A KINEI'IC MODEL FOR THE REACTIONS OF CD AND H2 TO CH4 AND Czflz !N A FLOW MICROWAVE PLASMA REACTOR Thesis for the Degreeof M. S. MIDI-HGAN STATE UNIVERSITY STEVEN F. MERTZ 197.5 E: BINEING av " . “DAB 8. SflNS' ' MA “a“. ‘ ‘ LIBRARY améz'fis , S'RING'URT. MICHIGRI lLM __ ‘ ,. -: ABSTRACT A KINETIC MODEL FOR THE REACTIONS OF CO AND H2 TO CH4 AND CZHZ IN A FLOW MICROWAVE DISCHARGE REACTOR BY Steven F. Mertz A kinetic model was developed to describe the reactions of CO and H2 to CH4 and C2H2 in a microwave plasma. The experimental system consisted of a 24 mm I.D. tubular quartz reactor which passed through a microwave cavity. A variable-incident power waveguide system could. supply up to 800 watts of incident microwave power to the cavity. The reactant gas mixture of H2 and CO flowed through the reactor, where a plasma was maintained under pressures of 20-100 mm Hg. The reactor effluent was analyzed by IR spectroscopy for CH4 and C2H2. Conversions of up to 5.3% CO to C2H2 and 7.2% CO to CH4 were observed. A 26-reaction kinetic model was developed and fitted to the experimental data. The plasma reactor was modeled in two zones: a discharge zone where electron-impact dissociations produce H, C, and O, and a downstream recombination zone where the atomic species from the discharge recombine. The discharge zone was modeled as a well-mixed reactor, and the recombination zone was modeled as a plug-flow reactor. The model was able to explain the asympotic shape of the observed conversion versus residence time data; the effect is due to a kinetic limitation. This also explains why the conversions obtained in the plasma cannot be predicted by thermodynamic equilibrium. A KINETIC MODEL FOR THE REACTIONS OF CO AND H2 TO CH4 AND CZHZ IN A FLOW MICROWAVE PLASMA REACTOR by Steven F. Mertz A THESIS Submitted to Mi chi gan State Unive r sity in partial fulfillment of the requirements for the degree of MAS TER OF SCIENCE Department of Chemical Engineering 1975 ACKNOWLEDGEMENT The author gratefully acknowledges the Detroit Edison Company and the Division of Engineering Research for their financial support of this project. Sincere appreciation is extended to Dr. Martin C. Hawley and Dr. Jes Asmussen for their assistance and guidance during the course of this effort. ii TABLE OF CONTENTS ABSTRACT.................. ACKNOWLEDGEMENTS . . . . . . . . . . . . INTRODUCTION. . . . . . . . . . . . . . . . REVIEW OF PREVIOUS WORK . . . . . . . . . DESCRIPTION OF EXPERIMENTAL PROGRAM . Reactor Flow System . . . . . . . . . . . . Plasma Cavity and Microwave System . . Experimental Procedure . . . . . . . . . . Experimental Results . . . . . . . . . . . . DISCUSSION AND MODELING. . . . . . . . . . Reactions of the Hz/CO Plasma System. . . . Reactor Model. . . . . . . . . . . . . . a) Plasma Zone . . . . . . . . . . . . b) Recombination Zone . . . . . . . . . . Data Fitting . . . . . . . . . . . . . Comparison of Model to Experimental Results. CONCLUSIONS . . . . . . . . . . . . . . . . NOTATION . . . . . . . . . . . . . . . . . . REFERENCES . . . . . . . . . . . . . . . . APPENDIX . . . . . . . . . . . . . . . . . . Fortran Listing of Regression Program and Simplified Reactor Model . . . . . . . . . . Fortran Listing of Complete Reactor Model. . Suggestions for Future Work . . . . . . . . iii Page ii \JNO‘O‘C‘Nt—I 11 11 l3 13 15 16 17 20 21 38 41 42 57 66 Osoooflchm-bwrvt— H y—u H 0 LIST OF FIGURES Plasma Reactor Flow System Plasma Cavity Effect of Space Time on Conversion of CO to CH4 Effect of Space Time on Conversion of CO to CZHZ Effect of Pressure on Conversion of CO to CH4 and CZHZ Effect of 6/1 and 2/1 HZ/CO Feeds on Conversion to CH4 Effect of 6/1 and 2/1 HZ/CO Feeds on Conversion to CZHZ Effect of Pressure on Shape of Plasma Conceptual Model of Plasma Reactor Effect of Residence Time upon Concentration of Atomic Species Leaving Discharge Concentration Profiles of Intermediates and Products in Recombination Zone iv AwNv—n LIS T OF TA BLES Reactions of Hz/CO Plasma System Reactions Used in Simplified Model Rate Expressions Used in Recombination Zone Comparison of Literature Estimates of Rates to Rates from Regression INTRODUCTION Several investigators have successfully modeled discharge reactor systems for relatively simple reacting gas mixtures. Brown and Bell1 have studied the reactions of an OZ/CO/COZ discharge; Bell2 has modeled H2 and Morris3 have analyzed 0 dissociation-recombination reactions.- 2 These investigators have simulated the behavior of simple reacting dissociation-recombination reactions; and Mearns chemical systems using straightforward plug flow, backmix, and non- :flow kinetic models. Their studies have defined the kinetic, discharge, and transport parameters that determine the chemical nature of plasma systems and have shown that such systems can be successfully described by kinetic models based on electron-impact dissociation and free radical recombination. The objective of the work described in this paper was to extend the previous work in plasma kinetic modeling to complex systems in- volving multiple parallel and consecutive free radical reactions. The papers of Brown and Bell formed the groundwork for our investigation of the reactions of H and CO to hydrocarbons in a microwave 2 plasma system. The kinetic constants for H and CO dissociation- recombination processes in discharges werze determined from their experimental studies. A flow-reactor model for HZ/CO plasmas was developed by combining the electron -impact dis sociation-recombination kinetics of H2 and CO plasmas with the recombination kinetics of hydrogen, carbon and hydrocarbon radicals. This model was success- fully fitted to experimental data. This paper describes the experimental plasma system and experimental procedure, develops a kinetic model for analyzing plasma reactions, and reports the kinetic parameters used to fit the data. The success of this model shows that the kinetic models of Bell and Brown can be extended to more complex reaction systems, and that plasma chemical reactions can be analyzed in terms of the kinetics of electron-impact dis sociations and free radical recombinations. PREVIOUS WORK In addition to the kinetic modeling of Bell and Brown, a number of researchers have experimentally investigated carbon -hydrogen dis- charge systems. Baddour and Blanchet4 have studied the reactions of carbon vapor with hydrogen and methane. The carbon vapor was pro- duced from the graphite electrodes of a 30 kw are at atmospheric pressure. The temperature of this are was estimated to be 3500-45000K. By withdrawing gas from the are through a water-cooled quench probe, mixtures of up to 26% acetylene in hydrogen could be obtained. Equilibrium calculations indicate that the maximum concentration of acetylene should be about 7 % ; however, it was explained that the high yield was due to the quench step. Blaustein and Fu5 have studied the reaction of HZ/CO and