MODELING OF CATALYTIC CHANNELS AND MONOLITH REACTORS PETER M. STRUK

Transcription

1 MODELING OF CATALYTIC CHANNEL AND MONOLITH REACTOR by PETER M. TRUK ubmitted in partial fulfillment of the requirements For the degree of Doctor of Philosophy Dissertation Adviser: Dr. James. T ien Department of Mechanical & Aerospace Engineering CAE WETERN REERVE UNIVERITY January 7

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3 DEDICATION This dissertation is dedicated first and foremost to my family who patiently waited for me over many long days and nights as I completed this wor. Also, this wor is dedicated to my parents who provided me the opportunity to pursue higher education. Finally, this dedication extends to my many friends and colleagues from NAA, CWRU, and the NCER who continually provided guidance and inspiration for this endeavor. iii

4 TABLE OF CONTENT DEDICATION...iii TABLE OF CONTENT... iv LIT OF TABLE... vi LIT OF FIGURE... vii ACKNOWLEDGEMENT... x NOMENCLATURE... xi ABTRACT... xiv INTRODUCTION... BACKGROUND Channel Geometry...5. Lumped versus Distributed Models...6. Gas-Phase Quasi-teadiness Chemistry Modeling olid-phase Heat Transfer....6 olution Methods....7 ummary... MODEL DECRIPTION Overview...5. Governing Equations OLUTION PROCEDURE Overview patial Integration in x Integration in Time olver DAPK REULT Case : teady-tate Comparisons Isothermal Platinum Tube Experiment Model Parameters Model vs. Experiment Discussion of Case Case : teady-tate Comparisons Monolith Experiment Model Parameters Model vs. Experiment...64 iv

5 5..4 Discussion of Case Case : Transient Propagation ingle Horizontal Platinum Tube Experiment Model Parameters Model vs. Experiment Discussion of Case DICUION... 7 CONCLUION... 8 RECOMMENDATION Physical Model Improvement Experimental Channel Configuration Improvements Further Recommended tudies Potential Improvements to the Performance of the Computer Program... Appendix A. DERIVATION OF GOVERNING EQUATION... Appendix B. TIMECALE & NON-DIMENIONAL EQUATION... Appendix C. THERMOPHYICAL AND TRANPORT PROPERTIE... 4 Appendix D. MONOLITH UPTREAM HEAT TRANFER... 6 Appendix E. PHYICAL CHEMITRY... 8 Appendix F. IR TEMPERATURE ANALYI OF PLATINUM TUBE BIBLIOGRAPHY v

6 LIT OF TABLE Table. Absolute (ATOL) and relative (RTOL) error tolerances used in the computations. The stable and radical species are defined in Appendix E... Table. Dry CO / O sub-mechanism on platinum from the wor of Deutschmann et al. [, 6, 64]... Table. ummary of experimental configurations modeled in this wor. Values denoted with an asteris (*) were not explicitly stated in the reference but were assumed. The ambient pressure surrounding the channel for case (denoted by ** ) was slightly less-than (.97 atm)... Table 4. Calculated herwood numbers for each species using the analogy of heat and mass transfer with Nu=4.64. The property values are based on the inlet mixture (% CO by volume in air with saturated water vapor) at 85K....4 Table 5. Calculated Reynolds numbers evaluated at the temperature extremes for the catalytic channel experiments of Khitrin and olovyeva...48 Table 6. Gas-phase timescales and Peclet numbers for various processes of the catalytic channel using the property values shown at the top...6 Table 7. olid-phase timescales and Peclet numbers for various processes of the catalytic channel using the property values shown at the top...7 Table 8. Volumetric ratios of the gas to solid heat capacities for all cases presented in the text...8 Table 9. Parameter used to gauge the importance of solid axial heat conduction to lateral heat transfer from the gas to the solid... Table. Thermophysical property values used in the calculations...4 Table. Homogeneous CO gas-phase mechanism [84]. The reactions used in the dry CO mechanism are shown in red. All reactions are used in the wet CO mechanism....4 Table. CH 4 / O mechanism on platinum []. The reactions shown in red are used in the dry CO calculation. All reactions are used in the wet CO mechanism....4 vi

7 LIT OF FIGURE Figure. Flowchart of basic solution algorithm...4 Figure. Discretization of catalytic channel into finite volumes. olid phase nodes (shown in blac) represent the entire cross-sectional volume. The inlet is at x=...6 Figure. Comparison of steady-state model results (including a plug-flow model) to the experiment of Khitrin and olvyeva[54] for channel velocities. The inlet gas consisted of % CO (by volume) with the balance being air...6 Figure 4. Computed steady-state profiles of CO mass fraction and select surface species along the length of the platinum channel for the 4 m/s case (Nu = 4.6) at 4 temperatures. The initial condition for the surface is Z O(s) =....4 Figure 5. Computed results showing the effect of varying the mass-transfer effectiveness. The herwood number is related to the Nusselt number via the heat and mass-transfer analogy. The surface initial condition for these calculations is Z O(s) =...4 Figure 6. Wall mass fraction, Y W (for CO and O ), and surface site fractions, Z (for CO(s) and O(s)), at the channel outlet versus Nu for 56K. Also shown is the product Z CO(s) Z O(s), which is proportional to the fuel conversion rate...45 Figure 7. Wall mass fraction, Y W (for CO and O ), and surface site fractions, Z (for CO(s) and O(s)), at the channel outlet versus Nu for 6K. Also shown is the product Z CO(s) Z O(s), which is proportional to the fuel conversion rate...45 Figure 8. Wall equivalence ratio, φ W, at the channel outlet versus Nu for four temperatures: 56K, 58K, 6K, and 6K. The dashed line corresponds to the approximate φ W just prior to lightoff (which would occur with any further temperature increase or Nu decrease)...48 Figure 9. Comparison of different values of Γ (surface site density) on the steady-state model results. The value of Γ =.7 x -9 mol/cm was estimated from the density of platinum[9, 6]. The initial surface condition was Z O(s) =....5 Figure. Comparison of different values of Γ (surface site density) on the steady-state model results. The value of Γ =.7 x -9 mol/cm was estimated from the density of platinum[9, 6]. The initial surface condition was Z CO(s) =....5 Figure. Global sensitivity study of the dry CO mechanism conducted by changing the preexponential constant or sticing coefficient by 5% and 75% relative to the published value. The data corresponds to 4 m/s and the blac lines are the unmodified inetics. In these calculations, the surface is initially covered by CO(s) Figure. Global sensitivity study of the dry CO mechanism conducted by changing the preexponential constant or sticing coefficient by 5% and 75% relative to the published value. The data corresponds to 4 m/s and the blac lines are the unmodified inetics. In these calculations, the surface is initially covered by O(s) Figure. Computed results showing the effect of saturated water vapor (using a wet CO mechanism) in the inlet feed on the steady-state conversion...57 Figure 4. chematic representation of monolith configuration tested in this section. The model represents the center channel of the monolith and is used to characterize the entire monolith performance. The left image is from Kee et al.[]...6 Figure 5. Measured CO concentration at the outlet of a commercial monolith from the wor of Ullah et al.[9]. The inlet velocity for an individual channel was 6. m/s at a temperature of 6K. The right axis shows the conversion calculated from the concentration measurements. 6 vii

