Operational and Structural Nonidealities in Modeling and Design of

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Ind. Eng. Chem. Res. 1997, 36, 3140-3148

Operational and Structural Nonidealities in Modeling and Design of Multitubular Catalytic Reactors Andrzej Cybulski* CHEMIPAN, R&D Laboratories, Polish Academy of Sciences, Kasprzaka 44/52, 01-224 Warszawa, Poland

Gerhart Eigenberger† Institut fu¨ r Chemische Verfahrenstechnik, Universita¨ t Stuttgart, Bo¨ blinger Strasse 72, 70199 Stuttgart, Germany

Andrzej Stankiewicz‡ DSM Research, P.O. Box 18, 6160 MD Geleen, The Netherlands

Idealized mathematical models for multitubular packed-bed reactors, based upon simplifying assumptions concerning symmetry, regularity, and homogeneity of the packing and the intertubular space, are commonly used for analysis and design. In reality, substantial deviations from these assumptions occur and give rise to maldistributions, both for the flow of reactants through a packed tube and for the flow of a heat-transfer medium through the intertubular space. These maldistributions and methods of how to deal with them in reactor design and simulation are reviewed in this paper.The following areas are covered: (i) porosity and flow patterns inside tubes, (ii) heat and mass transfer inside tubes, (iii) flow distribution over individual tubes, and (iv) distribution of coolant flow over the intertubular space. 1. Introduction The catalytic fixed-bed reactor is the dominating reactor type for large-scale, heterogeneously catalyzed gas-phase reactions. In case of strongly exothermic or endothermic reactions, the multitubular arrangement with catalyst-filled tubes and a liquid heat-transfer medium pumped around the tubes is the standard design (Froment and Bischoff, 1979; Eigenberger, 1992, 1996). For analysis, scale-up and design of fixed-bed reactors, mathematical modeling and simulation is one of the most useful tools. The hitherto developed “ideal” models describing phenomena occurring inside tubes in multitubular reactors, as classified first by Froment (1962, 1967, 1972, 1979), have been based upon simplifying assumptions such as homogeneous and isotropic packing of catalyst particles and plug flow of the reaction mixture within all individual tubes of the whole reactor. Moreover, uniform conditions of heat exchange between the tubes and a coolant flowing through the intertubular space are commonly assumed. In reality, maldistributions of all kinds occur in multitubular reactors. This gives rise to a number of problems when the commonly accepted mathematical models are used for design and analysis of industrial fixed-bed reactors. A general observation is that idealized models cannot represent the reactor behavior quantitatively, if independently determined, intrinsic kinetics and transport parameters are used (Hofmann, 1979a,b; Eigenberger and Ruppel, 1986). There have been many attempts to identify key problems and to refine and work out models accounting for nonuniformities, while preserving Froment’s concept of modeling based upon intrinsic and independently determined kinetics. In this paper, the inference of maldistributions in multitubular reactors and methods to deal with non* E-mail: [email protected]. † E-mail: [email protected]. ‡ E-mail: [email protected]. S0888-5885(96)00596-9 CCC: $14.00

uniformities will be reviewed. The following problem areas are discussed: porosity and flow patterns inside the tube, heat and mass transfer inside the tubes, flow distribution over individual tubes, and distribution of the coolant’s flow over the intertubular space. 2. Transport Phenomena inside the Tubes The present models of catalyst-filled reactor tubes assume a deterministic behavior based upon a regular and axisymmetric structure of the catalyst packing. This holds both for cell models which try to mimic the sequence of pellets and voids and for one- or two-phase continuous models where a continuous variation of temperature and concentration is assumed. In reality, a fixed bed is a random arrangement of nonuniform pellets, and the nonuniformity is apparent on each level of scale, from the active-site distribution in single pores over the pore distribution in single pellets to the arrangement of individual pellets in axial, radial, and circumferential directions in a single tube. This has led to the question whether deterministic models are able at all to describe the behavior of such a system with a sufficient degree of detail. The answer has been given by industrial practice, which showed that a carefully designed and operated multitubular fixed-bed reactor of some 20 000 parallel tubes has indeed about the same performance as a single test tube which was used for its design. The obvious explanation is that nonuniformities tend to level out even in single-catalyst tubes and for sensitive reactions like high-conversion partial oxidations if conversion and selectivity are considered not pointwise but over a larger reactor volume. A significant difference between the performance of a single tube and that of a multitubular reactor appears when the reaction mixture and/or the coolant are distributed nonuniformly. The main reasons of flow maldistributions in multitubular reactors and the means to deal with maldistributions are discussed in sections 3.1 and 3.2 of this paper. © 1997 American Chemical Society

Ind. Eng. Chem. Res., Vol. 36, No. 8, 1997 3141

Figure 1. Packing of particles in the core of the bed (a) and at the wall (b).

Figure 2. Void fraction distribution for spherical particles.