HZ/COZ of 12 and 50 torr and reaction times of 30 to 240 seconds were used. mixtures in a nonflow 2450 MHz discharge reactor. Pressures Without quenching, the maximum hydrocarbon yield was 17 -18 ‘70 con- version of CO to CH4 and CZHZ at 12 torr, and 24-25% at 50 torr. By freezing out the water formed during the reaction, the conversion was increased to 78%. With a liquid N2 hydrocarbon products, conversions of about 90% were observed. The cold trap which froze out all the predominant products are CH4, C H2, and C2H6; the ratio of products 2 depends upon quench temperature. Very little C H was observed. Lett, et. a1. , 6 have studied the reactionszof4an HZ/CO/Ar mixture in a flow plasma system. A 2450 MHz microwave source was used. Mass Spectrometry was used to analyze for hydrocarbon products. The major products were CH4 and CZHZ; negligible amounts of C2H4 and C2H6 were observed. The reaction appeared to have a first-order dependence upon CO concentration. Baddour and Iwasyk7 have also reported the results of C/H2 reactions in a high-intensity arc. As in the paper by Baddour and Blanchet, they have used an equilibrium approach to explain the formation of CZHZ in high -temperature systems; methane formation is not con- sidered. A key reaction step in Baddour's analysis is the formation of C‘2 radicals in the gas phase as a precursor of acetylene: 2C-->CZ C2 + H2 -> CZHZ McTaggert8 and Vastola et. a1. 9 have investigated low pressure reactions of H2, graphite, and CO in microwave (2450 MHz) flow reactors with low residence times. Only traces of hydrocarbon products (mainly CH4 and CZHZ) were observed in the gas collected from the HZ/CO dis- charge. A hydrogen conversion of about 10 % was observed when graphite rods were suspended in a hydrogen discharge. Spectroscopic evidence indicates that CH radicals are intermediates in the formation of hydro- carbons. Fu, Blaustein, and Wenderlo’ 11have conducted pyrolysis experiments by heating coal in Ar, C02’ and H2 discharges. The main products were H2, CO, and C02' Some methane and acetylene were also produced. A number of papers also deal with the kinetic analysis of plasma reactions. Kondratiev, et. al. 12have discussed the difficulties involved in calculating rate constants for plasma reactions from collision cross sections. Furthermore, there are problems associated with applying kinetic rates determined for molecules in the ground state to plasma reactions; while a large fraction of the molecules in a low- temperature plasma are not ionized or dissociated, it is likely that a significant number of these molecules will possess some excitation due to electron-molecule collisions. The deviation between the rate constants for excited molecules and for ground-state molecules can vary by 30 percent in some cases. Therefore, although the general principles of kinetic analysis apply to plasma reactions, the rate con- stants for plasma systems may vary from those measured for non- plasma gas phase reactions. Polakl3 has observed that one of the main difficulties in relating kinetic data to theory for plasma systems is that low -temperature plasmas are not in thermal equilibrium. The activated complex theory of gas -phase reactions assumes an equilibrium between reactant molecules and the excited activated complex (which has a fixed prob- ability of either proceeding to products or reverting to reactant molecules); the Arrhenius equation for rate variation with temperature is a result of this theory. However, the reacting mixture must be in thermal equilibrium (i.e. , have a Maxwell-Boltzman distribution of energies) in order to apply the activated complex theory. Hence, the temperature —rate variation for plasma reactions may not be adequately described by the Arrhenius equation. In order to rigorously treat low- temperature plasma processes, it is necessary to deal with several different temperatures: electron temperatures, ion temperatures, gas translational temperatures, gas rotational temperatures, and gas vibrational temperatures . Polak has also developed a kinetic model for a high temperature plasma arc converting methane to acetylene. This system, which operates at a pressure of l atmOSphere, probably approaches thermal equilibrium due to the high frequency of electron-gas collisions (due, in turn, to the high pressure). A classical multiple-reaction kinetic model using Arrhenius rates successfully described the performance of the plasma jet. Bell and others recombination reactions in plasma systems. Bell has develOped a 1’ 2’ 3’ 14 have successfully modeled dissociation- general model for plasma reactions14 that includes equations of continuity for electrons, ions, and uncharged species (free radicals and ground state molecules). He has applied this model to a hydrogen plasma2 for nonflow, backmix flow, and plug flow systems. Bell found that data for H.2 dissociation in a flowing microwave discharge was best described by a backmix model. Bell and Brown1 have modeled the discharge reactions of CO and O2 and the decomposition of C02 using a lS-reaction model. Their data were generated in a 13. 56 MHz RF discharge; pressure varied from 2-32 torr, flow rate varied from 2-30u moles/sec. , and power varied from 50-350 watts. Their model consisted of two plug flow reaction zones in series. In the first zone, electron impact dissociations form the reactive precursors of the products; in the second zone, these free radicals recombine to yield the final stable products. A correction was included for gas bypassing the plasma without reacting. 15,16 Bell and Kwong and Mearns and Morris3 have studied the dissociation-recombination reactions of oxygen discharges. 'Iheir experiments were performed in flow reactors using either radio frequency or microwave excitation. The proposed reaction mechanism consists of electron impact dissociation followed by four competing recombination reactions. The important variables in plasma systems were reported to be gas flow rate, power, and pressure. EXPERIMENTAL PROGRAM A. Reactor Flow System The plasma reactor flow system is shown in Figure l. The reactor was a 24mm ID quartz tube passing through the axis of a cylindrical microwave cavity. The reactor could be operated at pressures from 2-100 torr. The quartz tube was cooled by an air stream directed into the cavity. The reactor feed system consisted of a feed cylinder, flow meter, and feed valve. H2 and CO were premixed at the desired feed ratio and stored in the feed cylinder during each series of experiments. A pressure regulator delivered the HZ/CO mixture to the rotameter at a constant pressure of 5 psig. The feed valve controlled the flow rate through the rotameter and throttled the feed to the reactor pressure. The effluent from the reactor was passed through a 10- meter Perkin-Elmer gas IR cell for analysis. A silica-gel drying tube ahead of the IR cell removed any H O generated by the 2 reactions; the IR absorption of water would otherwise mask the ab- sorptions of CH4 and CZHZ' After passing through the IR cell, the gases passed through another valve which was used to regulate the reactor pressure. The gases were then exhausted to vacuum. B. Plasma Cavity and Microwave System The plasma is generated within the cavity and is contained by the 24 mm ID quartz tube located along the axis of the cavity (Figure 2). The cavity has an I.D. of 20. 