Velocity distribution in the turbulent

The velocity at the inner edge of the turbulent region must also be given by the equation for the velocity distribution in the turbulent region. 

Derive the relation between the friction factor and Reynolds number in turbulent flow for smooth pipe [Eq. (6-34)], starting with the von Karman equation for the velocity distribution in the turbulent boundary layer [Eq. (6-26)]. 

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

The modeling procedure can be sketched as follows. First an approximate description of the velocity distribution in the turbulent boundary layer is required. The universal velocity profile called the Law of the wall is normally used. The local shear stress in the boundary layer is expressed in terms of the shear stress at the wall. From this relation a dimensionless velocity profile is derived. Secondly, a similar strategy can be used for heat and species mass relating the local boundary layer fluxes to the corresponding wall fluxes. From these relations dimensionless profiles for temperature and species concentration are derived. At this point the concentration and temperature distributions are not known. Therefore, based on the similarity hypothesis we assume that the functional form of the dimensionless fluxes are similar, so the heat and species concentration fluxes can be expressed in terms of the momentum transport coefficients and velocity scales. Finally, a comparison of the resulting boundary layer fluxes with the definitions of the heat and mass transfer coefficients, indiates that parameterizations for the engineering transfer coefficients can be put up in terms of the appropriate dimensionless groups. 

The similarity of velocity and of turbulence intensity is documented in Fig. 12.29. The figure shows a vertical dimensionless velocity profile and a turbulence intensity profile measured by isothermal model experiments at two different Reynolds numbers. It is obvious that the shown dimensionless profiles of both the velocity distribution and the turbulence intensity distribution are similar, which implies that the Reynolds number of 4700 is above the threshold Reynolds number for those two parameters at the given location. 

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... 

The mean velocity distribution in the outer portion of the turbulent boundary layer on a flat plate is approximately given by ... 

The momentum, heat and species concentration equations (5.250), (5.266) and (5.274) may now be integrated to express the velocity, temperature, and concentration distribution in the turbulent boundary layer. 

The parameter k is called the von Karman constant, and the value that fits most of the data is 0.41. The corresponding value of B is 5.0. Intermediate between these layers is the buffer layer, where both shear mechanisms are important. The essential feature of this data correlation is that the wall shear completely controls the turbulent boundary layer velocity distribution in the vicinity of the wall. So dominant is the effect of the wall shear that even when pressure gradients along the surface are present, the velocity distributions near the surface are essentially coincident with the data obtained on plates with uniform surface pressure [82]. Within this region for a flat plate, the local shear stress remains within about 10 percent of the surface shear stress. It is noted that this shear variation is often ignored in turbulent boundary layer theory. 

No such simple analysis is possible in the transition zone nor is it possible to delineate the transition boimdaries for the three regions of flow. Based on experimental results, it is now generally beheved that the laminar sub-layer extends up to y+ 5 and the turbulent core begins at Cd 30. The following empirical correlation provides an adequately approximate velocity distribution in the transition layer (5 < y" " < 30) in smooth pipes ... 

In order to see how the boundary layer affects particle detachment, let us turn to Fig. X. 1. Depending on the flow velocity, the boundary layer may be either laminar (Fig. X. 1. a) or turbulent (Fig. X. 1. b). The laminar boundary layer is characterized by a linear velocity distribution in the layer. The adherent particles may be completely submerged in this layer if the particle diameter is smaller than the boundary layer thickness (see Position I in Fig. X.l.a). Position II shows the case in which the diameter of the adherent particle is greater than the boundary layer thickness. 

Laminar Flow Although heat-transfer coefficients for laminar flow are considerably smaller than for turbulent flow, it is sometimes necessary to accept lower heat transfer in order to reduce pumping costs. The heat-flow mechanism in purely laminar flow is conduction. The rate of heat flow between the walls of the conduit and the fluid flowing in it can be obtained analytically. But to obtain a solution it is necessary to know or assume the velocity distribution in the conduit. In fully developed laminar flow without heat transfer, the velocity distribution at any cross section has the shape of a parabola. The velocity profile in laminar flow usually becomes fully established much more rapidly than the temperature profile. Heat-transfer equations based on the assumption of a parabolic velocity distribution will therefore not introduce serious errors for viscous fluids flowing in long ducts, if they are modified to account for effects caused by the variation of the viscosity due to the temperature gradient. The equation below can be used to predict heat transfer in laminar flow. 

