Patent Publication Number: US-7908855-B2

Title: Fluidic oscillator

Description:
This invention relates to a fluidic oscillator. More particularly, but not exclusively, the invention relates to a fluidic oscillator in which there is provided means for giving rise to a phase shift between pressurisation and displacement in a fluid contained therein. Even more particularly, but not exclusively, the invention relates to a pump comprising such a fluidic oscillator. 
     The class of device referred to herein as a fluidic oscillator typically refers to a system in which a flow of energy is coupled to an oscillation in a fluid. 
     Fluidic oscillators provide a means to generate useful work from heat sources, pressurised fluid sources, or other types of energy source. Conversely, they may also transform work in order to perform a useful function, such as compressing gases, pumping liquids, heat pumping and cooling. Fluidic oscillators are distinguished from other similar devices by not relying on mechanical pistons, turbines, flywheels, springs, linkages and externally actuated valves. Examples of fluidic oscillators include liquid piston heat engines, air and water pulsejets, thermoacoustic engines and float valve actuated pumps. 
     In these devices a working fluid undergoes cyclic pressure variations which give rise to (e.g. heat engine), or originate from (e.g. heat pump) oscillatory displacement in a load. The characteristics of the load determine the ‘load phase angle’ between a displacement of a given part of the working fluid and a pressure thereof at a given position and time. 
     A means of arranging a ‘feedback phase angle’ between the displacement and the pressure supplied or dissipated by an energy source or sink must also be provided. The load and feedback phase angles must be approximately equal under non-transient conditions. The optimal value of the load and feedback phase angles is typically around 90°. 
     In thermofluidic oscillators (e.g. fluidic oscillators of the heat engine or heat pump classes), a second, thermal, feedback phase angle also exists. This thermal phase angle arises due to the time delay between the temperature at which heat is exchanged and the rate of flow of entropy between the working fluid and heat exchangers at a given position and time. The optimal value of the thermal phase angle depends on the nature of the working fluid, however it is typically close to 0°. 
     There are a number of types of fluidic oscillators, for example pulsejets (see for example GB2180299), water pulsejets (see for U.S. Pat. No. 3,898,800 and U.S. Pat. No. 4,057,961) and many thermoacoustic engines and thermoacoustic coolers (see for example U.S. Pat. No. 4,489,553 and U.S. Pat. No. 5,901,556). All of the aforementioned types of fluidic oscillators may be sub-classed as ‘LRC (Inductance-Resistance-Capacitance) feedback’ oscillators in which the phase angle between displacement and pressurisation is determined by the extent of an inductive process, ‘L’, a dissipative process, or resistance, ‘R’, and a capacitive process, ‘C’. Typically, the inductive process arises due to the inertia of the working fluid, the dissipative process is thermal resistance that causes the dissipation of exergy, or available work contained within the system and the capacitive process arises due to the compressibility of the working fluid. In some examples, the dissipative process is friction or viscous drag and the capacitive process is associated with a hydrostatic pressure increase due to a flow of fluid, usually but not necessarily the working fluid, (see for example GB2017227). An LRC thermal phase angle is always between 0° and 90°, and high values imply a very large dissipative loss and consequential poor energy efficiency. Therefore typical practical LRC thermal phase angles are between 30° and 60°. 
     Another class of fluidic oscillators may be sub-classed as ‘LC (Inductance—Capacitance) feedback’ oscillators, typical examples of such LC feedback oscillators are liquid piston Stirling engines (see for example GB1581748, GB1507678) and thermoacoustic Stirling engines (see for example U.S. Pat. No. 4,114,380). In these thermofluidic oscillators the feedback phase angle may be close to 90° and the thermal phase angle may be close to 0°, typically with a small dissipative contribution. 
     The advantage of LC feedback oscillators over LRC feedback oscillators is that no trade off between dissipative losses and the thermal and feedback phase angles is required. Thus, near 90° load and feedback phase angles and a 0° thermal phase angle, are realisable in practical systems. 
     The disadvantage of LC feedback oscillators is that only one running frequency can be attained practically with a given apparatus and load. It is also necessary to add significant reactance in the region of the load in order to achieve steady state oscillations, prevent the running frequency from being very high and the pressure amplitude from being very low. This has the further disadvantage of making LC feedback oscillators large and operable across only limited loads and pressure amplitudes. 
     There are also float actuated pumps (see for example U.S. Pat. No. 3,905,724, FR2758162) which cannot be considered as linear oscillators as they depend on rapid switching. In these devices, a float is situated within a chamber which is connected to suction and discharge lines via respective suction and discharge non-return valves. Further valves (e.g. slide or pitot valves) connected to high and low pressure sources cause the float to rise due to an intake of liquid through the suction valve when the further valves are arranged to place the chamber in communication with a low pressure source. The float rises unimpeded until it nears the top of the chamber, when it causes the further valves to switch out the low pressure source and switch in the high pressure source simultaneously. This causes the float to descend unimpeded as liquid is discharged until the float gives rise to the switching in of the low pressure source and to the switching out of the high pressure source as the float reaches the bottom of the chamber, thus the cycle repeats. 
     The disadvantages associated with these devices include the need to locate switching means at both ends of the chamber. Also, the float must move a large distance before switching if a large displacement is to be achieved without requiring that the further valves have a critically adjusted actuation pressure, or the chamber becoming undesirably large. 
     According to the present invention there is provided a fluidic oscillator comprising a vessel arranged to contain a working fluid, the oscillator being arranged to permit steady oscillations independently of the inertia of said working fluid. 
     According to an aspect of the present invention there is provided a fluidic oscillator arranged to permit steady oscillations independently of the inertia of a working fluid further comprising:
         a. first and second vessels arranged to contain a working fluid;   b. means for conjoining the first and second vessels so as to subject said two vessels to a common pressure;   c. means for coupling the working fluid in the first vessel to a load such that changes in the volume of the working fluid contained in said first and second vessels give rise to transfer of work between the first vessel and the load;   d. means for communicating the volume of working fluid contained within the first vessel to the second vessel;   e. means for giving rise to pressure changes in the working fluid, located substantially within the second vessel;   f. At least one time delay mechanism giving rise to a phase shift between the volume of working fluid contained within the first vessel and the pressure changes therein. The at least one time delay mechanism being arranged to be independent of the inertia of the working fluid.       