3 cm and can be adjusted to a maximum length of 35 cm. The cavity is water cooled by externally soldered tubing coils. The quartz tube itself is cooled by an air blast directed into the cavity. The plasma can be viewed through a screened window in the cavity side wall. The cavity is designed to operate in a number of different modes. Depending on the method of coupling energy into the cavity and the cavity length, either the TE* 112 or the TM 011 mode can be excited. A probe coupling is used to excite the TE* 112 mode, and a loop coupling is used to excite the TM 011 mode. The cavity length can also be adjusted to achieve maximum power absorption by the plasma. The optimum cavity length depends upon the cavity mode and the plasma pressure. During the experiments described in this paper, the cavity was Operated in the TE>1= 112 mode with a cavity length of 12—13 cm. Power is coupled into the cavity from a variable -incident -power 2450 MHz microwave system. The maximum incident power is 800 watts. Power meters monitor the incident and reflected power; the difference between the two powers represents the power absorbed by the plasma. The cavity and power system were described in detail in earlier papers. 19’ 20 C. Experimental Procedure Prior to operating the system, the feed tank was evacuated and filled with H2 and CO in a predetermined ratio. The reactor system was then evacuated. The plasma was ignited, and then the feed rate, reactor pressure, and absorbed power were adjusted to the desired levels. The system was allowed to operate for several minutes to guarantee that the IR cell was filled with steady-state reactor effluent, then the IR spectrum was scanned for CH4 and CZHZ' The concentrations of CH4 and CZHZ in the reactor effluent were determined from the strength of their IR absorptions. The areas under the CH4 abslorption at 3000 cm-1 and under the CZHZ absorption at 1300-1400 cm' were compared to the areas obtained for mixtures of CH4 and CZHZ in H2 and CO at concentrations corresponding to 5 and 10% conversion, and at the pressures of the experimental runs. The IR system was capable of detecting CH4 and CZHZ concentrations as low as 0.05% at pressures above 20 mm Hg. No absorptions were observed for other hydrocarbons such as C H and C H6’ although 2 4 2 mass spectrosc0py indicated that they were present in small amounts. D. Experimental Results Our experimental program was designed to determine the effects of residence time, feed composition, pressure, and power on the conversion of CO and Hz to hydrocarbons. Figures 3-7 show typical experimental results and will be compared to the results of the kinetic model described in the next section. A complete summary of the experimental results has been reported. Figures 3 and 4 show the effect of residence time on conversion. The space time reported in these figures is defined as V T=_2 V 0 where Vp is the observed plasma volume, and v0 is the volumetric flow of gas into the plasma zone. Due to the volumetric expansion that accompanies the partial dissociation of the reactant gases, the actual residence time is slightly less than the space time. As can be seen, the conversion increases rapidly at first, but then asymptotically approaches a limiting conversion of about 7. 3 per- cent to CH4 and 5. 3 percent to CZHZ. It should be noted that the total conversion of CO is the sum of the conversions shown in Figures 3 and 4, since both CH4 and CZHZ are produced simultaneously. Figure 5 illustrates the effect of plasma pressure on conversion. The data shown was taken at constant space time. As can be seen, for pressures above 20 mm Hg, conversion decreases as pressure increases. Figures 6 and 7 show the effect of variation in feed composition. As expected, a large increase in the Hz/CO ratio increases the conversion of CO to hydrocarbons. The yield of CZHZ shows some dependence upon power level (increasing the power level increases the conversion to CZHZ)’ but the conversion to CH4 does not depend strongly on the power level. During the experiments, the plasma was observed to decrease in size as the pressure was increased. At high pressures the plasma was not symetrical; rather than being cylindrical or spherical, it appeared to be somewhat flattened (Figure 8). At pressures below 40 mm Hg, the plasma appeared to extend to the tube walls. At pressures above 40 mm, the plasma did not extend to the tube walls. At pressures of 80-100 mm, the plasma zone is a flattened sphere about 1. 5 cm across. Because of the small size of the plasma zone, considerable bypassing occurs at high pressure. During high pressure Operation, a soft black deposit resembling soot appeared on the tube wall for a short distance immediately down- stream and upstream of the plasma zone. This deposit apparently is due to the migration of carbon atoms to the tube wall, where they condense to form an amorphous carbon deposit. The tube wall is subject to severe local heating around carbon deposits in the discharge zone. This may be due to an insulating effect of the carbon, or it may be due to exothermic recombination reactions occurring at the carbon surface. With an 112/CO plasma, the maximum feasible pressure range was found to be 80-100 mm Hg. Above 100 mm, the tube wall in the plasma zone overheated excessively in spite of the cooling air jet. A large fraction of the tube wall heating is due to exothermic recombination reactions that occur at the wall. With an H2 or an HZ/CO plasma, tube wall heating is observed as a dull red glow at pressures above 30 mm Hg; air cooling of the quartz tube is required under these conditions. With an argon plasma, no tube wall heating is observed for plasma pressures as high as 500 mm Hg. The reason for the difference in tube wall heating effects be- tween H2 and Ar is that no dissociation reactions occur in an Ar plasma, whereas in a hydrogen plasma, highly endothermic dissociations occur in the plasma zone, and highly exothermic recombinations occur at the tube wall. Bell has indicated that the wall recombination reaction is significant in H2 plasmas;2 this observation was confirmed by our experiments, and the results of our kinetic model are also consistant with this. The HZ/CO plasma has the same blue color as an H2 plasma. However, a very faint greenish -yellow afterglow is observed to extend 5-10 mm downstream of the plasma. This afterglow is probably due to recombination reactions that continue after the gas has left