The review is dedicated to the research and development work made in USSR and Russia in the area of catalytic processes with artificially created unsteady-state conditions. The paper discusses the reverse-flow operation of catalytic reactor, forced feed composition cycling and sorption enhancement of catalytic reactions. It is demonstrated that under proper choice of process concept and control strategy these approaches may result in creation of new technologies with improved efficiency, lower capital and operation costs, higher process stability under variation of process conditions and increased target product yield in thermodynamically limited reaction and complex reaction systems.
For more than a 100 years of chemical reaction engineering history researchers were looking for the sources of new technical ideas for creation and improvement of chemical reactors. In general, all these sources may be classified as follows:
chemical drivers – when the progress is based on discovery of new reaction pathways and application of new practically available reagents,
catalysis drivers – when the progress is based on application of new catalysts,
engineering drivers – when the progress is based on new engineering approaches to the process development.
With respect to widely distributed heterogeneous catalytic processes, applying solid catalysts for performance of catalytic reactions in gaseous or liquid media, such engineering-driven approaches mostly relate to various efforts aimed to provide more efficient interaction of reaction processes with heat and mass transfer phenomena or with separation processes, to improve process heat management, to minimize internal nonuniformities inside the reaction volume etc. Such efforts include the following:
development and optimization of different catalyst bed configurations (packed, fluidized, moving, tubular beds and so on) and their conjunction with technological environment,
development of new catalysts (not meaning new catalytic materials but focused on new catalyst pellet shapes with, e.g. improved heat and mass transfer properties),
reaction volume structuring (application of uniformly structured catalyst beds and microreactors),
development of multifunctional processes combining reaction and separation processes (reactive distillation, reactive adsorption, etc.),
application of dynamic properties of catalysts under forced nonstationary conditions.
The last approach attracts the attention of numerous researchers for many years. Actually, none of the known steady-state catalytic processes can be assumed as absolutely stationary, at least, due to catalyst deactivation or fluctuations of catalyst operation conditions in practice. Nevertheless, there are two opposite basic approaches (or, better to say, paradigms) to the process development: the steady-state one, tending to keep (as much as possible) all the operation parameters constant in time, and unsteady-state one, based on controlled forced dynamic changing of these parameters in time, targeted to achieve the new process quality and new technological benefits. This second paradigm is the subject of the given review.
As soon as the rate of chemical reaction is influenced mainly by temperature, pressure and composition of the reaction mixture, the unsteady-state processes are based on forced oscillations of these parameters. Obviously, these oscillations should interact with corresponding inertial capacitive properties of the catalyst, which should be sufficient to provide the long enough non-stationary cycles. To be applicable for practical purposes, the duration of such cycles should be preferably measured in minutes or hours and not in fractions of seconds.
2 Temperature cycling in catalyst beds
The reaction rates and other reaction parameters (conversions, selectivity, etc.) are highly sensitive to temperature in a strictly nonlinear manner so the forced temperature cycling regimes are obviously an effective tool for creation of unsteady-state conditions in the catalytic systems. Long operation cycles in this case may be provided by high heat capacity of solid catalysts.
2.1 Moving heat waves of exothermic catalytic reactions in fixed catalyst beds
The typical temperature profile along the reactor axis is shown in Figure 1.
Inlet gas is usually fed into the reactor with low (e.g. ambient) temperature without any preheating. In the inlet part of the bed this gas contacts the preliminary heated solid material, resulting in gas heating and solid material cooling. Then the heated reaction gas enters the area where the temperature is sufficient for occurrence of the chemical reaction between the gaseous reagents, which is accompanied by emission of reaction heat and heating of both solid material and gaseous flow. Finally, the heated gas passes through the outlet part of the packed bed, where gas cooling and solid material heating occur. The described phenomena lead to evolution of the temperature profile in the form of a moving heat wave. Normally, such wave is moving in the direction of the gas flow. However, for reactions with high heat effect the movement of the heat wave in direction counter-current to the gas flow is possible due to heat conductivity of the packed bed.
The established heat front is characterized by low movement velocity which is 2–3 orders of magnitude less than the velocity of the gas flow; this value depends upon the ratio of gas to catalyst heat capacities (Boreskov et al. 1979a,b, Kiselev and Matros 1980). In the co-current heat front, the difference between maximum and inlet temperatures may exceed the adiabatic heat rise, while the counter-current front may be characterized by sub-adiabatic temperature regime. The heat front may consist of two heat waves: the slowly moving backward one and the more fast forward one (Gerasev et al. 1997, Gerasev 2015). Though heat waves are usually associated with exothermic reactions, they may also exist in case of endothermic reaction under supply of additional energy, e.g. microwave heating of catalysts (Gerasev 2017).
2.2 Catalytic technologies based on heat non-stationarity of fixed catalyst beds
To provide the practical use of described nonstationary regimes, it is necessary to keep the moving heat wave inside the catalyst beds. This may be provided by redirection of heat wave in the system of fixed catalyst beds.
In the two-reactor system shown in Figure 2A the heat wave moves along one reactor into another one. The flow is periodically redirected by switching valves, keeping the same flow direction in each reactor but changing the sequence of reactors. Thus, the heat wave always remains inside the system. In the two-bed reactor (Figure 2B) in one semi-cycle there are two heat waves, moving in opposite directions. In the following semi-cycle, when the heat wave leaves the lower bed the flow in the upper bed is reversed and the upper heat wave starts travelling downward. When it reaches the entrance of the lower bed, initiating new heat wave in the lower section, it is again reversed upward and so on. A similar principle is used in three-bed reactor (Figure 2C), where the heat wave movement direction is periodically reversed in the central part, remaining constant in the side beds.
2.3 Reverse-flow catalytic reactors
Another method of keeping the heat wave inside the catalyst bed is periodical reversals of the reaction mixture flow direction to the opposite one.
The pioneering patent on the reverse-flow reactor was filed in 1935 (Cottrell 1935), followed in the 1950s by later patents on the flow-sheet versions with stationary catalyst and heat regenerative beds and switching valves (Houdry et al. 1955, Houdry 1956) as well as with rotating catalyst bed (Asker and Siggelin, 1958). The first theoretical description of the reverse-flow operation with respect to catalytic selective oxidation reaction was published by Frank-Kamenetskii (1955).
The first known attempt to create a commercial catalytic reverse-flow reactor for reduction of SO2 was made in the late 1960s to early 1970s (Watson and Aubrecht 1969, Watson 1970), though this attempt seemed to be unsuccessful due to some technical problems with switching valves.
The intensive theoretical and practical development of reverse-flow reactors started in the mid-1970s, pioneered by the team of Prof. Yurii Matros in Boreskov Institute of Catalysis (Novosibirsk, Russia). This research has led to formulation of fundamental process concept and development of the mathematical simulation methods for the reverse-flow operation of catalytic reactors (Boreskov et al. 1975, 1977, 1978, 1979c, 1980a,b,c,d, 1981, 1983, Matros et al. 1979, Boreskov and Matros 1983, Matros and Zagoruiko 1987, Bobrova et al. 1988, Chumakova and Zolotarskii 1990, Matros 1990, Gerasev and Matros 1991, Noskov et al. 1993, 1996, Bunimovich et al. 1995, Matros and Bunimovich 1996, Zagoruiko et al. 1996, Borisova et al. 1997, 1988a, 1993, 1994, Vernikovskaya et al. 1999a, Zagoruiko and Matros 2002). This work has also resulted in development, creation and startup of the first successful commercial reverse-flow reactors for SO2 oxidation (Krasnouralsk non-ferrous smelter, Russia, 1982) and incineration of volatile organic compounds (VOCs) (Novosibirsk Chemical Plant, Russia, 1984) (Matros and Bunimovich 1996). The contribution of Prof. Matros in this field appeared to be very significant, so sometimes the reverse-flow reactors are named Matros reactors (Gosiewski et al. 1996, Lintz and Wittstock 2001). The achievements in this area were discussed earlier in reviews (Matros and Bunimovich 1996, Zagoruiko 2012).
2.3.1 Working principle of the reverse-flow technology
The reverse-flow technology is mostly targeted for performance of the exothermic chemical reactions between gaseous reagents in the packed bed of solid catalysts or non-catalytic materials. The main distinctive working principle, common for all reverse-flow reactors, is periodic cyclically repeating reversals of the gaseous reaction mixture flow inside the packed bed of solid catalyst or non-catalytic heat regenerative material. The simplest flow-sheet of the reverse-flow reactor is given in Figure 3.
The reverse-flow reactor is equipped with the system of switching valves, providing the periodical gas flow reversals inside the reactor (the directions of the gas flow are shown in Figure 3 by solid and dashed arrows).
The flow reversal is performed when the high-temperature reaction zone reaches the outlet of the reactor. After the reversal, when the gas flow direction turns to opposite, the heat wave also starts to move in the backward direction. Such periodical reversals make it possible to keep the high-temperature reaction zone within the reactor for an unlimited time.
Figure 4 demonstrates the dynamics of the temperature profiles within the process cycles between flow reversals. It is seen that the final temperature profile in the operation cycle (Figure 4, curve C) is completely symmetric to the initial profile (Figure 4, curve A) with respect to the bed center. After the flow reversal the c profile will turn into a profile and the temperature evolution will be repeated in the next cycle. This indicates the established character of the cyclic regime in the reactor, meaning that such cycles may be repeated for an unlimited time (say, for years of continuous operation, if necessary).
Figure 5 shows the axial temperature profile in the reverse-flow reactor averaged per cycle duration in the established cyclic operation regime. It is seen that the difference between average outlet gas temperature and inlet temperature is equal to the adiabatic heat rise of the reaction, in strict agreement with the energy conservation law.
At the same time, the difference between the maximum temperature in the reactor and the inlet one may be significantly higher than the adiabatic heat rise. Such super-adiabatic behavior is known for the heat waves of exothermic reactions moving in the direction of the gas flow, when the energy is extracted by the gas flow from the heated heat regeneration material in the inlet part of the reactor and transported by this flow to the reaction area in the central part of the reactor. Such energy transfer from lower temperature area to the higher temperature one, being in formal contradiction to thermodynamics (which may be easily resolved in case of combined account of heat energy and potential energy of exothermic chemical reaction), actually leads to super-adiabatic heat concentration or “heat pumping”, resulting in high maximum temperatures in the reactor even in case of lean reaction mixtures with low potential adiabatic heat rise.
In other terms, the reverse-flow reactor may be represented by the high-temperature reaction zone, surrounded by regenerative heat exchangers for preheating of the gaseous reaction mixture. As shown by a special simulation study (Matros et al. 1994) the reverse-flow processes are more efficient in terms of heat regeneration, conversion, process stability, energy consumption and capital costs than the unsteady-state processes, based on permanent flow direction (see Chapter 3.2).
In general, the regenerative heat exchange here is a basis of the most important features of the reverse-flow reactors. This fact is reflected in the related terminology, when reverse-flow reactors are sometimes called regenerative reactors; for example, the catalytic and non-catalytic reactors for deep oxidation reactions are often termed regenerative catalytic oxidizers (RCO) or regenerative thermal oxidizers (RTO), respectively.