     According to another aspect of the present invention, said fluidic oscillator comprises two or more time delay mechanisms, the two or more time delay mechanisms each comprising a dissipative process comprising any one, or combination, of the following: viscous drag, thermal resistance or mechanical friction; and a capacitive process comprising any one, or combination of the following: hydrostatic pressure change due to a flow, fluid compressibility, thermal capacitance, or elasticity; and wherein, the magnitude of the pressure changes in the working fluid increases or remains constant with time due to at least one mechanism giving rise to a gain. 
     According to a further aspect of the present invention there is provided a fluidic oscillator comprising:
         a. A vessel arranged to contain a working fluid;   b. Liquid coupling means arranged to couple said working fluid to a load such that changes in the volume of working fluid contained within said vessel give rise to displacement of said liquid and transfer of work between said vessel and the load;   c. Means of connecting said vessel to reservoirs of high or low pressure giving rise to changes of pressure within said vessel;   d. Means of communicating the volume of working fluid within said vessel to said means of connecting said vessel to high or low pressure reservoirs;   e. Two or more time delay mechanisms arranged to give rise to a phase difference between the volume of working fluid contained within said vessel and said pressure changes therein, the time delay mechanisms each comprising a dissipative process and a capacitive process wherein one of said time delay mechanisms comprises the viscous drag between said pressure reservoir and said vessel and the compressibility of the working fluid and one or more other time delay mechanisms comprise any one, or combination, of the following: viscous drag, hydrostatic pressure change due to flow, thermal resistance, fluid compressibility, thermal capacitance, friction or elasticity;   f. Pressure sources having a pressure difference therebetween coupled to said pressure reservoirs.       

     In the aspects of the present invention that comprise a fluidic oscillator, the fluidic oscillator may further comprise the optional features described hereinafter. The fluidic oscillator may comprise float means having a lower density than that of said liquid. Typically, said float means may have approximately half the density of said liquid. Said float means may be arranged to give rise to a reservoir of high pressure being connected to said vessel or said first vessel when the volume of working fluid is substantially small therein and to a low pressure reservoir when the volume of working fluid therein is substantially large. 
     The float means may be situated within said vessel such that said float means is actuated by the liquid level therein. Said float means may be connected to said vessel or said first vessel such that mechanical friction between the vessel and the float means is arranged to delay the motion of said float means with respect to said liquid within said vessel over at least part of the range of motion of the float means. 
     Said float means may be situated within a or the second vessel. Said fluidic oscillator may further comprise means of permitting a flow of liquid between the said first vessel and said second vessel wherein said flow may be driven by the hydrostatic pressure difference at the bottom each said vessel, due to the liquid therein. Said means of permitting a flow of liquid may further comprise a viscous drag intended to give rise to a phase shift between the liquid levels in said first and second vessels or otherwise. 
     Said float means may be free to move substantially free from impedance due to mechanical friction in the mid range of the trajectory thereof, such that said float means gives rise to switching between high and low pressure reservoirs only when it is substantially high or low so as to cause hysteresis. 
     Said high pressure reservoir may contain a gas. The gas may be compressed air. 
     At least one of the dissipative processes which may give rise to a time delay in combination with a capacitive process may be due to the said load. The dissipative processes giving rise to a time delay may comprise thermal resistance. 
     Said load may be located between either of the first or second vessels to which it is coupled and another vessel which is arranged to provide load compliance. 
     Said means of communicating the volume of working fluid within said fluidic oscillator to the means of giving rise to pressure changes may involve the pressure or volume of said working fluid within said load compliance. 
     Said load compliance may comprise a second fluidic machine such as a fluidic heat pump substantially of the same type or otherwise. Said second fluidic machine may comprise a fluidic oscillator arranged to operate in antiphase with respect to the first fluidic oscillator. 
     According to a still further aspect of the present invention there is provided an oscillating thermofluidic heat engine or heat pump comprising a fluidic oscillator, and further comprising:
         a. First and second vessels arranged to contain a working fluid;   b. Means of conjoining said two vessels so as to subject them to a common pressure;   c. Means of coupling the working fluid in the first vessel to a load such that changes in the volume of working fluid contained in said first and second vessels give rise to transfer of work between said first vessel and the load;   d. Means of communicating the volume of working fluid contained within said first vessel to the second vessel;   e. Heat exchanger means located substantially within said second vessel intended to give rise to pressure changes in the working fluid by heating or cooling of part thereof;   f. Two or more time delay mechanisms arranged to give rise to a phase shift between the volume of working fluid contained within said first vessel and said pressure changes therein, said time delay mechanisms each comprising a dissipative process and a capacitive process wherein at least one of said time delay mechanisms comprises the thermal resistance of said heat exchangers and the compressibility of the working fluid and at least one of the time delay mechanisms comprises any of the following: viscous drag, hydrostatic pressure change due to flow, thermal resistance, fluid compressibility, thermal capacitance, friction or elasticity;   g. Thermal reservoirs having a temperature difference therebetween coupled to said heat exchanger means such that the temperature difference within the heat exchangers gives rise to the magnitude of said pressure changes in the working fluid increasing or remaining constant with time.       

