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Analysis of expansion waves appearing in the outlets of two-phase flow nozzles.

INTRODUCTION

Two-phase flow nozzles are used in total-flow systems of geothermal power plants (Mori and Suyama 1980) and in the ejector of ejector refrigeration systems (Nakagawa 2004). One of the most important functions of the nozzles is to convert thermal energy into kinetic energy. The kinetic energy of the two-phase flow exhausted from nozzles is used to rotate turbine wheels in geothermal power plants and to compress the suctioned vapor in the ejectors of ejector refrigeration systems. The sound speeds in those nozzles are usually very low (Landau and Lifshitz 1959), so the two-phase flows inside usually accelerate into a supersonic state. The energy conversion efficiency of the nozzles is too low because of the transport problems peculiar to two-phase flow, so the authors studied several ways to improve it (Nakagawa et al. 1998).

When the kinetic energy of the two-phase fluid flowing out of a nozzle is used, it is necessary to control the occurrence of shock waves and expansion waves. It is also necessary to design a nozzle that can work at any state. In particular, it is important to elucidate the mixing process of the two-phase fluid and the suctioned vapor in the mixing section of the two-phase ejector in a refrigeration cycle. There are many studies (Nakagawa and Mamiya 1986) that treat the expansion wave of two-phase flow with phase change with a one-dimensional approach. However, there are few that treat it with a two-dimensional approach (Marble 1963).

The expansion waves of liquid-vapor flow of water at a nozzle outlet (Nakagawa et al. 2007) were experimentally verified in this research. This study focuses on the momentum-relaxation phenomena between the phases of high-speed mist flow. The objective is to elucidate the theoretical characteristics of the expansion waves at the outlet of a supersonic two-phase flow nozzle. This paper was presented at the 12th International Refrigeration and Air-Conditioning Conference at Purdue University (Nakagawa et al. 2008) and is published in the conference proceedings. Modifications were made in order to include inch-pound (I-P) units of measurement, add more details, and improve coherence.

THE BASIC EQUATIONS

The momentum relaxation phenomena are very important in high-speed two-phase flow. In this paper, the momentum relaxation phenomena of vapor-mist flow are presented. The derivation is based on the equation of total momentum and the equation of motion of a droplet. Frictional drag was assumed to act on the droplet due to relative velocities of the vapor and the liquid phases. Phase change was neglected, but the compressibility of the vapor was considered for simplicity.

The continuity equations of the vapor and the liquid phases were expressed as follows:

[[partial derivative]/[[partial derivative]t]][alpha][[rho].sub.g] + [[partial derivative]/[[partial derivative][x.sub.j]]][alpha][[rho].sub.g][v.sub.gj] = 0 (1)

[[partial derivative]/[[partial derivative]t]](1 - [alpha])[[rho].sub.l] + [[partial derivative]/[[partial derivative][x.sub.j]]](1 - [alpha])[[rho].sub.l][v.sub.lj] = 0 (2)

The equation for total momentum conservation of two-phase flow is given by

[[partial derivative]/[[partial derivative]t]]{[alpha][[rho].sub.g][v.sub.gi] + (1 - [alpha])[[rho].sub.l][v.sub.li]} + [[partial derivative]/[[partial derivative][x.sub.j]]]{[alpha][[rho].sub.g][v.sub.gj][v.sub.gi] + (1 - [alpha])[[rho].sub.l][v.sub.lj][v.sub.li]} = [[[partial derivative]p]/[[partial derivative][x.sub.i]]]. (3)

In this study, the quality of the two-phase flow was given a value of 0.1. The diameter of a droplet was considered very small, and the frictional drag on the droplet was assumed to be governed by Stokes' law. The resulting equation of motion of the droplet is given by

[[rho].sub.l][[[pi][d.sup.3]]/6]([[partial derivative][v.sub.li]]/[[partial derivative]t] + [v.sub.lj][[partial derivative][v.sub.li]]/[[partial derivative][x.sub.ji]]) = 3[pi][[mu].sub.g]d([v.sub.gi] - [v.sub.li]). (4)

If the expression for the mass of the droplet on the left-hand side of Equation 4 is transferred to the right-hand side, the expression for the relaxation time shown below appears.