8 Figure 6. Computed results showing the effect of varying a* on the conversion of CO as a function of reactor length. The experimental data is from Ullah et al.[9]. The top graph shows the conversion while the bottom graph shows the corresponding solid temperatures...66 Figure 7. Temperature and select species mass and site fraction as a function of monolith length up to 4cm (the calculation used cm). The model parameters are a*= and Nu= Figure 8. Temperature and select species mass and site fraction as a function of monolith length up to 4cm (the calculation used cm). The model parameters are a*= and Nu= Figure 9. Effect of Nu / h on the steady-state conversion of CO as a function of reactor length. The herwood number comes from the analogy of heat and mass transfer. The computations use a*= and correspond to the mass-transfer limited or high-conversion solution...7 Figure. Ignition and propagation sequence of a CO / O (φ = ) catalytic reaction along the inside of a platinum tube (.74 mm ID,.95 mm OD). The inlet gas velocity is m/s...74 Figure. Leading edge of catalytic reaction versus time in a platinum tube for φ =. to with a dry CO / O mixture flowing at m/s. A hot-wire heats the outlet end of the tube for seconds (denoted by the red vertical line). There are up to tests plotted for each φ. The tube outlet is at.5 cm...76 Figure. Leading edge of catalytic reaction versus time in a platinum tube for φ =. to with a dry CO / O mixture flowing at m/s. A hot-wire heats the outlet end of the tube for seconds (denoted by the red vertical line). There are up to tests plotted for each φ. The tube outlet is at.5 cm...77 Figure. Leading edge of catalytic reaction versus time in a platinum tube for φ =. to with a wet CO / O mixture flowing at m/s. A hot-wire heats the outlet end of the tube for seconds (denoted by the red vertical line). There are up to tests plotted for each φ. The tube outlet is at.5 cm...78 Figure 4. Leading edge of catalytic reaction versus time in a platinum tube for φ =. to with a wet CO / O mixture flowing at m/s. A hot-wire heats the outlet end of the tube for seconds (denoted by the red vertical line). There are up to tests plotted for each φ. The tube outlet is at.5 cm...79 Figure 5. Measurements and model predictions of the propagation velocity for the catalytic reaction front versus φ for both dry and wet CO tests....8 Figure 6. Comparison of model and experiment looing at the propagation of a catalytic reaction front along a Pt tube for φ ranging from. to with a dry CO / O mixture flowing at m/s (Z CO(s) =). The model predictions show both the position of the 65K isotherm and location of maximum catalytic heat release versus time. The position.5 cm corresponds to the tube outlet...86 Figure 7. Effect of varying the input power on the transient evolution of the catalytic flame. The test condition was φ = using dry CO with Z CO(s) = initially...87 Figure 8. Model results with varying Nu (and h via the heat and mass-transfer analogy) that compare the propagation of a catalytic reaction in a Pt tube for φ ranging from. to with a dry CO / O mixture flowing at m/s (Z CO(s) =). The position.5 cm corresponds to the tube outlet...89 Figure 9. Comparison of model and experiment looing at the propagation of a catalytic reaction front inside a Pt tube for φ=. to with a wet CO / O mixture flowing at m/s (Z CO(s) =). The model predictions show the position of the 65K isotherm versus time. The position.5 cm corresponds to the tube outlet...9 Figure. Computations showing the effect on catalytic propagation of varying the external heat transfer coefficients. Two values are compared: h =W/m K and times the natural convection correlation of Churchill and Chu[74]...9 viii

9 Figure. External heat transfer coefficient, h, along the length of the platinum channel for 4, 6, and 8 seconds after the beginning of ignition. The test conditions correspond to φ = with dry CO. The values for h come from the correlation of Churchill and Chu[74]...94 Figure. Computed versus measured temperature profiles along the platinum channel at 4, 6, and 8 seconds after the beginning of ignition...95 Figure. Computed temperature profiles along the platinum channel comparing the effect of external heat transfer coefficient Figure 4. Control volumes of solid and gas phase... Figure 5. Overall mass conservation control volume for gas-phase...4 Figure 6. Gas-phase species conservation control volume....4 Figure 7. Gas-phase energy conservation control volume....6 Figure 8. olid phase control volume (including catalytic surface shown in red)...9 Figure 9. chematic of stagnation point region used to estimate heat transfer from front face of monolith...6 Figure 4. False colored image of the platinum tube during catalytic flame propagation taen with the IR camera. The platinum tube is shown schematically overlaid on the image. There are approximately pixels of resolution across the diameter of the tube...44 Figure 4. Transmission characteristics of the filter used in IR imaging of platinum tube...45 Figure 4. Relationship between Blacbody temperature to the camera counts and the parameter κ from equation ix

10 ACKNOWLEDGEMENT I wish to acnowledge my academic advisor, Professor James T ien, whose dedication to his students has left a long legacy of which I am proud to be a member. Also, my deepest gratitude goes to Dr. Daniel Dietrich of the NAA Glenn Research Center and Dr. Fletcher Miller of the National Center for pace Exploration Research, both of whom provided valuable guidance and support during the many years leading up to the completion of this wor. I also wish to than my friend and former colleague, Mr. Benjamin Mellish, for his hard wor and dedication in the laboratory. Finally, I wish to express my appreciation to my many friends and colleagues who stepped in to tae up my other responsibilities as I completed this wor. x

11 NOMENCLATURE a* - ratio of catalytic to geometric surface area A - channel open flow area (m ) A i - pre-exponential factor for reaction i (cm, mol, K, s) A - solid cross-sectional area (m ) A T - total cross-sectional area = A+A (m ) ATOL j - absolute error tolerance on solution component, f j C P - = Kg C = P Y, ; mean gas specific heat at constant pressure (J / g /K) C P,I - mean inlet gas specific heat at constant pressure (J / g /K) C P, - species gas specific heat at constant pressure (J / g /K) C P,ref - reference specific heat used in non-dimensional analysis (J / g /K) C - solid specific heat (J / g /K) d - channel inner (hydraulic) diameter (m) do - channel outer diameter (m) D m - species diffusion coefficient into mixture (m /s) E i - activation energy of the i th reaction (cal / mole or Joules / mole) f j - solution component j L - length of catalytic channel (m) h - mixture enthalpy (J/g) h - enthalpy of species (J/g) h D - convection mass transfer coefficient for specie in bul mixture (m/s) h T - internal convection heat transfer coefficient (W / m K) h - external convection heat transfer coefficient (W / m K) h U - convection heat transfer coefficient on upstream face of solid (W /m K) h - enthalpy of formation of species at 98K (J/g) f i G - gas-phase node index i - solid-phase node index - thermal conductivity of gas (J/m/s/K) or species index K - total number of species (K + K g ) ext - thermal conductivity of the exterior air at the film temperature (J/m/s/K) fi - forward rate constant for the ith reaction (units m, mole, s) K g - number of gas-phase species ri - reverse rate constant for the ith reaction (units m, mole, s) - thermal conductivity of gas (J/m/s/K) K - number of surface species Le - Lewis number for species (α / D m ) m - sum of all surface reaction stoichiometric coefficients for a given reaction m& - mass flow rate down the channel (g/s) n g - number of gas-phase reactions n - number of surface reactions Nu O - Exterior Nusselt number Nu U - Upstream face Nusselt number for monolith xi