     According to a yet further aspect of the present invention there is provided an oscillating thermofluidic heat engine or heat pump comprising a fluidic oscillator, and further comprising:
         a. First and second vessels arranged to contain a working fluid which is part liquid and part vapour within the vessels;   b. Means of cojoining said two vessels so as to subject them to a common pressure;   c. Means of coupling said working fluid to a load comprising a liquid such that changes in the volume of working fluid contained within said first and second vessels give rise to displacement of said liquid and transfer of work between said first vessel and the load;   d. Means of permitting a flow of liquid between the first vessel and the second vessel driven by the hydrostatic pressure difference at a lower portion of each said vessel due to the liquid therein;   e. Heat exchanger means located substantially within said second vessel intended to heat and thereby expand part of said working fluid when liquid level is high therein and cool and thereby contract part of said working fluid when liquid level is low therein;   f. A time delay mechanism comprising the viscous drag arising from said flow of liquid between said first and second vessels and the change in hydrostatic pressure therein, said time delay mechanism giving rise to a phase difference between the liquid levels in said two vessels;   g. A second time delay mechanism comprising the thermal resistance arising due to said heat exchanger means and the compressibility of the working fluid, said second time delay mechanism giving rise to a phase shift between the liquid level in said second vessel and the pressure of said working fluid;   h. Thermal reservoirs having a temperature difference therebetween coupled to said heat exchanger means such that the temperature difference within the heat exchangers gives rise to the magnitude of said pressure changes in the working fluid increasing or remaining constant.       

     In the aspects of the present invention that comprise a thermofluidic heat engine or heat pump, the thermofluidic heat engine or heat pump may further comprise the optional features described hereinafter. 
     In the aspects of the invention in which high or low pressure reservoirs are provided, said high pressure reservoir may comprise heat exchanger means arranged to give rise to heating of working fluid therein. Said low pressure reservoir may comprise heat exchanger means arranged to give rise to cooling of working fluid therein. 
     In all aspects of the invention that comprise heat exchanger means said heat exchanger means may be arranged to give rise to alternate evaporation and condensation of the said working fluid. Said hot and cold heat exchange means may be located in separate vessels or separate chambers within the same vessel. A hot heat exchanger may be located within a first chamber parallel to a second chamber in which a cold exchanger is located. Said thermofluidic oscillator may comprise regenerator or recuperator means. One of said dissipative process giving rise to a time delay may be thermal resistance. 
     All aspects of the invention that comprise a fluidic oscillator, a thermofluidic heat engine or a thermofluidic heat pump may further comprise the optional features described hereinafter. 
     The vessel, or first said vessel may be substantially greater in height than in width and waisted towards the centre thereof. A float in addition to any existing float means may be provided within said vessel containing working fluid coupled to said load. Said float in addition to any existing float means may be arranged to rest on top of said liquid such that thermal or other losses are reduced. A second liquid in addition to any existing liquid may be provided within said vessel containing working fluid coupled to said load. Said second liquid may be substantially immiscible therewith. Said second liquid may have a lower density than existing liquid therein so that it floats on top thereof, such that thermal or other losses are reduced, or ideally minimised. The vertical axes of the first and second chambers may be parallel. 
     Said working fluid may comprise two or more components. One or more of said components may be arranged to be active in giving rise to pressure changes in the working fluid. One or more of said components may be interspersed throughout said fluidic oscillator. One or more of said components may be passive and mainly occupy said vessel from which working fluid is coupled to a load so as to impede active components of said working fluid from entering therein. 
     An additional chamber may be situated between said means of giving rise to pressure changes and the one of the first or second vessels from which working fluid is coupled to a load. Said additional chamber may be arranged to separate, at least substantially completely, by diffusion, gravity or other means passive components from active components of said working fluid. 
     A flexible bag, diaphragm, membrane or other flow separation means may be located between said means of giving rise to pressure changes and said vessel or first vessel connected to said load. 
     An additional hydraulic or pneumatic work transfer fluid may be situated between said working fluid and said load such that said working fluid is substantially immiscible with said work transfer fluid and means of retaining a stable interface between said working fluid and said work transfer fluid are arranged. 
     A flexible bag, diaphragm, membrane or other flow separation means may be located between said vessel from which working fluid is coupled to a load and said load, such that fluid in the vicinity of said load is unable to mix with fluid in the vicinity of said vessel from which working fluid is coupled to said load. 
     Said load may comprise one or more further fluidic machines such as a fluidic heat pump substantially of the same type or otherwise. 
     Said one or more further fluidic machines may each comprise a fluidic oscillator substantially of the same type as said first fluidic oscillator or otherwise, and preferably being arranged to have a phase difference therebetween. 
     Said one or more other fluidic machines may each comprise a load. At least one said load may further comprise inter alia said first fluidic oscillator. 
     All aspects of the invention may comprise a pump, arranged to impart a velocity to a fluid to be pumped. 
     In the aspects of the invention that comprise a pump, the pump may comprise a fluidic oscillator, a thermofluidic heat engine or heat pump. The fluidic oscillator, thermofluidic heat engine or heat pump may further comprise first and second thermal reservoirs having a temperature difference therebetween. The first thermal reservoir may comprise a solar collector. Said first thermal reservoir may comprise an output from a heating apparatus. Said heating apparatus may comprise a boiler arranged to circulate heating fluid about a heating system. 
     A thermo-siphon or a heat pipe may connect said first thermal reservoir to the fluidic oscillator, the heat engine or the heat pump. The second thermal reservoir may comprise the fluid to which the said pump is applied. Said second thermal reservoir may comprise a subterranean heat sink. Said second thermal reservoir may comprise a fluid inlet or return to a heating apparatus. 
    
