Velocity profile natural convection

The wall boundary condition applies to a solid tube without transpiration. The centerline boundary condition assumes S5anmetry in the radial direction. It is consistent with the assumption of an axis5Tnmetric velocity profile without concentration or temperature gradients in the 0-direction. This boundary condition is by no means inevitable since gradients in the 0-direction can arise from natural convection. However, it is desirable to avoid 0-dependency since appropriate design methods are generally lacking. 

This section derives a simple version of the convective diffusion equation, applicable to tubular reactors with a one-dimensional velocity profile V (r). The starting point is Equation (1.4) applied to the differential volume element shown in Figure 8.9. The volume element is located at point (r, z) and is in the shape of a ring. Note that 0-dependence is ignored so that the results will not be applicable to systems with significant natural convection. Also, convection due to is neglected. Component A is transported by radial and axial diffusion and by axial convection. The diffusive flux is governed by Pick s law. 

For a number of flow situations, the mass-transfer rate can be derived directly from the equation of convective diffusion (see Table VII, Part A). The velocity profile near the electrode is known, and the equation is reduced to a simpler form by appropriate similarity transformations (N6). These well-defined flows, therefore, are being exploited increasingly by electrochemists as tools for the kinetic characterization of electrode reactions. Current distributions at, or below, the limiting current, transient mass transfer, and other aspects of these flows are amenable to analysis. Especially noteworthy are the systematic investigations conducted by Newman (review until 1973 in N7 also N9b, N9c, H6b and references in Table VII), by Daguenet and other French workers (references in Table VII), and by Matsuda (M4a-d). Here we only want to comment on the nature of the velocity profile near the electrode, and on the agreement between theory and mass-transfer experiment. 

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. 

Dimensionless velocity profiles in natural convective boundary layer on a vertical plate for various values of Prandtl number. 

As previously discussed, there are two limiting cases for natural convective flow through a vertical channel. One of these occurs when /W is large and the Rayleigh number is low. Under these circumstances all the fluid will be heated to very near the wall temperature within a relatively short distance up the channel and a type of fully developed flow will exist in which the velocity profile is not changing with Z and in which the dimensionless cross-stream velocity component, V, is essentially zero, i.e., in this limiting solution ... 

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. 

The simulations were performed assuming that the flow is laminar. Additionally, the contact angle is assumed to be known. The initial velocity is assumed to be zero everywhere in the domain. The initial fluid temperature profile is taken to be linear in the natural convection thermal boundary layer and the thermal boundary layer thickness, 5j, is evaluated using the correlation for the turbulent natural convection on a horizontal plate as, Jj. =1. 4(vfiCil ... 

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. 

Heat transfer to a laminar flow in an annulus is complicated by the fact that both the velocity and thermal profiles are simultaneously developing near the entrance and, often, over the length of the heated channel. Natural convection may also be a factor. It is usually conservative (i.e., predicted heat-transfer coefficients are lower than those experienced) to use equations for the fully developed flow. 

Consider the natural convection for Pr > 1 from a vertical plate at a temperature Tw iu an ambient at temperature T . Evaluate the local heat transfer for the following three cases, (a) linear temperature, parabolic velocity profiles, (b) parabolic temperature, linear velocity profiles, (c) parabolic temperature and velocity profiles. Compare the results with Eq. (5.104). 

Heat transfer for laminar flow in tubes with a parabolic velocity profile. (Does not include effects of natural convection or viscosity gradients.)... 

The convective contribution on the left side of (23-14) contains variable coefficients due to the viscous nature of the velocity profile. Hence, numerical methods are required to calculate Ca(x, y, z) discretely at selected grid points within the flow channel via finite-difference approximations for first and second derivatives of a continuous function (i.e., see Sections 23-3.5 and 23-5). 

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. ... 

Figure 4.7-1. Boundary-layer velocity profile for natural convection heat transfer from a heated, vertical plate.

The steady laminar flow of a liquid through a heated cylindrical pipe has a parabolic velocity profile if natural convection effects, and variation of physical properties with temperature are neglected [4], If the fluid entering the heated section is at a uniform temperature (Ti) and the wall is maintained at a crmstant temperature (T ), develop Graetz s solution by neglecting the thermal conductivity in the axial directiOTi. 

Note that we have divided the Darcy velocity by fractional porosity in the last step to have true velocity. The (8p/Bx) term was previously expressed in terms of (BT/Bx) and (Bxi/3x) (see Eq. (2.122)). Equation (2.139) applies to both thermal convection, where the convection is driven by (BT/Bx) as well as natural convection where flow is driven by (BT/Bx) and (BXi/Bx). As was stated before, convection may weaken or enhance composition variation. Figure 2.33 provides a simple explanation of the change in composition due to convection. In this figure, the diagram on the right (Fig. 2.33a) shows the composition variation vs. depth with zero convection at x = 0 assuming that = 0, and that and are not functions of temperature (see Eq. (2.125)). The thin line shows zero vertical compositional grading. Now allow for small values of pv (proportional to z) as shown by thick line B. Assume that pv is identically zero. Because of convection, the composition profile A cannot stay the same, otherwise the material balance for component 1... 

Middleman (1998) presents the natural convection generated velocity profile in a fluid between two vertical parallel plates separated by distance w, meters, across which a temperature gradient AT/w, in °K/m, exist. The fluid region near the hot plate experiences an upward flow and that adjacent to the cold plate, a downward flow. The maximum velocity in each region is... 

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...

Fig. 8. Perturbation of the axial velocity profile set up by natural convection due to introduction of feed and withdrawal of product and waste. TR = total reflux, E = enricher and S = stripper, with throughput.

Figure 6.30 (a) Velocity and (b) temperature profiles for natural convection of a fluid between two vertical plates maintained at different temperatures. Results from FEMLAB (www.comsol.com). 

Unstable Conditions In unstable conditions there is usually an inversion base height at z = Zi that defines the extent of the mixed layer. The two parameters that are key in determining Kzz are the convective velocity scale wr and Zi- We expect that a dimensionless profile Kzz = Kzz/w,z., which is a function only of z/z., should be applicable. This form should be valid as long as Kzz is independent of the nature of the source distribution. Lamb and Duran (1977) determined that Kzz does depend on the source height. With the proviso that the result be applied when emissions are at or near ground level, Lamb et al. (1975) and Lamb and Duran (1977) derived an empirical expression for Kzz under unstable conditions, using the numerical turbulence model of Deardorff (1970) ... 

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