Conversely, it has to be stated that point measurements in catalyst-filled tubes will always show a large degree of scattering and that only values which are averaged over a certain volume of the tube can be reasonably compared with the simulation results from deterministic models. With the above restrictions in mind, we will concentrate in the following on homogeneous one- or two-phase models since they seem to offer the best compromise between computational simplicity and the ability to incorporate additional modeling details (see sections 2.1 and 2.2 of this paper). 2.1. Flow Maldistribution and Exothermic Reaction. Considering the random flow-through catalystfilled tubes, there is one structural feature which is apparent from all measurements. It stems from the ordering influence of the tube wall. This is well seen in Figure 1 where the black areas correspond to void spaces between the particles; the volume fraction of those spaces is obviously larger at the tube wall. This difference is structural in character. It is superimposed by random nonuniformities which originate from the fact that the geometry and/or the filling procedure do not allow for a regular arrangement of pellets so that pellet clusters and larger voids are formed randomly. Averaging over a certain distance, however, gives a surprising regular and reproducible radial void distribution. The void fraction oscillates, decreasing from unity at the wall to a damped porosity of ca. 0.4 at a distance of 4 particles from the wall as shown in Figure 2 where the well-known experimental data of Benenati and Brosilov (1962) and of Schuster and Vortmeyer (1980) are given. The broken line is an approximation of the void fraction profile calculated from an equation derived by Chandrasekhara and Vortmeyer (1979). The mini-

mum voidage (ca. 0.2-0.25) is located at a distance of half a particle diameter from the wall. A voidage profile with a pronounced decrease of the voidage toward the tube center is characteristic for packings with more than 4-5 particles per tube diameter. The decrease is most pronounced for spheres and least for Raschig rings. For less than 3 particles per diameter, a voidage maximum in the tube center may develop due to a kind-of spiral arrangement of the pellets along the tube wall (Vortmeyer, 1996). A voidage profile with a maximum at the wall must give rise to a maximum of the axial flow profile close to the wall. Presumably, Schwartz and Smith (1953) were the first who found that the radial flow pattern in narrow tubes of catalytic reactors is nonuniform. The velocity profile is affected by the particle shape. Figure 3 shows measurements below the packed bed for spheres and Raschig rings for a sequence of bed repackings (Bey and Eigenberger, 1996). For each radial position, point measurements have been carefully averaged along the circumference. The differences after repacking result from the aforementioned unavoidable degree of scatter; the solid lines give an averaged profile. It is important to note that the maximum close to the wall which dominates the radial nonuniformity of the flow can be measured with the least degree of scattering. This is a direct consequence of averaging a large number of circumferential point measurements close to the wall. Measurements below or above packed beds like Figure 4 give only a qualitative indication of the nonuniform flow inside the packing since they are falsified by the rapid reorientation of the gas flow after the bed exit. So far, no reliable gas-flow measurements inside packed beds could be performed. The only possibilities are liquid-flow measurements with LDA in transparent column packings with a carefully adjusted refractory index of liquid and packing material (McGreavy et al., 1986; Vortmeyer, 1996). Since these measurements are delicate and time-consuming, published results are not yet available to a sufficient extent. Research has therefore been focused on a simulation of flow inside the packing based upon the independently measured voidage profiles and a consideration of flow reorientation behind the packing. The pioneering work has been performed by Vortmeyer and co-workers (Vortmeyer and Schuster, 1983; Haidegger et al., 1989), who introduced and used the extended Brinkmann equation (Brinkman, 1947). The Brinkmann equation consists of a momentum balance based upon the two-dimensional NavierStokes equation where the fluid-particle interactions are described in a semiempirical way through an Erguntype pressure drop relation. Vortmeyer’s approach has been adopted and somewhat modified by Daszkowski (1991), Daszkowski and Eigenberger (1992), and Bey and Eigenberger (1996). Figure 4 from the last mentioned publication shows simulated flow profiles inside a packing of spheres (solid lines) and below the packing (dashed lines). It can be seen that the velocity maximum of the predicted flow profile inside the packing exceeds the mean velocity by more than a factor 2. Similar results have been obtained for other packing materials. A procedure has been given of how to calculate the radially varying velocity profiles inside the packing from voidage profiles for random packing of spheres, cylinders, and rings. The problems of geometry and flow in packed beds have also been treated and reviewed by Zio´łkowska and Zio´łkowski (1988). Readers interested in more details are also referred to the book by Staneˇk (1994).

3142 Ind. Eng. Chem. Res., Vol. 36, No. 8, 1997

Figure 3. Velocity profiles measured 5 mm below a packing after repeated repackings (symbols) for an mean empty tube velocity of v0 ) 0.5 m/s. The solid lines represent the model predictions (from Bey and Eigenberger (1996)).

Figure 4. Simulated flow profiles inside (solid lines) and below the packing (circles) for spheres, dp ) 7.5 mm, at v0 ) 0.5 m/s (from Bey and Eigenberger (1996)).

