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Article Excerpt
INTRODUCTION
Previous researchers have analyzed the compression process of single-stage rolling-piston compressors. Several have focused on providing a detailed analysis of the compression chamber geometry and the motion of the rolling piston in the cylinder (Okada and Kuyama 1982; Yanagisawa et al. 1982). The motion analysis includes a force balance on the compressor, which proves essential for determining the mechanical efficiency based on frictional losses. Other papers explore the topic of frictional losses in rolling-piston compressors in more detail, but for this project a constant mechanical efficiency was assumed, eliminating the need for a force analysis.
Another topic that many researchers have explored is the refrigerant and oil leakage that occurs in the compressor. Because leakage interactions between the suction and compression chambers and the shell can have a large impact on the compressor efficiency, it is a very important topic. Yanagisawa and Shimizu (1985a, 1985b) focused on the leakage through the radial clearance between the roller and the cylinder and leakage across the roller face. Lee and Min (1988) combined a study of leakage losses and frictional losses to better understand sources of inefficiencies in the compressor. A similar study on optimal compressor design based on minimizing the effects of leakage and friction losses was performed by Costa (1986). Costa et al. (1990) also experimentally studied the flow patterns through leakage paths to develop a new leakage model.
Heat transfer from the cylinder to the refrigerant gas is also a significant source of inefficiencies in the compressor and, thus, is the focus of many other studies. No new correlations have been developed to characterize the heat transfer in a rolling-piston chamber, so researchers have proposed different methods of modeling this process. Shimizu et al. (1980) suggested using Dittus and Boelter's formula for the heat transfer coefficient, while Padhy and Dwivedi (1994) treated the suction chamber as a circular duct and used a correlation for reciprocating compressors in the compression chamber. Ishii et al. (2000a) focused on the heat transfer from the thrust plates on the top and bottom of the chamber to the gas. The correlation selected for this project, originally developed for spiral plate tube heat exchangers, was demonstrated for a scroll compressor by Chen et al. (2002a).
Though several researchers have combined the topics of friction, leakage, and heat transfer losses to develop models for single-stage rolling-piston compressors, analysis of two-stage compressors is limited. Mechanical friction losses have been considered (Jun 2002), but no analysis pulls together the friction, leakage, and heat transfer losses for a two-stage model. Because of the potential for energy savings through intercooling or economizing between stages, the development of a two-stage model that can consider these different system configurations is important. As the demand for energy-efficient air-conditioning and refrigeration equipment increases and companies seek to incorporate two-stage compressors into systems, engineers will need models that can be used to develop optimized two-stage compressor designs.
This paper presents a complete model for a hermetic two-stage rotary compressor. Results from the simulation model are compared to external compressor measurements, which were conducted using an available compressor load stand (Chen et al. 2002b). The model was also used to study the performance of the existing compressor in order to understand the relative importance of different leakage paths and the impact of intercooling on performance.
MODELING EQUATIONS
Volumes of the Chambers
The rolling-piston compressor uses an eccentric roller contained in a cylinder to form the suction and compression chambers, which are separated by a vane that extends from the cylinder wall to the roller surface. The geometry of the rolling piston and cylinder is shown in Figure 1, with the compression chamber shaded. In this diagram, the roller is drawn to follow a counterclockwise path, with the suction port located to the left of the vane and the discharge port to the right of the vane. The crankshaft angle, [theta], is defined as the angle between the vane slot and the point of contact between the rolling piston and the cylinder wall. The angle is measured across the suction chamber. Thus, at small crankshaft angles, which will be considered the beginning of the crankshaft revolution, the volume of the suction chamber is small.