[tau] = [[[[rho].sub.l][d.sup.2]]/[18[[mu].sub.g]]] (5)

The relaxation time indicates the time required for the vapor and the liquid phases to reach the same velocity or equilibrium in momentum. By using Equations 3, 4, and 5, the following momentum equations of the vapor and the liquid phases, respectively, were obtained:

[alpha][[rho].sub.g][[[partial derivative][v.sub.gi]]/[[partial derivative]t]] + [alpha][[rho].sub.g][v.sub.gj][[[partial derivative][v.sub.gi]]/[[partial derivative][x.sub.j]]] = - (1 - [alpha])[[rho].sub.l][[[v.sub.gi] - [v.sub.li]]/[tau]] - [alpha][[[partial derivative]p]/[[partial derivative][x.sub.i]]] (i = 1,2) (6)

(1 - [alpha])[[rho].sub.l][[[partial derivative][v.sub.li]]/[[partial derivative]t]] + (1 - [alpha])[[rho].sub.l][v.sub.lj][[[partial derivative][v.sub.li]]/[[partial derivative][x.sub.j]]] = (1 - [alpha])[[rho].sub.l][[[v.sub.gi] - [v.sub.li]]/[tau]] - (1 - [alpha])[[[partial derivative]p]/[[partial derivative][x.sub.i]]] (i = 1,2) (7)

The vapor phase was assumed to be a perfect gas and assumed to be changing isothermally, because the heat capacity of the liquid was very large. The state equation of the vapor phase was expressed as

[p/[[rho].sub.g]] = constant. (8)

Correspondingly, the compressibility of the liquid was much less than that of the vapor, so the liquid droplets were assumed to be incompressible. The state equation of the liquid phase was expressed as

[[rho].sub.l] = constant. (9)

It was assumed that the droplets were not colliding, not aggregating with each other, and not breaking up. The diameter of the droplets was assumed to be constant.

In Equations 6 and 7, the velocity difference between the vapor and the liquid phase becomes small as a result of a large exchange of momentum when [tau] is small. If [tau] is 0, equilibrium in velocity of the flow is established. The vapor and the droplets have the same velocity in this case. At larger [tau], the momentum exchange between the phases is negligible. If [tau] is infinity, frozen flow is established. There is no interaction between the phases. The vapor and the droplets flow independently from each other.

The differential terms of Equations 1, 2, 6, and 7, which control the momentum-relaxation phenomena of two-phase flow, were the same as the basic equations of frozen flow. The dimensionless parameters of the equations were controlled by the Mach number of the frozen flow.

Equations 4-9 are valid for vapor-mist flow. The liquid concentration of such regime is smaller than those of bubbly and slug flows.

SOUND SPEED

Sound speed is defined as the propagation speed of disturbance in static fluid. The propagation speed of the microscopic disturbance in two-phase flow was obtained from linearization of Equations 1, 2, and 6-9. The physical parameters shown in Equation 10 were divided into components of constant values and small disturbances, which are denoted by overbars and hats, respectively. The components with higher orders of disturbance were neglected. Then, the linearized Equations 11-14 were obtained.

[alpha] = [bar.[alpha]] + [^.[alpha]], [[rho].sub.g] = [[bar.[rho]].sub.g] + [[^.[rho]].sub.g],[v.sub.gj] = [[bar.v].sub.gj] + [[^.v].sub.gj], [v.sub.lj] = [[bar.v].sub.lj] + [[^.v].sub.lj] (10)

[bar.[alpha]][[[partial derivative][[^.[rho].sub.g]]]/[[partial derivative]t]] + [[bar.[rho]].sub.g][[[partial derivative][^.[alpha]]]/[[partial derivative][x.sub.j]]] + [bar.[alpha]][[bar.[rho]].sub.g][[[partial derivative][[^.v].sub.gj]]/[[partial derivative][x.sub.j]]] = 0 (11)

- [[[partial derivative][^.[alpha]]]/[[partial derivative]t]] + (1 - [bar.[alpha]])[[[partial derivative][[^.v].sub.lj]]/[[partial derivative][x.sub.j]]] = 0 (12)

