In this work, the lubrication regime of a rotary compressor is studied in the regions of the roller, cylinder, shaft, and bearings. A code is programmed so it can solve multiple cases with different suction-discharge pressures, angular velocity, viscosity, and geometrical conditions, among others. In order to study the lubrication regime, the balance of the different forces that act on the roller is computed. To calculate the oil pressure forces, the Reynolds equation coupled with an iterative method is used to find the minimal distance and pressure distribution that balances the external forces. This process is repeated for a discrete number of rotation angles to evaluate the overall rotation of a cycle. In addition, the effect that the thermal expansion of the different materials can have on the lubrication regime is studied. If the materials get too hot, the distances between surfaces might increase or reduce with respect to the distances at room temperature. These changes might lead to a point where the lubrication regime changes. The objective is to include this thermal expansion effect in the model and evaluate how it can affect the lubrication regime.The author Jordi Vera has been financially supported by the Ministerio de Educación y Ciencia (MEC), Spain, (FPI grant PRE2018-084017). The author E.Schillaci acknowledges the financial support of the Programa Torres Quevedo (PTQ2018-010060).Postprint (published version

Rigola, Joaquim

Schillaci, Eugenio

Vera, Jordi

English

Vera i Fernández, Jordi

Rigola Serrano, Joaquim

UPCommons. Portal del coneixement obert de la UPC

Evaluation of the Lubrication Regime for Rotary Compressors. Influence of Thermal Expansions on the Minimum Film Thickness.

