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J. Phys. Chem. 1980, 84, 2481-2484
momentum about the N-N direction. Note Added in Proof: We have recently observed the spectrum of the radical anion of Mn(NO)&O by sodium reduction of Mn(NO)3C0 in THF at 200 K: g = 1.9968, ~ p ~ (= 3 )16.5 IMHz, uMn = 33.0 MHz.
References amd Notes (1) N. M. Atherton, J. R. Morton, K. F. Preston, and M. J. Vuolle, Chem. Phys. Lett., 70, 4 (1980). (2) C. Couture, J. R. Morton, and K. F. Preston, J. Magn. Reson., to be submitted for publication. (3) g = 2.0061, aN(2)= 50.5 MHz, ah = 13.5 MHz, a+ ,', = 24.3 MHz.
(4) F. Seel, 2.Anorg. Allg. Chem., 269, 40 (1952). (5) R. Lefebvre and J. Maruani, J . Chem. Phys., 42, 1480 (1965). (6) We thank Dr. P. J. Krusic for a sample of Fe(CO), enriched to 90% in the isotope "C. (7) J. R. Morton and K. F. Preston, J. Magn. Reson., 30, 577 (1978). (8) A. R. Boate, J. R. Morton, and K. F. Preston, J . Magn. Reson., 29, 243 (1978). (9) D. Griller, P. R. Marriott, and K. F. Preston, J. Chem. Phys., 71, 3703 (1979). (10) A. R. Boate, J. R. Morton, K. F. Preston, and S. J. Strach, J. Chem. Phys., 71, 388 (1979). (11) A. Carrington and A. D. McLachian, "Introduction to Magnetic Resonance", Harper and Row, New York, 1967, p 138. (12) J. R. Brailsford, J. R. Morton, and L. E. Vannotti, J . Chem. Phys., 50, 1051 (1969).
Mass Transport and Chemical Reaction in Cylindrical and Annular Flow Tubes Henry S. Judelkls Chemistry and Physics Laboratory, The Ivan A. Getting Laboratories, The Aerospace Corporation, €1 Segundo, California 90009 (Received: October 22, 1979; In h a / Form: April 16, 1980)
Solutions of the mass transport equations, including axial and radial diffusion, are presented for cylindrical and annular flow systems. The solutions include first-order gas-phase and wall reactions, at either wall or both walls in the annular case. The solutions are applicable at large distances from the entry to the flow tube, Le., where pure exponential decay of the reacting species is observed. The solutions, which are in the form of infinite series, can be used to calculate both axial and radial concentration gradients under these conditions or, given experimentally measured gradients, to extract kinetic information on gas-phase and wall reaction rates.
Introduction The extensivle use of cylindrical flow tubes in the study of fast gas-phase reactions has led to a number of discussions on their limitations (e.g., ref 1 and 2). These limitations frequently are based upon the fact that solutions of the mass-transport equations applicable to flow systems generally cannot be obtained in closed form, necessitating laborious numerical integration of the equations. To circumvent the latter difficulty, one usually chooses experimental conditions such that certain terms can be neglected and approximate closed-form solutions obtained. Indeed, under ideal conditions, particularly simple approximations can be used (ref 1and 2). One such condition is the situation where mass transport by axial diffusion is negligible compared to mass transport by flow. In selected gas-phase studies, and especially in the study of wall reactions, it is not always possible to work under conditions where the simpler approximations can be utilized. In these cases, alternate approximations are desirable. Presented here are solutions to thle mass-transport equations, including first-order gas-phase and wall reactions, that are valid a t large distances from the entry to the flow tube. Solutions applicable to flow through a cylindrical tube are discussed, as well as those for flow through the annular space between concentric cylinders. Discussion A. Single-Tube Solutions. Under conditions of constant temperature and pressure, the steady-state concentration of a trace reacting species ( C ) in a tube with fully developed Poiseuille flow is described by eq 1,3-7 where D is the
0022-3654/80/2084-2481$01 .OO/O
diffusion coefficient, k is the (first-order) gas-phase reaction rate constant, r and z are the radial and axial dimensions, and V, is given by eq 2. In eq 2, V,, is the
average volume flow velocity and R is the radius of the flow tube. In many cases axial diffusion, Dd2C/dz2, can be neglected. Solutions of eq 1for these conditions have been d e ~ c r i b e d .Analogous ~~ solutions for related heat-transfer problems have also been given.&13 When axial diffusion cannot be neglected, the method of Graetz'* and others15is generally employed. This method involves substitution of m