[bar.[alpha][rho]].sub.g][[[partial derivative][[^.v].sub.gi]]/[[partial derivative]t]] + (1 - [bar.[alpha]])[[[[^.v].sub.gi] - [[^.v].sub.li]]/[tau]] + [bar.[alpha]][[bar.p]/[[bar.[rho]].sub.g]][[[partial derivative][[^.[rho]].sub.gi]]/[[partial derivative][x.sub.i]]] = 0 (i = 1,2) (13)

[[rho].sub.l][[[partial derivative][[^.v].sub.gi]]/[[partial derivative]t]] - [[[[^.v].sub.gi] - [[^.v].sub.li]]/[tau]] + [[bar.p]/[[bar.[rho]].sub.g]][[[partial derivative][[^.[rho]].sub.gi]]/[[partial derivative][x.sub.i]]] = 0 (i = 1,2) (14)

The small disturbances were expressed using [omega] and [k.sub.i], as shown in the equations below.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)

The above equations were substituted into Equation 11-14, and six linear equations were obtained. The linear equations were transformed into a single matrix equation. To satisfy the condition that all of the amplitudes must be nonzero, the determinant of the square matrix in the matrix equation can be taken as zero, as shown in Equation 16. The relationship between [omega] and [k.sub.i] can be obtained from Equation 16. The square of the sound speed in Equation 17 can be obtained by dividing the square of [omega] by the square of the magnitude of [k.sub.i].

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16)

By using the obtained relationship between [omega] and [k.sub.i] from Equation 16, the detailed expression for the square of the sound speed in Equation 17 was derived.

[a.sup.2] = [[[omega].sup.2]/[[k.sub.1.sup.2] + [k.sub.2.sup.2]]] = [[i(p/[[rho].sub.l]){(1 - [alpha])[[rho].sub.g] + [alpha][[rho].sub.l]}[omega][tau] - (p/[alpha])]/[i[alpha][[rho].sub.g][omega][tau] - {[alpha][[rho].sub.g] + (1 - [alpha])[[rho].sub.l]}]] (17)

When [tau] is 0 in Equation 17, the sound speed of equilibrium flow shown in the equation below is obtained, and it is the same as that of Campbell and Pitcher (1958).

[a.sub.e] = [square root of [p/[[alpha]([alpha][[rho].sub.g] + (1 - [alpha])[[rho].sub.l])]]] (18)

whereas, if [tau] is infinity, the sound speed of frozen flow is obtained as

[a.sub.f] = [square root of [[p/[[rho].sub.g]] + [p/[[rho].sub.l]][[1 - [alpha]]/[alpha]]]] (19)

If [[rho].sub.l] [much greater than] [[rho].sub.g], the frozen sound speed is approximately equal to the isothermal sound speed of the vapor. Therefore, a sound speed that is made nondimensional using the isothermal sound speed of the vapor, as shown in the equation below, approaches unity.

a' = [a/[(p/[[rho].sub.g]).sup.[1/2]]] (20)

The propagation speed or sound speed calculated using Equation 17 at a quality of 0.1 and density ratio [[rho].sub.l]/[[rho].sub.g] of 1000 was plotted in Figure 1. The values of physical properties were taken from the steam data. The vertical and the horizontal axes are the nondimensional sound speed a' and [omega][tau], respectively. When [omega][tau] was smaller than 1, the sound speed was nearly equal to the equilibrium sound speed. Conversely, when [omega][tau] was larger than 20, the sound speed was nearly equal to the frozen sound speed.

[FIGURE 1 OMITTED]

The equilibrium sound speed depends not only on [omega][tau] but also on [alpha]. When the liquid concentration (1 - [alpha])[[rho].sub.l] becomes very small and the void fraction approaches unity, the equilibrium sound speed becomes almost the same as the frozen sound speed. Such case is just similar to that of gas dynamics and is not the focus of the authors' research. If the liquid concentration becomes large, the difference between the frozen and the equilibrium sound speeds becomes large. The momentum transfer between the phases becomes significant. However, vapor-mist flow cannot be maintained anymore when the void fraction becomes less than 0.5. Then, Equation 4 will not hold anymore. In our calculation, the values of [alpha] were slightly higher than 0.5.

ANALYTICAL MODEL

In analyzing the expansion wave at the nozzle outlet, the analysis must begin in the nozzle, move to the nozzle outlet, and then after the nozzle. The two-phase flow in the nozzle was assumed to be fully developed, so the pressure gradient was taken as constant. Then, the inlet condition of the nozzle was determined using these parameters.