[FIGURE 1 OMITTED]
Because the suction port does not have a valve, gas continuously enters the suction chamber as its volume increases over an entire revolution of the rolling piston. The volume of the shaded compression chamber, located opposite the suction chamber, decreases as the suction volume increases. At the beginning of the crankshaft revolution, the compression chamber is open to the suction port and refrigerant can flow out of the compression chamber to the suction pipe. After the rolling piston rotates past the suction port, the refrigerant mass is sealed in the compression chamber, and the pressure increases until the valve in the discharge port opens. Refrigerant then flows through the discharge port to a muffler. From the first-stage muffler, the refrigerant is piped outside of the shell to enter the suction pipe of the second stage. If the compressor is operating with intercooling, the refrigerant will also pass through a cooling coil before entering the second-stage suction pipe. If the compressor is operating with economizing, saturated vapor or two-phase refrigerant mixes with the first-stage discharge gas before entering the second-stage suction chamber. Economizing was not considered in the study presented here. From the second-stage muffler, the refrigerant enters the high-side shell, flows over the motor, and exits through a discharge pipe at the top of the shell. For this analysis, the gas inside the shell is separated into two control volumes; the volume below the motor, which includes the volume surrounding the compression cylinders, will be called the lower cavity, while the volume surrounding and above the motor will be called the upper cavity. The shell is divided into these control volumes for the purpose of modeling the heat transfer to gas in the upper cavity due to motor and mechanical inefficiencies.
The volume of the compression chamber can be calculated using the known dimensions of the compressor and the calculated vane extension:
V = [pi][H.sub.c]([R.sub.c.sup.2] - [R.sub.r.sup.2]) - [[H.sub.c]/2][[R.sub.c.sup.2][theta] - [R.sub.r.sup.2]([theta] + [alpha])] + [[H.sub.c]/2]e([R.sub.r] + [r.sub.v])sin([theta] + [alpha]) - [[H.sub.c]/2][r.sub.v.sup.2]tan[alpha] - [[H.sub.c]/2]bx (1)
where the distance that the vane extends into the cylinder, x, can be calculated as
x = [R.sub.c] + [r.sub.v] - ([R.sub.r] + [r.sub.v])cos[alpha] - ecos[theta] (2)
and
[alpha] = [sin.sup.-1]([e/[[R.sub.r] + [r.sub.v]]]sin[theta]) (3)
The resulting variation of volume with crankshaft angle is shown in Figure 2. For both stages, the beginning of the crankshaft revolution, when the compression chamber volume is at a maximum, corresponds to an angle of 0[degrees]. However, it is important to note that the two stages are 180[degrees] out of phase. This must be taken into consideration when linking the two stages in the model.
[FIGURE 2 OMITTED]
It is also necessary to know the rate at which the chamber volumes change with respect to crankshaft angle, which can be determined by taking the derivative of Equation 1:
[dV/[d[theta]]] = [H.sub.c][[1/2](-[R.sub.c.sup.2] + [R.sub.r.sup.2] + [R.sub.r.sup.2][alpha]) + e([[R.sub.r] + [r.sub.v]]/2)cos([theta] + [alpha])(1 + [alpha]) - [[r.sub.v.sup.2]/[2[(cos[alpha]).sup.2]]][alpha] - [b/2]([R.sub.r] + [r.sub.v])(sin[alpha])[alpha] - ebsin[theta]] (4)
where
z = e[[sin[theta]]/[[R.sub.r] + [r.sub.v]]] (5)
[alpha] = [sin.sup.-1]z (6)
[alpha] = [1/[square root of (1 - [z.sup.2])]] (7)
Surface Area of the Chambers
The surface area of the suction and compression chambers must also be calculated for use in the heat transfer calculations. Figure 3 shows the suction and compression chambers with a new variable, [delta], defined to measure the distance between the cylinder wall and the roller at any angle, [phi].
[FIGURE 3 OMITTED]
This distance is both a function of the crankshaft angle, [theta], and the angle at which it is measured, [phi]:
[delta] = [R.sub.c] - [[2ecos([theta] - [phi])[+ or -][square root of (4[e.sup.2][cos.sup.2]([theta] - [phi]) - 4([e.sup.2] - [R.sub.r.sup.2])]]]/2] (8)
The area of the chamber on...
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