 63–73. Braga, V.M., Deschamps, C.J., 2017. Numerical analysis of gas leakage in the pistoncylinder clearance of reciprocating compressors considering compressibility effects. IOP Conf. Ser. Mater. Sci. Eng. 232, 012006. Cadafalch, J., Pérez-Segarra, C.D., Cònsul, R., Oliva, A., 2002. Verification of finite volume computations on steady-state fluid flow and heat transfer. J. Fluids Eng. 124 (1), 11–21 n. Davies, G.F., Eames, I.W., Bailey, P.B., Dadd, M.W., Janiszewski, A., Stone, C.R., Maidment, G.G., Agnew., B., 2010. Cooling microprocessors using vapour compression refrigeration. Thermal and thermomechanical phenomena in electronics systems (ITherm). In: Proceedings of the Twelfth IEEE Intersociety Conference, Las Vegas, pp. 1–8. Hülse, E.R., Prata, A.T., 2016. Numerical modeling of capillary compensated aerostatic bearing applied to linear reciprocating compressor. In: Proceedings of the International Compressor Engineering Conference at Purdue. West Lafayette Paper 2410. Jomde, A., Anderson, A., Bhojwani, V., Kedia, S., Jangale, N., Kolas, K., Khedar, P., 2018. Modeling and measurement of a moving coil oil-free linear compressor performance for refrigeration application using R134a. Int. J. Refrig. 88, 182–194. Lemmon, E.W., Huber, M.L., McLinden, M.O., 2002. NIST standard reference database 23: reference fluid thermodynamic and transport properties-REFPROP,version 8.0. National Institute of Standards and Technology, Gaithersburg, USA Standard Reference Data Program. Nishikawa, H., Nishikawa, T., 1998. Development of the non-lubricated four-stage compressor compressing up to 24.52MPa. In: Proceedings of the International Compressor Engineering Conference at Purdue, West Lafayette, pp. 183–188. Pérez-Segarra, C.D., Rigola, J., Oliva, A., 2002. Thermal and fluid dynamic characterization of hermetic reciprocating compressors. In: Proceedings of the International Compressor Engineering Conference at Purdue. West Lafayette Paper C4-2. Pérez-Segarra, C.D., Rigola, J., Sòria, M., Oliva, A., 2005. Detailed thermodynamic characterization of hermetic reciprocating compressors. Int. J. Refrig. 28, 579–593. Schreiner, J.E., Barbosa, J.R., Deschamps, C.J., 2010. Theoretical analysis of the volumetric efficiency reduction in reciprocating compressors due to in-cylinder thermodynamics. In: Proceedings of the International Compressor Engineering Conference at Purdue. West Lafayette Paper 1348. Suefuji, K., Shiibayashi, M., Tojo, K., 1992. Performance analysis of hermetic scroll compressor. In: Proceedings of the International Compressor Engineering Conference at Purdue, West Lafayette, pp. 75–84.
Contents lists available at ScienceDirect
International Journal of Refrigeration journal homepage: www.elsevier.com/locate/ijrefrig
Scrutinizing the sources of inefficiencies in the piston-cylinder clearance of an oil-free linear compressor Claudio J. Santos, Thiago Dutra, Cesar J. Deschamps∗ POLO Research Laboratories for Emerging Technologies in Cooling and Thermophysics, Federal University of Santa Catarina, CEP 88048-300, Florianopolis, SC, Brazil
a r t i c l e
i n f o
Article history: Available online 17 May 2019 Keywords: Linear compressor Piston-cylinder clearance Aerostatic bearing Leakage
a b s t r a c t The piston of linear compressors is subjected to low side loads and metal contact can be avoided by feeding high-pressure refrigerant gas from the discharge system into the piston-cylinder clearance. This paper reports a thorough study on the inefficiencies associated with leakage and viscous friction brought about by the transient compressible fluid flow in the piston-cylinder clearance of an oil-free linear compressor. The main objective is to understand the effect of design parameters, such as compressor speed, piston oscillation amplitude, pressure ratio and radial clearance, on the volumetric and isentropic efficiencies. The compressor efficiency was found to be mainly affected by the clearance and pressure ratio. A reduction of 2.5% in the volumetric efficiency and 4.2% in the isentropic efficiency, for instance, were predicted when the radial clearance was varied from 2.5 to 10.0 μm. © 2019 Elsevier Ltd and IIR. All rights reserved.
Examiner les sources d’inefficacités dans le jeu piston-cylindre d’un compresseur linéaire sans huile Mots-clés: Compresseur linéaire; Jeu piston-cylindre; Palier aérostatique; Fuite
1. Introduction The drive mechanism of a linear compressor consists of a linear actuator coupled to a resonant spring and an electric motor. Fig. 1 shows the main components of an oil-free linear compressor driven by a magnet motor. The linear drive mechanism with a resonant spring allows the use of part of the kinetic energy of the system, reducing the current required in the motor and, therefore, the power input. The piston of a linear compressor is subjected to low side loads and metal contact can be avoided without the use of lubricating oil, by feeding high-pressure refrigerant gas from the discharge system into the piston-cylinder clearance. An oil-free linear compressor can bring some benefits, such as an increase in the heat transfer in the evaporator and condenser, reduction in the friction between moving parts and flexibility in the mounting orientation of the compressor next to the refrigeration system.
