Enhanced Mass Transfer in Monolith Catalysts with Bumps on the

The use of promoters in monoliths for car exhaust has been investigated both .... Two mass flow controllers were used: 0−10 dm3/min (at 20 °C and 1...
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Ind. Eng. Chem. Res. 1999, 38, 2091-2097

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Enhanced Mass Transfer in Monolith Catalysts with Bumps on the Channel Walls Anna M. Holmgren† Department of Chemical Reaction Engineering, Chalmers University of Technology, SE-412 96 Go¨ teborg, Sweden

The mass-transfer rate in metallic triangular monolith channels with small protuberances on the channel walls was measured and compared with conventional, straight channels without bumps. The CO oxidation reaction was used to estimate the mass-transfer rates. Measurements were made at Reynolds numbers between 300 and 1400. The channels were 3.7 mm wide and 2.6 mm high and had a length of 96.5-152 mm. Two different bump heights were investigated: 1 and 1.2 mm. Moreover, the distance between the bumps as well as the distance from the inlet to the first bump was varied. It was found that the protuberances increase the mass-transfer rate, expressed as the Sherwood number. The increase was higher when the Reynolds number was increased. There was also a penalty of increased pressure drop. However, when the jD factor for mass transfer was compared with the friction factor f, it was found that the increase in mass transfer was generally higher than the increase in pressure drop. 1. Introduction The catalysts that clean the exhaust from vehicles are normally of the honeycomb-like monolith structure. This reactor design provides a large gas-solid contact area and a low pressure drop, with a minimum cost in engine power. The monolith substrate is either ceramic or metallic. It is coated with a thin layer of the so-called washcoat, which contains the catalytically active material. The ceramic substrates are most common, mainly because of their low manufacturing cost, but the metal monoliths are increasingly used. The best utilization of the metal monoliths is possibly as start-up catalysts, placed near the engine, where their high thermal conductivity and low thermal mass are advantageous. Furthermore, the corrugation technique of the metal monolith provides greater flexibility in design compared with the ceramic monolith, which is manufactured by extrusion. This flexibility will be exploited in the current investigation, concerning the presence of bumps in the channels of metal monoliths. The idea of introducing promoters into channel flow to improve heat- and mass-transfer properties is far from new; it has been tested or used in a wide range of applications. These include compact heat exchangers,1-7 gas-cooled nuclear reactors,8,9 gas turbine blade cooling,10 electrochemical reactors,11 electronic devices,12,13 and electrodialysis.14 The promoters have been used for both laminar and turbulent channel flows. The forms of the promoters are, e.g., ribs,8,12 cylindrical wires,11 baffles,7 wings or winglets,4-6 or corrugated or sinusoidal channel shapes.3,15,16 Such promoters have been found to improve the heat- and mass-transfer rates by up to several hundred percent. The disadvantage is the increased pressure drop, which raises the demand for pumping power. The roles of the promoters are (i) to disturb the boundary layers, (ii) to generate swirls and vortices, and †

Present address: Caran Automotive AB, Box 5445, SE402 29 Go¨teborg, Sweden. Phone: +46-31-335 60 04. Fax: +46-31-335 97 12. E-mail: [email protected].

(iii) to destabilize the flow or to intensify the turbulence.17 For fully turbulent flows, the first point is perhaps the most important, and the promoters are often placed close to the channel walls. It has been established that promoters may significantly reduce the critical Re number at which transition to turbulent flow takes place. Critical numbers of about 1000-120015 or even below 7003 have been reported, compared with about 2000, which is the value for flow in straight, smooth ducts. Already at Re of about 250-300, obstacles have been found to produce nonsteady disturbances in the flow.14,18 Most of the promoters cause a reduction in channel area. The mass/heat transfer rate is then also increased by the increase in flow rate due to the decreased channel area. An observation that is interesting for car-exhaust catalysts, which are exposed to unsteady and pulsed flow, is that the combination of channel promoters and pulsed flow has been found to be particularly beneficial for high heat- and masstransfer rates at some conditions.16,18,19 The use of promoters in monoliths for car exhaust has been investigated both computationally and experimentally. Andersson and Scho¨o¨n20 did computational fluid dynamics calculations of the CO conversion and the pressure drop in a sinusoidal-shaped monolith channel with two protuberances in the form of parallelepipeds; see Figure 1a. They obtained an improvement in the amount of converted CO by up to 50%, although at a large pressure drop penalty of up to 432%. Bella et al.21 did similar calculations for square and trapezoidal obstacles in a monolith channel. They too observed a higher mass-transfer rate for the channels with obstacles compared to those with a straight channel, mainly because the flow was directed toward the walls. The idea of disturbing the flow by wall inclinations has been commercialized in the Emitec TS, which is a metal monolith with triangular channels with small, rounded corrugations;22 see Figure 1b. These corrugations have been found to improve the mass-transfer rate by about 15%, but at the same time the pressure drop was increased by 14-18%.23 Finally, Figure 1c shows the

