The Flowing System Gasdynamics Part 1: On Static Head in The Pipe Flowing Element
The Flowing System Gasdynamics Part 1: On Static Head in The Pipe Flowing Element
The Flowing System Gasdynamics Part 1: On Static Head in The Pipe Flowing Element
The solution of problem on distribution of static head along the pipe flowing element is
submitted. The solution is reached on the basis of consideration of contact interaction of gas
and liquid stream with the wall of the pipe flowing element. The expression for distribution of
static head along the pipe flowing element is obtained on the basis of usage of three
fundamental laws in fluid dynamics: Torricelli formula, Weissbach-Darcy formula and
Bernoulli equation. The general solution is obtained for a gas stream. The special case of the
obtained solution is retrieved for liquid stream.
Nomenclature
g
hi
H
L
L
D
in
ex
p0, ph
pst
Vfw
1 Introduction
The creating of physically substantial bases of the fluid
medium motion theory envisions overcoming the set of
problems and solution of series of attendant questions.
The problem of determination of distribution of the
energy potential along the pipe flowing element or system
is one of fundamental in this area. The essence of this
problem lies in the fact to define an unitized mathematical
expression, which will physically adequately and
1
2 Approach
For the solution of the problem the authors utilized: the
formula for free fall setting by proceedings of Torricelli Galilei (1643) and Borda - Du Buat (1766), the formulae
for determination of friction losses at motion of liquid
stream in pipe of Weissbach-Darcy (1857) and local
losses of Weissbach (1865) and also the equation of
conservation of energy (density of energy is more exact)
for stream in pipe by Bernoulli principle (1738) in the
form that conform to the end of I century. The
sufficiency of these dependences for reaching an object is
determined by their stated below distinctions. Torricelli Galilei - Borda - Du Buat (TGBD) formula represents the
historically first equation of conservation of energy for
motion of solid or medium without contact interaction.
The path of development of this formula from the
moment of an experimental research up to physically
correct record was required almost 150 years (Du Buat)
and up to usage in equation of conservation of energy was
required additional almost 120 years. This formula has
passed inspection in general mechanics and then in
mechanics of fluid medium and it is one of fundamental
in mechanics. Weissbach-Darcys formula represents the
historically first modification of TGBD formula for fluid
2
medium motion with taking into account of contact
interaction (with pipe wall). This formula is obtained also
on the basis of experimental research and reflects the
approach to a viewing of motion indicated by Aristotle
(328 ..) as the third problem of his Mechanical
problems. The fundamentality of taking into account of
contact interaction called simplistically as hydraulic
friction is affirmed not only Aristotle's prevision but also
that this formula became an inalienable part of equation
of conservation of energy for fluid flow in pipe.
Weissbach's formula is modification of WeissbachDarcy's formula and allows to determine the different
shape local resistances at fluid medium motion in flowing
system.
The fundamental character of equation of conservation of
energy for stream of fluid medium in flowing element or
system, apparently, cannot call doubt, and furthermore
this equation is constructed on above-mentioned laws.
The principle of conservation of energy at motion without
contact interaction and the principle of consumptions of
energy for contact interaction at motion is harmonic
combined in this equation. The last reflects even more
severe sense, connected with self-organizing of stream
with minimum of energy consumption that applied to the
flowing system.
1 = Kl
L
p st (l ) h(l ) = L K l
2
V fw
2g
+ ph
(3)
po = p h +
2
V fw
2g
+ (L K l + in + ex )
2
V fw
2g
(4)
p st ( l ) = ( p0 p h )
L K l
+ p h (5)
1 + L K l + in + ex
3 Solution
So, TGBD formula may be written in the form:
V2
= H 1
H
2g
l V fw
p st (l ) h(l ) = L 1
+ ph (2)
L 2g
2
V fw
(l )=
2g
(p0 ph )
1
= const
1 + L K l + in + ex
(6)
p st ( l ) = ( p0 p h )
L K l
+ p h (7)
1 + L + in + ex
3
of a complex varying along the pipe length 0 K l 1
both in numerator and in denominator of fraction in a
right member of expression (5) specifies the nonlinear
character of static head change in gas stream in pipe. At
the same time the diminution of quantity of static head in
gas stream in direction of outlet from pipe happens more
and more intensively. Physically it means increasing of
velocity of gas stream to outlet from pipe with constant
values of cross section and coefficient of hydraulic
friction. Such differences of physical characteristics of
gas from liquid as a major elastic (reversible)
compressibility and a major kinematical viscosity in
combination with small heat capacity stipulate the
consecutive transformation of friction work to the heat,
which one, in turn, stipulates the consecutive increase of
gas stream velocity along the pipe of uniform cross
section. In distribution of static head along the pipe for
liquid and gas, the common is that the initial and final
values of static head can be the same for these two
mediums. The difference is, that at the mentioned
identity, the curve of static head of gas stream between
points st(0) = p0 and st(L) = ph is situated above a
straight line of static head of liquid stream.
In problem about free fall in gravity field, the velocity of
body is increased on the way of the fall under stationary
value of gravity acceleration. In problem about motion of
gas stream in pipe the flow velocity is increased under
stationary value of intensity of contact interaction of gas
stream with surface. Such acceleration of gas stream
motion is the frictional self-acceleration.
5 Congruence of results
The obtained expressions (5) and (7) are used as basis for
working out of mathematical algorithm and then VeriGas
program for computing of the state and motion
parameters of gas stream in the diverse flowing element
and its systems. The computational results are tested by
means of comparison with experimental data of a great
many of sources of the specialized literature.
In particular, it was determined, that overstated
experimental curve of static head approximately on 5% in
contrast to our computations, inserted in the book [2], is
explained that the pipe termed in the book text as a
smooth really had 19 static head measurement orifices.
Thus, the comparison has given result not for the benefit
of experiment. There is a lot of such examples in papers
and monographies [3,4,5].
6 Final remark
The expressions shown in this paper have a key nature,
therefore their deduction is submitted in one-dimensional
stationary statement. At the same time, results of
computational experiments that repeatedly realized does
not yield to experimental results on regularity and
precision.
4 Discussion
Lately, the analogous object was pursued by Duxbury in
his work [1]. The equation derived by this author contains
the link of pressure drop along the pipe with the mass
flow rate of fluid. The link is not immediate because it
requires to introduce the discharge coefficient as the
additional empirical factor defined experimentally.
Furthermore, the link of pressure drop with flow rate is
complicated by the necessity of determination of density
of fluid at the beginning and end of stream as well as an
average quantity in conditions of essential non-linear
nature of the change of the pressure drop along the pipe.
The mentioned imperfections is a consequence of the
physically inferior approach based on balance of the
forces but not the equation of conservation of energy.
Therefore equation for determination of pressure drop
along the pipe derived in [1] as many other attempts