Crypto-steady supersonic pressure exchange: A simple analytical model

Available online at www.sciencedirect.com

APPLIED ENERGY Applied Energy 85 (2008) 228–242 www.elsevier.com/locate/apenergy

Crypto-steady supersonic pressure exchange: A simple analytical model Hongfang Zhang, Charles A. Garris Jr.

*

Department of Mechanical and Aerospace Engineering, The George Washington University, Washington, DC 20052, USA Available online 26 November 2007

Abstract This paper is motivated by a long-range goal of developing a new class of turbomachine where fluid impellers, created through supersonic wave structures, replace mechanical impellers. This paper is an exploratory study to show how effectively supersonic wave structures can fulfill this goal, and to provide some fundamental insight as to the behavior. A comparison, at a fundamental level, of the spectrum of flow induction devices from mechanical turbomachinery to direct flow induction including steady flow ejectors, wave rotors, and pressure exchange devices is explored. While the authors have been studying various flow induction devices intended to lead to practical solutions, the details of the flow interactions are obscured by geometries and the complex interactions. This paper attempts to take a step backward and look at the simplest conceivable model that demonstrates the phenomena thought to be the key to a new technology. This model is a crypto-steady supersonic pressure exchange process occurring behind a supersonic frictionless semi-infinite flat plate having a pressure differential between both sides of the plate. It is found that very high compressor efficiencies and energy transfer rates are possible even with the presence of supersonic flow structure. Because of the high efficiencies, the analysis shows that there is a potential for the development of a new generation of direct flow induction devices far simpler and more compact than conventional turbomachinery, yet far more efficient than conventional ejectors and wave rotors. Fundamental information is provided which will facilitate exploiting supersonic pressure exchange in practical crypto-steady flow induction devices.  2007 Elsevier Ltd. All rights reserved. Keywords: Ejectors; Supersonic flow Induction; Pressure exchange; Compressor

1. Introduction This paper is motivated by a long-range goal of developing a new class of turbomachine where fluid impellers, created through supersonic wave structures, replace mechanical impellers. The process by which this might be accomplished at efficiencies comparable to, or even superior to, conventional turbomachinery is through crypto-steady pressure exchange. Such processes are non-steady in the laboratory frame of reference and yield to very low dissipation of energy since work is done by theoretically reversible ‘‘pd"’’ action. Dean [1] and Foa [2,3] discussed the benefits of non-steady mechanisms on flow induction. Our research has *

Corresponding author. Tel.: +1 202 994 6749; fax: +1 202 994 0238. E-mail address: [email protected] (C.A. Garris Jr.).

0306-2619/$ - see front matter  2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.apenergy.2007.06.016

H. Zhang, C.A. Garris Jr. / Applied Energy 85 (2008) 228–242

229

Nomenclature a A h H H_ m_ M p R T V " W MFI FDFI b c g

speed of sound (m/s) area (m2) enthalpy (J/kg) enthalpy per volume (J/m3) rate of total energy change (Watt) mass flow rate (kg/s) mach number pressure (Pa) specific gas constant (J/kg K) temperature (K) velocity (m/s) volume (m3) mechanical power (J/s) machine flow induction fluid dynamic flow induction shock angle ratio of specific heats adiabatic energy efficiency

Subscripts 0 total parameter (e.g., total pressure, total temperature, etc.) 1 inlet of compressor or of secondary fluid 2 outlet of compressor or of secondary fluid 3 outlet of turbine or of expanded primary fluid 4 inlet of turbine or of primary fluid c compressor t turbine x x direction (parallel to the moving flat plate) M mixed fluid discharge P moving flat plate Superscript 0 isentropic process

explored various devices which attempt to achieve this goal in a practical configuration. While the structures of these machines are fairly simple, in a three-dimensional crypto-steady environment, the supersonic flow structure in an enclosure can be quite complex, making it difficult to appreciate the physics of the pressure exchange process. This paper is an exploratory study which seeks to divorce the complex geometrical considerations from the physics of supersonic pressure exchange by studying the simplest possible two-dimensional configuration having crypto-steady supersonic flow. Foa [4] suggested such a process might be demonstrated in two-dimensions by studying the flow induction process provided by a supersonic semi-infinite flat plate separating a high energy primary fluid from a low energy secondary fluid. This paper will briefly discuss competing forms of flow induction in order to establish the context for flow induction by supersonic crypto-steady pressure exchange. In particular, the use of conventional turbomachinery at one end of the spectrum of complexity will be discussed, and the steady flow ejector at the other end of the spectrum will also be discussed. Within this spectrum of complexity, we seek to use pressure exchange technology to obtain intermediary machines demonstrating both the efficiency advantages of turbomachinery and the simplicity of ejectors.

230

H. Zhang, C.A. Garris Jr. / Applied Energy 85 (2008) 228–242

Important applications of this new technology include air-conditioning/refrigeration, engine turbo-chargers, power system topping cycles, water desalinization, and many others. Hong et al. [5] provides some preliminary research on the air-conditioning/refrigeration application. 2. Flow induction Flow induction is a process by which a fluid is caused to be transported by a means which generally imparts momentum and energy to the fluid. The flow induction can be achieved via vanes, pistons, screws, or mechanical elements of various kinds, or, alternatively, by the direct contact between a relatively low energy fluid and a high energy fluid of the same or different substance. The former we will call ‘‘machine’’ flow induction (MFI) since it encompasses all pumps, compressors, both positive displacement and turbomachinery devices. The latter, we will term ‘‘fluid dynamic’’ flow induction (FDFI) since it involves the direct transfer of energy and momentum from one fluid to another through fluid dynamic contact processes. Examples of this type of flow induction device include ejectors and wave rotors. MFI devices designed to capitalize on the work of non-steady pressure forces can be highly efficient since this process, in its ideal form, is reversible and conceptually equivalent to the pd" work of reversible thermodynamics. Two of the most common MFI devices are a turbine and a compressor. The most common way of expressing the efficiency of such machines is by comparing the work of an isentropic machine with that of the actual machine for the same mass flow rates and pressure differentials. In the case of a compressor, the adiabatic compressor efficiency is defined as the ratio of the isentropic work required to compress a certain mass of fluid with a certain pressure rise, to the work actually required, and is indicated in: gc ¼

