Structural optimal design of a swing vane compressor

Abstract

In this paper, a novel swing vane rotary compressor (SVC) was introduced, which had significant advantages—simple mechanism, reduced frictional loss, reliable operation, and a comparatively higher compression ratio. Based on the swing vane compressor geometry model, thermodynamic model and kinetic model, the mathematical model of optimum design was established, and further theoretical and experimental studies were conducted. The length of the cylinder, radius of the rotor and cylinder were defined as design variables and the reciprocal of EER as objective function. The complex optimization method was adopted to study the structure of the swing vane compressor. The theoretical model could provide an effective method for predicting compressor performance, which would also contribute to structural optimization of the SVC. The study shows that the friction loss of the compressor are greatly reduced by optimized design in a given initial value, and the EER increased by 8.55%.

Keywords

swing vane compressor / simulation / optimization design

Introduction

Rotary compressor has occupied a bigger share of the market in the field of refrigerator, air conditioner and gas-compressed device with compact size and better performance. For acquiring the optimization of structure, mathematical, theoretical and experimental studies have been conducted. Ooi and Teh [¹] introduced a new compressor, the improved revolving vane (RV-i) compressor and analyzed significant friction loss. Phang and Ooi [²] reported that simulation studies are related to the aspect of thermodynamic property of fluids, dynamic characteristics of valves and energy analysis of equipment. Teh and Ooi [³] presented a designed improvement to reduce the wear and friction at the vane side effectively by achieving a 30% reduction over its predecessor. Zong et al. [⁴] established the dynamic model of translational rotary compressor to discuss the natural frequency and vibration type by comparing the critical speed and actual speed. In order to solve the problem of rotary compressor friction loss more effectively and simplify the manufacturing process, Hu et al. [⁵] conducted a deep study of the swing vane compressor (SVC), which is characterized by simple structure, small size, light weight, stable operation, and efficient characteristics, compared with other rotary compressors.

To reduce the friction loss of the SVC and effectively improve the mechanical efficiency, simulations were conducted in this paper to optimize the structure of its main parameters. First, a thermodynamic model was established. Based on this model and the simulation result, the optimization design variables and objective function were chosen to get the ideal structural parameters by setting the constraint conditions.

Establishment of mathematical model

The mathematical model of the compressor was established considering the geometric parameters, the thermodynamic properties, the working flow characteristics, and the input power of the compressor.

Geometric model

As shown in Fig. 1, the working chamber of the SVC is surrounded by a cylinder, a rotor, swing vane and end covers, with the rotor in eccentric arrangement and the ex-circle of the rotor and the inner circle of the cylinder in the tangent. The tip of the vane is embedded in the cylinder by means of hinge joint, and the end inserted in the groove of the rotor. When the compressor operates, the motor drives the eccentric shaft to make the rotor revolve. Thus the chamber volume has a periodic change: suction, compression and exhaust, as illustrated in Fig. 2. During the operation, the chamber volume can be given as

(1)V(\theta )=H\left[{\displaystyle \frac{{{{\displaystyle R}}_{\mathrm{cy}}}^{\mathrm{2}}}{\mathrm{2}}}\theta -{\displaystyle \frac{{{\displaystyle R}}_{\mathrm{p}\mathrm{i}\mathrm{s}}^{\mathrm{2}}}{\mathrm{2}}}\left(\theta +\arcsin {\displaystyle \frac{\mathrm{e}\sin \theta }{\sqrt{{{{\displaystyle R}}_{\mathrm{c}\mathrm{y}}}^{\mathrm{2}}+{\mathrm{e}}^{\mathrm{2}}-\mathrm{2}\mathrm{e}{{\displaystyle R}}_{\mathrm{c}\mathrm{y}}\cos \theta }}}\right)-{\displaystyle \frac{\mathrm{e}{{\displaystyle R}}_{\mathrm{c}\mathrm{y}}}{\mathrm{2}}}\sin \theta -{\displaystyle \frac{b}{\mathrm{2}}}(\sqrt{{{{\displaystyle R}}_{\mathrm{c}\mathrm{y}}}^{\mathrm{2}}+{\mathrm{e}}^{\mathrm{2}}-\mathrm{2}\mathrm{e}{{\displaystyle R}}_{\mathrm{c}\mathrm{y}}\cos \theta }-{{\displaystyle R}}_{\mathrm{p}\mathrm{i}\mathrm{s}})\right],