Figure 2 shows the flow field used in the analytical modeling. The analytical area was divided into the diverging section of the nozzle and the back-pressure chamber. The boundary condition at the top was set to symmetric condition, because it was the center axis of the nozzle. The lower and the right parts were set as free outlets.

[FIGURE 2 OMITTED]

The inlet condition of the nozzle was determined by using the basic equations of one-dimensional two-phase flow, which were derived from Equations 1, 2, 6, and 7. The difference in velocities between the vapor and the liquid were expressed as

[v.sub.g] - [v.sub.l] = ([alpha]/[1 - [alpha]][[tau]/[[rho].sub.l]] - [[alpha]/[1 - [alpha]]][[[rho].sub.g]/[[rho].sub.l]][tau]/[rho])( - [[partial derivative]p]/[[partial derivative]x]), (21)

where [rho] is the average density of the two-phase fluid given by the equation

[rho] = [alpha][[rho].sub.g] + (1 - [alpha])[[rho].sub.l]. (22)

Equation 21 was derived by transforming Equation 6, which is the momentum-conservation equation of the vapor, into one dimension.

The pressure-gradient term in Equation 21 was written as

[1/p][[dp]/[dx]] = (1 - [1 - [alpha]]/[alpha])[[M.sub.e.sup.2]/[1 - [M.sub.e.sup.2]]][1/A][[dA]/[dx]]. (23)

The divergence angle of the nozzle was set to 1.5[degrees], which was the same as that of the nozzle used in the experiment. The width of the inlet was defined as the characteristic length L, and the length of the diverging section of the nozzle was set as 4L. The size of the back-pressure chamber after the nozzle was made 10L x 5L.

Calculations were carried out in nondimensional form. The coordinate length was nondimensionalized by L. The velocities and sound speed were nondimensionalized by the isothermal sound speed of the vapor phase [(p/[[rho].sub.g]).sup.[1/2]]. The relaxation time was nondimensionalized by dividing it with the time that the inlet isothermal sound of the vapor phase takes in travelling through the inlet width. It is expressed as

[tau]' = [[tau]/[L/[square root of [[p.sub.0]/[[rho].sub.g0]]]]]. (24)

Numerical calculation was made by applying the CIP method (Yabe et al. 2004) to the basic equations. Unsteady two-dimensional calculation was carried out, and the final solution was obtained when the pressure became constant.

RELATIONSHIP OF EXPANSION WAVES TO RELAXATION TIME

Figures 3 and 4 show the streamline distribution for each phase, calculated at different relaxation times. The given numerical parameters were inlet quality of 0.1, density ratio [[rho].sub.l]/[[rho].sub.g] of 1000, and nondimensional inlet velocity of 0.5. There is a relative velocity between the vapor and the liquid throughout the nozzle; however, steady-state assumptions were applied to the nozzle inlet condition. The nondimensional relaxation time was set to 0.1 and 1.0 in the simulation.

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

In Figures 3 and 4, dashed and solid lines show flow lines of the vapor and the liquid, respectively. The frozen sound speed governed the flow field and was almost equal to the sound speed of the vapor. Although the inlet velocity was lower than the frozen sound speed, the flow formed a supersonic flow field. When the relaxation time was short, the flow behaved as supersonic if the Mach number of the equilibrium sound speed was larger than unity, even though the flow was subsonic based on the governing equations.

In the calculation, expansion waves did not occur when the inlet velocity was smaller than the equilibrium sound speed. Therefore, the simulated two-phase expansion waves showed supersonic characteristics even if the Mach number of the frozen sound speed was subsonic.

When the relaxation time became long, the liquid streamlines became much more straight compared to the vapor streamlines. This was due to the increase in the inertial force of the droplets, because the droplet size became larger when the relaxation time was increased.

Figure 5 shows the pressure distributions along the centerline of the analytical area for relaxation times of 0.01, 0.1, and 1.0. The vertical and the horizontal axes were nondimensional pressure and distance from the inlet of the analytical area, respectively. Inlet velocity, which was three times the equilibrium sound speed, and divergence angle of the nozzle equal to 1.5[degrees] were used in the analysis. The pressure distributions throughout the nozzle were the same for different relaxation times, and they were almost linear. On the other hand, the pressure distributions after the nozzle became curves, and they were different from each other because of their different relaxation times.

