Single-phase and two-phase flows through helical rectangular channels in single screw expander prototype*

XIA Guo-dong (夏国栋), LIU Xian-fei (刘献飞), ZHAI Yu-ling (翟玉玲), CUI Zhen-zhen (崔珍珍)

Key Laboratory of Enhanced Heat Transfer and Energy Conservation, Ministry of Education, College of Environmental and Energy Engineering, Beijing University of Technology, Beijing, China, E-mail: xgd@bjut.edu.cn

Single-phase and two-phase flows through helical rectangular channels in single screw expander prototype*

XIA Guo-dong (夏国栋), LIU Xian-fei (刘献飞), ZHAI Yu-ling (翟玉玲), CUI Zhen-zhen (崔珍珍)

Key Laboratory of Enhanced Heat Transfer and Energy Conservation, Ministry of Education, College of Environmental and Energy Engineering, Beijing University of Technology, Beijing, China, E-mail: xgd@bjut.edu.cn

(Received December 5, 2012, Revised January 31, 2013)

The CFD simulations are carried out for the flows in a horizontally oriented helical pipe with various inlet sectional liquid holdups and coil pitches ()H. The development of the pressure fields for the single phase air flow and the air-water two-phase flow through the helical rectangular channels is studied. The points with a higher pressure often become the position of expansion leakage. The liquid phase distribution at these points can prevent the leakage of air. It is shown that the increase of the inlet sectional liquid holdup may increase the local liquid holdup at the outmost side of the helical channel. Based on the published pressure drop correlation, a new modified relation for predicting the pressure drop in the helical rectangle channel is proposed.

helical channel, numerical simulation, flow characteristics

Introduction

As important parts of industrial processes, the helically coiled channels are used in the power generation, the food industry, the environmental engineering, the heat recovery systems, the air-conditioning systems and the space industry[1]and they receive increasing attentions in recent years.

The single-phase fluid flow characteristics in the helical channels were widely studied both theoretically and experimentally. The numerical method has several advantages in scientific researches, e.g., a large amount of results can be obtained from parametric studies, with a low cost. In addition, due to certain complexity of the flow processes in the helically coiled tubes, the numerical investigations are superior to the experimental ones.

The flow characteristics in the helical channel were much studied. The researches are mainly concentrated on the effect of the torsion and the curvature of the helical channels[2,3]. The velocity contours, the turbulent kinetic energy, the turbulent intensity and the combined effects of rotation, torsion and curvature on the fluid flow through a helical pipe were addressed. In fact, the results obtained from a mathematical model must be validated by comparison with experimental data. Colorado et al.[4]presented an experimental validation of a computational model for a helically coiled steam generator. All flow variables (pressure, quality of steam, velocity, etc.) were evaluated at each point of the grid in which the domain was discretized. The proposed model was validated against the results of an experimental study carried out at SIET labs (Italy). Guo et al.[5]presented an experimental investigation of the frictional pressure drop of the single-phase water flow in a helical pipe. Experimental data show that the frictional pressure drop of smaller coils is higher than that of larger ones, and this difference is even more pronounced in the high Reynolds number region. Mandal and Nigam[6]carried out an experimental study of the pressure drop in a helical double-pipe heat exchanger. On the basis of the experimental results, new correlations for the friction factor in the inner as well as outer tubes were obtained.

A number of correlations for the two-phase flow frictional pressure drop could be found in literature.Most of studies of the two-phase flow in helically coiled tubes employed the correlations based on the Lockhart-Martinelli parameter[5,7-10]. On the basis of the experimental results, some new correlations for the frictional pressure drop correlation of two-phase flows inside a small helical pipe were proposed. Zhang et al.[11]investigated the oil-water separation mechanism inside the helical pipes. The results show that the separation efficiency of the helical pipes is a function of the gyration radius, the residence time, the pressure at the inlet and the outlet, the density difference between phases, the liquid droplet diameter, etc.. Vashisth and Nigam[12]investigated the local variables and the interfacial phenomena for two-phase flows through a helical pipe, the evolutions of the velocity, the local and average friction factors, the interfacial friction factor, the phase distribution and the entry length were obtained. Jayakumar et al.[13]presented the hydraulic characteristics of the air-water two-phase flow in helical pipes. The CFD analysis was carried out by varying these parameters and their influence on the thermal hydraulic characteristics of the two-phase flow was revealed.