W Compressor isentropic W

Compressor actual

¼

h020  h01 h02  h01

ð1Þ

2

2’

3 3’

Compressor

M

Turbine

Specific Enthalpy

4

p01

p02

p04

=p

03

In Fig. 1, a comparison of the isentropic compression process (from 1 to 2’) and the actual compression process (from 1 to 2) is shown. The subscript 0 indicates stagnation or total quantity. From Fig. 1, it can be readily seen that as entropy is created in the machine, the specific work required, (h02–h01), to provide the stated pressure differential at the stated mass flow rate increases, hence, the compressor efficiency decreases. If the working fluid is an ideal gas, this expression can be written [6]:   ðc1Þ T 01 ½p02 =p01  c  1 gc ¼ ð2Þ ðT 02  T 01 Þ

1

Specific Entropy

Fig. 1. Enthalpy–entropy diagram showing actual and isentropic expansions and compressions in a turbine and a compressor, respectively.

H. Zhang, C.A. Garris Jr. / Applied Energy 85 (2008) 228–242

231

In the case of a turbine, the ‘‘total-to-total’’ [6] turbine efficiency is similarly compared with an isentropic process and is typically expressed as gt ¼

W Turbine h04  h03 actual ¼ h W Turbine 04  h030 isentropic

ð3Þ

where this process is also shown in Fig. 1. It is noted from Fig. 1 for the case of the turbine, that the work obtained through an isentropic process is greater than that obtained through the actual irreversible process indicated by the dashed lines for the same mass flow rates and total pressure differentials. 3. Compressor–expander (turbomachinery analog) Since the object of this paper is to study flow induction processes whereby a high energy primary fluid imparts energy to a relatively low energy secondary fluid, it is of interest to show how this can be done with conventional turbomachinery. In Fig. 2 is shown a compressor–expander configuration whereby a high energy primary fluid can be expanded through a turbine which drives a compressor through a common drive shaft and thereby compresses a low energy secondary fluid. In both turbine and compressor, the processes are considered adiabatic. In the current analysis, both the turbine and the compressor discharge to a common reservoir, called the mixer, in which primary and secondary fluids are mixed. In general, in the mixer, heat is transferred between the primary fluid and the secondary fluid so that the process undergone by each respective stream is neither adiabatic nor isentropic, even though the overall mixing process is adiabatic since no heat is exchanged between the mixer and the surroundings. The second law requires that the net entropy change for an adiabatic mixing process must be positive. In this configuration, the mixed fluid discharges to a common duct. Fig. 1 also demonstrates a typical process of this Low Pressure Secondary

High Pressure Primary

1

COMPRESSOR

on mm Co

TURBINE

Dr

ive

Sh

aft

4

2 Compressor Discharge

Turbine Discharge 3

MIXER

M Intermediate Pressure Mixed Discharge

Fig. 2. Compressor–expander in the form of a Turbomachinery analog of a conventional steady flow ejector with adiabatic compression and expansion in the compressor and turbine, respectively, followed by constant pressure mixing in the mixer.

232

H. Zhang, C.A. Garris Jr. / Applied Energy 85 (2008) 228–242

kind. In operation, the turbine discharge (3) will mix with the compressor discharge (2) and leave the mixed fluids at a common thermodynamic state (M). Assuming adiabatic conditions in both the compressor and the turbine, and no friction in the shaft: m_ 1 ðh02  h01 Þ ¼ m_ 4 ðh04  h03 Þ

ð4Þ

The art of turbomachinery design is well developed. This is witnessed by the fact that the design of compressors and turbines has evolved over the last century to the point where very high efficiencies are possible and users can expect excellent and reliable performance. Compressor efficiencies and turbine efficiencies of 90% and higher are fairly common [6]. Nevertheless, there are important applications where machine flow induction devices have serious limitations. These limitations generally involve: (a) excessive size (volume and/or weight); (b) excessive capital cost of the high precision complex machinery; (c) materials limitations prevent operation at high temperatures or in corrosive environments, etc.; (d) stall and instability characteristics; (e) maintenance; (f) sealing difficulties at shafts; (g) energy dissipation through bearings and lubrication, and, (h) energy losses due to viscous effects boundary layer separation in flows over the mechanical vanes. The question arises as to whether or not all of the complex machinery is necessary. Can efficient flow induction be produced by the direct contact between a high energy primary fluid and a low energy secondary fluid without intervening machinery? The fundamental purpose of our research is to provide answers to this question. 4. Fluid dynamic flow induction With fluid dynamic flow induction (FDFI) processes, a high energy and high momentum primary fluid comes into direct contact with the low energy and low momentum secondary fluid and, in so doing, elevates its momentum and energy levels. Since FDFI devices do not involve all of the complex equipment required in machine flow induction devices, many of the aforementioned deficiencies can be overcome. In particular, fluid dynamic flow induction devices can be extremely compact, low in cost, capable of handling very high temperature environments, and potentially high reliability due to their relative simplicity. Although to date, higher efficiency has not been considered to be one of their advantages, we will show in this paper that there is reason to believe that a certain type of FDFI devices yet to be designed may offer superior efficiencies to MFI devices. Among the FDFI devices, there are two broad classes: mixing controlled, and pressure exchange controlled. 4.1. Mixing controlled flow induction Common ‘‘steady flow’’ ejectors are FDFI devices that directly transfer energy and momentum from a high energy primary fluid to a low energy secondary fluid by means of the work provided by turbulent shear stresses (turbulent entrainment), with corresponding turbulent mixing. An example is shown in Fig. 3. These ejectors are termed ‘‘steady flow’’ because the controlling mechanism does not depend on non-steadiness produced by moving interfaces, but rather the dragging effect of turbulent shear stresses that are generally time independent in the mean. A rule of thumb is that the more intense the mixing, the better the suction and pressure rise. As shown in Fig. 3, when used with gases or vapors, the primary fluid is accelerated through a supersonic nozzle

