Ind. Eng. Chem. Res. 2008, 47, 6579–6588
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Direct Partial Oxidation of Natural Gas to Liquid Chemicals: Chemical Kinetic Modeling and Global Optimization Christian Lund Rasmussen and Peter Glarborg* Department of Chemical and Biochemical Engineering, Technical UniVersity of Denmark, DK-2800 Kgs. Lyngby, Denmark
The direct homogeneous partial oxidation of methane to methanol is investigated through modeling and a limited set of experiments. The experiments are conducted in a laminar flow quartz reactor at 100 bar and 598-763 K under undiluted conditions. The modeling is performed with a detailed chemical kinetic model, previously validated against a range of high-pressure data on oxidation of CO/H2 and small hydrocarbons. Modeling predictions compare well also with the experimental data obtained in the present study. On the basis of the chemical kinetic modeling, the effect of the main process parameters is discussed and the process is optimized, using a numerical global optimization methodology based on interval analysis. The experimental data, as well as the optimization results, indicate yields of methanol in the upper range of those reported in literature. However, the values are well below the commercial target. Introduction Methanol is currently produced from natural gas or gasified coal in a complex two-step process, where the hydrocarbon feed gas is initially converted to synthesis gas (CO/H2) by steam- or autothermal reforming followed by catalytic reduction under high pressure to the desired product.1–3 The endothermic reforming reactions make it difficult to obtain high energy efficiencies unless large-scale installations are employed. In an alternative process, direct homogeneous partial oxidation of natural gas, reactants are converted to methanol or mixtures of methanol and other oxygenated hydrocarbons in a simple onestep process CH4 + O2 f CH3OH
(1)
This reaction, which occurs under fuel-rich, high pressure, and relatively low temperature conditions, is an exothermic process and could conceivably be superior in terms of energy efficiency compared to conventional methanol production. Investigations of the direct process date as far back as the early years of the 20th century,4,5 with the first high-pressure experiments reported in the 1930s.6,7 More recent studies have relied mostly on experimental work,8–22,28,29 even though also modeling studies have been reported, based on simplified kinetic schemes.20,30–35 In addition, some reviews have appeared in the field36–40 as well as some patents.41–45 An overview of the most important experimental achievements from the literature is presented in Table 1. The table shows large variations in the applied experimental conditions, as well as in the resulting methanol yields. The available studies indicate that relatively low reaction temperatures in the range of 550-800 K and high pressures between 30-100 bar, along with a high CH4/O2 ratio in the feed, are key parameters to obtain a high selectivity for methanol. However, reactor design, surface materials, residence time effects, and the presence of gas phase sensitizers such as higher hydrocarbons may also play a role. Figure 1 presents a comparison of experimental selectivities of methanol as function of the methane conversion. The * To whom correspondence should be addressed. Fax: +45 45 882258. E-mail:
[email protected].
selectivity for methanol, SCH3OH ) FCH3OH,out/(FCH4,in - FCH4,out), and the conversion of methane, XCH4 ) (FCH4,in - FCH4,out)/ FCH4,in, in a single pass of the reactor are two key figures for the comparison of the direct and indirect processes. Here F denotes the molar flow rate in and out of the reactor. The yield is then defined as YCH3OH ) SCH3OH × XCH4 ) FCH3OH,out/FCH4,in. In the figure, also recent results obtained from the direct catalytic conversion of methane to methanol, as well as technical economic assessments, are shown. The data shown in Figure 1 are not necessarily comparable in terms of applied experimental conditions, but the figure provides an overview of the most promising achievements in the field as of today. As indicated in Table 1 and Figure 1, previous results from the literature are characterized by significant scatter in terms of measured peak selectivities and fuel conversions; even when apparently