However, even at relatively low film Reynolds numbers, the assumption that the condensate layer is in laminar flow is open to some question. Experiments have shown that the surface of the film exhibits considerable waviness (turbulence). This waviness causes increased heat transfer rates. Better heat transfer correlations for vertical condensation were presented by Dukler in 1960. He obtained velocity distributions in the liquid film as a function of the interfacial shear (due to the vapor velocity) and film thickness. From the integration of the velocity and temperature profiles, liquid film thickness and point heat-transfer coefficients were computed. According to the Dukler development, there is no... 

The important hydrodynamic variables are the relative velocity. Vs, between the solids and the liquid (also know as slip velocity) and the rate of renewal of the liquid layer near the solid surface. The relative velocity, Vg, obviously varies from point to point within the vessel, and the average value is difficult to estimate. So, in practice, the relative velocity. Vs, is assumed equal to the free settling velocity, Vt. The renewal of the boundary layer depends on the intensity of turbulence around the solid particle as well as the convective velocity distribution in the vessel. 

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. 

Airborne contaminant movement in the building depends upon the type of heat and contaminant sources, which can be classified as (1) buoyant (e.g., heat) sources, (2) nonbuoyant (diffusion) sources, and (d) dynamic sources.- With the first type of sources, contaminants move in the space primarily due to the heat energy as buoyant plumes over the heated surfaces. The second type of sources is characterized by cimtaminant diffusion in the room in all directions due to the concentration gradient in all directions (e.g., in the case of emission from painted surfaces). The emission rare in this case is significantly affected by the intensity of the ambient air turbulence and air velocity, dhe third type of sources is characterized by contaminant movement in the space with an air jet (e.g., linear jet over the tank with a push-pull ventilation), or particle flow (e.g., from a grinding wheel). In some cases, the above factors influencing contaminant distribution in the room are combined. 

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. 

In the Lagrangian approach, individual parcels or blobs of (miscible) fluid added via some feed pipe or otherwise are tracked, while they may exhibit properties (density, viscosity, concentrations, color, temperature, but also vorti-city) that distinguish them from the ambient fluid. Their path through the turbulent-flow field in response to the local advection and further local forces if applicable) is calculated by means of Newton s law, usually under the assumption of one-way coupling that these parcels do not affect the flow field. On their way through the tank, these parcels or blobs may mix or exchange mass and/or temperature with the ambient fluid or may adapt shape or internal velocity distributions in response to events in the surrounding fluid. 

These methods hardly take spatial distributions of velocity field and chemical species or transient phenomena into account, although most chemical reactors are operated in the turbulent regime and/or a multiphase flow mode. As a result, yield and selectivity of commercial chemical reactors often deviate from the values at their laboratory or pilot-scale prototypes. Scale-up of many chemical reactors, in particular the multiphase types, is still surrounded by a fame of mystery indeed. Another problem relates to the occurrence of thermal runaways due to hot spots as a result of poor local mixing effects. 

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... 

Equations 2.58, 2.70 and 2.71 enable the velocity distribution to be calculated for steady fully developed turbulent flow. These equations are only approximate and lead to a discontinuity of the gradient at y+ = 30, which is where equations 2.70 and 2.71 intersect. The actual profile is, of course, smooth and the transition from the buffer zone to the fully turbulent outer zone is particularly gradual. As a result it is somewhat arbitrary where the limit of the buffer zone is taken often the value y+ = 70 rather than j + = 30 is used. The ability to represent the velocity profile in most turbulent boundary layers by the same v+ - y+ relationships (equations 2.58, 2.70 and 2.71) is the reason for calling this the universal velocity profile. The use of in defining v+ and y+ demonstrates the fundamental importance of the wall shear stress. 

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