# Modeling of a two-stage rotary compressor.

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 the top and bottom of the cylinder is then calculated by numerically integrating the distance between the cylinder wall and the roller across the entire chamber. This area is added to the area of the vertical surfaces to determine the total surface area of each chamber:

[A.sub.s] = [theta]([R.sub.r] + [R.sub.c]) + [[theta].summation over (i = 0)][[delta].sub.i](d[phi]) (9)

[A.sub.c] = (2[pi] - [theta])([R.sub.r] + [R.sub.c]) + [2[pi].summation over (i = [theta])][[delta].sub.i](d[phi]) (10)

Chamber Conservation of Mass

A mass balance can be written for the gas within the suction and compression chambers, mufflers, and upper and lower cavities of the shell as follows:

[[d([rho]V)]/dt] = [summation][m.sub.in] - [summation][m.sub.out] (11)

By application of the chain rule for differentiation, and assuming that density is a function of temperature and pressure, the mass balance can be rewritten in terms of the unknowns, dP/d[theta] and dT/d[theta]:

V([[delta][rho]]/[[delta]P][dP/[d[theta]]] + [[[delta][rho]]/[[delta]T]][dT/[d[theta]]]) = - [rho][dV/[d[theta]]] + ([summation][m.sub.in] - [summation][m.sub.out])[1/[omega]] (12)

Chamber Conservation of Energy

An energy balance for the gas in the suction and compression chambers, mufflers, and upper and lower cavities of the shell can be written as follows:

[d[E.sub.cv]/dt] = [summation][m.sub.in][h.sub.in] - [summation][m.sub.out][h.sub.out] + Q - W (13)

The enthalpy terms in the energy balance account for the energy transfer by gas flow through the suction pipe, discharge pipe, and leakage paths. Assuming that the compression process is a quasi-equilibrium process, the changes in kinetic and potential energy are negligible, and the specific internal energy is a function of pressure and temperature, the energy balance can be rewritten as follows:

(uV[[[delta][rho]]/[[delta]P]] + [rho]V[[delta]u]/[[delta]P])[dP/d[theta]] + (uV[[[delta][rho]]/[delta]T] + [rho]V[[delta]u]/[[delta]T])[dT/[d[theta]]] = -(u[rho] + P)[dV/[d[theta]]] + ([summation][m.sub.in][h.sub.in] - [summation][m.sub.out][h.sub.out] + Q)[1/[omega]] (14)

Chamber Leakage Model

Leakage occurs to and from the suction and compression chambers and the upper and lower cavities of the compressor at several locations, causing an overall decrease in efficiency due to the re-expansion of compressed gas. Figure 4 shows a sketch of a single stage of the compressor with arrows marking the flow paths, including leakage, inlet, and outlet flows. A description of each flow path is included in Table 1 along with a classification of the type of flow through the path. The types of flow are classified as either isentropic flow of compressible ideal gas, laminar viscous flow, or mixed plane Couette and Poiseuille flow.

[FIGURE 4 OMITTED]
```Table 1. Summary of Leakage Paths (1)

Path            From                                          Flow Type

[m.sub.ec]  From back of vane to compression chamber through      2
clearance between vane and slot

[m.sub.rb]  From shell to suction chamber through vertical        3
clearance between rolling piston and cylinder

[m.sub.rc]  From shell to compression chamber through             3
vertical clearance between rolling piston and
cylinder

[m.sub.vb]  From compression to suction chamber through           1
vertical clearance between vane and cylinder

[m.sub.vt]  From compression to suction chamber through           1
radial clearance between vane tip and
rolling piston

[m.sub.21]  From suction chamber to suction pipe                  1

[m.sub.31]  From compression chamber to suction pipe              1

[m.sub.32]  From compression to suction chamber through           1
radial clearance between cylinder wall and
rolling piston

[m.sub.42]  From clearance volume to suction chamber              1

[m.sub.46]  From clearance volume to muffler                      1

(1) Flow is designated as (1) isentropic flow of compressible ideal gas,
(2) mixed plane Couette flow and Poiseuille flow, or (3) laminar viscous
flow into the chamber and isentropic flow out of the chamber.
```

Isentropic Flow Model. For the case of isentropic flow, the mass flow rate is dependent on the pressure ratio across the flow path, [P.sub.r]:

[P.sub.r] = [[P.sub.l]/[P.sub.h]] (15)

However, the pressure ratio must be greater than or equal to the critical pressure ratio that occurs at choked flow. The critical pressure ratio depends on the specific heat ratio of the gas, k:

[P.sub.r][greater than or equal to][([P.sub.r]).sub.critical] = [(2/[k + 1]).sup.[k/[k - 1]]] (16)

The mass flow rate also depends on flow path area, A, and the high-side pressure and temperature, [P.sub.h] and [T.sub.h], respectively:

m = A[P.sub.h][square root of ([2k/[(k - 1)R[T.sub.h]]][([P.sub.r.sup.(2/k)] - [P.sub.r.sup.([k + 1]/k)])]]] (17)

Mixed Couette and Poiseuille Flow Model. The assumption of mixed Couette and Poiseuille flow is used to model the flow along the sides of the vane in the vane slot because the effect of the vane's motion on the leakage flow must be considered. The motion of the vane can increase or decrease the leakage flow rate either by acting in the same direction as the pressure-driven flow or by opposing the pressure-driven flow. Therefore, the velocity of the vane must be determined as follows:

[v.sub.vane] = esin[theta]([alpha] + [omega]) (18)

where

[alpha] = ecos[theta][[omega]/[([R.sub.r] + [R.sub.b])[square root of(z)]]] (19)

and

z = 1 - [([esin[theta]]/[[R.sub.r] + [R.sub.b]]).sup.2] (20)

Then the mass flow rate is a function of the vane velocity and the high- and low-side pressures, [P.sub.h] and [P.sub.l], respectively:

m = [rho]h([[c.sup.3]/12[micro]][([P.sub.h] - [P.sub.l])/l] + c[[v.sub.vane]/2]) (21)

where h is the height of the vane slot and c is the clearance between the vane and the vane slot.