Regenerative heat exchange is quite advantageous in case of gas-phase reaction mixtures with low concentration of reagents (and, thus, with low potential heat of reaction). This case is characterized by relatively low typical values of heat-exchange coefficients for the gas-solid heat transfer (under gas pressure closed to ambient) and low temperature gradients, resulting in increased requirements for the heat exchange surface area. The heat regenerative packing in the reverse-flow reactors is characterized by very high heat exchange area (up to few thousands of square meters per cubic meter of packing), being much higher than that for recuperative heat exchanger.
Another advantage of the reverse-flow reactor is its improved operation stability under fluctuation of the process external parameters (e.g. gas flow rate, inlet temperature and composition) in time. It is explained by the fact that these parameters mostly influence the velocity of the heat wave movement in the reactor, therefore, it is only necessary to provide the flow reversals at the optimum time moments. These moments may be rather easily defined by achievement of some fixed temperature in the fixed points of the reactor (for example, the usual control procedure for the catalytic reverse-flow reactor involves switching of the flow when the temperature at the interface between inlet heat regenerative bed and catalyst bed becomes lower than some fixed value).
2.3.2 Flow sheets of the reverse-flow processes
The catalytic reverse-flow reactor (Figure 6) most often contains a packed bed (C) of the catalyst (in the form of pellets or monoliths), surrounded by the packed beds of heat regenerative material (H) which is catalytically inert. Usually this material presents itself as ceramic pellets (cylinders, saddles, etc.) or monoliths. To avoid the undesired adsorption/desorption phenomena at the bed ends, it is recommended to use heat regenerative materials with zero or minimum adsorption capacity with respect to reagents (Zagoruiko 2009). Such type of reactors is usually applied in the processes for VOC combustion in waste gases.
Figure 7 demonstrates the reverse-flow reactor with improved heat management (Boreskov and Matros 1981). It may be applied for treatment of lean gases (requiring the additional supply of heat for provision of stable reverse-flow operation), as well as for treatment of gases with increased content of reagents (requiring the heat withdrawal for energy utilization purposes and in order to avoid the overheating of the catalyst beds). The gas cooling or heating is performed between the sections of the catalyst bed (by means of internal or external heat-exchangers) – such positioning provides the most efficient heat management of the process.
Such reactors are applied in a heat addition mode for combustion of lean VOC-containing waste gases (characterized by its own adiabatic heat rise below 20–30°C). Heat withdrawal mode may be realized in the case of higher VOC content (adiabatic heat rise above 100–150°C) with production of heat in the form of hot water or steam.
Majority of existing SO2 oxidation reverse-flow reactors with two or three catalyst beds use the heat withdrawal between the beds for control of the catalyst temperature, which defines the equilibrium level of SO2 conversion.
Figure 8 represents the reverse-flow reactor with the possibility to additionally feed one of the reacting streams into the central part of the reactor. Such reactor configuration is, for example, applied in the processes of selective catalytic reduction (SCR) of NOx by ammonia, when ammonia (in the form of ammonia water or urea water solution) is introduced into the reactor center (In2), while the main stream of NOx-containing waste gases (In1) is fed to the inlet switching valve. Such approach helps to avoid the undesired ammonia desorption from bed ends (which may happen in case of direct ammonia feeding into the inlet gas stream and result in significant ammonia slip due to ammonia desorption from the bed ends) and thus to provide high process efficiency.
The reactor configuration with partial gas stream withdrawal from the reactor may be used as a version of the process for heat withdrawal, when the extracted central stream is forwarded to the external heat exchanger.
Some flow sheets of the reverse-flow process may include more than one flow reversal contour, e.g. double reverse-flow process, proposed for performance of the Claus process (Zagoruiko and Matros 2002) (Figure 9).
Figure 10 demonstrates the design of the rotating reverse-flow reactor (Asker and Siggelin, 1958), where the flow direction is kept constant with respect to the reactor vessel body, while the periodical changing of the flow direction in the packed catalyst and heat regenerative beds is provided by continuous rotation of the packing. The volume above the packing may be used for addition/withdrawal of heat, as well as for side stream feeding or withdrawal. Such reactors are used for treatment of VOC containing waste gases.
Such design does not include switching valves, and it helps to minimize conversion losses occurring during flow switching, being a typical problem for valve-based reverse-flow reactors. On the other hand, rotating design may produce problems with sealing of the reaction volume under the rotating packing, resulting in some bypass of the inlet mixture directly to the outlet pipeline, leading to decrease in conversion.
2.3.3 Comparison with competitive technologies
Compared to the steady-state reactor, the following advantages of the reverse-flow reactor may be formulated:
184.108.40.206 Decrease of energy consumption
Regenerative heat exchange provides efficient utilization of the target reaction heat, therefore making it possible to autothermally (i.e. without supply of external energy/fuel) process the reaction mixtures with adiabatic heat rise of ~30°C and higher, as well as to process more lean mixtures with minimized energy consumption. Decrease of energy consumption at processing of lean gases, compared to the steady-state process, may reach 100–130 kJ per st.m3 of inlet gas or by 30–90% of the overall energy consumption.
220.127.116.11 Decrease of capital costs
Decrease of capital costs is provided mostly by elimination or minimization of the expensive recuperative heat exchangers. The value of capital cost saving may reach 30–40% compared to the steady-state process.
18.104.22.168 High process stability under variation of external conditions
Surprisingly, reverse-flow processes are much more stable in operation and much easier to control than steady-state processes in case of significant oscillation of reaction gas parameters (gas flow rate, inlet temperature, composition) in time, which is quite a typical situation at processing of various waste gases.
22.214.171.124 High thermal efficiency of reaction heat utilization
It is possible to produce high-potential heat from lean fuel mixtures. Super-adiabatic heat accumulation in the moving heat waves opens the way to achieving maximum temperatures much higher than adiabatic ones; therefore, the heat of reaction may be efficiently utilized in the form of high-potential heat (e.g. high-pressure steam) even from lean fuel mixtures. For example, it is possible to achieve maximum reaction gas temperatures up to 700–900°C from gaseous feedstock with typical adiabatic heat rise as low as 100–400°C, with possibility to utilize this heat in the form of hot water or steam with thermal efficiency up to 80–90%.
There also exists the widely spread opinion that a reverse-flow reactor may provide increased product output in the equilibrium-limited exothermic reactions due to the specific temperature distribution along the reactor – decrease of the catalyst temperature to the reactor outlet (see Figure 4). Such decrease is theoretically favorable for shifting the equilibrium and raising the equilibrium conversion. Actually, the temperature profile in the reactor is thermodynamically advantageous only in the first half of the process cycle (see Figure 3), while in the second half of the cycle the outlet temperature is even greater than the adiabatic one with consequent loss of equilibrium conversion. Another problem is that the reaction rate increases sharply with temperature, so the high maximum temperature in the reverse-flow reactor may impose a more significant negative influence on the equilibrium conversion than the low outlet temperature and, thus, the observed conversion may be even lower than in the steady-state process. In some cases, specific reasons may exist for the improvement of thermodynamically limited conversion in the reverse-flow regime (Zagoruiko and Matros 2002), but in general this declared advantage of the reverse-flow reactors does not look evident.
The catalyst lifetime in the reverse-flow reactors is sometimes intuitively estimated as decreased compared to steady-state operation due to possible negative influence of periodical temperature oscillations. Actually, commercial experience with reverse-flow reactors does not show any specific influence of nonstationary conditions, and the observed catalyst lifetime is not different from the conventional one. Moreover, this is directly confirmed by special experimental studies (Tsyrulnikov et al. 1998).
The main drawback of the reverse-flow reactor in comparison with the steady-state one is conversion losses during the flow reversal procedure resulting from a possible reaction mixture bypassing the reactor during valves switching and with replacement of unreacted mixture from pipelines, reactor void volume and inlet heat regenerative bed (before the catalyst bed) into the outlet stream immediately after flow switching. Losses from valve leakage may be minimized by application of quality valves with low duration of the switching, while the losses from mixture replacement requires changes in the reactor design (minimization of void volumes) and improvements in the process flow sheet, e.g. by application of purging circuits and intermediate vessels for gas storage during switching under appropriate control strategy (Matros et al. 1992, 1997).
Additional complications may be caused by reversible adsorption of reacting species in the low-temperature inlet part of the catalyst and inert beds in the end of the process cycle – after the flow switching this inlet part becomes an outlet one with temperature gradually rising in time, leading to the desorption of unreacted species into the outlet flow (Bunimovich et al. 1995, Gosiewski et al. 1996, Borisova et al. 1997, Smith and Bobrova 2002). If it is possible to feed the adsorbing components separately from the other reaction mixture (like reducing ammonia in de-NOx process), then it is reasonable to apply multi-bed reactors with feeding of this component between the beds (Noskov et al. 1993, Bobrova et al. 1997a, Lakhmostov et al. 2000). Optimization of the control strategy may also help to resolve the problem (Gosiewski et al. 1996).
2.3.4 Current reverse-flow technology status
Reverse-flow reactors are applied at the commercial scale since the early 1980s, so the general stage of development may be characterized as commercial level.
At the same time, the potential of the reverse-flow approach is far from depletion. There are numerous new application areas and promising directions for development of new modifications of reverse-flow reactors. The stage of development within these particular areas may differ from theoretical concept to pilot or semi-commercial level.
Such a wide range of development status reflects the fact that reverse-flow operation of chemical reactors is more of a global engineering approach being a basement for creation of various chemical technologies than some specific technology.
It is interesting that although the reverse-flow reactors have been under active development for more than 40 years (and it is a significant time in relation to the history of chemical reaction engineering branch lasting for not much more than a century), the reverse-flow approach is often considered in R&D and commercial practice as a still new approach. Partially it may be explained by the mentioned remaining technological potential, but to some extent, it may be a result of conflict between stationary and non-stationary paradigms in engineering. It seems that people normally tend to perceive the steady-state approaches as conventional by default, while dynamic methods are more easily accepted as non-traditional.
2.3.5 Known commercial applications
The main advantages of the reverse-flow reactors, as mentioned above, are the possibility to process lean feedstock with minimum energy consumption and the high operation stability under oscillations of external process parameters (feedstock flow rate, temperature and composition). Such conditions are generally typical for treatment of various waste gases.
The main current application area of reverse-flow reactors is incineration of VOCs in waste gases (Figure 11). Such gases with relatively low VOC content (from 10 to 50 ppm up to 5–10 g/m3) are met in very many branches of industry – chemistry and petrochemistry, painting, printing and lacquering facilities in machinery, wood processing, agriculture, paper production, food industries etc. The range of VOCs includes practically all classes of organic compounds (hydrocarbons, spirits, acids, ethers, sulfur- and halogen-containing compounds, dioxins, etc.).