10.1021/ie980597f CCC: $18.00 © 1999 American Chemical Society Published on Web 04/15/1999

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Figure 1. Some of the shapes of obstacles in monolith channels taken from the literature: (a) a parallelepiped shape suggested by Andersson and Scho¨o¨n; (b) commercial Emitec TS; (c) Behr MS.

Behr MS design, in which the channels are cut and displaced to generate repeated inlet disturbances of the flow.22 The present work consists of a first investigation of a new type of corrugated metal monolith, the EcoCat,24 which contains bumps to improve the mass-transfer properties in the channels. The mass transfer and pressure drop of this new type of monolith are investigated experimentally. 2. Experimental Methods Catalyst Preparation. The monolith is made from two types of metal sheets: one with grooves and the other corrugated and with “tongues” at the positions of the grooves to fit the grooved sheets. The whole monolith construction is shown in Figure 2a. The shapes of the metal foils are more clearly seen in Figure 2b. Figure

2c shows a cross section of the channel, viewed perpendicular to the flow direction. Figure 2d shows the cross section of two channels over the passage of a bump, viewed in the flow direction. This figure illustrates that, when the gas passes a bump, there is a slight increase or decrease in the cross-sectional area, depending on whether the protuberance is facing upward or downward at the base of the triangular channel. For the purpose of accurate measurements, test monoliths were constructed, consisting of one corrugated and two grooved foils. The test monoliths have six channels, of which four were plugged. In this way, an even flow distribution between the two open, central channels was ensured. Six such metallic monolith samples were prepared: five with bumps and one without. The dimensions of the samples are given in Table 1. The effective channel length, L*, includes the extra length that the gas passed because of the bumps. For all monoliths, the nominal channel height, channel width, bump length, and bump angle were h ) 2.6 mm, b ) 3.7 mm, a ) 2 mm, and θ ) 45°, respectively; see Figure 2c,d. The actual open area was measured and is also given in Table 1. The metal foil parts for the test monoliths were heated in air at 950 °C for 2 h to improve the washcoat adhesion. They were then washcoated with a slurry of 1% Pt on γ-alumina (Aldrich), mixed with alumina binder (Disperal), deionized water, and a disperser (Dispex A40). The mass proportions were 27.2% Pt on γ-alumina, 2.7% Disperal, 69.8% water, and 0.3% Dispex A40. After the washcoating, the monoliths were dried in air at 550 °C for 1 h. The resulting washcoat loading was 4.71-4.79 mg/cm2 of metal surface area. Reactor System. The monolith was placed in a cylindrical steel tube inside an oven. Two thermocouples measured the temperatures at the inlet and outlet. Two pressure taps in the reactor wall were connected to a U-tube ethanol manometer, which could measure pressure drops over the monolith in the range of 0-11.6 kPa, with an accuracy of (5 Pa. Two mass flow controllers were used: 0-10 dm3/min (at 20 °C and 1 atm) of air and 0-0.1 dm3/min of CO. The concentration of CO in the outflow was measured by an IR detector (ADC) with a relative accuracy of (1%. Activity and Pressure Drop Measurements. In the CO oxidation experiments, a mixture of 0.5% CO in air was passed through the two monolith channels. The CO conversion was measured at four different flow rates: 3, 5, 7, and 10 dm3/min (at 20 °C, 1 atm), corresponding to total mass flow rates of 0.0595, 0.0991, 0.1387, and 0.1982 g/s, respectively, and at three different temperatures of 300, 400, and 500 °C. For a full-scale cylindrical monolith with a diameter of 0.1 m, these flow rates correspond to total exhaust flow rates from the engine of 160-530 kg/h, which are typical exhaust flow rates for a car that is run at moderate to high speeds. Rather high flow rates were chosen for the