Laminar Viscous Flow Model. When leakage occurs across the top of the rolling piston into either the suction chamber or the compression chamber, the model of laminar viscous flow is used. This model is applied because of the high concentration of oil on the top of the rolling piston. The mass flow rate in this case depends on the vertical clearance between the top of the rolling piston and the cylinder, c; the ratio of the outer and inner radii of the rolling piston, [R.sub.r] and [R.sub.r, i], respectively; and the mean viscosity of the high- and low-side fluids, [[mu].sub.mean]:

m = 2[pi][[rho].sub.h][c.sup.3][[[P.sub.h] - [P.sub.l]]/[6[[mu].sub.mean]log([R.sub.r]/[R.sub.r, i])]]([2[pi] - [beta]]/[2[pi]]) (22)

where

[alpha] = [sin.sup.-1]([e/[R.sub.r]]sin[theta]) (23)

and

[beta] = [alpha] + [theta] (24)

Valve Model

The acceleration of the valve, used to solve for the valve position, is predicted using the pressure difference between the compression chamber and the muffler. The acceleration of the valve at the point in time t + [DELTA]t depends on its displacement, [y.sub.n], and velocity, [y'.sub.n.], at time t:

[y".sub.[n + 1]] = [F.sub.f][[[pi][(1.14[D.sub.dp]).sup.2]]/4][[[P.sub.muffler] - [P.sub.compression]]/[[m.sub.valve](1 - 0.5[[y.sub.n]/[y.sub.max]])]] - [2[C.sub.d][square root of([k.sub.valve]/[[m.sub.valve](1 - 0.5[[y.sub.n]/[y.sub.max]])]]]][y'.sub.n] - [[k.sub.valve]/[[m.sub.valve](1 - 0.5[[y.sub.n]/[y.sub.max]])]][y.sub.n] (25)

The maximum displacement of the valve, [y.sub.max]; the mass of the valve, [m.sub.valve]; and the diameter of the discharge port, [D.sub.dp] are all known properties of the compressor. The friction factor, [F.sub.f], and the damping coefficient, [C.sub.d], are determined experimentally. The spring constant for the valve, [k.sub.valve], depends on the position of the valve:

[k.sub.valve] = a*exp(b*[y.sub.n]) + c (26)

where the coefficients a, b, and c must also be determined experimentally. Then, the velocity of the valve is given as follows:

[y'.sub.[n + 1]] = [y'.sub.n] + [y".sub.n][DELTA]t (27)

[y.sub.[n + 1]] = [y.sub.n] + [y'.sub.n][DELTA]t + [[[y".sub.n][([DELTA]t).sup.2]]/2] (28)

If the displacement of the valve is nonzero, then mass can flow through the discharge port, and the valve displacement is used to calculate an effective cross-sectional area for the mass flow:

[A.sub.valve] = [[[pi][(1.14[D.sub.dp]).sup.2]]/2][{1.5[[[[pi][(1.14[D.sub.dp]).sup.2]]/2/[[pi][D.sub.dp]y]].sup.2]}.sup.[ - 1/2]] (29)

The isentropic flow model is then applied to calculate the mass flow rate based on the pressure across the discharge port and the effective cross-sectional area.

Heat Transfer Model

Heat transfer between the cylinder wall and the gas in the suction and compression chambers is calculated by determining an appropriate convection coefficient and applying Newton's law of cooling. The spiral heat exchanger model that has been applied to scroll compressors (Chen et al. 2002a; Yi et al. 2004) was selected for modeling the heat exchange in the rolling-piston suction and compression chambers. The spiral heat exchanger model relates the heat transfer coefficient to the Reynolds number, Re, and Prandtl number, Pr, by the following expression:

[h.sub.c] = 0.023[k/[D.sub.h]][Re.sup.0.8][Pr.sup.0.4][1.0 + 1.77([D.sub.h]/[r.sub.aver])] (30)

where [D.sub.h] is the hydraulic diameter of the chamber and [r.sub.aver] is the average radius of the chamber. The Reynolds number and Prandtl number must be calculated at each crankshaft position because of their dependence on temperature and pressure. The velocity of the gas, u, used in the Reynolds number can be approximated as the average surface velocity of the cylinder and the roller, which is constant:

u = [1/2](2[pi][R.sub.r]*rps) (31)

The hydraulic diameter of the chamber is a function of the volume, V, and surface area, A, of the chamber:

[D.sub.h] = [4V/A] (32)

Once the heat transfer coefficient is determined, Newton's law of cooling states that the heat transfer rate will be proportional to the temperature difference between the gas and the surface:

Q = [h.sub.c]A([T.sub.c] - [T.sub.g]) (33)

Newton's law of cooling also applies to the heat transfer from the outer surface of the compression cylinder to the gas in the shell. However, for this case the correlation for forced convection in an annulus is used to predict the heat transfer coefficient, [h.sub.c, outer], and the hydraulic diameter is defined as follows:

[D.sub.h, out] = [D.sub.shell] - 2[R.sub.c] (34)

The Reynolds number is used to determine the regime of flow according to the following transition points:

[Re.sub.lam] = 2089.26 + 686.15R * (35)

[Re.sub.turb] = 2963.02 + 3343.16R * (36)

where

R * = [[2[R.sub.c]]/[D.sub.shell]] (37)

