Direct PZT Fluid Flow Control Feasibility

1. Incompressible flow
in the round pipe.

a). In laminar flow
, fluid particles move along straight, parallel paths
in layers or laminae. Magnitudes of velocities of adjacent
laminae are not the same.

Laminar flow is governed by the law relating shear stress
to rate of angular

deformation, i.e., the product of viscosity of fluid
and velocity gradient or

In laminar flow, t
he viscosity of the fluid is dominant and thus suppresses
any tendency to turbulent conditions.

Flow became turbulent, when velocity has reached value
of the critical velocity
(is the velocity below which all turbulence is damped
out by the viscosity of the fluid). It is found that
the upper limit of laminar flow of practical interest
is represented by a Reynolds number of about 2000.

Velocity distribution
at a cross section will follow a
parabolic law of variation for
laminar
flow. Maximum velocity is at the center and is twice
the average velocity. The equation of the velocity
profile for laminar flow can be expressed as

for circle pipe velocity distribution is parabolic.

or

where hL is the
lost head:

Volumetric flow via round/circle pipe is:

using finite element approximation

and equation for velocity distribution obtain

famous Poiseuille equation.

In this case average velocity is:

and maximum velocity (r=0) is:

B.
Incompressible flow in the Rectangular pipe (duct).

Duct with dimension:
2a x 2b and b
- is a gap.

for rectangular pipe.

where:

For laminar flow between two flat plates

where b
is a gap

Friction factor f
can be derived mathematically for laminar flow, but
no simple mathematical relation for the variation of
f
with Reynolds number is available for turbulent flow.
Furthermore, Nikuradse and others found that the relative
roughness of a pipe (ratio of size of surface imperfections
e
to inside diameter of the pipe) affects the value of
f also.

(a) Thus,
for laminar flow in all pipes for all fluids,
the value of f
is

Re has a practical
maximum

value of 2000 for laminar flow.

A. Incompressible flow
in the round pipe.

Flow characterization without forcing from DPZTFFC device
(only viscosity changes which allowed change

character of turbulent flow) for steady, incompressible
flow in the round pipe only.

Basic equations for steady flow:

Laminar

Turbulent

Average velocity for laminar flow is 0.5 of maximum
velocity in the center-line (parabolic profile).

Average velocity for turbulent flow is a Re number dependent
and for Re=5,000 equal 0.77 and for Re=3*10^6 equal
0.87 of maximum velocity in center-line ( logarithmic
profile).

DPZTFFC device authority estimation:

In this case (round pipe flow) control authority would
be limited by 4-5%.

However, in assumption that the DPZTFFC will change
interaction between flow and wall surface we can change
the velocity to make flow turbulent earlier. For example,
for pipe with 10 mm dia. and water as media Re~10^4*V
and critical velocity is 0.2 m/sec. Presumably the
DPZTFFC could make the critical velocity smaller.
In this case ( laminar / turbulent flow transfer) control
authority would be 50-80% of flow.

B.
Incompressible flow between flat plates.

For laminar boundary layer, if Re high enough:

where : b - is boundary layer thickness;

l - character length (plate f.e.)

For our case
Re~10^4...10^5
and this number give us boundary layer with gap b~(0.003...0.010)*l

If device will used PZT plate with l=100 mm we will capable
to control gap
b~(0.3...1) mm.

Let's assume b=0.5
mm for l=100
mm. Let's assume also 2*a=100 mm.

According to equation ( ) for laminar flow in rectangular
duct with dimension: 2a x 2b (b
- is a gap) the average velocity could be calculated
as:

Device would be capable for maximum flow:

where:

We can see device with good possibility for get authority
on flow and with reasonable flow characteristics.

With drop of the pressure less then 1 atm volumetric
flow rate would be close to 30 m3 per hour.

Developed by Roman N Tunkel,

Research Engineer,
Ph.D.,

tunkelr@gmail.com