Flow in a pipe

When a fluid flowing at a uniform velocity enters a pipe, the layers of fluid adjacent to the walls are slowed down as they are on a plane surface and a boundary layer forms at the entrance. This builds up in thickness as the fluid passes into the pipe. At some distance downstream from the entrance, the boundary layer thickness equals the pipe radius, after which conditions remain constant and fully developed flow exists. If the flow in the boundary layers is streamline where they meet, laminar flow exists in the pipe. If the transition has already taken place before they meet, turbulent flow will persist in the 

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Fig. 2. (a) Particle concentration profile of liquid flowing in a pipe, where YjD = the ratio of the distance along the diameter to the diameter ( ) (b)... 

FIG. 6-10 Parabolic velocity profile for laminar flow in a pipe, with average velocity V. 

FIG, 6"22 Adiabatic compressible flow in a pipe with a well-rounded entrance. 

For homogeneous flow in a pipe of diameter D, the differential form of the Bernoulli equation (6-15) rearranges to... 

Water Hammer When hquid flowing in a pipe is suddenly decelerated to zero velocity by a fast-closing valve, a pressure wave propagates upstream to the pipe inlet, where it is reflected a pounding of the hne commonly known as water hammer is often produced. For an instantaneous flow stoppage of a truly incompressible fluid in an inelastic pipe, the pressure rise would be infinite. Finite compressibility of the flmd and elasticity of the pipe limit the pressure rise to a finite value. The Joukowstd formula gives the maximum pressure... 

EGL Energy grade line - a line that represents the elevation of energy head in feet of water flowing in a pipe, conduit, or channel. 

Heat transfer coefficient between a pipe and a wall. Water flows in a pipe d =15 mm) with a velocity of v = 1.0 m s. The mean temperature of water is 0 , = 15 °C, and the wall temperature 6 = 50 °C. Calculate the heat transfer coefficient away from the pipe inlet. For water the properties are... 

The pressure loss for pure gas flow in a pipe of length dl is... 

For flow in a pipe, a Reynolds nmiiber above 2100 is an indication of turbulent flow. Thus, witli a Reynolds number of 9769.23, Uie flow is in the turbulent region. 

This is the basis for establishing the condition or type of fluid flow in a pipe. Reynolds numbers below 2000 to 2100 are usually considered to define laminar or thscous flow numbers from 2000 to 3000-4000 to define a transition region of peculiar flow, and numbers above 4000 to define a state of turbulent flow. Reference to Figure 2-3 and Figure 2-11 will identify these regions, and the friction factors associated with them [2]. 

For gases/vapors flowing in a pipe system from point 1 with pressure Pi and point 2 with pressure Po, the Pj - Po is the pressure drop, AP, between the points [3]. 

Thus the maximum flow in a pipe occurs when the velocity at the exit becomes sonic. The sonic location may be other than the exit, can be at restrictive points in the system, or at control/safety relief valves. 

The theory of pressure losses can be established by developing Bernoulli s theorem for the case of a pipe in which the work done in overcoming frictional losses is derived from the pressure available. For a fluid flowing in a pipe, the pressure loss will depend on various parameters. If... 

Magnetic devices A magnet(s) is fixed onto, or plumbed into a system, along the parallel axis of water flowing in a pipe. It is claimed that, with careful sizing and fitting, these devices inhibit the formation of scale. 

Instantaneous surges of water under pressure caused by sudden interruptions in water flow in a pipe or water system, producing a hammering sound and leading to metal stress and possible eventual failure. Water hammer can develop where a steam main is incorrectly pitched, has un-drained pockets or where steam flows up and meets draining condensate flowing down causing a temporary interruption in both flows. 

It is seen that it is important to be able to determine the velocity profile so that the flowrate can be calculated, and this is done in Chapter 3. For streamline flow in a pipe the mean velocity is 0.5 times the maximum stream velocity which occurs at the axis. For turbulent flow, the profile is flatter and the ratio of the mean velocity to the maximum... 

For flow in a pipe of circular cross-section a will be shown to be exactly 0.5 for streamline flow and to approximate to unity for turbulent flow. 

Calculation of pressure drop for liquid flowing in a pipe... 

If it is necessary to calculate the flow in a pipe where the pressure drop is specified, the velocity u is required but the Reynolds number is unknown, and this approach cannot be used to give R/pu2 directly. One alternative here is to estimate the value of R/pu2 and calculate the velocity and hence the corresponding value of Re. The value of R/pu2 is then determined and, if different from the assumed value, a further trial becomes necessary. 

The velocity over the cross-section of a fluid flowing in a pipe is not uniform. Whilst this distribution in velocity over a diameter can be calculated for streamline flow this is not possible in the same basic manner for turbulent flow. 

Laminar flow ceases to be stable when a small perturbation or disturbance in the flow tends to increase in magnitude rather than decay. For flow in a pipe of circular cross-section, the critical condition occurs at a Reynolds number of about 2100. Thus although laminar flow can take place at much higher values of Reynolds number, that flow is no longer stable and a small disturbance to the flow will lead to the growth of the disturbance and the onset of turbulence. Similarly, if turbulence is artificially promoted at a Reynolds number of less than 2100 the flow will ultimately revert to a laminar condition in the absence of any further disturbance. 

Equation 3.152 provides a method of determining the relationship between pressure gradient and mean velocity of flow in a pipe for fluids whose rheological properties may be expressed in the form of an explicit relation for shear rate as a function of shear stress. 

Yooi24) has proposed a simple modification to the Blasius equation for turbulent flow in a pipe, which gives values of the friction factor accurate to within about 10 per cent. The friction factor is expressed in terms of the Metzner and Reed(I8) generalised Reynolds number ReMR and the power-law index n. 

Compressibility of a gas flowing in a pipe can have significant effect on the relation between flowrate and the pressures at the two ends. Changes in fluid density can arise as a result of changes in either temperature or pressure, or in both, and the flow will be affected by the rate of heat transfer between the pipe and the surroundings. Two limiting cases of particular interest are for isothermal and adiabatic conditions. 

In considering the flow in a pipe, the differential form of the general energy balance equation 2.54 are used, and the friction term 8F will be written in terms of the energy dissipated per unit mass of fluid for flow through a length d/ of pipe. In the first instance, isothermal flow of an ideal gas is considered and the flowrate is expressed as a function of upstream and downstream pressures. Non-isothermal and adiabatic flow are discussed later. 

The conditions existing during the adiabatic flow in a pipe may be calculated using the approximate expression Pi/ = a constant to give the relation between the pressure and the specific volume of the fluid. In general, however, the value of the index k may not be known for an irreversible adiabatic process. An alternative approach to the problem is therefore desirable.(2,3)... 

In an adiabatic process, Sq = 0, and the equation may then be written for the flow in a pipe of constant cross-sectional area A to give ... 

Jones, C. and Hermges, G. Brit. J. Appl. Phys. 3 (1952) 283. The measurement of velocities for solid-fluid flow in a pipe. 

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