    
     
       The invention will now be described, by way of example only, with reference to the accompanying drawings, in which: 
         FIG. 1  is a schematic representation of an embodiment of a fluidic oscillator according to an aspect of the present invention; 
         FIG. 2  is a diagram of an electrical circuit analogous to the fluidic oscillator of  FIG. 1 , in which the dissipative processes are represented by electrical resistors and the capacitive processes are represented by electrical capacitors; 
         FIG. 3  shows a heat engine incorporating an embodiment of a thermofluidic oscillator according to an aspect of the present invention; 
         FIG. 4  is a representation of a heat engine, incorporating an embodiment of a thermofluidic oscillator according to an aspect of the present invention, in which the heat exchangers giving rise to a thermal resistance are arranged in separate chambers; 
         FIG. 5  is a representation of a heat engine, incorporating an embodiment of a thermofluidic oscillator according to an aspect of the present invention, comprising a passive component in a working fluid and a diffusion region is included; 
         FIG. 6  is a representation of a heat engine, incorporating an embodiment of a thermofluidic oscillator according to an aspect of the present invention, in which a membrane, diaphragm or other obturator means separates an active component of a working fluid from a passive component thereof; 
         FIG. 7  is a representation of a pump, incorporating an embodiment of a fluidic oscillator according to an aspect of the present invention, in which pressurisation and depressurisation means are separated into additional vessels which are in selective communication with a central chamber by means of a float actuated valve; 
         FIG. 8  shows an alternative arrangement of the pump of  FIG. 7 ; 
         FIG. 9  shows a means of adding hysteresis to the float of  FIGS. 7 and 8 ; 
         FIG. 10  is a representation of a heat engine, incorporating an embodiment of a thermofluidic oscillator according to an aspect of the present invention, in which one dissipative process is nominal thermal resistance of a fluid and heat exchanger components and another is a load; 
         FIG. 11  shows a specific arrangement of the heat engine of  FIG. 10 ; 
         FIG. 12  is a representation of a solar irrigation pump comprising a heat engine according to an aspect of the present invention; and 
         FIG. 13  is a representation of a hot water pump, suitable for use in a home heating system, comprising a heat engine according to an aspect of the present invention. 
     
    
    
     Referring now to  FIG. 1 , a fluidic oscillator comprises first and second vessels  11 , 13  containing a working fluid, a connecting conduit  14 , means, typically a pipe or conduit  15  of communicating the mass of working fluid in vessel  11  to vessel  13 , a load  12 , pressurising means  27 , for example a hot heat exchanger or a source of pressurised gas, intended for raising the pressure within the vessels  11 , 13  and depressurising means  28 , for example a cold heat exchanger or a pressure release mechanism, of lowering the pressure therein. 
     The conduit  14  connects the top of the first and second vessels  11 , 13  so as to subject them to a common pressure by coupling the working fluid in the two vessels  11 , 13 . 
     The load  12  is connected to the working fluid within the first vessel  11  to enable changes in the mass of the working fluid in the two vessels  11 , 13  to give rise to a transfer of work between the first vessel  11  and the load  12 . 
     The pressurising means  27  is arranged so that the pressure is caused to rise within the vessels  11 , 13  due to a flow of heat, or further working fluid into vessel  13  when the mass of working fluid within vessel  11  is large. The rise in pressure causes further working fluid to be forced out of the first vessel  11  through the load  12 . The decreasing mass of working fluid in vessel  11  is communicated to vessel  13  by means  15 . As the mass of working fluid in first vessel  11  decreases, vessel  13  is no longer subjected to means of pressurisation  27 . When the mass of working fluid in first vessel  11  is sufficiently decreased, means  28  of depressurising the vessels  11 , 13  is applied to vessel  13  causing the pressure to fall within the vessels  11 , 13  due to a flow of heat or working fluid therefrom. The ensuing fall in pressure causes further working fluid to be sucked into vessel  11  through the load  12  and the process starts again. 
     Work is obtained from the load as the mass of working fluid in the two vessels fluctuates due to the flow of working fluid through the load and the pressure differential across the load. 
     Time delay mechanisms  16  associated with the working fluid are arranged to cause a phase shift between the mass of working fluid contained within first vessel  11  and the pressure changes therein. The time delay mechanisms each comprise a dissipative process, such as viscous drag, thermal resistance or mechanical friction, and a capacitive process, such as hydrostatic pressure change due to flow, fluid compressibility, thermal capacitance, or elasticity. 
     There is also at least one mechanism  17  associated with the working fluid that gives rise to a gain such that under normal operation, the magnitude of said pressure changes in the working fluid preferably increases or remains constant with time. 
     The nature of some of the dissipative and capacitive process at play in a fluidic oscillator according to the present invention will now be detailed. 
     Inertance (Inductance) 
     Consider an inviscid fluid flowing in a tube of constant radius. The rate of change in the mass flow rate {dot over (m)} (c.f current) due to a pressure difference ΔP (c.f potential difference) over a distance l, parallel to the flow is: 
     
       
         
           
             
               
                 ⅆ 
                 
                   m 
                   . 
                 
               
               
                 ⅆ 
                 t 
               
             
             = 
             
               A 
               ⁢ 
               
                 
                   Δ 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   P 
                 
                 l 
               
             
           
         
       
     
     So that if pressure is held analogous to voltage and current is held analogous to mass flow, the fluid inductance 
               L   =     l   A       ,         
behaves in the same was as inductance in an electrical circuit.
 
     For an incompressible fluid, mass flow rate is proportional to volumetric velocity, so the analogy holds equally well between pressure and volumetric velocity in the case of a liquid, but the inertance needs to account for density and thus becomes: 
     
       
         
           
             L 
             = 
             
               
                 ρ 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 l 
               
               A 
             
           
         
       
     
     Hydrostatic capacitance: For a flow of fluid into a vertically aligned chamber of cross section A, the rate of change of pressure P at the base is proportional to the flow in, i.e.: 
     
       
         
           
             
               
                 ⅆ 
                 P 
               
               
                 ⅆ 
                 t 
               
             
             = 
             
               
                 
                   ρ 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   gU 
                 
                 A 
               
               = 
               
                 
                   
                     m 
                     . 
                   