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Purdue University Purdue e-Pubs International Compressor Engineering Conference School of Mechanical Engineering 2022 Evaluation of the Lubrication Regime for Rotary Compressors. Influence of Thermal Expansions on the Minimum Film Thickness. Jordi Vera Eugenio Schillaci Joaquim Rigola Follow this and additional works at: https://docs.lib.purdue.edu/icec Vera, Jordi; Schillaci, Eugenio; and Rigola, Joaquim, "Evaluation of the Lubrication Regime for Rotary Compressors. Influence of Thermal Expansions on the Minimum Film Thickness." (2022). International Compressor Engineering Conference. Paper 2761. https://docs.lib.purdue.edu/icec/2761 This document has been made available through Purdue e-Pubs, a service of the Purdue University Libraries. Please contact epubs@purdue.edu for additional information. Complete proceedings may be acquired in print and on CD-ROM directly from the Ray W. Herrick Laboratories at https://engineering.purdue.edu/Herrick/Events/orderlit.html 2364 , Page 1Evaluation of the lubrication regime for rotary compressors.Influence of thermal expansions on the minimum film thickness.Jordi VERA1, Eugenio SCHILLACI2*, Joaquim RIGOLA11 Universitat Politècnica de Catalunya - Barcelona Tech (UPC),ESEIAAT, C/ Colom, 11 - 08222 Terrassa (Barcelona), Spain2 Termo Fluids S.L.,Carrer Magi Colet 8, Sabadell (Barcelona), Spain* Corresponding Author (eugenio@termofluids.com)ABSTRACTIn this work, the lubrication regime of a rotary compressor is studied in the regions of the roller, cylinder,shaft, and bearings. A code is programmed so it can solve multiple cases with different suction-dischargepressures, angular velocity, viscosity, and geometrical conditions, among others. In order to study thelubrication regime, the balance of the different forces that act on the roller is computed. To calculate the oilpressure forces, the Reynolds equation coupled with an iterative method is used to find the minimal distanceand pressure distribution that balances the external forces. This process is repeated for a discrete number ofrotation angles to evaluate the overall rotation of a cycle. In addition, the effect that the thermal expansionof the different materials can have on the lubrication regime is studied. If the materials get too hot, thedistances between surfaces might increase or reduce with respect to the distances at room temperature.These changes might lead to a point where the lubrication regime changes. The objective is to include thisthermal expansion effect in the model and evaluate how it can affect the lubrication regime.1. INTRODUCTIONThe rotary compressor is an element that compresses a refrigerant using rotary motion. An oil that is mixedwith the refrigerant is used to lubricate moving parts. A simplified diagram of the rotary compressor andstudied lubricated contact surfaces is shown in Figure 1.The objective of this work is to create a model capable of studying the lubrication regime of a rotarycompressor. This lubrication regime depends mainly on the forces acting on the compressor, the geometry,the rotating speed, and the oil viscosity. Other parameters also play an important role in the model. Severalmodels have been made regarding the study of the lubrication regime between surfaces. A common andsimple approach is to use the curves developed by (Raimondi & Boyd, 1958). These curves can give theminimum film thickness given a force and some geometrical and operational parameters. Another approachis to compute the forces and numerically solve the Reynolds equation for each case and determine thelubrication regime. Some examples of this approach are shown in (Ito, Hattori, & Miura, 2010), (Zhou,2012) and (Noh et al., 2016), among others. The complexity and accuracy of these types of models isincreased by adding more parameters. For example in (Mi & Meng, 2014) the movement of the roller is alsoevaluated with the equations of motion.19th International Refrigeration and Air Conditioning Conference at Purdue, July 10-14, 20222364 , Page 2CylinderSuction sideDischarge sideVaneyxωsCamShaftRoller(a) Rotary compressor geometry, front viewεyzBottom bearingTop bearingRollerCamShaft(b) Shaft and cam contact surfacesFigure 1: Rotary compressor geometry and lubricated partsThe main idea is that the pressure force of the oil compensates for the acting forces, however, as the forceincreases, the distance between surfaces decreases and the regime of lubrication might change, as shown inFigure 2.F⃗Hydrodynamic lubrication Λ > 3F⃗Mixed lubrication Λ ∼ 1− 3F⃗Boundary lubrication Λ < 1Figure 2: Oil lubrication and film thickness for different loadsThe regime of lubrication is defined as the relation between the minimum film thickness and the surface19th International Refrigeration and Air Conditioning Conference at Purdue, July 10-14, 20222364 , Page 3rugosity R of each surface.