Pseudodistributed Feed Configurations for Catalytic Plate

Article pubs.acs.org/IECR

Pseudodistributed Feed Configurations for Catalytic Plate Microchannel Reactors Richard C. Pattison, Forrest E. Estep, and Michael Baldea* McKetta Department of Chemical Engineering, The University of Texas at Austin, 200 East Dean Keeton Street, Stop C0400, Austin, Texas 78712, United States ABSTRACT: Catalytic plate microchannel reactors (CPRs) are one of the most successful implementations of the process intensification philosophy, and have been recognized as an important contributor in the path toward monetizing distributed natural gas resources through localized, small-scale processing. While their reduced geometric dimensions lead to welldocumented energy efficiency gains and lower capital costs, they also give rise to specific control and operation challenges, particularly in terms of preventing temperature excursions and the formation of hotspots that can compromise the integrity of the reactor. In conventional reactors, such issues are typically addressed through external cooling and by judiciously distributing the feed streams along the reactor to modulate the release of reaction heat. Owing to the difficulty of implementing distributed sensors and actuators at a small scale, this solution is not readily applicable to the design and operation of microchannel reactors. As a consequence, in this paper, we propose a novel approach to modulation of heat generation in microchannel reactors, via a segmented catalyst macromorphology consisting of alternating catalytically active and inactive (“blank”) reactor sections. We also introduce a novel optimization-based strategy for determining the number, length, and axial location of the active sections based on closely tracking a desirable, optimized temperature profile. Using the detailed model of an autothermal methane-steam reforming reactor, we demonstrate the efficacy of the optimized segmented macromorphology in ensuring a uniform axial temperature profile. Further, we demonstrate through simulations that, at the optimum, the temperature and conversion profiles resulting from the proposed segmented macromorphology are similar to those obtained in a microchannel reactor with multiple axially distributed reactant feed points.

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INTRODUCTION Advances in horizontal drilling and hydraulic fracturing technologies have increased production of natural gas and its use as an energy source. Although natural gas has a higher energy density per unit mass relative to other hydrocarbons, its low density at atmospheric conditions results in orders of magnitude less energy per unit volume. The distribution of such low-energy density fuels is a costly undertaking, as the construction of a pipeline for direct transport requires a major capital investment. Alternative distribution methods include the production and transport of compressed or liquified natural gas (CNG/LNG) and frozen natural gas hydrates (NGH).1 These methods necessitate the production of large transportation vessels equipped to maintain refrigerated conditions as well as capital intensive liquefaction and regasification processes. Gas-to-liquids (GTL) processes, which lead to increasing the energy density of the fuel through chemical, rather than physical transformations2−4 have also been explored. Existing GTL technologies are predominantly based on Fischer−Tropsch synthesis (along with the ancillary process of syngas production through, e.g., steam-methane reforming), and are typically economically viable only when implemented in large-scale facilities.5,6 However, the relatively modest quantities of feedstock provided by individual stranded and associated natural gas sources suggest that the GTL transformation be carried out on a small scale and as close to the point of production as possible to minimize logistic costs. Process intensification provides an avenue for the development of such miniaturized plants to facilitate localized production.7 Contrary to past trends in process development, which sought to reduce costs through economies of scale, process © 2013 American Chemical Society

intensification aims to utilize available resources more efficiently. Intensified processes minimize transfer and transport limitations such that processes are governed by their intrinsic rates. In this manner, process intensification reduces capital investments, energy use, and feedstock and product inventories while improving process flexibility, safety, and environmental performance.7,8 Catalytic plate microchannel reactors (CPRs) represent one of the most successful implementations of the process intensification philosophy, and autothermal CPRs have been repeatedly confirmed to be particulary apt at carrying out methane steam reforming (MSR) reactions for generating synthesis gas, the feedstock of Fischer−Tropsch synthesis.9−13 An autothermal CPR consists of stacked plates with alternating channels in which a set of exothermic reactions occurring in half the channels supports a set of endothermic reactions occurring in the remaining channels. The channels generally have heights in the range of millimeters, while the plates are coated with micrometerthick catalyst layers, with the benefit of both minimized heat transfer resistance and minimal intracatalyst diffusion resistance. This results in an order of magnitude reduction in overall size relative to conventional reactors, with no loss in conversion or production rate.13 Special Issue: David Himmelblau and Gary Powers Memorial Received: Revised: Accepted: Published: 5028

March 20, 2013 June 10, 2013 July 12, 2013 August 9, 2013 dx.doi.org/10.1021/ie4008997 | Ind. Eng. Chem. Res. 2014, 53, 5028−5037

Industrial & Engineering Chemistry Research

Article

Figure 1. Reactor structure diagram (not represented to scale). The model considered in this work comprises one-half reforming channel and one-half combustion channel, the corresponding catalyst layers, and the separating plate.

segmented macromorphology in ensuring a uniform longitudinal temperature profile. Further, we demonstrate through simulations that, at the optimum, the temperature and conversion profiles resulting from the proposed segmented macromorphology are equivalent to those obtained in a microchannel reactor with multiple, axially distributed reactant feed points.