Thermodynamic model

The change of the chamber volume causes the pressure, the temperature and the mass of fluid to change. If regarding the chamber volume as the control volume and the medium in the homogeneous state and taking the law of thermodynamics and mass conservation, the expression of pressure and mass can be given as

(2){\displaystyle \frac{\mathrm{d}p}{d\theta }}={\displaystyle \frac{{\displaystyle \frac{\mathrm{1}}{v}}[{\left({\displaystyle \frac{\partial h}{\partial v}}\right)}_T-{\left({\displaystyle \frac{\partial h}{\partial T}}\right)}_v{\left({\displaystyle \frac{\partial p}{\partial v}}\right)}_T/{\left({\displaystyle \frac{\partial p}{\partial T}}\right)}_v]{\displaystyle \frac{\mathrm{d}v}{\mathrm{d}\theta }}-{\displaystyle \frac{\mathrm{1}}{V}}\{{\displaystyle \sum [{\displaystyle \frac{\mathrm{d}{m}_{\mathrm{i}}}{\mathrm{d}\theta }}(h_{\mathrm{i}}-h)]}+{\displaystyle \frac{\mathrm{d}Q}{\mathrm{d}\theta }}\}}{\mathrm{1}-{\left({\displaystyle \frac{\partial h}{\partial T}}\right)}_v/\left[v{\left({\displaystyle \frac{\partial p}{\partial T}}\right)}_v\right]}},

(3){\displaystyle \frac{\mathrm{d}T}{\mathrm{d}\theta }}={\left({\displaystyle \frac{\partial T}{\partial v}}\right)}_p{\displaystyle \frac{\mathrm{d}v}{\mathrm{d}\theta }}+{\left({\displaystyle \frac{\partial T}{\partial p}}\right)}_v{\displaystyle \frac{\mathrm{d}p}{\mathrm{d}\theta }},

(4){\displaystyle \frac{\mathrm{d}{m}_{\mathrm{c}}}{\mathrm{d}\theta }}={\displaystyle \sum _{}^{}\frac{\mathrm{d}{m}_{\mathrm{i}}}{\mathrm{d}\theta }}-{\displaystyle \sum _{}^{}\frac{\mathrm{d}{m}_{\mathrm{o}}}{\mathrm{d}\theta }}.

In order to reduce the number of iterations and converge faster, the model is supposed to be ideally hermetic, ignoring the heat transfer and the leakage of the SVC operation. Besides, the medium flow through the valve is considered to be one- dimensional steady adiabatic state while discharging. The mass flow can be expressed as

(5){\displaystyle \frac{\mathrm{d}{m}_{\mathrm{2}}}{\mathrm{d}\theta }}={\displaystyle \frac{C_d\cdot A_{\mathrm{2}}\sqrt{\mathrm{2}(h_{\mathrm{1}}-h_{\mathrm{2}})}}{v_{\mathrm{2}}}}\omega ,

in which C_d indicates the discharge coefficient, representing the influence of non- isentropic flow loss.

Using the fourth order Runge-Kutta method to solve the above differential equations, the law of the working chamber pressure, temperature and flow mass can be obtained.

Analysis of energy consumption

The input energy of the SVC mainly contains of indicated work and friction consumption. According to the thermodynamics analysis, the indicated work of a working circle can be expressed as

(6)W_{\mathrm{i}}={\displaystyle \frac{n}{\mathrm{60}}}{\displaystyle {\int }_{\mathrm{0}}^{\mathrm{2}\pi }V(\theta )}\mathrm{d}p.

The indicated power can be calculated by

(7)P_{\mathrm{i}}={\displaystyle \frac{n}{\mathrm{60}}}{W}_{\mathrm{i}}.

One of the main factors affecting the performances of the SVC is the friction loss of the relative motion parts. In the simulation five kinds of friction losses are mainly considered, namely:

Frictional loss of sliding vaneL_{\mathrm{f,vs}}

(8)L_{\mathrm{f,vs}}={\eta }_{\mathrm{vs}}(\left|F_{\mathrm{1}}\right|+\left|F_{\mathrm{2}}\right|)U_{\mathrm{slide}}+(F_{\mathrm{vs,\tau }}+F_{\mathrm{vs,\tau }}^{\text{'}})U_{\mathrm{slide}},

where U_{\mathrm{slide}}is the velocity of the vane and can be given as

U_{\mathrm{slide}}={\displaystyle \frac{\omega R_{\mathrm{cy}}e\sin \theta }{\sqrt{R_{\mathrm{cy}}^2+e^{\mathrm{2}}-\mathrm{2}{R}_{\mathrm{cy}}e\cos \theta }}}.