Λ = hmin/√R21 +R22 (1)The chosen approach considers a known roller movement, and the balance of forces at the roller, shaft,and bearings is used to determine the minimum film thickness by solving the Reynolds equations. Usingthe developed model, a parametric study is carried out to study the influence of thermal expansions in thelubrication regime.The calculations are performed for each angle of rotation θ. This angle corresponds to the rotation positionof the shaft. The definition of this angle is shown in Figure 3θθFigure 3: Shaft angle θ definitionThe paper is distributes as follows: the numerical set-up is described in Sec. 2, the algorithm is explainedin Sec. 3, then an example test case is shown in Sec. 4, finally this case is modified to evaluate the influenceof thermal expansions at 4.2.2. NUMERICAL SET-UPThe objective is to evaluate the lubrication regime for each studied contact surface. The first step for eachsurface is to solve its pressure distribution a given hmin/c value. This is done solving the Reynolds Equation(Section 2.1). This pressure distribution has to be integrated to obtain the reacting force. The processis repeated for multiple hmin/c, obtaining a curve of the force as a function of the eccentric displacementF (hmin). Then performing a force balance, as shown in Section 2.2, the minimum film thickness can befound for each rotation angle θ.2.1 Reynolds equationThe Reynolds equation (2) gives us the pressure distribution.∇ ·(h3µ∇p)= 6Ω∂h∂θ′(2)Where h is the distance between surfaces at a given position, p the pressure, µ the viscosity, and Ω therelative angular velocity between surfaces.19th International Refrigeration and Air Conditioning Conference at Purdue, July 10-14, 20222364 , Page 4To solve the equations, they are discretized using the finite volume method, using the 2D Gauss theo-rem. ∫S∇ ·(h3µ∇p)dS =∫S6Ω∂h∂θ′dS →∫Ch3µ∇p · n⃗ dC =∫S6Ω∂h∂θ′dS (3)Considering the discrete form we have:h3eµpi+1,j − pi,j∆z′2+h3wµpi−1,j − pi,j∆z′2+h3nµpi,j+1 − pi,j(R∆θ′)2+h3sµpi,j−1 − pi,j(R∆θ′)2= 6Ω∂h∂θ′(4)Given a Nθ ×Nz mesh, Expression (4) gives a Nθ ×Nz linear system of equations that can be solved usinga linear solver (e.g. Gauss-Seidel).A special case has to be considered in the solution for the system of equations, when the pressure givesnegative values, the solution at that point is non-physical. In general, this zone is considered a cavitationzone, so the pressure will be artificially threshold to only positive values and set to zero negative values, asshown in Figure 4.x′y′ωCavitationregionx′y′ϕF⃗CavitationregionωFigure 4: Pressure distribution (left) and global force and angle with respect to hmin (right)The solution of the Reynolds equations give us a pressure distribution field on the surface as shown in Figure5.Figure 5: Pressure distribution field for a bearing. The line indicates the minimum film thickness position.19th International Refrigeration and Air Conditioning Conference at Purdue, July 10-14, 20222364 , Page 52.2 Force balancesIn order to calculate the pressure force of the refrigerant, the pressure is integrated over all the surface. Thepressure at the suction chamber is considered constant. The pressure at the discharge chamber is a functionof the instantaneous chamber volume, which is a function of the shaft angle θ.P2(θ) = Ps(Vθ0Vθ)kif P2 < Pd (5)P2(θ) = Pd otherwise (6)Where Ps is the suction pressure, Pd the discharge pressure, Vθ0 is the volume of the chamber at the initialposition (θ = 0), Vθ is the volume of the chamber at the current position, and k is the Polytropic compressionexponent of the refrigerant.The normal vane force can be easily calculated using Hooke’s law.F⃗vane = −k ·∆x (7)The inertial force F⃗ rI is given by the angular velocity, mass and eccentricity.Finally the reaction force is unknown, but can be obtained with the summation of all the other forces.F⃗R = −(F⃗vane + F⃗P1 + F⃗P2 + F⃗oil + F⃗rI)(8)Having the roller force, the reaction force of the bottom and top bearings can be calculated as follows:F⃗b + F⃗t = F⃗cmI − F⃗R (9)A diagram of acting forces is shown in Figure 6.F⃗oilF⃗RF⃗ rIF⃗vaneF⃗(n)vaneF⃗P1F⃗P2(a) Forces acting on the RollerF⃗RF⃗bF⃗t(b) Shaft applied forcesFigure 6: Acting