Table 1. Dimensions of the Monolith Samples

sample

channel length, L (mm)

effective channel length, L* (mm)

channel open area (mm2)

no. of bumps

distance from the inlet to the first bump, L1 (mm)

distance between the bumps, L2 (mm)

bump height, e (mm)

A B1520-1 B1520-1.2 B1520-1.2c B1530-1.2 B3130-1.2

152 150 150 96.5 150 150

152 153.3 154.0 97.5 154.0 154.0

4.24 4.42 4.72 4.46 4.49 4.45

0 4 4 4 4 4

15 15 15 15 31

20 20 20 30 30

1 1.2 1.2 1.2 1.2

Ind. Eng. Chem. Res., Vol. 38, No. 5, 1999 2093

Figure 2. Monolith structure investigated in the present study: (a) photograph of the full-scale monolith; (b) construction of the metal foils in detail; (c) cross section of a channel, viewed perpendicular to the flow direction; (d) cross section of two channels at the passage of the bump, viewed in the flow direction.

present study because a high mass-transfer rate is most important at high flow rates, for which a high conversion of pollutants is most difficult to achieve. The flow rates that were used resulted in Re numbers in the channels of 300-1400. These Re numbers are somewhat higher than what is normally encountered because the diameter of the channels is quite large. The pressure drop was measured at the above 12 conditions and also at 25 °C. On monoliths A and B15201, activity and pressure drop measurements were also carried out at 600 and 700 °C. Repeated runs were performed to determine the experimental deviations. The CO conversion and pressure drop without catalyst were also measured at all flow rates and temperatures and subtracted from the measurements with catalyst. 3. Theory

Hayes and Kolaczkowski suggested a value of Rxc above 100 for a reaction to be mass-transfer-controlled. Although our measurements were not carried out at fully laminar conditions, for which this criterion is valid, it seems as a fair approximation to assume complete masstransfer control. Details about these calculations are given in the Supporting Information. The mass transfer in the channel can be described by a film model. A simple mass balance of CO over the channel then implies, with the boundary condition of zero CO concentration at the wall, that the average film mass-transfer coefficient kc can be calculated from the conversion of CO according to

kc ) -

q ln(1 - conversion) A

(2)

Mass Transfer from Chemical Activity Measurements. Hayes and Kolaczkowski25 set up a criterion to determine whether a first-order reaction is mass-transfercontrolled or kinetically controlled. Because the CO concentration was low, oxygen was in large excess (O2: CO of about 40), and the temperature was above 300 °C, we may consider the CO oxidation reaction to be first order with respect to CO, which was also verified by varying the CO concentration. For our measurements, the value of the catalytic reaction number

The flow rate q was assigned a constant value through the channel, calculated from the average temperature between the channel inlet and outlet. This approximation was considered valid because the difference between the inlet and outlet gas temperatures was less than 25 K in all experiments. The average Sherwood number is then calculated as

Rxc ) kwdh/D

The diffusivity of CO in air, D, was calculated from Fuller et al.28 according to Hayes and Kolaczkowski:25

(1)

was approximately between 120 and 20 000 when using the rate expression for CO oxidation given by Voltz et al.26 with the parameters from Oh and Cavendish.27

Sh ) kcdh/D

T1.75 D ) 9.2635 × 10-5 P

(3)

(4)

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Figure 3. Sh versus Re × Sc(dh/L*) for monoliths B1530-1.2, B1520-1.2, B3130-1.2, B1520-1, B1520-1.2c, and A. Hawthorn’s correlation for laminar flow is also indicated.

Figure 4. Pressure drop per channel length at 400 °C versus mass flow rate. The lower curve represents the theoretical values for laminar flow.

Pressure Drop. The Fanning friction factor for flow in conduits is29

Sh of the straight monolith A as well as Hawthorn’s correlation for equilateral triangular channels32

f)-

∆P dh 2L Fu2

(5)

For fully developed laminar flow, f depends on Re according to

f)

C1 Re

(6)

where the constant C1 depends on the duct geometry.30 For equilateral channels, C1 equals 13.33, whereas for the geometry of our samples, C1 is approximately 13.31. Combining eqs 5 and 6 gives for the pressure drop of laminar flow in ducts

µu ∆P/L ) -2C1 2 dh

(7)

