Velocity distribution for turbulent flow in a pipe

The theory for the turbulent flow of fluids through pipes is far less developed than that for laminar flow. 

An approximate equation for the profile of the time-averaged velocity for steady turbulent flow of a Newtonian fluid through a pipe of circular 

Equation 2.40 is an empirical equation known as the one-seventh power velocity distribution equation for turbulent flow. It fits the experimentally determined velocity distribution data with a fair degree of accuracy. In fact the value of the power decreases with increasing Re and at very high values of Re it falls as low as 1/10 [Schlichting (1968)]. Equation 2.40 is not valid in the viscous sublayer or in the buffer zone of the turbulent boundary layer and does not give the required zero velocity gradient at the centre-line. The l/7th power law is commonly written in the form 

Equation 2.43 can be derived from the velocity profile given in equation 2.40, using the same method as for laminar flow. The volumetric flow rate is given by 

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Universal velocity distribution for turbulent flow in a pipe... 

Prandtl s one-seventh power law A relationship used to determine the velocity distribution for turbulent flow in pipes carrying fluids given as ... 

For steady flow in a pipe or tube the kinetic energy term can be written as m2/(2 a) where u is the volumetric average velocity in the pipe or tube and a is a dimensionless correction factor which accounts for the velocity distribution across the pipe or tube. Fluids that are treated as compressible are almost always in turbulent flow and a is approximately 1 for turbulent flow. Thus for a compressible fluid flowing in a pipe or tube, equation 6.4 can be written as... 

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. 

It is now required to obtain the spatial distribution of the concentration of the tracer for two cases—center injection and wall ring injection. In general, it is difficult to theoretically obtain the spatial distribution of the concentration of tracer, and the number of experimental results with regard to this subject is insufficient because a very long pipe is required for performing experiments. However, it is possible to estimate the distribution of the tracer concentration based on the mean velocity profile, the intensity of velocity fluctuations, and so on, during the turbulent flow in a circular pipe. 

A typical velocity distribution for a newtonian fluid moving in turbulent flow in a smooth pipe at a Reynolds number of 10,000 is shown in Fig. 5.3. The figure also shows the velocity distribution for laminar flow at the same maximum velocity at the center of the pipe. The curve for turbulent flow is clearly much flatter than that for laminar flow, and the difference between the average velocity and the maximum velocity is considerably less. At still higher Reynolds numbers the curve for turbulent flow would be even flatter than that in Fig. 5.3. 

With turbulent flow in a straight pipe, the velocity profile is blunter than with laminar flow but still quite different from the flat profile assumed for plug flow. The ratio of maximum velocity to average velocity is about 1.3 at Re = 10", and this ratio slowly decreases to 1.15 at Re = 10 . A pulse of tracer introduced at the inlet gradually expands, but the distribution of residence times at the exit is fairly narrow. The effect of the axial velocity profile is largely offset by rapid radial mixing due to the turbulent velocity fluctuations. 

The velocity distribution and frictional resistance have been calculated from purely theoretical considerations for the streamline flow of a fluid in a pipe. The boundary layer theory can now be applied in order to calculate, approximately, the conditions when the fluid is turbulent. For this purpose it is assumed that the boundary layer expressions may be applied to flow over a cylindrical surface and that the flow conditions in the region of fully developed flow are the same as those when the boundary layers first join. The thickness of the boundary layer is thus taken to be equal to the radius of the pipe and the velocity at the outer edge of the boundary layer is assumed to be the velocity at the axis. Such assumptions are valid very close to the walls, although significant errors will arise near the centre of the pipe. 

Liquid core temperature and velocity distribution analysis. BankofT (1961) analyzed the convective heat transfer capability of a subcooled liquid core in local boiling by using the turbulent liquid flow equations. He found that boiling crisis occurs when the core is unable to remove the heat as fast as it can be transmitted by the wall. The temperature and velocity distributions were analyzed in the singlephase turbulent core of a boiling annular flow in a circular pipe of radius r. For fully developed steady flow, the momentum equation is given as... 

