Boundary layer thickness natural convection

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 ... 

Natural convection to blunt bodies such as cylinders (2-dimensional) and spheres (3-dimensional) has been studied by Acrivos (9) and from his analysis one can show that these configurations are characterized by constant boundary-layer thicknesses. For 2-dimensional bodies,... 

Simple integration of Equations (A.4) or (A.5) and substitution of the boundary conditions (A.l2) and (A.9) shows that as stated in Chapter 1, the solutions of the equations are indeterminate. It can be shown by solving the time dependent equations (see later) and letting r that the only steady state solution is Co = 0 and 7=0. This is because with the model used, there are no steady concentration profiles until all the species 0 is removed from solution. In practice we know that a steady state current is easily obtained, and the experimental situation is readily predicted if we define a boundary layer, thickness 6, and we assume that outside this layer the concentrations of O and R are maintained constant by convection, either natural or forced. The boundary conditions to (A.4) and (A. 5) are then... 

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. 

In a hydrodynamically free system the flow of solution may be induced by the boundary conditions, as for example when a solution is fed forcibly into an electrodialysis (ED) cell. This type of flow is known as forced convection. The flow may also result from the action of the volume force entering the right-hand side of (1.6a). This is the so-called natural convection, either gravitational, if it results from the component defined by (1.6c), or electroconvection, if it results from the action of the electric force defined by (1.6d). In most practical situations the dimensionless Peclet number Pe, defined by (1.11b), is large. Accordingly, we distinguish between the bulk of the fluid where the solute transport is entirely dominated by convection, and the boundary diffusion layer, where the transport is electro-diffusion-dominated. Sometimes, as a crude qualitative model, the diffusion layer is replaced by a motionless unstirred layer (the Nemst film) with electrodiffusion assumed to be the only transport mechanism in it. The thickness of the unstirred layer is evaluated as the Peclet number-dependent thickness of the diffusion boundary layer. 

Because the surface-particle interaction forces are important only near to the surface of the collector, it is natural to divide the domain into two regions. The inner region, called the interaction-force boundary layer, has a thickness determined by the range over which interaction forces are important (less than about 10-5 cm). Because of the proximity of the surface, convection is negligible. Outside the... 

Hydrodynamic boundary layer — is the region of fluid flow at or near a solid surface where the shear stresses are significantly different to those observed in bulk. The interaction between fluid and solid results in a retardation of the fluid flow which gives rise to a boundary layer of slower moving material. As the distance from the surface increases the fluid becomes less affected by these forces and the fluid velocity approaches the freestream velocity. The thickness of the boundary layer is commonly defined as the distance from the surface where the velocity is 99% of the freestream velocity. The hydrodynamic boundary layer is significant in electrochemical measurements whether the convection is forced or natural the effect of the size of the boundary layer has been studied using hydrodynamic measurements such as the rotating disk electrode [i] and - flow-cells [ii]. 

The thickness of the Nernst layer increases with the square root of time until natural - convection sets in, after which it remains constant. In the presence of forced convection (stirring, electrode rotation) (see also Prandtl boundary layer), the Nernst-layer thickness depends on the degree of convection that can be controlled e.g., by controlling the rotation speed of a -> rotating disk electrode. See also - diffusion layer. See also Fick s law. 

As reported in Ref. , the spread rate of a flame moving up a vertical surface of a sufficiently thick PMMA sheet increases under the effect of an external heat radiation. Depending on the heat radiation intensity and exposure time, various effects on the flame spread rate are observed. Additional heating of the polymer surface by a radiative flux results, first of all, in a decrease of the temperature dilTerence (T — Tp) and, in accordance with Eq. (2.19), in an increase of v. The experimental relationship v (T — To)" at T = 363 °C is close to that predicted by theory. According to Femandez-Pello , an increase of the initial polymer surface temperature, Tp, cause a parallel enhancement of the natural convection in the boundary heat layer and heat radiation by the surface, leading to its partial cooling. Therefore, when the intensity of the external radiative heat flux is low, the flame spread rate increases with time, but only up to a certain constant value. 