The multi-component multiphase flow structure in the complex three-dimensional curve channel of an expander is one of the most important factors that affect the efficiency of the expander[14]. The previous researches were mainly focused on the processing and the performance of the prototype, the multi-component multiphase flow structure in the complex three-dimensional curve channel remains an issue for further studies. We do not have much knowledge about the local variables like the flow profiles, the local pressure distribution, the phase distribution and the entry length development. This paper aims to explore the evolutions of the pressure and the distribution of the liquid for the flows in the helical rectangle channel to help locating the position vulnerable to leakage and revealing the effect of the distribution of the liquid in sealing the leakage gap that affects the efficiency of the single-screw expansion. With few experimental results available about hydrodynamics of the fluid flow through helical rectangular channels, extensive numerical investigations might give us insight into the physics of the problem.

1. Theoretical and numerical methodology

1.1Characteristics of helical rectangular channel

In the present analysis, the helical rectangular channel is assumed to be horizontally oriented. A schematic diagram of the helical rectangular channel with its main geometrical parameters is shown in Fig.1. The side in the horizontal direction isain length, and that in the vertical direction isb. The coil radius (measured from the centre of the channel to the axis) isR. The distance between two adjacent turns is called the pitchH. The equivalent radius of the channel is defined asr=2A/c(whereAis the area for cross-section perimeter of helical channel). The side of the channel wall nearest to the coil axis is termed the inner side I and the farthest side is termed as the outer side O. The dimensionless curvature and torsion can be defined asδ=r/Randλ=H/2πR. The Reynolds numberRe=ρvd/μ(whereρis the density of fluid andμis the viscosity). The Dean number,De=Reδ0.5,is used to characterize the flow in the helical pipe. The centrifugal acceleration is defined asv2/R.

Fig.1 Schematic diagram of helical channel with its main geometrical parameter

1.2Mathematical foundation

1.2.1 For single-phase flow

With the air used as the working fluid, the numerical model for the fluid flow in the helical rectangular channel is defined under the following assumptions:

(1) Unsteady three-dimensional fluid flow.

(2)kε- model.

(3) Gravity as a factor in operating conditions.

Jayakumar et al.[3]presented an analysis of the single-phase fluid flow inside a helical channel. In another paper[15], an experimental research was conducted for the estimation of the CFD calculation results. The experimental results were found to be in agreement with the calculation results obtained by using the CFD package. The flow in the helical rectangle channel is calculated by us with three differentkεmodels. According to our calculations, there is no obvious difference between the results of these three models.

1.2.2 For two-phase flow

The numerical computations are carried out by solving the governing conservation equations along with the boundary conditions using the finite volume method. The following concepts and assumptions are adopted:

(1) Incompressible three–dimensional air-water two-phase flow.

(2) Turbulent flow andkε-model.

(3) Eulerian model is chosen as the multiphase model.

(4) Gravity is included in operating conditions.

The numerical methods for solving the governing equations and the closure laws of the two-phase flow are extremely complicated. In order to replace the single-phase model with the multiphase model in our simulations, additional conservation equations must be introduced and the original settings must be modified, including the introduction of the phase volume fraction and the mechanism of the momentum exchange between the phases. The volume fraction represents the space occupied by each phase. The momentum and continuity equations are solved for each phase.

The summation of the volume fractions is equal to unity.

wherepqvis the interphase velocity,pqRis the interaction between the phases,τis the stress-strain tensor,qFis the external body force,,vmqFis the virtual mass force andlift,qFis the lift force.

The hydrodynamic boundary conditions for the fluid flow in the helical rectangle channel are:

At the channel wall surface: no-slip and no-penetration.

At the inlet,in=vv.

At the outlet, =xH, pout=1atm.

1.3Validation of hydrodynamic modeling

The grid used in the present study is shown in Fig.2. The grid refinement test is carried out to find an appropriate grid for studying the flow in the helical rectangular channel.

