Flat plate equations

Flat plate equations are based on the following conditions ... 

A detailed mathematical analysis has been possible for a second situation, of a wetting meniscus against a flat plate, illustrated in Fig. X-16b. The relevant equation is [226]... 

Wlien describing the interactions between two charged flat plates in an electrolyte solution, equation (C2.6.6) cannot be solved analytically, so in the general case a numerical solution will have to be used. Several equations are available, however, to describe the behaviour in a number of limiting cases (see [41] for a detailed discussion). Here we present two limiting cases for the interactions between two charged spheres, surrounded by their counterions and added electrolyte, which will be referred to in further sections. This pair interaction is always repulsive in the theory discussed here. 

Let us consider the flow in a narrow gap between two large flat plates, as shown in Figure 5.19, where L is a characteristic length in the a and y directions and h is the characteristic gap height so that /z < L. It is reasonable to assume that in this flow field il c iq, Vy. Tlierefore for an incompressible Newtonian fluid with a constant viscosity of q, components of the equation of motion are reduced (Middleman, 1977), as... 

Under conditions of limiting current, the system can be analyzed using the traditional convective-diffusion equations. For example, the correlation for flow between two flat plates is... 

Example The equation dQ/dx = (A/f/)(3 6/3f/ ) with the boundary conditions 0 = OatA.=O, y>0 6 = 0aty = oo,A.>0 6=iaty = 0, A.>0 represents the nondimensional temperature 6 of a fluid moving past an infinitely wide flat plate immersed in the fluid. Turbulent transfer is neglected, as is molecular transport except in the y direction. It is now assumed that the equation and the boundary conditions can be satisfied by a solution of the form 6 =f y/x ) =j[u), where 6 =... 

For the turbulent flow of a fluid over a flat plate the Colburn type of equation may be used with a different constant ... 

Colburn equation, flat plates 434 Colorimetric methcxls for humidity determination 759 Comings, E. W, 745, 786 Compact heat exchangers 550 Compressed air for pumping 358 Compressibility factor 34, 35,161 Compressible fluids 30, 48... 

Equation (8.12) is a form of the convective dijfusion equation. More general forms can be found in any good textbook on transport phenomena, but Equation (8.12) is sufficient for many practical situations. It assumes constant diffusivity and constant density. It is written in cylindrical coordinates since we are primarily concerned with reactors that have circular cross sections, but Section 8.4 gives a rectangular-coordinate version applicable to flow between flat plates. 

Comparing this equation with Equation (8.34) shows that 3At/Y is the flat-plate counterpart of aAII - We thus seek a value for t/T below which diffusion has a negligible effect on the yield of a first-order reaction. 

This velocity profile is commonly called drag flow. It is used to model the flow of lubricant between sliding metal surfaces or the flow of polymer in extruders. A pressure-driven flow—typically in the opposite direction—is sometimes superimposed on the drag flow, but we will avoid this complication. Equation (8.51) also represents a limiting case of Couette flow (which is flow between coaxial cylinders, one of which is rotating) when the gap width is small. Equation (8.38) continues to govern convective diffusion in the flat-plate geometry, but the boundary conditions are different. The zero-flux condition applies at both walls, but there is no line of symmetry. Calculations must be made over the entire channel width and not just the half-width. 

The equivalent of radial flow for flat-plate geometries is Vy. The governing equations are similar to those for Vy. However, the various corrections for Vy are seldom necessary. The reason for this is that the distance Y is usually so small that diffusion in the y-direction tends to eliminate the composition and temperature differences that cause Vy. That is precisely why flat-plate geometries are used as chemical reactors and for laminar heat transfer. 

Derive the equations necessary to calculate Vz y) given iJi y) for pressure-driven flow between flat plates. 

Transient Heat Conduction. Our next simulation might be used to model the transient temperature history in a slab of material placed suddenly in a heated press, as is frequently done in lamination processing. This is a classical problem with a well known closed solution it is governed by the much-studied differential equation (3T/3x) - q(3 T/3x ), where here a - (k/pc) is the thermal diffuslvity. This analysis is also identical to transient species diffusion or flow near a suddenly accelerated flat plate, if q is suitably interpreted (6). 