The radial variation of flow velocity has originally been considered to be of minor importance for pure heattransfer applications. It was obvious early, however, that it should have a great influence on fixed-bed reactors with strongly exothermic reactions, cooled through the wall. If the gas flows preferentially along the wall, the residence time will be short where the cooling is best, resulting in low conversion close to the wall. Conversely, a long residence time in the poorly cooled tube center gives rise to high conversion with much more pronounced hot spots as compared to plug flow. This general behavior has been found by Choudhary et al. (1976), who introduced the concept of a two-region model, dividing the bed into a central core with the annular space between the core and the wall with porosities and, hence, velocities differing in both sections. A two-region model was also used by Borkink and Westerterp (1992). Lerou and Froment (1977) accounted for the distribution of the superficial velocity in calculations of a reactor for the partial oxidation of o-xylene. It was heuristically assumed that the velocity is inversely proportional to voidage and the effective heat conductivity consists of a static contribution depending upon voidage and a dynamic contribution depending upon velocity. The porosity profiles of Benenati and Brosilov (1962) were used in the calculations. The wall heat-transfer coefficient appeared to be higher than that for plug flow, while the radial thermal conductivities were lower. Peak temperatures in the cases of the radial velocity profiles were higher than those in plug flow, and the reactor model was more sensitive to the operating conditions. A similar influence was predicted by Eigenberger and Ruppel (1986). Delmas and Froment (1988) extended the model of Lerou and Froment to the dependence of the effective diffusivity on radial position. They found considerably different reactor performances for a number of realistic models. The differences were particularly pronounced for small Dt/dp ratios. A comparison of the model results based upon the Brinkmann equation with detailed

measurements was presented by Kalkhoff and Vortmeyer (1980), Daszkowski and Eigenberger (1992), and Vormeyer and Winter (1984). They proved the abovementioned differences between plug flow and radially varying flow models and showed that the latter lead to a considerably better agreement between experiment and simulation. However, most of the existing correlations between heat-transfer coefficients and flow conditions have been derived based upon plug flow. As such, these correlations should not be used for models accounting for nonuniform radial flow. New correlations for such models are needed. 2.2. Mass and Heat Transfer in the Bed. There are three levels where heat and mass transfer cause temperature and concentration gradients: the intraparticle level, where differences in temperatures and concentrations inside the particle exist; the interphase level, where differences in temperatures and concentrations between the bulk of the fluid and the catalyst surface exist; and the intrareactor level, where differences in temperatures and concentrations appear between different radial and axial positions in the bulk of the fluid or the catalyst particles. At all three levels, inhomogeneities in mass and heat transfer exist which are due to the inhomogeneity of particle sizes, shapes, pore structure, and packing as well as flow maldistribution around particles and over the entire bed. In this paper, nonidealities at the third level will be discussed. Generally, axial heat and mass dispersion can be neglected if the axial aspect ratio, L/dt, is greater than, say, 50, and this is usually the case for industrial reactors. Radial mass transport is usually considered to be fast enough to justify the plug-flow assumption. Therefore, the scope of this paper is limited only to inhomogeneities influencing heat transport over the cross section of tubes. Radial transport of mass and heat in catalyst-filled tubes is primarily due to convection since fluid flow splits before and recombines behind each pellet. For heat transport, heat conduction through the solid pellets and the narrow pellet interstices is added. It is generally agreed that radiation needs not be considered separately at temperatures below 400-500 °C. Obviously its contribution can be significant also at lower temperatures if the temperature gradients are steep. In spite of the dominating convective nature, radial heat and mass transport have been traditionally described by a conduction (radial heat conductivity) and diffusion mechanism (radial diffusivity). The radial diffusivity can be safely calculated from a superposition of molecular diffusion and convective flow splitting, given by Schlu¨nder and Tsotsas (1988). There has been some discussion on whether a radially varying radial effective conductivity or the traditional model with constant λr,eff and an additional wall heat-


transfer coefficient Rw is the better representation for radial heat transport (see Cheng and Vortmeyer, 1988; Delmas and Froment, 1988; Foumeny and Pahlevanzadeh, 1990). Based upon detailed analysis, Tsotsas and Schlu¨nder (1990) came to the conclusion that the λr,eff, Rw model is a reasonable approximation for large mass-flow velocities prevailing in industrial fixed-bed reactor operation. Correlations for the calculation of λr,eff and Rw have been proposed by numerous authors. However, in spite of advanced techniques used, there is a significant scatter of experimental data on heat transfer even for physically similar systems (see, e.g., the books by Froment and Bischoff, 1979; and Wakao and Kaguei, 1982; or a review by Stankiewicz, 1989) with no clear indication to which of the literature correlations to choose for reactor modeling. A significant portion of the literature correlations on radial heat transfer are presented as