In a later section, we will study the temperature dependence of the pressure drop. For a constant mass flow rate, the temperature dependence can be expressed as R

T (293 )

∆P(T) ) ∆P(293 K)

(8)

By using eq 7, the ideal gas law, and the temperature dependence of the viscosity of air between 20 and 500 °C,31 one obtains R ) 1.72 for laminar flow. For fully developed turbulent flow in circular ducts, the friction factor depends on Re according to29

f ) 0.0791/Re0.25

(9)

which, together with eq 5, the ideal gas law, and the temperature dependence of the viscosity of air, gives R ) 1.18 for turbulent flow. 4. Results Mass-Transfer Rate. The measured CO conversion over the different monoliths at different experimental conditions was between 81.2 and 99.6%. In Figure 3,

(

)

dh Sh ) 2.35 1 + 0.095Re × Sc L*

0.45

(10)

is shown. The measured Sh are higher than what the above correlation predicts and have a more exponential Re dependence. A regression analysis, testing different two-parameter models, gave the best fit with the following expression:

(

)

dh Sh ) 2.82 exp 0.056Re × Sc L*

(11)

The individual, approximate, 95% confidence intervals of the parameters were 2.82 ( 0.20 and 0.056 ( 0.008, and the correlation between the two parameters was -0.93. In Figure 3, Sh as a function of Re × Sc(dh/L*) is also given for the monoliths with bumps. It is seen that all of the monoliths with bumps have significantly higher Sh than the straight monolith A. The difference between the samples with bumps is quite small. Monoliths B1530-1.2, B1520-1.2, and B3130-1.2 have the highest mass-transfer rates. The similar Sh values of these samples indicate that the distance between the bumps is not an important factor. In contrast, the lower bump height of sample B1520-1 seems to be a disadvantage because this sample has lower Sh than B1520-1.2. The cut monolith B1520-1.2c reaches the highest Sh, but because Sh is plotted against the inverse of the channel length, its short length makes it less efficient for a given value of the group Re × Sc(dh/L*). Pressure Drop. The cost of the increased mass transfer is the increased pressure drop. Figure 4 shows the pressure drop per channel length at 400 °C as a function of the mass flow rate per channel, G. The theoretical curve is given as a comparison, according to eq 7 and C1 ) 13.31. The pressure drop of the straight monolith A corresponds well to the theoretical value at the lowest mass flow rates but is higher than predicted for higher flow rates. This indicates that there are significant turbulent features in the flow at higher flow rates, corresponding to Re above 400. The pressure drop is generally higher over the monoliths with bumps compared with the straight monolith A, although the increase is not dramatic. The cut monolith B1520-1.2c

Ind. Eng. Chem. Res., Vol. 38, No. 5, 1999 2095 Table 2. Temperature Dependence of the Pressure Drop, Given as the Exponent r in Equation 8 G (g/s) sample

0.0298

0.0496

0.0693

0.0991

A B1520-1 B1520-1.2 B1520-1.2c B1530-1.2 B3130-1.2

1.46 1.38 1.45 1.34 1.39 1.40

1.36 1.26 1.29 1.22 1.25 1.28

1.23 1.27 1.28 1.21 1.16 1.25

1.14 1.29 1.24 1.18 1.16 1.22

has the highest pressure drop per channel length, which is expected because it has the greatest number of protuberances per channel length. Still, the pressure drop over the cut monolith B1520-1.2c is unexpectedly high, almost twice as high per channel length as over the full-length monolith B1520-1.2. This is probably due to the fact that the cutting deformed the back of the monolith, which caused contraction of the flow and an accompanying pressure drop. The bump height does not dramatically affect the pressure drop; monoliths B15201.2 and B1520-1 have about the same pressure drop. It is even somewhat higher for B1520-1. One should note, however, that monolith B1520-1 has a smaller channel diameter (see Table 1) which will increase the pressure drop. Temperature Dependence of the Pressure Drop. To obtain information about the degree of turbulence in the monolith channels, we may investigate the temperature dependence of the pressure drop. In a previous section, it was shown that for fully developed laminar flow R in eq 8 should equal 1.72, and for natural turbulent flow it should equal 1.18. A nonlinear regression fit to eq 8 gave R values according to Table 2. The R values measured in this study are between 1.14 and 1.46, indicating that the flow has significant turbulent properties for all channels, straight as well as with bumps. The pressure drop dependence on the temperature is strongly correlated with the mass flow rate but does not vary much between the different samples. This indicates that the bumps do not generate any real natural turbulence and that it is mainly the direction of flow toward the active wall that causes the increased mass-transfer rate. Mass Transfer and Pressure Drop Trade-Off. A way to study the increased mass transfer and the pressure drop penalty is to compare the Chilton-

