Turbulent flow velocity profile

Thus the turbulent flow velocity profile is flatter than the corresponding laminar flow profile, as shown in Figure 1.24. 

The turbulent flow velocity profile for Newtonian fluids is arbitrarily divided into three regions the viscous sublayer, the buffer layer, and the turbulent core. To represent velocity profiles in pipe flow, friction velocity defined as... 

Heat removal from the sulfomass may occur through the wall of the tubular reactor. Generally, the heat transfer coefficient is defined by the geometric parameters of the equipment, the thermal physical properties of the sulfomass, and its flow conditions over the cooled wall. To ensure the turbulent flow conditions of the sulfomass intensify significantly, heat is removed from the flow volume, as the turbulent flow velocity profile of the microlayers is different from that of laminar flow. 

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... 

As will be shown later, the velocity profile for a Newtonian fluid in laminar flow in a circular tube is parabolic. When this is introduced into Eq. (5-38), the result is a = 2. For highly turbulent flow, the profile is much flatter and a 1.06, although for practical applications it is usually assumed that a = 1 for turbulent flow. 

In turbulent flow, properties such as the pressure and velocity fluctuate rapidly at each location, as do the temperature and solute concentration in flows with heat and mass transfer. By tracking patches of dye distributed across the diameter of the tube, it is possible to demonstrate that the liquid s velocity (the time-averaged value in the case of turbulent flow) varies across the diameter of the tube. In both laminar and turbulent flow the velocity is zero at the wall and has a maximum value at the centre-line. For laminar flow the velocity profile is a parabola but for turbulent flow the profile is much flatter over most of the diameter. 

In practice, there is always some degree of departure from the ideal plug flow condition of uniform velocity, temperature, and composition profiles. If the reactor is not packed and the flow is turbulent, the velocity profile is reasonably flat in the region of the turbulent core (Volume 1, Chapter 3), but in laminar flow, the velocity profile is parabolic. More serious however than departures from a uniform velocity profile are departures from a uniform temperature profile. If there are variations in temperature across the reactor, there will be local variations in reaction rate and therefore in the composition of the reaction mixture. These transverse variations in temperature may be particularly serious in the case of strongly exothermic catalytic reactions which are cooled at the wall (Chapter 3, Section 3.6.1). An excellent discussion on how deviations from plug flow arise is given by DENBIGH and TURNER 5 . 

Fig. 12 Boundary layer and turbulent core velocity profile relationships for Newtonian and DR flows. (From Ref...

Figure 10.1. Time-averaged velocity profiles in fully developed turbulent flow identifying the presence of a laminar sublayer, a buffer layer, and the turbulent core of the flow. Velocity profile calculated from Eqs. 10.2.14-10.2.16.

Structure of the flow. Velocity profile. The flow in the boundary layer on a flat plate is laminar until Rex = U[X/v 3 x 105. On a longer plate, the boundary layer becomes turbulent, that is, its thickness increases sharply and the longitudinal velocity profile alters. 

The tube is packed with catalyst pellets. Flow may be either laminar or turbulent. The velocity profile is assumed to be flat. Transfer of heat and mass in the radial direction is modeled using empirical diffusion coefficients that combine the effects of convection and true diffusion in the radial direction. There is no axial diffusion. Details are given in Chapter 9. This model is important only for nonisothermai reactors. It reduces to piston flow if the reaction is isothermal. 

The reactor is turbulent, the velocity profile is flat, radial mixing is complete, and there is some transfer of heat and mass in the axial direction. The tube can be either packed (e.g., with a catalyst) or open. This model provides an estimate of reaction yields in highly turbulent reactors that is more conservative than assuming piston flow. 

Transition from laminar to turbulent flow is identified by a departure from the laminar flow velocity profile and the presence of time-varying velocity component in the flow, especially near the wall. The wall shear stress becomes higher following the departure from the laminar flow. The transition occurs over a range of Reynolds number and is dependent on the local wall and flow conditions. Microchannels are defined as channels with the minimum channel dimension in the range from 1 to 200 pm [1]. 

In Chapter 2, the design of the so-called ideal reactors was discussed. The reactor ideahty was based on defined hydrodynamic behavior. We had assumedtwo flow patterns plug flow (piston type) where axial dispersion is excluded and completely mixed flow achieved in ideal stirred tank reactors. These flow patterns are often used for reactor design because the mass and heat balances are relatively simple to treat. But real equipment often deviates from that of the ideal flow pattern. In tubular reactors radial velocity and concentration profiles may develop in laminar flow. In turbulent flow, velocity fluctuations can lead to an axial dispersion. In catalytic packed bed reactors, irregular flow with the formation of channels may occur while stagnant fluid zones (dead zones) may develop in other parts of the reactor. Incompletely mixed zones and thus inhomogeneity can also be observed in CSTR, especially in the cases of viscous media. 