Despite their efficiency and economic benefits, microchannel reactors present a number of design and operational challenges. Microscale systems have faster dynamics due to their reduced physical dimensions, and complex designs often result in nonlinear responses that render simple, linear feedback controllers ineffective. An improvement in the dynamics and control properties of intensified systems should thus be sought at the design stage, rather than once the physical system has been constructed.14 In the case of autothermal CPRs, the primary concern relates to matching heat generation and consumption rates. Co-current flow, with both channels fed from the same side, gives the advantage of simultaneous axial reaction progress, but requires higher feed temperatures.15 Counter-current flow configurations simplify the reactor design by separating the channel feeds, but matching heat generation and consumption rates along the reactor becomes problematic.15−17 The exothermic and endothermic reactions propagate in opposite axial directions, and a poor synchronization of heat fluxes can result in either decreasing temperatures until reactor extinction occurs, or in the development of localized temperature extremes which can damage the reactor structure or the catalyst coatings. Several approaches to improving the operating performance of counter-current autothermal reactor designs have been proposed, including the addition of a layer of thermally conductive phase change material to prevent the formation of hot spots during flow and composition disturbances,2 periodic switching of the flow direction in the exothermic and endothermic channels,18,19 offset catalyst distribution to more effectively match heat generation and consumption,20,21 and varying the activity of the catalyst along the longitudinal coordinate of the reactor.16 Using multiple feed points for the exothermic reaction mixture has also been proposed as a strategy to modulate the rate of heat generation and to prevent the formation of hot spots.10 While theoretically effective in increasing design and operational flexibility, the implementation of variable-activity catalysts and the use of distributed feed points is hindered by practical considerations related to catalyst and reactor fabrication. Focusing on the prototype autothermal MSR CPR system described in previous work,13,20 this article proposes a novel reactor design featuring a segmented catalyst macromorphology, whereby catalytic and noncatalytic (“blank”) sections alternate in the combustion channel. This configuration aims to modulate the reaction progress along the axial coordinate of the reactor. By distributing the heat generation, this system emulates a distributed fuel feed configuration. Furthermore, we introduce a novel optimization-based approach for selecting the number, length, and axial location of the active segments based on closely tracking a desirable, optimized longitudinal temperature profile. Using the detailed model of an autothermal methane-steam reforming reactor, we demonstrate the efficacy of the optimized

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SYSTEM DESCRIPTION We assume that the following reactions occur in the reforming channels: Methane steam reforming (1) CH4 + H 2O ⇌ CO + 3H 2 ΔH = +206 kJ mol−1

Water gas shift (2) CO + H 2O ⇌ CO2 + H 2

ΔH = −41 kJ mol−1 Reverse methanation (3) CH4 + 2H 2O ⇌ CO2 + 4H 2

ΔH = +165 kJ mol−1

The reaction kinetics developed by Xu and Froment22 for a nickel-based catalyst accurately describe this process over a wide range of parameters23 and are used in our calculations. The catalytic combustion of methane occurs in the combustion channels: Methane combustion (4) CH4 + 2O2 → CO2 + 2H 2O

ΔH = −803 kJ mol−1

The catalytic combustion rate, as observed with supported noble metal catalysts such as Pd and Pt, is assumed to be first-order with respect to methane and zero-order with respect to oxygen. The activation energy for catalytic combustion is taken as 90 kJ mol−1. Homogeneous combustion plays an important role at high temperatures and is accounted for using a simplified rate law of order −0.3 and 1.3 with respect to methane and oxygen, an activation energy of 125.49 kJ mol−1, and a pre-exponential factor of 8.3 × 105 s−1.20 Reactor Model and Numerical Implementation. Figure 1 shows a diagram of a catalytic plate reactor. The reactors typically consist of hundreds of stacked steel plates, with reforming and combustion reactions occurring in alternating channels between the plates. Under the assumption that end effects (notably, the thermal interaction of the top and bottom plates and nearby 5029

dx.doi.org/10.1021/ie4008997 | Ind. Eng. Chem. Res. 2014, 53, 5028−5037


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channels with the environment, and preferential flow distribution at the channel inlets) are negligible, this system can be accurately described by considering a single catalyst-coated plate and the adjacent half channels. In this work, the gas phase in both channels, as well as the solid plate, are modeled using a 2D geometry. Symmetry boundary conditions are imposed at the channel centerlines, no-flux conditions at the channel outlets, and equal flux conditions at the catalyst surface. The catalyst layers are modeled in a single dimension with negligible thickness, an approach justified by the very small ratio of catalyst thickness to reactor length. The inlet velocity profile to each channel is assumed to be a fully developed laminar flow between two infinite parallel plates. When the diffusion coefficients are being calculated, a binary mixture is assumed between each component and either steam or air for the reforming and combustion channels, respectively. The complete set of equations modeling the reactor have been introduced elsewhere,13,20 and, for completeness, are also included in the appendix. The model parameters13,20 are included in Table 1. The mathematical model and optimization