the plasma zone. We may speculate that these reactions only occur downstream of the plasma; however, it is equally likely that they occur to some extent within the plasma, but their emissions are masked by the light emitted by the dominant excitations within the plasma zone. Several investigators of H /CO systems have stressed the 4,5,7 2 importance of quenching. To test this effect, the section of the quartz reactor tube immediately downstream of the plasma zone was packed with dry ice. In our quench experiment, the quench zone was about 3 cm downstream of the plasma zone; this restriction was dictated by the geometry of our microwave cavity. Because of the large diameter of the reactor tube, the contacting of the gases with the chilled walls was not as effective nor as fast as the quench method of Baddour,4' 7 who has used a water-cooled probe of very small diameter to withdraw gases directly from the plasma zone. No improvement in conversion was observed in our system, which indicated that our quench arrange- ment was ineffective. The kinetic model of the reactions in the HZ/CO system explains the importance of rapid quenching, as will be discussed later . lO DISCUSSION AND MODELING A. Reactions of the HZ/CO Plasma System Table 1 lists the chemical equations used to model the reactions of the discharge zone and of the zone immediately downstream of the discharge. The rate constants listed are those used in the kinetic model. As will be explained later, some of the key rate constants were adjusted to improve the fit of the model to our data. Some of the rate constants in Table 1 were reported as functions of temperature; to evaluate these, a temperature of 8000K was assumed for the plasma zone and for the region immediately downstream of the plasma. Bell has reported a temperature range of 6000K to 1000°K for a CO/COZ/O2 plasma. 1 For those rates that were reported for tem- peratures outside the range of temperature expected for the plasma, activation energies available for similar reactions were used to estimate the rates at 8000K. Bell has calculated rate constants for electron -impact dis- sociations of H2 and CO. 1’ 2 These rates are evaluated from reaction cross sections (0') with the assumption of a Maxwell-Boltzmann distri- bution of electron energies. Referring to Table l, the expressions for kl and k are given by 2 k = A( -8— (KT )'3/2 S 6 6(6) eXp(-€/KTe) d6 (1) fime e - 0 Bell has successfully used these rate constants to model the dissociation of H2 and CO in discharges. Electron impact ionizations also occur, but their influence on the products of the plasma reactions is small. The calculated concen- trations of uncharged species (C,H) in the plasma zone are in the range of 1015-1017 atoms/cm3. Charged species must exist at aPPI'OJ-Cimately the same concentration as electrons to maintain charge neutrality in the plasma. The electron density of our system was about 10 electrons/cmB. Ionic species would therefore not exceed concen- trations of 1012 ions/cm3; they would arise from reactions of the general type + e+H2—*H2 +2e e+co~ co++26 11 These ions do react with neutral molecules, but because they are pre- sent in concentrations several orders of magnitude smaller than the uncharged atomic species, their contribution to the total product yield was expected to be small. Bell has reached the same conclusion regarding ion reactions in H2 plasmas. The reactions of the plasma system include electron -impact dissociations to C and H (reactions 1 and 2), recombinations to the feed materials (reactions 3, 4, 5), recombinations to hydrocarbons or hydrocarbon intermediates (reactions 6-22), and reactions to CO and 2 H20 (reactions 23-26). All of the recombinations with the exception of reaction 4 are homogeneous gas-phase reactions. Reaction 4 represents the recombination of H to H2 at the surface of the tube. The rate expression for the disappearance of H is 2R4 = Zk4[H] (2) Bell has evaluated the recombination constant from the rate of H atom 1. migration to the tube wall and the recombination coefficent (y): k=—--r-y (3) where the random velocity and recombination efficiency are given by ’ 8161‘ Vr = '\I m (4) v = .114 exp(-1090/T) (5) This heterogeneous recombination is probably a major factor causing the tube wall heating observed in the plasma zone; nearly half the total hY 1, axial dispersion becomes a significant transport mechanism; axial dispersion results in internal mixing, causing a change from plug- flow reactor performance to backmix performance. In the case of the hydrogen reactions considered here, the effect of axial dispersion is significant -- Pe ranges from 0. 3 to 370 for the range of flows encountered in our experiment. This indicates that a backmix approximation should apply to the hydrogen dissociation reactions. For the carbon monoxide dissociation reactions, the smaller rate constant and diffusivity result in a Peclet number ranging 4 x 10-4 - 0. 6. This would indicate that the carbon reactions should follow a plug 14 flow model. However, experimental observations to be discussed later indicated that axial dispersion may cause a significant mixing effect for the carbon reactions also. Because of this, and to simplify the calculations and reduce the computation time required for the model, the backmix equations were applied to both the H2 and the CO reactions. Bell has compared backmix and plug flow models for H2 dissociation and has found that the backmix approximation be st simulates experimental data. 2 A correction for gas bypassing the plasma was also included. This was necessary because of the experimentally observed variations in plasma size, shape, and flow cross section at different pressures and power levels. As mentioned earlier, at pressures above 40 mm Hg, the plasma recedes from the tube walls and becomes asymmetrical. The bypassing factor was based on the ratio of the observed plasma cross sectional area to the reactor flow area. 2) Recombination Zone (Plug Flow Reactor) After the gases leave the plasma zone, they enter the recom- bination zone. Reactions 3-26 were applied in this region to calculate the conversion of H, C, and O to stable product molecules. Because the diffusivities of the radicals and molecules formed during the re- combination reactions are substantially lower than the diffusivities of H and C atoms, a plug flow model was applied to the recombination zone. The plug flow model consisted of rate equations for the forma- tion and consumption of each of the species appearing in the reaction scheme;1 these equations were solved numerically, using the concentrations of H, C, 0, H2’ and CO exiting the plasma zone (obtained from Equations 6 and 7) as the initial conditions for the plug flow zone. The following steady state plug flow material balance for any species in the recombination zone can be derived from a balance on a differential element of the recombination