identical reaction conditions have been applied. The work of Gesser and co-workers9,11,12 displays very promising results in terms of high methanol yields as do the results reported by Feng et al.19 and Zhang et al.22 However, these data are not supported by most other investigations conducted in the field. Burch et al.10 dedicated their work to reproduce the results of Yarlagadda et al.9 They investigated the homogeneous reaction at various temperatures and pressures from 648-723 K and 5-50 bar, as well as a broad range of O2 concentrations in the feed, but despite all their efforts to recreate the conditions applied by Yarlagadda et al., they remained unsuccessful in the sense that measured methanol selectivities and methane conversions were not even remotely close to those reported by Yarlagadda et al.9 Burch et al. tentatively attributed these discrepancies to minor differences in the reactor design. Even so, the lack of reproducibility emphasizes the need for a more fundamental process development strategy where a detailed understanding of the governing chemistry is in focus rather than a subjective experimental optimization. The objective of this paper is to evaluate the potential of the homogeneous partial oxidation of natural gas to methanol. Due to the limited knowledge of the complex chemistry involved in the process, previous attempts to optimize the process have mainly been experimental. However, in recent work,46–49 the authors have presented an updated detailed kinetic model for high pressure and medium temperature combustion of light hydrocarbons. In the present work, this model is further
10.1021/ie800137d CCC: $40.75 2008 American Chemical Society Published on Web 07/25/2008
6580 Ind. Eng. Chem. Res., Vol. 47, No. 17, 2008 Table 1. Overview of the Most Important Experimental Achievements Reported in the Literaturea reported by
T [K]
P [bar]
τ [s]
CH4/O2
O2 [%]
Newitt and Haffner, 1932 Lott and Sliepcevich, 1967 Yarlagadda et al., 1988
614 537 725 684 724 729 723 723 723 628 628 637 654 723 748 713 657 704 673 733 673 633 603 603 719-740 727-809 696-744 710-786 703 703 723 675-763 675-763
107.8 3450 25.2 36.0 50.7 66.2 50.0 50.0 50.0 30.4 50.7 60.8 60.8 50.7 50.7 20.0 50.7 50.7 50.0 41.5 10.0 30.0 50.0 70.0 40.0 40.0 60.0 59.9 34.0 50.0 50.0 100.0 100.0
720 420 82 284 211 236 200 200 200 174 264 42 25 9 9 24.6 187 5.9 6.0 1 38 126 220 308 1.3 1.2 1.3 1.3 60 60 33 100 K for experiment A, while the higher O2 feed concentration in experiment B resulted in an adiabatic temperature rise >200 K. This induced instabilities in the temperature and pressure control of the system and facilitated relatively large measuring uncertainties of the alcohols, in particular. Moreover, it makes the experimental results less suited for model validation. For this reason, the deviations between experimental and numerical results seen in Figure 2 do not come unexpected. Even though care should be taken to validate the model performance based on the undiluted data, it is notable that the methanol yield is predicted satisfactorily. Figure 2 shows that methanol is the dominating product from the reaction followed by CO, while CO2, C2H6, and C2H5OH are all limited side products. A calculation of the average CH3OH selectivity during experiment A with the lowest feed concentration of O2 (Figure 2, left) yields SjCH3OH ) 74.6% based on the measurements obtained from 675-763 K. The correj CH4 sponding experimental average conversion of CH4 yields X ) 2.2% throughout the same temperature interval. It is noted that the highest measured selectivity of CH3OH during experiment A is 79% at 1.8% CH4 conversion at 683 K. The lower CH4/O2 ratio in experiment B results in a somewhat j CH4 ) 5.3%), while the average higher CH4 conversion (X selectivity of CH3OH has decreased to SjCH3OH ) 53.7%. Even so, this implies a significant increase of the absolute concentration of CH3OH. Again, average values are based on the measurements conducted at 675-763 K. The peak selectivity of CH3OH during this experiment is also measured at 683 K yielding 74% at 4.8% CH4 conversion, but the general scatter
Figure 3. Main reaction pathways for CH4 conversion at the investigated conditions. The fractional contributions of competing pathways are dependent on the reaction conditions, but the thicker arrows denote the most important pathways.