Then the convection coefficient is determined for the laminar, transition, and turbulent regimes as follows:

N[u.sub.lam] = [[[0.186 + 0.029logR * - 0.008[(logR *).sup.2]].sup.-1] for Re < [Re.sub.lam] (38)

Nu = 0.025[Re.sup.0.78][Pr.sup.0.48][(R *).sup.-0.14] for Re > [Re.sub.turb] (39)

Nu = exp(log [Nu.sub.lam] + (log [Nu.sub.turb] - log [Nu.sub.lam])X[log Re - log [Re.sub.lam]]/[log [Re.sub.turb] - log [Re.sub.lam]]) for [Re.sub.lam]<Re<[Re.sub.turb] (40)

where

[Nu.sub.turb] = 0.025[Re.sub.turb.sup.0.78][Pr.sup.0.48][(R *).sup.-0.14] (41)

The heat transfer from both the interior and exterior walls of the cylinder depends on the cylinder temperature. For these calculations, the temperature of the cylinder surface, [T.sub.c], is assumed to have a linear distribution around the cylinder, varying from 5 K above the average cylinder temperature near the discharge port to 5 K below the average cylinder temperature near the suction port. However, since the average cylinder temperature is initially unknown, solving for the heat transfer is an iterative process. An initial average cylinder temperature is assumed and used to calculate the heat transfer; it is then adjusted between iterations until the heat transfer into the cylinder from the gas in the shell is equal to the heat transfer out of the cylinder to the gas in the suction and compression chambers.

Intercooling Model

When intercooling is incorporated, the model uses a specified degree of superheat at the second-stage suction to control the amount of intercooling. The model calculates the saturation temperature corresponding to the pressure in the first-stage muffler and adds the specified superheat to determine the temperature of the gas available to the second stage. If [T.sub.int] represents the temperature of the gas entering the second stage, [P.sub.int] is the intermediate pressure, and [DELTA][T.sub.sh] is the degree of superheat:

[T.sub.int] = [T.sub.sat] + [DELTA][T.sub.sh] (42)

where [T.sub.sat] = T([P.sub.int], X = 1).

The required capacity of the intercooler to achieve the specified superheat is determined by multiplying the change in the enthalpy of the fluid across the intercooler by its mass flow rate:

[Q.sub.int] = m[DELTA][h.sub.int] (43)

Shell Energy Balance

A simple shell energy balance is applied to the compressor to estimate the temperature of the gas in the shell. For these calculations, it is assumed that all of the compressor power lost due to mechanical and electrical inefficiencies is transferred to the gas in the shell as heat:

[Q.sub.in]] = [P.sub.overall](1 - [[eta].sub.mech][[eta].sub.motor]) (44)

It is also assumed that the outer surface of the compressor is at the same temperature as the gas in the shell. Therefore, the amount of heat lost to the environment due to natural convection is dependent on the temperature of the gas in the shell according to Newton's law of cooling:

[Q.sub.out] = [h.sub.c]A([T.sub.shell] - [T.sub.amb]) (45)

The heat transfer coefficient for natural convection from a vertical cylinder can be approximated as follows:

[h.sub.c] = [k/[H.sub.shell]][{0.825 + 0.387[Ra.sub.L.sup.[1/6]][[1 + [(0.492/Pr).sup.[9/16]]].sup.[-8/27]]}.sup.2] (46)

where the properties of air are evaluated at the film temperature, which is the average of the ambient and shell temperatures. The Raleigh number, [Ra.sub.L], is determined according to the following expression:

[Ra.sub.L] = [[g([T.sub.shell] - [T.sub.amb])[H.sub.shell.sup.3]]/[T.sub.film][v.sub.film][[alpha].sub.film]]] (47)

Because the enthalpy and mass flow rate of the gas entering the shell from the second-stage compression chamber are known, the energy balance can be solved for the enthalpy of the gas leaving the shell:

[h.sub.out] = [[[(mh).sub.in] + [Q.sub.in] - [Q.sub.out]]/[m.sub.out]] (48)

The mass flow rate of gas leaving the shell is known because it must equal the mass flow rate of gas entering the shell at steady-state conditions. However, both the enthalpy of the gas leaving the shell and the rate of heat transfer from the shell due to natural convection depend on the shell temperature. Therefore, determining the shell temperature is an iterative process.

NUMERICAL SOLUTION OF THE MODEL

For each control volume, combining the final mass and energy balance equations in matrix form yields the following:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (49)

Solving this set of equations using Cramer's Rule and replacing the internal energy terms with the relation u = h - Pv results in the following expression for the differential temperature and pressure, respectively:

[dT/[d[theta]]] = [[( -[[rho].sup.2][[[delta]h]/[[delta]P]] + [[rho].sup.2]P[[[delta]v]/[[delta]P]] + [rho] + P[[delta][rho]]/[[delta]P])([[delta]V]/[[delta][theta]] - v[[delta]m]/[[delta][theta]]) + [[[delta][rho]]/[[delta]P]][[[delta][m.sub.in]]/[[delta][theta]]([h.sub.out] - [h.sub.in]) - [1/[omega]]Q]]/[V[([rho][[[delta]h]/[[delta]P]] - 1)[[[delta][rho]]/[[delta]T]] - ([rho][[[delta]h]/[[delta]T]] - [[delta]P]/[[delta]T])[[delta][rho]]/[[delta]P]]]] (50)

