Velocity profile forced convection

Bulk displacement by fluid flow is much more common. When a multicomponent mixture moves in a vessel or in a region of a vessel at a certain bulk velocity, the fluid motion carries all the components, in general, at the same velocity. In the case of a flat velocity profile with convective transport being dominant over diffusive transport, the fluid flows like a "plug (Froment and Bischoff, 1979), and every species has the same displacement and velocity. If no forces are present to impart different displacement rates to different components, all components will he non-selectively carried hy hulk fluid motion and there would not he any separation. ... 

Gibaldi et al. [45] postulated that convective forces may be present in the GI tract during in vivo dissolution. This study took advantage of the well-defined hydrodynamics of the rotating disk, incorporating the solutions for the velocity profile and transport equations of Cochran [50] and Levich [51] to obtain... 

Fig. 5. Gas flow by forced convection through a wetted-wall tower, (a) with flat velocity profile, and (b) with parabolic velocity profile.

As was done in dealing with forced convective flow over a uniform temperature plate, it is assumed that the velocity and temperature profiles are similar at all values of x, i.e., that ... 

In using these equations, the forms of the velocity and temperature profiles in the boundary layer are assumed. Now, in turbulent forced convective boundary layer flows it has been found that the velocity profile is well described by ... 

The external electric field is in the direction of the pore axis. The particle is driven to move by the imposed electric field, the electroosmotic flow, and the Brownian force due to thermal fluctuation of the solvent molecules. Unlike the usual electroosmotic flow in an open slit, the fluid velocity profile is no longer uniform because a pressure gradient is built up due to the presence of the closed end. The probability of the particle position is obtained by solving the Fokker-Planck equation. The penetration depth is found to be dependent upon the Peclet number, which is a measure of significance of the convective electroosmotic flow relative to the Brownian diffusion, and the Damkohler number, which is a ratio of the characteristic diffusion-to-deposition times. 

Consider the vertical flat plate shown in Fig. 7-1. When the plate is heated, a free-convection boundary layer is formed, as shown. The velocity profile in this boundary layer is quite unlike the velocity profile in a forced-convection boundary layer. At the wall the velocity is zero because of the no-slip condition it increases to some maximum value and then decreases to zero at the edge of the boundary layer since the free-stream conditions are at rest in the free-convection system. The initial boundary-layer development is laminar but at... 

This is the equation of motion for the free-convection boundary layer. Notice that the solution for the velocity profile demands a knowledge of the temperature distribution. The energy equation for the free-convection system is the same as that for a forced-convection system at low velocity ... 

As in the integral analysis for forced-convection problems, we assume that the velocity profiles have geometrically similar shapes at various x distances along the plate. For the free-convection problem, we now assume that the velocity may be represented as a polynomial function of y multiplied by some arbitrary function of x. Thus,... 

The foregoing analysis of free-convection heat transfer on a vertical flat plate is the simplest case that may be treated mathematically, and it has served to introduce the new dimensionless variable, the Grashof number, which is important in all free-convection problems. But as in some forced-convection problems, experimental measurements must be relied upon to obtain relations for heat transfer in other circumstances. These circumstances are usually those in which it is difficult to predict temperature and velocity profiles analytically. Turbulent free convection is an important example, just as is turbulent forced convection, of a problem area in which experimental data are necessary however, the problem is more acute with free-convection flow systems than with forced-convection systems because the velocities are usually so small that they are very difficult to measure. Despite the experimental difficulties, velocity measurements have been performed using hydrogen-bubble techniques [26], hot-wire anemometry [28], and quartz-fiber anemometers. Temperature field measurements have been obtained through the use of the Zehnder-Mach interferometer. The laser anemometer [29] is particularly useful for free-convection measurements because it does not disturb the flow field. 

The velocity and temperature profiles for natural convection over a vertical hot plate are also shown in Fig. 9 -6. Note lhat as in forced convection, the thickness df the boundary layer increases in the flow direction. Unlike forced convection, however, the fluid velocity is zero at the outer edge of the velocity boundary layer as well as at the surface of the plate. This is expected since the fluid beyond the boundary layer is motionless. Thus, the fluid velocity increases with distance from the surface, reaches a maximum, and gradually decreases to zero at a distance sufflciently far from (be surface. At the. surface, the fluid temperature is equal to the plate temperature, and gradually decreases to the temperature of the surrounding fluid at a distance sufficiently far from the surface, as shown in the figure. In the case of cold surfaces, the shape of the velocity and temperature profiles remains the same but their direction is reversed. 

In the process under study here, where S02 oxidation is the limiting half reaction, the electrolyte flow rates are controlled by volumetric pumps to ensure forced convection. Moreover, both the anode and cathode compartments are provided with a plastic mesh turbulence promoter. The flow is therefore assumed fully turbulent and a uniform velocity profile is assumed at the inlet. However, for simplification, these devices are not represented in the simulation domain. Although the turbulence promoter should actually influence the bubble population, no reference has been found on its effects. 

