Section 1

 

Linear ( streamline) Flow and Flow  Power Amplification

 

 

Abstract:   Part 1 is a brief review of air flow fundamentals which  lead to a way of  enormously increasing   the power of an air flow by efficiently accelerating it. This leads to two new inventions for extracting the increased flow power to do useful work. The new methods use ordinary air and involve no fossil fuel heat input or noxious exhaust gases. If desired, the new air motor can potentially provide a self-sustaining mode of operation ( Perpetual Motion).

 

 

Contents

1.1 Fluid Flow Principles: Linear  (streamline)  flow

 

1.2 Flow Losses and Inefficiencies

 

1.3  Flow Velocity Changes/ Acceleration  and Pressure Changes

 

 1.4   Flow Power Amplification: Accelerating the Linear Flow of a Gas to Increase its Flow Power

 

 1.5 Accelerating the Basic Linear Mass Flow

 

 References   

 

 

 1.1   Fluid  Flow Principles:  Linear ( streamline) flow:

 

Before we describe the new acceleration and energy production  process, we need a few basic fluid flow principles. The general subject of air flow is a part of fluid mechanics [Ref.1] which deals with both incompressible and compressible flows. Air is compressible, so that its density is not constant but varies with the flow speed V, the pressure p and the temperature T. The changes in air density ρ with flow speed are quite small up to speeds of about Mach 0.3 (about 90 m/s), so that for low wind speeds, air is often treated as being incompressible with the density ρ being taken as constant  (at about  1.2 kg  per cu. m. at m.s.l.).With the density constant, only the pressure and temperature changes need be considered.  In this case, for the low speed  incompressible case, the two principal flow equations are the Bernoulli  equation and the equation of mass continuity which are as follows: ( units are SI, that is to say, kilogram/meter/second  units)

 

First, the Bernoulli equation, which is related to the energy, is:

 

∫dp/ρ + ½  V2 = constant along a streamline                                     (1)

 

 For incompressible flow, this  becomes

 

p/ ρ +  ½ V2 = constant along a streamline; in energy units (joules)                 (1a)

 

or   p + ½ ρV2 = constant along a stream line;  in pressure units (Pascals)                  (1b)

 

The second flow equation is the Equation of Continuity of Mass:

 

ρ1 V1 A1  = ρ2 V2 A2  = dm/dt  = m-dot =  constant  ( kilograms/sec)              (2)

 

where the subscript numerals refer to values of the density ρ, velocity V and cross-sectional area A at different cross sections along the flow path.

 

It is seen that Equation  (2) represents the mass m of air flowing per second (dm/dt),  which in SI units  would be expressed as  kilograms of air passing  through any given area per second. Thus, a reduced cross-sectional area A, means an increased velocity V  and  vice versa..  The velocity increase in a flow passing through a converging conical nozzle can thus be formulated from (2) as

V2 = V1 [A1/ A2] [ ρ1/ ρ2]                                                                              (3)                           

 

From this we can see that, if the air density ρ were to be constant, the velocity increase in a converging nozzle is inversely proportional to the cross-sectional area decrease. This simplifying constant density assumption  could apply for example to air  speeds of less than about Mach 0.3 ( 90 m/s) . Above that speed compressibility becomes important and density changes must  explicitly be taken into account  through the isentropic relationships. 

 

 

In an isentropic flow [Ref. 1, 2, 3]  the changes in the thermodynamic variables of the  gas, i.e. in the pressure p, density ρ, temperature T, speed of sound c, and flow velocity V,  all take place  according to the isentropic relationships:

 

(p/po)(k-1)/k  = ( ρ/ ρo)k-1 = T/To  = (c/co )2 =  1 – (V/co)2 /n                          (4)                   

 

where  n = 2/(k-1) and k is the ratio of specific heats. Expressed in terms of the number of ways  n that the energy is divided ( n = 2/(k -1)) we also have

 

(p/po)2/(n+2)  =  ρ/ ρo)2/n  = T/To  = (c/co )2 =  [1 –  (V/co)2 /n ]                     (4a)

 

Thus, for example, a drop in any one of pressure, density or temperature brings about  an increase in the flow velocity.  The subscripted values are for stagnation or zero velocity initial conditions. For air, the value of  k, the  ratio of specific heats ( k = cp/cv) , has the value 1.4 . Here c is the speed of sound and co is its stagnation ( no flow) speed. 

 

As an example of the magnitude of the isentropic effects, we may note that that at the flow  speed  of approximately 313 m/s  ( i.e. at the sonic speed), the pressure will have dropped by 47.2% to almost half an atmosphere, the density  will have dropped  by 37 % and  the temperature by 17%,. The isentropic values are readily available from tables ( Ref. 1,2,3]  or from various commercially available computer programs for isentropic flow calculations.