Both catalytic and non-catalytic reverse-flow reactors (called also as RCOs and RTOs, respectively) are applied in this area. The number of industrial installations here is measured in hundreds. The range of capacities (with respect to the flow rate of waste gases) varies from 500 to 800,000 m3/h.
Another significant area of reverse-flow reactors application is oxidation of SO2 in production of sulfuric acid (Figure 12). The most attractive area here is processing of waste gases from non-ferrous smelters, characterized by relatively low SO2 content (1–4 vol.%) and expressed oscillations of gas flow rate, composition and temperature, providing the maximum advantages of the reverse-flow technology over the conventional steady-state one. More than 15 installations with maximum unit capacity (with respect to the wastes gas flow rate) up to 110,000 m3/h were put into industrial operation since 1982.
One industrial application is known for the reverse-flow reactor for NOx reduction by ammonia (see Chapter 5.2.1).
2.3.6 Potential future applications of the reverse-flow reactors
Though the SO2 oxidation process appeared to be one of the most successful application areas for reverse-flow reactors, this area keeps new various challenges for further development. The patent (Zagoruiko 2002) describes the multi-bed SO2 oxidation reactor, where the flow reversals are performed in the inert heat-regenerative beds only, while the flow direction in the catalyst beds is kept constant. As shown by a modeling study (Matros et al. 1994), in such partial reverse-flow mode the heat regeneration efficiency is lower in comparison with the full reverse-flow regime. This leads to the lower maximum temperature in the catalyst beds, thus improving the processing of the gas streams with increased SO2 content (above 2–3 vol.%).
Another problem in processing of waste gases from non-ferrous smelters is possible presence of CO in these gases with concentration varying from 0 to 1–2 vol.% CO oxidation may lead to significant overheating of the catalyst beds thus tightening the equilibrium limitations and decreasing the SO2 conversions. Moreover, the fluctuations of CO concentrations may lead to dramatic decrease of process operation stability. It was proposed (Zagoruiko and Vanag 2014) to place the additional beds of glass-fiber catalysts for low-temperature CO oxidation at the border between beds of conventional vanadia-based SO2 oxidation catalyst and heat-regenerative material. In such system, the CO oxidation is completed at temperatures below the ignition point for SO2 oxidation at V2O5. As shown by a simulation study (Zagoruiko and Vanag 2014), such temperature separation of reactions prevents the increase of maximum temperature and improves the process stability.
The reverse-flow reactors may also be applied to a wide range of other gas-phase exothermic reactions. The following potential applications are reported in literature:
Application of the reverse-flow reactors for these processes may lead to decrease of capital costs and looks especially attractive at application for processing of purge gases at conventional ammonia and methanol plants.
These plants may also use the reverse-flow reactors for partial oxidation of the natural gas to syngas (Boreskov et al. 1984):
This process may be applied for efficient energy production from non-fossil methane, contained in landfill gas, biogas and coal mine waste gases with significant CO2 sequestration effect.
Simulation studies (Matros and Zagoruiko 1987, Zagoruiko and Matros 2002) show that the reverse-flow reactors in different configurations may significantly decrease capital costs and increase degree of sulfur recovery from sulfur-containing gases in oil and natural gas processing. Reverse-flow process may be efficiently used here for replacement of the basic catalytic stage of the Claus process as well as for replacement of the basic process with tail gas cleanup facilities.
Another potential application of the reverse-flow process is selective oxidation of H2S to sulfur (Zagoruiko 1994a):
applicable for desulfurization of natural gas and biogas with moderate H2S content (up to 1–2%), as well as for tail gas cleanup in Claus units.
In addition to the reverse-flow reactors for oxidation of VOCs in waste gases, it is also reasonable to mention the reverse-flow processes for abatement of VOCs in wastewater. In this process the heat regenerators are used for evaporation and condensation of waste water, while the oxidation reactions occur in the catalyst bed in the vapor phase (Zagoruiko 1994b).
It is necessary to understand that in the case of processes with heat non-stationarity of the catalyst beds we talk more about the technological paradigm, put in their basis, than about the catalyst stationary state in terms of its chemical composition.
3 Pressure cycling of catalytic reactors
Another alternative method for creation of unsteady-state conditions in the fixed catalyst beds is pressure cycling. Reaction pressure may affect the reaction rates, though this influence is less significant than in the case with temperature – the dependence of reaction rates on pressure is usually close to linear compared to the exponential one for temperature.
The processes with relatively long (tens of seconds and longer) pressure alteration cycles combine the principles of pressure swing adsorption processes and catalytic reactions. Though such approach looks promising, it has not resulted in any serious practical applications yet.
Another type of pressure cycling with higher alteration frequency (from decimal fractions to tens to hundreds of hertz). Theoretically, such pressure cycling, in addition to forcing the reaction rates, may lead to intensification of mass transfer in the catalyst pores, thus increasing the apparent rates of simple reactions and changing the selectivity and products yield in complex reaction systems.
The pressure impulses may have a natural character, sometimes not targeted for any positive effect on catalytic reaction. For example, this may be observed in automotive catalytic converters under pressure oscillations of exhaust gas from engine.
Creation of the forced pressure oscillations at the reactor entrance may be not efficient due to fast dumping of pressure impulses in the catalyst bed. At the same time, high-frequency pressure oscillations, with frequency of hundreds and even thousands of hertz, were theoretically predicted in the channels of catalytic monoliths due to fluid self-turbulization in case of some roughness and local geometry defects at the channel entrance (Zakharov et al. 2003). Similar effects may be observed in the structured cartridges with glass-fiber catalysts (Zagoruiko et al. 2017) due to interaction between moving flow and flexible catalyst fabric.
4 Composition cycling in catalytic reactors
4.1 Forced feed composition cycling in isothermal reactors
The influence of the composition of the reaction mixture on the reaction parameters is very significant, and it may be not only quantitative but qualitative as well. Therefore, the forced feed composition cycling may be used for creation of new catalytic technologies.
In the simple reaction systems containing one reaction route, like in reactions of CO (Bykov et al. 1981, Veniaminov 1993, Slinko and Jaeger 1994, Tomilov et al. 1996) or SO2 (Strots et al. 1992) oxidation, the main result of feed composition fluctuation may be the rise of the average reaction rate compared to the equivalent steady-state regime (with parameters equal to averaged per cycle parameters of the unsteady-state regime). However, for the majority of studied reactions such rise does not exceed 1.5–2 times, so it can hardly be accounted as significant. At the same time, in the SO2 oxidation reaction (Strots et al. 1992) under periodical alteration of SO2-containing mixture and air feeding, the observed average conversion of sulfur dioxide exceeds the equilibrium value for the equivalent steady-state regime.
It seems more interesting to apply the forced feed composition cycling approach to reaction systems with complex networks of parallel and consecutive stages, where the issues of selectivity and product yield appear.
Theoretical model studies (Zolotarskii and Matros 1982) showed the possibility to achieve a gain in selectivity in cyclic unsteady-state regimes, and the value of this gain significantly depends upon the reaction mechanism, as well as from strategies used for creation of nonstationary conditions. Besides, it was shown that for adequate description of such systems it is necessary to use the nonstationary kinetic models, accounting for dynamics of adsorption processes and catalyst surface changing in the course of reaction.
The possibility to increase the selectivity of reactions and product yield and to decrease the formation of undesired byproducts was demonstrated in experiments and simulation studies performed in Boreskov Institute of Catalysis for complex reaction systems, such as oxidative and steam methane reforming (Sadykov et al. 2002), propylene oxidation into propylene oxide (Balzhinimaev et al. 1984) and anaerobic oxidative dehydrogenation of butylene into butadiene (Tomilov et al. 1999).
Another sub-class of catalytic processes also based on dynamic interaction between adsorption and reaction processes in the nonstationary conditions is the reactive-chromatography processes. This approach was proposed by Professor Simon Roginskii from the Moscow Institute of Chemical Physics in the early 1960s (Roginskii and Rosental 1962, Roginskii et al. 1962, 1963). Such processes utilize the adsorption properties of catalysts giving way to simultaneously performing the reaction and separation of products due to the difference in their adsorption properties. This makes possible to obtain the pure individual products and also to achieve the over-equilibrium product yield in reversible reactions, due to dynamic separation of products in the catalyst bed space. On the other hand, the reactive-chromatographic process requires increased catalyst loading, this leading to low unit catalyst productivity and decreasing the efficiency of the technology.
In summary, the forced feed composition cycling and other process approaches based on cyclic changing of the catalyst surface composition and dynamic interaction of reactive and adsorptive phenomena have a significant potential for creation of novel highly efficient catalytic technologies. However, the direct transfer of experimental data obtained in lab scale isothermal reactor is impossible; the development of the large-scale process should be based on deep understanding and adequate account of both the catalyst temperature and composition dynamic changing and their interaction in the conditions of real fixed catalyst beds.
4.2 The existing and prospective catalytic processes with nonstationary catalyst state
This chapter is dedicated to catalytic technologies using forced unsteady state of the catalyst. These technologies are based on combined action of heat and adsorptive capacitive factors. In literature this class of technologies is also known as adsorption-catalytic or sorption-enhanced catalytic processes.
Figure 13 shows the flowsheets of the processes, making possible to realize the periodical forced feed composition cycling or separate feeding of reactants. In the simplest case (Figure 13A) the composition cycling is provided by periodical alteration of initial mixture composition or by separate feeding of reactants. To provide the overall continuous performance of the process, it is possible to use the “swing-reactor” system with two or more reactors (Figure 13B) with cyclic alteration of feeding the different reactants in co-current or counter-current directions. Alternatively, the process may be performed in the system of two reactors, each of them fed by different reactants with catalyst circulating between these reactors (Figure 13C); this approach is currently widely used in “chemical looping combustion” technologies. One more option is the rotating catalyst bed, where reactants are fed into different parts of the bed (Figure 13D).
4.2.1 Selective catalytic reduction of nitrogen oxides
The interesting example of the process based on the artificially created nonstationarity of the catalyst is the Reverse-NOx process, developed in the Boreskov Institute of Catalysis for SCR of nitrogen oxides by ammonia in the reverse-flow mode (Noskov et al. 1993, Matros and Bunimovich 1996):
The first commercial Reverse-NOx installation was created in 1989 at Byisk Oleum Plant in Russia for reduction of NOx in waste gases of weal nitric acid manufacturing. Ammonia water in this installation was fed into the central part of the catalyst bed in the reverse-flow reactor (see flow-sheet in Figure 8 and reactor assembly in Figure 14). Initially, such approach was based only on the technical simplicity of ammonia water evaporation in a high-temperature bed zone. However, the experience of commercial operation showed that the efficiency of the process exceeded the simulated values: the conversion of nitrogen oxides was very high, and the ammonia slip with outlet gases was practically absent. It was unusual for conventional approach with ammonia feeding into the inlet gas, where ammonia desorption from catalyst bed after flow switching was a serious problem.