12 Nu - Interior Nusselt number downstream of entry length Q i - heat evolved from the ith reaction (J / mole) " Q - heat flux into solid on downstream side (W/m ) D " U Q - heat flux into solid on upstream side (W/m ) q i - rate of progress of the i th gas reaction (mole / m /s) or rate of progress of the i th surface reaction (mole / m /s) R - gas constant (J / g / K) Re D - inlet Reynold s number based on channel hydraulic diameter RTOL j - relative error tolerance on solution component, f j R U - universal gas-constant (J / mol / K) s& - net production rate of specie due to all surface reactions (mole / m /s) - circumferential length of channel cross section = πd (m) t - heat-transfer tanton number t m - mass-transfer tanton number t - time (s) t - time integration between successive gas-phase integrations (s) T - bul gas temperature (K) T I - inlet gas temperature (K) T - temperature of species entering or leaving control volume (if s& > then T =T otherwise T =T) T ref - reference temperature used in non-dimensional analysis (K) T - solid temperature (K) T U - upstream gas temperature (K) T - reference temperature (98 K) T - temperature of ambient gas surrounding channel (K) u - average (bul) velocity of flow (m/s) U I - inlet velocity at x= (m/s) U U - average velocity upstream of monolith (m/s) W - molecular weight of specie (g/mole) x - axial coordinate from channel inlet (m) x - axial coordinate control volume length (m) [X ] - molar concentration of specie (mole / m ) Y - mass fraction of specie in the bul mixture Y W - mass fraction of specie at the wall Z - site fraction of surface specie Gree Letters α - thermal diffusivity (m /s) βi - temperature exponent in the rate constant of the i th reaction γ i - sticing coefficient of i th reaction Γ - surface density of site on solid (.76 x -5 mol / m for platinum) µ - dynamic viscosity (N s/m ) ν i - net stoichiometric coefficient of the th specie in the ith reaction; ν i - stoichiometric coefficient of the th reactant specie in the ith reaction xii

13 ν i - stoichiometric coefficient of the th product specie in the ith reaction ρ - mass density of gas (g/m ) ρ ref - reference density used in non-dimensional analysis (g/m ) ρ - mass density of solid (g/m ) σ - site occupancy number - number of sites that species occupies on surface τ - characteristic time scale (s) τ D - characteristic radial mass transport time (s) τ - characteristic solid heat-up time (s) τ L - characteristic residence time for channel (s) τ Ldx - characteristic residence time for control volume length dx (s) τ RXi - characteristic reaction time for reaction i (s) φ - Equivalence ratio φ W - Equivalence ratio using wall mass fractions ω& - gas-phase production rate of specie due to all gas reactions (mole/m /s) xiii

14 Modeling of Catalytic Channels and Monolith Reactors Abstract by PETER M TRUK This dissertation presents a lumped two-phase (solid and gas) model of transient catalytic combustion suitable for isolated channels or monolith reactors. The gas-phase is quasi-steady relative to the transient solid. Axial diffusion is neglected; lateral diffusion, however, is accounted for using transfer coefficients. The solid phase includes axial heat conduction and external heat loss due to convection and radiation. The combustion process uses detailed gas-phase and surface reactions. The gas-phase model becomes a system of stiff ordinary differential equations while the solid-phase reduces, after discretization, into a system of stiff ordinary differential-algebraic equations. The time evolution of the system came from alternating integrations of the quasi-steady gas and transient solid. The model simulates three experiments using CO fuel: () steady-state conversion through an isothermal platinum tube, () steady-state conversion through a commercial monolith, and () transient propagation of a catalytic reaction inside a small platinum tube including temperature measurements. This wor presents detailed parametric / sensitivity studies on important parameters in each case including internal transfer coefficients, catalytic surface site density, and external heat-loss (for the third case). For xiv

15 the second case, the model uses a parameter a* (ratio of apparent to geometric surface area) appropriate for enhanced surface-area washcoats in monoliths. The model requires internal mass-transfer resistance to match experiment at lower residence times. Under mass-transport limited conditions, the model reasonably predicted exit conversion using global mass-transfer coefficients. Near light-off, the model results did not match experiment precisely even after adjustment of mass-transfer coefficients. Agreement improved for the first case after adjusting the surface inetics such that the net rate of CO adsorption increased compared to O. The CO / O surface mechanism came from a sub-set of reactions in a popular CH 4 / O mechanism. For the third case, predictions improved for lean conditions with increased external heat loss or adjustment of the inetics as in the first case. Finally, the results show that different initial surface-species distribution (first case) as well as heating profile (second case) leads to multiple steady-states under certain conditions. These results demonstrate the utility of a lumped two-phase model of a transient catalytic combustor with detailed chemistry. xv

16 INTRODUCTION Catalytic combustion has been extensively studied over the past several decades largely because it offers lower ignition temperatures, can reduce pollutants, and operates with a wide range of fuels and fuel-air ratios. The many potential applications for catalytic combustion are typically divided into primary and secondary processes. Examples using a catalyst for primary combustion, whose main objective is to generate heat for the process, include various heaters and gas turbines with ultra-low NOx emissions and very high combustion efficiencies. Primary catalytic combustion processes can further be subdivided into a category involving catalytically stabilized combustion where the catalytic (heterogeneous) reaction stabilizes or supports gas-phase (homogeneous) reactions. econdary combustion processes, which are almost always purely heterogeneous, typically involve removing harmful emissions from many sources (e.g. the automobile catalytic converter or organic emissions from industrial processes). Hayes and Kolaczowsi [] recently published a textboo which offers an excellent overview of the many potential applications as well as a description of the fundamental aspects of catalytic combustion. Additional review articles on the topic of catalytic combustion are listed in the references [, ]. Recently, catalytic systems have been proposed for micro-combustion devices [4-8] and space micro-propulsion [9]. In some cases, the processes are endothermic as in reforming of fuels for yngas generation [- 4], fuel cell applications [5, 6], and in-situ resource utilization for future space missions [7]. While many practical applications exist using catalytic combustion, there still are many aspects where fundamentals are not completely understood. Among active areas of

17 research in catalytic systems are the physical surface chemistry [8], development of surface inetic mechanisms suitable for combustion studies [9-], physical model development [] and numerical simulations [, ]. Notable among the latter include understanding the transient behavior of catalytic systems, particularly light-off which is of high importance for many catalytic systems that routinely start and stop or see transient inlet feed conditions. Moreover, surface reactions can stabilize homogeneous combustion on scales smaller than the quenching distance the latter receiving recent attention in the literature [4-8]. Predicting transient phenomena in combustion, especially ignition and extinction, require accurate nowledge of the chemical inetics (i.e. detailed chemistry) of the reactions taing place. Detailed homogeneous chemical mechanisms are prevalent in the literature. Heterogeneous chemical mechanisms, however, have only recently become available. While the quantitative aspects of many of these mechanisms are largely untested or only applicable to very controlled circumstances, the inclusion of such mechanisms into existing models offers the potential for better understanding of complex phenomena such as catalytic light-off. A rigorous physical model of a transient catalytic combustion system can be computationally expensive even without the inclusion of detailed or full chemistry. Thus, models of catalytic monolith reactors simulate a single (or just a few) channel(s) to characterize the behavior of the entire reactor. Despite this reduced geometry, detailed flow treatment can require D codes because the channel cross-sections can be noncircular. While some studies in catalytic combustion have begun to couple both multidimensional flow fields with detailed homogeneous and heterogeneous chemistry (e.g.