Fig.2 The whole grid of the helical rectangular channel used for analysis

Fig.3 Pressure drop of air-water at different cross-sections with different interval sizes

Figure 3 shows the pressure drop for different grid systems. The predicted results for different systems for the fully developed flow are compared. The deviations of the pressure drops using the interval size of 0.005, 0.003, 0.0025 and 0.002 with respect to that obtained by using the interval size of 0.0018 are 5.59%, 2.97%, 0.159% and 0.04%, respectively. It is shown that the grid arrangement with the interval size of 0.002 gives satisfactory results for studying the fluid flow in the helical rectangular channel. The model geometry is built with ProE, and then imported into GAMBIT for meshing. The governing equations are solved using the commercial code Fluent 6.3.26. The velocity boundary condition is used at the inlet and the pressure boundary condition at the outlet. For the momentum equation, the walls are specified as in no-slip condition.

1.3.1 For single-phase flow

The pressure based solver with the Green-Gauss Cell based gradient option is used. The pressure-velocity coupling is realized by using the SIMPLE scheme.The pressure is discretized by using the standard scheme (which is one of the interpolation schemes for calculating the cell-face pressure while using the segregated solver in the Fluent). The First Order Upwind of discretization is used for the momentum, turbulent kinetic energy and dissipation rate equations.

1.3.2 For two-phase flow

The phase coupled SIMPLE scheme is used for the pressure-velocity coupling. The momentum and void fraction are discretized by using the QUICK scheme. The first order upwind discretization is used for the turbulent kinetic energy and dissipation rate equations. In addition, the time step, the maximum number of iterations per time step, and the relaxation factors are carefully adjusted to ensure convergence. Convergence criterion used is 10-5for continuity, velocities of phases,k,εand water void fraction.

Fig.4 Comparison of present predictions with Eq.(5)

2. Results and discussions

2.1Comparison of numerical predictions and formulas

2.1.1 For single-phase flow

The pressure drop formula for the single-phase turbulent flow through a small horizontal helically coiled tube is given by Zhao et al.[1]as

wherePΔ is the total pressure drop in the helical channel,Sfis the friction factor andGis the mass flow rate. The measurement distance for different pressures isL.dis the diameter of the cross-section andDis the coil diameter of the helical channel. Figure 4 shows a comparison between the present predicted pressure drops and those obtained by Eq.(5). The pressure drop measured by the CFD is in good agreement with the values estimated by using formula. The data are predicted within a maximum error of 16%. These results have validated the hydrodynamic model used in the analysis.

2.1.2 For two-phase flow

The pressure loss in different helical pipes is given by Murai et al.[10], where the static pressure difference due to gravity is not involved.

Fig.5 Comparison of present predictions with Eq.(7)

Figure 5 shows a comparison between the present predicted pressure drops and those obtained by Eq.(7). The pressure drops measured by the Fluent is in good agreement with those estimated by using Murai et al. formula. The data are predicted within a maximum error of 24.3%. The empirical formulas for the pressure drop in the helical channel are obtained for the low Reynolds number flow from experimental data. The deviation of the numerical prediction from the calculation results by using the empirical formulas increases with the increase of the velocity. Due to the difference in the section structure of the helical rectangular channel and the helically coiled channel, the frictional pressure drop within the helical rectangular channel might differ from the calculation results by Eqs.(5)and (7). The formulas (Eqs.(5) and (7)) are modified to correlate the frictional pressure drop in the helical rectangular channel as

(1) For single-phase air flow

Fig.6 Comparison of predictions with the modified formula

Figures 6(a) and 6(b) show the comparison between the present predicted pressure drops and those obtained by the modified Eq.(9) for the single phase flow and by Eq.(10) for the air-water two-phase flow, respectively. The pressure drops from the simulations are in excellent agreement with those estimated by using the modified formulas. The maximum errors for the single-phase flow and the two-phase flow are found to be 1% and 2.5%, respectively.

2.2Velocity field, pressure profile and distribution of liquid holdup

2.2.1 For single phase flow

In this section, the flow characteristics of the air single phase and air-water two-phase flows through the helical rectangular channel are analyzed. At the inlet of the helical channel, the superficial velocity is given as 25 m/s for the single phase air. For the twophase mixture, the superficial velocity of the air is 22 m/s and that of the liquid is 3 m/s. The inlet sectional liquid holdup used in the analysis is 0.12. The curvature of the helical channel is 0.16. The other parameters are:a=0.03 m,b=0.035 m,R=0.1 m, andH=0.4396 m.