In a pioneering investigation, Serizawa [127, 128] measured the lateral void distribution as well as the turbulent axial liquid velocity fluctuations for bubbly air/water up-flows in a vertical pipe of diameter 60 (mm) inner diameter. They used electrical resistivity probes to measure the local void fraction, the bubble impaction rate, the bubble velocity and its spectrum. Turbulence quantities, such as the liquid phase mean velocity, and the axial turbulent fluctuations were measured using a hotfilm anemometer. A supplementary... 

The velocity distribution is different under these two regimes. Under laminar conditions the velocity profile is a parabola (see Fig. 5.1) and the mean velocity of flow is half the velocity at the centre of the tube (the maximum velocity). Under turbulent conditions the velocity profile is no longer parabolic but is as shown on Fig. 5.2. For turbulent flow the mean velocity of the fluid in the pipe is 0.82 x the velocity at the centre. 

There are no "nozzles for an even SOj/air distribution over all pipes. The pressure drop over the 6 m pipe can be calculated with the well-known Fanning equation for turbulent flow. For an SOa/air velocity of 30 m/s in an empty reactor pipe a value of 0.025 bar is found. In experiments with air and organic film (without any sulphonation reaction) a pressure drop of 0.1 bar was measured at an air velocity of 30 m/s. The interaction of the air with liquid film ripples and waves causes an increase of about a factor 4 in the pressure drop compared with the empty tube. Under real sulphonation conditions the pressure drop measured becomes even greater at 0.25 bar, due to the sharp increase of viscosity at high conversions. 

In some convection equations, such as for turbulent pipe flow, a special correction factor is used. This factor relates to the heat transfer conditions at the flow inlet, where the flow has not reached its final velocity distribution and the boundary layer is not fully developed. In this region the heat transfer rate is better than at the region of fully developed flow. 

Almost all flows in chemical reactors are turbulent and traditionally turbulence is seen as random fluctuations in velocity. A better view is to recognize the structure of turbulence. The large turbulent eddies are about the size of the width of the impeller blades in a stirred tank reactor and about 1/10 of the pipe diameter in pipe flows. These large turbulent eddies have a lifetime of some tens of milliseconds. Use of averaged turbulent properties is only valid for linear processes while all nonlinear phenomena are sensitive to the details in the process. Mixing coupled with fast chemical reactions, coalescence and breakup of bubbles and drops, and nucleation in crystallization is a phenomenon that is affected by the turbulent structure. Either a resolution of the turbulent fluctuations or some measure of the distribution of the turbulent properties is required in order to obtain accurate predictions. 

Thus for a 50-mm- (2-in.-) ID pipe and a Reynolds number of 1500 the transition length is 3.75 m (12.3 ft). If the fluid entering the pipe is turbulent and the velocity in the tube is above the critical, the transition length is nearly independent of the Reynolds number and is about 40 to 50 pipe diameters, with little difference between the distribution at 25 diameters and that at greater distances from the entrance, For a 50-mm-ID pipe, 2 to 3 m of straight pipe is sufficient when flow... 

The lifetime tj of a drop of volume Vin a gas flow enters the expression for the rate of change of the drop distribution, 4, that takes place because of the deposition of liquid on the pipe wall. The lifetime can be interpreted as the time it takes for the drop of volume V to reach the pipe wall. This time is averaged over all possible initial positions of the drop in the cross section of the flow. To determine ti, we should first estimate the velocity of transversal motion of the drop in the gas flow. As shown in [69], the transversal drift of drops inside the core of the turbulent flow occurs due to the gravitational force, and the drift velocity is equal to... 

The shear stresses within the fluid are responsible for the frictional force at the walls and the velocity distribution over the cross-section of the pipe. A given assumption for the shear stress at the walls therefore implies some particular velocity distribution. In line with the traditional concepts that have proved of value for Newtonian fluids, the turbulent flow of power-law fluids in smooth pipes can be considered by dividing the flow into three zones, as shown schematically in Figure 3.12. 

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