Detailed investigation of the reaction kinetics and process simulation studies (Noskov et al. 1993, 1996, Borisova et al. 1997, Popova et al. 1998a,b, Smith and Bobrova 2002) provided the explanation of the observed phenomena (see more detailed explanation in Chapter 6.1). It was shown that ammonia is strongly adsorbed at the catalyst surface and interacts with nitrogen oxides already in a chemisorbed form. In the reverse-flow process with central ammonia feeding, the chemisorption occurs in the outlet part of the bed. Simultaneously, in the inlet part of the bed the entering NOx interacts with ammonia, chemisorbed in the previous process cycle. This picture is repeated after each flow reversal. In this case, the adsorbed ammonia is mostly accumulated in the central part of the bed, thus preventing the NH3 slip from bed ends. The additional positive effect from ammonia feeding in the high-temperature central part of the bed is prevention of possible direct contact of NH3 and NOx at low temperatures, thus avoiding the formation of explosive ammonia salts.
4.2.2 Deep oxidation of VOCs
The industrial waste gases purification from VOCs is an actual problem in atmospheric air protection. Such impurities (hydrocarbons, alcohols, acids, esters, aldehydes, etc.) are present in waste gases of various industries, such as chemicals and petrochemicals, machinery, wood processing, printing and others.
Whereas for emissions with high VOC concentrations, it is cost-effective to use the methods involving the separation and recycling of impurities (e.g. absorption, adsorption, condensation and membrane methods) (Matros et al. 1991), for dilute gases, it is expedient to use thermal and catalytic oxidation of VOCs with oxygen (usually atmospheric oxygen) to harmless products – carbon dioxide and water vapor. From economic and environmental viewpoints, the best methods for low-concentration emissions treatment are catalytic reverse-flow processes that provide autothermal (i.e. without the supply of extra energy and fuels) treatment of gases with a VOC content above 0.6–0.8 g/m3 or higher. For gases with lower VOC content, an additional energy input is required.
In practice the widespread problem is the treatment of waste and vent gases with very low VOC concentration (below 0.1 g/m3) and with high volumes of gases to be purified. These factors are the reason of high energy consumption and capital cost of purification units, significantly limiting their wide practical application. In addition, because of the wide distribution of VOC emission sources among various branches of industry with the possible lack of qualified service personnel, such technologies should be simple and safe to use.
From the standpoint of meeting these requirements, of particular interest are adsorption-catalytic processes that combine adsorption and catalytic technologies. A large number of such processes are based on a simple combination of adsorbers and catalytic reactors. In these processes, VOCs are adsorbed in an adsorbent bed; the adsorbent is then regenerated (usually by steam or hot air), and the desorbed impurities undergo catalytic oxidation in a separate unit. Of special interest, however, are adsorption-catalytic processes based on the adsorption of VOCs directly on the active surface of deep oxidation catalysts at ambient temperature with periodic regeneration of the catalyst by oxidation of the adsorbed impurities at elevated temperatures (Samoilov et al. 1985, Suprunov 1986, Rabinovich et al. 1988, Kalinkina et al. 1990, Matros et al. 1991, Orlyk et al. 1995, Zagoruiko 2007a).
Adsorption-catalytic processes have significant advantages over traditional catalytic technologies for deep oxidation of VOCs, particularly in the treatment of low-concentration gases, where their use eliminates the need for constant heating of purified gases and thus substantially reduces the power consumption of the purification process. Other important advantages of these processes are high technological flexibility due to the possibility of treating gases with widely varying initial concentrations of VOCs and the possibility of quick startup of the purification unit without long and energy-consuming preheating.
However, practical experience and its evaluation by mathematical modeling have shown (Zagoruiko et al. 1996, 1997) that these processes also have disadvantages, such as the risk of overheating of the catalyst during regeneration of the bed and partial desorption of unoxidized VOCs or the products of their incomplete oxidation during heating of the catalyst. This reduces the purification degree and requires additional purification of the desorption gases.
The described disadvantages can be overcome by optimizing the technological processes, operational parameters and the adsorbent-catalyst parameters.
One of the ways to minimize the desorption losses is the gradual stepped heating of the catalyst during regeneration (Rabinovich et al. 1988). Such a gradual heating way allows to convert the most part of reversibly adsorbed impurities into the irreversible chemisorbed form at the beginning of the regeneration cycle. But this approach is effective for a fairly limited number of VOCs and, moreover, significantly complicates the regeneration process and increases its duration.
The other way to reduce the desorption losses is using relatively large adsorbent-catalyst pellets that inhibit intra-diffusion desorption (Vernikovskaya et al. 1999b).
The process may be organized by way of the adsorption-catalytic reverse-process (Zagoruiko et al. 1995, 1996, 1997, Zagoruiko and Noskov 1997, Zagoruiko 2006a). At the adsorption stage the inlet gases are fed into two parallel reactors, and the regeneration is carried out by heating the gas with simultaneous air supply and periodic flow reversals. So during the regeneration VOCs are found to be adsorbed mainly in the inlet bed parts, which leads to minimization of their losses during desorption. In addition, the efficient heat recovery in the reverse flow mode allows to reduce the energy consumption for the adsorbent-catalyst regeneration.
We should note that the developed mathematical model of adsorption-catalytic process allowed taking a fresh look on the conventional VOC deep oxidation reverse process. The simulation (Zagoruiko 2009) showed that at moderate temperature in the bed inlet part at the cycle ending before the flow reversal, the VOCs adsorption may occur and the considerable part of them adsorbs reversibly. After the flow reversal this bed part becomes the outlet and the temperature in it starts to increase. As a result, some part of the adsorbed impurities may desorb and reduce the overall purification degree. It can be concluded that to avoid such emission it is necessary to use reasonably long beds of an inert heat-regenerating material.
One of the other technological schemes of adsorption-catalytic process is implementing catalyst regeneration under conditions of the adsorbed VOCs combustion heat wave propagation in the opposite direction of the gas flow due to the thermal conductivity of the catalyst bed (Kalinkina et al. 1990). Such a mode is of considerable interest as the desorbed impurities get into the preheated bed part where they can be oxidized to the deep oxidation products.
Another scheme (Dobrynkin et al. 2000) concerns the process where the catalyst regeneration and adsorbed VOCs oxidation occur at simultaneous temperature and pressure increasing in the closed reactor-adsorber due to the compressed air delivery. The process advantage consists in the fact that there are no outflows before the regeneration ending; therefore, there are no desorption losses in the reactor. But such a process is technologically complex and requires high staff skills, which limits its application.
For the effective realization of adsorption-catalytic principle the multidispersed adsorption-catalytic system can be used (Zagoruiko 2004, Zazhigalov et al. 2012). It consists of relatively large adsorbent-catalyst pellets with the space between them filled with a microfiber catalyst (e.g. catalysts based on glass or metallic fiber support). Because the characteristic geometrical size of the catalyst microfibers (not more than 10 μm) is much smaller than the pellet size (more than 1 mm), the external specific surface area of the fibers is considerably larger than that of the pellets. As a result, during the system heating for regeneration, the microfibers are heated in contact with the hot gas flow much faster than the pellets. Accordingly, the VOCs desorbed during regeneration enter the already heated microfiber catalyst, which provides their effective oxidation.
Turning back to the other adsorption-catalytic process problem – the possible catalyst overheating above the threshold of its thermal stability, we should note that the most efficient solution is the optimization of adsorbent-catalyst maximal adsorption capacity. For example, as simulation showed (Zagoruiko et al. 1996, 1997) for the case of aromatic hydrocarbons oxidation on alumina-copper-chromium catalyst (with the operation temperature limit of 750°C), the adsorption capacity should be in the range 1–2 wt.%. It may be concluded that the optimum adsorption capacity often does not correlate with the maximal one.
In spite of relatively low energy consumption of adsorption-catalytic process in case of low VOCs concentration, there are some process modifications that allow to save even more energy. One of the solutions is the location of the heater for the regeneration initiation directly inside the inlet part of the adsorbent-catalyst bed (Figure 15B) (Zazhigalov et al. 2017). In such configuration, there is no idle energy consumption for heating the feeding pipelines and reactor shell as the energy is spent for the catalyst heating directly. It was shown that intra-bed heater positioning leads to significantly (by at least 2 orders of magnitude) decreasing the specific energy consumption for regeneration of the adsorbent catalyst and the required heater power compared to the conventional adsorption-catalytic process (Figure 15A).
Figure 16A confirms this statement and also shows that conventional adsorption-catalytic process (red dashed line) decreases energy consumption compared to not only thermal and catalytic oxidation processes but also the catalytic reverse-process for the low inlet VOC concentration. And the required heater power for the adsorption-catalytic process with internal heater location is about a few kilowatts (Figure 16B).
In the case of internal heater location the process efficiency may be increased also by the catalyst bed shape optimization and location of the heater in the narrow inlet bed zone (Zazhigalov et al. 2018). It was shown that the beds with truncated cone entrance with cone angle α=60° and higher (Figure 17) allow effective bed regeneration and continuous cyclic operation mode due to the balance of the heat wave propagation in the axial and radial directions (Figure 18). Nevertheless, we should note that the optimal cone angle depends on many process factors such as inlet gas flow rate, catalyst properties and others.
The other interesting modification of adsorption-catalytic process is the division of the adsorbent-catalyst bed into several non-communicating parallel sections (Figure 15C) (Zazhigalov et al. 2017). Each section is an independent adsorption-catalytic system with its own heating element that has a common gas inlet and outlet. These systems can be located either in one reactor shell with the inner partitioning thermo-insulating walls or in separate shells. During the adsorption cycle the inlet flow is evenly distributed between the sections and goes in parallel through all of them. The regeneration is performed in the sections by turn: when one section becomes completely regenerated, the regeneration starts in the next one and so on, in a cyclic way. It was shown that the mixing of all outlet gas flows leads to reduction of the peak outlet VOC concentration and gas temperature without any loss of the abatement efficiency.
4.2.3 Oxidation of sulfur dioxide
The reversibility of reaction of sulfur dioxide oxidation
creates serious limitations in achievement of extra-high SO2 conversion. The sorption-catalytic approach may be used to shift the reaction equilibrium towards SO3 formation.
This reaction is performed in industry with vanadia catalyst. Under reaction conditions, vanadia is present at the catalyst surface in a form of a melt, so it is possible to use the sorption properties of this melt with respect to SO2 as a basis for novel process approach. As soon as such sorption occurs in the liquid melt, it is possible to apply the term absorption-catalytic process.
A modeling study (Strots et al. 1992) has revealed that alteration of the reaction mixture (SO2 in air) and pure air may lead to significant rise of the sulfur dioxide conversion. It was explained by over-saturation of active component melt with oxygen during the air feeding cycle. During the following reaction mixture feeding cycle, the oxygen content in the melt exceeds the equilibrium level (with respect to oxygen content in reaction mixture), and it may lead to SO2 conversion, exceeding the equilibrium limit for the steady-state case.