forces on the rotary compressor19th International Refrigeration and Air Conditioning Conference at Purdue, July 10-14, 20222364 , Page 62.3 Roller dynamicsThe shaft movement is an imposed parameter, it consists of a rotation along its axis characterized by theangle of rotation θ. It will rotate at a given speed θ̇ = −ωs. This speed is known.The roller has two degrees of freedom, one would be the angle θ (shaft rotation) and the other would be anangle ψ (angle between the roller and the shaft). The evolution of this parameter ψ is unknown and willdepend on the forces acting on the roller. We will call the angular speed of the roller ψ̇ = −ωr.The proper way to determine the roller movement would be to formulate the movement equations, solvethis equations numerically, and get ψ(t), ψ̇(t), ψ̈(t). However the oil film forces are also unknown, and theydepend on the roller movement, so we would have to make a coupled solver that solves the movement andthe forces together. This is possible and could be archived using iterative methods, as it is done in (Mi &Meng, 2014), however, this would increase the complexity of the code and formulation, so this approach isdiscarded.From some papers that performed numerically and experimental analysis (Yanagisawa, Shimizu, Chu, &Ishijima, 1982) regarding the roller movement, we can conclude that we will have at most ωr ≈ ±0.1ωsdepending on the angle and the dynamics. To simplify the model and avoid solving the equations of motionof the roller, which would lead to the need of coupling the movement with the Reynolds equations and do aniterative algorithm, we will consider that the roller angular velocity ωr = 0. The relative velocity betweenthe roller and the shaft will be ωs (Shaft angular velocity).3. ALGORITHMA first preprocess described in Algorithm 1 is performed for each lubrication surface. This will save theacting force as a function of the displacement on a bearing solving the Reynolds equations.Algorithm 1 Finding S(hmin) curves1: Start in equilibrium condition (displacement δ = 0)2: while δ < 1 do3: Solve the Reynolds equations to get the pressure distribution p(θ′, z)4: Integrate the pressure distribution to get the force F (δ) and angle ϕ(δ)5: Calculate Sommerfeld number6: Advance a displacement δi = δi +∆δi7: end whileThen the main algorithm of the code is19th International Refrigeration and Air Conditioning Conference at Purdue, July 10-14, 20222364 , Page 7Algorithm 2 Main algorithm1: Input physical parameters2: Input numerical parameters3: Find S(hmin) for bottom bearing/shaft to save F1(δ1) and ϕ1(δ1)4: Find S(hmin) for top bearing/shaft to save F2(δ2) and ϕ2(δ2)5: Find S(hmin) for the roller/cam to save F3(δ3) and ϕ3(δ3)6: δi = δi +∆δi7: Join curves from bottom (line 3) and top (line 4) bearings into F⃗12(δ12)8: θ = 09: while θ < 360 do10: Calculate known acting forces (F⃗ rI , F⃗cmI F⃗P1, F⃗P2 and F⃗vane)11: Calculate roller force F⃗r and find δ3 interpolating from F3(δ3) (line 5)12: Calculate total bearing forces (F⃗b + F⃗t) and find δ12 interpolating from F12(δ12) (line 7)13: Decompose the bearing displacements δ12 → ξ12 → (δ1, δ2)14: Calculate bottom bearing force F⃗b interpolating from F1(δ1) (line 3)15: Calculate top bearing force F⃗t interpolating from F2(δ2) (line 4)16: Align the forces and find ξ⃗s, ξ⃗rs, ξ⃗r17: Calculate F⃗oil from ξ⃗r and verify is small enough (check hypothesis)18: Store δmin(i) and calculate the lubrication regime for each surface Λ(i)19: Print results20: Advance the shaft angle θ = θ +∆θ21: end while22: Print final results and post-processing4. RESULTSIn this section first the results of a chosen reference case are shown. Then this same case but with theparametric variation is studied.4.1 Reference case resultsIn Figure 7 the forces acting on the roller are presented. Figure 8 shows the lubrication curve for the Roller-Cam interface. Finally, Figure 9 shows the lubrication regime for the bottom bearing, top bearing, roller,and cylinder. As it can be observed, the main driving force is the pressure force.Figure 7: Acting forces on the roller as a function of the rotation angle19th International Refrigeration and Air Conditioning Conference at Purdue, July 10-14, 20222364 , Page 8Figure 8: Lubrication regime Λ for the Roller-Cam interfaceFigure 9: Lubrication regime Λ for the Roller-Cam interface4.2 Parametric case resultsFor the Roller, made of a gray cast iron, a thermal expansion coefficient of α2 = 10.5 µm/(m◦C) is considered.For the Shaft, made from a ductile cast iron, a thermal expansion coefficient of α1 = 