[dP/[d[theta]]] = [[([[[rho].sup.2][[[delta]h]/[[delta]T]]] - [[[rho].sup.2]P[[[delta]v]/[[delta]T]]] - [[rho][[[delta]P]/[[delta]T]]] - [P[[delta][rho]]/[[delta]P]])([[delta]V]/[[delta][theta]] - [v[[delta]m]/[[delta][theta]]]) - [[[delta][rho]]/[[delta]T]][[[delta][m.sub.in]]/[[delta][theta]]([h.sub.out] - [h.sub.in]) - [1/[omega]]Q]]/[V[([rho][[[delta]h]/[[delta]P]] - 1)[[[delta][rho]]/[[delta]T]] - ([rho][[[delta]h]/[[delta]T]] - [[delta]P]/[[delta]T])[[delta][rho]]/[[delta]P]]]] (51)

where

[dm/[d[theta]]] = ([summation][m.sub.in] - [summation][m.sub.out])[dt/[d[theta]]] (52)

and

[[d[m.sub.in]]/[d[theta]]] = [summation][m.sub.in][dt/[d[theta]]] (53)

The differential terms in Equations 50 and 51 can be approximated using the centered-difference formula. If [f.sub.x] is the derivative of property f with respect to x,

[f.sub.x]|n = [[[f.sup.[n + 1]] - [f.sup.[n - 1]]]/[2[DELTA]x]] (54)

Before applying the mass and energy balance in the form shown in Equations 50 and 51, it is necessary to have an initial guess of the variation of temperature and pressure in the chambers with crankshaft angle for evaluating the mass flow rates through the various leakage paths and estimating the heat transfer. To obtain the initial guess values, the model assumes constant temperatures and pressures in the suction chambers and mufflers. The first-stage muffler pressure is estimated as the intermediate pressure for an ideal two-stage compressor:

[P.sub.int] = [square root of ([P.sub.suction][P.sub.discharge])] (55)

Then, the temperature in the first-stage and second-stage mufflers are estimated by assuming an isentropic process:

[T.sub.int] = [T.sub.suction][([P.sub.int]/[P.sub.suction]).sup.[(k - 1)/k]] and [T.sub.d] = [T.sub.suction][([P.sub.d]/[P.sub.suction]).sup.[(k - 1)/k]] (56)

The model starts by assuming that when the crankshaft angle, [theta], is zero (at the end of the suction process and the start of the compression process), the first-stage compression chamber is at suction temperature and pressure. The remaining properties, such as density, enthalpy, and viscosity, can be evaluated using the temperature and pressure. Then, the derivatives of temperature and pressure with respect to crankshaft angle at the crankshaft angle of zero can be estimated by neglecting leakage and heat transfer:

[dT/[d[theta]]] = [[(-[[rho].sup.2][[[delta]h]/[[delta]P]] + [[rho].sup.2]P[[[delta]v]/[[delta]P]] + [rho] + P[[delta][rho]]/[[delta]P])([[delta]V]/[[delta][theta]])]/[V[([rho][[[delta]h]/[[delta]P]] - 1)[[[delta][rho]]/[[delta]T]] - ([rho][[[delta]h]/[[delta]T]] - [[delta]P]/[[delta]T])[[delta][rho]]/[[delta]P]]]] (57)

[dP/[d[theta]]] = [[([[rho].sup.2][[[delta]h]/[[delta]T]] - [[rho].sup.2]P[[[delta]v]/[[delta]T]] - [rho][[[delta]P]/[[delta]T]] - P[[delta][rho]]/[[delta]P])([[delta]V]/[[delta][theta]])]/[V[([rho][[[delta]h]/[[delta]P]] - 1)[[[delta][rho]]/[[delta]T]] - ([rho][[[delta]h]/[[delta]T]] - [[delta]P]/[[delta]T])[[delta][rho]]/[[delta]P]]]] (58)

The derivatives of pressure and temperature with respect to crankshaft angle [theta] evaluated at angle are then used to predict the pressure and temperature at angle [theta] + [DELTA][theta] using the modified Euler method. The modified Euler method first calculates a predicted temperature and pressure at angle [theta] + [DELTA][theta]:

[T.sup.P]([theta] + [DELTA][theta]) = T([theta]) + [DELTA][theta][[[delta]T]/[[delta][theta]]([theta])] (59)

[P.sup.P]([theta] + [DELTA][theta]) = P([theta]) + [DELTA][theta][[[delta]P]/[[delta][theta]]([theta])] (60)

The derivatives of pressure and temperature with respect to crankshaft angle are then reevaluated using the predicted pressure and temperature at angle [theta] + [DELTA][theta]. The pressure and temperature at angle [theta] + [DELTA][theta] are then recalculated using an average of the derivatives:

T([theta] + [DELTA][theta]) = T([theta]) + [[[DELTA][theta]]/2][[[[delta]T]/[[delta][theta]]([theta])] + [[[delta]T]/[[delta][theta]]]([theta] + [DELTA][theta])] (61)

P([theta] + [DELTA][theta]) = P([theta]) + [[[DELTA][theta]]/2][[[delta]P]/[[delta][theta]]([theta]) + [[[delta]P]/[[delta][theta]]]([theta] + [DELTA][theta])] (62)

Using the modified Euler method, it is possible to start with the initial guess of temperature and pressure at the crankshaft angle of zero in the compression chamber and step through the crankshaft revolution to predict the temperature and pressure at each angle. The valve subroutine is also called at each angle to determine when the discharge process begins; it uses the predicted temperature and pressure in the compression chamber along with the muffler pressure and temperature to determine if the valve will open. As soon as the valve opens, it is assumed that a constant pressure discharge process occurs. The process is repeated for the second stage.