Note that the distributions are cardinally different from those over smooth surfaces like in the previous cases. Again, two parts of the profiles have to be discussed. Over the top SCS s level z = h = 6 m, the wind velocity distributions grow monotonically in the case of a strong wind the temperature diminishes, as a rule. Few cases where the wind velocity diminishes over the SCS are characterized by a weak external wind so that the horizontal forced convection is perhaps comparative with the intense natural convective motion rising up from the heated and wetted air layer within SCS. 

Figure 9-12. The self-similar temperature profile given by Eq. (9-240) for forced convection heat transfer from a heated (or cooled) solid sphere in a uniform velocity field at small Re and large Pe. The function g( i]) represents the dependence of the thermal boundary-layer thickness on // and is given by (9-237).

For forced convection of liquid metals (Pr 1) over a horizontal flat plate subject to a uniform heat flux qm, evaluate the Nusselt number based on a uniform velocity and linear temperature profiles (Eg. 5P-2). 

The heat transfer immediately downstream of the location where heating begins will be dominated by forced convection and will depend on the velocity profile. For a parabolic inlet profile, the forced convection Nusselt number can be approximated by [249] ... 

Shell Momentum Balance for Flow Down an Inclined Plane. Consider the case of a Newtonian fluid in steady-state laminar flow down an inclined plane surface that makes an angle 6 with the horizontal. Using a shell momentum balance, find the equation for the velocity profile within the liquid layer having a thickness L and the maximum velocity of the free surface. Hint The convective momentum terms cancel for fully developed flow and the pressure-force terms also cancel, because of the presence of a free surface. Note that there is a gravity force on the fluid.)... 

An important heat-transfer system occurring in process engineering is that in which heat is being transferred from a hot vertical plate to a gas or liquid adjacent to it by natural convection. The fluid is not moving by forced convection but only by natural or free convection. In Fig. 4.7-1 the vertical flat plate is heated and the free-convection boundary layer is formed. The velocity profile differs from that in a forced-convection system in that the velocity at the wall is zero and also is zero at the other edge of the boundary layer since the free-stream velocity is zero for natural convection. The boundary layer initially is laminar as shown, but at some distance from the leading edge it starts to become turbulent. The wall temperature is T K and the bulk temperature T. ... 

The rotating disc electrode (RDE) is the classical hydrodynamic electroanalytical technique used to limit the diffusion layer thickness. However, readers should also consider alternative controlled flow methods including the channel flow cell (38), the wall pipe and wall jet configurations (39). Forced convection has several advantages which include (1) the rapid establishment of a high rate of steady-state mass transport and (2) easily and reproducibly controlled convection over a wide range of mass transfer coefficients. There are also drawbacks (1) in many instances, the construction of electrodes and cells is not easy and (2) the theoretical treatment requires the determination of the solution flow velocity profiles (as functions of rotation rate, viscosities and densities) and of the electrochemical problem very few cases yield exact solutions. 

Fluid flow in small devices acts differently from those in macroscopic scale. The Reynolds number (Re) is the most often mentioned dimensionless number in fluid mechanics. The Re number, defined by pUL/jj, represents the ratio of inertial forces to viscous ones. In most circumstances involved in micro- and nanofluidics, the Re number is at least one order of magnitude smaller than unity, ruling out any turbulence flows in micro/nanochannels. Inertial force plays an insignificant role in microfluidics, and as systems continue to scale down, it will become even less important. For such small Re number flows, the convective term pu Vm) of Navier-Stokes equations can be dropped. Without this nonlinear convection, simple micro/nanofluidic systems have laminar, deterministic flow patterns. They have parabolic velocity profile in pressure-driven flows, plug-like velocity profile in elec-froosmotic flows, or a superposition of both. One of the benefits from the low Re number flow is that genomic material can be transported easily without shearing in Lab-... 

Solution. For pedagogical purposes we solve this problem using the forced convection assumption in which it is assumed that the velocity profile is unaffected by changes in the viscosity as a result of changes in temperature. With... 

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...

C.4. Instead of assuming a given Sherwood nnmber for the external mass transfer coefficients, use FEMLAB to compute the laminar velocity profile aronndthe cylinder to directly model the effect of forced convection on the external mass transfer rate. Report your results for Reynolds numbers of 10 ", 10, 10, 1, 10, using a viscosity jx = 10 Pas and density p = lO kg/m ... 

In electrochemical reactors, the externally imposed velocity is often low. Therefore, natural convection can exert a substantial influence. As an example, let us consider a vertical parallel plate reactor in which the electrodes are separated by a distance d and let us assume that the electrodes are sufficiently distant from the reactor inlet for the forced laminar flow to be fully developed. Since the reaction occurs only at the electrodes, the concentration profile begins to develop at the leading edges of the electrodes. The thickness of the concentration boundary layer along the length of the electrode is assumed to be much smaller than the distance d between the plates, a condition that is usually satisfied in practice. 

---