Comparison of Cylindrical and 3-D Cubic B-Spline Curve of Shroud-Diffuser for a Supersonic Pressure Exchange Ejector in a 3-D, Non-Steady, Viscous Flow

KHALED ALHUSSAN* and CHARLES GARRIS**

* Research Assistant Professor, Space Research Institute, KingAbdulazizCity for Science and Technology, P.O. Box 6086Riyadh11442, KSA

** Professor, Mechanical and Aerospace Engineering Department, GeorgeWashingtonUniversity, 801 22 Street N. W., Washington DC, USA

Abstract:-This paper will compare the different shapes of the shroud-diffuser of the supersonic pressure exchange ejector in regard to the flow induction between two fluids. Two different shapes of shroud-diffuser are used in this analysis; one is a cylindrical shape and the other is a 3-D cubic B-spline curve.

The work to be presented herein is a Computational Fluid Dynamics investigation of the complex fluid mechanisms that occur inside a non-steady, three-dimensional, supersonic pressure exchange ejector, specifically with regard to the pressure exchange mechanisms and the induction processes between a “driving” primary fluid and a “driven” secondary fluid and how this is related to the shape of the aerodynamic shroud- diffuser surface.

The results will show, that the 3-D cubic B-spline curve is capable of producing the desired aerodynamics flow structure of the flow induction in a three-dimensional supersonic, non-steady, viscous flow. Results of contour plots of static pressure ratio will demonstrate that the shape of the shroud-diffuser is a critical element in flow induction mechanism, especially when a pressure recovery is needed. Results of velocity vectors of the two different shroud-diffuser shapes will show the structure of flow induction mechanism in a complex three-dimensional conical surface.

Key- Words: Ejector, Pressure Exchange, Ejector Refrigeration, Aerodynamic Shroud Surfaces, CFD.

1

Nomenclature

Symbol Meaning

AR Primary inlet area to secondary inlet area

DSHD,E Shroud exit diameter

DP Primary inlet diameter

D* Rotor throat diameter

LR Rotor length

M Mach number of primary fluid at the

discharge nozzle

Sshroud Shroud length

SL Ratio of the shroud length to the rotor

length

XLE Position of the leading edge of the vane

 Ratio of shroud exit diameter to

the primary inlet diameter

 XLE/LR

 Ratio of shroud exit diameter to rotor

throat diameter

 Rotor throat diameter to reference

diameter

 Mass flow ratio of secondary fluid to

primary fluid

1 Introduction

The work to be presented herein is a Computational Fluid Dynamics investigation of the complex fluid mechanisms that occur inside a non-steady, three-dimensional, supersonic pressure exchange ejector, specifically with regard to the pressure exchange mechanisms and the induction processes between a “driving” primary fluid and a “driven” secondary fluid [1].

This research is based upon utilizing a non-steady flow field resulting in the work of pressure forces acting at a fluid interface between primary fluid and secondary fluid. These interfaces are produced through the aerodynamic design of a flow field, non-steady in the laboratory frame of reference, consisting of rotating oblique shock waves and expansion fans. Minimizing the possible losses from the oblique shocks and boundary layers offer the potential of achieving adiabatic efficiencies approaching those of turbo-machines[2-4].

Pressure exchange is a designation applied to any process whereby contiguous fluid bodies or flows exchange mechanical energy through the work of mutually exerted pressure forces at their interfaces [5]. Pressure exchange cannot take place in steady flow, because no work is done by pressure forces acting on a stationary interface [6]. Therefore, pressure exchange is always a non-steady process.

An ejector is a direct contact flow induction device, which exchanges mechanical energy and momentum between a high-energy primary “driving” fluid and a relatively low-energy secondary “driven” fluid to produce a discharge of an intermediate specific energy level. The transfer of momentum gives rise to an increase in the stagnation enthalpy of the secondary fluid and enables the ejector to function as a compressor [7-9].