18 [9]), few studies have looed at simplified flow fields with the inclusion of detailed chemistry. A significant and notable exception to this is the plug-flow model (PLUG) developed by andia National Laboratories []. One primary assumption in the PLUG model is that diffusive transport perpendicular to the main flow direction is infinitely fast - see pg. 65 of Kee et al. for a comprehensive description of plug-flow reactors []. The model also assumes diffusion is small compared to convection in the primary flow direction. While there are conditions where the PLUG model assumptions are valid [], many catalytic combustion systems are mass-transport limited perpendicular to the main flow direction and thus require the inclusion of finite rate species transport. Incorporation of heat and mass-transfer coefficients can potentially improve the accuracy of PLUG models while still taing advantage of the general D flow-field. There exists an extensive body of literature using simplified chemistry in which catalytic combustion systems have been modeled using simplified flow-fields, i.e. the socalled lumped models, which capture finite rate diffusion perpendicular to the main flow direction through the use of heat and mass-transfer coefficients [, -8]. These models, which are less expensive computationally, have successfully predicted a wide variety of catalytic systems but hinge to various degrees on nowledge of accurate heat and mass-transport coefficients. There have been several studies which explore the use of heat-and-mass transport coefficients in catalytic systems [, 7, 9-4]. A logical next step would be to include detailed chemistry in a lumped catalytic model. Finally, many studies have shown that heat transfer along the solid structure of the reactor is important and in some cases dominant [8, 4, 4]. The heat transfer can be in The chemical engineering literature often refers to lumped and distributed models. Lumped models typically average some spatial distribution (i.e. radial species profiles) whereas distributed models do not.

19 the form of heat conduction along the length of the solid (parallel to the primary flow direction) as well as radiant heat transfer. Recently, a catalytic reaction front has been observed to propagate along a platinum channel due to solid-phase axial conduction [44, 45]. In such a case, the model requires inclusion of heat transfer along the solid. The goal of this wor is to develop a two-phase (gas & solid) transient catalytic combustor model using a simplified flow field inside a single channel with detailed chemistry. The flow field model includes axial convective transport with transverse energy and mass exchange via heat and mass transfer correlations. The solid is a thermally thin shell along which finite-rate heat conduction occurs in the axial direction. uch a model, with the inclusion of detail chemistry, is largely absent from the literature. The advantage of this approach would be improved model predictability (compared to simpler models) with potential for reasonable computational times (compared to fully distributed models) allowing parametric study. This wor explores the advantages and limitations of using such a model by comparing predictions to three separate experimental catalytic configurations using CO fuel with O on Platinum: () steady-state catalytic oxidation in an isothermal channel, () steady-state catalytic oxidation in a monolith reactor, and () transient catalytic oxidation propagating along a channel. The first two cases use data taen from the literature. The rd case uses recent experimental data using a pure platinum channel provided by NAA [44, 45]. In all these experiments, the important reactions are heterogeneous but the model includes homogenous reactions in anticipation of future studies. This modeling wor supports a NAA funded effort which is studying the fundamentals of catalytic combustion in small scale channels. 4

20 BACKGROUND This section reviews specific topics on modeling of catalytic combustion reactors in channels and monolith reactors as applicable to the present wor. It is not intended to be a comprehensive review of this topic; however, examples and literature references are cited often. For more comprehensive reviews, the reader is directed to articles and textboos in the literature dealing with modeling of catalytic channels and monolith reactors [,,, ].. Channel Geometry Typical geometries in which surface reactions occur is that of a paced-bed or monolith reactor. The catalytic paced-bed reactor is more prevalent in applications where pressure loss is less of a concern (e.g. chemical processing plants) while the monolith reactor is more suited to applications where pressure loss must be avoided (e.g. faster velocity applications such as automobile catalytic converters and pre-burners for gas-turbines). A monolith reactor minimizes the pressure loss by arranging multiple catalytic channels in parallel. The walls of the channels are usually very thin to maximize the surface area for catalytic reactions and to minimize pressures loss and weight of the system. Typical cross-sectional geometries of the open areas in monolith reactors vary from circular, square (or rectangular), to even triangular shapes. This wor models a single catalytic channel. Models of catalytic monoliths often use a single channel to characterize the behavior of the entire monolith since every channel within a monolith structure should behave alie. There are, of course, exceptions. Researchers have explored heat loss effects near the periphery [, 6] and 5

21 non-uniform feed effects [46]. In general, however, it is simpler and very reasonable to study monolith behavior using a single channel model. The present wor loos not only at monolith reactors but isolated channels the latter is amenable to temperature profile measurements which are not easily obtainable for monoliths. The primary difference between a channel in a monolith reactor and an isolated channel is inclusion of external heat loss in the latter. The model includes external heat transfer to the surroundings via radiation and convection and is discussed in section.5.. Lumped versus Distributed Models Physical models of catalytic channels and monolith reactors have been classified into two categories in the chemical engineering literature: lumped and distributed []. Lumped models average spatial variations within the reactor s flow-field to a single representative value while distributed models account for spatial variation within the flow field. Groppi et al. [] compares both lumped and distributed models of monolith catalytic combustors. Examples of the lumped category include radial-averaged models [4-6] and even axially averaged models [7]. The popular plug-flow model, PLUG [], is an example of a lumped model. There are many distributed models in the literature and many of these include calculated bul heat and mass transfer coefficients suitable for use with lumped models [, 4, 47-5]. Raja et al. [] evaluated the various flow modeling assumptions including Navier-toes, boundary-layer, and plugflow assumptions. The authors point out that mass-transfer coefficients can be used to improve the accuracy of the (one dimensional) plug-flow model by providing a mass- 6

22 transfer resistance between the bul gas and reacting surface this is the approach taen in this wor. Heat and mass transport coefficients are typically expressed in dimensionless form in terms of the Nusselt (equation ) and herwood (equation ) numbers, respectively. Many engineering applications use a constant value of Nu and h to estimate the heat and mass transfer between a bul flow and a surface (usually assuming fully developed conditions). For example, in the fully developed region of a circular tube, the classical Graetz-Nusselt analysis shows that Nu =.655 for a constant temperature wall boundary condition. For a constant heat flux wall boundary condition, the same analysis yields Nu = Nu ht d = () h d D h = () Dm For catalytic monoliths and channels, the wall boundary conditions are neither a constant heat flux nor a constant temperature condition. Rather, the wall boundary condition varies along the length of the channel (as well as in time for transient calculations). This is a major limitation of a lumped parameter model using heat and mass transfer correlations. Nonetheless, Nu and h numbers correlations have been routinely used in the past and have provided reasonable agreement between model and experiment. Hayes and Kolaczowsi [] as well as others [,, 7-4, 5] discuss the selection of heat and mass-transfer coefficients for catalytic monoliths. 7

23 The majority of computations in this wor use a constant Nu corresponding to the constant flux boundary condition. This is a reasonable approximation up to the catalytic light-off point (discussed below) but might slightly over predict heat transfer further downstream []. The value of the herwood number comes from the heat and mass transfer analogy shown in equation [5]. In this way, there is some distinction between the relative effectiveness of diffusion based on the species Lewis number. A major focus of this wor was to explore the sensitivity of the model results to variations in Nu and h. Nu h = Le (). Gas-Phase Quasi-teadiness While steady-state models date bac to the wor of Khitrin and olovyeva [54], transient models began appearing almost years later. Ferguson and Finlayson [55], Young and Finlayson [47, 48], and T ien [56] were among the first to model the transient behavior of a catalytic combustor. A major assumption in these wors was a quasisteady gas-phase relative to a transient solid. A scaling analysis in Appendix B shows that the solid-phase timescales are significantly longer than any gas-phase process including the channel residence time. Using this observation, gas-phase transients can be neglected (i.e. quasi-steady assumption) while still accurately resolving the solid transients. Furthermore, the ratio ρc P A / ρ C A is typically very small so that neglecting the accumulation term in the overall energy balance on system causes minimal error [55]..4 Chemistry Modeling Up until the early 99 s, the combustion community was primarily using simple global reactions for both gas-phase and surface chemistry [57]. Where reaction mechanisms are not well understood, or if the heterogeneous reaction is completely mass- 8