Figure 2. An offset catalyst geometry results in a temperature peak near the center of the reactor. The inlet of the reforming channels (solid line) is on the left-hand side, the inlet of the combustion channels (dashed line) is on the right-hand side.

Table 1. Reactor System Details parameter

value

reactor length reforming channel half-height combustion channel half-height plate thickness reforming catalyst height combustion catalyst height reforming inlet temperature combustion inlet temperature reforming inlet velocity combustion inlet velocity reforming inlet composition (mass fraction)

63.4 cm 1.0 mm 1.0 mm 0.5 mm 20 μm 20 μm 793.15 K 793.15 K 4.0 m/s 3.2 m/s 19.11% CH4 72.18% H2O 2.94% CO2 0.29% H2 5.48% N2 5.26% CH4 22.09% O2 72.65% N2

combustion inlet composition (mass fraction)

decreasing reforming reaction rate (as a majority of the reactants have been spent) and the maximum combustion reaction rate (when the combustion mixture first contacts the catalyst).

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SEGMENTED CATALYST MACROMORPHOLOGY The rate at which heat is generated by the combustion reactions could be modulated by distributing the fuel feed along the reactor, which would, in turn, result in a reduction of the peak reactor temperature. This solution has been suggested in the literature (see, e.g., the work by Kolios et al.10) but its implementation is hindered by practical considerations: creating the appropriate physical mechanisms (i.e., flow channels) for uniformly distributing reactant flow to all channels at several given axial coordinates is a challenge. Furthermore, given the intricate microchannel structure of CPRs, the potential for high thermal stresses arising at the feed points due to the temperature difference between the feed stream and the mixture already present in the reactor represents a concern. In this paper, we propose a segmented catalyst macromorphology as an alternative approach to axially distributed feed points. Intuitively, alternating catalytically active and catalytically inactive (“blank”) sections in the combustion channels emulates the distributed feed by modulating the reaction progress along the reactorcatalytic combustion will occur over the catalytic sections, while only homogeneous combustion will occur in the blank segments. From a practical perspective, the segmented macromorphology can be implemented by a slight alteration of the current catalyst preparation techniques (e.g.,25) consisting of selectively impregnating the reactor wall plates (carrying an alumina washcoat) with catalyst precursors to create the desired catalytic/blank section pattern. Figure 3 illustrates the distributed feed and segmented catalyst concepts. Both systems modulate reaction progress along the reactor length. Evidently, identifying the optimal number, location, and length of the catalytic segments is a challenge akin to determining the optimal location and flow rate of the aforementioned distributed feed points. However, it is important to observe that, unlike distributed-feed systems, in which the feed flow rates and the associated heat generation rates can, in principle, be adjusted or tuned (possibly in closed-loop) online,

problem were solved using gPROMS.24 The axial domains of the reforming channel and catalyst were discretized using backward finite differences; forward finite differences were used to discretize the axial domain of the combustion channel and catalyst, and central finite differences were used to discretize the axial domain of the plate. Orthogonal collocation of finite elements was used to discretize the transverse domain in every layer. The base case reactor considers the optimal offset catalyst arrangement established in previous work.20 The offset arrangement (diagram in Figure 1) greatly improves reactor performance compared to full catalyst coverage in both channels by synchronizing heat generation and consumption. However, as seen in Figure 2, temperatures in the middle of the reactor are elevated because the majority of the methane fuel is consumed at the beginning of the catalyst-coated zone. Indeed, 90% of the fuel is consumed in a span of less than 30% of the reactor length. The rate at which the released thermal energy is absorbed by the reforming reactions is limited by the reforming reaction rate, and any excess heat contributes to increasing the reactor temperature. The location of the peak temperature corresponds to both a 5030



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Figure 3. Distributed feed configuration and the proposed segmented catalyst macromorphology.