 

Isentropic velocity increases readily take place, for example, in a converging nozzle through which the flow is directed; the velocity reaches its maximum value at the narrowest cross-section of the nozzle called the ‘throat’. If the flow is then passed on  through a diverging nozzle, it will decelerate efficiently without losses  and drop back to normal velocity, pressure and density value at the exit.  The governing equations for compressible flow of air in a nozzle  are the isentropic flow relationships as given in Eqns.4 and 4a  plus  the equation of  continuity  of mass in Eqn.2

 

1.2  Flow Losses and Inefficiencies

 

The Bernoulli equation, mass continuity equation and isentropic equations  given above for linear flow all assume that no flow turbulence, viscosity or friction is present  This amounts to assuming that the flow is 100 % efficient. In practice, of course, some turbulence, heat flow ,viscosity  and so on will be present,  but the losses can still with care be kept as low as a few  percentage points. We also note that fluid acceleration can quite easily  be made nearly 100% efficient; deceleration, on the other hand, is more difficult to achieve efficiently, that is to say without losses.[ Ref. 1]. We shall consider such accelerations in detail shortly.

 

1.3    Flow Velocity Changes/ Acceleration  and Pressure Changes

From isentropic considerations, we have [Ref. 2]:

 

dp = −ρV dV

 

Also, in incompressible flow the familiar Bernoulli equation

 

 p  +1/2  ρ V2     = constant

 

gives the same  result,  namely dp = − ρV dV

 

Thus, an acceleration in flow velocity must result in a decrease in pressure, whereas deceleration must bring about an increase in pressure.

 

We can now apply the above flow principles to explain several interesting experimental problems.

 

Since an acceleration increases the kinetic energy of the flow, it is natural to consider how the kinetic energy and power of a flow may be increased and then extracted to do useful work.

 

 

Energy can be  added to a fluid in various ways. Heating a gas adds internal energy and  the temperature and pressure are thereby raised. This process is the basis for the operation of steam engines and internal combustion engines of all sorts, and its thermodynamics and dynamics are well understood. 

 

Another way of adding energy is to increase the flow velocity, so as to add kinetic energy to the flow. This process is also very well understood, for example in  rocket nozzle propulsion, gas dynamics, and so on.

 

Here we shall be concerned with adding kinetic energy and power to a flow of gas ( specifically to air)  and then extracting a portion of the power increase to do useful work. The basic  mass flow of gas can be supplied  (1) by a vacuum motor or an air compressor ( and this leads to a first invention to be described in Section 2, or (2) by the wind or a moving vehicle  ( which  is the basis for a second invention to be described in Section 3).

 

        

1.4   Flow Power Amplification: Accelerating the Linear Flow of a Gas to Increase its Power

 

The starting point here is a basic mass flow of air. This is supplied to a fluid in several ways, for example  by a “pull” or vacuum source, a “push” or compressor source,  by the wind, or by the air flowing past a  moving  vehicle  on which the flow apparatus is mounted.

 

 

Mass flow rate = ρ1 V1 A1  = ρ2 V2 A2  = dm/dt  = m-dot =  constant  ( kg/s)

 

(which is  the Equation of Mass Continuity  ( Eq. 2) of Section 1.1 above)

 

The power of this basic mass flow at the source is usually low since the flow velocity initially is low. The formula for air flow power is

 

P = ½ x (mass flow rate ) x V2

 

Thus a mass flow of 0.5 kg/s at a velocity of 10 m/s has a power of P = ½ x 0.5 x 102 = 25 watts

 

In connection with supplying and maintaining a given mass flow, it will be well to point out that any flow of  air from whatever source through an apparatus will involve the pressure difference Δp = pin – pout  across the apparatus from inlet to outlet; it is this pressure difference  which drives and maintains the mass flow.  In the case of a vacuum pump flow source (Fig.1), the pressure difference will be that between the ambient pressure at the inlet to the apparatus and the vacuum pressure maintained at the vacuum pump. In the case of a compressor, it will be the difference between the compressor exit pressure and the ambient pressure at the apparatus flow exit port. In the case where the source of the mass flow is the wind, the pressure differences across the apparatus from entrance to exit are very much smaller,  being just the dynamic or flow pressure of the air, p = 1/2  ρ V2 . 

 

For example, a flow speed of  V = 10 m/s will have a dynamic pressure of p = ½ ρ V2  = ½ x 1.2 x 102 =  60 Pascals.  This is  the  pressure differential Δp across the apparatus from entrance port to  exit port needed to maintain the  flow  of 10 m/s through the apparatus. Inefficiencies in the flow through the apparatus, such as turbulence and frictional losses will lower the pressure differential,  flow speed, flow power and mass flow rate.