24 transfer limited, global chemistry can be a reasonable assumption. Multi-step mechanisms (not detailed) can further improve agreement with experiment [56, 58, 59]. The years 98 through 99 saw the development of the CHEMKIN [6, 6] and urface CHEMKIN [6] software which provided a basic formalism for detailed homogeneous and heterogeneous chemistry to the combustion community. Detailed surface chemistry models were subsequently presented by Warnatz et al. [9], Deutschmann et al. [, 6, 64], Bond et al. [], Chou et al. [65], and Mhadeshwar and Vlachos [66]. While significant uncertainty still exists with surface mechanisms, detailed chemistry offers tremendous insight into the process of catalytic combustion, the most important being simply an accurate description of the physiochemical process taing place. Heterogeneous reactions occur on a solid surface and exchange molecules with an adjacent fluid. In this wor, the adjacent fluid is a gas. In general, surface reactions involve the adsorption of gas-phase species onto surface sites. The adsorbed species may then react to form a product, usually at a temperature lower than required in a pure homogeneous reaction. The product, which is still adsorbed to the surface, subsequently desorbs bac into the gas leaving a vacant surface site. Furthermore, unreacted surface species may also desorb bac into the gas-phase. Kee et al. [] provide an excellent description of this process from a combustion perspective. The textboo uses the framewor set forth by urface CHEMKIN [6] and is the convention adopted in this wor. The specific chemical mechanisms used in this study are presented and discussed in Appendix E. 9

25 The number and types of active sites on a catalyst may vary significantly by the inherent surface structure and manufacturing processes as well as through time after use (e.g. catalyst deactivation or poisoning). There are many types of surface sites which include effects of surface irregularities such as dislocations, edges of crystals, cracs along the grain boundary, etc. (e.g. [67]) all of which can affect the physics of the chemisorption process. As an example, Feibelman et al. [68] discusses the complexities that surface chemists face in describing the adsorption of a simple CO molecule on platinum. Most state-of-the-art surface mechanisms used in combustion simulations today, however, do not distinguish between these different types of sites and represent the number of active catalytic sites by a single number Γ which is constant. For example, Warnatz et al. [9] and Deutschmann et al. [6], estimated a value of Γ =.766 x -9 mol / cm (.6 x 5 sites / cm ) from the atomic surface density of polycrystalline platinum and this value has been used subsequently by numerous studies [,,, 64, 69]. Hwang et al. [7], however, claimed that this value has not been verified for accuracy and goes on to use a value of Γ =.757 x - mol / cm in their modeling wor claiming that this value of Γ produced better agreement when comparing their model to experiment. The authors, however, used a different surface mechanism [65] than the previous studies which all used the mechanism by Deutschmann et al. [64]. The present wor uses the CO sub-mechanism from Deutschmann s CH 4 and O mechanism on Pt [64]) and is presented in Appendix E. The model calculations investigate the sensitivity of the solution to individual steps within the mechanism as well as the parameter Γ. In addition to manipulating Γ directly to account for variations in catalytic surface area, the model investigates a parameter a* (discussed further in the Model

26 Description) which is defined as the ratio of effective catalytic surface area to the geometric area and is applicable to situations with a high-surface area washcoat. One final concept related to surface chemistry is that of catalytic light-off. Lightoff is usually the onset of significant reactant conversion and is often (but not always) associated with a rapid rises in temperature []. Furthermore, light-off is frequently characterized by the (typically abrupt) transition from inetic to mass-transfer controlled surface reactions [4]. Exceptions to latter are cases where a plug flow model is applicable and the reactions remain inetically controlled. Hayes and Kolaczowsi [] provide some rough guidelines to help identify inetic and mass-transfer limited reactions which are easily applied to catalytic models. The guidelines suggest that if the surface concentration of the limiting reactant is greater than 95% of the bul concentration then the reaction is inetically controlled. imilarly, if the surface concentration of the limiting reactant is less than 5% of the bul then the reaction is mass-transfer controlled. In between, both mass-transport and inetics are important. The present wor explores regimes which are inetically or mass-transfer limited as well as cases which liely are affected by both inetics and mass-transfer..5 olid-phase Heat Transfer olid-phase models of monoliths and catalytic channels can include (or exclude) various modes of heat transfer. While the solid walls of the channel are typically thermally-thin (i.e. heat conduction is very fast in the radial direction compared to the axial direction), finite-rate axial heat conduction along the walls of the channel is often important [4, 8, 4, 4]. Recently, a catalytic flame has been observed to propagate along a platinum channel due to solid-phase axial conduction [44, 45]. Ramanathan et al.

27 [8] defined an axial heat conduction Peclet number, Pe h, and showed its effect on the light-off behavior of a monolith including both transient and steady-state results. imilar to a traditional gas-phase Peclet number, the importance of axial heat conduction in the solid increases as Pe h increases. Appendix B presents a scaling analysis which defines a similar parameter. Internal heat exchange by radiation (solid to solid) can also play an important role within the catalytic combustor [4, 7-7]. The general consensus from the literature is that both solid-phase heat conduction and radiation exchange tend to affect the solid temperature by lowering pea values and broadening profiles. This in turn can affect the performance of the monolith. Hayes et al. [4] studied a typical ceramic monolith and showed that solid-phase axial conduction had a greater affect on the solution compared to radiation. With metal channels and monoliths, whose thermal conductivities increase and emissivities decrease, axial conduction is expected to be even greater importance relative to radiation. The heat-transfer along the channel exterior can vary from adiabatic to specified heat fluxes. An interior channel of a monolith reactor will typically have adiabatic boundary conditions for the tube exterior whereas a single tube reactor requires an appropriate heat transfer boundary condition. For the isolated circular channels, Churchill and Chu [74] provide correlations for external heat loss from natural convection. Heat loss from thermal radiation also occurs..6 olution Methods Typically, transient models (both lumped and distributed) require the solution of a system of partial differential equations (PDE) which can be computationally expensive.

28 The addition of detailed chemistry can be computationally prohibitive especially in cases where the model includes multi-dimensional fluid flow, detailed chemistry, and heat transfer to another phase. Essentially all methods employed today transform the PDEs to systems of ordinary differential equations (ODEs) or differential-algebraic equations (DAEs) using techniques such as the method of lines [75]. To solve such systems of equations, early models have employed methods such as orthogonal collocation [55] and Runge-Kutta methods [56]. With the inclusion of detailed chemistry, the resulting systems of ODEs / DAEs are stiff and require special algorithms. Many modern programs for solving catalytic systems employ codes such as LODE [76], LIMEX [77], DAL [78] and DAPK [79]. Despite these recent advances in computational algorithms and increased computer speeds, computational times for transient simulations with detailed chemistry are still often quoted in durations of days or longer depending on the dimensionality of the problem. A substantial reduction of numerical effort can be obtained by means of a lumped one-dimensional spatial model as well as the assumption of gas-phase quasi-steadiness..7 ummary The goal of this wor is to develop a transient model which is applicable to both single catalytic channels as well as monolith reactors. Based on the bacground presented, there are basic elements required to adequately model the relevant physics of a general catalytic channel. These include () mass-transfer effects from the gas to the surface, () a solid phase which includes axial conduction, and () external heat loss for an isolated channel. Furthermore, the current understanding of surface chemistry allows the inclusion of detail heterogeneous mechanisms which is a significant improvement

