Influence of External Mass Transfer on Catatytic Reaction Rates on Metals Daniel 0. Loffler and Lanny D. Schmidt' Department of Chemical Engineering and A.4ateriab Science, University of Minnesota, Minneapolis, Minnesota 55455
External mass transfer limitations on catalytic reaction rate measurements are examined and compared with some typical fast reaction rates on metal wires and foils. The dependences on pressure, temperature, gas composition, and size and shape of the surface are considered, and the mass transfer limited rate is found to be usually between 1020and IO2' molecules cm-2 s-l over a wide range of these variables. Literature data and experiments on NH3 decomposition on Pt are analyzed which indicate that the kinetics of many reactions can Only be studied over restricted ranges of pressure and temperature.
Introduction Wires and foils have been used for many years to measure reaction kinetics on metals. Langmuir (1921) studied CO and H2 oxidation on Pt in a low-pressure batch reactor, while Hinshelwood (1925a,b) studied N2O and NH3 decomposition at -200 Torr in a similar reactor. More recently, Cardoso and Luss (1969) analyzed stability aspects of catalytic reactions occurring on a wire in a flow reactor, and Rader and Weller (1974a,b)suggested a method to determine kinetic parameters from experimental measurements of the ignition temperature. The kinetics of NH3 oxidation (Pignet and Schmidt, 1974, 1975), NH3 decomposition (Loffler and Schmidt, 1976a,b), and CO oxidation (Hori and Schmidt, 1975) have been studied in this laboratory on different metal wires at pressures between 10-2 and 10 Torr using flow reactors. However, the importance of mass transfer resistances in these reactor geometries has not been quantitatively assessed. We have shown (Loffler and Schmidt, 1975) that even small resistances can impair kinetic studies on catalytic surfaces. In the present work mass transfer coefficients are calculated from heat transfer measurements on a catalytic wire. In batch reactors heat is transferred to the gas by conduction and natural convection, while forced convection can also be important in flow and stirred batch reactors at high pressures. Experimental conditions are usually chosen to minimize heat conduction along the wire. Radiation heat losses can be important a t very high temperatures, as shown by Langmuir (1912). Mass Transfer Mass transfer coefficients are usually estimated from heat transfer data using the heat and mass transfer analogy (Bird et al., 1960),that is, to measure NUH= F(GrPr) (in the case of natural convection) and then estimate the Nusselt number for mass transfer, NUM,as NUM = F(GrSc) assuming the function F is the same in both cases. The Reynolds number, Re, replaces Gr in similar expressions for forced convection. Natural Convection. The transfer of heat from an electrically heated wire to a surrounding gas was studied by Langmuir (1912). Nusselt (1929) proposed a dimensionless correlation (Figure 1)for free convection heat transfer from horizontal cylinders, presenting his data as NUH= F(GrPr) for values of GrPr down to Kyte et al. (1953) extended the correlation to GrPr = lo-'. We measured the power input to a Pt wire during a study of NH3 decomposition (Lijffler and Schmidt (19761, and we calculated the Nusselt number for heat transfer from the expression 362 Ind. Eng. Chem., Fundam., Val. 16, No. 3, 1977
QD NUH= kAT
The net heat flux density, Q, can be calculated as the difference between the measured total heat flux density and the radiant heat flux density. The latter is calculated using values for emissity of heated platinum wires presented by Langmuir (1912). Radiation heat transfer accounts for a fraction of the total heat flux varying between 3% (at low temperature, high pressure) and 50% (at high temperature, low pressure). Under our experimental conditions (low conversion and low pressure) chemical reaction heat effects are negligible as is the contribution of forced convection. Figure 1shows NUHas a function of GrPr for values of GrPr between and Physical properties were estimated using methods proposed by Reid and Sherwood (1966). For values of GrPr > lom8our data points lie close to the values predicted by the correlation by Kyte et al. (1953). Data of Brown et al. (1970) for heat transfer from Pt wires to an N2 atmosphere are also shown. We conclude that our data are consistent with observations of other workers. This indicates that natural convection accounts for most of the heat losses in this reaction system except perhaps at low pressures, where axial conduction along the wire can be important and at high temperatures where radiation may be significant. Figure 2 shows plots of k,, the mass transfer coefficient for self-diffusion of "3, as functions of pressure and temperature. These curves were calculated from the data in Figure 1. Values of k , for different gases are presented in Figures 3,4, and 5 for pressures between 1 and IO3 Torr and temperatures between 500 and 2000 K.