In an ejector, momentum can be imparted from the primary fluid to the secondary fluid by two mechanisms. The first mechanism is by shear stresses at the tangential interfaces between primary and secondary fluids as a result of turbulence and viscosity. The second mechanism is by pressure-exchange; that is, the work of interface pressure forces acting across normal interfaces separating the primary and secondary fluids. All of the attempts at improving ejector performance have involved variations on the conventional design of the “steady-flow” ejector. The physical principle upon which this device functions is that of entrainment of a secondary fluid by an energetic primary fluid by virtue of the work of turbulent shear stresses. Thus, the relatively low-energy secondary flow is “dragged” by the relatively high-energy primary flow through tangential shear stresses acting at the interface between the two contacting streams. These turbulent stresses are a result of mixing that occurs between primary and secondary streams and therefore exchange of momentum. Inherent in this highly irreversible process is a large entropy rise [10-13].

While this mechanism is quite effective and has been widely adopted in many applications, an inherent characteristic of the mixing process is to dissipate valuable mechanical energy. This results in a substantial entropy rise, which is intimately connected with ejector performance and, consequently, refrigeration system performance.

Unlike the steady flow conventional ejector, the non-steady pressure exchange ejector is designed to utilize an entirely different physical principle, which is based on the pressure exchange phenomenon. Pressure exchange utilizes the reversible work of interface pressure forces, which exist only in non-steady flows. This mode of energy exchange is inherently non-dissipative. The utilization of this mode of energy-transfer is of interest, because of its potential to produce high efficiency [14-17]. The overarching goal is to create a flow induction machine which utilizes the work of interface pressure forces available in non-steady flows through direct contact of two fluids.

A number of important conclusions follow from the current research. First, study of the actual flow configuration inside the complex three dimensional supersonic pressure exchange ejector offers some insight into the complex flow phenomena that occur inside the ejector. Second, designing the aerodynamic shroud surface for the supersonic pressure exchange ejector that produces the right environment for pressure exchange to take place, which results in a high adiabatic efficiency. Adiabatic efficiency is related to both the ratio of mass flow rate and the ratio of temperature of secondary fluid to primary fluid [11 & 18]. The efficiency of the energy exchange is increased if it is made to take place, at least in part, through the non-dissipative work of interface pressure forces.

2 Computational Fluid Dynamics Analysis

The flow induction in the supersonic pressure exchange ejector is so complex that there exist direct fluid-fluid interactions, oblique shock waves, expansion fans, slip surfaces, and shock wave interactions and reflections. The flow is non-steady, viscous, compressible, and high-speed supersonic.

The governing equations are a set of coupled nonlinear, partial differential equations. In order to formulate or approximate a valid solution for these equations they must be solved using computational fluid dynamics techniques. To solve the equations numerically they must be discretized. That is, the continuous control volume equations must be applied to each discrete control volume that is formed by the computational grid. The integral equations are replaced with a set of linear algebraic equations solved at a discrete set of points.

CFX-TASCflow is used in the current research to model the flow in the supersonic pressure exchange ejector. The CFD code is an integrated software system capable of solving diverse and complex multidimensional fluid flow problems. The fluid flow solver provides solutions for incompressible or compressible, steady-state or transient, laminar or turbulent single-phase fluid flow in complex geometries. The code uses block-structured, non-orthogonal grids with grid embedding and grid attaching to discretize the domain. The code system has additional capabilities that can predict subsonic, transonic and supersonic compressible flows, including temperature solutions in solid regions of the domain for laminar or turbulent flow.

CFX-TASCflow is a finite volume method, but is based on a finite element approach of representing the geometry. Thus, the method used here possesses much of the geometric flexibility of finite element methods as well as the important conservation properties of the finite volume method.

It should be possible to model the interaction of the shock waves and expansion fans around the rotating vanes and describe how the secondary flow is drawn into the interstices. It should be possible to study the mutual deflection and pressure exchange processes between primary and secondary flows using the CFD analysis.