in the CH3OH concentrations makes this value of SCH3OH,max quite uncertain. Comparison of the present results with literature data (Figure 1) shows that our data, while in the upper range for peak selectivity and fuel conversion, are consistent with the bulk of data reported. Our results fall short compared to the commercial target area and do not lend support to the most promising literature data.9,11,12,19,22 Figure 3 shows a reaction path diagram for the conditions of Figure 2, based on the present chemical kinetic model. Methane is converted to the methoxy radical (CH3O) through a sequence of reactions that is similar to that proposed for dilute conditions48 CH4 + OH f CH3 + H2O
(R1)
CH3 + O2(+M) f CH3OO(+M)
(R2)
CH3 + HO2 f CH3O + OH
(R3)
CH3OO + CH3 f CH3O + CH3O
(R4)
The methoxy radical is a key intermediate in the hydrocarbon oxidation chain as well as a precursor of methanol and the selectivity for forming methanol is very sensitive to the fate of this species. Reaction of CH3O with CH4 CH3O + CH4 f CH3OH + CH3
(R5)
is the dominating source of CH3OH in the system, while thermal dissociation CH3O(+M) f CH2O + H(+M)
(R6)
yields formaldehyde (CH2O), which is easily oxidized further to CO (Figure 3). Compared to the dilute conditions investigated previously,48 the increased CH4 concentration of the practical
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Figure 4. Influence of the pressure on the selectivity of CH3OH (SCH3OH), yield of CH3OH (YCH3OH), and conversion of CH4 (XCH4). Results are based on adiabatic plug-flow simulations with the kinetic model using an inlet temperature of 643 K and a constant CH4/O2 ratio in the feed of 23.6. Residence times corresponds to τ[O2]f0 + 1 s.
process serves to enhance formation of CH3OH by favoring reaction R5 compared to R6. However, this increase in selectivity is partly counterbalanced by the considerable temperature increase caused by the higher reactant levels. Process Parameters. In the following, the importance of the major process parameters are discussed, based on calculations with the chemical kinetic model, as well as results from literature. The main variables are temperature, pressure, and CH4 /O2 ratio. In addition, also the natural gas composition and potential additives to the process are discussed. Temperature and Pressure. As discussed above, the homogeneous oxidation of methane is governed by a complex freeradical mechanism, which is difficult to control. The very stable methane molecule must be converted to methanol without promoting further oxidation to undesired carbon oxides. The rate limiting step in methane combustion is the breaking of the very stable H-C bond. At atmospheric pressure, this typically requires temperatures above 1000 K. Such temperatures have an adverse effect on the selectivity for the desired products. Instead reaction is promoted by increasing the pressure. A high pressure causes a shift in the CH4 initiation temperature toward lower values.48 The selectivity of oxygenated hydrocarbons is greatly improved by applying relatively low reaction temperatures combined with high pressure. Following Le Chatelier’s principle, high pressure also favors the overall conversion of methane and oxygen to oxygenated hydrocarbons, e.g. CH4 + 1 /2O2 f CH3OH, in order to decrease the total number of molecules. Figure 4 shows how the pressure affects the selectivity of CH3OH (SCH3OH), the yield of CH3OH (YCH3OH), and the conversion of CH4 (XCH4) at 643 K, according to the chemical kinetic model. The selectivity for forming CH3OH increases rapidly with pressure in the range 1-100 bar and then approaches asymptotically a constant value as the pressure increases above 150 bar. The yield of CH3OH also increases with pressure up to about 150 bar and then reaches a constant value, even though the conversion of CH4 slowly decreases over this pressure range. The effect of pressure can largely be understood in terms of the competition between thermal dissociation of CH3O (R6) and reaction with CH4 (R5). Reaction R6 is a pressure dependent reaction that is in the falloff region under the conditions of interest for the process (Figure 5). As the pressure increases above atmospheric, the resulting rate constant for reaction R6 asymptotically approaches the high-pressure limit k∞, where it becomes a first-order reaction. In this way, an increased pressure favors the CH3O + CH4 reaction (a second order reaction) over
Figure 5. Falloff behavior for the thermal dissociation of CH3O (R6) at 643 K.