Fig. 3. Conventional steady flow ejector having a high energy primary fluid entering at 4, a relatively low energy secondary fluid entering at 1, and a common discharge, M. (Fox Valve Development Corp.)

H. Zhang, C.A. Garris Jr. / Applied Energy 85 (2008) 228–242

233

having a coaxial chamber through which the low energy secondary fluid is introduced. By means of turbulent mixing between primary and secondary fluids, the secondary fluid acquires momentum and energy while the primary fluid correspondingly loses it. The mixture discharges from the ejector at an intermediate level of specific energy. It should be observed that the conventional ejector of Fig. 3 performs the identical function as the turbomachinery compressor–expander shown in Fig. 2. For this reason, the configuration shown in Fig. 2 is often called the ‘‘turbomachinery analog’’ of an ejector. However, the conventional steady flow ejector is much simpler mechanically because it utilizes direct fluid–fluid interaction, thereby eliminating all of the rotating machinery. As can readily be seen in Fig. 3, the steady flow ejector has no moving parts and no dynamic seals. By virtue of its extreme simplicity, it is very compact in comparison to MFI devices such as shown in Fig. 2. It is highly reliable, cost-effective, and capable of providing very high vacuums. Furthermore, it can handle very large volumes of primary and secondary fluids in a compact unit. Conventional ejectors therefore find many applications in the processing and power industries. For example, steam ejector refrigeration is well known and has been used for over a century, particularly in applications having waste heat or low cost energy. It is also widely used in water desalinization and in jet engine thrust augmentation applications. A common way of characterizing the performance of an ejector is to compare it with that of ideal turbomachinery in the analog of Fig. 2. The processes shown in Fig. 1 relate to those that occur in an ejector in that the primary fluid does work as does the primary fluid in the turbine, and the secondary fluid is worked upon, as does the secondary fluid in the compressor. However, since the process by which work is performed is through turbulent mixing, there is considerable entropy generation, the impact of which is suggested in Fig. 1. Because of its reliance on turbulent mixing, it is inherently a generator of entropy and a dissipater of energy during its operation. For a given secondary mass flow rate, suction pressure, and discharge pressure, the primary mass flow rate, m_ 4 , for a steady flow ejector would be substantially higher than that required by an ideal turbomachinery analog configuration of Fig. 2 producing the same flow induction effect on the secondary fluid. Thus, the operating cost and energy requirements for the ejector would be much higher than for the corresponding turbomachinery. Nevertheless, ejectors can produce very low suction pressures, p01, and fairly high discharge pressures, p0M, if one is willing to provide a primary with enough mass flow, m_ 4 . For that reason, and their extreme simplicity, conventional steady flow ejectors are very commonly used in industrial applications where efficiency is not crucial. The concept of ‘‘ejector efficiency’’ has been utilized in the literature to provide a measure of performance by introducing ideal turbomachinery. A common way of expressing the ejector efficiency is by taking the product of the adiabatic compressor efficiency, Eq. (1), and the adiabatic turbine efficiency, Eq. (2), assume that at discharge, p03 = p02 = p0M, and assume that the fluid is an ideal gas. After some manipulation, one obtains the simple expression:   ðc1Þ m_ 1 T 01 ½p0M =p01  c  1 gejector ¼ ð5Þ   ðc1Þ m_ 4 T 04 1  ½p0M =p04  c Because the mixing processes inherent in steady flow ejectors are not adiabatic, even if the overall process is, efficiency expressed in this way is not a rigorous measure of efficiency and the actual physical meaning of the parameter is not entirely clear. Yet, ejector efficiency does provide a value which is easy to determine experimentally for an ejector, does have some value as a crude figure of merit, and is reported in slightly varying forms in the literature. Hedges [7] reports experimental values of ejector efficiency in air as high as 32%. Watanabe [8] obtained maximum values of ejector efficiency of 16.8% in air for a high area ratio, low pressure differential ejector. By further lowering the pressure ratio and increasing the size, he was able to obtain a maximum ejector efficiency of 31%. These maximum ejector efficiencies were obtained from research grade ejectors in controlled laboratory settings and are believed to be considerably higher than the normal operating efficiencies of conventional ejectors. The main shortcoming of the steady flow ejector is that the mechanism through which it operates is turbulent mixing, a most effective process for generating entropy and dissipating mechanical energy. Nevertheless, there are many important areas of technology which yearn for a direct flow induction device having the sim-