After the initial guesses of pressure and temperature with crankshaft angle are obtained, the model starts again with the first-stage suction chamber, this time applying the mass and energy balance in Equations 50 and 51 instead of assuming constant temperatures and pressures in the suction chambers and mufflers. At each angle, the leakage flow rates are evaluated using the pressure and temperature in the chamber and the surrounding control volumes. The instantaneous heat transfer rate is also evaluated and then substituted into the calculations for the derivatives of pressure and temperature with respect to crankshaft angle.

The suction processes no longer occur at a constant temperature or pressure. The mass and energy balance is applied to the first-stage suction chamber and then the first-stage compression chamber, assuming a constant temperature and pressure in the first-stage muffler. The mass flow rate into the first stage is then compared to the mass flow rate exiting the first stage, and the temperature in the muffler is adjusted. Without changing the first-stage muffler pressure, the model continues to loop through the first-stage calculations until the first-stage inlet and outlet mass flow rates agree within a specified tolerance. This fixes the first-stage muffler temperature.

The model then repeats the process for the second stage using the constant temperature and pressure in the first-stage muffler as an input. The second-stage muffler pressure is held constant at the discharge pressure while the second-stage muffler temperature is varied. The model loops through the second-stage calculations until the muffler temperature is such that the inlet and outlet mass flow rates for the second stage agree within a specified tolerance.

The temperature of the gas in the shell is assumed to be equal to the discharge temperature from the second stage. Because leakage occurs from the shell to the first stage, and the heat transfer to the first stage depends on the shell temperature, the calculations must now be repeated with the updated discharge temperature. The muffler pressure is not updated until the discharge temperature does not change significantly between iterations.

When the discharge temperature converges, the muffler pressure is updated to attempt to satisfy an overall mass balance. The first-stage muffler pressure is adjusted until the mass flow rate through the two stages agrees within the specified tolerance. As mentioned previously, the temperature of the gas in the shell must also be determined in an iterative process. Therefore, the model will only converge when the change in the shell temperature between iterations is less than a specified tolerance. Figure 5 provides a flowchart of the program showing the iterative process.

[FIGURE 5 OMITTED]

After the program has converged, the calculated pressure and temperature variations are used to determine the power consumption of each stage. The suction and compression chambers can be approximated as closed systems with constant pressures over the very small time steps used in the program. Therefore, the work required for each stage is the sum of the boundary work for each time step over one revolution:

[W.sub.actual] = [[[W.sub.s] + [W.sub.c]]/[[eta].sub.total]] = [summation over (i)][[[P.sub.s](i)[[V.sub.s](i + 1) - [V.sub.s](i)] + [P.sub.c](i)[[V.sub.c](i + 1) - [V.sub.c](i)]]/[[[eta].sub.motor][[eta].sub.mechanical]]] (63)

where the total efficiency, [[eta].sub.total], is the product of the motor and mechanical efficiencies, [[eta].sub.motor] and [[eta].sub.mechanical], respectively. Then, the isentropic efficiency can be calculated by comparing the actual power input to each stage to the power required for an isentropic compression process between the same pressures:

[[eta].sub.s] = [m([h.sub.2s] - [h.sub.1])]/[W.sub.actual]] (64)

The volumetric efficiency, [[eta].sub.v], can also be calculated from the model results. The actual mass flow rate predicted by the model is compared to the maximum theoretical mass flow rate, which is based on the maximum suction volume of the stage, [V.sub.max]; the density of the gas entering the stage, [[rho].sub.s]; and the number of revolutions per second made by the crankshaft, rps:

[[eta].sub.v] = [m.sub.actual]/[[V.sub.max]*[[rho].sub.s]*rps]] (65)

RESULTS AND DISCUSSION

Model Results

The model was tuned to accurately predict discharge temperature, mass flow rate, and power consumption based on the external measurements taken at test conditions similar to those encountered in air-conditioning applications. Test Condition 1 will be used to reference an evaporating temperature of 10.1[degrees]C, a suction gas temperature of 33.5[degrees]C, and a condensing temperature of 55.3[degrees]C for the remainder of this paper. The parameters that were tuned include the damping coefficients of the valve model, the combined motor and mechanical efficiency, the friction coefficient for the largest leakage path, and the heat transfer coefficient between the compressor and the surroundings.

Without tuning, the model predicted mass flow rates about 3% lower than those measured. To address this discrepancy, the mass flow rate through the most significant leakage path, the predicted flank leakage from the compression chamber to the suction chamber, was multiplied by a correction factor of 0.43. This value was selected to achieve agreement between the experimental and modeled mass flow rates through the compressor at Test Condition 1. Although this is a significant reduction in the leakage flow rate, several variables contribute to inaccuracy in the leakage model, including the assumption that the effect of oil is insignificant in most of the leakage paths. Only the calculations for leakage over the top of the rolling piston and leakage from the back of the vane slot consider the effect of oil on the density and viscosity of the fluid in the flow path, but the presence of oil will affect the flank leakage as well. Therefore, the tuning factor approximates the reduction in leakage achieved when oil seals the contact between the rolling piston and cylinder. The damping coefficients in the valve model also influence the model's ability to predict mass flow rate. These coefficients were selected so that the valve opened at the angle expected by the compressor designers and exhibited the damping that would be expected in the valve's motion.

Before tuning, the model predicted power consumptions approximately 2% lower than the measured values. Modifying the model to increase the mass flow rate improved the model's power consumption calculations. To further improve the model's accuracy at Test Condition 1, the combined mechanical and motor efficiency of the compressor was modified from 76.5% to 78.2%. These two modifications also improved the model's ability to predict the discharge temperature, which was over-predicted by 4 K before tuning.