A numerical analysis must start with breaking the computational domain into discrete sub-domains, which is the grid generation process. A grid must be provided in terms of the spatial coordinates of grid nodes distributed throughout the computational domain. At each node in the domain, the numerical analysis will determine values for all dependent variables such as pressure and velocity components.

Creating the grid is the first step in calculating a flow. A 180-deg sector was chosen to model the flow. The grid is refined near the surface of the vanes in order to model the large gradient in that region. For computation, a Mach Number of 3.0 is used as the Nozzle Exit Mach Number. Solution parameters and fluid properties are defined in the parameter file. For the steady-state solution, the time step is selected equal to 0.00001. The advection discretization scheme selected is the Modified Linear Profile Skew scheme with the Physical Correction. The convergence criterion is 10E+04 and the maximum number if iteration is 300. The maximum number of iteration is increased form 300 to 1000 to allow the solver to run until a converged solution is found.

A computational model that illustrates the physics of flow induction through non-steady shock waves and expansion fans was developed. This provides strong insight into the mechanisms through which the supersonic pressure exchange ejector operates.

Through this computational analysis, a better interpretation of the physical phenomenon of the non-steady pressure exchange ejector can be achieved. The results from the numerical analysis are used to study the flow structure inside the complex geometry of the pressure exchange ejector and to develop a design methodology so as to predict optimal ejector performance.

3 Scope of the Analysis

It is computationally possible to study the real behavior of the supersonic pressure exchange ejector. A complete investigation of the actual behavior of the flow inside the supersonic pressure exchange ejector is analyzed. The assumptions in this analysis are listed below:

A)Free spinning rotor.

B)Isentropic expansion of primary flow inside the primary nozzle.

C)Matching the static pressure at the discharge nozzle.

D)Steady flow in the rotor frame of reference.

E)Laminar flow.

F)Compressible and viscous flow.

G)Initial guess:

1-using a coarse grid for the first run then improving the solution with a finer grid.

2-starting with subsonic flow, increasing to low-speed supersonic flow and finally to the high speed supersonic flow (Mach number=3.0).

H)Real flow boundary conditions that specify total pressure and total temperature for the inlet conditions and static pressure for the outlet condition.

4 Discussions and Results

This paper describes the geometrical analysis that produces the flow configuration for the supersonic pressure-exchange ejector. To arrive at such design, it is necessary to consider the effect of several important geometrical parameters such as, size and shape of the aerodynamic shroud surface, diffuser, rotor length and diameter, and inlet area ratio.

The domain of interest is a complex, three-dimensional, bounded conical shape. The flow is non-steady and compressible. In this situation, one should expect oblique shock waves, expansion fans, shock wave interactions, and slip surface generation.

The results of the numerical data from this section, such as velocity vectors and mass flow ratio of secondary fluid to primary fluid, were used to determine the possible flow requirement conditions for the pressure exchange mechanism inside the supersonic pressure exchange ejector.

A study was conducted to analyze the best design configuration of the aerodynamic shroud surface. Three-dimensional cubic-spline curves were used to design a number of shroud surfaces. Figure 1 shows a sketch of some geometrical parameters that were varied in the design analysis such as throat diameter, and diffuser out let diameter, inlet area ratio, and shroud exit diameter.

In designing the aerodynamic shroud, one divides the domain into four regions:

1-The subsonic flow region. This zone sees only the secondary fluid. The function of this region is to direct and accelerate the secondary fluid into the interaction zone in the vicinity of the rotor and the vanes.

2-The expansion zone. This region sees both the secondary and the primary fluids. The function of this region is to initiate the pressure exchange process and not to deflect the primary stream.

3-The compression zone. The pressure exchange continues in this region. The function of this region is the initiation of the mixing process of primary and secondary fluids.

4-The deceleration and diffusion region. In this region the flow is mixed.