λr,eff ) C1 + C2RePr λf

(1)

Nup ) C3 + C4ReC5PrC6

(2)

where λf is thermal conductivity of the fluid, Re is the Reynolds number, Pr is the Prandtl number, Nup is Nusselt number defined as Rwdp/λf, and Ci are adjusting coefficients: C1 and C3 express static contributions to effective conductivity and the wall heat-transfer coefficient. The values of coefficients in the above equations can be found in the books by Froment and Bischoff (1979) and by Wakao and Kaguei (1982) or in the review by Stankiewicz (1989). All those correlations have been derived for plug flow. This is justified only for larger ratios of tube-to-particles diameters (Dt/dp) (see Zio´łkowska and Zio´łkowski, 1988). As shown by Daszkowski and Eigenberger (1992), this leads to a 20-40% difference compared with heat-transfer coefficients estimated for real velocity profiles. Real velocity profiles were considered by Legawiec and Zio´łkowski (1995) in their study on heat transfer in packed beds. The differences between heat-transfer coefficients for plug flow and for a two-region model were also found by Borkink and Westerterp (1992). Besides of the neglectance of radial flow variation, a second source for the differences in published heattransfer data may stem from deficiencies in the measuring procedure. It was pointed out that measurements will always contain a large degree of random scattering. Cybulski (1971) found that angular differences in temperature at the same radial position were up to about 20 °C when heating air from ca. 20 to 100 °C (see Figure 5). This was confirmed by Cresswell (1986), who found a 10 °C spread between angular replicates at a total gradient of 40 °C. Reprocessing the data of Cybulski and Głowacka-Lesńiak (1974) on angular temperature differences resulted in an apparent but significant relationship between the estimates of heat-transfer coefficients in the pseudohomogeneous deterministic model and the angular position of thermocouples (see Figure 6). Zio´łkowski and Legawiec (1987) repeated temperature measurements repacking the bed and estimated heat-transfer coefficients from those profiles. They found differences between λr,eff’s ranging from 50 to 75%. Wijngaarden and Westerterp (1992) applied the same procedure and found a significant scatter of temperatures at the same angular and radial positions after the bed was repacked. Borman and Westerterp

Figure 5. Temperature vs angular position of thermocouples at various radial positions; Dt ) 54 mm, L ) 95 mm, dp (lead spheres) ) 3.1 mm, G ) 2558 kg/(m2 h) (air flow) (from Cybulski (1971)).

Figure 6. Heat-transfer coefficients for radial temperature profiles measured at different angular positions; for configuration and operation conditions, see Figure 5.

(1992) attributed a large scatter of their temperature profiles to angular temperature variations. In a number of cases, the differences between heattransfer coefficients in reacting and nonreacting systems were found (Kim et al., 1966; Emig et al., 1972; Cybulski et al., 1973; Hofmann, 1979a,b; Clement and Jørgensen, 1983). On the other hand, Daszkowski (1991) and Daszkowski and Eigenberger (1992) could show in a number of carefully conducted experiments that good agreement between experiments and model simulations based upon independently determined heat-transfer parameters and kinetics could be obtained. In their model, they used the detailed radial flow variation both for the reactor simulation and for the determination of their heat-transfer parameters from independent heatup experiments without reaction. Similar observations have been made by Vortmeyer and Winter (1984). This gives rise to the hope that a better quantitative prediction of packed-bed reactor operation and scale-up will be possible if more detailed information on radially varying velocity profiles for different catalyst shapes and heat-transfer parameter correlations based upon these velocity profiles become available. However, heattransfer coefficients in packed beds can depend, among others, on the location and intensity of the heat source. Therefore, especially in view of the above literature discrepancies, it might be useful to find whether differences between heat-transfer coefficient measured under process and nonprocess conditions are caused by the influence of reaction on heat-transfer coefficients by variation in packing and flow distribution. Whether, in addition, a mixed deterministic-stochastic model for heat transfer is necessary to take the unavoidable randomness of packings into account remains a question of further research. Randomness of packing leads to randomness in flow distribution, and this, in turn, results in local differences of heat-transfer coefficients.


Figure 7. Cumulative Gaussian distribution for flow rates in individual tubes of a multitubular reactor; points represent a number of experimental data.

Figure 8. Relative yield of maleic anhydride vs normalized flow rate; V-Mo catalyst KMD-32; benzene-to-air ratio of 1:66.5; (b) coolant of 415 °C; (O) coolant of 425 °C (from Cybulski and Głowacka-Lesńiak (1974)).