The process may be realized in a two-reactor flow-sheet (see Figure 13B). While one of the beds is fed by reaction mixture, containing SO2 and oxygen, another bed is flushed by pure air stream. In the reaction mixture feeding cycle, SO2 is absorbed in the melt of active component, converting these into SO3. Sulfur trioxide is then removed from the outlet stream in acid absorber.
Process simulation studies (Vernikovskaya et al. 1999a) performed on the base of one-dimensional model of a plug flow adiabatic catalyst bed (Bunimovich et al. 1995) showed that it is possible to achieve extra-high SO2 conversion (above 99.9%) under such forced feed composition cycling. The comparison of processes with co-current and counter-current feeding of SO2 and air streams showed that the counter-current version provides stable operation in a wider range of inlet SO2 concentrations and inlet gas temperatures.
Such process may be realized within reasonable values of process characteristics (cycle duration, catalyst loading, gas temperature, etc.) at the second stage of the conventional “double contact-double absorption” systems in sulfuric acid production plants (Goldman et al. 1997). The overall conversion of sulfur dioxide in this system may reach 99.999%, corresponding to outlet SO2 content in waste gases ~50 ppm, this agreeing with the modern requirements of environmental protection regulations. This theoretical conclusion was successfully confirmed by the pilot-scale experiments (Zagoruiko 2006a).
The specific problem of such technology is possible desorption of SO2 and SO3 into the air flow during air flushing cycle. To avoid discharge of SOx into the atmosphere, it is possible to use this stream as a feed air in sulfur furnaces in the sulfuric acid unit using elemental sulfur as a feedstock. In case of processing of waste gases from non-ferrous smelters (Zagoruiko 2006b), this air flow may be used for dilution of waste gases with high SO2 content (up to 40 vol.%) from autogenous melting furnaces.
4.2.4 Sorption-enhanced steam reforming of hydrocarbons
Active development of hydrogen-based “green” energy technologies is limited by the absence of efficient and low-cost hydrogen production technologies. Manufacturing of H2 from water using solar, wind and renewable energy sources still cannot compete economically with hydrocarbon feedstock. Existing technologies of hydrogen production from fossil hydrocarbons have significant carbon footprint in terms of global climate change impact. Besides, these technologies are feasible at high productivity scale only.
Therefore, there is a strong need for the development of a novel inexpensive, easily scalable technology which may use low-grade, non-conventional or renewable feedstock (stranded methane, gaseous and liquid hydrocarbon wastes, biogas, bioethanol, etc.).
The novel process is based on known approach of sorption-enhanced catalytic steam reforming of hydrocarbons, e.g. methane:
with simultaneous adsorption of CO2 by solid sorbent, e.g. CaO:
CO2 removal from reaction mixture shifts the equilibrium in reactions (IX) and (X) giving high yield of pure hydrogen in one conversion stage. Exothermal reaction (XI) provides direct internal supply of energy. This approach was proposed by professor August Brun-Tsekhovoy (Kurdyumov et al. 1996), and later it was developed by many research groups worldwide.
Despite all these efforts, the technology is still not used in practice. The reforming cycle (reactions IX and X) is well studied, but sorbent regeneration (backward decomposition of carbonate) causes serious complications: the direct external heating of the fixed sorbent beds leads to inefficient energy losses and is possible only in reactors with relatively small diameter.
It was proposed (Zagoruiko and Okunev 2007, Zagoruiko et al. 2008) to perform the sorbent regeneration under super-adiabatic conditions, using oxidation of available combustible substances (e.g. methane) in the air flow directly in the fixed sorbent-catalyst bed:
Superadiabatic heat wave is formed in the bed under filtration of the air-fuel flow with low inlet temperature. As described above, the maximum temperature in such wave is known to significantly exceed the adiabatic one, thus providing fast and complete decomposition of carbonates. The emission of the reaction heat directly inside the catalyst-sorbent bed minimizes the possible heat losses and provides efficient regeneration of sorbent in the reactor of any size.
The process may be realized in a two-reactor scheme (Figure 13B) with counter-current feeding of the feedstock, which cyclically alternates between steam/methane and air/methane mixtures. The process has minimum heat exchange infrastructure, low capital and operation cost, good scalability in wide range of production capacities and high flexibility with respect to feedstock type.
4.2.5 Anaerobic oxidation and oxidative dehydrogenation of hydrocarbons
Forced feed composition cycling may find very interesting applications in the complex reaction systems where selectivity and products yield are of importance. For instance, it may relate to processes of selective oxidation or oxidative dehydrogenation of paraffins to olefins (propane to propylene, butane to butylene and butadiene, etc.).
The process approach here is based on separate feeding of reagents (hydrocarbons and oxygen); it may be realized in a multi-bed flow sheet shown in Figure 13B. In the reduction cycle hydrocarbons are oxidized in the catalyst bed by lattice oxygen in the absence of molecular oxygen:
while in the reoxidation cycle the lattice oxygen storage is replenished by contact of catalyst with air:
where [Ox] and [Red] are oxidized and reduced sites at the catalyst surface, respectively.
It was proposed that forced feed cycling regime in isothermal conditions may enable an increased selectivity itself, but actually this was not confirmed by experimental data. At the same time, the unsteady-state approach enables manipulation of the heat regimes in the adiabatic catalyst bed, thus potentially leading to selectivity gain. The reactions of selective oxidation are exothermic, and, in most cases, the selectivity of partial oxidation decreases with rise in temperature. In the steady-state catalytic process this imposes significant limitations on the maximum concentrations of reactants in the initial feedstock or requires the application of complicated tubular or fluidized bed reactors for efficient heat management in the process. Obviously, in the unsteady-state process it is necessary to provide effective conjunction of heat and concentration front in the catalyst bed.
Earlier model studies of the system of parallel exothermic reactions (Zagoruiko 2005a,b) showed that feeding of preheated hydrocarbon into a cold fixed catalyst bed results in formation of a heat front with maximum temperature much lower than that for the equivalent steady-state process. It is caused by consumption of a part of the reaction heat for heating of the catalyst. Simulation studies also showed potential existence of the “reversed” heat front, which moves counter-currently to the direction of the reaction mixture flow in the cold catalyst after heat initiation in its inlet part of the bed due to the bed heat conductivity. In this case, the maximum temperature may be decreased even more significantly, thus enabling the improvement of reaction selectivity and use of the undiluted hydrocarbon as a feedstock.
More detailed analysis of the reaction enthalpies showed that the reduction stage of anaerobic oxidation reactions is endothermic due to energy consumption for breaking of the lattice oxygen bonding with catalyst surface. For example, a simplified mechanism of propane oxidative dehydrogenation to propylene may be represented in case of anaerobic oxidation by vanadia as follows (Zagoruiko 2007b, 2008)
Estimation of the heat effects of the reaction stages showed that anaerobic oxidation stage (XV) has practically zero heat effect (−4.6 kJ per mole of propane), while the total high exothermic effect of selective oxidation of propane into propylene (116.8 kJ/mol) is caused by high heat effect of catalyst reoxidation stage (XVIII) – 121.4 kJ per mole of V2O4. At the same time, the stage of deep oxidation of propylene (XVI) and propane (XVII) have very high heat effect of ~840 kJ/mol.
Obviously, the bonding energy of the lattice oxygen is different for different active oxides in the catalyst composition. As seen from Figure 19, oxides with relatively high oxygen bonding energy (like iron oxide) may have much more significant heat effect, while catalysts with lower bonding energy (like ceria) may even turn this effect into an exothermic one. Therefore, we may conclude that the temperature regime in the anaerobic process will depend upon not only its activity and selectivity but also on its chemisorption capacity with respect to oxygen and to thermodynamic properties of the catalyst; this is quite untypical for steady-state processes.
Process simulation studies (Zagoruiko 2007b, 2008) showed that under feeding of undiluted propane with inlet temperature of 300°C into the cold catalyst bed there occur the formation of heat front moving in the direction of gas flow. Due to the practically zero heat effect of the target reaction and negligible formation of deep oxidation products, the maximum temperature in such front practically does not exceed the inlet gas temperature.
Besides, the process may be realized in the continuous regime with periodical alteration of propane and air feeding into the preheated bed (e.g. using the flow-sheet shown in Figure 13B). As shown by simulation studies, the counter-current air/propane feeding is more advantageous for establishment of autothermal cyclic regime, which does not need to use external energy and any preheating of air or propane flows.
According to preliminary technological evaluations, the use of undiluted propane enables us to significantly increase the process productivity. Unique thermal specifics of the unsteady-state process permit hydrocarbon oxidation reaction at moderate temperatures, thus providing high selectivity of partial oxidation and minimum formation of deep oxidation products. The excessive heat is transferred to the catalyst reoxidation stage, where there are no hydrocarbons and no related issues with selectivity decrease at elevated temperatures. Other potential advantages may include the following:
Technological simplicity and low capital costs due to use of simple and cheap fixed bed reactors with minimized heat transfer infrastructure.
Higher process safety due to absence of direct contact between air and hydrocarbons.
Possibility to use air instead of pure oxygen with no need to separate reaction products from nitrogen.
Potential possibility to increase the catalyst lifetime due to efficient coke accumulation prevention resulting from coke incineration in the catalyst reoxidation stage.
The described approach may be applied to a wide range of selective oxidation and oxidative dehydrogenation reactions.
4.2.6 Chemisorption-catalytic decomposition of hydrogen sulfide
Active development of hydroprocessing technologies and hydrogen energy require the development of new efficient ways for production of hydrogen, preferably from non-hydrocarbon feedstocks.
Hydrogen sulfide seems to be a very attractive feedstock for this purpose. First, it is the conventional waste from oil and natural gas processing facilities. Second, the bonding energy of hydrogen in H2S is the lowest among all natural hydrogen-containing compounds.
Unfortunately, the reaction of hydrogen sulfide decomposition
is characterized by severe equilibrium limitations. The complete H2S decomposition requires extra-high temperatures (above 1500°C), leading to high energy consumption, necessity to apply expensive thermostable materials and risk of backward element recombination at the cooling stage. In (very typical) case when the carbonaceous compounds (CO2 and hydrocarbons) are present in the gas feedstock, such temperatures may also cause side reactions with formation of undesired products (coke, CO, COS and CS2). Because of these, still there is no feasible technology for H2S decomposition in wide practical application.
The alternative process approach (Govorov et al. 1993, Startsev et al. 2002, 2004, Zagoruiko 2019) is based on the chemisorption enhancement of the decomposition reaction (XIX). The process, involving metal sulfide chemisorbent-catalyst, includes cyclic alteration of two reaction stages technologically separated in time and space:
As shown by thermodynamic calculations (Zagoruiko 2019), the reaction (XX) is exothermic and the corresponding equilibrium conversion of H2S may reach 100% at ambient temperature. Reaction (XXI) requires higher increased temperatures (up to 600–800°C). The whole process may be thus realized with achievement of complete H2S decomposition; the backward recombination of elements is prevented by technological separation of stages (XX) and (XXI), and hydrogen is not present at stage (XXI).