∗
Corresponding author. E-mail address: [email protected] (C.J. Deschamps).
https://doi.org/10.1016/j.ijrefrig.2019.05.015 0140-7007/© 2019 Elsevier Ltd and IIR. All rights reserved.
Most of the models developed to simulate linear compressors are comprised of submodels for the piston dynamics, compression cycle and leakage in the piston-cylinder clearance (Davies et al., 2010; Bradshaw et al., 2011; Jomde et al., 2018). In these models, the leakage in the piston-cylinder clearance is analytically solved for a combined Couette-Poiseuille incompressible flow. However, this type of approach is not suitable for predicting the leakage of gas through the clearance due to presence of compressibility effects (Braga and Deschamps, 2017). Nishikawa et al. (1998) developed a finite volume model to solve the compressible fluid flow in the piston-cylinder clearance of a four-stage compressor operating without lubricating oil. In addition to accuracy, the results should be presented in a form that is convenient for analysis. Pérez-Segarra et al. (2002) and Pérez-Segarra et al. (2005) proposed a convenient method to identify the thermodynamic inefficiencies of each process of the compression cycle (suction, compression, discharge and expansion). The authors also isolated the main inefficiencies affecting the volumetric efficiency, such as losses due to the suction and discharge processes, re-expansion of the cylinder clearance volume,
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Nomenclature cv Dcap Dpist f h Lamp Lcap Lcfo Lcyl Linstop Lpist m m˙ p T Tcond Tevap Tsuc Tw v
∀
W˙ W˙ f W˙ ind W˙ s
specific heat at constant volume (m2 s−2 k−1 ) restrictor channel diameter (m) piston diameter (m) operating frequency (s−1 ) specific enthalpy (j kg−1 ) piston oscillation amplitude (m) length of the restrictor channel (m) length of the feeding hole (m) cylinder length (m) distance from datum position to valve plate (m) piston length (m) mass (kg) mass flow rate (kg s−1 ) pressure (Pa) temperature (K) condensing temperature (K) evaporating temperature (K) suction line temperature (K) wall temperature (K) specific volume (m3 kg−1 ) volume (m3 ) compressor power consumption (W) piston friction power consumption (W) indicated power (W) isentropic power (W)
Greek letters γ specific heat ratio (-) δ piston-cylinder clearance (m) ɛv volumetric inefficiency (-) ɛs isentropic inefficiency (-) ηc overall efficiency (-) ηele electrical efficiency (-) ηmec mechanical efficiency (-) ηs isentropic efficiency (-) ηv volumetric efficiency (-) ρ density (kg m−3 ) ϕ rate of viscous dissipation in the piston-cylinder clearance (W) Subscripts cyl cylinder dis discharge exp expansion fo feeding orifice gap piston cylinder clearance gsk piston skirt gtp piston top i compression chamber in entering the control volume ini initial manc aerostatic bearing out exiting the control volume pist piston s isentropic sh superheating suc suction vol volume wall wall backflow in the valves and leakage in the piston-cylinder gap. Schreiner et al. (2010) suggested that the suction gas superheating could be split into two contributions: the first in the suction system and the second during the suction process inside the cylinder.
Fig. 1. Schematic showing the main moving parts of a linear compressor.
The concept of an oil-free linear compressor has many advantages, but the effect of the aerostatic bearing formed by the pistoncylinder clearance on the compressor efficiency is not yet fully understood. This paper reports an assessment of the thermodynamic inefficiencies brought about by the transient compressible fluid flow in the piston-cylinder clearance of an oil-free linear compressor. First, a simulation model is developed by combining three models: (i) a three-dimensional finite-volume model for the fluid flow in the clearance; (ii) a lumped-parameter approach for the compression cycle; and (iii) analytical determination of the mass flow rate through the restrictor channels. The different sources of volumetric and isentropic inefficiencies in the piston-cylinder clearance are then identified and quantified for the baseline operating conditions. Finally, a sensitivity analysis is performed to evaluate the effect of design parameters on these inefficiencies. 2. Simulation model Three submodels were developed to form the simulation model adopted to analyze the inefficiencies in the piston-cylinder clearance of an oil-free linear compressor. These models take into account different phenomena: i) compression cycle in the cylinder; ii) transient compressible fluid flow in the piston-cylinder clearance; and iii) compressible fluid flow through the restrictor channels. These submodels are detailed in the following sections. 2.1. Compression chamber model The compression chamber model (CC) is used to solve the compression cycle. The indicated power and the mass flow rate predicted by this model are required to evaluate the thermodynamic inefficiencies. The CC model was developed via a lumped formulation for the mass and energy conservation equations. Applying the ideal gas assumption, the energy conservation equation applied to the compression chamber assumes the following form:
dTi 1 = dt mi cv,i
−hi
d mi − m˙ j h j dt
− Ti
R d ∀i d mi −R vi dt dt
, (1)
C.J. Santos, T. Dutra and C.J. Deschamps / International Journal of Refrigeration 104 (2019) 513–520
515
Fig. 2. Piston-cylinder clearance solution domain.
