Flow of Incompressible Fully Developed steady
Fluid Flow in Pipe
Idrees Majeed Kareem
Mechanical engineer, Department
Faculty of Engineering
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.
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.