Low operation temperatures at H2S chemisorption stage provide decrease of heat losses to environment, resulting in increased energy efficiency of the process. The process may be based on cheap standard equipment from conventional materials, thus significantly minimizing the capital costs. Besides, it makes processing of feedstock gases (containing carbonaceous admixtures) possible without risk of their involvement in reaction and formation of undesired side products.
5 Mathematical modeling of unsteady processes in heterogeneous catalytic reactors
5.1 Mathematical description of ongoing processes and phenomena
Mathematical models for the simulation of unsteady-state processes have some peculiarities with respect to models for steady-state processes. Disturbances in the gas composition involve not only a change in the temperature inside the catalytic bed but also a modification of the catalytic surface. There are a number of physical/chemical phenomena whose role is negligibly small in the steady-state conditions. However, they may turn out to become decisive in unsteady-state regimes. These factors may include the heat transfer along a bed and inside a particle, as well as unequal accessibility of external particle surface. At steady state, many factors may affect the system behavior independently and often additively. It is also important to determine the correlation between the parameters. Quite often, the same factors should be taken into account separately, when constructing a model for unsteady-state processes. Thus, e.g. in the steady-state regime it is sufficient to take into account the heat dispersion effect along an adiabatically operating catalyst bed by means of the effective heat conduction coefficient. It is inadmissible to do in modeling the unsteady-state processes. The heat transfer along the catalyst bed frame and the heat exchange between the reaction mixture, as well as the external particle surface and, sometimes, the heat transfer inside a porous particle, have to be accounted for separately (Matros 1985).
In the reverse flow operation, concentration and temperature gradients in the inter- and intraparticle fields of the catalyst bed are generally higher than those that occur in steady state. This can result, i.e. in the absence of a proportionality between temperature and conversion, in a short significant overheating of the particle surface. Through the forced unsteady state mode, local overheating of the catalyst can greatly exceed an adiabatic temperature rise at complete conversion. Nevertheless, the principles being used in constructing and analyzing mathematical models for reactors operating either in stationary or non-stationary conditions have more common features. Let us consider them in close connection with each other.
To solve the problem of sizing of a reactor – scaling up a process from laboratory concept into an industrial-scale plant – a similarity concept was first applied in Germany and then in the US since 1925. It was believed that laboratory installations and full-scale plants behave similarly if they have geometric similarity, kinematic similarity, dynamic similarity, thermal similarity and chemical similarity. The application of the theory of similarity was rather successful for the hydrodynamic and thermal processes. However, it was impossible to reach all similarities simultaneously for catalytic reactors operated continuously. The differences in dimensions and typical operating conditions between the reactors of different scales resulted in differences between the obtained conversions and product yields, because the flow patterns and heat and mass transfer effects entered the arena (Thomas and Thomas 1997, Zlokarnik 2002). A great advantage over the similarity theory is the method based on fundamental knowledge of chemical transformations and physical processes of matter, heat and momentum transfer. The basis of scaling up of a reactor is the construction of mathematical models of catalytic processes by combining computational and full-scale experiments using modern methods of mathematical analysis. Therefore, the chemical process is studied via a model that has another physical basis but whose mathematical description is identical to the mathematical description of the reactor (Boreskov and Slinko 1961).
The founder of the theory of mathematical modeling of catalytic processes and reactors in Russia was M.G. Slinko. His creative activity from 1935 to 2008 was closely connected with the development of catalytic processes engineering and, first of all, with the development of a theoretical base for the mathematical modeling of catalytic processes and reactors (Slinko 1968, 2007). The principles of mathematical models of fixed bed reactors operating under forced unsteady-state conditions were proposed by Prof. Yu. Sh. Matros and developed by his group (Matros 1985, 1989).
The first translation of our understanding of chemical reactions into mathematics was kinetic rate expression. The theory of adsorption, desorption, and surface reaction on ideal surfaces developed by Irving Langmuir (New York) and Cyril Hinshelwood (Oxford) was extended to nonuniform surfaces by Mikhail Temkin, who worked at the Karpov Institute of Physical Chemistry, Moscow. It was a key milestone on the historical way of the successful development of mathematical modeling. The widely applied Temkin-Pyzhov kinetic equation for ammonia synthesis was suggested in 1939 (Temkin and Pyzhov 1939). One can find a comprehensive historical survey on kinetic modeling in catalysis in the book of G. Yablonskii (1991). The kinetic models reflect the dependence of the reaction rate on the composition and pressure of the gas phase and on the temperature and state of the catalyst at any time. The unsteady-state processes on the internal surface of the catalyst can be divided into two classes (Matros 1985). The first one includes the processes for which the unsteady state of the catalyst arises via periodic variations in the composition, temperature and pressure. This results in the change of the catalyst state. The second one concerns unsteady state processes with changing properties and composition of the catalyst surface even if the gas-phase conditions do not vary with time. These are processes with irreversible changes in the catalyst activity.
A kinetic model of the nonstationary process of oxidation of sulfur dioxide on a vanadium catalyst is an example of a kinetic model taking into account the nonstationary state of a catalyst being dependent on the gas phase condition. The proposed kinetic model (Ivanov and Balzhinimaev 1987, 1990) includes both catalytic reaction steps and side processes involving changes in the catalyst’s state. Sulfur dioxide oxidation reaction on alkali-promoted vanadium catalyst is known to take place in the melt phase of an active component distributed throughout the porous catalyst pellet. The catalytic cycle involves the following steps:
where the intermediates were assumed to be coordinatively unsaturated binuclear vanadium (V) , and sulfite complexes. The sulfite complex can further decompose to sulfur trioxide and an inactive vanadium complex
Interphase mass exchange by the reactants, SO2, O2 and SO3, is assumed to be so fast that concentrations of SO2 and O2 in the melt are given by Henry’s law:
where 1 is SO2, 2 is O2, and Hj is Henry’s constant of jth reactant dissolution (mol/m3/atm). The SO3 concentration in the melt correlates to its mole fraction in the gas phase in a more complex way:
This equation arises because of the fast reactions between dissolved sulfur trioxide and sulfate anions in the melt:
where C0 is the limit of sulfur trioxide solubility (mol/m3), is the concentration of jth reactant in the melt (mol/m3), and H3 is the observed Henry’s constant for SO3 (mol/m3/atm).
Each of the binuclear vanadium complexes Zm taking part in reactions (IXX)–(XXII) can transform into a coordinatively saturated form, which is inactive in reactions (IXX)–(XXII) by interaction with the pyrosulfate anion:
The m-indices correspond, respectively, to the following: 1 is 2 is 3 is and 4 is Since reactions (XXVI) and (XXVII) occur much quicker than steps (XXII)–(XXV), they may be accounted for using equilibrium relationships. Such an assumption results in the following equation for the concentration of the active complexes, θa:
The rate equations for steps (IXX)–(XXII) are in accordance with the mass-action law:
ki is the rate constant of the forward reaction for ith reaction step, 1/(s·atm), (1/s for k4); k−i is the rate constant of the reverse reaction for ith reaction step, 1/(s·atm), (1/s for k−3); and KL is the equilibrium constant of reaction (XXVI) (m3/mol).
The equations for vanadium intermediates are as follows:
The expressions for the observed rates of the generation of SO2, O2 and SO3 are written as follows:
The observed rates have to be included into the equations of material and heat balances for a catalyst particle and/or packed bed. The model was shown to be capable of describing both dynamic experimental results of reverse-flow SO2 oxidation in a fixed-bed reactor (Bunimovich et al. 1995, Vernikovskaya et al. 1999a) as well as known steady-state data. Silveston et al. (1994) have used this model to explain the results of periodic concentration forcing experiments performed by Briggs et al. (1977). The time scale of reactions (XXII)–(XXV) is about ~0.1–10 s.
Another example of a kinetic model taking into account the dynamic properties of the catalyst surface is the kinetic model of selective nitrogen oxides reduction by ammonia on vanadium-containing catalysts (SCR). This model considers the dynamic character of interaction of NOx and ammonia on the catalyst surface. The model parameters were elucidated by Bobrova L.N. with coworkers (Noskov et al. 1993, 1996, Borisova et al. 1997, Popova et al. 1998a,b, Smith and Bobrova 2002).
The basic reactions of the process are the following:
The kinetic model involves the following steps:
This mechanism makes it possible to take into account the adsorption capacity of vanadium catalyst surface. For typical SCR catalysts this capacity was estimated to be about 10 ammonia volumes per catalyst volume. The change in ammonia surface coverage (θ) is described by the following equation:
where is the rate of reversible NH3 adsorption; W2=k2CNOθ is the rate of adsorbed ammonia interaction with NO; W3=k3θ is the rate of adsorbed ammonia oxidation; and a is the adsorption capacity of the catalyst surface. The reaction and desorption time scales are ~0.1 s and 10−5 s, correspondingly (Smith and Bobrova 2002).
The determined rates were included into the equations of material and heat balances of the mathematical model of unsteady-state SCR process in the fixed catalyst bed (Noskov et al. 1993, 1996, Borisova et al. 1997, Smith and Bobrova 2002). The model in particular describes the high degree of gas conversion of NOx obtained in the industrial unit.
The next step was a coupling of transport phenomena with reaction kinetics. The methods of mathematical modeling in catalysis have proved themselves in Russia in the early 1930s, during the industrialization period, when the Russian physicist D.A. Frank-Kamenetskii developed a theory for coupled chemical reactions and mass transfer on non-porous solid surfaces in connection to combustion processes. This work became known in the Western world only in 1955 after translation into English (Frank-Kamenetskii 1955). Especially important landmarks were three pioneering treaties on the interaction of chemical and physical phenomena: one by G. Damkohler in Germany at the end of the 1930s, the next one by E.W. Thiele in the United States and the third one by Ya.B. Zeldovich in Russia developed the theory of simultaneous reaction and diffusion in porous catalyst particles (Roginskii and Zeldovich 1934, Zeldovich 1935).
There is an essential difference between the conditions of the strong (or weak) effect of one or another parameter on the steady- and unsteady-state conditions inside the catalyst particle. The unsteady-state conditions are strongly affected by the ratio of the total heat capacity of the solid phase and the gas phase in the catalyst particle to the heat capacity of the gas phase (Matros 1985). Increasing the heat and mass transfer between the fluid and catalyst surface due to an increase of the maximum temperature results in a decrease in the thickness of the reaction zone. The effect of coupling transport phenomena in a porous catalyst with reaction kinetics with respect to unsteady processes in adiabatic reactors was considered in (Noskov et al. 1982, Matros et al. 1989).