                   ⁢ 
                   g 
                 
                 A 
               
             
           
         
       
     
     So that if current is held analogous to mass flow, the hydrostatic capacitance is 
             C   =     A   g           
or if the analogy is between current and volumetric velocity, U:
 
             C   =     A     ρ   ⁢           ⁢   g             
Compliance:
 
     Consider a compressible fluid flowing into a closed volume, being forced into that volume by pressurised fluid from underneath/behind which may or may not be compressible itself. 
     The rate of change of pressure in that closed volume is related to the mass flow of fluid forcing it in from behind. If the volume is adiabatic and the compression is isentropic: PV γ =const. where V is the volume of the closed volume, γ is the ratio of the specific heats at constant pressure and constant volume. 
     Differentiating we have: 
     
       
         
           
             
               
                 
                   V 
                   γ 
                 
                 ⁢ 
                 
                   
                     ⅆ 
                     P 
                   
                   
                     ⅆ 
                     t 
                   
                 
               
               + 
               
                 P 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 γ 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   V 
                   
                     γ 
                     - 
                     1 
                   
                 
                 ⁢ 
                 
                   
                     ⅆ 
                     V 
                   
                   
                     ⅆ 
                     t 
                   
                 
               
             
             = 
             0 
           
         
       
     
     Such that, rearranging and dividing through by V γ . 
     
       
         
           
             
               
                 ⅆ 
                 P 
               
               
                 ⅆ 
                 t 
               
             
             = 
             
               
                 - 
                 
                   
                     γ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     P 
                   
                   V 
                 
               
               ⁢ 
               
                 
                   ⅆ 
                   V 
                 
                 
                   ⅆ 
                   t 
                 
               
             
           
         
       
     
     P and V are not constants however. Considering small changes: 
     
       
         
           
             
               ( 
               
                 
                   P 
                   0 
                 
                 + 
                 dP 
               
               ) 
             
             = 
             
               
                 
                   
                     P 
                     0 
                   
                   ⁢ 
                   
                     V 
                     0 
                     γ 
                   
                 
                 
                   
                     ( 
                     
                       
                         V 
                         0 
                       
                       + 
                       dV 
                     
                     ) 
                   
                   γ 
                 
               
               = 
               
                 
                   
                     P 
                     0 
                   
                   
                     
                       ( 
                       
                         1 
                         + 
                         
                           dV 
                           / 
                           
                             V 
                             0 
                           
                         
                       
                       ) 
                     
                     γ 
                   
                 
                 ≈ 
                 
                   
                     P 
                     0 
                   
                   ⁡ 
                   
                     ( 
                     
                       1 
                       - 
                       
                         
                           γ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           dV 
                         
                         
                           V 
                           0 
                         
                       
                     
                     ) 
                   
                 
               
             
           
         
       
     
     So that to a first order approximation: 
                 ⅆ   P       ⅆ   t       =         -       γ   ⁢           ⁢     P   0         V   0         ⁢       ⅆ   V       ⅆ   t         =           γ   ⁢           ⁢     P   0         V   0       ⁢   U     =         γ   ⁢           ⁢     P   0         ρ   ⁢           ⁢     V   0         ⁢     m   .                 
wherein isentropic fluid capacitance becomes:
 
             C   =       V   0       γ   ⁢           ⁢     P   0               
if pressure is held analogous to voltage and current is held analogous to mass flow, or the ‘isentropic compliance’ becomes
 
             C   =       ρ   ⁢           ⁢     V   0         γ   ⁢           ⁢     P   0               
if pressure is held analogous to voltage and current is held analogous to volumetric velocity. Similar analogies can be made for isothermal spaces, polytropic spaces and all sorts of boundary conditions.
 
