Plate geometries

Since we have parallel-plate geometry, the viscosity of the liquid layer is given by... 

For the simplest one-dimensional or flat-plate geometry, a simple statement of the material balance for diffusion and catalytic reactions in the pore at steady-state can be made that which diffuses in and does not come out has been converted. The depth of the pore for a flat plate is the half width L, for long, cylindrical pellets is L = dp/2 and for spherical particles L = dp/3. The varying coordinate along the pore length is x ... 

Turn now to the flat-plate geometry. The coefficients A, B, and C, and the mixing-cup averaging technique must be revised. This programming exercise is left to the reader. Run the modified program with ki = I but without... 

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. 

Surface forces measurement directly determines interaction forces between two surfaces as a function of the surface separation (D) using a simple spring balance. Instruments employed are a surface forces apparatus (SFA), developed by Israelachivili and Tabor [17], and a colloidal probe atomic force microscope introduced by Ducker et al. [18] (Fig. 1). The former utilizes crossed cylinder geometry, and the latter uses the sphere-plate geometry. For both geometries, the measured force (F) normalized by the mean radius (R) of cylinders or a sphere, F/R, is known to be proportional to the interaction energy, Gf, between flat plates (Derjaguin approximation). 

With the critical exponent being positive, it follows that large shifts of the critical temperature are expected when the fluid is confined in a narrow space. Evans et al. computed the shift of the critical temperature for a liquid/vapor phase transition in a parallel-plates geometry [67]. They considered a maximum width of the slit of 20 times the range of the interaction potential between the fluid and the solid wall. For this case, a shift in critical temperature of 5% compared with the free-space phase transition was found. From theoretical considerations of critical phenomena... 

Figure 2.27 Streamline patterns in a channel with sinusoidal walls (left) and Nusselt number as a function of Reynolds number for the same channel (right), taken from [120]. For comparison, the triangles represent the Nusselt number obtained in parallel-plates geometry.

Gel formation was monitored using a controlled-stress rheometer (Carri-Med CS 50, TA Instruments, Guyancourt, France) with cone-and-plate geometry (cone diameter 4 cm, angle 3°58 ). The bottom plate was fitted with a Peltier temperature controller that... 

Deviation from laminar shear flow [88,89],by calculating the material functions r =f( y),x12=f( Y),x11-x22=f( y),is assumed to be of a laminar type and this assumption is applied to Newtonian as well as viscoelastic fluids. Deviations from laminar flow conditions are often described as turbulent, as flow irregularities or flow instabilities. However, deviation from laminar flow conditions in cone-and-plate geometries have been observed and analysed for Newtonian and viscoelastic liquids in numerous investigations [90-95]. Theories have been derived for predicting the onset of the deviation of laminar flow between a cone and plate for Newtonian liquids [91-93] and in experiments reasonable agreements were found [95]. 

Flow irregularities at gap angles of 30° were observed in viscoelastic liquids [94]. It has been indicated in theoretical treatments that the possibility of secondary flows [96,97] in rotational devices is to be expected if the gap angle is much greater than 5°. For viscoelastic fluids deviations from laminar flow have only been reported in cone-and-plate geometries with gap angles above 10°. 

In the Couette cell the shear stress varies signficantly with radial position across the gap as r2. Should a more uniform stress environment be required then the cone-and-plate geometry may be used [17]. An example apparatus is shown in Figure 2.8.7. 

The unperforated area can be calculated from the plate geometry. The relationship between the weir chord length, chord height and the angle subtended by the chord is given in Figure 11.32. 

The process has been commercially implemented in Japan since 1977 [1] and a decade later in the U.S., Germany and Austria. The catalysts are based on a support material (titanium oxide in the anatase form), the active components (oxides of vanadium, tungsten and, in some cases, of molybdenum) and modifiers, dopants and additives to improve the performance, especially stability. The catalyst is then deposited over a structured support based on a ceramic or metallic honeycomb and plate-type structure on which a washcoat is then deposited. The honeycomb form usually is an extruded ceramic with the catalyst either incorporated throughout the stmcture (homogeneous) or coated on the substrate. In the plate geometry, the support material is generally coated with the catalyst. 

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... 

The samples were cured with 0.2, 0.4, 0.8 and 1.6 wt.% dicu-myl peroxide. In this way, we obtained twelve different networks with great variations in relaxation Intensities. Dynamic mechanical measurements were performed In torsion in the linear region (deformations smaller than 5 %) with a mechanical spectrometer, using the parallel-plate geometry. The frequency ranged from 0.01 to 15 Hz and the temperature was usually between 300 and 435 K. 

Such a generalization must reproduce, for example, the known results for the Casimir effect in the case of the parallel plates geometry at finite temperatures. Since energy is an additive quantity, we expect to have L- and T-dependent contributions plus a mixed (LT-dependent) contribution representing the interference of the two effects. In the next Section, we will show that the proper extension of expressions (16) and (21), for this case, is... 

Figure 8.10 (a) Representation of flat-plate geometry (b) concentration profile z) (dimensionless) for various values of Thiele modulus (j>... 

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. 

Repeat Example 9-1 and problem 9-1 for an isothermal particle of flat-plate geometry rep-... 

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