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. 

For the heated vertical plate and horizontal cylinder, the flow results from natural convection. The stagnation configuration is a forced flow. In each case the flow is of the boimdai7 Kiyer type. Simple analytical solutions can be obtained when the thickness of the du.st-free space is much smaller than that of the boundary layer. In this case the gas velocity distribution can be approximated by the first term in an expansion in the distance norroal to the surface. Expressions for the thickness of the dust-free space for a heated vertical surface and a plane stagnation flow are derived below. 

The thin-layer approximation fails because natural convective boundary layers are not thin. From the interferometric fringes in Fig. 4.2ft (which are essentially isotherms), the thermal boundary layer around a circular cylinder is seen to be nearly 30 percent of the cylinder diameter. For such thick boundary layers, curvature effects are important. Despite this failure, thin-layer solutions provide an important foundation for the development of correlation equations, as explained in the section on heat transfer correlation method. 

The equation for the laminar Nusselt number Nut is obtained in a two-step procedure. In the first step, not only is the flow idealized as everywhere laminar, but the boundary layer is treated as thin. There results from this idealization the equation for the laminar thin-layer Nusselt number Nur. As already explained, natural convection boundary layers are generally not thin, so the second step is to correct Nur to account for thick boundary layers. This correction uses the method of Langmuir [175]. The corrected Nusselt number is the laminar Nusselt number Nuc. 

In the papers of Ulrich et al. (1990) and Kuszlil (1990), it was shown that when diffusion coefficients are low, viscosities are high therefore natural convection is low. This means that the boundary layers are thick and only poor separation effects can be achieved. This is especially true in cases of rapid crystallization rates. In these cases, however, a mixed process, e.g., by pulsation, can reduce the impurity by as much as threefold compared to a process with a stagnant melt (see Kehm (1990)). The apparatus is shown in Figure 7.13. 

Mass transfer coefficient (fe) A measure of the solute s mobility due to forced or natural convection in the system. Analogous to a heat transfer coefficient, it is measured as the ratio of the mass flux to the driving force. In membrane processes the driving force is the difference in solute concentration at the membrane surface and at some arbitrarily defined point in the bulk fluid. When lasing the film theory to model mass transfer, k is also defined as D/S, where D is solute diffusivity and d is the thickness of the concentration boundary layer. 

The zone close to the electrode surface where the concentrations of 0 and R are different from those in the bulk is known as the diffusion layer. In most experiments its thickness increases with time until it reaches a steady state value, approximately 10" cm thick, as natural convection stirring the bulk solution becomes important. It takes of the order of 10 s for this boundary layer to form. This also means that for the first 10 s of any experiment, the concentration changes close to the electrode are the result of only diffusion. Thereafter the effects of natural convection must be taken into account. 

It is worthwhile to note that Equations (A.22) and (A. 23) have genuine steady state solutions without the need to introduce a boundary layer. This is because the chemical reaction (A. 19) causes the formation of a steady state kinetic layer. Only within this boundary layer is R present in solution, and the concentration of 0 perturbed from its initial concentration. The thickness of the kinetic layer depends on k the larger k the thinner the layer. Certainly for high values of k, the kinetic layer will lie well within the normal steady state diffusion layer defined by natural convection. The time required to reach the steady state (form the kinetic layer) also depends inversely on k. 

To get to the surface, the heat in-flow is first absorbed by a process of natural convective heat transfer creating an upward flow of less dense superheated liquid there is no boiling and also no evaporation at the point where the heat is absorbed. At a vertical wall, the flow of superheated liquid assumes the form of a boundary layer inomediately adjacent to the wall, in a layer about 1-5 mm thick. Heat transfer from the heated wall to the liquid by such a boundary layer flow is very effective and is well documented in many texts on heat transfer [6,7]. 

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