 

Figure 1.       Linear  “Pull” Flow Vacuum Motor

Also, to give a specific example, a  vacuum source motor capable of delivering a pressure differential  to sustain a mass flow of say, 0.074  kg/s,  would typically would require a  an input  power of about 1700 watts to run it.   However, conventional vacuum pumps are  inefficient ; for example, the 1700 watts input cited would produce only 600 to 100 watts air flow power, so that the vacuum pump efficiency would only be from 35% to about 6%.

 

 

1.5 Accelerating the Basic Linear Mass Flow

 

This can efficiently be accomplished by passing the mass  flow  through a converging nozzle. The nozzle can be of various shapes, such as conical, bell- shaped, parabolic, etc., but all must be smooth walled.

 

 

 

 

Figure 2.  Flow Acceleration in a converging/diverging nozzle ( De Laval Nozzle)

                                                                            

 

The appropriate mass flow equation here is, again,  the Equation of Continuity of Mass

 

 ρ1 V1 A1  = ρ2 V2 A2  = dm/dt  = m-dot =  constant                                          (2)

 

where the subscript numerals refer to values of the density ρ, velocity V and flow duct cross-sectional area A at different cross sections along the flow path.

 

It is seen that Equation  (5) represents the mass m of air flowing per second (dm/dt),  which in SI units  would be expressed as  kilograms of air passing  through any given area per second.

 

Thus, a reduced cross-sectional area A, means an increased velocity V  and  vice versa..  The velocity increase in a flow passing through a converging conical nozzle can thus be formulated as

V2 = V1 [A1/A2] [ ρ1/ ρ2]                                                               (3)

 

If the density is constant, so that  ρ1  =  ρ2 , then we have

 

V2 = V1 [A1/A2]

 

and  we see  that the velocity increase in a converging nozzle is inversely proportional to the cross-sectional area decrease. This simplifying assumption of constant density applies to air  speeds of about Mach 0.3 ( 90 m/s) . Above that speed, the decrease in density with increasing velocity becomes important for compressible fluids and must be explicitly be taken into account  through the isentropic relationships. 

 

If the ratio of cross-sectional area is made large enough (that is to say, if the nozzle  throat diameter is made small enough) then  the throat  velocity can be increased to the sonic speed (313 m/s).  At this speed the flow rate ( mass flow ) cannot  increase further at the given throat diameter and  the flow is then said to be “choked” [ Ref. 1,2,3]. 

 

It is interesting to calculate the increase in the flow  power that this acceleration to sonic speed has  caused. The power of the flow is

 

P =  ½ (m-dot) V2 = ½  (ρ V A) V2                                                               (8)

 

 

Let us take as an example, a converging cone of inlet area 0.0198 m2  (diameter 15.9 cm); and throat  area  of  .000998 m2 (  diameter 3.56 cm). Then, for an inlet flow of 10 m/s we have a mass flow of m-dot = (ρ V A) = 1.2 x 10 x .0198 = 0.2376 kg/s., which will be the same at inlet, throat and exit.

 

The power of the air flowing in at the  inlet is then

 

P = ½ x  0.2376 x 102 = 11.9 watts.

 

However the power at the throat, where the velocity is now sonic or 313 m/s and has the same mass flow of 0.2376 kg/s, becomes 

 

P = ½ x 0.2376 x 3132 = 11,639 watts

 

or an increase in flow power of 978 times!

 

Furthermore, since the process is isentropic, this all has happened without any addition of energy from the outside.

 

 What has happened is that the internal energy of the gas itself has been lowered and has reappeared in the form of increased kinetic energy of the flow.

 

All that is required here is that we maintain the mass flow by maintaining the pressure differential from inlet to exit;   the acceleration through the nozzle  then takes place automatically. There  is no input or output of outside of heat and so the process is called isentropic. Isentropic nozzle acceleration of air flow is truly an amazing phenomenon.

 

The natural question now is:  Q. Can we extract any of this astonishing increase in flow power to do useful work?  This will be answered in Sections 2 and 3.

 

References

 

1.  Munson, Bruce, R., Donald F. Young, and Theodore H. Okiishi, Fundamentals of Fluid Mechanics,  Wiley and Sons, New York, 1990.

 

2. Shapiro, Ascher, H., The Dynamics and Thermodynamics of Compressible Fluid Flow,  2 Vols.,  John Wiley & Sons, New York.1954.

 

3.  R. Courant and K.O. Friedrichs,  Supersonic Flow and Shock Waves. Interscience, New York, 1948.

 

 

Copyright   Bernard A. Power,  May 2011

 

 

Section Links:

                                            

Section 2: Invention No. 1:  A New Isentropic Air Motor and  Clean Energy Source

 

Section 3:  To be posted in near future

 

Section 4:  Flow Acceleration and Centrifugal Force as a Possible Cause of the  Observed Temperature Rise Anomaly in the Ranque-Hilsch Vortex Tube

 

Section 5:  A Note on Isentropic Flow ‘Potential Motion’

 

 

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