The inertia of the processes on the internal surface of a catalytic particle also influences the dynamic properties of the catalyst and vice versa. For example, the dynamics of hydrocarbons’ deep oxidation in the catalyst pellet with account of intraparticle diffusion limitations under alternating adsorption and oxidation stages was studied by means of mathematical modeling in Vernikovskaya et al. (1999b). Namely, the isopropyl benzene oxidation by oxygen on Cu/Cr/Al oxide catalyst was considered. The kinetic scheme of the reaction was formulated in the following form:
where A is the molecule of the original organic compound, [O] is the oxidized active site of the catalyst, [AO] is the compound A being reversibly adsorbed on the catalyst surface, and [P] is the compound irreversibly chemisorbed on the catalyst surface. To describe the processes inside the sphere-shaped catalyst pellet, the non-isothermal mathematical model was used. The model accounts for transient heat and mass transfer in a catalytic pellet, transient equations for the surface compounds, heat and mass transfer resistance between the gas and solid phases at the outer surface of the catalyst pellet, and transient equations for the temperature and the concentrations in the gas phase in the continuous stirred tank reactor. It was shown that intraparticle diffusion limitation significantly affects both adsorption and oxidation stages. Simulation results have demonstrated the existence of an optimal size for the catalytic particle that provides the maximum value for purification efficiency in the adsorption-catalytic process with VOCs. The adsorption, oxidation and diffusion time scales were estimated to be about ~10−6 s, 10−1 s and 101–102 s, correspondingly.
The mathematical modeling of the processes in the catalytic bed and in the reactor are the next steps of mathematical description of heterogeneous catalytic reactors. By using mathematical approach for steady processes, many design problems were solved in Russia in 1935–1940, when a number of chemical reactors and other apparatuses (e.g. for production of sulfuric acid, nitrogen-containing substances and synthetic rubber) were put into industrial operation (Slinko 2007). General chemical reaction engineering approach to a wide range of topics, all dealing with chemical reactors, was first formulated in 1956, when a group of European chemical engineers came together in Amsterdam for a symposium. Professor van Krevelen suggested that they name the theme of their conference chemical reaction engineering (1958). Four interrelated quantities – flow pattern, kinetics, heat transfer and mass transfer – were formulated to govern reactor behavior. Their relationships in the form of mathematical expressions (mathematical model) describing the dependence of a process output (yield, product properties, etc.) upon process variables should include the following: (1) design parameters (size and shape of equipment and of individual parts of equipment), (2) operating (process) conditions (temperature, pressure, flow rates and compositions) and (3) physicochemical properties (thermodynamic data, transport properties and kinetic constants) (Levenspiel 1999, Cybulski et al. 2001).
The behavior of a catalytic reactor is multi-scaled with respect to both time and space, spanning from quantum scales to macroscopic scales. The space scale of chemical systems ranges from 10−10 to 103 m, including those of atoms (~10−10 m), metal crystals and active component particles (~10−9–10−7 m), catalyst particles (~10−4–10−2 m), the reactor (~10−4–101 m) and the plant (>101 m). The associated time scale ranges from 10−15 to 108 s. Hierarchical multiscale modeling approach, when prediction of large-scale process performance based on small-scale information was termed upscaling, was first applied in 1946, when a plant for heavy water production by multistage electrolysis in combination with catalytic isotope exchange of deuterium between hydrogen and water vapor was very rapidly designed in Russia. At the molecular level, thermodynamic and kinetic problems were solved; second, at catalyst-grain level, kinetic equations taking into account diffusion processes were determined. At the third level, the structure of a separating element was determined, and at the fourth, the optimal cascade was calculated. The interdependence and interrelationship of these levels was characterized by the integrity and specificity of the catalytic separation process (Slinko 2007). Later, this approach was summarized and developed by Mikhail Slinko in the book (1968).
The hierarchical multiscale modeling approach is also applicable to the mathematical modeling of nonstationary processes. Different levels of the mathematical model of the reactor, i.e. an internal surface of isolated catalytic pellet, the volume of voids between the pellets, etc., can be considered as the elementary dynamic units or groups of units (Matros 1989). Every unit has its own inertial properties determining the variation of its state in time. A time scale of an unsteady-state process can be evaluated as the ratio of the unit capacity to the intensity of its outer link. Time scales for model components are determined by the time scales of the units comprising the links between them. Relationships between the units often have both distributed and reversed characters. This results in a complex dependence on the time scales of all units. The construction of a specific mathematical model of a reactor requires evaluating this dependence since it allows summarization of the basic properties of those elements, while each of them affects decisively on both the static and dynamic characteristics of the reactor as a whole.
A detailed theory of mathematical modeling of the unsteady processes in catalytic reactors has been presented by Matros (1989), including an analysis of mathematical models for two limiting cases: a stationary heat front in an infinitely long catalyst bed (Boreskov et al. 1979a,b, Kiselev and Matros 1980) and a relaxed steady-state process corresponding to frequent flow reversals (Boreskov et al. 1983). The results referred to in the cited studies allow for satisfactory explanation of some regularities observed in the process with the flow reversals and help to determine control strategy. One can find the results of mathematical modeling of the unsteady processes in the fixed bed reactors, for example, in Agar and Ruppel (1988), Borisova et al. (1997), Budhi et al. (2004), Bunimovich et al. (1995), Eigenberger et al. (2007), Fissore and Barresi (2007), Gosiewski (2004), Li et al. (2018), Matros (1989), Muñoz et al. 2015, Noskov et al. (1996), Salomons et al. (2004), Smith and Bobrova (2002), Snyder and Subramaniam (1993), Somani et al. (1997) and Van De Beld and Westerterp (1996).
As an example, we present the results of mathematical modeling of SCR of nitrogen oxides by ammonia under reverse flow operation in catalytic bed (Vernikovskaya et al. 1993, Bobrova et al. 1997b). A two-dimensional, two-phase mathematical model with steady-state kinetic equations was used in order to examine the spatial temperature nonuniformities in the reactor. A similar model was previously used in modeling of temperature inhomogeneities in a flow reversal reactor with first-order reaction (Bunimovich et al. 1990). Rates of NO reduction (WNO) and ammonia oxidation (WNH3) were described in the following way:
where CNO and are nitrogen oxides and ammonia concentrations; kNO and are the apparent rate constants; K is adsorption equilibrium constant.
The two-dimensional heterogeneous model with reaction kinetics (9) was used with assumption of a low heat capacity of the reactor walls. The model assumes temperature nonuniformity arising from heat losses through the wall and the void fraction distribution. The monotonous approximation of the radial porosity was applied, describing an increase in porosity at a peripheral zone of 5–10 catalyst pellets. The catalyst pellets were assumed to be isothermal. This assumption is permissible in most cases, since the effect of the finite rate of heat transfer in a catalyst pellet on the characteristics of nonstationary regimes must be taken into account only for large pellet sizes, high linear gas velocities and significant adiabatic temperature rises. The longitudinal heat and mass transfer both in the gas phase and in the solid phase is assumed to be negligibly small. The model takes into account heat and mass transfer between the surface of catalysts particles and gas phase, longitudinal convection, effective radial diffusivity and thermal conductivity.
The conservation equations can be written as follows:
in solid phase
in gas phase
where l∈(0, L), r∈(0, R0) and t are the axial, radial and temporal coordinates; aV is the specific outer surface of the grain; Ts, Tf, Tm are the temperatures of solid phase, gas phase and environment; are concentrations of NH3 and NO in solid and gas phases; are the effective radial conductivities in gas and solid phases and gas phase diffusivity; αsf, β are the effective solid-gas heat and mass transfer coefficients; v0 is the superficial fluid velocity; αfm, αsm are the effective coefficients of heat transfer from gas and solid phases through the reactor wall towards the environment; are the gas and catalyst heat capacity per unit volume; and ε, εg are the bed void fraction and catalyst grain porosity.
The set of coupled partial differential equations (PDEs) (10)–(14) with the boundary (15) and initial (16) conditions was solved numerically. The algorithm was described in Drobyshevich and Ilyin (1984). Simulation of the start-up regime shows that for several hours of the reactor preheating, the cold zone is formed near the reactor wall. After the gas feeding, NOx conversion at the walls can be about 10% below and then is in the reactor center. The maximum temperature near the wall reduces as well. At these conditions ammonium salts may deposit in the catalyst and inert beds, thus giving undesirable side effect.
Temperature profiles formed at quasi-steady state mode along the axis (1) and near the wall (2) in the reverse flow reactor are shown in Figure 20. Typical radial evolution of the temperature is observed along the whole reactor. The high gradients, more than 100°, may be observed in the catalyst bed. At a low adiabatic temperature rise, even a small heat loss can lead to the reaction extinction near the wall. Although spatial nonuniformity brings about moderate loss in efficiency of unsteady-state SCR process (within 1–2%), the cold zones being formed can lead to undesirable effects.
At some value of the temperature gradient NOx conversion even can rise compared to the reactor axis (curve 2, Figure 21) as a result of increasing ammonia fraction for NOx reduction. The observed drop in the NH3 conversion can be more than in NOx. The rate of ammonium salts formation depends in direct proportion to ammonia slip. It is required to be prevented at designing the reactor.
Nowadays, a hierarchical multiscale approach is intensively employed in reactor modeling and simulations to take into account the complexity of reactions at the different scales, such as catalyst surfaces and pellets, atomic or molecular levels.
5.2 Methods and algorithms
The development of mathematical modeling has always been closely related to significant achievements in computational mathematics. The profession of chemical engineers was one of the first to use computers to model heat and material balances in a routine way. The dependence of the individual rates of reaction on temperature is highly nonlinear. Moreover, more than one species and more than one mechanism of reaction, each one incorporating a numerically differing, non-linear and somewhat uncertain dependence on temperature must be considered, leading to a set of non-linear algebraic equations. These algebraic equations are strongly coupled with the equations of conservation in partial differential or integro-differential form. The specification of proper boundary and initial conditions are incorporated in the mathematical description in order to fully determine a reactor configuration. For every PDE, a set of discretized equations equal to the number of grid cells has to be solved. For every grid cell, the governing mass, momentum, and energy equations describing the direct interactions between chemical reactions, fluid mechanics, heat transfer, and mass transfer are assumed to hold.
In the modeling of nonstationary processes in fixed-bed reactors, various methods and algorithms have been used (Noskov et al. 1982, Drobyshevich and Ilyin 1984, Drobyshevich 1988, Matros et al. 1988b, 1989, Drobyshevich et al. 1990, Silveston et al. 1994, Bunimovich et al. 1995, Vernikovskaya et al. 1999b). Among them are algorithms based on implicit balance difference approximations (Noskov et al. 1982, Drobyshevich and Ilyin 1984, Drobyshevich 1988, Matros et al. 1989, Drobyshevich et al. 1990, Vernikovskaya et al. 1999b), algorithms based on the collocation method for numerical modeling of nonstationary processes of heat and mass transfer on the catalyst grain (Vernikovskaya et al. 1993), (1999a,b), algorithms based on the method of straight lines for transforming PDE into ordinary differential equations (ODE) and Rozenbroke’s algorithm for solving the system of nonlinear ODEs (Matros et al. 1989), (Silveston et al. 1994), (Bunimovich et al. 1995), (Vernikovskaya et al. 1999a,b).
5.3 Multiscale flexible simulation approach to modeling and upscaling
Currently, there is explosive growth in multiscale modeling as a method to seamlessly and dynamically link models and phenomena across multiple length and time scales.