Figure 6. Influence of the CH4/O2 ratio on the selectivity of CH3OH (SCH3OH), yield of CH3OH (YCH3OH), and conversion of CH4 (XCH4). Results are based on adiabatic plug-flow simulations with the kinetic model using an inlet temperature of 643 K and a constant pressure of 97.4 bar. Residence times corresponds to τ[O2]f0 + 1 s.
thermal dissociation of CH3O (approaching first order), yielding an increase in the selectivity for formation of methanol. However, since the atomic hydrogen formed in (R6) is more reactive than the methyl radical from (R5), the conversion of CH4 decreases with increasing pressure, thereby limiting the overall yield of CH3OH at high pressures. From a practical point of view, it is unfortunate if pressures higher than the wellhead pressure obtained at gas recovery sites (typically 80-150 bar66) must be applied, since gas compression is an expensive unit operation. Fortunately, reports from the literature13,20,37,38,40,67 indicate that methanol formation reaches a maximum value at pressures between 100 and 150 bar with the largest gain occurring below 100 bar. This observation is consistent with the present findings (Figure 4). It is also in line with the results of experiments carried out at extremely high pressures (3450-13800 bar); these yielded a peak methanol selectivity of 40.1% at 6.3% methane conversion.8 CH4/O2 Ratio. Figure 6 shows how the CH4/O2 ratio affects SCH3OH, YCH3OH, and XCH4, according to the model. A high CH4/ O2 ratio in the feed favors a high selectivity of methanol. The reason is found in the conversion of the intermediate methoxy radical. As discussed above, reaction of CH3O with CH4 CH3O + CH4 f CH3OH + CH3
(R7)
is the dominating source of CH3OH in the system, while thermal dissociation CH3O(+M) f CH2O + H(+M)
(R8)
yields formaldehyde and is an important step toward deep oxidation. To be significant compared to thermal dissociation, reaction R5 requires that CH4 is present in sufficient amounts to become the most frequent collision partner of CH3O. As a
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consequence, high CH4/O2 ratios combined with a high absolute concentration of CH4 promote high selectivities of CH3OH, while suppressing alternative reaction channels. At the chosen temperature of 643 K, the selectivity of CH3OH approaches 100% when CH4/O2 > 35 (Figure 6), but too high ratios require very low absolute concentrations of O2 and thereby limit the fuel conversion and the product yield. High absolute O2 concentrations, consistent with low CH4/O2 ratios, will, on the other hand, result in a substantial heat release and hence, promote deep oxidation to CO and CO2. This trend of inverse proportionality between methanol selectivity and methane conversion is recognized among the experimental series depicted in Figure 1. Previous investigations listed in Table 1 cover CH4/ O2 ratios from 8.1 to 39, but considering the large scatter in these results, it is difficult to deduce a more narrow optimal range. Gas Phase Sensitizers. Certain gaseous components, present in the raw gas or additives to the process, may be able to catalyze the low-temperature conversion of methane to oxygenates. This can be achieved indirectly if the species promote fuel conversion at lower temperatures, which favors oxygenated products with low thermal stability. The selectivity of desired products may also be directly promoted if the components can enable new and more attractive reaction pathways. Compounds with these abilities are called initiators and promoters, respectively, or gas phase sensitizers in general, and they may prove to open up new possibilities in the optimization of the partial oxidation process. It has been demonstrated that natural gas is more susceptible to partial oxidation to oxygenated hydrocarbons than pure methane.36,68 This indicates that the higher hydrocarbons (C >1) present in natural gas have a promoting effect. These compounds typically represent 1-10% of raw natural gas with C2H6 as the major constituent. Compared to CH4, bond dissociation energies are lower in higher hydrocarbons, which are therefore expected to react more easily and form radicals that can promote ignition of the main hydrocarbon fuel CH4. Recent experimental and modeling investigations of CH4/ C2H6 blends under relevant conditions48 have indicated a minor enhancement of the yield of alcohols compared to pure CH4 oxidation experiments. On the other hand, corresponding kinetic modeling predictions suggest that the oxidation pathways of C2H6 do not permit more than maximum 50% of the carbon content enclosed in C2H6 to be converted to CH3OH. Additives of interest include nitrogen oxides (NOx, i.e. NO and NO2). The promoting effect of nitrogen oxides on hydrocarbon oxidation chemistry was reported already in the early thirties69,70 and has been the subject of numerous investigations, e.g., refs 46, 49, and 71–78, with some specifically dedicated to the direct GTL process.79–83 Nitrogen oxides have the ability to reduce the initiation temperature of the hydrocarbon conversion, either through direct reactions with intermediate species from the hydrocarbon oxidation chain, or indirectly through a number of radical reactions between NO/NO2 and the H/O radical pool. Both mechanisms share the common feature that NO and NO2 are continuously recirculated in so-called autocatalytic cycles. As a consequence, even trace amounts of NOx present in the reaction mixture can have a substantial impact on the oxidation chemistry of CH4 and other hydrocarbon fuels. The influence of NOx on CH3OH formation from partial oxidation of CH4 were recently investigated by the authors49 under conditions relevant to the GTL process. Here, experimental and modeling investigations documented the sensitizing effects of NOx on the general hydrocarbon oxidation chemistry,
Figure 7. Temperature and concentration profiles as a function of the residence time from an adiabatic plug-flow simulation of experiment B (100 bar, φ ) 40.9) based on the detailed chemical kinetic model from Rasmussen and co-workers.46–48 Experimental conditions are found in Table 1. The initial reactor temperature is 683 K. The figure only shows the initial section of the reactor where the main fuel conversion takes place. The arrow indicates the peak concentration of CH3OH.