where T is temperature and the subscript i refers to the gas in the compression chamber, m is mass, cv is specific heat at constant volume, h is specific enthalpy, v is specific volume and R is the gas constant. Eq. (2) neglects the convective heat transfer at the cylinder walls. The specific volume is determined from vi = ∀i /mi and the instantaneous pressure is calculated from the ideal gas equation, pi = R Ti /vi . The term dmi /dt in Eq. (1) represents the rate of change of the mass inside the compression chamber, which is obtained from the mass conservation equation:
d mi = m˙ in,i − m˙ out,i + m˙ out ,gt p − m˙ in,gt p dt
(2)
where m˙ in,i , m˙ out,i , m˙ out ,gt p and m˙ in,gt p represent the mass flow rate entering and exiting the cylinder through the valves and cylinder clearance, respectively. Finally, the term m˙ j h j corresponds to the net balance of energy entering and leaving the control volume:
m˙ j h j = − m˙ in,i hin,i − m˙ out,i hi + m˙ out ,gt p hgt p − m˙ in,gt p hi .
(3)
Leakage at the top clearance, m˙ gt p , can occur in two directions, from the clearance toward the compression chamber and in the opposite direction. When the leakage occurs toward the clearance, m˙ in,gt p , the enthalpy of the gas in the cylinder is considered in the energy balance, Eq. (3). When the leakage occurs toward the compression chamber, m˙ out ,gt p , the enthalpy of the gas coming from the top clearance is considered. The leakage at the top clearance is calculated from the finite-volume model adopted for the clearance, as will be detailed in Section 2.2. The conservation Eqs. (1) and (2) are solved by the Euler method. The suction and discharge processes are modeled as isobaric, i.e., pressure drops in the valves and mufflers are not taken into account. Therefore, the mass flow rates m˙ in,i and m˙ out,i are calculated instantaneously for a fixed value of pi and satisfying the mass and energy conservation in the compression chamber. 2.2. Piston-cylinder clearance model The solution domain of the piston-cylinder clearance model (PCC) consists of the clearance region shown in Fig. 1. Perfect alignment between the piston and the cylinder was considered. The dimension of the piston-cylinder radial clearance is 5 μm, much smaller than the piston diameter (by a factor of < 1/10 0 0). Therefore, curvature effects can be disregarded, and the solution domain can be simplified to the geometry of two parallel plates. The solution domain is further simplified due to planes of symmetry in the circumferential direction (Fig. 2). The mass, momentum and energy conservation equations are required to solve the transient compressible fluid flow. The dynamic viscosity is obtained using data from the library REFPROP® v8.0 (Lemmon et al., 2002). The hypothesis of ideal gas is adopted and the viscous dissipation rate, , is calculated at each time step from the results of velocity field. The governing equations were
solved via the finite-volume method available in the computational fluid dynamics software Fluent v14.0.0. An implicit formulation was used for time discretization and a second-order upwind scheme was employed to evaluate advective terms in the transport equations. The SIMPLE algorithm was adopted for the coupling between the pressure and velocity fields during the iterative solution procedure. The boundary conditions for the flow through the piston-cylinder clearance are those indicated in Fig. 2: (a) Compression chamber interface: pressure and temperature given by the CC model at each piston position. Gas flow is allowed in both directions across the interface. (b) Cylinder surface: no-slip condition and prescribed temperature (T = 328.15 K). (c) Restrictor channels interface: mass flow rate analytically determined in the restrictor channel. The gas temperature is assumed to be equal to that in the discharge system. (d) Symmetry boundary condition: normal velocity components and normal gradient components of any other property are set to zero. (e) Piston surface: no-slip condition and prescribed temperature (T = 328.15 K). (f) Internal environment interface: Evaporating pressure and suction gas temperature are prescribed. It should be mentioned that the temperature of 328.15 K prescribed to the cylinder wall was obtained from measurements for the evaporating and condensing temperatures of 249.85 and 313.65 K, respectively. Naturally, the wall temperature varies with the pressure ratio, but a sensitivity analysis revealed that the volumetric and isentropic inefficiencies (ɛv and ɛs) for this compressor design virtually does not change when this temperature is varied from 328.15 to 373.15 K. Simulations are initialized with the piston positioned at the top dead center (TDC). The velocity field is set to zero and the temperature field is set to 328.15 K. At the TDC, interfaces a) and f) are submitted to condensing and evaporation pressure conditions, respectively. A linear variation in the pressure between these two surfaces is set as the initial condition. 2.3. Restrictor channels model The restrictor channels are grooves positioned on the outer wall of the cylinder. The mass flow rate leaving the restrictor channels is estimated by assuming the hypothesis of laminar, isothermal compressible fluid flow, using the following expression:
m˙ f o = ρ f o U¯
fo
Acap ,
(4)
where ρ fo and U¯ f o are the density and mean velocity at the channel inlet conditions, and Acap is the cross-section area of the channel. The mean velocity U¯ f o is calculated from the Mach number at
