Flow of Incompressible Fully Developed steady

Fluid Flow in Pipe

Idrees Majeed Kareem

Mechanical engineer, Department

Faculty of Engineering

Zakho University

Abstract

Pipe flow, a type of hydraulics and fluid mechanics of liquid flow contained by a closed conduit. The 2nd kind of flow within a channel

is open channel flow. This 2 types of flow are similar in many method,

but different in one chief part. The channel flow have a free surface which is not found in pipe flow. Flow in

pipe, being limited within closed channel, does not apply direct atmospheric

pressure, but it can concern hydraulic pressure on the channel.

Dimensions

of skin friction and mean-speed profiles have been made in fully developed

flows in pipes and channels in the Reynolds number range 1000 < Re <
10000. These measurements, and explanation of hot-wire signals, indicate rather
remarkable differences between two-dimensional and axially symmetric flows and
also make it difficult to give a precise definition of the, full term, fully
developed turbulent flow1.
Observe
Reynolds first established that the Reynolds number in fact
characterizes the state of the flow – laminar, transition, and fully turbulent.
The exact values of the Reynolds number that classify these three region vary
depending on the geometry under study. For smooth walled circular pipe flow we
havec2.
We can
also find with, Navier-Stokes equations are use to investigate the laminar
viscous flow inside fully developed pipe with different angles. The assumptions
of the flow are: steady, fully developed, one-dimensional and incompressible
flow.
Keywords:
laminar flow, pipe
flow, fully developed, Steatd flow, friction factor.
Introduction
The conduct experiment described in this review paper were
undertaken in connetion with a study of reverse transition in turbulent
boundary layers, the results, although, appear to be of adequate general notice
to warrant separate publication. Measurements of skin friction and
mean-velocity profile include made in fully developed pipe and channel flows
over a range of Reynolds number that includes the conversion from fully laminar
to fully turbulent flow. The results indicate considerable, and
until now unsuspected1.
The laminar flow is a
kind flow in which the streamlines are not crossing each other, that is, they are
parallel to one another. What determines a flow if it is laminar in nature or
not is the value of its Reynolds number, if the Reynolds number is less than
2000, the flow is still measured to be laminar flow. The laminar flow still relics
an important form of flow in engineering. The incompressible flow
finds its applications in the area of pipe flow in which the pipe length may be
too short of achieving fully developed conditions, such as in a short length
heat exchangers.
The incompressible
flow has its density remains constant. While in the incompressible region the
flow parameters changes with temperature change and this may result in a
significant drop in the pressure. When the pressure drop due to the flow of the
gas is large enough, causing a considerable decrease in density, then the flow
may be considered to be compressible, and appropriate formulas that take into
consideration changes in both density and velocity must be used to describe the
flow. As the fluid flows through a pipe many things happen, heat is generated
and the flow pressure also reduces, this is due to the friction existing
between the flow fluid and the wall of the pipe. Estimation of the value of the
head loss hl is very
important for proper engineering design. One important formula for calculating
the value of head loss hl is given by Darcy-Weisbach. This is shown in Equation (1)
below:
(1)
hl is given as the head loss, f
represents the Darcy friction factor, L stands for the length of
pipe, V represents the flow velocity, D is the internal diameter
of the pipe and g represents the gravitational.
In the study of external flow over a body, the relation-ship
between the wall heating and the change of skin friction drag, which is caused
by the difference in viscos-ity and density of a fluid when it is heated, can
easily be seen to be proportional to the temperature ratio taken to the power
of ?2/3. On the other hand, simple calculations on the momentum equation of
incompressible gas flow through a pipe show that for a constant pressure drop,
the mass flow rate is a function of inflow temperature taken to the power of
?2.5. This means that by increasing the inflow temperature, the mass flow rate
will be decreased considerably3.
The conveyance from laminar flow to turbulent flow depends on
roughness of surface and the temperature of the surface, geometry and velocity
of fluid flow. In the (1880) Osborne Reynolds discovered that the flow regime fundamentally
depends on the ratio of inertia forces to viscous forces. This ratio is called
Reynolds Number which is expressed as an equation (2) for circular pipec.4
(2)
Dimensional Analysis for Pipe Flow
With dimensional
analysis under our belt lets apply it to pipe flow and see what we can learn.
Let's consider a horizontal pipe, since no component of the flow direction is
in line with gravity (and there is no free surface as the pipe is full) we can
ignore gravity. Hence the only variables that can effect the flow are the
velocity V, the diameter D, the viscosity ?, the pipe length, L and the density ?. What is being affected? The pressure! We expect that the viscous action
on the walls will dissipate energy and induce a head loss, hence we must
include.
?P. Thus we write:
?P = f(V,D, ?, L, ?)
Clearly k=6, r=3, and we must have 3 ?'s. By visual inspection we found ?2 = ReD,
?3 = L/D, and recalling from the Bernoulli equation that we can form the dynamic
pressure as 1/2?V2 we have ?1 = ?P/( 1/2?V2) = Eu = Euler Number. Hence we now
write:
Eu = g
ReD,
)
This seems to be
as far as we can go. However, if we consider steady state then the
inertial terms
are all zero and we should have a solution that is independent of density,
if this is the
case we can write
?P = f?(V,D, ?, L)
Now we have k ? r = 2 with ?2 = L/D, therefore, ?1 = ?PD/(?V ), or
How does this
help? Well, we expect that ?P is directly related to the head loss (from
the energy
equation applied to the flow) and hence we expect that if the pipe length is
doubled _P doubles. If this is
true then we know our function g? is just a constant (as
this is the only
way to preserve the linear relationship between ?P and L). Hence we
write
Now, we expect
that the constant (C) will depend on Reynolds number as our intuition
is that
turbulent and laminar flow are fundamentally different. Let's rewrite our
equation
solving for ?P
Now, recalling
our original solution that involves Eu, Re, and the aspect ratio, we divide
through by 1/2?V2 and get
where C?is a new constant that has absorbed a factor of 2 but may still be a
function
of Reynolds
number (despite Re showing up formally in the equations the constant still
may depend on Re as
well). Hence we have:
All that remains
is to determine the constant C?. It turns out that
pipe flow yields an analytic solution to the Navier-Stokes Equations from which
C? can be found (we may work this as an example of Navier - Stokes analysis
during Wednesday's review section if you do not come armed with enough
questions). Alternatively, we could study the constant C? via experiment. We would find that our analysis stands up to both
laminar and turbulent flows, however the constant is in fact a function of Re.
If the flow is laminar C? = a constant = 64. In
fact the ratio C?/Re
is given a particular name, the friction factor or the Darcy friction factor. We write
for laminar flow. Note
that F in the above expression
(the Darcy friction factor) is often written as f and should not be confused with the f used earlier as the
general function relating ?P to V,D, …..and L/D. They are two
completely different f'sc5.