Frictional loss of sliding vane-end

(9)L_{\mathrm{f,vt}}={\omega }_{\mathrm{vt}}{\eta }_{\mathrm{vt}}{R}_{\mathrm{vt}}\sqrt{F_{\mathrm{vt,\tau }}^{\mathrm{2}}+F_{\mathrm{vt,r}}^2}.

Radial frictional loss between rotor and cylinder

(10)L_{\mathrm{f,cl}}={\displaystyle {\int }_{{\theta }_{\mathrm{suc}}}^{{\theta }_{\mathrm{com}}}\mu {\displaystyle \frac{U(\theta )}{{\delta }_{\mathrm{cl}}(\theta )}}{U}_{\mathrm{cl}}{R}_{\mathrm{pis}}{L}_{\mathrm{cy}}}d\theta .

Frictional loss of rotor bearing

(11)L_{\mathrm{f,be}}=(\omega -{\omega }_{\mathrm{pis}})\left[{\displaystyle \frac{\mathrm{2}\pi \mu (\omega -{\omega }_{\mathrm{pis}})L_{\mathrm{be}}{R}_{\mathrm{ec}}^{\mathrm{3}}}{{\delta }_{\mathrm{be}}\sqrt{\mathrm{1}-{\epsilon }^{\mathrm{2}}}}}-{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{\delta }_{\mathrm{be}}\epsilon F_{\mathrm{ec}}\sin \varphi \right].

Frictional loss of end face

(12)L_{\mathrm{f,ef}}={\displaystyle \frac{\pi \mu }{{\delta }_{\mathrm{ef}}}}[(R_{\mathrm{pis}}^{\mathrm{4}}-R_{\mathrm{ec}}^{\mathrm{4}}){(\omega +{\omega }_{\mathrm{pis}})}^{\mathrm{2}}+\mathrm{2}{e}^{\mathrm{2}}{\omega }^{\mathrm{2}}(R_{\mathrm{pis}}^{\mathrm{2}}-R_{\mathrm{ec}}^{\mathrm{2}})].

The sum of the above losses exactly equals the total friction loss

(13)P_f={\displaystyle \frac{n}{\mathrm{60}}}{\displaystyle \sum L_{f,i}},

and the total power consumption can be respectively expressed as

(14)P_{\mathrm{tot}}=P_i+P_f.

Structural optimization

Obviously, the optimization of the SVC is actually a complex problem under multiple constraints, the nature of which is the process of looking for the best results in the feasible region. Therefore, the complex method can be adopted to optimize the calculation.

Mathematical description

According to the engineering optimization theory, the structural optimization of the SVC can be converted into a mathematical description as

(15)\{\begin{array}{c}\min f\left(X\right)={{\displaystyle \left[{{\displaystyle x}}_1,{{\displaystyle x}}_2\dots ,{{\displaystyle x}}_n\right]}}^T,x\in {{\displaystyle E}}_n,\\ {{\displaystyle g}}_i\left(x\right)\le 0\left(i=1,2,\dots ,m\right),\\ {{\displaystyle l}}_j\le {{\displaystyle x}}_j\le {{\displaystyle u}}_j\left(j=1,2,\dots n\right).\end{array}

As a result, the optimization design of the SVC is transformed into seeking for a set of variables X_n in the feasible design multi-dimensional space to make the objective function f(x) achieve the minimum value. The geometric description can be demonstrated in Fig. 3.