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Quantity
Compression chamber (CC) Piston-cylinder clearance (PCC)
m˙ in,i , m˙ out,i , hout,i , Ti , hi , pi , W˙ ind m˙ gsk , m˙ in,gt p , m˙ out ,gt p , hgsk , W˙ f , Q˙ gap , hgt p , Tgap , pgap
Restrictor channel Suction system (energy balance) Discharge system (energy balance)
m˙ f o Q˙ suc Q˙ dis
m˙ = m˙ th − 1 − ηv,idc m˙ th − m˙ gt p(exp) − m˙ gt p(suc ) − (ρini,comp − ρsuc )
Fig. 3. Interaction between the models. Table 1 Input quantities required in the simulation. Model
Quantity
Compression chamber (CC) Piston-cylinder clearance (PCC) Restrictor channel Suction system (energy balance) Discharge system (energy balance)
hin,i , m˙ out ,gt p , hgt p , pevap , pcond hi , pi , pevap , m˙ f o , h f o pcond , pgap m˙ , hsuc , m˙ gsk , hgsk , m˙ in,i , hin, i m˙ , hdis , m˙ out,i , hout,i , m˙ f o
the inlet, as suggested by Hülse and Prata (2016):
⎡ 1−
p pfo
2
⎢1 γ ln p f o 2 + p
Mfo = ⎣
4 fD L Dh
⎤ 12 ⎥ ⎦ ,
(5)
where γ is the ratio of specific heats (= c p /cv ), fD is the Darcy friction factor ( fD = 64/Re), L is the length of the restrictor channel, pfo is the pressure at the inlet of the channel, which is equal to the pressure in the discharge system (pcond ), and p is the pressure at the outlet of the channel, calculated by the PCC model in the feeding hole (surface c of Fig. 2). The mass flow rate in each restrictor channel is provided as a boundary condition for the PCC model. 3. Analysis of thermodynamic inefficiencies The parameters commonly adopted to evaluate the performance of compressors are the volumetric efficiency and isentropic efficiency. In the following sections, a method is presented to identify the main volumetric and isentropic inefficiencies associated with the flow in the piston-cylinder clearance of a small oil-free linear compressor. In order to support the next sections, Fig. 3 shows a diagram of the mass and energy fluxes between different regions of the compressor and Tables 1 and 2 indicate the input and output quantities associated with each model during the simulation. 3.1. Volumetric inefficiency The mass flow rate provided by the compressor (m˙ ) can be written as a function of the ideal mass flow rate, m˙ th = ∀sw ρsuc f , where ∀sw is the swept volume and ρ suc is the gas density in the suction line at the compressor entrance, and reductions associated with different effects:
∀BDC f − m˙ gsk
(6)
The second term on the right-hand side of Eq. (6) corresponds to the reduction in the flow caused by the dead volume (ηv, idc ). The terms m˙ gt p(exp) and m˙ gt p(suc ) represent the mean flow rates through the piston top clearance during the expansion and suction processes, respectively. A positive value for m˙ gt p indicates mass input into the compression chamber, i.e., leakage occurs from the gap to the compression chamber. This would represent a reduction in the mass flow rate, since a smaller amount of mass would enter the compression chamber during the suction process due to the mass intake from the piston-cylinder clearance. The fourth term expresses the loss due to the difference between the gas densities in the suction line and inside the cylinder at the beginning of the compression process (ρsuc − ρini,comp ). The variable ∀BDC is the volume of the compression chamber when the piston is positioned at the bottom dead center (BDC). The last term states for the mass flow loss concerning the leakage through the piston skirt. Eq. (6) disregards losses in the valves, since ideal suction and discharge processes were assumed in the present analysis, which focused on inefficiencies of the fluid flow through the piston-cylinder clearance. Dividing Eq. (6) by m˙ th and introducing the parameter ɛvk to represent the volumetric inefficiencies, i.e., the fraction of the total mass flow rate reduction due to a specific effect k (ε vk = m˙ k /m˙ th ), one obtains:
ηv = ηv,idc − ε vgt p,exp − ε vgt p,suc − ε vsh − ε vgsk ,
(7)
where ηv (= m˙ /m˙ th ) is the volumetric efficiency. The total volumetric inefficiency is
ε v = ε vgt p,exp + ε vgt p,suc + ε vsh + ε vgsk ,
(8)
3.2. Isentropic inefficiency The isentropic inefficiencies can be determined by calculating the amount of heat that should be withdrawn from the control volume (dashed line in Fig. 3) to make the entropy of the gas at the outlet equal to the entropy of the gas at the inlet. This means that the heat transfer rates indicated in Fig. 3 can be regarded as energy transfer rates that make the actual process different from the isentropic process. It should be noted that our study was restricted to evaluate the inefficiencies brought about by the fluid flow in the pistoncylinder clearance. Therefore, the electric motor losses were not taken into account in the analysis. As a result, the compressor power input is defined herein as W˙ = W˙ ind + W˙ f , where W˙ ind is the indicated power and W˙ f is the bearing loss due to the viscous friction on the piston, and the energy balance in the control volume of Fig. 3 yields:
W˙ ind + W˙ f + m˙ (hsuc − hdis ) = Q˙ gap + Q˙ suc + Q˙ dis ,
(9)
where hsuc and hdis represent the specific enthalpy at the inlet and outlet of the control volume, respectively. The term Q˙ gap is the heat rejection rate in the piston-cylinder clearance calculated
C.J. Santos, T. Dutra and C.J. Deschamps / International Journal of Refrigeration 104 (2019) 513–520
Fig. 4. Linear compressor geometric parameters – piston at the TDC.