-For the wire diameters commonly used ( < D < lo-' cm) and pressures above 1Torr, k , only varies between lozoand loz1 molecules cm-2 s-l. Since the viscosity, thermal conductivity, and diffusion coefficients of low molecular weight gases do not vary much with gas composition, we can expect 1020< k , < loz1molecules cm-2 s-l to be reasonable bounds for most gases. For pressures below 1 Torr, k , should decrease rapidly with decreasing pressure and merge with the line in Figure 2 representing the calculated gas flux of NH3 to the surface. k , values corresponding €0 the lowest pressures are observed to lie slightly above the flux line. This is probably due to axial conduction on the wires affecting heat transfer data used to calculate k , because the maximum rate obviously should merge with the gas flux at very low pressures. Forced Convection. Although our experiments were conducted in a flow system, heat transfer was mainly due to natural convection because Reynolds numbers were very small. For Re = 4 X the largest value we observed, an extrapolation of data presented by McAdams (1954) gives
I
I
T
I
I
I
I
I
I
I
I
wire
I
I
Nusselt (1929)
800K 0
1000
A
1200
A
1400
0
1600
Dato cf Brown et a1 (1970) Kyte et a1 (1953)
oa4P
d* A >
IV
10-1i
10-10
IO-*
10-9
10-7
IO-^
10-5
10-6
IO-'
10-2
ioi
ioo
io2
Gr Pr
Figure 1. Experimental and calculated Nusselt number for heat transfer NUHVS. GrPr.
I
I
1
1 1 1 1 l 1 [
I
I
I
I Ill1
I
I
I
I I Ill1
I
I 1 1 I I 1 1 .
Calculated NH3 flux
10''
///
/ / / loo
P
"3
1
/
(torr)
IO'
io3
lo4
Figure 2. Experimental reaction rate for NH3 decomposition on Pt wire vs. pressure at temperatures indicated. Mass transfer coefficients (dashed lines) a t 800 and 1600 K show the rate becomes influenced by mass transfer only for P > 10 Torr and T > 1500 K. Also shown is the calculated NH3 flux to the surface at 300 K. NUH~ 0 . 0 7Under . the same conditions, natural convection calculations predict a value for NUHten times larger. Heat transfer to wires under forced convection has been studied recently by Cardoso and Luss (1969) and by Rader and Weller (1974a,b) for values of Re numbers up to 15. Although there is no theoretical bound for Re numbers, very high values are not likely to be obtained in standard laboratory experiments. Furthermore, given the weak dependence of the mass transfer coefficient on Re (k, Re1/3), no substantial
-
increase in k , can be obtained by increasing Re. Rader (1974) estimates, elaborating on the Cardoso and Luss results, a value of k, 5 X lozomolecules cm-2 s-l for Re = 15.3 and a mass flow rate of 25 g ern+ min-I. Increasing the mass flow rate by a factor of lo4in the experiment of Cardoso and Luss, a value of k, = 8 X loz1 molecules cm-2 should be obtained, an increase of only an order of magnitude a t the unlikely mass flow rate of 250 kg cm-2 min-'. We conclude that under usual laboratory conditions k, for forced convection should also be Ind. Eng. Chem., Fundam., Vol. 16, No. 3, 1977
363
T = 1000 K D = 0.025cm
1-
P=tOO Torr Dz0.025crn
0.5
1.0
-I 1.5
2.0
io3 ( K - I I Figure 3. Mass transfer coefficient k, vs. temperature for several I T x
gas mixtures. I
T = 800 K
I
I
I
I
P (Torr)
Figure 5. Mass transfer coefficient k,
vs. pressure for
(X
several gas mixtures.
Da = k R 6 / a ) should be less than 0.1. For a first order reaction
10-2
to-1
Since values of r d k , > x j b are physically impossible, Da will vary between zero and infinity. W$en Da > 3, the reaction becomes mass transfer controlled. Any rate measured under these conditions should represent a measurement of k,. It is interesting to consider the apparent orders of reaction and activation energies that would be observed in mass transfer limited reactions and how rates might be affected by wire diameter D . Using the heat and mass transfer analogy and the correlation presented by Kyte et al. (1953), we can write the mass transfer coefficient for natural convection to horizontal wires as 100
D
2PN0 k, = RTD
D (cm)
Figure 4. Mass transfer coefficient k, vs. wire diameter D for several
(4)
-
gas mixtures.