Hence, a mixed deterministic-stochastic model for heat transfer in packed beds seems to be a reasonable solution. 3. Transport Phenomena outside the Tubes It has been found that the yield of the desired product in processes of partial oxidations of hydrocarbons in single-tube reactors can differ from that in multitubular reactors and that it can also vary in time (Cybulski and Głowacka-Lesńiak, 1974; Eigenberger and Ruppel, 1986). Several reasons for this have been proposed: (1) the differences in activity of various batches of the catalyst, (2) catalyst aging, (3) nonuniform distribution of the reaction mixture over individual tubes, and (4) nonuniform heat exchange between the tubes and the coolant. The last two points will be addressed in the following. 3.1. Flow Distribution over Individual Tubes. In multitubular reactors with random packings, the pressure drop in the tubes usually exceeds the pressure drop in the inlet and exit hood by orders of magnitude. If the tubes are packed carefully and the pressure drop is adjusted to a common value in each individual tube, an equal flow distribution to all tubes can be safely assumed, irrespective of the specific hood construction. Only if very short tubes and a low pressure drop packing (thin walled rings or monoliths) are used must both hoods be carefully designed to avoid an unequal flow distribution (Eigenberger and Nieken, 1991). It has been observed in industrial operation that an initially uniform flow distribution tends to deteriorate with time on stream. Cybulski and Głowacka-Lesńiak (1974) measured the flow rate distribution in an industrial reactor with 8540 tubes of 1-in. diameter filled with catalyst particles ca. 5 mm in diameter. The gas was introduced tangentially into the hood. The pressure drop in all tubes was thoroughly equalized at the same flow rate to a standard value before the reactor was started up. Flow-rate measurements were repeated in some tubes after the reactor had been shut down. The distribution turned out to be close to the Gaussian one (see Figure 7) with considerable deviations at the lowflow-rate range (the lowest flow rates were observed near the reactor shell). The deviations can be caused by either inadequacy of the normal distribution model to fit experimental data and/or greater weight of experimental points at the higher flow-rate range where each point represents more measurements than these for the low-flow-rate region. Similarly, significant deviations of the pressure drop which developed during the reactor operation were reported by Eigenberger and Ruppel (1986). The highest pressure drop was observed near the reactor centerline, just opposite to the central hood inlet pipe. An explanation which was substanti-

Figure 9. Coolant cross-flow (A) and coolant parallel-flow design (B) for multitubular fixed-bed reactors with molten salt cooling (from Eigenberger (1992)).

ated by visual inspection was a nonuniform deposition of dust particles on the top layer of the catalyst. Such a deposition will be, of course, strongly dependent upon the flow conditions in the inlet hood, where a tangential inflow favors deposition at the peripheral and at central inflow a deposition below the feed pipe. Hence, care must be taken in industrial fixed-bed reactors to completely remove any dust particles or droplets from the feed preparation stage before the gas enters the reactor. A nonuniform flow rate through individual tubes of a multitubular reactor can easily result in low yields and an increased danger of temperature runaway if a temperature-sensitive reaction is considered. Particularly sensitive are tubes with a reduced throughput because of their higher conversion and reduced heat removal. This has been shown clearly by Westerterp and Ptasinski (1984). On the other hand, if flow is too high, conversion decreases because residence time is too short. This is illustrated in Figure 8, where data on the partial oxidation of benzene to maleic anhydride (Cybulski and Głowacka-Lesńiak, 1974) are presented. An apparent plateu in Figure 8 is likely to be caused by commensurability of experimental errors with rather low yield sensitivity to flow rate near the maximum yield area. Obviously for lower flow rates, yield decreases, but no replicated data from that region were available. A 10% increase of the flow rate resulted in a yield drop by 3-5%. Thus, both factors contribute to lowering the yield of the desired product. 3.2. Distribution of Coolant Flow in the Intertubular Space. Generally there are two main classes of multitubular reactors as far as design of the intertubular space is concerned: the parallel-flow and crossflow units (Figure 9). In the reactors of the first category (Figure 9B), the coolant flows in parallel to


Figure 10. Distribution of shell-side heat-transfer coefficients in a leakage-proof exchanger of rectangular cross section (11.7 × 6 in.), containing 144 tubes of 1/2-in. diameter on 3/4-in. triangular pitch (form Stachiewicz and Short (1963)).