12 µm/(m◦C) is used.The changes of diameter due to temperature differences are given by equation (10)L′ = L(1 + α∆T ) (10)Let’s assume that the Roller-Cam interface have a diameter of 20 mm with a clearance of 0.02 mm (20 µm)Table 1: Clearance variation as a function of temperatureTemperature Inner diameter D1 Outer diameter D2 Clearance c c variation c/c020 ◦C 19.980 mm 20.000 mm 0.0200 mm 100 %40 ◦C 19.985 mm 20.004 mm 0.0192 mm 96 %60 ◦C 19.990 mm 20.008 mm 0.0185 mm 92 %80 ◦C 19.995 mm 20.012 mm 0.0177 mm 88 %100 ◦C 20.000 mm 20.017 mm 0.0170 mm 85 %19th International Refrigeration and Air Conditioning Conference at Purdue, July 10-14, 20222364 , Page 9As shown in Table 1, in this case we could get a 15% reduction of the clearance distance. In the resultsshown in Table 2 the following changes of lubrication regime and friction forces are observed:Table 2: Results of the lubrication regime Λ and friction force on the roller as a function of temperatureTemperature Lubrication regime Λ Friction force20 ◦C 0.98 0.757 N40 ◦C 1.00 0.767 N60 ◦C 1.02 0.776 N80 ◦C 1.05 0.786 N100 ◦C 1.05 0.796 NIn this case in particular we observe that the lubrication regime improves from boundary lubrication (Λ < 1)to mixed lubrication. The changes are about 7%. We can also observe this behaviour in Figure 10. As theclearance decreases, the Sommerfeld number increases as it is proportional to 1/c2, so the working regimecurve is displaced to the right. Even if the lubrication regime improves, we can observe that the frictionforce between the roller and cam increases up to a 5%. This simulation was performed for some specificconditions, so other conclusions might arise if testing another case, but the general conclusion is that theeffect of thermal expansion is a parameter that will have some effect on the simulation results. For simplicityin most models it can be neglected, but including this variation would improve the accuracy without theneed of increasing the computational cost or complexity of the model.Figure 10: Change of working regime from thermal expansion between room temperature (left) and surfacesat 100 ◦C (right) keeping the rest of variables constant.5. CONCLUSIONSIn general, the acting forces on the roller are mainly driven by pressure forces. Other forces are also consideredbut could be neglected if a simpler model was to be made. In the specific case studied case, it was observedthat the increase in temperature improved the lubrication regime. This is because the smaller the clearance,the more pressure a bearing can support. However, even if the lubrication regime improves, an increase inthe friction force was observed. In conclusion, the thermal expansion can have some effect on the lubricationregime and friction between surfaces, and its addition to the model can be done straightforwardly whenassuming known and constant element temperatures.19th International Refrigeration and Air Conditioning Conference at Purdue, July 10-14, 20222364 , Page 10REFERENCESIto, Y., Hattori, H., & Miura, K. (2010). Numerical analysis for rotating motion of a rolling piston in rotarycompressors-effective factors for characteristics of rotating motion of a rolling piston.Mi, J., & Meng, Y. (2014). Numerical analyses of hydrodynamic lubrication and dynamics of the rollingpiston and crankshaft in a rotary compressor. Tribology Transactions, 57 (6), 1136–1147.Noh, K.-Y., Min, B.-C., Song, S.-J., Yang, J.-S., Choi, G.-M., & Kim, D.-J. (2016). Compressor efficiencywith cylinder slenderness ratio of rotary compressor at various compression ratios. International journalof refrigeration, 70 , 42–56.Raimondi, A., & Boyd, J. (1958). A solution for the finite journal bearing and its application to analysisand design: Iii. ASLE Transactions, 1 (1), 194–209.Yanagisawa, T., Shimizu, T., Chu, I., & Ishijima, K. (1982). Motion analysis of rolling piston in rotarycompressor.Zhou, X. (2012). Minimum viscosity for bearing reliability in r290 rotary compressors.ACKNOWLEDGMENTThe author Jordi Vera has been financially supported by the Ministerio de Educación y Ciencia (MEC),Spain, (FPI grant PRE2018-084017). The author E.Schillaci acknowledges the financial support of thePrograma Torres Quevedo (PTQ2018-010060).19th International Refrigeration and Air Conditioning Conference at Purdue, July 10-14, 2022

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Evaluation of the Lubrication Regime for Rotary Compressors. Influence of Thermal Expansions on the Minimum Film Thickness.

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