While tuning the model improved the accuracy of its predictions of discharge temperature, mass flow rate, and power consumption, it is interesting to note that the important parameters of isentropic and volumetric efficiency are less sensitive to the tuning factors. Figures 6 through 8 investigate the effect of motor efficiency, the flank leakage coefficient, and the shell heat transfer coefficient on isentropic efficiency, volumetric efficiency, and compressor discharge temperature. The tuning factors were varied over different ranges based on the expected accuracy of the parameter as well as the sensitivity of the model to the parameter. For example, because the combined motor and mechanical efficiency can be estimated with reasonable accuracy and is expected to have a large impact on the results, it was only varied by [+ or -]15%, whereas the heat transfer coefficient, which can be predicted with less confidence, was varied by [+ or -]50%. Therefore, the horizontal axis uses a variable multiplier that represents the percentage of the range over which the tuning factor is varied. For example, the results for multiplying the combined efficiency by 0.85 are shown as -100% of the variable multiplier, as are the results for multiplying the heat transfer coefficient by 0.50.

[FIGURE 6 OMITTED]

[FIGURE 7 OMITTED]

[FIGURE 8 OMITTED]

As expected, Figure 6 shows that increasing the flank leakage coefficient results in lower volumetric efficiencies. However, varying the flank leakage by [+ or -]75% only results in a [+ or -]4% variation in volumetric efficiency. In addition, the effects of combined motor and mechanical efficiency and heat transfer on volumetric efficiency are negligible. Therefore, the model's prediction of volumetric efficiency can be stated with confidence. As seen in Figure 7, the isentropic efficiency is more sensitive to tuning factors, showing a direct dependence on motor and mechanical efficiency. To a much lesser degree, the isentropic efficiency also depends on flank leakage, decreasing by about 4% as the flank leakage multiplier is increased from 0 to 1.18. Figure 8 shows that the variation in the motor and mechanical efficiency has the greatest impact on the compressor exit temperature, causing the temperature to decrease by 18 K as the efficiency increases from -15% to +15% of its nominal value. In addition, the exit temperature decreases with increasing heat transfer and increases with increasing flank leakage, but these variations are less than [+ or -]2 K.

The effect of the convergence criteria on the model results was also studied. For the model to converge, the percent difference between the mass flow rates into and out of the compressor shell has to be less than the specified percent tolerance. While the model results in this paper were obtained using a 0.1% tolerance, raising the tolerance to 5% does not change any of the output parameters by more than [+ or -]0.5%. Thus, the results of the model can be used with reasonable confidence.

The variation of pressure with crankshaft angle in the suction and compression chambers predicted by the tuned model is shown in Figure 9. The results are plotted with the beginning of the compression process for each stage shown as 0[degrees]. However, it is important to remember that the two stages operate 180[degrees] out of phase. The variation of temperature with crankshaft angle for the same compressor at the same operating conditions is shown in Figure 10. The graph shows that the gas in both the first- and second-stage compression chambers experiences a significant rise in temperature at the beginning of the suction process (at a crankshaft angle of approximately 10[degrees]) due to heat transfer to the small amount of gas that is present in the suction chamber before the suction port is open to the chamber. The temperature in the chamber drops quickly as soon as mass begins to flow in through the suction pipe. Leakage from the high-pressure shell also contributes to the rise in temperature.

[FIGURE 9 OMITTED]

[FIGURE 10 OMITTED]

The heat transfer to the cylinder gas is a function of the heat transfer coefficient, the temperature difference, and the surface area. The instantaneous heat transfer rate is plotted as a function of crankshaft angle in Figure 11, and the average heat transfer rate, obtained by integrating the instantaneous rate, is summarized in Table 2. The heat transfer rate to the gas in the suction chambers increases with crankshaft angle as expected due to the increase in surface area. Similarly, the heat transfer rate to the gas in the compression chambers decreases with crankshaft angle due to both decreased surface area and a decrease in the temperature difference driving the heat transfer. The heat transfer even becomes negative at the end of the second-stage compression process, indicating that heat is transferred from the gas to the cylinder wall, due to the high gas temperature.

[FIGURE 11 OMITTED]
```Table 2. Average Heat Transfer Rate to Refrigerant over One Revolution
Predicted for Prototype Compressor at Test Condition 1

Chamber       Average Heat Transfer Rate to Refrigerant, W

First Stage

Suction                          9.95
Compression                     -8.74

Second Stage

Suction                         20.3
Compression                    -19.8
```

Leakage also has a significant impact on compressor performance. Figure 12 provides a comparison of the total mass flow rate through each path in the first stage over an entire revolution. It can be seen that the most significant source of leakage is through the contact between the roller and cylinder wall, [m.sub.32]. The leakages across the top and bottom of the vane, [m.sub.vb], are also significant. By summing the mass flow rate through all of the leakage paths in Figure 12, it can be seen that the leakage accounts for about 3.95% of the total mass flow rate. The most significant leakage path in the compression chamber is flank leakage at the point of contact between the roller and cylinder wall.

[FIGURE 12 OMITTED]

When intercooling is incorporated, the mass flow rate through the first stage does not change significantly because the first-stage suction state has not changed. Therefore, the mass flow rate through the second stage and, thus, the density of the gas entering the second stage should not change significantly when intercooling is incorporated. To achieve the same gas density at the second-stage suction when the intermediate temperature is lower due to intercooling, the intermediate pressure becomes lower. Figure 13 shows that the intermediate pressure predicted by the model when intercooling is used to maintain a 10 K superheat is lower than the case without intercooling, as expected. The temperature profiles with and without intercooling are compared in Figure 14.