29 compared to global chemistry. The model can include the following simplifying assumptions and still adequately capture the relevant physics: () lumped onedimensional model with transverse heat and mass transfer coefficients and () gas-phase quasi-steadiness relative to a transient solid. uch a model with detailed chemistry is notably lacing in the current literature. The advantage of such an approach would be improved predictability with reasonable computational times allowing parametric study. An additional advantage is that such a model would be generally applicable to a variety of cross-sectional configurations (where appropriate transfer coefficients exist). Appendix B provides further assumptions based on a detailed time-scale analysis of the physical processes in a catalytic channel. 4

30 MODEL DECRIPTION. Overview The model consists of a channel geometry and considers both a gas and solid phase. For the gas, a mixture of fuel, oxidizer, and optional inert enters the channel with a prescribed velocity and pressure. There is negligible pressure drop along the channel and the gas obeys the ideal gas law. The gas-phase is quasi-steady relative to the transient solid. Bul temperatures and species describe the gas and solid along the channel with lateral gradients captured using the surface temperature and by defining a gas-phase mass-fraction adjacent to the wall (i.e. a two-layer gas-phase model). The lateral diffusion rate is quantified via heat and mass transfer coefficients. Heat and mass diffusion in the axial direction is neglected because the Peclet number based on typical gas velocities is much greater than unity (Appendix B). Transport properties are calculated using the bul temperature and mass fraction at each axial location. The model uses detailed inetic mechanisms for both the gas and surface reactions. The transient, thermally-thin 4 solid includes heat transfer to and from the gas inside the tube, external heat transfer to the surroundings via convection and radiation, heat generation terms due to catalytic reactions and resistive heating (to simulate applied heat), and axial heat conduction. The model accounts for varying coverage (due to catalytic reactions) of adsorbed surface species along the inner surface of the channel adjacent to the gas-phase the amount of adsorbed species is assumed to be sufficiently small so as not to affect bul material properties. There are K total surface species including vacant surface sites. 4 Thermally-thin implies infinitely fast heat conduction in the radial direction because of the small physical distance and large thermal conductivity. 5

31 . Governing Equations This section presents the governing equations for the transient catalytic channel model. Appendix A provides the detailed derivations of each equation which come from mass, species, and energy balances from differential control volumes along the channel. In the present formulation, pressure drop along the reactor is neglected. The control volumes include both gas and solid phases detailed diagrams for each control volume are in Appendix A. Equation 4 shows the steady-state, overall mass conservation equation. This equation is valid for purely catalytic systems in which surface reactions do not supply or remove net mass from the flow. Hence, the mass flow rate is constant down the channel. m& = ρua = constant (4) The gas-phase energy equation (Equation 5) comes from an enthalpy formulation although it explicitly involves temperature. This formulation of the energy equation accounts for variable specific heats. The st term corresponds to energy advection down the channel while the second term is due to heat release from homogeneous reactions. The rd term of Equation 5 represents the sensible enthalpy change between adsorbing and desorbing species and involves the reaction rate, s&, of gas-phase species by surface reactions. A positive s& represents the production (desorption) of gas-phase species due to surface reactions these species enter the control volume at the surface temperature. Thus, h is evaluated at the surface temperature ( T = T ) if s& >. A value of s& < represents the consumption (adsorption) of gas-phase species due to surface reactions these gas-phase species leave the control volume (via mass diffusion) at the bul gas temperature ( T = T). The parameter a* accounts for the potential surface area 6

32 enhancement of a catalytic washcoat per geometric area. This parameter is discussed in detail later. The final term in Equation 5 represents heat transfer between the gas and the solid channel (i.e. perpendicular to the flow) and is described using a heat-transfer coefficient. Kg Kg T * ρ uac P + A &ω Wh + a s& W ( h ( T ) h ( T ')) + ht [ T T ] = x = = (5) T if s& ' T = T if s& > Equation 6 shows the conservation of mass species. This equation states that the change in mass of gas-phase species along the length of the channel (term ) comes from two sources: surface reactions, s& via lateral/radial mass-diffusion into or out of the control volume (Equation 7) and gas-phase reactions, ω. Y ρ ua + ρhd W = x ( Y Y ) ω W A * ( Y Y ) = s& W a ; (= K g ) (6) ρ h ; (= K g ) (7) D W In equation 6, there are two values of gas-phase species: a bul flow value, Y, and a value adjacent to the catalytic surface, Y W, but still in the gas-phase. This latter value exists only in an infinitesimally thin layer at the exterior of the control volume and does not affect the bul mass properties of the control volume. The flux of each species diffusing to (or from) the surface must be balanced by the rate of adsorption and desorption by surface reactions (Equation 7). In Equations 6 and 7, the surface reaction rates, s& (adsorption or desorption), of gas-phase species are evaluated using the wall concentrations and surface temperatures as opposed to the bul value this is an important distinction between this model and general plug-flow models. The latter is described in detail by Kee et al. on pages []. 7

33 Equation 8 shows the transient solid phase energy balance and includes internal heat transfer from the gas-phase; external heat transfer to the surroundings; heat generation terms due to catalytic reactions and Joule heating; and axial heat conduction in a thermally thin solid. The internal gas-solid heat transfer (term ) is equal in magnitude to term 4 of Equation 5 using appropriate heat transfer correlations. External heat loss due to natural convection (term ) and radiation (term 4) allows modeling of a single catalytic channel. For an isolated horizontal channel, the external heat loss via natural convection comes from correlations of Churchill and Chu [74]. For a monolith, terms and 4 are set to zero. Term 5 represents the enthalpy of absorbing and desorbing gas species and is analogous to the gas phase (term in Equation 5). Because the solid is thermally thin, the surface catalytic reactions (those involving surface species only) are modeled as heat generation (term 6) in the solid. The solid-phase includes internal heat generation, term 7, to simulate ignition (via solid heating) and initiate reactions. Term 8 represents the axial heat transfer due to conduction. The solid density, ρ, heat capacity, C, and thermal conductivity,, are all constant in the model. ρ C a A K * g = T t = h s& W h ( T ) + a T 4 4 ( T T ) h ( T T ) εσ ( T T ) K * = s& W h ( T ) + q& O O gen A O + x A T x (8) The effective catalytic surface area (often due to a high-surface area washcoat) can be different from the geometric surface area x (see exploded view in Figure 4 of Appendix A). This can be achieved, for example, by a thin deposition of catalytic washcoat on a catalytically inert substrate material. For thicer depositions, pore diffusion effects (e.g. see [, 5, 8]) need to be considered but are not accounted for in 8

34 this model. To account for an area enhancement due to the use of a thin catalytic washcoat, the geometric surface area is multiplied by a surface area adjustment factor, a*, for terms dealing with surface reactions but not terms dealing with gas-to-surface transport. The parameter a* can also be thought of as the ratio of effective catalytic surface area to the geometric area. While a* can represent a surface area enhancement, it is mathematically equivalent to accelerating (with a*>) each inetic pre-exponential factor by the factor a*. In this study, a* is constant. Oh and Cavendish [4] employed a similar parameter but varied the catalytic loading along the length of the channel. In reality, the specification of a * is difficult since it is highly dependent on the manufacturing process and data is not readily available. In this wor, the parameter a * is adjusted to best match experimental data for cases with a washcoat. For a pure metal channel, the parameter a * is set to. The parameter a* may appear analogous to adjusting the active surface site density, Γ. Comparison of equations 5, 7, and 8 to the expressions for evaluating the surface reaction rates (equations, and 5 below) shows that a* and Γ are not mathematically equivalent. Furthermore, the manipulation of Γ can be problematic owing to its non-linear nature in chemical mechanisms. In the model, the surface species vary along the inner portion of channel and are accounted for via the surface species site fraction, Z (ratio of adsorbed species to the total number of active sites at a specific axial location). urface species can only adsorb from or desorb to the adjacent gas there is no surface mobility of species accounted for in the current model. For a particular species, the surface site fraction varies during the transient portion of the calculation as shown in Equation 9. In this equation, the number 9