Figures 2.5 and 2.6 reveal that deterioration is caused by a different mechanism at low flow rates. The calculation results at G = 39 kg m s and 7 = T, which gives the Reynolds number 10,000, are rearranged in terms of the Grashof number and the Nusselt number in Fig. 2.8. Nu has a minimum value at Gr = 2 x 10. Nu is constant when Gr is lower than it, which means forced convection is dominant. On the other hand, Nu increases linearly when Gr is larger than the minimum point, which implies that natural convection is dominant. The minimum point emerges at the boundary between the two convection modes. Flow velocity and turbulence energy profiles are depicted in Fig. 2.9. When the heat flux is enhanced, the flow velocity increases near the wall and the profile becomes flat. Since turbulence energy is produced by the derivative of flow velocity, it is reduced. Hence, heat transfer is deteriorated. When the heat flux is enhanced above the minimum point, the flow velocity profile is more distorted and turbulent heat transfer is then enhanced. This type of heat transfer deterioration is attributed to acceleration as well as buoyancy. In the present analysis, buoyancy force is dominant. The computational results without the buoyancy force term in the Navier-Stokes equations are...

A low Reynolds number indicates laminar flow and a paraboHc velocity profile of the type shown in Figure la. In this case, the velocity of flow in the center of the conduit is much greater than that near the wall. If the operating Reynolds number is increased, a transition point is reached (somewhere over Re = 2000) where the flow becomes turbulent and the velocity profile more evenly distributed over the interior of the conduit as shown in Figure lb. This tendency to a uniform fluid velocity profile continues as the pipe Reynolds number is increased further into the turbulent region. 

Most flow meters are designed and caHbrated for use on turbulent flow, by far the more common fluid condition. Measurements of laminar flow rates may be seriously in error unless the meter selected is insensitive to velocity profile or is specifically caHbrated for the condition of use. 

In general, V For laminar Newtonian flow the radial velocity profile is paraboHc and /5 = 3/4. For fully developed turbulent flow the radial... 

Here, h is the enthalpy per unit mass, h = u + p/. The shaft work per unit of mass flowing through the control volume is 6W5 = W, /m. Similarly, is the heat input rate per unit of mass. The fac tor Ot is the ratio of the cross-sectional area average of the cube of the velocity to the cube of the average velocity. For a uniform velocity profile, Ot = 1. In turbulent flow, Ot is usually assumed to equal unity in turbulent pipe flow, it is typically about 1.07. For laminar flow in a circiilar pipe with a parabohc velocity profile, Ot = 2. 

In turbulent flow, the velocity profile is much more blunt, with most of the velocity gradient being in a region near the wall, described by a universal velocity profile. It is characterized by a viscous sublayer, a turbulent core, and a buffer zone in between. 

Vfjp is the friction velocity and =/pVV2 is the wall stress. The friction velocity is of the order of the root mean square velocity fluctuation perpendicular to the wall in the turbulent core. The dimensionless distance from the wall is y+ = yu p/. . The universal velocity profile is vahd in the wall region for any cross-sectional channel shape. For incompressible flow in constant diameter circular pipes, = AP/4L where AP is the pressure drop in length L. In circular pipes, Eq. (6-44) gives a surprisingly good fit to experimental results over the entire cross section of the pipe, even though it is based on assumptions which are vahd only near the pipe wall. 

Perforated Plates and Screens A nonuniform velocity profile in turbulent flow through channels or process equipment can be smoothed out to any desired degree by adding sufficient uniform resistance, such as perforated plates or screens across the flow channel, as shown in Fig. 6-38. Stoker Ind. Eng. Chem., 38, 622-624 [1946]) provides the following equation for the effect of a uniform resistance on velocity profile ... 

The universal turbulent velocity profile near the pipe wall presented in the preceding subsection Tncompressible Flow in Pipes and Channels may be developed using the Prandtl mixing length approximation for the eddy viscosity,... 

The dispersion that takes place in an open tube, as discussed in chapter 8, results from the parabolic velocity profile that occurs under conditions of Newtonian flow (i.e., when the velocity is significantly below that which produces turbulence). Under condition of Newtonian flow, the distribution of fluid velocity across the tube... 

The distribution of tracer molecule residence times in the reactor is the result of molecular diffusion and turbulent mixing if tlie Reynolds number exceeds a critical value. Additionally, a non-uniform velocity profile causes different portions of the tracer to move at different rates, and this results in a spreading of the measured response at the reactor outlet. The dispersion coefficient D (m /sec) represents this result in the tracer cloud. Therefore, a large D indicates a rapid spreading of the tracer curve, a small D indicates slow spreading, and D = 0 means no spreading (hence, plug flow). 

To calibrate larger sensors/instruments such as vane anemometers, a wind tunnel is required. A calibration wind tunnel consists of an open or closed tunnel, a fan to deliver the air, a nozzle to shape the velocity profile, and a mesh arrangement to uniform and reduce the flow turbulence. It may be necessary to control the air temperature in the tunnel by means of a heating/cooling sys-... 

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