[FIGURE 5 OMITTED]

The axial locations where the expansion waves reached the centerline of the analytical area for the first time moved upstream, although the inlet velocity was the same. This trend of the expansion waves showed that the Mach number decreased with increasing relaxation time.

COMPARISON WITH THE EQUILIBRIUM THEORY

In gas dynamics, the semi-infinite flow that turns around an edge is known as the Prandtl-Meyer expansion fan. The pressure of the expansion wave in this case is shown as a function of the angle only. The relationship between the pressure and the expansion angle from the nozzle outlet was investigated in order to determine the expansion characteristics of two-phase flow.

An equation relating the expansion wave of equilibrium two-phase flow to the compressible characteristic of the vapor was derived. It was assumed that the pressure was proportional only to the density and that the velocities of the vapor and the liquid were the same. The velocity potential was expressed as

[phi] = f([theta]). (25)

Then, the expression

[([df]/[d[theta]]).sup.2] = [[dp]/[d[rho]]] (26)

was obtained from the continuity and the momentum equations of two-phase flow in the r-[theta] coordinate. The radial coordinate originates from the edge of the nozzle outlet.

In the case of potential flow, the mass-conservation equation of two-phase flow can be integrated. The integrated equation was written as

[[partial derivative]/[[partial derivative][x.sub.j]]]{[[v.sub.i.sup.2]/2] + [[p + Klogp]/[K([[rho].sub.g0]/[p.sub.0]) + [[rho].sub.l]]]} = 0, (27)

where the expression in the braces is conserved.

The following equation was obtained by combining the continuity equation (Equations 1 and 2) and the state equations (Equations 8 and 9).

[v.sub.j][[partial derivative]/[[partial derivative][x.sub.j]]]([[alpha]p]/[1 - [alpha]]) = 0 (28)

This equation was obtained because the expression in the parentheses was conserved along the flow line. The term [alpha]p/(1 - [alpha]) in Equation 28 was constant, so it was written as a constant K. Then, Equation 26 was expressed as

[[dp]/[d[rho]]] = [[(p + K).sup.2]/[K{K([[rho].sub.g0]/[p.sub.0]) + [[rho].sub.l]}]]. (29)

Eventually, Equation 27 became

[[f.sup.2]/2][[[(p + K).sup.2]/(2K) + p + Klogp]/[(K[[rho].sub.g0])/[p.sub.0] + [[rho].sub.l]]] = constant. (30)

In Equation 30, p became a function of f only. Therefore, Equation 26 was resolved, and p was obtained as a function of [theta].

Figures 6 and 7 show the pressure profiles obtained by using the Prandtl-Meyer equilibrium theory and those obtained by using the nonequilibrium theory. Inlet velocity, which was three times the equilibrium sound speed, and divergence angle of the nozzle equal to 1.5[degrees] were also used in this analysis. The selected values of nondimensional relaxation times [tau]' were 0.01 and 10.0. The vertical axis is the nondimensional pressure and the horizontal axis is the angle from the corner of the nozzle outlet.

[FIGURE 6 OMITTED]

[FIGURE 7 OMITTED]

When [tau]' was 0.01, the pressure profiles numerically simulated using the equilibrium theory and those using the nonequilibrium theory were almost the same. Therefore, the flow could be considered as equilibrium flow when nondimensional relaxation time was smaller than 0.01. Conversely, the numerical results using nonequilibrium theory did not agree well with those using the equilibrium theory when [tau]' was 10.0. In Figure 7, the pressure decreased while increasing the distance r/[L.sub.out] from the edge. Then, the pressure profiles could not be expressed as a function of [theta] only.

When the nondimensional relaxation time was large, the frictional force between the vapor and the liquid was also large because of the high velocity difference between the phases. Therefore, the attenuation of pressure could be attributed to the increase in the internal friction of two-phase flow.

COMPARISON WITH THE PREVIOUS EXPERIMENTAL RESULTS FOR LIQUID-VAPOR FLOW OF WATER

Two-phase expansion waves that were obtained from experiment and from calculation considering relaxation phenomena are presented in this section. The comparison was categorized by considering phase change first and then by neglecting phase change. However, the overall characteristics of expansion waves in terms of relaxation phenomena from all results were also compared.