Colburn mass-transfer factor jD with the Fanning friction factor f. For fully turbulent flow at Re >10 000 in long, smooth pipes, the following empirical correlation is valid:29

jD ) f/2

(12)

We therefore plot jD/(f/2), which thus for turbulent flow should equal 1, versus Re. This is done in Figure 5. The strong correlation between the pressure drop and the mass-transfer rates is seen by the moderate gradients of the curves in Figure 5. None of the monoliths reaches the value of one for turbulent flow, but the monoliths with bumps generally have higher jD/(f/2) ratios, and they are thus more efficient than the monolith without bumps. The most efficient choice seems to be monolith B1530-1.2, which combines a high mass-transfer rate with a low pressure drop. 5. Discussion This paper presents the first experimental investigation of a new monolith structure with improved masstransfer properties. The experiments were done on a laboratory scale, using a small monolith, steady flow, and a simple test reaction. The main uncertainties with the experiments are that, when only two channels are used, the exact shape of these channels and their packing is crucial to the results. Still, the results seem reasonable, in that the Sherwood numbers and pressure drops of the straight monolith are relatively close to values from the literature, at least at low Re. Moreover, the results from the monoliths with bumps are similar to each other. The differences that occur can be related, for instance, to the bump height. The part of the results that may seem most surprising is the high degree of turbulence that seems to prevail in all monoliths, even in the straight monolith. This result is seen when studying the temperature dependence of the pressure drop in Table 2. The degree of turbulence is strongly dependent on Re but is not significantly higher in the channels with bumps. This indicates that other disturbances of the flow are important already at Re of about 300. As Re increases, these disturbances survive longer, which gives increased turbulent properties of the flow. The disturbances may be caused by the expansions and contractions in the

Figure 5. Gain in mass transfer relative to the loss in pressure drop: jD/(f/2) versus Re for (a) monoliths B1530-1.2, B1520-1.2, and A and (b) monoliths B1520-1, B3130-1.2, and B1520-1.2c. The dotted line at jD/(f/2) ) 1.0 represents the expected value for fully developed turbulent flow (only valid for Re > 10 000).

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cross section in front of the monolith samples. The Reynolds numbers here are between 170 and 1960. There are also disturbances when the flow enters the channels and from the surface roughness of the washcoat. Because the bumps do not seem to generate turbulence to any large extent, it is likely that their main role is to disturb the boundary layers and to generate vortices that direct the flow toward the walls, thereby achieving an enhanced mass transfer. Such an effect on the mass transfer by obstacles has been seen in earlier investigations.21,33 The high degree of turbulence that seems to be present in the flow implies that conventional fluid dynamic calculations, assuming laminar flow, are dubious. Examples of such calculations for monolith channels are available.20,21,34 These are likely to underestimate the real mass-transfer rates. In accordance, Holmgren and Andersson34 measured higher masstransfer rates than the simulated ones. Simulations of flows that undergo transition to turbulence seem to require special methods; see, for instance, Johnson et al.35 Alternatively, direct numerical simulations, resolving all scales of the flow, can be made. This is, however, for almost all practical applications, still too timeconsuming for today’s computers. The gain in mass transfer of the channels with bumps is most significant at high flow rates. This is very valuable, because the conversion normally drops dramatically as the flow rates reach high values, for example, during high-speed driving on the German Autobahn. Such high flow rates are presently not included in the standard FTP test. An alternative solution to the problem of low conversion at high flow rates is to increase the mass-transfer rate by decreasing the channel diameter. However, in addition to an increased demand in active surface area, this will cause a linear increase in pressure drop as well as in masstransfer rate. The monoliths with bumps that are investigated in the present study have the advantage of increasing the mass transfer more than they increase the pressure drop. For the future, measurements on larger, full-scale monoliths of the type presented here are necessary. Also, measurements with pulsed flow from a real engine are of interest, because it is possible that the effect of the bumps is different in steady and pulsed flow. Pulsed flow has been found to increase the positive effect of promoters in channel flow.16,18,19 Finally, slightly different geometries will have to be tested to evaluate the optimal design in mass transfer/pressure drop properties for each flow rate. 6. Conclusions Bumps on the channel walls of metallic monoliths were found to significantly increase the mass-transfer rate. The increase is largest at the highest flow rates. The bumps also increase the pressure drop, but the increase in mass transfer is greater than the increase in pressure drop. Thus, the new catalyst presents a promising alternative to today’s metallic and ceramic monolith catalysts. Acknowledgment The author thanks Dr. Sven M. Nilsson for providing the catalyst samples and for numerous valuable discussions and suggestions.

