Centerline deposition rate

These numerical solutions account for advective transport of the reactant gases, as well as both ordinary and Knudsen diffusion. Finally, we consider the influence on the optimum pressure of a constraint on the deposition uniformity. In this case, the optimum pressure maximizing centerline deposition rates is obtained in closed form using the method of Lagrange multipliers. 

To calculate centerline deposition rates, we consider the region within a preform, as shown... 

A second useful normalization of the deposition rate is the simple modification of S given by S = Note that the normalized centerline deposition rate can be obtained from Eqs.22a, b simply by taking/ =f where/ is the normalized reactant fraction at z = 0. The pressure, temperature and all other variables in these expressions are uniform over the 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. 

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 conditions giving rise to the maximum centerline deposition rate do not yield the maximum surface rate, nor do they yield the maximum mean deposition rate. This is shown in Figure 4, where the centerline, surface and mean normalized deposition rates are shown as a function of the normalized temperature. Here we see that both the surface and mean rates increase smoothly with increasing temperature and that only the centerline rate exhibits a maximum. For the conditions shown, this maximum occurs at 7 0.0491, corresponding to P 4.453 and a deposition uniformity of [/=/ -0.239. 

Fig. 4 Surface, mean and centerline deposition rates. For a given pressure, the maximum centerline deposition rate occurs at a specific temperature and corresponding value of the deposition moduius.

This equation can be solved easily by successive substitution. Using a guessed value of the temperature in the first factor on the left of Eq.37a, the exponential term is inverted to obtain an improved value for the guess. When repeated, this procedure converges in just a few cycles for all conditions of practical importance. Eor = 10, the temperature obtained from Eq. 37b is T = 0.04898. The maximum centerline deposition rate can be computed from Eqs. 19 and 22b once the pressure and temperature have been obtained. 

Sample calculations of the optimum pressure and optimum temperature and the maximum centerline deposition rate are shown in Eigure 5 as a function of the normalized preform thickness, These results are all for the special case of negligible reaction yield, y = 0. As noted... 

The conditions yielding the unconstrained maximum centerline deposition rate give a deposition uniformity of only about 25%. While this may well be acceptable for some fiber coating processes, there are likely applications for which it is not. We now consider the problem of maximizing the centerline deposition rate, subject to an additional constraint that the deposition uniformity satisfies some minimum requirement. Assuming that the required uniformity is better than that obtained in the unconstrained case, the constrained maximum centerline deposition rate should occur when the uniformity constraint is just marginally satisfied. This permits replacing the inequality constraint of a minimum uniformity by an equality constraint that is satisfied exactly. 

Thus, we find that the optimum pressure maximizing the centerline deposition rate for a... 

Note that even the small normalized reaction yield of our sample problem sigiuficantly affects the optimum conditions and maximum deposition rate. If we take / = 0, but leave all other problem parameters unchanged, the resulting optimum temperature increases to 551 °C and the maximum centerline deposition rate increases to 0.92 um/hr. Thus a normalized reaction yield of only / = 0.33 has reduced the maximum deposition rate by over 15%. 

The optimum conditions yield a weak onesided maximum of the centerline deposition rate. The deposition rate falls linearly at pressures well below the optimum value, but remains nearly constant at all values above. The reason for this behavior lies in the tradeoff... 

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