Since D P-1T1.5 and since the logarithmic term will be rather insensitive to changes in GrSc, this becomes expected to be between lozoand 1021molecules ern+ s-l for most gases. Mass Transfer a n d Chemical Reaction When a chemical reaction occurs on a surface, the rate of mass transport to the reactive surface is in steady state equal to the chemical reaction rate. Thus for a given component j we have
kxj(xjb-xjs) (2) The transfer of reactants to the reactive surface and the chemical reaction correspond to two resistances in series. In order to measure r R , ( x i - x j s ) must be small; in other words, the mass transfer resistance must be small compared to the surface reaction resistance. We have shown (Loffler and Schmidt, 1975) that in order to measure reaction rates with only a 10%error due to mass transfer the Damkoeler number rRj =
364
Ind. Eng. Chem., Fundam., Vol. 16, No. 3, 1977
k,
-
(5)
TO.5D-l
Figure 3 shows that k, varies as -TO.5 at high temperatures, corresponding to an activation energy of E 1000 cal/g-mol at T 1500 K . Figure 4 shows that k, depends strongly on wire diameter, the exponent for D is close to -1 for D < lo-' cm; thus small diameter wires are more suitable for measuring rates of fast reactions. Correlations presented by McAdams (1954) for heat transfer to spheres and flat plates suggest that k, decreases with increasing characteristic length in spheres and foils. Finally, Figure 5 shows that k, depends weakly on pressure ( k , P0.3) for P < 1 atm. When the temperature difference between gas and catalyst is small, forced convection is the main mechanism for mass transfer. Following correlations presented by Bird et al. (1960), we can write
-
N
-
k,
,0.3TO.5D-O.7P-O.3
(6)
Table I. Reactions on Pt Wires with Mass Transfer Influence
Reaction
Author
Pressure, Torr
CzHs oxidation C3Hs oxidation C4H!O oxidation C4H10 oxidation C4H10 oxidation
Hiam et al. (1968) Hiam et al. (1968) Hiam et al. (1968)
760 760 760
Table 11. Reactions with Small Mass Transfer Influence
Temp, K Da 600 600 600
0.66 137 414
Hiam et al. (1968)
760
773
16.7
Rader and Weller
760
830
48
10
1500
(1974) "3
decomposition
Loffler and Schmidt
0.13
(1976a) "3
decomposition
Loffler and Schmidt
20
1500
0.22
(1976a)
Thus the mass transfer coefficient for forced convection will be slightly less dependent on wire diameter than in the case of natural convection. The exponent of the gas velocity u in eq 6 is verified by an elaboration of data by Cardoso and LUSS presented by Rader (1974). The negative exponent of wire diameter D can also be qualitatively confirmed from data presented by that author. Examples Tables I and I1 list some fast reactions on metal wires and films for which rate parameters have been published. The reactions in Table I have been reported to be strongly influenced by mass transfer resistances under the given conditions by the individual authors, while the reactions listed in Table I1 were not assumed to be affected by mass transfer resistances. Since our calculations predict Da > 0.1 for the reactions listed in Table I and Da < 0.1 for most reactions listed in Table 11, we conclude that experimental evidence seems to confirm the accuracy of our calculations. Hydrocarbon Oxidation on Pt. Catalytic oxidation of hydrocarbons are among the fastest heterogeneous reactions. Rates quoted in Table I have been obtained with hydrocarbon-oxygen mixtures a t atmospheric pressure in excess oxygen on wires. Under these conditions reactions are first order in hydrocarbon partial pressure. Rate expressions have been obtained by measuring the heterogeneous ignition temperature; a t the temperatures quoted in the table all these reactions are reported to be mass transfer controlled. Table I1 shows hydrocarbon oxidation rate data obtained on deposited metal films. In these systems mass transfer to the catalytic surface is due mainly to molecular diffusion and the rate of mass transfer will be a function of time as well as of composition and reactor geometry. We can roughly estimate a mass transfer coefficient k , averaged over a period of time At as 1 (C/CO) (7) At where C/CO is the ratio of mean concentration of reactant at time end and the beginning of the period. At can be estimated from the equations presented by Crank (1967). NH3 Decomposition on Pt. We have presented in a previous publication (Loffler and Schmidt, 1976a) a rate expression for NH3 decomposition on Pt wires. Dashed curves in Figure 2 correspond to this rate expression. For pressures
k , = -In
Reaction
Author
Para-H2 conversion on W wire Para-H2 conversion on Pt wire Ethylene oxidation on Pd film Ethylene oxidation on Pt film Propane oxidation on Pd film Propylene oxidation on Pd film CO oxidation on Pt wire
Patterson and Kemball
"3
Hinshelwood and Burk
Couper et al.