Supporting Information Available: Details of calculations on the estimation of the degree of masstransfer control. Ordering information is given on any current masthead page. Nomenclature A ) active wall surface area, m2 C1 ) geometry-dependent constant for the relation between f and Re dh ) hydraulic diameter, 4 (cross-sectional area)/(wetted perimeter), m D ) diffusivity, m2/s G ) mass flow rate per channel, g/s kc ) gas-solid mass-transfer coefficient, m/s kw ) reaction rate coefficient based on the washcoat volume, m/s L ) channel length, m L* ) effective channel length, m P ) pressure, Pa ∆P ) pressure drop, Pa q ) flow rate, m3/s T ) temperature, K u ) average linear velocity, m/s Dimensionless Groups f ) Fanning friction factor, f ) -(∆P/2L)(dh/Fu2) jD ) Chilton-Colburn factor for mass transfer, jD ) Sh/ (Re × Sc1/3) Re ) Reynolds number, Re ) Fdhu/µ Rxc ) catalytic reaction number, Rxc ) kwdh/D Sc ) Schmidt number, Sc ) µ/FD Sh ) Sherwood number, Sh ) kcdh/D Greek Letters R ) exponent for the temperature dependence of the pressure drop µ ) dynamic viscosity, Pa s F ) density, kg/m3

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(24) Nilsson, S. M. Turbulence Inducer in Chemical Reactor. Patent PCT Int. Appl. WO 9721489 A1, 1997. (25) Hayes, R. E.; Kolaczkowski, S. T. Mass and Heat Transfer Effects in Catalytic Monolith Reactors. Chem. Eng. Sci. 1994, 49 (21), 3587. (26) Voltz, S. E.; Morgan, C. R.; Liederman, D.; Jacob, S. M. Kinetic Study of Carbon Monoxide and Propylene Oxidation on Platinum Catalysts. Ind. Eng. Chem. Prod. Res. Dev. 1973, 12 (4), 294. (27) Oh, S. H.; Cavendish, J. C. Mathematical Modeling of Catalytic Converter Light-Off. Part II. Model Verification by Engine-Dynamometer Experiments. AIChE J. 1985, 31 (6), 935. (28) Fuller, E. N.; Schettler, P. D.; Giddings, J. C. A New Method for the Prediction of Gas-Phase Diffusion Coefficients. Ind. Eng. Chem. 1966, 58, 19. (29) Bird, R. B.; Stewart, W. E.; Lightfoot, E. N. Transport Phenomena; Wiley: New York, 1960. (30) Shah, R. K.; London, A. L. Laminar Flow Forced Convection in Ducts; Academic Press: New York, 1978. (31) Weast, R. C., Astle, M. J., Beyer, W. H., Eds. Handbook of Chemistry and Physics; CRC Press: Boca Raton, FL, 1986. (32) Hawthorn, R. D. Afterburner CatalystssEffects of Heat and Mass Transfer between Gas and Catalyst Surface. AIChE Symp. Ser. 1974, 70 (137), 428. (33) Suzuki, K.; Suzuki, H. Instantaneous Structure and Statistical Feature of Unsteady Flow in a Channel Obstructed by a Square Rod. Int. J. Heat Fluid Flow 1994, 15, 426. (34) Holmgren, A.; Andersson, B. Mass Transfer in Monolith CatalystssCO Oxidation Experiments and Simulations. Chem. Eng. Sci. 1998, 53 (13), 2285. (35) Johnson, C.; Rannacher, R.; Boman, M. On Transition to Turbulence and Error Control in CFD. Preprint NO 1994:26, ISSN 0347-2809, Department of Mathematics, Chalmers University of Technology, 1994.

Received for review September 17, 1998 Revised manuscript received February 12, 1999 Accepted March 2, 1999 IE980597F