OCR Text |
Show 1.7.7 by an infinite series, each term of which is the product of a function of the radius and an exponential function of the axial distance. Sufficiently far downstream, the series may be approximated by its first term. Values for the axial distance at which this approximation yields an error of 5% in the axial concentration gradient were computed by Judeikis [143. These values indicate that the approximation is sufficiently accurate for present conditions and purposes. The one-term approximate solution is C(R,Z) = A g(R) exp(-K*Z), 9 < R ) « 2 n a o - B n R 2 n ( B 0 = 1, Bj = - ^/4, B n = (2nr2[(K*/D)Bn-2 - ^Bn-13, 0 = K*2 + (K*/D) - K/(2D), where all quantities are nondimensional. They are given by C = concentration of the trace species, A = normalization constant, R = r/a = radius, Z = z/a = axial distance, D = Dc/(2au) = diffusion constant, K = ak/u = gas phase rate constant, K* = ak*/u = effective rate constant, representing the combined effect of wall and gas phase reactions, and Kw = ^ / u = wall rate constant. Here, a is the tube radius, u the average gas velocity, D c the dimensional diffusion coefficient, while k*, k and k^ are the dimensional effective, gas phase and wall rate constants, respectively; all of these rate constants represent first order reactions. The solution for C is subject to the boundary condition DcoC/dr(r = a) = k^Cfr = a). This leads to the following relation between K*, K, K w and D: F(K#, K, Kw, D) = 2 n = 0* <2n * Kw/2D)Bn = 0. A computer program for solving this relation is listed in [133. The relations just given can be applied in an approximate way to the present experimental results by noting that the initial decomposition of N H 3 is the rate-limiting step. The influence of wall reactions then is limited by the rate at which N H 3 reacts at the wall. An upper limit for this rate is obtained by assuming that the wall is infinitely reactive: k w • ». The nondimensionalized diffusion coefficient of N H 3 in helium at 850 K under the conditions in the reactor is found to be D = 0.0164. Using Brown's computer program [133, the resulting e-folding length for the disappearance of N H 3 is found to be a/K° = 53 cm; the corresponding reduction is i - i/e = 63%. The reduction observed at a distance of 45 cm from the inlet port of N H 3 is about 20%. It can be concluded that the reaction rate is limited by the reactivity of the wall. In order to estimate this reactivity, use can again be made of Brown's program. With u = 137 cm/s and K = 0, the remaining parameter at T = 850 K and helium as the diluent is K* = 1.8 x 10~3. It is found from these values that K w = 8.9 x 10"4. With argon as the |