     Referring now to  FIG. 2 , an electrical circuit  200  analogous to the fluidic oscillator of  FIG. 1  is shown. The load  12  is modelled as a resistor  12 ′, and the first vessel  11  is modelled as a capacitor  11 ′. The second vessel  13  is modelled by the circuit denoted by  13 ′, where the time delay mechanisms  16  comprise dissipative processes modelled as resistors  260 , 262 , and capacitive processes modelled as capacitors  261 , 263 . The voltage (pressure) source is analogous to pressure in the fluidic oscillator of  FIG. 1  and is equal to the voltage across capacitance  261  (i.e.: hydrostatic pressure across vessel  13 ) multiplied by the gain. The means of pressurising and depressurising,  27  and  28  are modelled as a variable voltage source  27 ′, the value of the voltage being linked to the pressure across capacitance  261 . 
     Referring now to  FIG. 3 , the fluidic oscillator is a heat engine embodiment of the fluidic oscillator described with reference to  FIG. 1  or ‘thermofluidic’ oscillator. Common components will be accorded the same reference numerals. The working fluid is part vapour and part liquid within the first and second vessels  11 , 13 . The vapour is free to flow between the vessels  11 , 13  via the conduit  14 . The liquid passes between the vessels through the restriction, or throttle,  364  wherein the flow rate of liquid is determined by the difference in the hydrostatic pressure of the liquid  365  at the respective bases of the first and second vessels  11 , 13 . 
     The second vessel  13  is provided with heat exchangers  366  giving rise to a flow of heat into vessel  13  when the liquid level is high and flow of heat therefrom when the liquid level is low therein, the flow of heat permits alternate evaporating and condensing. The evaporation and condensation of the fluid within vessel  13  causes pressure changes within the vessels  11 , 13 . The rate of change of pressure within the vessels  11 , 13  is determined, amongst other factors, by the volume of the conduit  14 , the volume of any dead space in vessels  11 , 13 , thermal resistance due to the heat exchangers  366  and the volume of a saturator  367 . 
     The purpose of the saturator  367  is not normally to add extra compliance, which should generally be minimised, but rather to provide a reservoir for the liquid piston to fill should its amplitude be very large under certain running conditions. The term saturator is used, because it causes the hydrostatic pressure to saturate in the power cylinder, i.e. once the liquid starts filling the reservoir, hydrostatic pressure is substantially invariant due to the large cross sectional area of the reservoir. 
     The changes in the pressure of the working fluid give rise to displacement of liquid in vessel  11  through the load  12  such that a ‘load phase shift’ between said pressure changes and said displacement arises due to dissipation within the load. Under normal operating conditions, the load phase shift is matched by a ‘feedback phase shift’ between the displacement of liquid in vessel  11  and the vapour pressure therein. The feedback phase shift is due to, inter alia, the viscous drag in restriction  364 , the thermal resistance due to heat exchangers  366  and the rate of change of hydrostatic pressure due to hydrostatic capacitance  365  and the compressibility of the working fluid in  14  and  367  due to the compliance therein. 
     Ideally, the load phase shift is related to the feedback phase shift such that the phase shifts normally have equal magnitude being close to 90°. Furthermore, the total phase shift between the flow of heat through heat exchangers  366  and the saturation temperature in vessel  13  is ideally close to the value required to give the maximum difference between mean heat addition and heat rejection temperatures for a given pressure amplitude. 
     The first vessel  11  is preferably significantly narrower in the centre region than at the ends thereof which are joined to the load  12  and saturator  367 . The total compliance contained within saturator  367  and conduit  14 , is preferably minimised. The load is preferably arranged to comprise inertance with reactive impedance having a magnitude equal to that of the compliance contained within saturator  367  and conduit  14  at the frequency of the oscillations. 
     The cross sectional area of the second vessel  13  in the centre section is usually minimised within limits set by the surface tension of the working fluid. The ideal ratio of the cross sectional area of the first vessel  11  in the centre section to the cross sectional area of the second vessel  13  is determined by a trade off between heat exchanger coverage with the thermal phase angle and the load phase angle and various loss mechanisms, usually including undesired cyclic transfer of heat to and from the walls of vessel  11 , and viscous drag in vessel  11 . The ideal ratio is typically, although not necessarily between 1:2 and 2:1. The second vessel  13  is typically provided with an annular insulator  368  or other means of maximising the area of the heat exchanger available to effect heat exchange within the vessel  13 . 
     Referring to  FIG. 4 , the fluidic oscillator is similar to that described with reference to  FIG. 3  and common components will be accorded the same reference numerals. The second vessel  13  is arranged in two parts side by side, one containing a hot heat exchanger  41  and the other containing a cold heat exchanger  42 . Ideally this arrangement minimises oscillatory heat transfer to and from said liquid and vapour not giving rise to evaporating or condensing thereof, thereby optimising the efficiency of the heat exchanger. 
     Referring now to  FIG. 5 , a quantity of passive gas and a diffusion column  467  are added to the working fluid such that the passive gas is situated substantially within the saturator  367 , the conduit  14 , and the upper region of the diffusion column  467 , whilst the vapour mainly occupies the second vessel  13  and the lower region of the diffusion column  467 . The passive gas is intended to limit cyclic transfer of heat to and from the first vessel  11  and the working fluid contained therein. 
     It is understood that diffusion column  467  will add further compliance to that of the saturator  367  and the conduit  14 , with an effect on the performance of the embodiment which must be accounted for in design calculations. Typical negative effects include a drop in frequency, a lower difference between mean heat addition and rejection temperatures and higher exergy, or available work dissipation in heat exchangers and typical positive effects of the additional compliance include an improvement in stability and the ability of the fluidic oscillator to self-start. 
     Referring now to  FIG. 6 , the fluidic oscillator is similar to that described with reference to  FIG. 1  and common components will be accorded the same reference numerals. The diffusion column of  FIG. 4  is replaced by a separation chamber  567  intended to prevent mixing of said passive gas situated substantially within the saturator  367 , the conduit  14  and the upper part of the separation chamber  567 , with said vapour which mainly occupies the second vessel  13  and the lower part of the separation chamber  567 . A flexible bag, diaphragm, bellows or other suitable fluid separator  61  is located within the separation chamber  567  to limit cyclic transfer of heat to and from the first vessel  11  and the working fluid contained therein. 
     The main advantage of a fluid separator over a diffusion column is that a separation chamber may be arranged to have substantially less volume than a diffusion column, lowering the compliance therein and the negative effects associated therewith. This is possible because there is a much sharper concentration gradient across the fluid separator than can exist due to diffusion alone as diffusion is no longer the process which limits the mixing of gas and vapour. The main disadvantage of a fluid separator is that it is liable to rupture due to mechanical stress or fatigue. 