Temp, K Da
1.2
150 0.003
1.2
150 0.0004
60
465 0.01
26.3
390 0.008
26.3
390 0.0006
26.3
390 0.0002
1
000 0.04
200
500 0.44"
300
800 1.52O
(1958)
Couper e t al. (1958)
Moss and Thomas (1967)
(1963)
Patterson and Kemball (1963)
Patterson and Kemball (1963)
Hori and Schmidt (1975)
decomposition on Pt wire "3
Pressure, Torr
(1925)
Schwab and Schmidt,
decomposition (1929) on Pt wire a Significant mass transfer influence is predicted.
above 10 Torr the curves are extrapolated since our expression was tested experimentally for pressures between and 10 Torr. It is clear that at high pressures and temperatures T R approaches k,, and the reaction becomes mass transfer controlled. At low pressures, P < 1Torr, the mass transfer coefficient is orders of magnitude larger than the rate of reaction; thus in this pressure range mass transfer resistances seem to be unimportant. Deviations from the rate expression observed at P = 10 and P = 20 Torr have been attributed to the reverse reaction or to mass transfer resistances. The calculations shown in Table I show that only for P 2 20 Torr should mass transfer resistances have been significant. Shown in Table I1 are data from some experiments by Hinshelwood (1925) and by Schwab (1929). Both authors worked a t high pressures and temperatures, and our calculations indicate that mass transfer resistances should have severely affected their rate experiments since Da >> 0.1 in both cases. Summary Several approximations have been made in calculating mass transfer coefficients from heat transfer data. Ideal gas'behavior was assumed in all cases, Stefan flow and thermal diffusion were neglected, and diffusion coefficients were calculated ignoring components in excess of 2. The methods used to predict properties of gases are reported to have a maximum error of 20%; from data presented by Rader (1974) we estimate the heat and mass transfer analogy to have a 40% error under forced convection conditions. Although we have not measured mass transfer coefficients independently, indirect experimental evidence available in the literature agrees with our calculations. According to this, ?IOrates higher than lozomolecules cm-2 s-l can be measured Ind. Eng. Chem., Fundam., Vol. 16, No. 3, 1977
365
on catalytic wires of diameter cm < D < 10-l cm for gases other than hydrogen. Reaction rates up to 1021molecules cm+ s-' can be observed in reactions iiivolving H2 or its isotopes. In order to obtain rate data free of mass transfer resistances, reactions should be carried out under conditions such that no rates higher than 1019molecules cm-2 s-l (lozofor reactions involving H2) are obtained. Nomenclature c = total molar concentration, m0l/L3 YJ = binary diffusivity, L2/t D = wirediameter Da = Damkoeler number, k&YJ Gr = Grashof number, D3pg@AT/p2 h = heat transfer coefficient, M/t 3T k = thermal conductivity, ML/t 3T kH = reaction rate constant, t - l k, = mass transfer coefficient in a binary system, moleculesltL2 No = Avogadronumber NUH = Nusselt number for heat transfer, hD/k NUM = Nusselt number for mass transfer, k,D/cD P = pressure, M/Lt2 P r = Prandtl number, C p p / k Q = heat flux density, ML 2 / t3 R = gas constant, ML2/t2Tmol Re =Reynolds number, Dup/p r R = reaction rate, molecules/L2t Sc = Schmidt number, p / p B u = gas velocity, L/t x, = mole fraction of species j in gas bulk x,8 = mole fraction of species j in equilibrium with adsorbed j Greek Letters @ = thermal coefficient of volumetric expansion, T-'
6 = boundary layer thickness, L p = gas density, M/L3 p
= viscosity, M/Lt