af ter the reactor has been constructed, a segmented-catalyst reactor must be configured at the design stage, as no further adjustments to the reactor layout are possible once the system has been built. The optimal design of such a segmented configuration is addressed in the following section. Design Optimization Problem Formulation and Solution Strategy. In its simplest form, the design optimization problem for the proposed segmented catalyst macromorphology seeks to compute the number, length, and location of the catalytic zones in the combustion channel, which maximize a performance objective, subject to operating and safety constraints. Such constraints ensure, e.g., that the reactor temperature profile is such that reactor extinction or the formation of hotspots are avoided. Furthermore, constraints must be used to guarantee acceptable conversion in both the combustion channel and the reforming channel, and prevent the maximum reactor temperature from reaching unsafe levels. Through its ability to modulate energy generation along the reactor, the proposed segmented catalyst macromorphology also offers an opportunity to shape the axial temperature profile of the reactor. Intuitively, the objective of the design optimization problem should be to eliminate temperature hotspots and large local temperature gradients. Thus, we propose to formulate the design optimization problem in terms of both identifying the target optimal temperature profile Tsp(z) and determining the geometry of the segmented catalyst that results in the heat generation rate distribution that establishes the desired profile. It is important to note that, assuming that Tsp(z) is a continuous variable, the above statement involves an inf inite dimensional optimization problem. To make this problem tractable we propose a parametrization approach,26 dividing the target temperature profile in three segments as illustrated in Figure 4. The proposed profile (i) rises from the feed temperature at the inlet of the reforming channels to (ii) a constant-temperature region (plateau) followed by (iii) a decrease in temperature toward the outlet of the reforming channels. Loosely speaking, the three sections are intended to provide reactant preheating, carry out the reaction and cool the reaction effluent, respectively, which bears some similarity to the conceptual reactor design proposed by Kolios et al.9an observation that will be discussed further in the paper. The profile can be characterized mathematically by the plateau temperature, the exit temperature, and the location and length of the plateau. These variables also constitute the decision variables for optimizing the target temperature profile. Note that this approach is similar to the control vector parametrization26 strategy used in the numerical

Figure 4. Temperature profile parametrization.

solution of dynamic optimization problems. The optimal profile can then be “tracked” by selecting the number, length, and location of the catalytic sections in the combustion channel that minimize the difference between the desired temperature profile and the actual reactor temperature. The mathematical formulation of the optimization problem described above is given in eq 1: min

L combi , Lref , Tsp

J=

∫0

L

(T s(z , x s = δ s/2) − Tsp(z))2 dz

s.t. X1(L) ≥ X min X 2(0) ≥ X min max(T s) ≤ Tmax L combi ≥ Lmin model equations (1)

where Ts(z,xs = δs/2) is the plate center temperature along the reactor and Tsp(z) is the desired temperature profile. X1(L) and X2(0) are the outlet methane conversions for the reforming and combustion profiles, respectively. Lcombi is a vector of locations and lengths of the combustion catalyst segments, and Lref refers to the offset length in the reforming channel. The system is constrained to yielding a minimum conversion of Xmin = 98.0%. The local conversion is calculated from the local methane concentration, ωgCH4j as R

X j (z ) = 1 −

g dx jg|z ∫0 j ρjg u zjωCH 4j R

g dx jg|z = 0(j = 1), z = L(j = 2) ∫0 j ρjg u zjωCH 4j

(2)

With manufacturing considerations in mind, additional constraints impose a minimum length Lmin = 1.2 cm for the catalytic sections. Finally, the maximum plate temperature along the reactor (Tmax) is constrained to an upper bound of 975°C. Intuitively, a finer segmentation (i.e., increasing the number and decreasing the size of the catalytic sections) improves the tuning of the heat generation rates and, consequently, the number of catalytic segments is an important variable. Using this variable directly in the optimization would transform (1) into a 5031



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Figure 5. Target temperature profiles (dashed) and actual optimal temperature profiles (solid) for the offset base case (top left), two segment-catalyst configuration (top right), three segment-configuration (bottom left), and four segment-configuration (bottom right). Below each plot is a graphical representation of the optimal catalyst arrangement (combustion catalyst configuration is displayed above reforming catalyst configuration; the inlet of the combustion channel is on the right-hand side, the inlet of the reforming channel is on the left-hand side).

minimize the maximum temperature in the reactor and (ii) minimize the total integral squared temperature along the reactor while imposing, in each case, lower bounds on conversion in both channels. These formulations have, however, the disadvantage of a nonconvex objective function (in case i) and were verified to result in solutions featuring hotspots and large local axial temperature gradients.