To study the effect of the torsion, a comparison of the evolution of a dimensionless number,v*=vmax/vavg, for the air flow in the horizontal helical channel is made. Figure 7 shows the evolution ofv*for the dimensionless torsionλ=0.1, 0.5 and 0.7, to see its influence onv*. With the increase of the torsion, the flow becomes more distorted due to the torsional force of the helical channel and this leads to the increase ofv*. The effect of the wall friction is to reduce the fluid velocity at different axial locations for the flow in the helical channel, and the magnitude ofv*is also decreased. The evolution ofv*can be described in two stages. In the early stage (Φ＜180o), the dimensionless numberv*drops sharply for the flow in the helical channel due to the rapid development of the flow boundary layer. In the later stage (Φ＞180o), the dimensionless numberv*varies smoothly with the increase ofΦuntil a fully developed flow is established. The velocity drops down greatly at the early stage with the increase of the torsion. Because of the increase of the torsion, the flow becomes more distorted and the length of the fluid flow at the same angle is also increased.

Fig.7 Effect of pitch on the evolution of dimensionless numbermaxavg/vv

Figure 8 shows the pressure profiles at different sections for the single phase air flow. In each figure the left side indicatesx/a=0 and the right sidex/a=1. The evolution of the pressure as the fluid flows downstream is shown. Due to the torsion, the highpressure region is mainly located at /=x a0 in the entrance of the helical channel. As the fluid flows downstream, the centrifugal force acting on the main flow leads to the forming of the high pressure region at the outer side of the helical channel. As the flow is developed, the pressure contours are almost symmetrical with respect to the left side and the right side. This shows that the effect of the torsion is negligibly small for a fully developed flow in the helical channel.

Fig.8 The pressure profiles at different sections

Fig.9 Pressure for both sides at =yb at different sections

The pressure on both sides of the helical channel at =ybis the main driving force for the air expansion leak. Figure 9 shows the pressure on both sides of the helical channel at =ybat different sections. When the fluid flows through the helical pipe, the pressure on the left side is higher than that on the right side due to the effect of the torsional force. It can be seen from Fig. 9 that the pressure on both sides of the helical rectangle channel at =ybincreases as the torsion increases from 0.1 to 0.7. This is because the effect of the torsional force on the flow in the helical channel is increased with the increase of the torsion. As the fluid flows downstream, the effect of the torsion on the pressure on both sides of the helical channel at =ybcan also be neglected.

2.2.2 For air-water two-phase flow

Figure 10 shows the pressure on both sides of the helical channel at =ybat different sections for the air-water two-phase flow. The pressure variation for the air-water two-phase flow in the leakage clearance can make the fluid flow turn from one flow pattern into another. The change of the flow pattern in the leakage clearance is important for the performance of the single screw expander. Due to the influence of the torsion on the fluid flow, the second flow is induced in the -xaxis direction. The pressure for the air-water two-phase flow in the helical rectangular channel is almost in the form of a cosine wave.

Fig.10 Pressure on both sides at =yb at different sections for two-phase flow

Fig.11 The liquid distribution at different sections

Figure 11 shows the development of the local liquid holdup at different sections. In each figure the left side indicates /=x a0 and the right side as /=x a1. The local liquid holdup at one point is the liquid volume fraction within the small region. The volume fraction equation will not be solved for the primary phase. The volume fraction of the primary phase is computed based on the constraint of Eq.(1), which is defined as the volume of a constituent divided by the volume of all constituents of the mixture. Due to the effect of the torsion, the local liquid holdup on the left side is much higher in the entrance of the helical channel. During the development of the two-phase flow, the effect of the centrifugal force gradually becomes significant, by which the liquid tends to move along the outer side ofthe channel. In this set of simulations, the centrifugal acceleration is much higher than the acceleration due to the gravity. This leads to a high liquid holdup along the outer side of the helical rectangular channel. The distribution of the liquid phase on both sides of the helical channel plays an important role in preventing the air leakage in the leaked clearance. With the increase of the local liquid holdup on both sides of helical rectangle at =yb, the blocking effect of the liquid is enhanced and the air leakage can be greatly reduced. If the liquid holdup is not enough to seal the leak clearance, the flow pattern changes from the liquid leakage into the air-water leakage. When the fluid flows through a helical channel, the torsion has an influence on the secondary flow, which leads to the liquid phase oscillating on the outmost side of the helical rectangular channel.

