Slab geometry

Finally, the long-range contribution to the energy (see Eq. (6.27c)] gives rise to a stress contribution 

The reader should appreciate the difference between the previous expression and the last term on the right side of Eq. (F.71). This difference arises because, for a dipolar system, = JZili / independent of Sy, whereas for a Coulomb system, My depends on Sy as one can verify from Eq. (F.69). The diagonal component of the total stress tensor is then obtained by adding the three contributions given in Eqs. (F.91), (F.93), and (F.94) [see Eqs. (F.89) and (F.90)]. 

The reader should appreciate the diflFerence between the previous expression and the last term on the right side of Eq. (F.71). This difference arises because, for a dipolar system, M-y = Cy is independent of Sy, 

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In the case of slab geometry, this is to be interpreted as the component of N along the coordinate axis normal to the faces, of course. 

To find the effectiveness under poisoned conditions, this form of the modulus is substituted into the appropriate relation for effec tiveness. For first-order reaction in slab geometry, for instance,... 

If the same quantity of active ingredient is concentrated in an outside shell of thickness 0.015 cm, one obtains y> = 2.27. This would yield an effectiveness factor of 0.431 in a slab geometry, and the apparent kinetic constant has risen to 99.2 sec-1. If the active ingredient is further concentrated in a shell of 0.0025 cm, one obtains y> = 0.38, an effectiveness factor of 0.957, and an apparent kinetic constant of 220 sec-1. These calculations are comparable to the data given in Fig. 15. This analysis applies just as well to the monolith, where the highly porous alumina washcoat should not be thicker than 0.001 in. 

One of the most accurate approaches to solve the LDF equations for the single slab geometry is the full-potential linearized augmented plane wave (FLAPW) method (10). Here, we highlight only the essential characteristics of this approach for further details the reader is referred to a recent review article (11). 

A slab geometry of the polymer matrix with center line at x=0 and outer surface atx=a. See Figure 1. 

A moving front is usually observed in swelling glassy polymers. A diffusion-controlled front will advance with the square root of time, and a case II front will advance linearly with time. Deviations from this simple time dependence of the fronts may be seen in non-slab geometries due to the decrease in the area of the fronts as they advance toward the center [135,140], Similarly, the values of the transport exponents described above for sheets will be slightly different for spherical and cylindrical geometries [141],... 

For slab geometry as an example, the material and energy balances over a pore are... 

Material balance on a slab geometry, Input - Output = Sink dCa... 

For first order reaction in slab geometry, the pore equation is... 

For first order reaction in slab geometry, evaluate the ratio of effectiveness with uniform poisoning, 7)un> and pore mouth poisoning, T)pm, in terms of fractional poisoning and the Thiele modulus. 

Develop a model and program for a rectangular slab geometry with diffusion from both sides and taking the thickness as Vbead/Abead=Rp- Compare the results for the slab with those for the bead. 

When the internal diffusion effects are considered explicitly, concentration variations in the catalytic washcoat layer are modeled both in the axial (z) and the transverse (radial, r) directions. Simple slab geometry is chosen for the washcoat layer, since the ratio of the washcoat thickness to the channel diameter is low. The layer is characterized by its external surface density a and the mean thickness <5. It can be assumed that there are no temperature gradients in the transverse direction within the washcoat layer and in the wall of the channel because of the sufficiently high heat conductivity, cf., e.g. Wanker et al. 

There exists a relation between the volume of catalytic washcoat layer (represented in the ID model by the volume fraction q>s in the solid phase) and the characteristic thickness of the layer 8 used in the spatially 2D model. This relation depends on the chosen washcoat geometry—for slab geometry used here it is... 

Fig. 9.4. (a) The dependence of the stationary-state concentration of reactant A at the centre of the reaction zone, a (0), on the dimensionless diffusion coefficient D for systems with various reservoir concentrations of the autocatalyst B curve a, / = 0, so one solution is the no reaction states a0i>8 = 0, whilst two other branches exist for low D curves b and c show the effect of increasing / , unfolding the hysteresis loop curve d corresponds to / = 0.1185 for which multiplicity has been lost, (b) The region of multiple stationary-state profiles forms a cusp in the / -D parameter plane the boundary a corresponds to the infinite slab geometry, with b and c appropriate to the infinite cylinder and sphere respectively. 

Similar reasoning (Ref 24) leads to the following Eqn for end-projectors which are a variant of the slab geometry examined in Section II ... 

For uniform poisoning, the effectiveness is obtained by simply replacing k with kv(l-/3) in the definition of For pore mouth poisoning the equation for if is in P7.06.07. These lesuLts are for first order reaction in slab geometry. 

The catalyst packing of the reactor consists of an iron oxide Fe20s, promoted with potassium carbonate K COo, and chromium oxide Cr O-s,. The catalyst pellets are extrudates of a cylindrical shape. Since at steady state the problem of simultaneous diffusion and reaction are independent of the particle shape, an equivalent slab geometry is used for the catalyst pellet, with a characteristic length making the surface to volume ratio of the slab equal to that of the original shape of the pellet. 

The kinetics of release from a monolithic solution system have been derived for a number of geometries by Crank [20], For a slab geometry, the release kinetics can be expressed by either of two series, both given here for completeness... 

Adsorption in ultramicroporous carbon was treated in terms of a slit-potential model by Everett and Powl51 and was later extended by Horvath and Kawazoe.52 They assumed a slab geometry with the slit walls comprised of two infinite graphitic planes. Adsorption occurs on the two parallel planes, as shown in figure 2.7. 

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