mixed-integer nonlinear program (MINLP), which brings additional solution challenges. However, since in this case the problem formulation is limited to a single integer variable, its solution is amenable to an iterative approach whereby the effect of incrementally finer segmentations is considered until the effect of increasing the number of segments on the objective function is sufficiently small. The results reported in the following section follow this approach. Remark 1. Note that the optimization formulation simultaneously selects the optimal temperature profile for the reactor (which is dictated by the reforming reaction) and the catalyst configuration in the combustion channel that leads to a realization of the desired profile. Manipulating the catalyst configuration to adjust the temperature profile lends itself to a feedback control interpretation. Thus, the problem formulation is analogous to the hierarchical control implemented in largescale chemical processes,27 whereby the set points of a layer of lower-level (distributed) controllers are calculated by an upperlevel optimization layer. However, in control applications, the optimization calculations and their implementation are carried out independently and over separate time horizons, with the distributed controllers acting in a faster time scale. By contrast, since in the present problem we are interested only in the steadystate solution, the “optimization” and ”control” layers (i.e., the computation of the temperature profile and, respectively, of the catalyst geometry) are resolved simultaneously. Remark 2. There are several potential alternatives to the proposed formulation of the design optimization problem: (i)

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RESULTS AND DISCUSSION The optimization calculations for the prototype system were solved in gPROMS using the time relaxation-based approach for optimizing the steady-state performance of dynamical systems introduced in previous work.28,20 The algorithm consists of (i) determining a set of initial estimates for the decision variables and initial conditions for the state variables of the model, (ii) integrating the model for a sufficiently long time interval until a steady state is reached, (iii) computing a new set of decision variables at steady state, (iv) propagating the steady-state values of the model states as initial conditions for a new iteration, along with the new decision variables, (v) returning to step ii and repeating the process until a convergence criterion (e.g., the change in objective function between two successive iterations is below a given threshold) is satisfied. The results of the optimization calculations are presented in Figure 5. The plots display the temperature set point profiles and the actual temperatures as a function of the reactor length. The configuration of the segmented catalyst macromorphology is presented at the bottom of each plot. Note that, in each case, 5032



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the length of the combustion catalyst segments increases from the inlet to the outlet of the combustion channel. This is because the combustion reaction is first order in methane concentration, so significantly more catalyst is needed to complete the reaction as the reactants are exhausted and the methane mass fraction in the fuel stream approaches zero. The single-segment optimized offset geometry presented in the top-left plot serves as the baseline for comparisons with the segmented macromorphology results. With the addition of a second catalytic combustion segment, the maximum temperature for the optimal arrangement is reduced to 973 °C, which is 64 °C less than the base case, and the conversion constraint of 98% is satisfied in both channels. The maximum reactor temperature with three catalytic segments is 939 °C, or 98 °C less than the base case. This reactor configuration exhibits a close match between the desired temperature profile and the actual temperature (Figure 5, bottom left). Finally, by increasing the number of segments to four, the maximum reactor temperature reaches = 928 °C. Clearly, the temperature profiles obtained using three and four segments are very similar, and we did not consider it necessary to explore a finer catalyst segmentation. The complete results of the optimization calculations are presented in Table 2. We note that, while a segmented reforming

Figure 6. Conversion in the optimal four-segment reactor. The catalyst macromorphology is displayed in the lower section of the figure.

support the endothermic reactions. Rather, this section serves as a preheater, raising the temperature of the reforming mixture to the plateau temperature. Although less than 10% of the methane is consumed in this section, the length of the section is necessary as the rate of reaction decreases proportionally with the methane concentration in the combustion channels. The two median combustion catalyst sections overlap axially with the reforming catalyst starting point at the dimensionless coordinate ζ = 0.46. This results in the formation of two temperature peaks. The temperature decrease between the peaks corresponds to the start of the reforming reaction. This plateau is the primary reaction section, with 75% reforming conversion being achieved within its bounds. The first catalyst segment in the flow direction in the combustion channel has the highest heat generation rate. To reduce temperatures in this zone, the optimization indicates that the length of this section should be at the lower bound of the allowable catalyst length, and thus this section provides sufficient heat to support the reforming reactions to approach complete conversion. Remark 3. It is interesting to note that the optimal configuration mirrors the autothermal catalytic plate reactor concept discussed by Kolios et al.,9 whereby a central catalytic area is bordered on each side by blank, inactive sections that act as heat exchangers. Our optimization results suggest, however, that in this case the heat exchangers should feature a catalyst coating in one of the channels (namely, in the channel corresponding to the stream exiting the reactor) to ensure complete conversion of both the reforming and combustion streams. Comparison between Segmented Catalyst Macromorphology and Distributed Feed Configuration. To further evaluate the advantages of the proposed segmented catalyst macromorphology, we compared the performance of the optimized four-segment configuration described above with that of a distributed-feed reactor. We assumed that the feed stream to both the combustion and reforming channels can enter at four different axial locations. The feed streams were assumed to have the same temperature, composition, and pressure as the feed to the segmented-catalyst reactor at all the feed points, and we further assumed that the addition of mass to the reactor does not disrupt the parabolic velocity profile of the channel flow. To determine the optimal location and feed flow rate of each