C = C fi(r)e-biz i=l
(3)
into eq 1, where m
fi(r) = C uimrm m=O
(4)
This results in an infinite set of ordinary differential equations that, when coupled with the appropriate boundary conditions, can be solved for the aimcoefficients and the pi eigenvalues. Walker3 has described these solutions for a single tube including gas-phase and wall reactions. At sufficiently large distances from the entry to the flow tube (ref 3-71, only the first term (i = 1) of eq 3 is important in the solution of eq 1. Under these conditions, a single ordinary differential equation is obtained. In such cases, analysis of data from flow-tube experiments is greatly simplified. Derivation of a model for wall reactions only (k = 0) under such conditions is given in ref 16. 0 1980 American Chemical Society
The Journal of Physical Chemistry, Vol. 84, No. 19, 1980
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Judeikis
For completeness, the case of first-order gas-phase reaction only (no wall reaction) will be considered here. From this point on, eq 3 and 4 will be used without the i subscript, since only the first term in eq 3 is of interest here. Substitution of eq 2, 3, and 4 into eq 1 and rearrangement results in the recurrence relation 1 a, = - vamJ m = 2, 4, 6, ... (5)
I000 I I
m2
where a 10, the distance at which a 5% slope error would occur through use of eq 12 becomes a significant fraction of the typical flow-tube length of -100 cm. Calculations carried out for a y of 1 X 10" and .$ = 1,10, and 100 gave values for Z* of 2.71,8.23, and 77.4 cm, respectively, compared to corresponding values of 2.62, 8.35, and 71.9 cm for y = 1. The models described here, as well as in ref 3 and 16, are summarized in Table I for a single-tube configuration.
The Journal of Physical Chemistty, Vol. 84, No. 19, 1980 2403
Mass Transport and Chemical Reaction in Flow Tubes
TABLE 11: Summary of Annular Modelsa reactions gas phase
inner wall
no no no no no no no no no Yes Yes
outer wall no no
Yes yes no no Yes Yes Yes Yes
Yes yes Yes Yes Yes Yes yes no no no
yes no
soln 71=yo
YI
no no no no
O [ ( h / D- P 2 ) x (al + a z ) + d 2 a l + 3ao) - a@e(al+ %ad] - %as (20)
a4
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where, as before, = 0. Application of the boundary conditions expressed by eq 8 and 10 can lead to a variety of solutions, depending upon whether the inner or outer walls are reactive, or both or neither is reactive. These boundary conditions can be applied in the manner indicated in the previous section. Of course, for no gas-phase reaction, k = 0. The general forms of the solutions for various scenarios are given in Table 11. In all of the solutions in Table 11, either a. or al is equal to unity. This comes about as in the single-tube case; i.e., these quantities appear as arbitrary constants in the solutions and can therefore be eliminated or, the equivalent, set equal to one. Similar to the single-tube case, values of Z* vs. ,$‘ [= 2V,,/[7rD(Ro2 - R t ) ] ]have been plotted in Figure 1 for an annular flow tube with no gas-phase reaction, yI = yo = 1, Ro = 2.5 cm and RI = 0.5 cm. (Slopes at large distances from the entry, as calculated from the models described here, were compared to slopes obtained from numerical integration of the appropriate differential equation by a finite difference method.) Here, to within a few percent, the plot of Z* vs. can be represented by eq 22. Z* = 0,79/[1 - exp(-6.60/(’)] (22) Additional results, illustrating radial concentration gradients, are shown in Figure 2 for selected conditions. For the examples illustrated in Figure 2, the following conditions were chosen: Ro = 2.5 cm, RI = 0.5 cm, V,, = 20 L/s (average linear velocity = 1.06 m/s), D = 30 cm2/s, k , = 8100 cm/s, and k = 2 s-l. In addition various conditions of wall reactivity are illustrated, for cases of no wall reactivity, either wall reactive, and both walls reactive. Figure 2, a and b, illustrates results for wall reactivities ( 4 ) of 1 X and 5 X respectively. The figure illustrates radial concentration gradients for these various conditions, while the figure caption indicates the axial concentration gradients (0). Both the radial and axial concentration gradients increase with increasing y as would be expected. Note however, that a slight radial concentration gradient exists even for yI = yo = 0. This results from mass transport to the walls by diffusion being slightly less than the first-order gas-phase reaction rate.