     Working fluid can be supplied to the second vessel  13  by a high pressure reservoir, typically an evaporator, and removed by a low pressure reservoir, typically a condenser. However, it will be understood that the high and low pressure working fluid reservoirs can be provided by means other than an evaporator or condenser, for example, an air compressor, or a vacuum pump. 
     Referring now to  FIG. 7 , the second vessel  13  is supplied with a float  72  intended to actuate a valve  73  such that the high pressure reservoir  70  is in fluid communication with vessel  13  when the float  72  is high in the vessel  13  and low pressure reservoir  71  is in fluid communication with vessel  13  when the float  72  is low therein. The time varying flow of working fluid between vessel  13  and pressure reservoirs  70 , 71  gives rise to pressure changes within the vessels  11 , 13 , wherein the rate of change of pressure is determined by, inter alia, the volume of conduit  14 , the dead space in vessels  11  and  13 , the viscous drag due to conduits  369  and the saturator  367 . 
     The changes in the pressure of the working fluid give rise to displacement of liquid in vessel  11  through the load  12  such that a ‘load phase shift’ between said pressure changes and said displacement arises due to dissipation within the load. Under normal operating conditions, the load phase shift is matched by a ‘feedback phase shift’ between the displacement of liquid in vessel  11  and the vapour pressure therein. The feedback phase shift is due to, inter alia, the dissipation in restriction  364  and conduits  369  and the capacitance or compliance in hydrostatic pressure changes  365  and compressibility of working fluid in the saturator  367 . 
     Ideally, the load phase shift is related to the feedback phase shift such that the phase shifts normally have equal magnitude being close to 90°. 
     In the case that high pressure reservoir  70  is an evaporator and low pressure reservoir  71  is a condenser, it is desirable to provide a means of replenishing pressure reservoir  70  with liquid and removing liquid from pressure reservoir  71 . This can be achieved by inserting feed-conduits between the base of vessel  11  and the said pressure reservoirs. It is desirable to add non-return valves  74  to the feed-conduits, preferably having a low cracking pressure so that a flow of liquid can be induced between the base of vessel  11  and the said pressure reservoirs by the hydrostatic pressure difference therein. 
     A similar arrangement can be conceived of in the case that the embodiment is a heat engine in which the working fluid does not undergo a phase change, wherein the replenishment of working fluid to the hot and cold heat exchange means is provided for by diverting work from the load. 
     Instead of arranging the embodiment in  FIG. 7  to comprise two separate working vessels, it may be desirable to eliminate said second vessel  13  as shown in  FIG. 8 . It is understood that the embodiment of the invention shown in  FIG. 8  functions substantially as the embodiment shown in  FIG. 7  except that the phase shift between the volume of liquid in the vessel  11  and the pressure of the working fluid therein is generated in different manner. In the embodiment of  FIG. 8 , the valve  73  has means of giving rise to a predetermined actuation force, for example friction between the valve seals and body  370  enabling a phase shift between the height of the float and the liquid level within vessel  11 . 
     The changes in the pressure of the working fluid give rise to displacement of liquid in vessel  11  through the load  12  such that a load phase shift between the displacement and said pressure changes arises due to dissipation within the load. Under normal operating conditions, the load phase shift is matched by a ‘feedback phase shift’ between the displacement of liquid in vessel  11  and the pressure therein. The feedback phase shift is due to, inter alia, the means of giving rise to a pre determined actuation force  370 , the viscous drag or dynamic pressure loss due to conduits  369 , the capacitance or compliance in hydrostatic pressure changes  365  and compressibility of working fluid in the compliance  367 . 
     In the embodiments of the invention described in relation to  FIGS. 7 and 8 , it is generally desirable to generate hysteresis in the actuation of said valve means  73  by separating the float  72  from the valve means in accordance with the embodiment shown in  FIG. 9 . 
     In another embodiment of the invention described in relation to  FIG. 10 , the invention is embodied into another heat engine. In the embodiment of the invention in  FIG. 10 , one dissipative process contributing to the feedback phase angle comprises the dissipation of work within the load due to the pressure difference across the load and the flow of fluid therethrough. The loss of work to the load behaves as the loss of work in a viscous drag or the loss of available work through a thermal resistor. The first vessel  11  is connected to the second vessel  13  by a conduit  14  such that the first and second vessels  11 , 13  contain working fluid at a common pressure. 
     The first vessel  11  is connected to the load  12  such that changes in the mass of working fluid contained within first and second vessels  11 , 13  give rise to a displacement of fluid between the vessels  11 , 13  and the load  12 . A compliant vessel  101  is connected to the opposite side of the load  12  to said first vessel  11 . Displacement of fluid between the first vessel  11  and the load  12  gives rise to the displacement of additional fluid between the load and the compliant vessel  101  such that a phase shift exists between the mass of working fluid contained within the vessels  11 ,  13  and the conduit  14 , and the pressure of fluid contained within the compliant vessel  101 . Means  15  of communicating the pressure contained within the compliant vessel  101  to the second vessel  13  is provided. 
     The second vessel  13  is provided with time delay mechanisms  16 , each comprising a dissipative process and a capacitive process. The processes that constitute the time delay mechanism might for example include the dissipation of exergy, or available work due to a thermal resistance, or a viscous drag and thermal capacitance or compliance due to fluid compressibility. Ideally, the total phase shift between the displacement of working fluid between the load  12  and the first vessel  11  and the pressure therein is determined by the dissipation in the load, the compliance of vessel  101 , the time delay mechanism  16  and the compliance of vessel  11  and conduit  14 . It will be appreciated that the time delay mechanisms  16  may be replaced by other means of giving rise to an RC time delay such as a pressure sensor going through an amplifier and electrical RC circuit connected to a compressor in communication with the compliant vessel  101 . 
     It will be understood that means of giving rise to a gain is provided such that the amplitude of pressure oscillations within vessels  11 ,  13  and the conduit  14  is greater than the amplitude of pressure oscillations within vessel  101  such that oscillation amplitude remains substantially constant with time under normal operating conditions. 
     A specific arrangement of the embodiment described in  FIG. 10 , in which non-return valves are provided with the intention that fluid flows in a single direction is shown in  FIG. 11 . Referring now to  FIG. 11 , a compliant sub-vessel  11   a  is connected to a second vessel  13   a , which comprises more than one chamber, by means of conduit  14   a . The conduit  14   a  is provided with a non-return valve  113   a  such that working fluid flows from the second vessel  13   a  to said compliant sub-vessel  11   a  when the pressure is greater within the second vessel  13   a  than the pressure in the compliant sub-vessel  11   a.    