Table 2. Optimization Results variable

2 segments

3 segments

4 segments

objective function value 153500 combustion catalyst 0−30.1 cm segments 35.9−38.1 cm

61900 0−20.6 cm 26.6−36.9 cm 44.4−45.7 cm

reforming catalyst offset plateau temperature exit set point temperature end coordinates of plateau maximum temperature reforming conversion combustion conversion

22.8 cm 824 °C 639 °C

29.2 cm 864 °C 654 °C

57500 0−17.9 cm 25.3−32.9 cm 34.2−38.1 cm 45.7−46.9 cm 29.2 cm 860 °C 652 °C

19.0−44.4 cm

25.0−48.9 cm

25.4−49.6 cm

974 °C 98.1% 98.7%

939 °C 98.0% 98.6%

928 °C 98.0% 98.1%

catalyst (in addition to the combustion catalyst) could in principle assist with further tuning the reactor performance, this is not the case in practice. We carried out several sets of simulations to test a segmented reforming catalyst (in addition to the offset layout introduced in our previous work20) along with the combustion catalyst segmentation, which showed no improvement and in effect resulted in a decrease in conversion in the reforming channel (the results are not presented for brevity). This outcome is easily understood since segmenting the catalyst in the reforming channels reduces the catalytic surface area available to the reforming reactions and thus lowers conversion if the overall reactor length is not increased. The optimized four-segment configuration discussed above is clearly successful at reducing the maximum reactor temperature (as shown in Figure 5) and increasing conversion (see Figure 6). In this optimal case, we note that the combustion catalyst segment nearest the channel outlet is significantly longer (nearly a third of the reactor) than the others. This section precedes the start of the offset reforming catalyst. Consequently, the heat generated through combustion in this section does not, in fact, 5033



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feed point, we solved the following optimization problem aimed at maximizing conversion: max

Lm , u1m , u 2m

J = X1(L) + X 2(0)

s.t. max(T s(z , x)) ≤ Tmax Lm ≥ Lmin

∑ Lm = L m

∑ u1

m

= u10

m

∑ u2

m

= u 20

m

model equations

(3)

where the indices m ∈ {1,2,3,4} correspond to the feed points, Lm are the lengths between the four feed points, L is the total reactor length, u1m and u2m are the average inlet velocities at each inlet for the reforming and combustion channels, respectively, and u01 and u02 are the total average inlet velocities. We ensured that the total mass flow rates through the channels are equal for the distributed feed reactor and the segmented catalyst reactor by equating the total average inlet velocities (u01 and u02) in the distributed feed reactor and the average inlet velocity in the segmented catalyst reactor (assuming that the feed points at every inlet position have similar geometric dimensions to the channel inlet). The parameter Lmin sets a minimum distance between feed points. The maximum temperature constraint of 975 °C was implemented as well. The optimization problem was solved using a similar approach as in the segmented case, and the results are presented in Table 3.

Figure 7. Comparison of reactor temperatures in a distributed feed reactor (dashed) and the optimal segmented catalyst reactor with four catalyst segments (solid).

Table 3. Optimal Distributed Feed Reactor variable

length

flow fraction

combustion feed points

23.8 cm 30.9 cm 35.9 cm 63.4 cm 0 cm 23.8 cm 30.9 cm 915 °C 98.8% 99.7%

6.0% 58.0% 32.8% 3.2% 28.9% 61.1% 10.0%

reforming feed points

maximum temperature reforming conversion combustion conversion

Figure 8. Comparison of conversions in a distributed feed reactor (dashed) and the optimal segmented catalyst reactor (solid).

temperature profiles in the two reactors are similar, and it is apparent that the segmented catalyst macromorphology is a valid approach for emulating a distributed-feed reactor and modulating heat generation and reaction rates in the absence of conventional distributed actuators for temperature regulation.

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Comparisons of the reactor temperatures and conversions for the four-segment segmented catalyst reactor and for the distributed feed reactor are shown in Figures 7 and 8 respectively. Note that the distributed feed reactor achieves slightly higher conversions (99%) in both channels, and has a higher average temperature. Importantly, the temperature at the combustion inlet is significantly higher than the segmented catalyst reactor; in turn, this results in a higher temperature at the outlet of the reforming channel and a potential decrease in the overall energy efficiency of the reactor. Owing to the numerical challenges and practical implementation difficulties of dealing with zero-flow conditions, the optimization problem formulation allowed for a small fraction of the total combustion flow (3.2% at the optimum) to be fed to the inlet of the reaction channels at z = L. However, it is interesting to note that conversion and