Fig. 5. Comparison between the results obtained with the PCC model and experimental data reported by Suefuji et al. (1992).
Table 3 Geometric parameters. Parameter
Value
Parameter
Value
Parameter
Value
Lpist /Lcyl Lcfo1 /Lcyl Lcfo2 /Lcyl
0.82 0.42 0.78
Linstop /Lcyl Lcap1 /Dcap Lcap2 /Dcap
0.12 5.15 × 103 6.53 × 103
Dfo /Lcyl Dcap /Lcyl Dpist /Lcyl
1.60 × 10−2 8.40 × 10−4 0.31
4.1. Model validation
with the PCC model. The heat transfer rate Q˙ suc is evaluated by assuming that the specific enthalpy at the compression chamber entrance (hin, i ) is equal to that at the compressor inlet (hsuc ), which is known from the evaporating pressure, pevap , and suction line temperature, Tsuc , both prescribed. Finally, Q˙ dis is the heat rejection rate required to reach the thermodynamic state at the compressor outlet represented by the condensing pressure (pcond ) and specific entropy equal to that at the compressor inlet. Q˙ suc and Q˙ dis are determined from energy balances:
Q˙ suc = m˙ hsuc + m˙ gsk hgsk − m˙ in,i hin,i ,
(10)
Q˙ dis = m˙ out,i hout,i − m˙ f o hdis − m˙ hdis ,
(11)
where m˙ gsk and hgsk are predicted with the PCC model, m˙ in,i , m˙ out,i and hout, i are calculated with the CC model and m˙ f o is obtained from the restrictor channels model. Replacing the definitions of isentropic power, W˙ s = m˙ (hdis − hsuc ), and isentropic efficiency, ηs = W˙ s /W˙ , in Eq. (9) and introducing the parameter ɛsk to represent the increase in power input, i.e., isentropic inefficiency due to the specific effect k (ε sk = Q˙ k /W˙ s ), gives:
εs =
1
ηs
− 1 = ε sgap + ε ssuc + ε sdis
517
(12)
where ɛs is the total isentropic inefficiency. 4. Results Fig. 4 shows a schematic of the linear compressor, highlighting the most important geometric parameters. The line located at a distance Linstop from the valve plate represents the position around which the piston oscillates with an amplitude Lamp . Table 3 shows the values of these geometric parameters. The nominal dimension of the piston-cylinder clearance is δ = 5 μm. All simulations carried out in the present study considered the evaporating pressure of 114.8 kPa. Considering R134a as the refrigerant fluid, the evaporating temperature is 249.85 K. Moreover, based on experimental data, the temperature of the gas at the compressor inlet was prescribed as 305.15 K and the temperatures of the piston and cylinder walls were set at 328.15 K.
A preliminary study was carried out to verify the time and space discretizations required for the PCC model and the number of cycles necessary to reach the solution for the fully periodic operating condition. The Richardson extrapolation method was adopted (Cadafalch et al., 2002), with three levels of mesh refinement and a growth rate of 2. The truncation errors were analyzed based on the results for the viscous dissipation () and the total volumetric and isentropic inefficiencies (ε v and ε s). For the simulations, the following parameters were adopted: pcond = 1030.3 kPa; Tcond = 313.65 K; δ = 5 μm; f = 120 Hz; and Lamp /Lcyl = 0.215. It was found that the time step size should be set at 5.79 × 10−7 s and that 15 cycles should be simulated to reach the fully periodic operating condition. The best compromise between computational cost and solution accuracy was found with a mesh of 20 elements in the radial direction. We found that mesh refinements in the other directions are much less important for solution accuracy. The results of the PCC model were validated through comparison with the experimental data of Suefuji et al. (1992) for mass flow rate of R22 through a channel with a clearance (h). The length and width of the channel were 4.5 and 126 mm, respectively, whereas the clearance value and the inlet pressure were varied in the tests. The temperature was set at 300 K and the outlet pressure was kept at 100 kPa. The PCC model was adapted to match the fluid flow condition of the experiments, i.e. the restrictor channels and piston motion were not considered in the simulations. Fig. 5 presents results of mass flow rate for three clearances (h = 3.6 μm, 9.0 μm and 21.0 μm) as a function of the pressure ratio. As can be seen, the leakage mass flow rate predicted by the PCC model and the measurements of Suefuji et al. (1992) are in reasonable agreement, with a mean difference of 14%, and a maximum deviation of 35% occurring for the most challenging case for measurements (h = 3.6 μm). 4.2. Baseline operating condition The baseline operating condition is defined by the following parameters: pressure ratio = 8.97; pcond = 1030.3 kPa; Tcond = 313.65 K; δ = 5 μm; f = 120 Hz; and Lamp /Lcyl = 0.215. Data presented in Table 4 indicate a reduction of 54.0% in the mass flow rate due to the dead volume brought about by the small amplitude of Lamp /Lcyl for the baseline operating condition. On the other hand, the volumetric inefficiency resulting from the flow in the pistoncylinder clearance (ɛv) is only 0.79%, mostly attributed to the inef-
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Table 4 Mass flow rates and different sources of the volumetric inefficiency. Parameter
Value
Parameter
Value (%)
Parameter
Value (%)
m˙ th [kg/s] m˙ [kg/s] ηv [−]
2.20E-03 9.96E-04 45.25%
ɛvidc [-] ɛvgsk [-] ɛvgtp, suc [-]
53.96 0.21 0.18
ɛvgtp, exp [-] ɛvsh [-] ɛv [-]
0.39 0.01 0.79
Fig. 7. Rates of energy flow through the top clearance and heat transfer through the piston and cylinder walls.