The expansion waves of liquid-vapor flow of water at the outlet of a nozzle were previously investigated by the authors through experiment (Nakagawa et al. 2007). The nozzle was designed for geothermal powerplants. Figure 8 shows the calculated and experimental pressure distributions along the center of the channel using dash-dot and line-circle curves, respectively. Phase change was considered in obtaining both profiles. Based on subsonic flow in gas dynamics, the pressure must be increased from the outlet due to the rapid increase in the flow area. Conversely, the experimental curve shows a decrease. On the other hand, based on supersonic flow in gas dynamics, the pressure after the outlet must remain constant first before it decreases at the start of expansion wave. The calculated profile based on equilibrium theory is the same as the profile that is expected from supersonic flow in gas dynamics. The pressure distribution right after the nozzle outlet that was predicted by using the equilibrium theory does not appear in the experimental pressure distribution. The results predicted by using the equilibrium theory do not match the experimental results because of nonequilibrium phenomena in two-phase flow. The experimental situation can also occur at the outlet of ejector nozzles in refrigeration systems. The pressure profiles of expansion waves can be explained by our study of supersonic two-phase flow. Similar situations can occur at the outlet of the ejector nozzle for refrigeration cycle. Those pressure profiles can be explained using our study on the dynamics of supersonic two-phase flow presented in this paper.

[FIGURE 8 OMITTED]

The analytical result, which considered nonequilibrium phenomena by momentum relaxation, was plotted using a solid curve, as shown in Figure 9. On the other hand, the analytical result, which considered equilibrium phenomena, was plotted using a dash-dot curve. Phase change was not considered in obtaining both profiles. The nondimensional relaxation time used in the analysis was 10, based on our experiment for the liquid-vapor flow of water corresponding to Figure 8. Variable d was in the order of 10 [micro]m (4 x [10.sup.-4] in.), [[rho].sub.l] was in the order of 1000 kg/[m.sup.3] (4 x [10.sup.-5] lb/[in..sup.3]), [[mu].sub.g] was in the order of [10.sup.-5] Pa*s (1 x [10.sup.-9] psi*s), L was in the order of 10 mm (0.4 in.), and the inlet isothermal sound speed of the vapor phase [([p.sub.0]/[[rho].sub.g0]).sup.[1/2]] was about 300 m/s (984 ft/s). The value of [tau]' around 10 was obtained from those parameters. We also obtained a Mach number of around 3 from the experiment, and it was also used in calculating both profiles in Figure 9. The pressure profile from the analysis considering nonequilibrium momentum transfer does not have a flat upper part and is similar to the nonconstant part of the experimental decompression curve. This pressure distribution shows that the start of expansion moves upstream relative to the start of expansion for the equilibrium theory because of the large pressure drop. The pressure drop on the other hand is due to the energy loss created by the interphase momentum transfer at the outlet. Moreover, Mach number decreasingly approaches unity due to the increasing sound speed of two-phase flow with momentum relaxation.

[FIGURE 9 OMITTED]

It is very important to predict the two-dimensional pressure and velocity profiles from the outlet of a nozzle in an ejector refrigeration system or in a geothermal powerplant. The ejector or the geothermal system must be operated not only to obtain the optimized pressure profile of the nozzle but also to avoid underexpansion or overexpansion after the nozzle outlet. Avoiding underexpansion or overexpansion can help in obtaining a designed coefficient of performance of an ejector refrigeration system or a designed efficiency of a powerplant. The present study shows how to calculate the pressure and the velocity fields of two-phase flow by only one simple parameter, which is the relaxation time. This study is useful in designing an ejector refrigeration system or a geothermal powerplant.

CONCLUSION

The characteristics of the expansion wave in high-speed vapor-mist flow with velocity relaxation phenomenon were presented in this paper. It was shown that there were two types of sound speed in the high-speed two-phase flow. One was the equilibrium sound speed, and the other was the frozen sound speed. The sound speed was closer to the equilibrium sound speed when the relaxation time was short, but it was closer to the frozen sound speed when the relaxation time was long. The start of expansion moved upstream with increasing relaxation time.