Literature Cited Bird, R. B.. Stewart, W. E., Lightfoot, E.N., "Transport Phenomena," Wiley, New York, N.Y., 1960. Brown, F. T., Boo-Goh. Ong, Mason, D. M., J. Chem. Eng. Data, 15, 556 (1970). Cardoso, M. A. A.. Luss, D.. Chem. Eng. Sci., 24, 1699 (1969). Chim. Belg., Couper. A., Eley, D. D.. Hulatt, M. J., Rossington, D. R., Bull. SOC. 87, 343 (1958). Crank, J.. "The Mathematics of Diffusion," Clarendon Press, London, 1967. Hiam, L., Wise, H., Chikins, S..J. Cafal., 10, 272 (1968). Hinshelwood, C. N., Prichard, C. R., J. Chem. SOC. (London), 127, 327 (1925). Hinshelwood, C. N., Burk, R. E.,J. Chem. SOC.(London), 127, 1105 (1925). Hori, G. and Schmidt, L. D., J. Catal., 38, 335 (1975). Kyte, J. R., Madden, A. J., Piret, E. L., Chem. Eng. Prog., 49, 653 (1953). Langmuir. I., Phys. Rev., 34, 40 1 (19 12). Langmuir. I., Trans. Faraday SOC., 17, 621 (1921). Loffler, D. G., Schmidt, L. D., A.I.Ch.E. J., 21, 786(1975). Loffler, D. G., Schmidt, L. D., J. Catal., 41, 440 (1976a). Loffler, D. G.. Schmidt, L. D.. J. Cafal., 44, 244 (1976b). McAdams, W. H., "Heat Transmission," McGraw-Hill, New York, N.Y., 1954. Moss, R. L., Thomas, D. H., J. Catal., 8, 162 (1967). Nusselt, W. Z., 2. Ver. Deut. lng., 73, 1475 (1929). Patterson, W. R., Kembal, C., J. Cafal., 2, 465 (1963). Pignet, T. P., Schmidt, L. D., Chem. Eng. Sci., 29, 1123 (1974). Pignet. T. P., Schmidt, L. D., J. Catal., 40, 212 (1975). Rader, C. G., Ph.D. Thesis, State University of New York at Buffalo, 1974. Rader, C. G., Weller. S. W., A.l.Ch.E. J., 20, 515 (1974). Reid, R . C., Sherwood, T. K., "The Properties of Gases and Liquids, 2nd ed, McGraw-Hill, New York, N.Y., 1966. Schwab, G. M.. Schmidt, H., 2.Phys. Chem. (Leipzig), 38, 337 (1929).
Received f o r reuiew November 12, 1976 Accepted April 25, 1977
This work was partially supported by the National Science Foundation under Grant No. ENG 75-01918.
Stability of Centrifugally Stratified Helical Couette Flow Larry A. Glasgow and Rlchard H. Luecke' Department of Chemical Engineering, University of Missouri, Columbia, Missouri 6520 1
Helical Couette flow, stabilized by viscous and centrifugal forces, undergoes catastrophic transition at particular combinations of axial pressure gradient and angular velocity of the outer cylinder. The dependence of the limit of stability upon flow parameters is predicted with an inviscid simplification of the method of small disturbances. Experimental evaluations of a laboratory-scale Couette device verify the disturbance equation within the limitations of the apparatus.
Introduction The stability of tangential flow in the annulus of rotating concentric cylinders has been considered both theoretically and experimentally by scores of investigators over the past 85 years. The flow field studied in this paper, however, represents a substantial difference from that case in that an axial pressure gradient was imposed upon the angular velocity of the fluid. Thus, the fluid in the annulus is subject simultaneously to both tangential and axial velocity components; the resultant motion is, of course, helical. The addition of the axial velocity alters the impact of centrifugal stratification upon stability. 366 Ind. Eng. Chem., Fundam., Vol. 16, No. 3, 1977
We became interested in this type of flow as part of studies of the influence of hydrodynamic shear upon floc. There have been a number of studies of particle coagulation dynamics in Couette devices but none have been reported that include an axial component of flow. This flow is important to our work in that large representative samples may be analyzed externally for particle size distribution. In general, the work described in this paper should prove useful to investigators requiring a well-defined shear field within the context of a recirculating system. This type of configuration is often a requisite for viscometry, lubrication research, and particle aggregation studies.