Fig. 6. Instantaneous mass flow rates through the piston-cylinder clearance. Table 5 Power input and different sources of the isentropic inefficiency. Parameter
Value
Parameter
Value
Parameter
Value
W˙ s [W] W˙ ind [W] W˙ f [W]
60.65 61.37 0.13 98.61%
ηc [-] ηmec [-]
98.81% 99.79% 0.14% 0.12%
ɛsgap [-] ɛs [-] [W]
1.15% 1.41% 0.69
ηs [−]
ɛssuc [-] ɛsdis [-]
ficiencies ɛvgtp, suc and ɛvgtp, exp associated with leakage through the top gap (m˙ gt p ) during the expansion and suction processes. Fig. 6 shows the results for the mass flow rate through the restrictors and leakage through the top and skirt gaps throughout the compression cycle. Negative values indicate that the gas leaves the piston-cylinder clearance domain. The instantaneous leakage through the top gap reaches magnitudes much greater than those in the skirt gap or feeding orifices. In fact, the piston-cylinder clearance acts as a reservoir, receiving, accumulating and delivering mass to the compression chamber periodically. This transient behavior is relevant with regard to the volumetric efficiency. For instance, leakage from the top gap to the compression chamber during the expansion and suction processes shown in Fig. 6 reduce the amount of mass of refrigerant that can enter the compression chamber through the suction valve. On the other hand, a positive net value of m˙ gt p during the cycle contributes to an increase in m˙ gsk and hence a decrease in the volumetric efficiency. The results for the compressor power, isentropic efficiency and inefficiencies are reported in Table 5. The total inefficiency, ɛs, increased the power consumption (W˙ ind + W˙ f ) by 1.41% with reference to the isentropic power W˙ s . Only 0.21% of this value is due to the piston friction power W˙ f , also referred to as mechanical loss.
On the other hand, the compression efficiency, ηc = W˙ s /W˙ ind , undergoes a more significant change, greater than 1%. Table 5 also shows that the main factor responsible for the increase in the energy consumption is the viscous dissipation released as heat in the piston-cylinder clearance (ɛsgap ). Fig. 7 shows energy flow rates through the boundaries of the PCC indicated in Fig. 2. According to Fig. 7, the instantaneous values for the energy entering the top gap and the heat transfer at the walls are similar in magnitude but opposite in sign. This means that a great part of
Fig. 8. Instantaneous viscous dissipation rate integrated throughout the domain.
the energy entering the piston-cylinder top gap (positive value) is transferred to the walls and does not return to the compression chamber when the flow through the top gap changes direction. The results for the viscous dissipation rate () integrated throughout the solution domain of the PCC model during the compression cycle are shown in Fig. 8. The peak for occurs close to the TDC when the pressure is highest and hence the same occurs with the velocity magnitudes and gradients in the pistoncylinder clearance. The mean value for calculated over the cycle is 0.69 W, which corresponds to 75% of the energy that enters the gap (= 0.92 W) during the compression cycle. 4.3. Parametric analysis The compressor performance and thermodynamic inefficiencies were analyzed by varying four parameters: pressure ratio ( ), piston-cylinder radial clearance (δ ), operating frequency (f), and amplitude of oscillation (Lamp ). Each of these parameters was varied individually, while for the others the values adopted for the baseline operating conditions were maintained. It should be mentioned that experiments to validate the results of our parametric analysis is quite an intricate task because several phenomena affect the compressor efficiency and take place in a coupled manner, such as valve dynamics, suction gas superheating and motor losses. In fact, in order to isolate the inefficiencies brought about by the fluid flow in the piston-cylinder clearance from the inefficiencies associated with other phenomena, the simulations
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Fig. 9. (a) Volumetric and (b) isentropic inefficiencies as a function of pressure ratio.
Table 6 Numerical results as a function of pressure ratio.
Table 7 Numerical results as a function of radial clearance.