234

H. Zhang, C.A. Garris Jr. / Applied Energy 85 (2008) 228–242

plicity of a steady flow ejector, but vastly improved efficiency. Yet, over a century of research on improving flow induction through turbulent mixing has brought this technology to complete maturity, and further improvements in energy efficiency are limited by the fundamental dissipative mechanism to which the process relies. The goal of our research is to find another mechanism which will enable us to design a device which will retain some of the simplicity of the conventional steady flow ejector, but have a vastly higher efficiency. The mechanism we are exploring is supersonic pressure exchange. 4.2. Pressure exchange controlled flow induction Foa [2] defines ‘‘pressure exchange’’ as ‘‘any process whereby a body of fluid is compressed by pressure forces that are exerted on it by another body of fluid which is expanding’’. As seen from this definition, pressure exchange involves: (1) direct fluid–fluid contact, and (2) an energetic fluid which does work on a less energetic fluid by the reversible work of interface pressure forces, or pd" work. Since pressure exchange utilizes reversible pd" work, pressure exchange processes can potentially have very high energy transfer efficiencies, and since the flow induction process is direct from one fluid to another, pressure exchange processes can be very compact and simple. Thus it is possible that a new direct flow induction technology based on pressure exchange can provide flow induction devices rivaling the steady flow ejector in simplicity, yet achieving energy efficiencies of a different order of magnitude. In the following paragraphs, we will demonstrate the potential for achieving a high energy efficiency by means of a simple example of ‘‘crypto-steady’’ supersonic pressure exchange in two-dimensions: the supersonic frictionless plate. 5. Crypto-steady pressure exchange The concept of one dimensional pressure exchange in a non-steady environment has been extensively investigated and the patent literature reveals many inventions based on this principal in the form of the ‘‘waverotor.’’ The wave rotor is a very ingenious device which was commercialized, most notably by Brown Boveri, and is the subject of current research. Although the wave rotor is indeed ingenious, the presence of valving, dynamic seals, mechanically driven components, and normal shock waves inherently give rise to losses which cannot be eliminated. Thus, while commercially acceptable performance has been obtained, the high efficiency aspired with pressure exchange was never realized. Foa [2,4] was the first to observe that the principles of pressure exchange can be realized in a ‘‘crypto-steady’’ flow without the presence of valves and their associated loss mechanisms. He defined a ‘‘crypto-steady’’ flow as one which was steady in a special moving frame of reference, but was non-steady in the laboratory frame of reference. Foa’s concept involved the use of rotating jets emanating from self-driven canted nozzles to produce direct energy exchange between a high energy primary fluid and a low energy secondary fluid. The rotor, in which the rotating nozzles were embedded, was self-driven and intended to be free-spinning such that, ideally, the rotor does not exchange energy with the fluid but merely serves as a vehicle to generate rotating fluid interfaces. While the concept proved very useful for thrust augmentating ejectors for jet propulsion where the static pressure rise is small and the secondary/primary mass flow ratio is large, major practical difficulties were encountered in attempting to apply it for a high pressure rise. Our previous research painfully revealed that the primary difficulties encountered in subsonic cryptosteady pressure exchange were in thrust management, frictionless bearing design, and sealing. This prevented its adoption to high pressure rise applications required in air conditioning, fuel cell, and other important domestic applications. An important outcome of our research was the realization that direct flow induction, in a manner analogous to the one-dimensional shock tube/wave rotor, can be achieved in a crypto-steady flow using supersonic flow phenomena. The use of supersonic flow structure in a crypto-steady flow induction process is novel and promises enormous potential benefits. Our research has shown that the use of a supersonic flow field enables the complete elimination of the dynamic seals which caused the severe problems associated with prior attempts at exploiting pressure exchange with rotating jet configurations because the momentum of the supersonic stream itself provides separation between primary and secondary fluids in the introductions stage. The thrust management problems of subsonic crypto-steady pressure exchange flow processes, as previously studied, is avoided by the presence of low static pressures in the supersonic flow field. It is our hypothesis that the valve

H. Zhang, C.A. Garris Jr. / Applied Energy 85 (2008) 228–242

235

losses of the wave rotor can be eliminated by the use of crypto-steady flow, while the shock losses can be minimized by the use of weak oblique shock waves rather than the strong normal shocks present in the shocktube/wave rotor. This offers the possibility of obtaining substantial static pressure rises with a high compressor efficiency. Some recent concepts are described by Garris [9,10]. Our basic hypothesis is that a multi-dimensional crypto-steady flow can be established which utilizes supersonic flow structure to provide a highly efficient transfer of energy and momentum with the direct contact between a high energy primary fluid and a low energy secondary fluid. In order to better understand the behavior of such energy transfer systems, we will analyze in detail a very simple example of a crypto-steady supersonic flow induction process. 6. Two dimensional pressure exchange: supersonic flat plate 6.1. Statement of the problem Consider the semi-infinite moving plate of infinitesimal thickness as shown in Fig. 4, which is moving into stagnant compressible fluids on both sides of the plate. In the laboratory frame of reference, the plate moves to the right with a supersonic speed. The fluid is considered to be inviscid so that the plate does not exert any force on the fluid, and consequently does no work on the fluid. Mixing and heat conduction are neglected. Since no force is exerted by the plate, it cannot impart momentum to the fluid. In the laboratory frame of reference, the upstream fluid is at rest on both sides, and the static pressure above the plate is higher than that below. The static temperatures and the molecular weights of the fluid above and below the plate in the upstream regions can assume any arbitrary specified values. It will be seen that when the static pressure ratio, P4/P1, is less than a certain critical value but greater than unity, an attached shock will form on the low pressure side of the trailing edge of the plate as shown in Fig. 4. Also indicated is an attached Prandtl Meyer expansion fan on the high pressure side of the plate, and an attached contact line separating the fluid originating on the high pressure side from that originating on the low pressure side. Under the conditions postulated, the supersonic frictionless flat plate flow field is indeed crypto-steady since the flow is steady relative to a reference frame moving with the plate, but is non-steady in a laboratory reference frame. Since momentum in the direction parallel to the plate cannot be imparted to the inviscid fluid by the plate, any momentum changes to either fluid in the direction of the plate motion (positive x-direction) can only be a result of x-momentum exchanges between the high pressure fluid above the plate, and the low pressure fluid below the plate. Thus, positive x-momentum is imparted to the secondary fluid in region 2 by the primary fluid in region 3 which has expanded from region 4 and therefore acquires negative x-momentum relative to the laboratory. Since the plate is frictionless, and both primary and secondary fluids are at rest upstream, the total downstream x-momentum of the system must be zero. This exchange of momentum occurs across the contact line as shown in Fig. 4, P3/P2 = 1 results in ‘‘pressure exchange’’ as previously defined: ‘‘the process whereby a body of fluid is compressed by pressure forces that are exerted on it by another body of fluid which is expanding.’’