Judeikis
Conclusions Models have been described for the analysis of data from flow tube experiments utilizing single flow tubes or the annular space between concentric cylinders. These models can also be used in the design of experiments and apparatus. They include mass transport by diffusion and flow, as well as various combinations of first-order gas-phase and wall reactions. They are applicable at large distances from the entry to the flow tube, i.e., where pure exponential decays of the reacting species are observed. When techniques described elsewhere3J4J5are used, the models can be extended for the determination of concentrations at shorter distances. The solutions are in the form of infinite series that, given experimental parameters and dimensions, depend only upon the gas-phase and wall reaction rate constants. They are readily amenable to computation on small computers employing general root-solving techniques. In some cases, parameters of interest appear explicitly, or, for selected experimental conditions, series converge rapidly so that calculations can be carried out with hand calculators. References and Notes F. Kaufman, Prog. React. Klnet., 1, 3 (1961). C. J. Howard, J . Pbys. Chem., 83, 3 (1979). R. E. Walker, Phys. Flulds, 4, 1211 (1961). H. A. Lauwerier, Appl. Sci. Res., Sect. A , 8, 366 (1959). E. H. Wissler and R. S. Schechter, Appl. Sci. Res., Sect. A , 10, 198 (1961). R. V. Poirier and R. W. Carr, Jr., J . Phys. Chem., 75,1593 (1971). P. J. Ogren, J. Phys. Chem., 79, 1749 (1975). M. Abramowitz, J. Math. Phys. (CambrMge, Mass.), 32, 184 (1953). J. Schenk and J. M. DumorB, Appl. Sci. Res., Sect. A , 4,39 (1953). J. R. Seliars, M. Tribus, and J. S. Klein, Trans. Am. SOC. Mech. Eng., 78, 441 (1956). S. N. Singh, Appl. Sci. Res., Sect. A , 7, 325 (1958). S. Sideman, D. Luss, and R. E. Peck, Appl. Sci. Res., Sect. A , 14, 157 (1965). C. Hsu, Appl. Sci. Res., Sect. A , 17, 359 (1967). L. Graetz, Ann. Pbys. Chem., 25, 337 (1885). M. Jakob, “Heat Transfer”, Vol. 1, Wiley, New York, 1949, and references therein. H. S. Judeikis and M. Wun, J . Chem. Phys., 68, 4123 (1978). H. F. Paneth and K. Herzfeld, Z. Elektrochem., 37, 577 (1931). H. Wise and C. M. Ablow, J. Chem. Phys., 29, 634 (1958). K. Murakawa, Int. J . Heat Mass Transfer, 2, 240 (1961). R. E. Lundberg, P. A. McCuen, and W. C. Reynolds, rnt. J. Heat Mass Transfer, 6,495 (1963). H. S.Heaton, W. C. Reynolds, and W. M. Kays, Int. J. Heat Mass Transfer, 7, 763 (1964). 6. S.Petukhov and L. I. Rolzen, Heat Mass Transfer, 1, 56 (1967). P. I. Puchkov and 0. S. Vinogradov, Heat Mass Transfer, 1, 65 (1967). H. Berg and W. Beyer, Chem. Tech. (Serlln), 8,235 (1956). A. E. Cassano, P. L. Silveston, and J. M. Smith, Ind. Eng. Chem., 59, 18 (1967). D. BrkiE, P. Forsatti, J. Pasquon, and F. Trifirb, J. Pbotochem., 5, 23 (1976). 0.BrkiE, P. Forsatti, and F. Trifir6, Chem. Eng.‘ Sci., 33, 853 (1978). R. 6. Bird, W. E. Stewart, and E. N. Lightfoot, “Transport Phenomena”, Wiley, New York, 1960, pp 51-4.
