The turbulent core

It is now possible to consider each of these regions in turn and to develop a series of equations to represent flie velocity over the whole cross section of a pipe. Together, they constitute the Universal Velocity Profile. 

3 apples in the turbulent cor except near the axis of the pipe Where the shear stress is markedly different from that at the walls. Inserting the value of 0 4 for K  

Equations 12.37 and 12.38 correlate experimental data well for values of exceeding 30. 

In the laminar sub-layer, turbulence has died out and momentum transfer is attributable solely to viscous shear. Because the layer is thin, the velocity gradient is proximately linear and equal to Ubjbi, where Ub is the velocity at the outer edge of a laminar sub-layer of thickness 5 (see Chapter 11). 

This relationship holds reasonably well for values of y+ up to about 5, and it applies to both rough and smooth surfaces. 

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For turbulent flow of a fluid past a solid, it has long been known that, in the immediate neighborhood of the surface, there exists a relatively quiet zone of fluid, commonly called the Him. As one approaches the wall from the body of the flowing fluid, the flow tends to become less turbulent and develops into laminar flow immediately adjacent to the wall. The film consists of that portion of the flow which is essentially in laminar motion (the laminar sublayer) and through which heat is transferred by molecular conduction. The resistance of the laminar layer to heat flow will vaiy according to its thickness and can range from 95 percent of the total resistance for some fluids to about I percent for other fluids (liquid metals). The turbulent core and the buffer layer between the laminar sublayer and turbulent core each offer a resistance to beat transfer which is a function of the turbulence and the thermal properties of the flowing fluid. The relative temperature difference across each of the layers is dependent upon their resistance to heat flow. 

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. 

For rough pipes, the velocity pronle in the turbulent core is given by... 

Dodge and Metzner (1959) deduced the velocity profile from their measurements of flow rate and pressure gradient for turbulent flow of power law fluids in pipes. For the turbulent core, the appropriate equation is... 

In the planning of a FAMS for operation at 1.0-atm pressure, the advantages of turbulent flow as summarized above would be extremely useful. The radial mixing of neutrals added to the tube would be very fast, and ions within the turbulent core would be protected from contact with the wall. Under these conditions, ion-ion or ion-electron recombination reactions, alone, would provide the only physical... 

In addition, the behavior in the turbulent core (S5) was included in Fig. 5. For this region Deissler (D2) suggests the following expression ... 

In the turbulent core it has been conventional to employ the velocity deficiency (B2) as a single-valued function of the relative position in the channel in order to correlate the velocity distribution outside the boundary flow. The velocity deficiency is defined by... 

Equation (31) is similar to Eq. (7) except that it takes into account the effect of kinematic viscosity on the eddy properties near the wall. An iterative solution is required for the solution of Eqs. (30) and (31). Throughout Deissler s recent analysis (D3) the Reynolds analogy has been assumed, and throughout the turbulent core Deissler shows that... 

Figure 2.4b shows, conceptually, the velocity distribution in steady turbulent flow through a straight round tube. The velocity at the tube wall is zero, and the fluid near the wall moves in laminar flow, even though the flow of the main body of fluid is turbulent. The thin layer near the wall in which the flow is laminar is called the laminar sublayer or laminar film, while the main body of fluid where turbulence always prevails is called the turbulent core. The intermediate zone between the laminar sublayer and the turbulent core is called the buffer layer, where the motion of fluid may be either laminar or turbulent at a given instant. With a relatively long tube, the above statement holds for most of the tube length, except for... 

Velocity distributions in turbulent flowthrough a straight, round tube vary with the flow rate or the Reynolds number. With increasing flow rates the velocity distribution becomes flatter and the laminar sublayer thinner. Dimensionless empirical equations involving viscosity and density are available that correlate the local fluid velocities in the turbulent core, buffer layer, and the laminar sublayer as functions of the distance from the tube axis. The ratio of the average velocity over the entire tube cross section to the maximum local velocity at the tube axis is approximately 0.7-0.85, and increases with the Reynolds number. 

The transfer of heat and/or mass in turbulent flow occurs mainly by eddy activity, namely the motion of gross fluid elements that carry heat and/or mass. Transfer by heat conduction and/or molecular diffusion is much smaller compared to that by eddy activity. In contrast, heat and/or mass transfer across the laminar sublayer near a wall, in which no velocity component normal to the wall exists, occurs solely by conduction and/or molecular diffusion. A similar statement holds for momentum transfer. Figure 2.5 shows the temperature profile for the case of heat transfer from a metal wall to a fluid flowing along the wall in turbulent flow. The temperature gradient in the laminar sublayer is linear and steep, because heat transfer across the laminar sublayer is solely by conduction and the thermal conductivities of fluids are much smaller those of metals. The temperature gradient in the turbulent core is much smaller, as heat transfer occurs mainly by convection - that is, by... 

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 . 

Alternatively, a single laminar boundary layer, with u+ = y+ reaching up to y+ 11.6, is often used, beyond which there is the turbulent core. 

Assume that the velocity distribution in the turbulent core for tube flow may be represented by... 

We use an universal velocity distribution obtained by Churchill (2001) to approximate the fully developed velocity profile u( l2) across the turbulent core,... 

Turbulent flow over a flat plate is characterized by three re-gions f l (a) a viscous sublayer often called the laminar sublayer, which exists right next to the plate, (b) an adjacent turbulent boundary layer, and (c) the turbulent core. Viscous forces dominate inertial forces in the viscous sublayer, which is relatively quiescent compared to the other regions and is therefore also called the laminar sublayer. This is a bit of a misnomer, since it is not really laminar. It is in this viscous sublayer that the velocity changes are the greatest, so that the shear is largest. Viscous forces become less dominant in the turbulent boundary layer. These forces are not controlling factors in the turbulent core. 

According to the analysis of many experimental data (M31, U5), the thickness 8 is usually much smaller than the column radius R, so that Mg essentially equals the peripheral velocity m of the turbulent core. As a... 

By introducing Eqs. (3-2) and (3-10) into Eq. (3-1) and neglecting vm in comparison with vt, the following basic equation is obtained for the turbulent core ... 

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