Table 1 shows the geometrical parameters for the supersonic pressure exchange ejector of design A & B. Design A is the 3-D cubic B-spline curve. Design B is the cylindrical shape. Table 1, also, shows the mass flow ratio of the secondary fluid to the primary fluid. Analyzing this table one should consider both the mass flow ratio and the structure of the expansion fan because the objective of the current research is to maximize the pressure exchange mechanism. However, one should not assume that low mass flow ratio is an advantage, because in refrigeration applications the higher the mass flow ratio of the secondary fluid to the primary fluid the less is the entropy rise; low entropy production means less losses.

Table 1: Classification and geometrical data of aerodynamic shroud surfaces: Air, M=3.0,
AR=0.30, =0.20.
Some Geometrical Parameters / Flow Parameter
Shroud /  /  / SL /  / 
A-1 / 1.00 / 5.19 / 1.00 / 1.75 / 0.75
A-2 / 0.80 / 5.19 / 1.00 / 1.75 / 0.62
A-3 / 0.90 / 5.19 / 1.00 / 1.75 / 0.66
A-4 / 1.10 / 5.19 / 1.00 / 1.75 / 0.87
A-5 / 1.30 / 5.19 / 1.00 / 1.75 / 0.97
A-6 / 1.50 / 5.19 / 1.00 / 1.75 / 1.02
B-1 / 1.00 / 3.30 / 1.00 / 1.00 / 0.50
B-2 / 1.10 / 3.30 / 1.00 / 1.00 / 0.60

Fig.1: Sketch of some geometrical

parameters, shroud-diffuser

Figure 2 shows a sketch of the geometry for the three dimensional view of the aerodynamic shroud of supersonic pressure exchange ejector.

Fig.2: Partial sketch of the geometry (shroud)

Figure 3 shows velocity vectors for different aerodynamic shroud surfaces (note that the ejectors used here have no diffusers). Note that the cylindrical surface is shown in the lower right corner in figure 3. By analyzing this figure, one can observe the existence of expansion fans over the vanes and the direction of the velocity vectors when they pass over the vanes. One can see the primary fluid expanding and pulling the secondary gas with it. From figure 3 one can compare the velocity vectors of ejector types A&B. In ejector type A the primary and the secondary flows interact with each other more than they do in ejector type B. This is represented by the direction and the magnitude of the velocity vectors of the entrained secondary fluid. In the same figure, one can see that the velocity vectors are longer and more curved in ejector type A than in ejector type B.

Fig.3: Velocity vectors, planes passing over the vanes side view

The velocity vectors in planes passing between the vanes are plotted in figure 4. Note that the cylindrical surface is shown in the lower right corner in figure 4. In this figure one should only see the primary flow filling up the passage and this is accomplished for the design A. But this is not true for ejector type B since one can see both the primary fluid and the secondary fluid in the passage.

From this analysis, namely table 1 and figures 3 &4, one can conclude that the shroud shape is very critical in the design analysis of the supersonic pressure exchange ejector. The results showed that the cylindrical shape does not produce the desired requirements for the pressure exchange mechanism. Therefore the 3-D cubic-spline is used for the design analysis of the aerodynamic shroud surface of the supersonic pressure exchange ejector.

Fig.4: Velocity vectors, planes between the vanes side view

A further study was conducted to investigate the effect of changing some geometrical parameters of the 3D cubic-spline curve, such as throat diameter and exit diameter. Because the optimal goal of this design analysis is to study an ejector that could be used in refrigeration systems where pressure recovery is important, a diffuser was added.

Table 2 shows the geometrical data for different shroud-diffuser shapes. Table 2, also, shows the mass flow ratio of the secondary flow to the primary stream as a function of the shroud geometry (design C). This table shows that for the same shape of aerodynamic shroud-diffuser the mass flow ratio is changing from 0.62 to 1.16.

Table 1: Classification and geometrical data of aerodynamic shroud surfaces: Air, M=3.0,