The pressure profiles simulated using nonequilibrium theory with relaxation phenomena resembled the pressure profiles from the experiments. Conversely, the pressure profiles simulated using the equilibrium theory were far from the pressure profiles obtained from the experiments. The nonequilibrium flow theory with relaxation phenomena could predict the behavior of expansion waves in supersonic two-phase flow more satisfactorily than the equilibrium flow theory.

The present study shows how to calculate the pressure and velocity fields of two-phase flow by only one simple parameter, which is the relaxation time. This study is useful in designing an ejector refrigeration system or a geothermal powerplant.

NOMENCLATURE

a = sound speed, m/s (in./s)

d = droplet diameter, m (in.)

f = function of the turn angle

k = complex propagation number,--

k = component of complex propagation number,--

p = pressure, Pa (psi)

r = radial coordinate, m (in.)

t = time, s

v = velocity, m/s (in./s)

x = coordinate axis, m (in.)

A = cross-sectional area, [m.sup.2] ([in..sup.2])

K = assigned constant, Pa (psi)

L = characteristic length, m (in.)

M = Mach number,--

Subscripts

0 = inlet

e = equilibrium flow

f = frozen flow

g = vapor phase

i = direction

j = direction

l = liquid phase

out = outlet

Superscript

' = nondimensional

Overbar

- = constant-value component

Hat

^ = small-disturbance component

Greek Symbols

[alpha] = void fraction,--

[theta] = turn angle, rad, degrees

[micro] = viscosity, Pa*s (psi*s)

[rho] = average density, kg/[m.sup.3] (lb/[in..sup.3])

[tau] = relaxation time, s

[phi] = velocity potential

[omega] = frequency, Hz

REFERENCES

Campbell, I., and A. Pitcher. 1958. Shock waves in a liquid containing gas bubbles. Proceedings of the Royal Society, A: Mathematical, Physical and Engineering Sciences, London, UK, 243(1235):534-45.

Landau, L., and E. Lifshitz. 1959. Fluid Mechanics. Oxford: Pergamon Press.

Marble, F. 1963. Dynamics of a gas containing small solid particles. Proceedings of the 5th AGARD Combustion and Propulsion Colloquium, Braunschweig, Germany, pp. 175-213.

Mori, Y., and J. Suyama. 1980. Geothermal Energy Reader. Tokyo: Ohmsha, Ltd.

Nakagawa, M. 2004. Refrigeration cycle with two-phase ejector. Refrigeration 79(925):856-61.

Nakagawa, M., A. Harada, and M.S. Berana. 2008. Analysis of expansion waves appearing in outlets of two-phase flow nozzles. Proceedings of the 12th International Refrigeration and Air Conditioning Conference at Purdue, West Lafayette, Indiana, USA, 2327:1-8.

Nakagawa, M., and N. Mamiya. 1986. Propagation of rarefaction waves in bubble flow with evaporation. Transactions of the JSME (B) 52(474):650-56.

Nakagawa, M., H. Miyazaki, A. Harada, and Z. Ibragimov. 2007. Expansion waves at the outlet of the supersonic two-phase flow nozzle. Journal of Thermal Science and Technology 2(2):291-300.

Nakagawa, M., H. Takeuchi, and A. Yokozeki. 1998. Performance of the two-phase nozzles using the refrigerant R134a. Transactions of the JSME (B) 64(626):3407-13.

Yabe, T., H. Mizoe, K. Takizawa, H. Moriki, H. Im, and Y. Ogata. 2004. Higher-order schemes with CIP method and adaptive Soroban grid towards mesh-free scheme. Journal of Computational Physics 194(1):57-77.

Masafumi Nakagawa, PhD

Atsushi Harada

Menandro Serrano Berana

Received October 23, 2008; accepted May 21, 2009

Masafumi Nakagawa is an associate professor, and Atsushi Harada and Menandro Serrano Berana are doctoral students in the Department of Mechanical and Structural System Engineering, Toyohashi University of Technology, Toyohashi City, Aichi, Japan. Menandro Serrano Berana is also an assistant professor, currently on study leave, in the Department of Mechanical Engineering, University of the Philippines Diliman, Quezon City, Philippines.
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Author:Nakagawa, Masafumi; Harada, Atsushi; Berana, Menandro Serrano
Publication:HVAC & R Research
Article Type:Report
Geographic Code:1USA
Date:Nov 1, 2009
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