A general equation for the thickness of a flat plate required to resist a given pressure load can be written in the form ... 

The design equations used to determine the thickness of flat ends are based on the analysis of stresses in flat plates Section 13.3.5. 

By modifying Berenson s equation (2-151) for film boiling on a flat plate, Sci-ance et al. (1967) suggested the correlation... 

The diffusivities thus obtained are necessarily effective diffusivities since (1) they reflect a migration contribution that is not always negligible and (2) they contain the effect of variable properties in the diffusion layer that are neglected in the well-known solutions to constant-property equations. It has been shown, however, that the limiting current at a rotating disk in the laminar range is still proportional to the square root of the rotation rate if the variation of physical properties in the diffusion layer is accounted for (D3e, H8). Similar invariant relationships hold for the laminar diffusion layer at a flat plate in forced convection (D4), in which case the mass-transfer rate is proportional to the square root of velocity, and in free convection at a vertical plate (Dl), where it is proportional to the three-fourths power of plate height. 

The analysis of simultaneous diffusion and chemical reaction in porous catalysts in terms of effective diffusivities is readily extended to geometries other than a sphere. Consider a flat plate of porous catalyst in contact with a reactant on one side, but sealed with an impermeable material along the edges and on the side opposite the reactant. If we assume simple power law kinetics, a reaction in which there is no change in the number of moles on reaction, and an isothermal flat plate, a simple material balance on a differential thickness of the plate leads to the following differential equation... 

For flat-plate geometry wher only one side of the plate is exposed to reactant gases, one may proceed as in previous subsections to show that for mechanistic equations of the form... 

We then wish to discover how tj depends on reaction and particle characteristics in order to use equation 8.5-5 as a rate law in operational terms. To do this, we first consider the relatively simple particle shape of a rectangular parallelepiped (flat plate) and simple kinetics. 

For a flat-plate porous particle of diffusion-path length L (and infinite extent in other directions), and with only one face permeable to diffusing reactant gas A, obtain an expression for tj, the particle effectiveness factor defined by equation 8.5-5, based on the following... 

Equation 8.5-11 applies to a first-order surface reaction for a particle of flat-plate geometry with one face permeable. In the next two sections, the effects of shape and reaction order on p are described. A general form independent of kinetics and of shape is given in Section 8.5.4.5. The units of are such that is dimensionless. For catalytic reactions, the rate constant may be expressed per unit mass of catalyst (k )m. To convert to kA for use in equation 8.5-11 or other equations for d>, kA)m is multiplied by pp, the particle density. 

Loh and Femandez-Pello [8] have shown that the form of Equation (8.30) appears to hold for laminar wind-aided spread over a flat plate. Data are shown plotted in Figure 8.17 and are consistent with... 

The temperature of a liquid metal stream discharged from the delivery tube prior to primary breakup can be calculated by integrating the energy equation in time. The cooling rate can be estimated from a cylinder cooling relation for the liquid jet-ligament breakup mechanism (with free-fall atomizers), or from a laminar flat plate boundary layer relation for the liquid film-sheet breakup mechanism (with close-coupled atomizers). 

When heat and mass are transferred simultaneously, the two processes interact through the Gr and Gq terms in Eq. (10-12) and the energy and diffusion equations. Although solutions to the governing equations are not available for spheres, results should be qualitatively similar to those for flat plates (T4), where for aiding flows (Gr /Gq > 0) the transfer rate and surface shear stress are increased, and for opposing flows (Gr Gq < 0) the surface shear stress is predicted to drop to zero yielding an unstable flow. 

The transition to a turbulent boundary layer for a flat plate has been experimentally determined to occur at an Rcx value of between 3 x 10 and 6 x 10. For this example, the transition would occur between 15 and 30 cm after the start of the plate. Thus, the computations for a laminar boundary layer at 0.6 and 1 m are not realistic. However, the Blasius solution helps in the analysis of experimental data for a turbulent boundary layer, because it can tell us which parameters are likely to be important for this analysis, although the equations may take a different form. 

---