tubes in the region where the reaction occurs. Hence, active catalyst is restricted to the space between the upper distribution and the lower collection plate; the rest of the tubes may be filled with inert material. The coolant is pumped through the reactor by a strong circulation pump which nowadays is usually placed outside the reactor for better service. Part of the coolant is led through a steam generator to remove the heat of reaction, and the coolant temperature is adjusted via this side stream. For properly designed distributing and collecting plates, the flow is uniform over the cross section of the reactor. Heat transfer due to the forced convection at flow parallel to the tubes is rather poor for typical heat exchangers. Accordingly, the parallelflow reactors are considered to be demanding for the higher flow rates of the coolant to ensure a sufficiently high rate of heat removal. In the cross-flow design (Figure 9A), baffles are installed to force the coolant to flow mainly perpendicularly to the tubes. This flow pattern is expected to enhance heat exchange between the tubes and the coolant. Baffled reactors contain segmental baffles or disk-and-doughnut baffles, the latter being the most commonly used in industrial practice. Obviously, dead zones for the coolant can be formed where the coolant motion changes from parallel to perpendicular. Dead zones are particularily pronounced in the center and at the peripheral ifscontrary to Figure 9Asthe central part and the peripheral part where the coolant flow changes direction are also equipped with tubes. In these local “dead zones”, the coolant heat transfer can be much worse than in the parallel-flow reactors. We (A.C. and A.S.) witnessed the occurrence of such zones when an industrial reactor of the Rheinstahl-Von Heyden-Deggendorfer type containing ca. 10 000 tubes was dismantled for repair. The areas of the tubes near the potential dead zones located at the region of expected hot spots were reddish violet, indicating overheating of the tubes from inside. Sintering of the catalyst particles in those tubes might result in an increase of pressure drop and, consequently, a decrease of flow rate. This can be a further explanation for the development of nonuniform flow through individual tubes of the bundle during time on stream. These observations of the dismantled reactor tubes are in agreement with results of Stachiewicz and Short (1963). The distribution of local shell-side heat-transfer coefficients in a cross-flow heat exchanger as found by Stachiewicz and Short is shown in Figure 10. Measurements were carried out with the aid of a specially

developed heat-transfer probe consisting of a closely wound fine wire coil of the same diameter as that of the heat-exchanger tubes. Local heat-transfer coefficients were determined by accurately positioning the short-length coil in the test section in place of a tube. Care was taken that no leakage between the tubes and the holes in the buffle plates occurred. Nevertheless, considerable differences up to 400% have been observed. To determine the distribution of the coolant flow, the coefficients of hydraulic resistance in all elementary geometrical configurations are needed. These configurations include the tube bundle (for both parallel flow and cross-flow), baffle windows, gaps between tube bundle and the shell, and finally the baffle-shell and tube-baffle clearances. These hydraulic resistances have been reviewed by Stankiewicz (1989). A model consisting of continuity and momentum balance equations containing these resistances gives the flow distribution over the intertubular space. The model for flow in the intertubular space was experimentally verified (Adamska-Rutkowska et al., 1983a,b, 1984; Stankiewicz et al., 1986; Cybulski et al., 1987) using an apparatus consisting of two chambers with cross-flow and one chamber with parallel flow, containing 28 rows of tubes. The agreement between the calculated overall pressure drop values and the experimental ones was sufficient for the design purposes. Heat-transfer coefficients are evaluated by summing up contributions for both parallel flow and cross-flow. The correlations for these heattransfer coeffficients have also been extensively discussed in the review paper by Stankiewicz (1989). The required heat-transfer conditions hrequired can be obtained from the model of a single tube. If hcalculated,min < hrequired, the overall flow rate of the coolant must be increased and/or the design of the intertubular space must be modified. As mentioned above, the prime objective of modeling of the intertubular space is to properly arrange the geometrical configuration of this space and to determine the minimal coolant flow rates which will guarantee a sufficiently high heat transfer. Literature data on heattransfer coefficients in bundles of tubes are commonly used in such procedures. However, literature correlations do not take the occurrence of a heat source of high intensity (region near the temperature peak) into account. Under such conditions, turbulent natural convection can contribute considerably to the overall heattransfer coefficient. This has been observed by Cybulski et al. (1986, 1987). Coolant flow rates in a single-tube reactor were measured by using radioactive tracers. Heat removal in processes of partial oxidation of benzene and o-xylene appeared to be high enough at coolant velocities as low as ca. 2 cm/s, i.e., deep in laminar regime. The yields of the desired products maleic and phthalic anhydrides were as high as those for the higher coolant flow rates, indicating that a heat-transfer mechanism other than forced convection was prevailing. Further work on coolant heat transfer under high heat flux conditions seems necessary to better understand and predict turbulent natural convection in multitubular reactors. The first models combining the tube- and the shellside phenomena in a multitubular reactor were developed for publication at the University of Leeds (Adderley, 1973; Dunbobbin, 1976; McGreavy and Dunbobbin, 1976; Biscaia, 1980). In those early models, ideal coolant cross-flow with constant velocity through the tube bundle in a reactor of rectangular cross section had been assumed. Thus, the shell-side effects in those models were limited to the continuous increase in the


Figure 11. Two-dimensional cell model of the baffled shell side in a multitubular reactor (from Stankiewicz (1989)).

Figure 12. Axial temperature profiles in catalytic tubes at several locations in intertubular space.

coolant temperature during its flow through the intertubular space. A detailed calculation of heat transfer in the outer tubular space is contained in Eigenberger and Ruppel (1986). It is restricted, however, to the heat transfer from a hot flue gas to the tube bundle for an endothermic dehydrogenation reaction.