[FIGURE 13 OMITTED]

[FIGURE 14 OMITTED]

The main benefit of intercooling is reduced power consumption, which is summarized in Table 3 by comparing the cases with and without intercooling. There is almost a 10% reduction in power requirement for intercooling at these conditions. The primary improvement results from increased density of the suction gas for the second stage that leads to lower isentropic work along with an improvement in the volumetric efficiencies. The isentropic efficiencies actually decrease slightly for the intercooled case, a trend that can be explained by differences in leakage flow rates between the standard and intercooled cases. However, the lower isentropic efficiencies are more than offset by reduced isentropic work.
```Table 3. Summary of Model Results for Prototype Compressor at Test
Condition 1 with and without Intercooling to Maintain a 10[degrees]C
Superheat

Without         With
Intercooling  Intercooling

Suction pressure, kPa                     1087          1087
Suction temperature, [degrees]C             33.2          33.2
Intermediate pressure, kPa                1745          1463
Intermediate temperature, [degrees]C        69.2          69.4
Discharge pressure, kPa                   3453          3453
Discharge temperature, [degrees]C          112.8          89.7
Power consumption, kW                        2.08          1.89
Mass flow rate, kg/h                       133.4         133.9

Isentropic efficiency
First stage                               64.8%         59.4%
Second stage                              69.3%         68.2%

Volumetric efficiency
First stage                               95.4%         96.2%
Second stage                              94.6%         94.9%
```

Comparison of Model and Experimental Results

The model was validated by testing a prototype compressor on a load stand at 17 different operating conditions. A detailed description of the load stand can be found in the paper by Chen et al. (2002b) that documents experimental testing of a scroll compressor. Table 4 compares the model results at Test Condition 1 without intercooling to the experimental results. Because the power required for each stage of compression was not measured experimentally, the actual power consumption term in the isentropic efficiency calculation (Equation 64) had to be estimated as the product of mass flow rate and enthalpy change. This resulted in a slightly different modeled isentropic efficiency compared to the values in Table 3. However, the tuned model predictions agree well with the experimental results.
```Table 4. Summary of Model Results for Prototype Compressor at Test
Condition 1 Compared to Experimental Results

Uncertainty in  Error in
Experimental    Modeled
Modeled   Experimental  Measurement     Result
Compared to
Experimental
Result

Suction        1087         1087       [+ or -]1%          --
pressure,
kPa

Suction          33.2         33.2     [+ or -] 2.2        --
temperature,                           [degrees]C
[degrees]C

Intermediate   1745         1780       [+ or -]1%           2.0%
pressure,
kPa

Intermediate     69.2         70.9     [+ or -]2.2          1.7
temperature,                           [degrees]C           [degrees]C
[degrees]C

Discharge      3453         3453       [+ or -]1%             --
pressure,
kPa

Discharge       112.8        111.2     [+ or -]2.2          1.6
temperature,                           [degrees]C          [degrees]C
[degrees]C

Power             2.08         2.08    [+ or -]0.2%         0.02%
consumption,
kW

Mass flow       133.4        133.2     [+ or -]0.7%         0.18%
rate, kg/h

Isentropic
efficiency

First stage      51.4%        51.0%    [+ or -]6.7%         0.8%

Second stage     76.8%        84.2%    [+ or -]12.5%        9.6%

Volumetric
efficiency

First stage      95.4%        95.0%    [+ or -]1.8%         0.4%

Second stage     94.6%        92.9%    [+ or -]1.4%         1.8%
```

Figures 15 and 16 compare the model predictions to the experimental data for intermediate pressure and temperature, respectively. Error bars are used in the figures to indicate that the experimental temperatures are measured with an accuracy of [+ or -]2.2 K, while the pressure measurements have an accuracy of [+ or -]1%. While the intermediate temperatures show a greater discrepancy between the modeled and experimental results than the intermediate pressures, the majority of the results are within [+ or -]10 K. The temperatures predicted by the model tend to be lower than the measured temperatures. This could be related to the model's tendency to underestimate the intermediate pressure. If the model were to correctly predict the density of refrigerant entering the second stage to satisfy the mass balance but predict a low intermediate pressure, it would also have to predict a low intermediate temperature to achieve this density. Comparing the model results to internal measurements will help to identify the source of this error.

[FIGURE 15 OMITTED]

[FIGURE 16 OMITTED]

For the majority of the cases, the discharge temperatures are predicted within [+ or -]10[degrees]C of the measured values, as shown in Figure 17. The error in the discharge temperature would partially result from the error in the intermediate temperature predicted by the model. However, uncertainty in the calculation of heat loss by natural convection from the shell would also contribute to this error.

[FIGURE 17 OMITTED]

Figure 18 compares the predicted and measured power consumptions of the compressor. The power meter used for the experimental measurements has an accuracy of [+ or -]0.2%, so the error bars for the experimental points are not visible in this case. All of the model results agree with the experimental results within [+ or -]5%, but the model seems to be most accurate for cases with high power consumption. This could indicate that the constant combined mechanical and motor efficiency of 78.3% that the model assumes is more accurate for the higher power consumption cases. The motor may operate less efficiently in the cases corresponding to lower torques.

[FIGURE 18 OMITTED]

The mass flow rate through the compressor predicted by the model is also predicted within [+ or -]5% of the experimental values, as shown in Figure 19. The accuracy of the mass flow measurement depends on the mass flow rate, but the accuracy is at least [+ or -]0.9% for all of the cases shown. The modeled flow rate is generally overpredicted at lower mass flow rates, which correspond to cases with the largest ratio between discharge and suction pressure. This indicates that the leakage is slightly underpredicted by the model for these cases.