35 of active surface sites per unit area, Γ, remains constant (i.e. no transient catalyst deactivation or poisoning is modeled). Equation 9, which is written for each solid control volume and species, describes the rate of change of surface species starting with the initial condition. The surface species (as well as mass fractions) sum to during the entire integration provided that the initial condition sums to and the reaction mechanism is balanced (hence s& = ). Ks = Z = s& ; ( = K ) (9) t Γ Equations and show the molar rates of production of species due to gasphase and surface reactions, respectively. The rate of production of species comes from summing the rate of progress of each reaction (forward and reverse) involving species (Equation ). For the gas-phase, the reaction rate constants, fi and ri, are evaluated at the bul temperature while for surface reactions, fi and ri are evaluated at the surface temperature. The various expressions for fi and ri, which typically tae on an Arrhenius form, can be found in Chapters 9. and.6 of Kee et al. [] for both homogeneous and heterogeneous reactions, respectively. Included in this reference are the expressions for handling pressure dependent homogeneous reactions as well as expressions which convert catalytic sticing coefficients to the Arrhenius form. q i K n g ω = ν q () n = i= i= i i i i s& ν q () i [ ] [ ] i X ri X fi = ν ' K = () = ν ''

36 [ X ] [ X ] ρy W ρy = ; for homogeneous reactions using bul species () W W = ; for surface reactions using gas species adjacent to surface, (4) [ X ] ΓZ σ = ; for surface reactions involving surface species (5) For gas-phase reactions, the concentrations in Equation are computed using the bul mixture temperature and mass-fractions (equation ). For surface adsorption and desorption reactions, the gaseous and surface concentrations are evaluated using the surface temperature and wall mass-fractions (equations 4 and 5, respectively). The use of the surface temperature for the wall gas-phase concentration may seem in violation of our gas-phase control volume formulation which extends the mixture bul temperature to immediately adjacent to the surface. The justification for use of the surface temperature for wall concentration is simply that the gas immediately adjacent to our surface, in physical situations, should be in thermal equilibrium with the solid and thus should more accurately represent the surface adsorption and desorption process. The choice of surface temperature for the wall concentration affects only the rates of surface chemistry and has no impact on any of the conservation principles discussed earlier in this section.

37 4 OLUTION PROCEDURE 4. Overview In general, the dependent variables (T, T, Y, Y W, and Z ) as well as all the property values, transport coefficients, and reaction rate terms are functions of axial position and time. Equations 5-6 represent the gas-phase while Equations 7-9 represent the solid-phase and surface. Equation 4 substitutes directly into these equations and is not explicitly solved. Both equations 7 (surface flux balance) and equations 8 (solidphase energy balance) are directly coupled to the gas-phase variables Y and T, respectively. The method of lines [75] transforms Equation 8 from a PDE into a system of ODEs. Equations 7-9 are written separately for each solid control volume (e.g. blue dashed box in Figure ) thus forming a large system of ordinary differential-algebraic equations (DAE) for the solid. ince the gas-phase is quasi-steady, it responds instantly to changes on the solid surface. Thus, the gas-phase must be solved simultaneously with the changing solid phase and surface variables. This fact is especially important when transitioning from a inetically controlled regime to a mass-diffusion limited regime and the latter becomes the rate limiting step. In principal, the gas-phase energy (5) and species balance equations (6) can reduce to pure algebraic equations via the method of lines and be included as algebraic constraints in transient solution of the solid. This technique, however, too excessive CPU time and was problematic for cases with light-off occurring near the outlet. The difficulty stemmed from the specifying an appropriate outlet boundary condition for the gas-phase. Other difficulties occurred when trying to resolve the gas-phase ignition region which can occur over small spatial distances

38 requiring excessively fine grids. Techniques, such as an adaptive grid, proved difficult to implement in the current solution scheme. Instead, the model separately integrated gas-phase equations 5 and 6 (ODEs) along the length of the reactor at specific time intervals. For this integration in x, the spatial distributions of the solid and surface parameters (T, Z, Y W, and s& ) were fixed at the particular instant in time. The integration, which proceeds across each gas control volume beginning at the inlet, taes the necessary spatial step to handle stiff regions (i.e. ignition). uch a technique allows a coarser grid to be stored in memory (i.e. at each axial face) but allows the necessary numerical resolution to capture gas-phase ignition, which may occur over a small distances. The inclusion of detailed gas-phase and surface chemistry introduces numerical difficulty via mathematical stiffness. tiffness in DAE systems essentially means that the time (or spatial) step required to solve the equations is much smaller than that required to obtain an accurate solution. Thus, stiff DAE systems require special numerical routines to solve them efficiently. A routine which is specifically designed to solve stiff DAE systems is the software pacage DAL [78, 8]. For larger scale problems (i.e. those involving many equations, such as this case), the DAPK code [79], which was derived from DAL, is more appropriate. Because of the potential for a large number of equations arising from the solid phase discretization, the model used DAPK as the solver. DAPK discretizes the differential variables of the ODEs using up to a 5 th order bacward Euler method. DAPK then solves the resulting system of nonlinear algebraic

39 equations using a preconditioned General Minimal Residual iterative method [79]. pecify solid/surface I.C. pecify gas-phase inlet B.C. t = Integrate Gas Phase Equations (From x= to L). olid Phase Parameters Constant. Gas-Phase Converged or t =? yes tore data at t no Revert solid to values at t = t Integrate olid Phase Equations (From t = t to t+ t). Gas Phase Parameters Constant. Figure. Flowchart of basic solution algorithm. Figure shows the basic solution algorithm for the computer program. After specifying the surface / solid initial condition, DAPK integrates the gas-phase equations (5-6) along the channel from x= to L (solid/surface-phase constant). Except after the very first spatial integration (t = ), the program checs whether or not the gas-phase has converged (compared to the previous values checed point-by-point along the reactor). If the gas-phase has not converged then the surface / solid parameters revert to the values at t = t. DAPK then performs an integration of the transient surface / solid equations (7-9) from t =t to t + t (gas-phase constant). ubsequently, the gas-phase is again integrated (this time with the newer solid values at t = t + t). If the gas-phase is now converged then the solution is permanently stored otherwise the iteration continues. The series of temporal and spatial integrations are continued to some forward time (usually steady-state). This method of solution holds the gas-phase constant for the entire 4

40 transient integration interval t. The interval solution s dependence, or more specifically independence, on t is checed for calculations which require time accuracy. For the calculations presented in this dissertation, a t =. τ provided a time-accurate (i.e. independent of t) solution. The parameter τ is a characteristic time-scale of the solid based on internal heat transfer and is typically on the order of milliseconds (Appendix B). 4. patial Integration in x The catalytic tube is discretized spatially as shown in Figure. There are a total of K g + gas-phase ODE equations. That is one gas-phase energy equation (5) and K g species equations (6). The gas-phase integration begins at the inlet (x= or i G =) with specified boundary conditions for T and Y. The DAPK routine requires that the initial conditions (or boundary conditions in this case) be consistent, that is, satisfy equations 5 and 6 at the beginning of the calculation. The consistent initial conditions require the computation of the initial derivative terms, dt/dx and dy /dx, at the inlet which can be non-zero, particularly if reactions are taing place near the inlet. DAPK computes these automatically using special algorithms and is discussed below. The spatial integration proceeds from x= (i G =) to x= x (i G =) using the spatial distribution of wall parameters T, Y W, Z (at i = )as well as s&, ht, and h D from the previous time step. These wall parameters are constant and the code interpolates them to match the current x location of the spatial integration. The solver DAPK chooses the spatial step until reaching x= x (i G =), where the solution components for T and Y are output to a file. The x-integration proceeds in this fashion to the next node at x= x + x (i G =) and repeats until reaching the end of the channel x=l (i G =n). For all computations presented, tests with various x verified that the solution was grid independent. 5