as described above, but the direct impact on CH3OH formation was limited in terms of enhanced concentrations due to side reactions between NOx and the methanol precursor CH3O at low temperatures. However, NOx addition proved to exhibit a stabilizing effect on CH3OH formation over a wide temperature span. Global Optimization using Chemical Kinetic Modeling. Approach. The detailed chemical kinetic model provides a reliable prediction of the GTL process and the effect of the main process parameters. In the following, this will be utilized for a semitheoretical determination of the optimal conditions for high methanol yields (YCH3OH) in terms of the most important independent parameters: temperature, pressure, and CH4/O2 ratio in the feed for a single pass of a flow reactor. Considering the low degree of conversion during a single pass of the reactor, it is reasonable to assume that the molar flow rate is constant throughout the reactor, which means that YCH3OH can be expressed directly as the ratio between the molar fractions of CH3OH and CH4 at the outlet and inlet respectively, i.e. YCH3OH ) FCH3OH,out/FCH4,in ≈ yCH3OH,out/yCH4,in. In the present work, it is assumed that the reaction takes place in a flow reactor. Hence, the kinetic modeling are carried out as adiabatic plug-flow calculations. Moreover, it is assumed that the reactor s/V ratio is sufficiently low to justify disregarding potential surface effects and that a critical residence time (τcrit) is exceeded, long enough to allow full conversion of the O2 load. The application of gas phase sensitizers are neglected in the present optimization. For simplicity, it is further assumed that the feed is composed solely of CH4 and O2, which means that the feed composition is defined solely by the CH4/O2 ratio. The optimization of the GTL process is now defined as the task of maximizing the function YCH3OH(T, P, CH4/O2), i.e. arg maxT,P,φˆ{YCH3OH(T, P, CH4/O2)}. Unfortunately, the problem cannot be considered as an ordinary nonlinear least-squares problem, because the function YCH3OH(T, P, CH4/O2) (based on the DCKM) probably has several local maxima from which only the global maximum YCH3OH(Topt, Popt, (CH4/O2)opt) is of interest. In order to meet this requirement, a global optimization methodology must be employed. The approach used in the present work, using interval analysis, is described in Appendix A. Figure 7 shows an example of calculated concentration profiles of selected species as a function of the residence time in a flow reactor during the time period of the main fuel conversion. The figure is based on an adiabatic plug-flow
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simulation of experiment B (Figure 2 (right), 100 bar, φ ) 40.9) using the DCKM with an initial reactor temperature of 683 K. A maximum concentration of CH3OH is obtained at the time of full O2 conversion (indicated by the small arrow) followed by a minor conversion until an almost constant level is obtained shortly after. The maximum CH3OH concentration is very sharp making it practically impossible to balance an industrial process at the corresponding conditions. However, as also shown in Figure 7, the removal of CH3OH is modest even though the final temperature is high and the example indicates less than 10% loss of CH3OH after 1 s from full conversion of the O2 feed. Still, calculations indicate that higher O2 loads, consistent with an increasing temperature rise during the reaction, will enhance the postconversion of desired product. This has an unfortunate implication in relation to the current process optimization strategy. The global optimization routine optimizes the reactor conditions in order to yield this peak concentration of CH3OH at the exit of the reactor (τexit ) τcrit) and thereby avoid any subsequent conversion of CH3OH. However, this condition will be difficult to obtain in a practical application where surely τ > τcrit. A minor resulting loss of the desired product may be acceptable. However, since the overall energy efficiency of the process is constrained by the selectivity of the recovered oxygenated hydrocarbons, it is costly to allow too high losses consistent with low CH4/O2 ratios. The problem is circumvented by including a penalty factor to the objective function in the form of the square root of the adiabatic temperature rise resulting from the reaction. ∆Tadb increases with decreasing values of CH4/O2 provided full conversion of O2. The weighting of ∆Tadb by the power of 1/2 is an estimate based on preliminary simulations and a maximum acceptable loss of CH3OH of ∼10% during postconversion of products. The modified objective function is defined by: q (T, P, CH4/O2) ) YCH 3OH