zone: dni U’dT' :Ri (14) 01‘ dni Tt_ 2R1 (15) 15 A tabulation of the rate expressions used in the model is shown in Table 3. The solution of this system of equations generates concentration profiles for all the chemical species in the recombination zone. A simplified model was also developed; it included only the reaction paths leading to CH4 and CZHZ' The reactions used in this model are listed in Table 2. This model was streamlined as much as possible to minimize the computation time required while still closely matching the results of the complete model. Both models were tested and were shown to give nearly identical results. The execution time of the simplified model was considerably less than that of the complete model. One reason for this was that the short model did not include the reactions leading to HZO' Because these reactions occur at an extremely high rate, they were not rate controlling; however, they required a very small step size in the numerical integration. When they were removed, the model (which had a floating step size) could run much faster. The total reactor model thus consists of the backmix plasma zone and the plug flow recombination zone (including either the complete set of recombination reactions shown in Table l, or the simplified set shown in Table 2). The model of the plasma zone calculates the con- centrations of reactive intermediates H, C, and 0 generated in the discharge. The model of the recombination zone calculates the con- centrations of products that result from the recombination of these intermediates . C. Data Fitting The simplified model was used to fit the experimental data shown in Figures 3 and 4 in a regression which adjusted some of the key reaction rate constants. A fast but accurate model was required in order to do the regres - sion using a reasonable amount of computational time. The reaction rates that were adjusted during the regression were k1, k2, k3, k4, k5, k9, kll’ and k20' The rates kl through k5 were allowed to float because the total conversion to hydrocarbon products depends directly upon the amount of atomic carbon formed in the plasma (compare Figures 3, 4, and 10). and the CH4/CZH2 ratio depends upon the amount of H generated. The CH4/C2H2 ratio also depends upon k9, k”, and k20' The regres- sion routine used Marquardt's method;36 it was set up to select the rate constants that would minimize the squared deviation between observed and calculated conversions to CH4 and C2H2° Table 4 com- pares the results of the regression to the estimates of the rate constants from literature sources. 16 D. Comparison of Model to Experimental Results The rates determined from the data shown in Figures 3 and 4 were used in the model; numerical results from the model are compared with our experimental data in Figures 3 - 7. An important observation is that the conversion versus residence time curves (Figures 3, 4) asymptotically approach a limiting value; however, the conversions observed are not equilibrium conversions. Equilibrium studies by the authors and others4’ 7 have indicated that the observed concentrations of CZHZ and CH4 are not possible at equilibrium in a system at the temperature of our discharge. CH4 will not be found at equilibrium in a system at a temperature high enough to have the observed concentration of CZHZ' The results of our experiments represent a kinetic steady state rather than thermodynamic equilibrium. This result is readily explained by the proposed kinetic model. For a given yield of C atoms leaving the plasma zone, the CH4/CZH2 ratio is largely determined by the recombination kinetics. The region downstream of the plasma zone is clearly not at chemical equilibrium. As soon as the gases leave the plasma, they begin a quench process in which heat is rapidly lost by radiation. The radicals that are present recombine very rapidly to form stable molecules (Figure 11); there is in- sufficient time for equilibrium to be established at the low temperatue of the system. The result is that the recombination process is irreversible, and the resulting product concentrations do not represent equilibrium. Instead, they represent the relative ratios of competing irreversible reaction paths. The asymptotic shapes of the curves of CH4 and CZHZ concentra- tion versus residence time,which resemble an approach to equilibrium, are actually due to kinetic phenomena within the plasma zone. In the plasma zone, the reactions of CO dissociation and recombination reach a_ steady state independent of flow rate for residence times in excess of 0. 20 seconds (see Figure 10); this has the appearance of a system at equilibrium. How- ever, for higher flow rates, the steady state is flow-rate (or residence time) dependent. ‘ To illustrate this, consider a simplified version of the CO dissociation ~recombination process: k2 CO+e-+C+O+e k co+Méc+o+M l7 By a steady-state material balance for the plasma zone: v[c] = Vp(k1ne[CO] - k5M[C][O]) (16) 1:ka ne [co] [C] - (17 Vpksto] + v ) k n [co] k 11 [co] [C] : 1 e : 1 e 1 (18) k5M[O] + v/Vp k5M[O] + ? If V/VP<< kZM [0] (low flow rate), then the conversion to carbon atoms will be independent of flow rate, and the system will appear to be at equilibrium: k 11 CO 1 [c] e K[ 1; K: e (19) [0| k5M However, if v/vpis significant in comparison to kSM [O] , then the conversion to carbon atoms will be flow rate dependent. This is the case for residence times less than 0.20 seconds (Figure 10). By this mechanism, a state of balance between the CO dissocia- tion and recombination reactions gives the appearance of a system at equilibrium, when in fact, most of the reactions occurring in the total reaction system are kinetically limited and are not at equilibrium. For this reason, the product composition cannot be predicted by thermodynamic equilibrium. The kinetic analysis of our model indicates that the conversion of CO to hydrocarbon products is strongly dependent upon the amount of atomic carbon formed in the plasma zone. This is in contrast to earlier speculation20 that the conversion was controlled by the amount of H formed in the plasma. The relatively low conversions observed in our system indicate that there is not a significant amount of free radical chain propagation via H such as the following: H+CO->CH+O H+CO—~OH +C If this were a significant reaction path, higher conversions of CO to hydrocarbons would be expected. Figure 5 illustrates that the model shows the same dependence upon pressure as was experimentally observed. Figures 6 and 7 show the effect of changes in feed Hz/CO ratio. The model predicts the 18 conversion of CH very well for 6/1 and 4/1 ratios, and shows the correct 4 trend for a 2/1 ratio. The model fails to predict the correct feed ratio effect for CZHZ yields. This indicates that the proposed reaction paths to CZHZ are possibly incomplete, or the reactor model is somewhat in error. The model does correctly predict low conversions to C2H4 and C2H6' as was experimentally observed and has been reported by others. 