In non-stationary processes, due to external processes and/or changes in the internal properties of the catalytic process and the reactor, the temperature and concentration fields in the catalyst layer vary with time. They can also vary in the catalyst grain. In this case, the catalyst can be in a nonstationary state, and the observed rates of chemical reactions can differ substantially from quasi-stationary ones. The time scales of the various processes in the reactor can differ greatly from one another (Matros 1989). This difference means that fast and slow processes occur independently, and in mathematical modeling, there is no need to consider these processes simultaneously. Then, when studying the process on the time scale of the “fast” element, the process in the “slow” element can be considered unchanged, and when studying the process on the time scale of the “slow” element, the process in the “fast” element will be quasi-stationary. However, there are processes in which the time scales of different processes are comparable or when there is a need to simultaneously consider the nonstationarity of these processes at strongly differing time scales. In this case there is a well-proven approach, based on the application of an identical set of numerical tools such as integro-interpolation method, method of straight lines, a special case of a second-order Rosenbrock method, tridiagonal matrix algorithm or Thomas algorithm on each scale of a multiscale reactor model (Vernikovskaya 2017). Step size control is implemented with account for the rate of change of the variables on each scale.
The scope of this approach is limited. In the case of multiscale modeling of unsteady processes in heterogeneous catalytic reactors, the system of PDEs on each scale can contain both first- and second-order derivatives with respect to one of the coordinates and the first-order derivatives with respect to time. The dimensionality of the model on each scale of a multiscale reactor model has to be less than or equal to 2D.
Put down the equations for a reactor-scale model. For clearness and simplification of the algorithm description, assume that these equations have the next simple form:
where u=(u1, …, uM) is, for example, the vector of the concentrations of substances and temperature; and ai, bi, ci are the coefficients depending nonlinearly on the solution, i.e. on the concentrations of substances and temperature.
5.3.1 Integro-interpolation method
Discrete analogues of the model equations were constructed by using the integro-interpolation method. Transition to discrete models cannot be made in a purely formal way. Such a transition must leave in force as many basic properties of the source object models as possible (Samarskii and Mikhailov 2001, Vernikovskaya 2017). With respect to the equations of system (17) with initial (18) and boundary (19) conditions, such a transition means that for the corresponding difference scheme it is necessary that the discrete analog of the mass and energy conservation laws be fulfilled. A common approach to the construction of such difference scheme is based on writing this law in an integral form for cells xj−1≤x<xj, xj=j·hj, j=1, N of the difference grid:
Substitution of the obtained integrals and derivatives with approximate difference equations gives the next system for each ui:
5.3.2 Method of straight lines
We do not introduce discrete formulas for derivatives with respect to “x”, thus transforming PDE for each ui into a set of ODEs for nodal values along the reactor radius. The process of transformation of equation (17) into a system of ODEs for each variable ui is an example of application of the method of straight lines or the method of semi-discretization (Vernikovskaya 2017). By successively applying this procedure to all inner nodes and taking account of the boundary conditions (21) in the relation at j=1 and j=N, one gets a system of equations for each variable ui=(u1, …, uM) written in the following form:
where tridiagonal matrix Ai and vector Ci depend on the solution in a non-linear manner.
The method of straight lines is advantageous because to solve the semidiscrete form of initial differential equation in partial derivatives, it is possible to use different methods for ODEs.
Let us put down system (22) for one variable ui, for simplicity, skipping index “i”:
where A is a tridiagonal matrix of dimensionality N×N; C is a vector of dimensionality N which is nonlinearly dependable on the solution; and u, f(u) are vectors of dimensionality N.
5.3.3 A special case of a second-order Rosenbrock method
When solving the problems with initial data described by ODEs the best results are normally obtained with either linear multistep or Runge-Kutta methods. In 1963, Rosenbrock (1963) proposed the methods that differ from the explicit Runge-Kutta schemes by the regularization of the right part of a differential problem. To solve the system of equations (23) let us make use of a second-order L-stable Rosenbrock-type formula as follows (Novikov 1990, Vernikovskaya 2017):
In this case the Jacobi matrix is of dimensionality N×N and systems of linear equations (26) and (27) can be solved using some laborious algorithms such as lower-upper (LU) decomposition with the number of operations proportional to N3. As applicable to our system of equations (23), in which matrix A is tridiagonal, it may be preferable to use the tridiagonal matrix algorithm concurrently with the iteration process. To avoid the limitations associated with the setting and application of the iteration process, the integration step is put under control to attain the convergence of the iteration process in one iteration.
5.3.4 Step size control
The step size adjustment algorithm is based on the computation of the values of the right part and difference between kn1 and kn2 for each equation of system (24):
Then maximum values of vectors R, RE, RQ, are determined for each variable of system (22):
5.3.5 Representation of the reaction term
The reaction term in the mass balance equations of system (22) was represented as a sum of two terms to extract a linear part with respect to the ith component. This allows the assigned linear part to be transferred to the diagonal of the equations system, thus increasing the monotony of the system and securing non-negativity of the concentrations (Rapatskii and Yausheva 1994, Vernikovskaya 2017).
5.3.6 Distinctive features of the method of solution for a two-phase model
In the case of a two-phase model the equations for the second phase are added to system (17). The combination of numerical methods remains the same. In some cases, when there is a lack of terms describing convective and diffusive transfer in the second phase, it is possible to implement a special run algorithm (tridiagonal matrix algorithm or Thomas algorithm) securing the solution at step n+1 in both phases.
5.3.7 Distinctive features of the method of solution on the second and first scales
The combination of numerical methods used for the solution of the equations of a second-scale model (catalyst grain) will be identical to that for the solution of equations of a reactor-scale model. The solution algorithm can be simplified only at modeling the processes on the first (atomic-molecular) scale when heat- and mass-transfer processes are not considered. For this scale the equations of the dynamics of adsorbed and intermediate species are put down as follows:
The system of equations (29) can be solved with an L-stable formula ensuring the second-order accuracy (24)–(27). The dimensionality of the system is equal to the number of adsorbed and intermediate species, Nθ. As a rule, for heterogeneous catalytic reactors this number is sufficiently small, at least in comparison with the system dimensionality when using the method of straight lines. Therefore, the dimensionality of a Jacobi matrix and matrix Dn, which is equal to Nθ×Nθ, is also small so it is possible to apply LU decomposition method to linear equations (26) and (27).
5.3.8 Distinctive features of the method of solution for a multiscale reactor model
The identical set of numerical tools (17)–(29) can be used on each scale of a multiscale reactor model (Vernikovskaya 2017). Step size control is implemented with account for the rate of change of the variables on each scale.
Simultaneous solution of a system of equations, including all the scales under consideration, which is characterized by high dimensionality, is extremely time-consuming because bulky data storage is required. The introduced approach of numerical solution significantly saves computation time and provides adequate accuracy and stability. The developed approach and the combination of numerical methods suggested for the solution of equations for the model on each scale proved effective for multiscale modeling of unsteady processes in heterogeneous catalytic reactors. This approach, for example, was used for modeling transient behavior of the methane partial oxidation in a short residence time reactor (Vernikovskaya et al. 2007, Bobrova et al. 2012, Vernikovskaya 2017). A monolith catalytic reactor has been analyzed by means of a dynamic one-dimensional two-phase reactor model with account for both transport limitations in the boundary layer of a fluid near the catalyst surface (reactor scale) and detailed kinetics via elementary reactions (molecular scale). The reaction mechanism contains 32 elementary steps describing methane oxidation on platinum, 14 gaseous compounds and 13 intermediates. The developed approach can also be applied to multiscale modeling of stationary processes in heterogeneous catalytic reactors of different types, including nonstandard ones. In this case the system of PDEs on each scale may contain both first- and second-order derivatives with respect to one of the coordinates and the first-order derivatives with respect to another coordinate.
Summarizing the discussed material, we may state that purposeful use of unsteady-state conditions in the packed catalyst beds may help to significantly increase the efficiency of many catalytic technologies. The most significant positive result may be achieved in case of combination of thermal and composition non-stationarity of the catalysts.
In fact, unsteady-state approach opens new vision of process design and new degrees of freedom for creative process developers. In particular, it gives the possibility to create the non-adiabatic regimes in the formally adiabatic catalyst beds. This is caused by regenerative heat exchange and moving heat fronts in combination with possibility to separate the reaction stages (and corresponding heat emission) in time and space. Depending upon the chosen control strategy, it is possible to provide the regime with maximum temperature much lower than that for equivalent steady-state process (like in the process of anaerobic oxidation of hydrocarbons), or vice versa, the super-adiabatic performance (as in the adsorption-catalytic process for VOC oxidation or in the regeneration of the chemisorbing in sorption-enhanced steam reforming process).
The use of sorption capacity of the catalyst surface makes possible to realize the separate reaction stages in different phases of the process cycle, thus opening wide possibilities for manipulation with surface concentrations of adsorbed reactants. In combination with temperature non-stationarity and proper process concept, it permits to overcome the equilibrium limitations in different reactions or obtain higher product yield in complex reaction systems in comparison with similar steady-state process.
Notably, in many cases the performance of unsteady-state processes is significantly influenced by the factors that are insignificant or have indirect effect in case of steady-state processes. These factors may include heat capacity of the catalyst, oxygen bonding energy at the catalyst surface, shape of the reactor etc. In general, in comparison with stationary processes the unsteady-state approach may provide the following:
Decrease of energy consumption at processing of lean feedstocks (reverse-flow reactors, adsorption-catalytic process for VOC incineration)
Increased conversions (reverse-flow reactor for NOx reduction)
Over-equilibrium yield of target products in the thermodynamically limited reactions (oxidation of SO2, sorption-enhanced methane steam conversion)
Increase of selectivity and product yield in reactions of partial oxidation and oxidative dehydrogenation (anaerobic oxidation of hydrocarbons).
The list of advantages should also include the significant decrease of capital costs of catalytic plants and, though sounding paradoxical at first sight, simplification of process flow-sheets and increase of the operation stability under variable conditions.
Of course, all these benefits are not provided automatically; in each case the unsteady-state process must be properly arranged to achieve the required targets. The phenomena involved in unsteady-state catalytic processes are numerous and complicated, especially accounting for their interaction at different scale levels. Due to this complexity, the main tool for development and optimization of nonstationary catalytic processes is mathematical modeling on the base of deep understanding of process fundamentals. The Russian school in modeling of catalytic processes, created by professor Mikhail Slinko and developed by professor Yurii Matros and his team, have made a significant contribution to the world scientific knowledge in this area. We may state that these theoretical achievements were finally confirmed by successful practical applications in Russia and worldwide.
The authors are sincerely grateful to professor Yurii Shaevich Matros, our teacher along all our lives in science. The work was performed within the framework of Russian budget project no. 0303-2016-0017, Funder Id: http://dx.doi.org/10.13039/501100002674 for Boreskov Institute of Catalysis.
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