[-]
3.61
5.15
8.97
12.8
δ [μm]
2.5
3.0
4.0
5.0
7.5
10.0
m˙ [kg/s] W˙ ind [W] W˙ f [W]
1.78E-03 60.74 0.13 80.81% 81.10% 99.65% 99.86%
1.55E-03 68.77 0.13 70.26% 70.70% 99.46% 99.66%
9.96E-04 61.50 0.13 45.25% 46.04% 98.61% 98.81%
4.68E-04 35.20 0.12 21.22% 22.39% 95.69% 96.00%
m˙ [kg/s] W˙ ind [W] W˙ f [W]
1.01E-03 61.84 0.21 45.73% 46.04% 99.13% 99.47%
1.01E-03 61.76 0.18 45.64% 46.04% 99.07% 99.36%
1.00E-03 61.63 0.15 45.45% 46.04% 98.87% 99.11%
9.96E-04 61.50 0.13 45.25% 46.04% 98.61% 98.81%
9.80E-04 61.36 0.11 44.47% 46.04% 97.19% 97.37%
9.54E-04 61.20 0.11 43.28% 46.04% 94.91% 95.08%
ηv [-] ηv, idc [-] ηs [-] ηc [-]
were carried for ideal valves and without considering convective heat transfer in the compression chamber. 4.3.1. Pressure ratio The effect of the pressure ratio was analyzed by fixing the evaporating pressure and varying the condensing pressure. Simulations were carried out for the following pressure ratios ( ): 3.61, 5.15, 8.97 and 12.8. The results for the mass flow rate, indicated power, friction power and volumetric, compression and isentropic efficiencies are reported in Table 6. As expected, the increase in the pressure ratio gives rise to a decrease in ηv , because the portion of the swept volume required to expand the vapor left in the dead volume after the discharge process also increases. Therefore, vapor from the suction line is only able to fill the portion of the swept volume that has not been filled with the expanded vapor. The leakages in the piston-cylinder clearance also increase with increasing pressure ratio, contributing to the reduction in ηv . The indicated power (W˙ ind ) reaches a maximum when = 5.15, since the mass flow rate increases and the specific work of compression decreases with the pressure ratio. The compression and isentropic efficiencies, ηc and ηs , decrease in the same proportion (approximately 4% in the range analyzed), since the friction power is very small when compared to the indicated power. According to Fig. 9(a), the total volumetric inefficiency caused by the aerostatic bearing formed by the piston-cylinder clearance, ɛv, increases by almost 1% when varies from 3.61 to 12.8. The effect of leakage through the skirt gap, ɛvgsk , is significant. The volumetric efficiency is also reduced by the leakage into the cylinder during the expansion process, ɛvgtp, exp , which occurs because the pressure difference in the gap increases with the pressure ratio. The results in Fig. 9(b) show that the heat rejection rate in the piston-cylinder clearance, ɛsgap , is the most important source of isentropic inefficiency. For instance, a reduction of 4.3% in the isentropic efficiency is observed when the pressure ratio is
ηv [-] ηv, idc [-] ηs [-] ηc [-]
increased from 3.61 to 12.8. This effect is significant and has to be taken into account if oil-free linear compressors are designed for operating conditions with a high-pressure ratio. 4.3.2. Piston-cylinder radial clearance The effect of the piston-cylinder radial clearance of the linear compressor was evaluated for six values: δ = 2.5, 3.0, 4.0, 5.0, 7.5 and 10.0 μm. According to Table 7, the volumetric efficiency undergoes a reduction of 2.5% (ηv − ηv,idc ) and the compression and isentropic efficiencies decrease by more than 4% when the clearance is varied from 2.5 to 10.0 μm. Fig. 10(a) shows that all volumetric inefficiencies increase with the piston-cylinder clearance. In particular, the increase in the leakage through the skirt gap (ɛvgsk ) is a direct result of the greater leakage at the top gap (ɛvgtp, exp and ɛvgtp, suc ), since the mass flow rate through the feeding orifices does not vary with δ . The results in Fig. 10(b) indicate that the effect of the heat dissipated inside the piston-cylinder clearance, ɛsgap , is the most significant isentropic inefficiency. The leakage through the skirt gap is considerably increased with the clearance and therefore the same occurs with the heat rejection rate in the suction system (Q˙ suc ) and the inefficiency ɛssuc . Small clearances are associated with greater frictional power, W˙ f , due to the resistive viscous force on the piston wall. However, the effect of greater leakage as the radial clearance δ is increased results in a negative trade-off and the compressor efficiency is decreased. 4.3.3. Operating frequency and piston oscillation amplitude The inefficiencies were found to be less affected by the variation of the operating frequency and piston oscillation amplitude. For instance, when the operating frequency is varied from 60 to 240 Hz, the total volumetric inefficiency (ɛv) and isentropic inefficiency (ɛs) are decreased by only 0.5% and 0.9%, respectively. It is worth mentioning that the operating frequency is a fundamen-
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Fig. 10. (a) Volumetric and (b) isentropic inefficiencies as a function of radial clearance.