[FIGURE 19 OMITTED]

Finally, Figure 20 shows that there is good agreement between the energy efficiency ratio (EER) calculated from the model and experimental results. The EER was calculated assuming that a vapor-compression cycle operates with 5 K of subcooling exiting the condenser. Employing an uncertainty analysis proposed by Beges et al. (2002), the accuracy of the experimental EER is within [+ or -]0.8%.

[FIGURE 20 OMITTED]

In summary, the model accurately predicts the mass flow rate through the compressor and its power consumption. However, there is a wider range of error in the predicted intermediate and discharge temperatures and in the intermediate pressure. This is to be expected as a result of the tuning to one particular set of experimental results at one test condition.

CONCLUSION

The agreement between external measurements and model results suggests that the model provides a good estimate of compressor performance (within 5% for mass flow rate and power). The model not only predicts the pressure and temperature distributions in the suction and compression chambers but also provides estimates of the mass flow rates through the various leakage paths and the heat transfer from the cylinder walls. Therefore, the effect of leakage and heat transfer on the performance of the compressor can be investigated. However, the leakage analysis is limited by the assumption of a constant flank leakage coefficient to account for the presence of lubrication. The assumption of a leakage coefficient also limits the model's ability to calculate exit temperatures, which were predicted within [+ or -]10 K of the experimental measurements. The incorporation of a lubrication analysis is an area for future work. However, the model does provide a tool for comparing the performance of an existing compressor prototype with and without intercooling. For a single case, intercooling resulted in an approximately 10% improvement in compressor EER.

ACKNOWLEDGMENTS

The authors would like to acknowledge the support of Dr. Kwan-Shik Cho, vice president and director of the Digital Appliance Company (DAC) Laboratory of LG Electronics. The authors would also like to acknowledge the contributions of Seung-Jun Lee and Young-Ju Bae of the DAC Research Laboratory Compressor Group.

NOMENCLATURE

[A.sub.c] = compression chamber surface area, [m.sup.2]

[A.sub.s] = suction chamber surface area, [m.sup.2]

b = vane thickness, m

c = leakage path clearance, m

[c.sub.v] = constant volume specific heat, J/kg*K

[D.sub.h] = hydraulic diameter, m

e = eccentricity of roller, m

[E.sub.cv] = total energy in the control volume, J

h = specific enthalpy, J/kg

[h.sub.c] = convective heat transfer coefficient, W/[m.sup.2]*K

[H.sub.c] = cylinder height, m

[DELTA][h.sub.int] = change in specific enthalpy of the refrigerant across the intercooler, J/kg

k = specific heat ratio (isentropic compression calculations)

k = thermal conductivity, W/m*K (heat transfer calculations)

[m.sub.in] = mass flow rate into the control volume, kg/s

[m.sub.out] = mass flow rate out of the control volume, kg/s

P = pressure, Pa

[P.sub.discharge] = pressure of gas leaving shell, Pa

[P.sub.h] = high-side pressure, Pa

[P.sub.int] = first-stage muffler pressure, Pa

[P.sub.l] = low-side pressure, Pa

[P.sub.overall] = shaft power, W

[P.sub.r] = ratio of low- to high-side pressure

[P.sub.suction] = pressure of gas in accumulator, Pa

Pr = Prandtl number

Q = heat transfer rate, W

[Q.sub.int] = heat transfer rate from refrigerant in the intercooler, W

R = gas constant, J/kg*K

[r.sub.aver] = average radius of curvature of chamber, m

[R.sub.r, i] = inner radius of roller, m

[r.sub.v] = vane tip radius, m

Re = Reynolds number

rps = rotational speed of crankshaft, revolutions per second

t = time, s

T = temperature, [degrees]C

[T.sub.amb] = ambient temperature, [degrees]C

[T.sub.c] = temperature of cylinder wall, [degrees]C

[T.sub.discharge] = temperature of gas exiting the shell, [degrees]C

[T.sub.g] = temperature of gas in control volume, [degrees]C

[T.sub.int] = temperature of gas entering second stage, [degrees]C

[T.sub.sat] = saturation temperature at specified pressure, [degrees]C

[T.sub.shell] = temperature of the gas in the compressor shell, [degrees]C

[T.sub.suction] = temperature of gas entering first stage, [degrees]C

[DELTA][T.sub.sh] = degree of superheat at inlet to second stage, [degrees]C

u = specific internal energy, J/kg (energy balance calculations)

u = average velocity of gas in control volume, m/s (Reynolds number calculations)

[U.sub.cv] = total internal energy of the control volume, J

v = specific volume, [m.sup.3]/kg

V = volume, [m.sup.3]

[v.sub.vane] = vane velocity, m/s

W = power, W

x = vane extension, m

Greek Symbols

[alpha] = angle between vane slot and line connecting vane tip center to roller center (geometry calculations)

[alpha] = thermal diffusivity, [m.sup.2]/s (heat transfer calculations)

[delta] = distance between the cylinder wall and roller, m

[[eta].sub.mech] = mechanical efficiency of the compressor

[[eta].sub.motor] = motor efficiency

[[eta].sub.s] = isentropic efficiency

[[eta].sub.v] = volumetric efficiency

[theta] = crankshaft angle

[mu] = dynamic viscosity, N*s/[m.sup.2]

[rho] = density, kg/[m.sup.3]

[psi] = angle at which distance [delta] measured

v = kinematic viscosity, [m.sup.2]/s

[omega] = rotational speed of crankshaft, degrees/s

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Margaret M. Mathison

Student Member ASHRAE

James E. Braun, PhD, PE

Fellow ASHRAE

Eckhard A. Groll, PhD

Fellow ASHRAE

Margaret M. Mathison is a graduate student and James E. Braun and Eckhard A. Groll are professors in the School of Mechanical Engineering, Purdue University, West Lafayette, IN.
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