Stationary Primary Gas

Ex

pa

ns

(High Static Pressure)

io

n

Fa

n

3

4 Supersonic Flat Plate

e)

ary

(Prim

c Line id Interfa tact Con dary Flu n o c / Se

e

2

k

S

c ho

av W

1 Stationary Secondary Gas (Low Static Pressure)

Fig. 4. Schematic of crypto-steady supersonic flat plate flow field when shock wave is attached to trailing edge.

236

H. Zhang, C.A. Garris Jr. / Applied Energy 85 (2008) 228–242

6.2. Characteristics of the solution It has been found that the type of solution to the supersonic flat plate problem varies depending on the Mach number, pressure ratios, and temperature ratios. Depending on the location in parameter space, solutions can be single-valued or double valued, attached or detached. A schematic of the flow field when the shock is attached is shown in Fig. 4, and a schematic of the flow field when the shock is detached is shown in Fig. 5. Fig. 6 demonstrates why the nature of the solution is different in at various locations in parameter space by plotting the simultaneous solution of the equations on the primary fluid side of the plate and on the secondary fluid side of the plate. Since the conditions to be followed on each side of the contact line are: static pressure ratio, P3/P2 = 1; and, deflection angles must be equal, solutions are shown by the crossing of the curves in Fig. 6. Note that Fig. 6 shows a plot for a specific plate Mach number, MP, static temperature ratio, T4/T1, and with air as the working fluid. The characteristic curve for the fluid in region 1 passing through the oblique shock wave into region 2 is indicated, as well as a family of characteristic curves for the fluid in region 4 which expands into region 3. Solutions exist where the curves intersect. Stationary Primary Gas

Ex

pa

ns

(High Static Pressure)

io

n

Fa

n

4

3

MP

Supersonic Flat Plate )

e ac

rf

te

ne

y ar

d ui

In

1

2

Li l ct y F ta dar n Co con e /S

Stationary Secondary Gas (Low Static Pressure)

e

ck

av W

o

rim

(P

d

Sh

e ch

ta

De

Fig. 5. Schematic of crypto-steady supersonic flat plate flow field when shock wave is detached from trailing edge of plate.

Expansion Fan Characteristics

10

4

P4 1

e

ta

gl

ub

ch

t lu

l)

al

itic

)

6

Cr

n io

ut

n) io

ica

l So

rit

ed

So

-c

Working Fluid: Air T4/T1 = 1.0 MPLATE = 2.0

At

in

(S

1

o (N

0

(S

=1

2

0

7

1

=1

/P

=2

1

P4

8

/P

/P P4

it S Lim

Oblique Shock Wave Characteristic

2

0

1= 4.5 Strong Shock Solution (Crit ical 5 (W forT woeak Solu Sho tion ck s) Solu tion Only )

n

P4/P P4/P

1

0

tio

4

olu

Static Pressure Ratio, P2/P1 ; P3/P1

3

=3.

5

10

15

Weak Shock Solution

20

25

Fluid Deflection Angle, Degrees Fig. 6. Example simultaneous solution map for expansion region 4 and compression region 1 for a plate Mach number of 2.0, equal static temperatures on each side of plate, Air as the working fluid.

H. Zhang, C.A. Garris Jr. / Applied Energy 85 (2008) 228–242

237

Fig. 6 reveals that for the subject conditions, when 4.5 < (P4/P1) < 17.0, two solutions exist: a strong shock solution corresponding to the upper branch of the oblique shock wave characteristic curve, and a weak shock solution corresponding to the lower branch of the oblique shock wave characteristic curve. A schematic of the flow field for either the weak shock or the strong shock attached solutions is indicated in Fig. 4, where the actual angles of the expansion fan, contact line, and shock would differ for the strong and weak shock solutions. At a pressure ratio of approximately 17.0, both solutions combine into a single solution. However, for pressure ratios greater than 17.0, attached shock solutions do not exist. Under these conditions, the shock would detach from the trailing edge of the plate and move upstream along the low pressure side of the plate, as indicated in Fig. 5. Preliminary attempts to simulate this condition using CFD suggested that the detached shock wave may be oscillatory in this region, and not able to find a stable upstream position. Fig. 6 also reveals that when 1.0 < (P4/P1) < 4.5, there is no strong shock solution and only a single attached weak shock solution exists, corresponding to Fig. 4. As in Fig. 6, the lowest static pressure ratio on the oblique shock characteristic corresponds to unity whereby the static pressures, and the total pressures, on both sides of the plate are the same, and the fluids behind the plate merge uniformly with zero deflection. Thus, under this condition, the stagnant inviscid fluid is totally unaffected by the presence of the supersonic plate. The precise values of pressure ratio which determine the nature of the solutions to the supersonic flat plate problem are generally functions of Mach number of the plate, MP, temperature ratio, T4/T1, and working fluids on either side of the plate. The Mach number of the plate is defined as VP M P ¼ pffiffiffiffiffiffiffiffiffiffi cRT 1