     The compliant sub-vessel  11   a  is provided with heat exchanger means giving rise to heating of working fluid entering from said second vessel  13   a , and a consequential rise in pressure within the compliant sub-vessel  11   a . The rise in pressure in the compliant sub-vessel  11   a  further gives rise to a flow of working fluid through a load  12   a  into a second compliant sub-vessel  101   a . The flow of fluid into the compliant sub-vessel  101   a  results in a pressure increase therein. The second compliant sub-vessel  101   a  is provided with a heat exchanger  112   a  that cools the working fluid entering the second compliant sub-vessel  101   a  from the load  12   a . A pressure increase in second compliant sub-vessel  101   a  is communicated to vessel  13   a  by a flow of cool working fluid, via a return conduit  15   a.    
     The vessel  13   a  comprises an identical set of components to those already herein described comprising a sub-vessel  11   b , a load  12   b , a vessel  13   b , means  14   b  of conjoining the sub-vessel  11   b  and the vessel  13   b , a non-return valve  113   b , heat exchanger means  111   b  and  112   b , means of communicating the pressure within second compliant sub-vessel  101   b  to the vessel  13   b , and a return conduit  15   b . Vessels  13   a  and  13   b  comprise time delay means wherein said time delay is caused by the thermal resistance in heat exchangers  111   a ,  111   b ,  112   a  and  112   b , the compressibility of the working fluid in sub-vessels  11   a ,  11   b  and the said conduits, the dissipation in the loads  12   a  and  12   b  and the compressibility of working fluid within the compliant sub-vessels  101   a , 101   b.    
     It is understood that the embodiments herein disclosed may be arranged in a variety of geometric configurations, for example in which the said two vessels are concentric about the same axis, or in which the said two vessels are combined. Furthermore, it is understood that the said working fluid may be made up of more than one component. In the heat engine embodiments herein described, regenerator or recuperator means may be applied so as to improve thermal efficiency. 
     It is preferable, in all embodiments of fluidic oscillators according to the present invention, that the vessels  11 ,  13  and the conduit  14  are constructed from a material or materials having a low product of specific heat capacity, thermal conductivity and density with the intention that heat transfer between the working fluid and the vessels  11 , 13  and the conduit  14  is minimised. 
     Referring now to  FIG. 12 , a solar irrigation pump  1200  comprises a heat engine  1202 , a solar thermal collector  1204 , typically a solar panel, a heat pipe or thermo-siphon  1206 , and a tube  1210  placed in a reservoir  1212  of fluid to be pumped, typically water. 
     The heat engine  1202  can be any one of the heat engines as described hereinbefore, or a heat engine operating according to similar principles thereto. 
     The solar collector  1204  typically comprises the hot end of the heat pipe or thermo-siphon  1206 . Evaporation or convective heating at the hot end of the heat pipe or thermo-siphon  1206  gives rise to condensing or convective cooling on the outside of a hot heat exchanger  1207  of the heat engine  1202  so that the solar collector acts to heat the hot heat exchanger  1207 . 
     The tube  1210  is connected to the outside of a cold heat exchanger  1208  of the heat engine  1202  such that fluid exiting the reservoir  1212  acts as a coolant of the cold heat exchanger  1208 . The temperature difference between the hot and cold heat exchangers  1207  and  1208  of the heat engine  1202  causes the working fluid  1214  of the heat engine to oscillate. In one embodiment, the working fluid  1214  is immiscible with the fluid to be pumped, for example if water is to be pumped, a hydrocarbon working fluid may be used. The interface between the working fluid and the fluid to be pumped  1218  acts as a piston face in a conventional pump, this can obviate the necessity for moving parts in such a pump. 
     The outstroke of the working fluid  1214  forces the fluid to be pumped so as to force open a non-return outlet valve  1215  to output the pumped fluid. The outlet valve  1215  closes as the outstroke finishes and the return stroke commences. 
     On the return stroke of the pump a non-return inlet valve  1216  opens and the volume left by the retreating interface is filled by fluid extracted from the reservoir  1212  through tube  1210 . This fluid is then used to further cool the cold heat exchanger  1208  before it is subsequently used, for example for irrigation. 
     In a further embodiment of the pump  1200  a flexible bag, diaphragm or other flow separation means separates the working fluid and the fluid to be pumped. The diaphragm is used to transfer the work between the working fluid and the fluid to be pumped in order to affect pumping of the fluid to be pumped. This separation of the working fluid from the fluid to be pumped allows a wider variety of working fluids to be used as it ensures no mixing of the fluids. 
     Referring to  FIG. 13 , a domestic hot water circulation system  1300  comprises a heat engine  1302 , a water heater, typically a gas or oil fired water heater  1304  and radiators  1310 . The water heater  1304  is connected to the heat engine via conduit  1306 . The difference in temperature between the primary heat exchanger of the boiler  1304  and the temperature of the water returning from radiators  1310  can be used to heat a hot heat exchanger  1307  and cool a cold heat exchanger  1308  of the heat engine  1302  respectively, and the work generated by the heat engine can be used to pump the water around the circulation system. It is understood that other available sources and sinks of heat, such as hot flue gases and mains water entering the building respectively, may be preferred. 
     The water heater  1304  supplies hot water to the outside of the hot heat exchanger  1307  of the heat engine  1302 . The radiators  1310  supply cooled water to the outside of the cold heat exchanger  1308 . The temperature difference between the hot and cold heat exchangers  1307  and  1308  of the heat engine  1302  causes the working fluid to oscillate between the hot and cold sides of the heat engine, and generates work in the load of the heat engine. The load comprises a pump  1318  to pump water from the water heater  1304 , around the hot heat exchanger  1307  and on to radiators  1310  distributed around the system  1300 . After the water has passed through all of the radiators  1310 , it has cooled sufficiently to provide a cooling effect to the cold heat exchanger  1308 . 
     The main advantage of using the heat engine according to the present invention in domestic heating systems is that electricity is not required to pump water around the system. The amount of electricity used to pump water around a domestic heating system is typically 10% of net domestic consumption, and the present invention therefore offers a significant reduction in electricity consumption. The heat engine is also capable of self-starting in the presence of a heat source, which offers the possibility of eliminating costly and unreliable control systems. 
     Any system that has an existing temperature differential can be exploited to generate work by using a heat engine according to the present invention. Such systems can include using the heat engine to pump liquid around refrigeration/air conditioning systems, power generation systems and other systems in which thermally driven pumping is relevant.