5’ 6 The low yields of C2H4 and C2H6 occur because the reaction paths to these products require bimolecular collisions between species that do not reach high concentrations in the recombination zone. C and CH are largely depleted before much CH 4 has been formed, hence reactions 12 and 15 produce little C2H4. The reaction paths to CH4 favorably compete with the paths to C2H6 because they require CH3-H2 or CH3 -H collisions, which are more likely than CH3-CH3 collisions (Figure 11)- The appearance of narrow bands of soot on the reactor both upstream and downstream of the plasma indicate that the backmix approximation does apply to the carbon reactions as well as to the hydrogen reactions. If there is sufficient axial diapersion to allow carbon atoms to diffuse upstream from the plasma zone, then the plasma zone is probably well mixed. The importance of rapid quench is evident from the rate of depletion of reactive intermediates in the recombination zone (Figure 11). Carbon atoms disappear very rapidly; nearly all the carbon-containing radicals are depleted with 2milliseconds of recombination time; H atoms persist somewhat longer. This means that the product distribution is determined with the first 2 milliseconds after the gases leave the plasma region. In our system, this corresponds to about 2 cm of flow distance at high flow rate. For a quench system to have any effect at all, the plasma gases must impinge immediately upon the quench section, and the contacting must be intimate. The high rate of recom- bination explains why our dry ice quench was ineffective; the recombina- tion reactions were essentially complete before the gases reached the quench zone . l9 CONCLUSIONS A kinetic model was developed to describe the dissociation and recombination processes occurring in a complex discharge reaction system. The model was based on two reaction zones: an electron- impact dissociation zone, followed by a zone where free radicals recom- bine to form hydrocarbon products. The model successfully simulates the residence time and pressure effects observed experimentally. It also explains why discharge reactions appear to approach steady state compositions that are not predicted by thermodynamics. Although it contained many simplifications and approximations, the model was based on a fundamental analysis of the flow processes and chemical reactions occurring in the plasma. The success obtained with this model indicates that kinetic modeling provides a valid analysis of plasma processes. With refinement of the model and more extensive experimental data (such as direct electron density and plasma temperature measurements, accurate measurements of trace products, and perhaps measurements of concentrations of recombination intermediates in a fast -flow experiment), it would probably be possible to completely analyze and quantify the processes occurring in the plasma reactor. 20 )(z -112 H-HZ W923 Flo GH”+-3PU"‘ <1 NOTATION diffusivity of (I through HZ, crud/sec diffusivity of H through H2, cmz/sec constant, dimensionless rate constant length, cm total gas density, particles/cm3 particle mass, g electron mass, g electron density, e/cm3 concentration of species i, atoms/cm3 reactor radius, cm rate, atoms/cm3sec temperature, 0K residence time, sec electron temperature, 0K velocity of gas through plasma, cm/sec volume of gas flowing out of plasma zone volumetric flow of reactants into plasma, measured at plasma temperature and pressure, cm /sec plasma volume , cm3 random velocity, cm/sec conversion of CO in plasma zone conversion of H2 in plasma zone mole fraction CO in feed mole fraction H2 in feed recombination efficiency energy, eV Boltzmann's constant, erg/molecule oK cross section for reaction, cm2 space time in plasma zone, sec fl" acumen 30$ acuooon «£33m .. a 6.3th ZS. 5:333 91) WT w 222 239$ WM 3. 1" 32. L4 0223 TXT . X I «Chan: <$m<= {)\\\ 391) mus—30m as u>._<> aOuhZOU mh<¢ Gum“. m<0 £5.03 KL )._. any—saws £53m: x24» 8: cu - a: chateau mammfinw 1N esswmrm g $me 02. ”.08 3.23 . 2.0.. 02—265 OmngUm ‘0‘ we: <22: 2:30 50...... easy 3:... ($343.. Nh¢<30 2.0.. #22. $30.. 23 Percent CO converted to CH4 Figure 3 - Effect of space time on conversion of CO to CH 4 O ' A . 300 watts 72 350 watts Space time in plasma zone, sec 0 A Percent CO converted to CZHZ Figure 4 - Effect of space time on conversion of CO to CZHZ results of model 01> 300 watts Q l )— 350 watts A 0 l i I 0.1 0.2 0.3 Space time in plasma zone, sec 25 Percent CO converted F Figure 5 - Effect of pressure on conversion of CO to CH4 and C2H2 l l J l 20 40 60 80 Plasma pressure, mm Hg 26 100 Percent CO converted to CH4 Figure 6 - Effect of 6/1 and 2/1 feeds on conversion to CH4 10 )- o 3_ I 9 .— O 8 . I Plasma Pressure: 40 mm Hg 300 watts 400 watts 7__ 86%H2, 14% CO C I 66% H2, 33 % CO O E Results of model l l 2 0 0.1 0.2 Space time in plasma zone, sec 9‘7 Rad mvtuvh-N 0U>F~ Hi (i fibulV un~t~ -UL aha — Percent CO converted to CZHZ Figure 7 - Effect Of 6/1 and 2/1 HZ/CO feeds on conversion to C H 10 2 2 Plasma Pressure: 40 mm Hg 300 watts 400 watts 86%H 14% CO C I 2, 67%H 34% CO O B 2’ results of model 0.2 0.3 28 Figure 8 - Effect of pressure on shape of plasma » 2a 1' . 3 . (a) (b) (C) Note 8 : (a) Plasma at pressure of 20 mm Hg (b) Plasma at pressure of 40 mm Hg as seen in direction of coaxial microwave probe (c) Plasma at pressure of 40 mm Hg as seen at right angles to coaxial microwave probe N.oo wee .emmo qvaU HO MQUdHu. .00 .Nm .ONm .Nmno.emo econ noflofinfiooon osou em Rosomwp "l" g mnwmmmmewo. oo .Nmfi N .O .0 .HIH OD . HIH Motown 30C mam Motown 5830.2“. R0909...» magma mo Epoch $530230 1 o ousmrm 30 Peak concentration of atomic species in plasma zone, atoms/cm3 1x10; 9-)- But- 7‘. 10‘ Figure 10 - Effect of residence time on concentration of atomic species leaving discharge ML- 1x10l II -17. (x1017 Figure 11 - Concentration profiles of intermediates and E products in recombination zone [x1016 _ [CH4] _ [Csz] 1x1015_ [H] _ [CZH6] r- xlOl‘l— H l l l (I '( [CZH4] (10l " l I (' [CH ] [CH1 [CH2] [C] 3 :101 1 1 1 n 0 0.2 0.4 0.6 0.8 1.0 1.2 10. ll. 12. 13. 14. 15. 16. 17. l8. 19. 20. 21. 22. 23. 24. 25. 26. Table 1. Reactions of HZ/CO Plasma System Reaction Hz+e —>2H+e CO+e—’C+O+e 2H+M—>H2+M 2H —.H2 (at wall) c+o+M~co+M C+HZ—’CH+H C+OH-—-CH+O C+H+M—*CH+M 2C+M-CZ+M C2 + H2 -. CZHZ C+H2+M—-CH2+M C + CH4 -. C2H4 CH+H2+M—>CH3+M ZCH —> C2H2 4-eC2H4 +H CH+O-OH+C CH +CH M+CH2+Hz-CH4+M CH2 +H2 —>CH3 +H CH2 + CHZ --C2H4 CH3+H+M—-CH4+M CH3+HZ-CH4+H ZCH3 - CZH6 O+HZ -’OH+H M+OH+H~HZO+M + _. OH H2 H20 + H CO+O+M->COZ+M 2.4x 7.4 x 1.0x 2.8x 9.5x 8.3x 2.0x 1.2x 4.0x 7.0x 1.0x 2.1x 2.0x 2.5x 8.8x 2.1x 1.9x 2.0x 3.9x 1.2x 1.2x 2.8x 4.0x 2.0x 5.2x k. 1 cm3/sec* 10‘9 10-12 10-31 cm3/sec* 6 cm /sec* 103/sec* 10-33 10-17 10-17 10—32 10-30 10-13 10-32 10-15 10-30 10'10 10-12 10-12 10-30 10-13 10-12 10-34 10'15 10-10 10-13 10-25 10'12 10-35 cm6/sec* cm3/sec cm3/sec cm6/sec cm6/sec* cm3/sec’i‘ cm6/sec cm3/sec cm6/sec cm3/sec cm3/sec cm3/sec cm6/sec cm3/sec cm3/sec cm6/sec‘:< cm3/sec cm3/sec cm3/sec cm6/sec cm3/sec cm6/sec Reference (2) (1) (2) (Z) (1) (27) (27) (22) (22) (est) (20,22) (20,22) (25) (22,25) (25) (27) (29,30) (29) (33) (est) (21.23.29.30) (24,26,29) (27,31) (31) (28,31) (32) :1: These rate constants were determined by a regression on the experimental data (see Table 4). 