tal parameter regarding the motor efficiency of linear compressors. However, in the present study its effect was only analyzed regarding the inefficiencies associated with the fluid flow in the pistoncylinder clearance. As far as the oscillation amplitude is concerned, the total volumetric inefficiency remains approximately constant when the piston oscillation amplitude (Lamp /Lcyl ) is varied between 0.21 and 0.24. In turn, the total isentropic inefficiency is found to decrease by almost 1%, mainly due to the reduction of pistoncylinder clearance losses (ɛsgap ). 5. Conclusions A study was successfully carried out on the thermodynamic inefficiencies associated with the transient compressible fluid flow in the piston-cylinder clearance of an oil-free linear compressor. A convenient method was adopted to identify different sources of the thermodynamic inefficiency in the piston-cylinder clearance, such as viscous friction force on the piston surface, viscous dissipation associated with the flow in the clearance, and leakage at the piston skirt. The leakage through the skirt gap causes a portion of the refrigerant to circulate continuously inside the compressor, reducing the flow of gas supplied to the refrigeration system. Another significant leakage occurs from the top gap to the compression chamber during the expansion and suction processes, which reduces the amount of mass of refrigerant that can enter the compression chamber through the suction valve. This phenomenon was found to be the most significant source of volumetric inefficiency of the flow in the clearance. Regarding the isentropic efficiency, the viscous friction force on the piston had an almost negligible effect on the isentropic efficiency. On the other hand, the mechanical energy dissipated by viscous friction in the clearance is the most influential source of isentropic inefficiency. The compressor performance and thermodynamic inefficiencies were analyzed by varying the pressure ratio, piston-cylinder radial clearance, operating frequency and piston oscillation amplitude. The compressor efficiency was found to be mainly affected by the clearance and pressure ratio. A reduction of 2.5% in the volumetric efficiency and 4.2% in the isentropic efficiency, for instance, were predicted when the clearance was varied from 2.5 to 10.0 μm.
Acknowledgements This study forms part of a technical-scientific cooperation program between the Federal University of Santa Catarina and EMBRACO. Support from CNPq (grant 465448/2014–3) and EMBRAPII Unit POLO/UFSC is also acknowledged. References Bradshaw, C.R., Groll, E.A., Garimella, S.V., 2011. A comprehensive model of miniature-scale linear compressor for electronics cooling. Int. J. Refrig. 34, 63–73. Braga, V.M., Deschamps, C.J., 2017. Numerical analysis of gas leakage in the pistoncylinder clearance of reciprocating compressors considering compressibility effects. IOP Conf. Ser. Mater. Sci. Eng. 232, 012006. Cadafalch, J., Pérez-Segarra, C.D., Cònsul, R., Oliva, A., 2002. Verification of finite volume computations on steady-state fluid flow and heat transfer. J. Fluids Eng. 124 (1), 11–21 n. Davies, G.F., Eames, I.W., Bailey, P.B., Dadd, M.W., Janiszewski, A., Stone, C.R., Maidment, G.G., Agnew., B., 2010. Cooling microprocessors using vapour compression refrigeration. Thermal and thermomechanical phenomena in electronics systems (ITherm). In: Proceedings of the Twelfth IEEE Intersociety Conference, Las Vegas, pp. 1–8. Hülse, E.R., Prata, A.T., 2016. Numerical modeling of capillary compensated aerostatic bearing applied to linear reciprocating compressor. In: Proceedings of the International Compressor Engineering Conference at Purdue. West Lafayette Paper 2410. Jomde, A., Anderson, A., Bhojwani, V., Kedia, S., Jangale, N., Kolas, K., Khedar, P., 2018. Modeling and measurement of a moving coil oil-free linear compressor performance for refrigeration application using R134a. Int. J. Refrig. 88, 182–194. Lemmon, E.W., Huber, M.L., McLinden, M.O., 2002. NIST standard reference database 23: reference fluid thermodynamic and transport properties-REFPROP,version 8.0. National Institute of Standards and Technology, Gaithersburg, USA Standard Reference Data Program. Nishikawa, H., Nishikawa, T., 1998. Development of the non-lubricated four-stage compressor compressing up to 24.52MPa. In: Proceedings of the International Compressor Engineering Conference at Purdue, West Lafayette, pp. 183–188. Pérez-Segarra, C.D., Rigola, J., Oliva, A., 2002. Thermal and fluid dynamic characterization of hermetic reciprocating compressors. In: Proceedings of the International Compressor Engineering Conference at Purdue. West Lafayette Paper C4-2. Pérez-Segarra, C.D., Rigola, J., Sòria, M., Oliva, A., 2005. Detailed thermodynamic characterization of hermetic reciprocating compressors. Int. J. Refrig. 28, 579–593. Schreiner, J.E., Barbosa, J.R., Deschamps, C.J., 2010. Theoretical analysis of the volumetric efficiency reduction in reciprocating compressors due to in-cylinder thermodynamics. In: Proceedings of the International Compressor Engineering Conference at Purdue. West Lafayette Paper 1348. Suefuji, K., Shiibayashi, M., Tojo, K., 1992. Performance analysis of hermetic scroll compressor. In: Proceedings of the International Compressor Engineering Conference at Purdue, West Lafayette, pp. 75–84.