ð6Þ

where VP is the velocity of the plate relative to the laboratory. In performing the calculations, a strong dependence in determining the type of solution on plate Mach number was observed. This dependence is shown in Fig. 7 for a temperature ratio of unity and with air. Note that the plot is semi-log. It is seen that as the plate Mach number increases, the detached solution requires very high pressure ratios and the double-solution domain broadens relative to the single-solution domain. 6.3. Local behavior of supersonic flat plate flow field Fig. 8 is a series of polar plots which show the local variations of the various thermodynamic variables for the case of the plate moving at a Mach number of 2.0 relative to the laboratory, pressure ratio across the plate, P4/P1 = 5.0, and a temperature ratio across the plate of 1.0. The static temperature ratio results show that even though the temperature on both sides of the plate is initially equal, there is a substantial temperature reduction of the primary fluid because of the expansion

Pressure Ratio, p4 /p 1

1000 , ns te tio on olu m pla s luti d So he es fro k c c a ho att ach kS No k det ea c W lution o k So Sh Shoc g n o Str ions olut nly 2S Solution O eak Shock W n tio 1 Solu

100

10

1 Fluid: Air T4/T1 = 1.0

1.5

2.0

2.5

3.0

Plate Mach Number, MP Fig. 7. Effect of plate Mach number on the existence of solutions to the supersonic flat plate problem.

238

H. Zhang, C.A. Garris Jr. / Applied Energy 85 (2008) 228–242

Fig. 8. Local variation on non-dimensional flow variables for a supersonic flat plate with MP = 2.0, P4/P1 = 5.0, T4/T1 = 1.0, air.

fan, and a substantial temperature increase of the secondary fluid effect due to the shock wave. Such phenomena would be useful for heat pump applications. The local Mach number results show that, in the laboratory frame of reference, the upstream fluid is at rest on both sides of the plate. However, downstream of the expansion fan and downstream of the shock, the local Mach number assumes different values which are dependent on both the downstream velocities and static temperatures. It is of interest to note that even though the plate Mach number of 2.0 is supersonic, the local Mach numbers for both primary and secondary fluids for the example shown are subsonic. This has important implications for the utilization of pressure exchange since by proper design, non-steady highly supersonic flow relative to the crypto-steady reference frame can be exploited, even though the local flow relative to the laboratory is subsonic. Clearly, subsonic flows relative to the laboratory facilitate pressure recovery and can ameliorate dissipation in the diffusers required. The polar plot for the horizontal component of local Mach number, in the laboratory frame of reference, is shown in Fig. 8. A schematic of the corresponding fluid structure and geometry is shown in Fig. 4. Let us define the x-direction as being in the same direction as the velocity of the plate in the laboratory frame of reference. Fig. 8 is particularly interesting because it shows how x-momentum is transferred from the driving fluid to the driven fluid through the mechanism of pressure exchange. Since the fluid is at rest upstream on both sides of the moving plate, the x-component of Mach number in region 1 and region 4 is zero. However, note that behind the expansion fan in region 3, the x-component is negative, while behind the shock in region 2, the x-component is positive. Thus, positive x-momentum is acquired by the fluid in region 2, while negative x-momentum is acquired by the fluid in region 3. This acquisition of x-momentum cannot be caused by the plate since the fluid is assumed inviscid and the plate is therefore incapable of imparting x-momentum. Thus,