CONCLUSIONS This paper explores a method for emulating distributed feed configurations in microchannel reactors via a segmented catalyst macromorphology, consisting of alternating active and catalytically inactive (blank) sections. An optimization-based approach was presented to determine the optimal catalyst configuration, consisting of (i) selecting an optimal parametrized temperature profile and (ii) determining the catalyst configuration that ensures that the real reactor temperature follows the optimal axial profile. We used the detailed model of an autothermal steam-methane reforming microchannel reactor as a testbed for validating these theoretical concepts. Our results confirmed that increasing the number of sections in the segmented morphology allowed for 5034



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more precise tuning of the temperature profile, which results in a lower maximum reactor temperature with no conversion penalty. Further simulations confirmed that the optimized segmented

geometry in effect successfully emulates a reactor layout featuring axially distributed feed streams, a desirable but practically challenging autothermal reactor configuration. The concepts and design optimization methods presented are generally applicable to other distributed-parameter systems (e.g., membrane separation modules) where temperature or composition profiles must be adjusted and distributed sensors and actuators are not a viable option.

Table 4. Nomenclature symbol

units

description

cp D Deff H k L Lref Lcombi

J/(kg/K) m2/s m2/s W/m2 W/(m/K) m m m

Heat capacity diffusion coefficient effective diffusion coefficient reaction heat flux thermal conductivity reactor length reforming catalyst offset combustion catalyst locations and lengths

Lm Lmin M J P r rhomog R Rg t T Tmax Tsp uz ujm

m m kg/mol A.U. Pa mol/(m3/s) mol/(m3/s) m J/(mol/K) s K K K m/s m/s

distributed feed points minimum catalyst/distributed feed length molecular weight objective variable pressure reaction rate homogeneous reaction rate radial channel thickness gas constant time temperature maximum allowable plate temperature set point temperature profile axial velocity distributed inlet feed velocities

u0 x X Xmin z δ ΔHrxn ηeff ν ρ ω superscript cat g s subscript i j k m numbering j=1 i= k=1 k=2 k=3 j=2 i= k=1

m/s m

m m J/mol

kg/m3

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APPENDICES

A. Model Equations

The model equations have been developed in previous studies13,20 and are included for completeness. Gas Phase. Mass balances: ∂ρjg ωig, j

+

∂t

ρjg u zj

∂ωig, j

∂z ⎛ ∂ωig, j ⎞ ∂ωig, j ⎞ ∂ ∂⎛ g ⎜ρj Di − mix j ⎟ = g ⎜⎜ρjg Di − mix j g ⎟⎟ + ∂x j ⎝ ∂x j ⎠ ∂z ⎝ ∂z ⎠ +

∑ (νi ,k ,jrhomog,k ,jMi ,j) k

Energy balances: ρjg cpjg =

inlet velocity radial coordinate local conversion minimum exit conversion axial coordinate radial thickness heat of reaction effectiveness factor stoichiometric coefficient density mass fraction description catalyst gas phase solid

∂T jg ∂t

ρjg u zjcpjg

+

g ∂ ⎛⎜ g ∂T j ⎞⎟ k j ∂x jg ⎜⎝ ∂x jg ⎟⎠

∂T jg ∂z

+

g ∂ ⎛ g ∂T j ⎞ ⎟+ ⎜k j ∂z ⎝ ∂z ⎠

∑ (rhomog,k ,jΔHrxn

j,k

)

k

Continuity equation: ∂ρjg ∂t

+

∂ρjg u zj

=0

∂z

Equation of state for ideal gas: ρjg

−1 Pj ⎛ ωig, j ⎞ ⎜ ⎟ = ∑ R gT jg ⎜⎝ i Mi , j ⎟⎠

Boundary Conditions. Inlet Conditions. Inlet composition:

ωig, j = ωi0, j

chemical species channel reaction distributed inlet feed points

Inlet temperature: T jg = T j0

reforming CH4,H2O,CO,CO2,H2,N2 methane steam reforming reaction CH4 + H2O ⇌ CO + 3H2 water gas shift reaction CO + H2O ⇌ CO2 + H2 Reverse methanation reaction CH4 + 2H2O ⇌ CO2 + 4H2 Combustion CH4,O2,CO2,H2O,N2 Methane catalytic and homogeneous combustion CH4 + 2O2 → CO2 + 2H2O