The first attemps to include the coolant maldistribution phenomena into the multitubular reactor model, originating also from Leeds, were made by Stankiewicz (1985, 1989) and consisted in dividing the intertubular space into a number of two-dimensional cells, in which both cross and parallel components of the local coolant flow were considered, as it is shown in Figure 11. Knowing the relations for hydraulic resistances in all individual elements of the intertubular space (tube bundle, tube-to-baffle clearances, baffle windows, etc.), it was possible to estimate the local parallel- and crossflow components of the coolant velocity in each shellside cell. Combination of the coolant heat balance in each cell with the balances on the tube side allowed for determination of the tube-side profiles as a function of the tube position in the bundle. Stankiewicz found that the parallel leakage flows of the coolant through the tube-to-baffle clearances may lead to substantial differences in the temperature profiles in individual tubes. An example of such differences is presented in Figure 12, in which “j” indices are the column numbers in the two-dimensional cell matrix, as shown in Figure 11, with j ) 1 and j ) 4 denoting the column adjacent to the coolant inlet and the column next to the reactor axis, respectively. The results shown in Figure 12 concern a multitubular unit for benzene oxidation to maleic anhydride, 4.652 m in diameter containing 12 144 tubes and fitted with five disk-and-doughnut baffles. In the same paper, Stankiewicz proposed five characteristic factors for multitubular fixed-bed reactors. The factors describe the uniformity of the tube bundle operation, thermal uniformity of tube and shell sides, and power consumption necessary for pumping the coolant through the intertubular space. Attention has also been given to the stability of multitubular reactor operation. Modeling studies (Dunbobbin, 1976; Stankiewicz, 1985; Stankiewicz and Leszczyn´ski, 1986) indicated that the cocurrent coolant-flow configuration presents a safer option than the countercurrent one, which, under certain conditions (e.g., coolant pump failure), is likely to move the reaction into the multiple steady-state region with accompanying thermal runaway in the catalyst beds. This phenomenon is due to the heat feed-back loop in countercurrent systems and can also occur locally in a part of the tube bundle being caused by maldistribution of the coolant flow (existence of stagnant zones).

Figure 13. Response of the reactor with disk-and-doughnut baffles (a) and of the parallel flow reactor (b) to a step increase in reactant concentration. Temperature profiles in the tubes positioned at the reactor axis.


A group from the Industrial Chemistry Research Institute, Warsaw, used subsequently the above-described two-dimensional cell approach to investigate the operational characteristics of the so-called mixed-flow co/countercurrent reactors (Stankiewicz et al., 1986) and the regions of steady-state multiplicity in large-scale industrial reactors for maleic anhydride production (Adamska-Rutkowska et al., 1988, 1990). Modeling strategies similar to that proposed by Stankiewicz have also been used by other authors, either to study the general characteristics of multitubular units (Baptista and Castro, 1993) or to investigate the influence of the configuration on the reactor controller design (Maciel Filho and McGreavy, 1993). Another, this time dynamic, model of a multitubular catalytic reactor was proposed by Stankiewicz and Eigenberger (1991). The main assumption and a novel feature of this model was a “porous body” approach, which implied that the entire intertubular space was regarded as a pseudohomogeneous continuum. The steady-state momentum and the unsteady-state heat and mass balances were written both for the tube and the shell sides in form of a system of mixed partial differential and algebraic equations and then discretized and solved using a semiimplicit Euler’s method. Stankiewicz and Eigenberger used the model to simulate the dynamic behavior of a reactor with parallel coolant flow as compared to a cross-flow unit with disk-and-doughnut baffles. The latter design appeared to be far less sensitive to diverse disturbances, both on the reactants as well as on the coolant side. Differences were most pronounced in the tubes positioned near the reactor axis. A typical example of the observed differences is presented in Figure 13, in which the response of the tube-side temperature profiles to a step increase in the reactant concentration is shown for both designs considered. 4. Concluding Remarks and Recommendations In modeling of multitubular reactors for fast and highly exothermic reactions, the first steps should be the selection of the most appropriate “ideal” model as classified by Froment (1979). Parametric sensitivity of this model should then be studied. If the model appears to be sensitive to flow rate, mass and heat-transfer coefficients maldistributions in these areas should be specifically included in the model. Considering the radial flow distribution in packed tubes, there is still a lack of reliable models and model parameters for different packings and operating conditions. Since radial heat evolution and wall heat transfer are strongly affected by the radial flow profile, radial heat transfer in packed-bed reactors can only be put on a sound basis if the heat-transfer parameters are determined in connection with the respective flow conditions. A second source of maldistribution in packed tubes originates from the essentially random character of a catalyst packing. Point measurements in packed beds will therefore always exhibit a large scatter and a weak reproducability. Larger differences therefore have to be accepted if such point measurements are compared with simulation results from conventional models. These models assume an averaging (e.g., around the priphery of a tube radius or a cross section of a reactor). A better agreement can therefore only be expected if the model results are compared with measurements which are also averaged in a similar way. It is so far not clear whether,