H. Zhang, C.A. Garris Jr. / Applied Energy 85 (2008) 228–242

239

what has occurred is the transfer of x-momentum from the driving fluid, originating in region 4 as shown in Fig. 4, to the driven fluid originating in region 1, through the work of interface pressure forces exerted across the contact line. This work is achieved through pressure exchange, i.e., the compression of one body of fluid by pressure forces exerted on it by another body of fluid which is expanding. The polar plot for total pressure ratio in the laboratory frame of reference is also shown in Fig. 8. It is seen that the total pressure on the driving side, region 4, is reduced as it expands to region 3. The work of this expanded fluid causes the driven fluid in region 2 to have a considerably higher total pressure than initially in region 1. Note that total pressure in the laboratory frame decreases across the expansion fan as a result of the non-steady flow which enables work to be done by the fluid in region 3, on the fluid in region 2. In a steady flow, the total pressure would be constant across the expansion fan, and no work would be done by one fluid against the other since pressure forces acting against stationary interfaces can do no work. The polar plot for static pressure ratio also shown in Fig. 8 is identical to the plot for total pressure in the upstream region 1 and region 4 since the fluid is at rest in these regions whereby total and static pressures are equal. However, downstream of the expansion fan and downstream of the shock in region 3 and region 2, respectively, the static pressures are equal, whereas total pressures are not. This corresponds to the standard boundary condition across a contact line whereby the static pressure must be constant. The polar plot for total enthalpy ratio in the laboratory frame of reference shown in Fig. 8 demonstrates how work is done by one body of fluid on the other. Note that in the example shown, the upstream values of total enthalpy are equal on both sides of the plate. However, after undergoing expansion, the total enthalpy reduces, while on the low pressure side, the total enthalpy increases across the shock. It is also noted that these changes in total enthalpy are manifestations of the non-steady flow in the laboratory frame of reference where work is done across a moving fluid interface (contact line). In the relative frame of reference fixed to the plate, the flow is crypto-steady, and no work is done by one fluid upon the other, and the total enthalpy relative to the plate (not shown) is invariant both across both the shock and across the expansion fan. 6.4. Compression performance of the supersonic flat plate flow field The primary goal of this paper is to obtain a fundamental understanding of how supersonic pressure exchange can be used to affect compression of a low energy fluid. Figs. 9 and 10 show a variety of parameters which reflect the performance of the supersonic flat plate in compressing the low energy fluid in region 1 to its final state in region 2 as in Fig. 4. Particular attention is given to the effects of plate Mach number, MP, driving to driven pressure ratio, P4/P1, and driving to driven temperature ratio, T4/T1. The results shown are for air as the working fluid for both driving and driven fluids. Fig. 9 shows the results of solutions for air under various plate Mach numbers, MP, temperature ratios, T4/T1, and pressure ratios, P4/P1. The compressor efficiency is defined as in Eq. (2). The turbine efficiency is considered to be unity since the expansion process in isentropic. The non-dimensional energy rate is defined as    H2 h02 ¼ M P sin b 1 Aa1 H 1 h01

ð7Þ

where H_ 2 = rate at which total energy of secondary fluid increases H = qh0, enthalpy per unit volume b = shock angle measured from the plate a = local speed of sound A = area normal to the plate It is seen in Fig. 9 for the case of the weak shock solutions, that as the plate Mach number increases, which would tend to strengthen the shock, the shock angle decreases, which tends to weaken the shock. Hence, the two effects partially cancel each other. This does not occur with the strong shock solution (solid lines). As seen in Fig. 9, as the plate Mach number increases, the shock angle for the strong shock solution increases, and

240

H. Zhang, C.A. Garris Jr. / Applied Energy 85 (2008) 228–242

5

0.9

10 15 15

10 5

0.8

MP = 2.0

Non-Dim Energy Rate

0.7 0

0.5

1

1.5

Compressor Efficiency

1

p4/p 1

2 5

2

10 15

p4/p 1

1.5 15 10

1

5

0.5

p4/p 1

5 10 15

0.9 15

0.8 10

0.7

T4/T1 = 1.0 0.6 1.5 10

Non-Dim Energy Rate

Compressor Efficiency

1

8

2

2.5

3

T4/T1 = 1.0

6 4

p4/p 1 10 15

2

p4/p 1 15 10 5

0.5

1

1.5

2

20

15

p4/p 1

10 15

15 5

10

10

MP = 2.0

5

5

0 0 90

0.5

1

1.5

2

0 1.5 25

2

2.5

3

p4/p 1

20

15 10

15

5

10 15

5

10

T4/T1 = 1.0 0 1.5

2

2.5

3

5 10

75

15 15

60

p4/p 1 10 5

45 30 15 0 0

Flow Deflection Angle, degrees

0 0 25

MP = 2.0 0.5

1

1.5

Temperature Ratio, T4/T1

Shock Angle, degrees

Shock Angle, degrees

Flow Deflection Angle, degrees

MP = 2.0

10 15

80

p4/p 1 60

15 10 5

40

20

T4/T1 = 1.0 2

0 1.5

2

2.5

3

Plate Mach Number

Fig. 9. Effects of plate Mach number, static pressure ratio, static temperature ratio, on pressure exchange and other characteristics of the flow over a supersonic flat plate moving in still air at zero angle of attack. Dashed lines refer to the weak shock solution, and solid lines refer to the strong shock solution. Detached solutions are not shown.

both effects combining to increase the strength of the shock. This leads to a substantial drop in compressor efficiency for the strong shock solution with increasing plate Mach number, while total pressure ratio, P02/P01, total enthalpy ratio, h02/h01, and energy rate increase rapidly with the plate Mach number. It is interesting to note that despite the increased energy dissipation and reduction of compressor efficiency in the strong shock solutions in comparison with the weak shock solutions, the magnitude of the work done through the pressure exchange process is substantially increased leading to higher total pressure rises and higher mass induction. These are certainly good characteristics for a compressor. From Fig. 9, it is seen that for the supersonic flat plate with the weak shock, when the temperature ratio, T4/T1 , is less than 1 (cool driving fluid, hot driven fluid), the compressor efficiencies can exceed 95%. As the

5 10

8

15 15

6

10

4 5

2

MP = 2.0 0 0

Total Enthalpy Ratio, h02/h01

p4/p 1

0.5

1

1.5

p4/p 1

5 10 10

2 15

1.5

5

1 0.5

MP = 2.0 0 0

0.5

1

1.5

Temperature Ratio, T4/T1

2

T4/T1 = 1.0

15

30

p4/p 1

20

10

p4/p 1

10

0 1.5

2

2.5

241

40

Compression Ratio, p02/p01

10

Total Enthalpy Ratio, h02/h01

Compression Ratio, p02/p01

H. Zhang, C.A. Garris Jr. / Applied Energy 85 (2008) 228–242

15 10 5

2

2.5

4

3 15

T4/T1 = 1.0 p4/p 1

3

10

p4/p 1

2

15 10 5

1

0 1.5

2

2.5

3

Plate Mach Number

Fig. 10. Effects of plate Mach number, static pressure ratio, and static temperature ratio on the compression ratio and the enthalpy ratio of the driven fluid. Total pressure and total enthalpy are given in the laboratory frame of reference.

