__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 + ˝ V^{2} = constant along a
streamline
(1)

For incompressible flow,
this becomes

p/ ρ + ˝ V^{2} = constant along a
streamline; in energy units (joules) (1a)

or
p + ˝ ρV^{2} = constant along a stream line; in pressure units (Pascals) (1b)

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

ρ_{1} V_{1} A_{1} = ρ_{2} V_{2} A_{2} = 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

V_{2} = V_{1} [A_{1}/
A_{2}] [ ρ_{1}/ ρ_{2}]
(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

(p/p_{o})^{(k-1)/k} = ( ρ/ ρ_{o})^{k-1}
= T/T_{o} = (c/c_{o} )^{2}
= 1 – (V/c_{o})^{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/p_{o})^{2/(n+2)} =
ρ/ ρ_{o})^{2/n} = T/T_{o} = (c/c_{o} )^{2} = [1 –
(V/c_{o})^{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 = c_{p}/c_{v}) , has the
value 1.4 . Here c is the speed of sound and c_{o} 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 V^{2} = 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 = ρ_{1} V_{1} A_{1} = ρ_{2} V_{2} A_{2} = 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 = ˝ x (mass flow rate ) x V^{2}

Thus a mass flow of
0.5 kg/s at a velocity of 10 m/s has a power of P = ˝ x 0.5 x 10^{2} =
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 = p_{in} – p_{out} 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 V^{2} .

For example, a flow speed
of V = 10 m/s will have a dynamic
pressure of p = ˝ ρ V^{2} =
˝ x 1.2 x 10^{2} = 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

ρ_{1} V_{1} A_{1} = ρ_{2} V_{2} A_{2} = 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

V_{2} = V_{1} [A_{1}/A_{2}]
[ ρ_{1}/ ρ_{2}]
(3)

If the density is constant, so that ρ_{1} =
ρ_{2} , then we have

V_{2} = V_{1} [A_{1}/A_{2}]

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 i*sentropic* 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 i**ncreased 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 =
˝ (m-dot) V_{2} = ˝
(ρ V A) V^{2}
(8)

Let us take as an example, a converging cone of inlet area 0.0198
m^{2} (diameter 15.9 cm); and
throat area of
.000998 m^{2} ( 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 = ˝ x
0.2376 x 10^{2} = 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 = ˝ x 0.2376 x 313^{2} = 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,

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