Deposition modulus

The usual interpretation of the parameter P, referred to here as the deposition modulus, is that it is the square of the ratio of the characteristic time for diffusion to the characteristic time for surface deposition. In this view it is equivalent to the square of the Thiele modulus commonly appearing in analyses of porous-bed catalysis. Another useful interpretation of this parameter is that it is the ratio of two rates - the rate of deposition on the preform fiber surfaces, Ss a, to the maximum rate of diffusive transport, pDDf/a. Thus when P is small, the actual rate of diffusive transport will be less than this maximum, and the mean gradient of the reactant fraction will be smaller than the maximum value off/a. Under any of these interpretations, small values of P are associated with high uniformity of both the reactant fraction and coating thickness. 

No such closed-form solution exists for the more general case v / 0. The general form of Eq. 18 can be solved for small values of the deposition modulus, [3, though as we will see later, such solutions are not applicable to the problem of interest." Fortunately, very accurate numerical solutions to the boundary value problem posed by Eqs. 8 and 18 are readily obtained using a numerical shooting technique. 

Finally, Eq.l2 may be rewritten to obtain an expression for the deposition modulus that depends only on the normalized pressure, temperature and preform thickness. 

The deposition modulus at low temperatures is small, and the profile of the reactant concentration through the preform thickness is very uniform. In this case, the deposition rate at the center is nearly as large as that at the preform surface. With increasing temperature, the deposition modulus increases and the reactant concentration at the preform center falls. In this case, the centerline deposition rate becomes small relative to that at the surface. This behavior is illustrated in Figure 2. Here we see that the normalized centerline reactant fraction falls monotonically with increasing values of the deposition modulus. The centerline reactant fraction does not exhibit any sort of maximum, as is well known, and the deposition uniformity, U = /, falls smoothly as the deposition modulus is increased. 

Fig. 2 Distribution of normalized reactant fraction through the preform thickness. Reactant fractions depend only on the dimensionless reaction yield, /, and the deposition modulus, p.

Figure 3 shows sample calculations of the normalized deposition rate through the preform thickness. We see that the deposition rate at the preform surface increases monotonically with increasing values of the deposition modulus. At the preform center, however, the deposition rate increases only up to a value of P 4. At still larger values the centerline rate begins to fall, gradually approaching zero as P o. Thus for the conditions shown, the centerline deposition rate exhibits a maximum when the deposition modulus is about P 4. 

Fig. 3 Normalized deposition rate through preform thickness. Maximum centerline deposition rate occurs at a specific value of the deposition modulus, P its value is p 4.453 for the conditions shown.

The derivatives of the deposition modulus with respect to pressure and temperature can now be obtained from Eq. 23. They are... 

Finally, using the definitions of Eqs. 27 and 28 for the derivatives of the deposition modulus, Eq.44 becomes... 

Optimum temperatures are obtained from the analytical expressions describing the optimum conditions, along with the derivative of the centerline reactant fraction with respect to the deposition modulus. Using an analytical solution to provide the derivative of the centerline reactant fraction, a closed-form implicit expression for the optimum temperature is obtained for the special case of no normalized reaction yield. For the more general case, this derivative is computed from numerical solutions to the equations governing transport and deposition. Optimum temperatures are presented graphically for a very wide range of the normalized preform thickness and normalized reaction yield. 

U Deposition Uniformity U = fjf = f ) f) Reactant Mean Molecular Speed X Transverse Position z Normalized Transverse Position (z = x/a) a Ratio of Diffusivities P Deposition Modulus A Reactant Mean Free Path Normalized Preform Thickness p Total Molar Density o Reactive Species Molecular Diameter ( ) Reaction Probability / Net Molar Yield of Deposition Reaction... 

Many materials deposited by CVD have a high elastic modulus and a low fracture toughness and are therefore affected by residual film stresses. 

Sihcon carbide fibers exhibit high temperature stabiUty and, therefore, find use as reinforcements in certain metal matrix composites (24). SiUcon fibers have also been considered for use with high temperature polymeric matrices, such as phenoHc resins, capable of operating at temperatures up to 300°C. Sihcon carbide fibers can be made in a number of ways, for example, by vapor deposition on carbon fibers. The fibers manufactured in this way have large diameters (up to 150 P-m), and relatively high Young s modulus and tensile strength, typically as much as 430 GPa (6.2 x 10 psi) and 3.5 GPa (507,500 psi), respectively (24,34) (see Refractory fibers). 

Although a variety of test methods, eg, Dk, modulus, and tear strength, exist to determine key properties of potential contact lens materials, a number of properties, eg, wettabihty and deposition, have no predictive methodology short of actual clinical experience. 

Another way in which catalyst deactivation may affect performance is by blocking catalyst pores. This is particularly prevalent during fouling, when large aggregates of materials may be deposited upon the catalyst surface. The resulting increase in diffu-sional resistance may dramatically increase the Thiele modulus, and reduce the effectiveness factor for the reaction. In extreme cases, the pressure drop through a catalyst bed may also increase dramatically. 

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