temperature ratio increases, the compressor efficiency decreases. This is significant in that most applications involving thermal sources have a driving fluid at a higher temperature than the driven fluid. This decrease in efficiency is due to the strengthening of the shock wave as a result of the increased speed of sound in the high temperature fluid. This strengthened shock results in greater dissipation, which, in turn, lowers the compressor efficiency. For the weak shock solutions of the flat plate problem, the energy rate increases with the temperature ratio, again due to strengthening of the shock. It is also seen that for the weak shock solutions, the compressor efficiency decreases with increasing pressure ratio, again as a result of the strengthening of the shock wave. The results for the strong shock solution (shown with solid lines) of the flat plate problem are different from the weak shock solutions. It can be seen from Fig. 9 that the strong shock compressor efficiency increases with the temperature ratio, while energy rate decreases, resulting from weakening of the shock. It is also interesting to note that for the strong shock solutions, as the temperature ratio increases, the deflection angle increases even though the shock angle decreases. The differences in behavior between the weak shock solutions and the strong shock solutions are strongly influenced by the respective changes in the shock strength. In Fig. 10, as expected, the trends for compression ratio, P02/P01, and for total enthalpy ratio, h02/h01, follow closely to each other for both weak shock and strong shock solutions. For the weak shock solutions, it is seen that both compression ratio and total enthalpy ratio increase with the temperature ratio, T4/T1. Since in most applications, the driving fluid in region 4 would be at a much higher temperature than the driven fluid in region 1, this is a favorable trend. However, in the case of the strong shock solutions, the behavior is the opposite. Figs. 9 and 10 show a comparative analysis of the pressure exchange process from the perspective of a direct fluid–fluid compression process. The results are quite promising. Thus, the goal of using a supersonic cryptosteady pressure exchange device for achieving high energy efficiencies with a direct contact process without all of the mechanical complexity of conventional turbomachinery appears possible and worthy of pursuit. Not only does the pressure exchange process offer the possibility of attaining a quantum jump in performance as compared with turbulent entrainment type flow induction as in a conventional ejector, but it shows that the oblique shock structure of the flat plate type crypto-steady pressure exchange process offers a substantial

242

H. Zhang, C.A. Garris Jr. / Applied Energy 85 (2008) 228–242

improvement over the one-dimensional technology if suitable machinery based on these principles can be developed. Having said this, however, it should be kept in mind that the one-dimensional pressure exchange process has been commercialized, and the benefits of pressure exchange are well known. Furthermore, the two-dimensional supersonic flat plate is not a commercially viable configuration. What is needed is a three-dimensional crypto-steady pressure exchange process which is commercially viable. This will require considerable research. Fundamental knowledge is lacking on how to generate suitable three-dimensional supersonic flow structures. Our research seeks to develop such devices and future publications will discuss our results in detail. 7. Conclusions This paper has explored from a fundamental perspective of both machine flow induction and direct fluid– fluid flow induction. It is observed that direct flow induction offers the possibility of developing extremely compact, robust, low cost, and highly efficient flow induction devices although current technology has not attained this level of performance. In particular, the paper has explored the behavior of direct flow induction based on pressure exchange utilizing supersonic wave structure in a crypto-steady mode. It is found from simple examples that very high compressor–expander efficiencies are possible even in the presence of strong supersonic wave structure. Insight is provided for the design of practical crypto-steady supersonic flow induction devices exploiting pressure exchange. As previously mentioned, the over-arching goal of this research is to develop a new class of turbomachine whereby conventional mechanical vanes are replaced by supersonic wave structures. This research shows that the dissipative effects associated with supersonic wave structures certainly do not preclude using three-dimensional crypto-steady supersonic structures in practical devices. However, there remains a need for extensive research in obtaining the optimal structures. Future papers will explore our efforts in this connection. References [1] Dean RC. On the necessity of unsteady flow in fluid machines. Trans ASME 1959:24–8. [2] Foa JV. Elements of flight propulsion. New York: John Wiley & Sons; 1960, p. 83, 190. [3] Foa JV, Garris CA. Crypto-steady modes of direct fluid–fluid energy exchange. Chapter in Sladky JF. Machinery for direct fluid–fluid energy exchange, ASME, vol. AD-07, New York; 1984. [4] Foa JV. Method of energy exchange and apparatus for carrying out the same. US Patent 3,046,732, July. [5] Hong WJ, Alhussan K, Zhang H, Garris CA. A novel thermally driven rotor-vane/pressure-exchange ejector refrigeration system with environmental benefits and energy efficiency. Energy, vol. 29. UK: Elsevier; 2004, p. 2331–45. [6] Lakshminarayana B. Fluid dynamics and heat transfer of turbomachinery. New York: John Wiley & Sons; 1996, p. 51–9, p. 123, 402. [7] Hedges KR, Hill PG. Compressible flow ejectors: part II – flow field measurements and analysis. J Fluids Eng ASME 1974:283–8. [8] Watanabe I. Experimental investigations concerning pneumatic ejectors. In: Symposium on jet pumps and ejectors, Inst. of Chemical Engineers, London, November 1972, Paper 7, p. 97–120. [9] Garris CA. Pressure exchanging ejector and methods of use. US Patent 6,138,456, October 31; 2000. [10] Garris CA. Pressure exchange ejector. US Patent Application No. 11/231,083; Publication no. US 2006/0239831 A1; October 26; 2006.