Parabolic inlet velocity profile: ⎡

u zj =

⎛ x jg ⎞2 ⎤ − ⎜⎜ ⎟⎟ ⎥ ⎝ R j ⎠ ⎥⎦ ⎣

1.5uj0⎢1 ⎢

Outlet conditions. Zero flux (reforming): ∂ωig,1 ∂z 5035

= z = L ; ∀ x1g

∂T1g ∂z

=0 z = L ; ∀ x1g



Article

Zero flux (combustion): ∂ωig,2

∂T2g ∂z

=

∂z

Laplace equation

z = 0; ∀ x 2g

ρs cps ∂T s

=0

ks

z = 0; ∀ x 2g

∂x jg

=

∂T jg

=

∂x jg

x jg = 0; ∀ z

x jg = 0; ∀ z

∂u zj

∂T s ∂z

=0

∂x jg

x jg = 0; ∀ z

∂ωig, j

= −ρjg Deffi, j

∂x jg

∂T s ∂z

∂ωicat ,j ∂xjcat

x jg = R j ; ∀ z

j = 1:

j = 2: k 2g

∂T2g ∂x 2g

= H1 + ks

j = 1: T s|x s= 0; ∀ z = T1g|x1g = R1; ∀ z

= H2 − k s x 2g = R 2 ; ∀ z

j = 2: T s|x s= δ s; ∀ z = T2g|x2g = R 2 ; ∀ z

s

∂T ∂x s

∂T s ∂x s

B. Nomenclature x s = 0; ∀ z

The symbol, units, and description of the abbreviations and variables used in this paper are given in Table 4.

■

x s= δ s;∀ z

*E-mail: [email protected]. Notes

u zj|xjg = R j ; ∀ z = 0

The authors declare no competing financial interest.

■

Catalyst Layer. Energy balance (isothermal in transverse direction):

ACKNOWLEDGMENTS Professor Himmelblau’s legacy and work have inspired and motivated many generations of researchers exploring the field of process systems engineering, both at the University of Texas at Austin and around the world. We dedicate this work to his memory. We also acknowledge with gratitude the financial support provided by the American Chemical Society−Petroleum Research Fund through Grant 52335-DNI9.

T jcat = Tjg |xjg = R j Mass balance: ⎞ ∂ωicat ∂ ⎛⎜ g ,j ⎟ = −∑ (νi , j , krj , kMi , j) ρ D eff, i , j ⎜ ∂xjcat ⎝ j ∂xjcat ⎟⎠ k

■

Boundary conditions. Plate conditions. Zero flux: ∂ωicat ,j

=0 xjcat = 0; ∀ z

ωig, j|xjg = R j ; ∀ z = ωicat , j |xjcat = δjcat ; ∀ z

Effectiveness factor: ηeff, k , j =

1 δjcat

δ cat

∫0 j rk , j dxjcat rk , j|xjcat = δ cat

Reaction heat flux:

∑ (−ΔHk ,j ∫ k

0

δjcat

REFERENCES

(1) Gudmundsson, J.; Hveding, F.; Borrehaug, A. Transport of natural gas as frozen hydrate. Proceedings of the 5th International Offshore and Polar Engineering Conference; ISOPE: Mountain View, CA, 1995; pp 11− 16 (2) Pattison, R. C.; Baldea, M. A thermal-flywheel approach to distributed temperature control in microchannel reactors. AIChE J. 2013, DOI: 10.1002/aic.13991. (3) Ogugbue, C.; Chukwu, G.; Khataniar, S. Economics of GTL technology for gas utilization. Hydrocarbon Economics and Evaluation Symposium, Dallas, TX, 2007. (4) Hall, K. R. A new gas to liquids (GTL) or gas to ethylene (GTE) technology. Catal. Today 2005, 106, 243−246. (5) Van Bibber, L.; Shuster, E.; Haslbeck, J.; Rutkowski, M.; Olsen, S.; Kramer, S. Baseline Technical and Economic Assessment of a Commercial Scale Fischer−Tropsch Liquids Facility. US Dept. of Energy Report DOE/NETL-2007/1260, 2007. (6) Van Bibber, L.; Shuster, E.; Haslbeck, J.; Rutkowski, M.; Olsen, S.; Kramer, S. Technical and Economic Assessment of Small-Scale Fischer−Tropsch Liquids Facilities. US Dept. of Energy Report DOE/NETL-2007/1253, 2007. (7) Reay, D. A.; Ramshaw, C.; Harvey, A. P. Process Intensification: Engineering for efficiency, Sustainability and Flexibility; ButterworthHeinemann: Amsterdam, 2008.

Catalyst surface conditions. Material transfer:

Hj =

AUTHOR INFORMATION

Corresponding Author

No slip:

∂xjcat

=0 z = L; ∀ x s

Catalyst surface. Temperature continuity: xjcat = δjcat ; ∀ z

Energy transfer ∂T g k1g 1g ∂x1 x g = R ; ∀ z 1 1

=0 z = 0; ∀ x s

Outlet condition. No flux:

Catalyst surface conditions. Material transfer: ρjg DG , i − mix j

∂ 2T s ∂ 2T s + 2 ∂z 2 ∂x s

Boundary conditions. Inlet condition. No flux:

Channel center. Symmetry: ∂ωig, j

∂t

=

rk , j dxjcat)

Plate. 5036



Article

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Pseudodistributed Feed Configurations for Catalytic Plate

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