YCH4,in
∆T YCH3OH,out √ adb
(2)
All the involved terms are dependent on T, P, and CH4/O2. The optimization problem can be defined as: arg min(T,P,φˆ){YqCH3OH(T, P, CH4/O2)}. On the basis of experiences from the literature (Table 1), it is expected to find the global minimizer within the ranges: T ∈ [550, 800] K, P ∈ [50, 110] bar, and CH4/O2 ∈ [10, 60], where T represents the initial reactor temperature. A relatively short constant residence time of τ ) 3 s is applied in the simulations. As discussed above, this residence time will, in practice, correspond to τcrit. Optimal Conditions. The optimal conditions listed below were located after 82 iterations: Topt ) 643.3 K
(3)
Popt ) 97.4 bar
(4)
(CH4/O2)opt ) 23.63
(5)
for τ g 3 s. The value of (CH4/O2)opt corresponds to [CH4]0 ) 95.94% and [O2]0 ) 4.06%. Evaluation of the optimal conditions at a more realistic residence time τ ) τcrit + 1.0 s based on the kinetic model yields: SCH3OH,opt ) 75 . 4%
(6)
XCH4,opt ) 5 . 59%
(7)
YCH3OH,opt ) 4 . 21%
(8)
Calculation of the adiabatic temperature rise gives ∆Tadb ) 193 K. The key figures in 6–8 include a loss of 6.2% of the predicted
maximum concentration of CH3OH due to product conversion after O2 depletion. The optimum conditions in 3–5 are given with a relatively high numeric accuracy that will be difficult to sustain in a practical application. More realistic operating conditions are expected to be Topt ( 10 K, Popt ( 10 bar, and (CH4/O2)opt ( 1. A sensitivity analysis indicates that variations of T, P, and within these limits can have an overall impact on SCH3OH,opt of up to (10% at the expense of (9% of XCH4,opt with an opposite sign. However, the calculations are much more sensitive to changes in CH4/O2 than T and P. Thus, if (CH4/O2)opt is held constant, calculations with Topt ( 10 K and Popt ( 10 bar only indicate a maximum impact on SCH3OH,opt and XCH4,opt of (2-3% with an opposite sign of ∆SCH3OH,opt and ∆XCH4,opt. Practical Implications. Technical economic evaluations of the direct homogeneous process by Kuo et al.26 predicted that a selectivity of methanol of 90% at 7.5% conversion of methane will make the direct process competitive with conventional techniques. This evaluation assumed the process to run at 50 bar with a recycling ratio of 30 for unconverted hydrocarbons, and an estimated energy efficiency of 70% compared to 65% for the conventional syngas-based process. Kuo et al. further stated that higher product selectivities are more desirable than high fuel conversions based on a sensitivity analysis of their calculations. A similar study by Lange and Tijm27 estimated a methanol selectivity of 80% to be competitive at 10% methane conversion based on derived correlations between capital costs and energy efficiencies, energy loss, and momentum transfer duties (transfer of heat and matter) for different related chemical processes. These economic evaluations date more than 10 years back and, consequently, do not reflect the latest technological developments of process equipment and design or the changes in the markets of methanol and natural gas. Moreover, both studies only consider large-capacity facilities where conventional syngas-based methanol plants are very competitive. It is expected that units of smaller scale are able to compete with conventional techniques at significantly lower methanol yields than indicated above, but updated technical economic calculations are needed for a quantitative assessment. Figure 1 shows that the predicted value of YCH3OH,opt is higher than most previous results from the literature except for the experiments reported by Gesser and co-workers9,11,12 and Feng et al.19 However, YCH3OH,opt is not sufficiently close to the “commercial target” area to guarantee industrial feasibility of the homogeneous GTL process. Still, there may be possibilities for a commercial exploitation of the technology under specific circumstances, e.g. in