33 All others were estimated from literature sources. Table 2 . 10. ll. l3. 14. 17. 18. 20. 21. Reactions Used in Simplified Model H2 + e —' 2H2 C0 + e —> C + O + e 2H+M—.H2+M 2H —» H2 (at wall) c+o+M»co+M C+H2~CH+H H+C+M-+CH+M 2C+M—~C2+M C2 + H2 ~CZH2 C+H +M-eCH +M 2 2 CH+H2+M~CH3+M ZCH—>C2H2 CHZ+H2+M~CH4+M CH2+H2-CH3+H CH3+H+M~CH4+M CH3 +H2—>CH4+H 34 Table 3 . c121 dt Rate Expressions Used in Recombination Zone. -Zk3[H]ZM - 2k4[H] + k6[C][HZ] - k8[C][H]M + k15[CH][CH4] + k18[CH2][HZ] .. k20[CH3][H]M + k21[CH3][HZ] - k24[OH][H]M + k25[OH][H2] 2 -k5[C][O]M — k6[C][H2] - k7[C][OH] - k8[C][H]M -2k9[C] M - kll[C][H2]M - k12[C][CH4] + k16[CH][O] - -k5[C][O]M + k7[C][OH] - kl6[CH][O] - k23[0][H2] - k26[CO][O]M k9[C]ZM - k10[C2][H2] k6[C][H2] + k7[C][OH] + k8[C][H]M - k13[CH][HZ]M - 21.1 4km]?- - k15[CH][CH4] - k16[CH][O] kll[C][Hz]M - k17[CHZ][H2]M - k18[CH2][HZ] - 2 k19[CHZ]2 kl3[CH][H2]M + k18[CHZ][H2] - k20[CH3][H]M - k21[CH3][H2] 2 - 2 k22[CH3] -klz[C][CH4] - k15[CH][CH4] + k17[CHZ][HZ]M + k20[CH3][H]M + k21[CH3][H2] klO[CZ][HZ] + k14[CH]2 k12[C][CH4] + k15[CH][CH4] + 1.14012)?- kzzmfis]2 k5[C][O]M - k26[CO][O]M k3[H]2M + k4[H] - k61c11H2) - k10[cz][H2] - k111c11H21M - kl3[CH][H2]M - kl7[CHZ][H2]M - k18[CH2][H2] — k21[CH3][H2] - k23[01[H2] 35 -k7[C][OH] + k16[CH][O] + k23[0][H2] _ k24[OH][H]M - k25[OH][H2] = k24[OH][H]M + k25[OH][HZ] k2 6[co][o]M 36 Table 4 . ll. 20. from Regression Reaction HZ+e—>2H+e CO+e—-C+O+e 2H+M-‘H2+M 2H —. H2 (at wall ) c+o+M»co+M 2C+M—~CZ+M C+H2+M—>CH2+M CH3+H+M~CH4+M Lite rature Estimate 37 k. 1 1.0 x 10'9 7.0x10-11 1.2x10-31 3.0 x 103 2.9x 10'33 1.2x10-28 7.1x 10'32 Comparison of Literature Estimates of Rates to Rates k. 1 Re gre s sion Re sult 2.4 x10.9 cm3/sec 7.4 x 1.0x 2.5x 2.8x 10 10 -12 -31 cm3/sec cm6/sec 103/ sec 10 10 10 10 -33 -30 -32 ~34 cm6/sec cm6/sec cm6/sec cm6/sec 1. 10. 11. 12. 13. REFERENCES Brown, L.C. , and A. T. Bell, "Kinetics of the oxidation of carbon monoxide and the decomposition of carbon dioxide in a radio frequency electric discharge, " Ind. Eng. Chem. Fundam. , 131(3), 203-218 (1974). Bell, A. T. , "A model for the dissociation of hydrogen in an electric discharge," Ind. Eng. Chem. Fundam., 11(2), 209-215 (1972). Mearns, A. M. , and A. J. Morris, "Oxidation reactions in a microwave discharge: factors affecting efficiency of oxygen atom production," Chem. Eng. Progr. Symposium Series, 61(112), 37 -46 (1971). Baddour, R. F. , and J. L. Blanchet, "Reactions of carbon vapor with hydrogen and with methane in a high intensity arc, " Ind. Eng. Chem. Proc. Des. Develop. 3 (3), 258 -266 (1964). Blaustein, B. D. , and Y.C. Fu, "Hydrocarbons from HZ' + CO and H2 + CO in microwave discharges. LeChatelier' s principle in discharge reactions, " Adv. Chem. Ser. , 80, 259 --,271 R. F. Gould, ed., Am. Chem. Soc.: Washington D. C. (1969). Lett, R.G., W.A. Steiner, and Y.C. Fu, ”Mass spectrometry study of microwave discharges in H - CO - Ar mixtures," J. Phys. Chem.. 22(21), 2941-2946 (1972). Baddour, R.F., and J. M. Iwasyk, "Reactions between elemental carbon and hydrogen at temperatures above 28000K, " Ind. Eng. Chem. Proc. Des. Develop. 1(3), 169-176 (1962). McTaggart, F. K. , ”Reactions of carbon monoxide in a high- frequency discharge," Aust. J. Chem., 17_, 1182-7 (1964). Vastola, F.J., P.L. Walker, and J.P. Wightman, "The reaction between carbon and the products of hydrogen, oxygen, and water microwave discharges," Carbon, 1, 11-16 (1963). Fu, Y.C. , and B.D. Blaustein, ”Pyrolysis of coals in a micro- wave discharge," Ind. Eng. Chem. Proc. Des. Develop., _8_(2), 257-262 (1969). Fu, Y.C., B.D. Blaustein, and I. Wender, "Gasification of solid fossil fuels in a microwave discharge," Chem. Eng. Prog. Symposium Series, (1(112), 47-54 (1971). Kondratiev, V.N., E.E. Nikitin, and V.L. Talrose, "Problems connected with the investigation of elementary processes in low-temperature plasma," Pure Appl. Chem. 11(3), 367 -387 (1967). Polak, L.S. , “Chemical Processes in low temperature plasmas, " Pure Appl. Chem., 13(3), 345-360 (1967). 14. 15. l6. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. Bell, A. T. , "Models for high frequency electric discharge reactors, " Chem. Eng. Prog. Symposium Series, 61(112), 1-11 (1971). Bell, A. T. , and K. Kwong, "A model for the kinetics of oxygen dissociation in a microwave discharge," Ind. Eng. Chem. Fundam., 12_(1), 90-94 (1973). Bell, A. T. , and K. Kwong, ”Dissociation of oxygen in a radio- frequency discharge," A.I.Ch.E. Journal, _l__8_(5), 990-998 (1972). Bell, A. T. , ”Spatial distribution of electron density and electric field strength in a high frequency discharge, " Ind. Eng. Chem. Fundam., 2(1), 160-164 (1970). Maksimov, A. I. , "Electron density and energy in a microwave helium discharge," Sov. Phys. -Tech. Phys. , _l_l_ (10), 1318-1321(1967). Asmussen, J., R. Mallavarpu, J.R. Hamann, and H.C. Park, "The design of a microwave plasma cavity, " Proc. I.E.E.E. , 62(1), 109-117 (1974). Mertz, S.F. , J. Asmussen, and M.C. Hawley, “An experimental study of reactions of CO and H2 in a continuous flow microwave discharge reactor," IEEE Trans. Plasma Science, PS-Z (4), 297-307 (1974). Husain, D. , and L. J. Kirsch, “Reactions of atomic carbon by kinetic absorption spectroscopy in the vacuum ultra -violet, " Trans. Faraday Soc., 6_7'(7), 2025-2035 (1971). Shapiro, J.S. , and R.E. Weston, Jr., "Kinetic isotope effects in the reaction of methyl radicals with molecular hydrogen, " J. Phys. Chem., 26(12), 1669-1679 (1972). Martinotti, F.F., M.J. Welch, A.P. Wolf, "The reactivity of thermal carbon atoms in the gas phase," Chem. Comm. (J. Chem. Soc. D), (3), 115-116 (1968). Walker, R. W. , "Activation energies of the reversible reaction between hydrogen atoms and methane to give hydrogen and methyl radicals," J. Chem. Soc. (A), 2391-2398 (1968). Basco, N., D.G.L. James, and R.D. Suart, "A quantitative study of alkyl radical reactions by kinetic spectroscopy: Part1," Int. J. Chem. Kine, 2.) 215-234 (1970). Brown, W., J.R. McNesby, A.M. Bass, "Flash photolysis of methane in the vacuum ultraviolet. II: absolute rate constants for reactions of CH with methane, hydrogen, and nitrogen," J. Chem. Phys., 46(6), 2071-2080 (1967). 