Structure optimization

The purpose of this paper is to reduce the energy consumption of the SVC after optimization, thus to seek an optimum combination of structural parameters with a higher EER at given conditions. The corresponding mathematical description is shown as

(16)

In order to obtain the maximum EER of the SVC, the reciprocal of the EER of any design variables is considered the objective function, namely

(17)F_{\mathrm{obj}}={\displaystyle \frac{\mathrm{1}}{\mathrm{E}\mathrm{E}\mathrm{R}}}={\displaystyle \frac{\mathrm{1}}{Q_{\mathrm{0}}/P_{\mathrm{tot}}}}={\displaystyle \frac{P_{\mathrm{tot}}}{Q_{\mathrm{0}}}},

where P_tot is shown in Eq. (14) and Q₀ is the cooling capacity of the compressor.

x(i),i=\mathrm{1},\mathrm{2},\mathrm{3} in Eq. (16) of the constraint conditions, as design variables, are supposed to meet a certain scope, which are the limits of the upper and lower parameters in Eq. (16). Table 1 lists the limit values of the basic design parameters for a refrigerating capacity of 2200 W.

Simulation and optimization analysis

According to the established thermodynamic model, the numerical simulation of the SVC has been completed. Because the changes of the working chamber pressure, temperature and mass flow rate are quite small with the rotational angle of the rotor in the process of suction, this paper just conducted a study of the compression process and the exhaust process.

As depicted in Fig. 4, the pressure variation with the rotation angle and pressure fluctuation in the process of exhaust is caused by the movement characteristic of discharge valve plate. Figure 5 displays the change law between the chamber pressure and volume. The area surrounded by the curve and axis is just equal to the indicated work of the SVC. The diagram of temperature-angle (Fig. 6) demonstrates that the changing characteristic of temperature is consistent with the pressure. The variation of mass in the working chamber is shown in Fig. 7. The model is assumed to be ideally hermetic during compressing, and the mass stays almost unchanged. When the drive angle is at about 258°C, the mass of the chamber gradually decreases with an opened valve.

The purpose for the SVC optimization design is to find the optimal structural parameters in order to achieve the best performance. In the model, the structural parameters R_cy, R_pis, and H are chosen to be design variables. Table 2 tabulates the initial parameter values and calculated results. The feasible initial parameter values can be randomly set between the upper and lower limits of the feasible region. In each step of the computation, the difference between the design variables and objective function, namely the difference between the best and worst point in the compound shape, constitutes the residual of computation\left|\right|x_{\max }-x_{\min }|{|}_{\mathrm{2}}, and|f_{\max }-f_{\min }|is defined as the calculation error. When the calculation error is less than 10^-4 in the simulation, the optimization design can be considered to be converged. According to the result, the optimal design parameters can be gained when the number of iteration reaches 86, with the corresponding EER improved by 8.50%, from 1.9476 to 2.1132.

Figure 8 indicates that the values of objective function change with the number of computing iteration. It is obviously that along with the iteration, the EER has been improved with the value of objective function decreasing, especially before the number of iteration adds up to 30. Subsequently, the result change of computation becomes smaller and design variables also tend to be a set of constant values, R_cy=64.6 mm, R_pis=55.9 mm, and H=29.6 mm. Figures 9 and 10 show the change of computing residual which eventually converges to a certain value. It can be seen in Fig. 11 that the values of the EER during the optimization vary that with the values increasing, and the EER finally reaches the optimized variable of 2.1132 from the initial values of 1.9476.

Conclusions

According to the basic structure of the SVC and the relevant theory of thermodynamics and kinetics, the basic optimization parameters of R_cy, R_pis, and H can be determined. After choosing the appropriate objective function and the corresponding computing method under the restricted reasonable conditions, the mathematical model of optimum design is established. The friction loss of optimized compressor is reduced dramatically. Based on the given constants and variable initial values, it can be concluded that the EER increases by 8.55%, which provides theoretical foundation to improve the performance of the SVC.

References

[1] Ooi K, Teh K. Design improvements of revolving vane compressors, RV-i. Fortune, 1987: 51–69

[2] Phang G, Ooi K T. Simulations of reciprocating compressors: the state of the art. In: International Conference on Mathematical Modelling Brunei. 1995: 334–338

[3] Teh Y, Ooi K. Design and friction analysis of the improved revolving vane compressor. Behavioural Brain Research,  2002,  128(2): 161–168

[4] Zong W B, Qin Y. Mao X M, QuZ C, Ma X H. The force analysis and study of the critical speed for sliding vane of translational rotary compressor. Compressor Technology, 2015: 11–16

[5] Hu X, Qu Z, Yang X, Sun J. Theoretical study on frictional losses of a novel automotive swing vane compressor. International Journal of Refrigeration,  2013,  36(3): 758–767

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Higher Education Press and Springer-Verlag Berlin Heidelberg