small-scale units or in situations where conventional utilization of natural gas resources is economically unattractive. Conclusions The direct homogeneous partial oxidation of methane to methanol was investigated through modeling and a limited set of experiments. The experiments were conducted in a laminar flow quartz reactor at 100 bar and 598-763 K under undiluted conditions. The modeling was performed with a detailed chemical kinetic model, previously validated against a range of high-pressure data on oxidation of CO/H2 and small hydrocarbons. Modeling predictions compared well also with the experimental data obtained in the present study. On the basis of the chemical kinetic modeling, the effect of the main process parameters was discussed and the process is optimized, using a numerical global optimization methodology based on interval
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analysis. The experimental data, as well as the optimization results, indicated yields of methanol in the upper range of those reported in literature. However, the values were well below the commercial target. Acknowledgment The work is part of the CHEC (Combustion and Harmful Emission Control) research program. It was financially supported by the Technical University of Denmark and the Danish Technical Research Council. The authors acknowledge the contributing work of Anja Egede Rasmussen. Appendix A: Global Optimization Procedure A global optimization methodology is employed. The algorithm utilizes the features of interval analysis to identify the global minimum of a given objective function within a finite region of the independent variables. Interval analysis84 is a tool for automatic control and quantification of the general sources of error in numerical computations, which are rounding errors, truncation errors, and input errors. The latter may be viewed in terms of uncertainty limits associated with a given measurable quantity obtained in the laboratory, e.g. temperature, concentration, dimension, time, etc. An important feature of interval analysis is the ability to enclose ranges of functions. This can be utilized to e.g. solve equations with interval coefficients, prove existence of certain solutions, and to solve global optimization problems, which has found wide use within rigorous mathematics and computer sciences.84,85 In the present work, all computational aspects of interval analysis are carried out in Matlab using the INTLAB (INTerval LABoratory) toolbox.86,87 The applied global optimization routine is based on the work of Henriksen and Madsen.88 A detailed description of the methodology is found elsewhere,89 and only a brief outline of the computational approach is provided here. Consider a continuous nonlinear function of the kind f: D f R, where D ⊆ Rn. Using interval analysis, the current method aims to determine the global minimum of f, i.e. f/ ) infimum{f(x)|x ∈ D}, and the points where it is attained: X/ ) {x ∈ D|f(x) ) f/}. A candidate set C is defined as a finite set of subregions C(j) ⊆ D where ∪j C(j) contains the set of minimizers X/ of f. The interval extension of f is denoted by F, and the upper and lower bounds of F(C(j)) are given by U(C(j)) and L(C(j)), respectively. Notice that min{L(C(j))} e f/ e min{U(C(j))} ≡ ξ j
j
(9)
which means that if L(C(j)) > ξ then C(j) does not contain a global minimizer, and it can safely be discarded from C. Narrow function value bounds can be obtained by splitting C(j) into smaller subregions. In the present work, this is done by simple bisection. When j > 1, the interval extension containing minj{L(C(j))} is selected for the split, since this is the most likely extension to contain X/; even though it is not guaranteed. Moreover, when considering multiple dimensions, the bisection is conducted in the direction of the largest radius of C(j). The candidate set C is divided into a work set S and a result set R, where X/ ∈ S ∪ R. If w(F(S(j))) e δ
(10)
then S(j) is moved from S to R. The algorithm is now based on continuous bisection of S(j) followed by updates of ξ, in accordance with eq 9 and subsequent evaluation of the solution
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ReceiVed for reView January 25, 2008 ReVised manuscript receiVed April 6, 2008 Accepted June 6, 2008 IE800137D