in addition, the development of stochastic reactor models is necessary to cope with the unavoidable randomness. For the design and operation of multitubular fixedbed reactors of industrial dimensions, special care must be taken to ensure sufficient and equal coolant conditions on the shell side and equal throughput through all individual tubes. This task, which presented substantial problems in the past, has been alleviated to some extent, on the one hand, through the availability of a great number of reliable heat-transfer data for the design of the shell side in shell-and-tube heat exchangers (here, the data of, e.g., VDI, ASME, or HTRI can be recommended for engineering calculations) and, on the other hand, through the development of sufficiently flexible and powerful computational fluid dynamics programs in recent years. Literature Cited Adamska-Rutkowska, D.; Cybulski, A.; Stankiewicz, A.; Leszczyn´ski, Z. A Mathematical Model of a Flow of a Heat Carrier in an Intertubular Space of Multitubular Reactors, Inz. Apar. Chem. 1983a, 22 (6), 3-7. Adamska-Rutkowska, D.; Cybulski, A.; Stankiewicz, A.; Leszczyn´ski, Z. A Mathematical Model of a Flow of a Heat Carrier in an Intertubular Space of Multitubular Reactors. Proc. 4th Natl. Symp. Chem. React. Eng., Warsaw 1983b; March, 2339. Adamska-Rutkowska, D.; Cybulski, A.; Stankiewicz, A.; Leszczyn´ski, Z. A Heat Carrier Flow in an Intertubular Space of a Multitubular Reactor (Eng.). 8th CHISA Congress, Prag, Sept 1984. Adamska-Rutkowska, D.; Stankiewicz, A.; Leszczyn´ski, Z. Simulation of the Operational Characteristics of the Large-scale Multitubular Reactor for Maleic Anhydride Production, Comput. Chem. Eng. 1988, 12, 171-175. Adamska-Rutkowska, D.; Stankiewicz, A.; Cybulski, A.; Leszczyn´ski, Z. The Simulation of Operational Characteristics of Different Constructions of Multitubular Reactors. Wiss. Z. Tech. Hochsch. 1990, 32, 737. Adderley, C. I. Ph.D. Dissertation, University of Leeds, 1973. Baptista, C. G.; Castro, J. A. Cell Models of the Shell-Side Flow in Multitubular Reactors. Ind. Eng. Chem. Res. 1993, 32, 10931101. Benenati, R. F.; Brosilov, C. B. Void Fraction Distribution in Beds of Spheres. AIChE J. 1962, 8, 359-361. Bey, O.; Eigenberger, G. Fluid flow through catalyst filled tubes. Chem. Eng. Sci. 1997, 52, 1365-1376. Biscaia, E. Ch., Jr. Ph.D. Dissertation, University of Rio de Janeiro, 1980. Borkink, J. G. H.; Westerterp, K. R. Determination of Effective Heat Transport Coefficients for Wall-cooled Packed Beds. Chem. Eng. Sci. 1992, 47, 2337-2342. Borman, P. C.; Westerterp, K. R. An Experimental Study of the Selective Oxidation of Ethene in a Wall Cooled Tubular Packed Bed Reactor. Chem. Eng. Sci. 1992, 47, 2541-2545. Brinkman, H. C. A Calculation of the Viscous Force Exerted by a Flowing Fluid on a Dense Swarm of Particles. Appl. Sci. Res., Sect. A1 1947, 27-34. Chandrasekhara, B. C.; Vortmeyer, D. Flow Model for Velocity Distribution in Fixed Porous Beds Under Isothermal Conditions. Wa¨ rme-Stoffu¨ bergang 1979, No. 12, 105-111. Cheng, P.; Vortmeyer, E. Transverse Thermal Dispersion and Wall Channeling in a Packed Bed with Forced Flow. Chem. Eng. Sci. 1988, 43, 2523-2532. Choudhary, M.; Szekely, J.; Weller, S. W. The Effect of Flow Maldistribution on Conversion in a Catalytic Packed-bed Reactor. AIChE J. 1976, 22, 1021-1032. Clement, K.; Jørgensen, S. B. Experimental Investigation of Axial and Radial Thermal Dispersion in a Packed Bed. Chem. Eng. Sci. 1983, 38, 835-842. Cresswell, D. L. Heat Transfer in Packed Beds. NATO ASI Ser. Ser. E 1986, 110, 687-728. Cybulski, A. Ph.D. Dissertation, Industrial Chemistry Research Institute, Warsaw, 1971. Cybulski, A. Multitubular Reactors. Inz. Apar. Chem. 1981, 20 (3), 6-12.

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Received for review September 30, 1996 Revised manuscript received February 23, 1997 Accepted March 8, 1997X IE960596S

X Abstract published in Advance ACS Abstracts, June 15, 1997.

Operational and Structural Nonidealities in Modeling and Design of

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