39 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. Umanski, U.M., and A.D. Stepukhovich, "The recombination of radicals at low pressures," Russ. J. Phys. Chem., 131(8), 1148-1151(1969). Mayer, S. W. , and L. Schieler, ”Computed activation energies and rate constants for forward and reverse transfers of hydrogen atoms," J. Phys. Chem., 22(1), 236-240 (1968). Dean, A.M. , and G. B. Kistiokowsky, "Oxidation of carbon monom'de/ methane mixtures in shock waves," J. Chem. Phys. , 54(4), 1718-1725 (1971). Bell, J.A. , and G.B. Kistiakowsky, "The reactions methylene. VI: the addition of methylene to hydro gen and methane, " J. Am. Chem. Soc., 84(18), 3417 -3425 (1962). Powell-Wiffen, J. W. , and R.P. Wayne, "The reactions of methylene with hydrogens," Photochem. Photobiol. . é. 131-140 (1968). Bahn, G.S. , Reaction Rate Compilation for the H-0 -N System, Gordon and Breach, Science Publishers, 74-99 (1968). Brabbs, T.A. , and F.E. Belles, "Recombination of carbon monoxide and atomic oxygen at high temperatures, " Syrnp. (Int.) Combus., LL, 125-135 (1967). Lee, P.S.T., R.L. Russel, and F.S. Rowland, "The reactions of triplet methylene with alkyl radicals in ordinary photolysis systems," Chem. Comm. (J. Chem. Soc. D). (3), 18-19 (1970). Petersen, E.E. , Chemical Reactions AnalysiAs, Prentice-Hall, Englewood Cliffs, N. J., p. 1977(I965). Kuester, J.L. , and J.H. Mize, Optimization Techniques with Fortran, McGraw-Hill, New York, pp. 240-250 (1973). 40 APPENDIX 41 Fortran Listing of Regression Program and Simplified Reactor Model 42 PROGRAM FITITNFUT,OUTPUT) C«--THIS PPOGRAM DUES A WULTIVARIADLF NON-LINEAR REGRESSION. C‘--THE PROGRAM PARAMETERS ARE AS FOLLOWS . . . NN=NUM8FR OF DATA POINTS KK=NUMDFR OF UNKNONNS T0 3F FOUND B=VECTOD 0F UNKNDNNS BMIN=VFCTOR OF MINIMUi VALUES OF B BMAX=VECTOR 0F MAXIMJW VALUES OF B X=VFCTOP OF INDE’FNDEVT VARIABLE DATA POINTS Y=VECTOR 0F DEPENDENT VARIAJLE DATA POINTS PH=LEAST SQUARES DBJESTIVE FUNCTION Z=COMPUTED VALUES 0? THE INDEPENDENT VARIABLE DV=CONTRDL PAIAMETER--SET TO 1 FOR NUMERICAL DERIVATIVES, SET TO -1 FOR ANALYTICAL DERIVATIVES 0000630 LDOCO C3 ---THE REGPESSION EDJATION IS OF THE FORM Y=F(X,B) 300 C) C---THE MAIN PROGRAM AND THE SUBROUTINES SHOULD BE DIMENSIONED AS C---FOLLONS . . . C DIMENSION P(NN‘K()1A((K9KK+2).ACTKK,KK+2).X(NN)g3(KK),Z(NN), C Y(NNl98V(KK),DMIN(KK).DMAXTKKlaFVTKK)oDVIKK) C C---THE INPUT VARIABLES ARE AS FOLLOWS . . . X(1.I)=INLFT MOLE FRACTION H2 X129T1=INLET MOLE FRACTION CD X13,I)=ROTAMETER VOLUMETRIC FLOW, CM3/SEC AT SPSIG, 300 DEG K XIA.I)=PLASTA VOLUME, 3M3 X(5.I)=FRACTIOV OF GAS PASSING THRU PLASMA X(6oI)=ELECTROV DENSITY X 10-121 ELECTRONS/CM3 X(7oI)=PLASMA 3RESSJRE, ATM X18.I)=PLASMA TEMPEiATJREo DEG K YOBSlloIl=PERCENT CONVERSION C0 TD FHA YOUS(2,I)=PERCZNT SDVVERSIDN CD TO CZHZ DIMENSION P(7OI.A(7.9),AC(7,9),B(7),Z(10),Y(10),BV(7).BWIN(7), L BMAXI7)oFV(7).DV(7)9YOBS(2910)9K(16)yX(8q10) EXTERNAL FUNC COMMON X COMMON/BFUNC/YOSSspRFLAS,KgM REAL K,M ICNT=0 C---RFAD NUMBER OF DATA POINTS (MN), NUMBER OF UNKNOHNS (KKI, AND C---PRINT PARAMETER (PRFLA3)-- t1 GIVES DIAGNOSTIC PRINTOUT, -1 SUPPRESSES READ 8011NN9KK93RFLAG 801 FORMAT121109F10.0) C"'READ INITIAL GUESSES 0F UNKNOHNS (B) READ 805,(d(J)yJ=19K() 805 FORMAT(8E10.A) C~--READ LTMTTS 0N VARIABLES READ 805,(BMIN(J),J=1,K<) READ 8059(8MAX(J’0J=19K() PRINT 808 808 FORMAT(’1’.AX,*42’a9X,‘30“,8X,‘2VF*,9X,4PV*,8X,’FTP¥,QX,*NE*,9X, 1 ‘Pp‘.9X.’PT'99X.’CHA*.8X.‘C2H2*) C---READ INDEPENDENT AND DEPENDENT VARIABLES. EACH CARD CORRESPONDS 00000053000 43 C"-TO A DATA POINT AND GIVES INFORMATION ON PLASMA CONDITIONS AND YIELD DO ()7 TtlgNN 808 FODMAI(IDE8.n) C°-'CONVERT ELECTRON FROM INPJT UNITS TO ACTUAL UNITS OF E/CM3 97 X(59T)=X(6gI)'1-E12 C"'REAU IN KNOWN VALJES O: READ 8069(KTI19I31,15) DO 98 T=19NN PRINT 809,1X1J9II’J31981,YOOSLLoIleYODS(ZyIl 809 FORMATTIX9101610o3olXTT 98 Y(I’=00 PRINT 810 810 FORMATT‘D‘) FNU=0o FLA=00_ TAU=0o EF5=0o PHMINZO. I=0 KDzKK DO 130 J=19KK FV(J):UO . RV(J)=1 100 CONTINUE TCON=KK C"'ICON IS THE NUMBER 3: JNSONVERGEO UNKNONNS TIER =0 200 CALL OSOLVE(KKyOgNN,ZgY,PH,FNU,FLA9TAU9EPS,PHNIN,IgICONyFVg 1 OV,3V,SHINyBMAxgngJNC9052IV9K39A9AC,DAMN) ITER=ITERtl PRINT 8079ITFRvICOngH 307 FORMATT’ AFTFR‘oIhg" ITERATIDiVSg‘gIQg‘ UNKNOHNS ARE NOT CONVERGED. I THE VALUE OF THE DJJESTIVE FUNCTION IS'.E15.8) C"-CHECK TO SEE IF ALL UNKNOJNS ARE CONVERGEO pRFSAVZPRFLAG IFTITER/Z‘Z .NE- ITER) GO TO 202 RRFLAGzlo CALL FUNCTKKvaNN929FV) pRFLAG=PRFSAV 202 CONTINUE IFCICON)1095001200 10 IF (ICON+I) 209509200 20 IFTICONtZI 309739200 30 IF‘ICOI‘HS) (000800200 #0 IFTICONtk) 509909200 50 GO TO 95 60 PRINT 820 320 FORMATT/I‘ NO FJNCTIJN IMPROVEMENT POSSIBLE‘) GO TO 300 70 PPINT 821 821 FORMATT/I,’ THERE ARE MORE UNKNOHNS THAN FUNCTIONS‘) GO TO 300 80 PRINT 822 822 FORMATT//‘ TOTAL VARIABLES ARE ZERO‘) RATE CONSTANTS 44 GO TO ‘00 90 PRINT 823 823 FORMATTl/92Xv‘CDRQECTIOVS SATTRFY CONVERGENCE RFQUIREMENTS, BUT LA 1MUA PACTOR (FLAT STI-L -APGEo*) GO TO 500 05 PRINT 82A 82“ FORMATT/l,’ THIS TS VOT POSSIaLE*) GO TO 300 300 PRINT 925 3?5 FORMATTI/g' THE SOLJTIOVS OF THE EQUATIONS ARE AS FOLLOWS o o 0‘) 00 A00 J=1oKK PRINT 0269J,HTJT 32F) FORMATTIQZXQTBTTQTEQT’ 3 ‘9E18.8) Q00 CONTINUE PRFLAG=1.0 CALL FUNCTKKvOyNN979:V) 1000 CONTINUE END 45 FUNCTION ARCOSTZ) X=Z KFY=0 IFTX.LT.('1.TT K='1. IF(XOGT.1.) x310 IFTX.GF.(‘1.) .AVJ. K.LT.0.0)KEY = 1 IFTX.E0.0.0) GO TO 10 APCOS=ATANTSQRTT1.‘X‘XTIXT IFTKFYoEQol) AR3OS=3olb159269-A9009 GO TO 999 10 APCOS=Io5707363 999 RETUPN FND RUOROUTINE UGOLJF(KK939VNong.PH,FNU.FLApTAU.CPS,PHHIN,I,ICONgFV, 1 OV.UV.1MIN.3MAX,°,FUNC,OC{IV9KD,AyncgfinMN) OIMFNSTON PT70)ollL7o9TyACTT,QT,TTL7),Z(10)9Y(10’,FHI(7T,BHIN(7), 1 UMAXT7),FV(7).0V(7’9YORS(3910T,X(8,10) N=NN K=KK KP1=K*1 KP2=K+2 KBT1=K‘N KBTZ=KBII4K KZI=KBT2*K IFTFNU.LE.U.) cVU3100 IFTFLA.LE.0.) F-A=Co01 IFTFPS.LE. 0.) EPS?0.00002 IFtpHMIN.LE.0.) PHHIV=0o 120 KF=0 130 00 160 Tl=1oK 190 IF(BV(I1).NE.0.|KE=KE*1 IFTKG.GT.0) GO TO 170 162 ICON=-? 163 GO TO 2120 170 IFTN.GL.KE) GO TO 500 180 ICON=-? 190 GO TO 2120 500 11=1 530 IFTI.GT.0) GO T) 1330 550 DO 560 J1=19K J2=KRII+J1 PTJ21=BTJ1) J3=KOI2*J1 560 PTJ3D=ABSTBTJ1)T+.01 GO TO 1030 590 IFTPHMIN.GT.PH.AND.I.GT.1) GO TO 625 00 620 J1=1oK N1=TJ1-1)‘N TF(BV(J1)) 601,520,605 601 CALL OFRIV‘KonvaZvPTNifl)9FV90V9J19JTFST) IFTJTLST.NE.(-1)) GO TO 620 BVTJ1T=lo 605 DO 605 J?=19K J3=KBI1+J2 606 PTJ3T=PTJZT J3=KOI1+J1 JA=KBIZ+J1 C“'INCPEMFNT VARIABLES TO TEST SLOPE OF OBJECTIVE FUNCTION OEN=0.01‘A4AX1(P(J4|oARSTPTJT))1 IFTPTJ3T+OEN .LE. BVAXTJiT) GO TO 95 PTJ3)=P(J3T-OEN OEN=-OFN GO TO 56 55 PTJ3T=PTJ3T+OFN 56 CALL FUNCTKngK9I1*11yNyPTN1+1TaFV) 00 610 J2=19N 47 610 620 JO=J7+N1 P1JBT=(PTJJ)-Z(JZD)/JEN CONTINUE C---SET UP CORRECTION FQUATIOVS 625 630 BAD 650 660 665 670 672 67k 680 692 69k 695 729 729 OO 72: J1=19K N1=TJ1’1)’N ATleKP1T=0o IFTRVTJ1))6309632g63J 00 6A0 J2=19N N2=N1+J2 A(leKP1)=A(J11(P1)§3(N2)*(YTJ2)-Z(J2)T DO 680 J2=11K ATJI,J2)=0. N2=TJ2'1)'N DO 600 J3=19N N3=N1+J3 NQ=N2+J3 ATJ19J2T=ATJ11J2T*PTN3T'PTNA) IFTATJloJlToGToloOE'ZOT GO TO 725 00 69h J2=19K°1 A(J1yJ7)=0o ATJ19J1)=100 CONTINUE GN=OO OO 729 J1=19K GN=GN+ATJ19K°1)”2 C---SCALE CORRECTION :ACTOQS 726 727 730 800 810 820 830 8&0 DO 726 J1=1vK ATJloKP2)=SQQT(A(J19J1)) 00 727 ~11:vi AIJ19KP1)=ATJ19