Fig.12 Development of local liquid holdup under different torsions

Figure 12 shows the development of the local liquid holdup for both sides of the helical rectangle aty=bat different sections under different torsions. In the figure, the left side indicatesx/a=0 and the right sidex/a=1. In the entrance of the helical channel, the local liquid holdup on the right side decreases with the increase of the torsion, quite contrary to the case on the left side. With the development of the flow, the local liquid holdup on both sides aty=bincreases with the increase of the torsion. The liquid holdup saturates at the later developing stage (Φ≥200o), and the value of the local liquid holdup is almost equal to 1. The effect of the torsion on the development of the local liquid holdup under different torsions is important to understand the sealing effect of liquid in a single screw expander prototype.

2.3Analysis of the development of pressure and local liquid holdup under different inlet sectional liquid holdups

In this section, the pressure and the local liquid holdup on both left and right sides aty=bare analyzed for the air-water two-phase flow through the helical rectangle channel where the inlet sectional liquid holdups are varied. In all these cases, the inlet velocity of the mixture is specified to be 25 m/s. The curvature and the torsion of the helical channel are 0.7 and 0.16, respectively. The inlet sectional liquid holdups are chosen as 0.12, 0.15, 0.18 and 0.2.

Fig.13 Development of the pressure (a) and liquid holdup (b) under different inlet sectional liquid holdup

Figures 13(a) and 13(b) show the development of the pressure and the local liquid holdup in the helical rectangle channel under different inlet sectional liquid holdups at different sections, respectively. With the increase of the inlet sectional liquid holdup, the pressure and the local liquid holdup on both left and right sides at =ybin the entire flow process are increased. Due to the effect of the centrifugal force, the increase of the inlet sectional liquid holdup makes more liquid go to the outmost side of the helical rectangle channel. This plays an importance role in preventing the air leakage in the process of air expansion. These observations provide some insight to improve the efficiency of the single screw expander and the necessary theory for further improving the performance of the single screw expander. The increase of the inlet sectional liquid holdup will decrease the volume of the air within the helical channel. This reduces the advantage of the bigger expansion ratio of the single screw expansion.

3. Conclusions

The 3-D CFD simulations of the single-phase air and air-water two-phase mixture flows through helical rectangular channels with finite pitch are carried out by using the commercial CFD package. The present predicted pressure drops are compared with those calculated by the experimental formulas. Hydrodynamics of the single-phase air and air-water two-phase flows through the helical rectangular channels are validated, and the results are in good agreement with experimental ones. Based on published pressure drop formulas, a new modified formulas are proposed for predicting the pressure drops in the helical rectangle channel.

The CFD simulations are carried out for the horizontally oriented helical channel with various inlet sectional liquid holdups. The effect of the coil pitch ()His also investigated. Unlike the flow through a straight pipe, due to the centrifugal force induced by the curvature of the helical channel, the heavier phase tends to move along the outer side of the channel. Higher pressure is also observed along the outer side. The torsion caused by the pitch of the helical channel makes the flow unsymmetrical about the plane of the helical channel. With the increase of the inlet sectional liquid holdups, the pressure and the local liquid holdup on both left and right sides at =ybduring the entire flow process are increased. These observations provide insight into the physics of the fluid flows in the helical rectangular channels

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10.1016/S1001-6058(14)60013-5

* Project supported by the National Basic Research Program of China (973 Program, Grant No. 2011CB710704), the National Natural Science Foundation of China (Grant No. 51176002) and the Research Fund for the Doctoral Program of Higher Education (Grant No. 20111103110009).

Biography: XIA Guo-dong (1965-), Male, Ph. D., Professor