41 x = = x L i j= x j C L C L w x i e C L i G = i G = i G = i - i G = i i G = i + i G = n bul gas gas at the wall " Q U " Q D i = i = i - i i + i = n solid phase x i- x i Figure. Discretization of catalytic channel into finite volumes. olid phase nodes (shown in blac) represent the entire cross-sectional volume. The inlet is at x=. 4. Integration in Time The solid / surface equations form a large matrix, n- times the sum K g + K +, of ordinary differential-algebraic equations. DAPK integrates these equations forward 6

42 in time by an interval t assuming that the bul gas-phase values remain constant during that interval. For the time integration, the gas-phase values Y, and T are evaluated at the center of the adjoining control volume (midpoint between i G = i and i G- = i -) as shown in Figure. The user specifies the time interval t while the solver DAPK adjusts the number of steps necessary to integrate from t to t+ t (depending on the behavior / stiffness of the solution). For all cases, the model started from a cold initial condition (T = K). This procedure was necessary because the surface site fraction distribution (and corresponding wall mass-fraction) was not nown initially for reacting conditions. As discussed in the next section, the model requires initial conditions that satisfy all the governing equations (i.e. consistent initial conditions) and the solver had difficulty in computing consistent initial conditions unless they were already close to the actual values. When heterogeneous reactions are significant, the surface site fractions and corresponding wall gas-phase mass fractions can vary significantly along the reactor maing it difficult to now the distribution a priori. At low temperatures, the integration by DAPK was largely insensitive to the specification of initial surface site fraction unless a large fraction of unoccupied sites was specified (i.e. Z Pt(s) >>, which is physically unrealistic at low temperatures) causing integration errors to occur. In this wor, calculations started with either complete surface coverage by O(s) or CO(s). 7

43 4.4 olver DAPK The DAPK 5 code solves both the gas-phase system of ODEs and the solid / surface system of DAEs using the Krylov iterative method [79]. DAPK is designed for general index- (or index-) DAE systems but can handle simpler ODE systems. The index of a system is loosely defined as the number of differentiations of the system of equations required to yield an explicit ODE system for all of the unnowns [8]. In the present system of equations, the gas-phase equations (5-6) are all ODEs, hence index-. For the solid-phase, equations 8 and 9 are ODEs while equation 7 requires only a single differentiation (with respect to t) to form an explicit ODE system (hence index-). DAPK integrates large scale systems of DAEs that are mathematically stiff. Other researchers utilized this code previously to handle method of line solutions of PDEs, similar to the present problem [79]. The code solves the non-linear system that arises from the discretized DAEs (up to a 5 th order bacward difference) at each time step in an iterative fashion using a preconditioner 6. In this wor, the program used a generalpurpose preconditioner (subroutines DBANJA and DBANP) supplied with the DAPK code which was designed for banded or approximately banded matrixes. The solid / surface equations are arranged in node, then species & temperature order thereby banding the matrix such that only adjoining control volumes (i + and i - ) influence any given solid control volume, i (see Figure ). The solver DAPK taes advantage of this banded structure eliminating unnecessary calculation and improving code speed. For the gas-phase, no special order was required since this group of equations did not utilize the method of lines. While these general purpose preconditioners wored for the solution 5 The DAPK code (this wor used version ) can be found on a web site sponsored by the Computational cience and Engineering department of the University of California, anta Barbara[9]. 6 A preconditioner is an approximation to the iteration matrix, used in Newton s method, which may lead to a computationally less-expensive solution. 8

44 of the problem presented herein, more complex preconditioners are available (even specific for reaction-diffusion equations) and should be explored in the future to improve code performance. The error tolerances for each of the solution components, f j, are split into absolute, ATOL j, and relative, RTOL j, tolerances as shown in equation 6 where f j = T, T, Y, Y W, or Z. DAPK uses the tolerances in a local error test at each time step which requires, approximately, that the absolute value in the local error be less than or equal to err j. More specifically, the root-mean-square norm is used to compare the size of the vectors (both local error and tolerances) where the local error uses the magnitude of the solution at the beginning of the time step. A mixed test with non-zero RTOL j and ATOL j corresponds roughly to a relative error test when the solution component is much bigger than ATOL j and to an absolute error test when the solution component is smaller than the threshold ATOL j. Chapter 5. of Brennan et al. [8] recommends that the value of ATOL j be the value where the solution component is essentially insignificant and the value of RTOL j =.x -(m+) where m is the number of significant digits desired for the solution. The values of ATOL j and RTOL j used for the computations are based on these recommendations and are presented in Table. The algebraic variables, Y W, used more significant digits than the differential variables which seemed to facilitate the calculation of consistent initial conditions as discussed below. err = RTOL * abs( f ) + ATOL (6) j j j j 9

45 olution Component ATOL RTOL T, T.x - K Y - stable species.x -7.x -4 Y - radical species.x -.x -4 Y W - stable species.x -7.x -5 Y W - radical species.x -.x -5 Z - CO (s).x -.x -4 Z - others.x -.x -4 Table. Absolute (ATOL) and relative (RTOL) error tolerances used in the computations. The stable and radical species are defined in Appendix E. Values of T, T, Y, Y W, and Z as well as their derivatives at the initial time must be given as input. These values should be consistent, that is, they should satisfy equations 5-8. The boundary conditions of the bul gas are specified (e.g. T and Y ). The gradients at the inlet, however, are not nown (and not-necessarily zero). imilarly, the initial conditions are specified for the differential variables of the solid (T and Z ). Their initial derivatives as well as the algebraic variable, Y W, also are not nown. The wall mass-fractions cannot be the same as the bul when surface reactions are taing place ( s&, see Equation 7). DAPK automatically calculates consistent initial conditions (i.e. dt/dx, dy/dx, dt /dt, dz /dt, and Y W ) given the initial differential variables [8]. Early results showed that DAPK requires good initial guesses for the automatic routine to calculate consistent initial conditions, particularly for the algebraic quantities Y W. In the solution procedure, the initial guessed values for Y W come from equation 7 where the inlet bul mass-fractions, Y, are used to compute s&. For non-reacting conditions, the values of YW Y.

46 5 REULT The numerical model presented herein predicts both steady-state and transient phenomena from different experimental configurations all using CO as fuel. Table presents a summary of the modeled cases. Cases and are steady-state results which provide model vs. experiment comparisons of the total fuel conversion at the end of the channel. The primary difference between Cases and is the experimental configuration: Case is an isothermal horizontal channel while Case is a monolith reactor. Case, which involves the propagation of a catalytic reaction front along a platinum channel, compares both transient and steady phenomenon. The transient parameters are the propagation velocity of the catalytic reaction front along the channel as well as temperature measurements of the solid. The steady-state upstream anchor point of the reaction front is also compared. For each case, the model explored the solution s sensitivity to various parameters which were also adjusted to try and best match the model with the particular experiment. All cases investigated the effects of initial surface species distribution and internal heat/mass mass-transfer coefficients. In cases and, the model examined the effect of water vapor, present in these experiments, by comparing both a wet and dry CO mechanism 7 the dry mechanism neglects all hydrogen chemistry and is presented in Table. Case further presents a global sensitivity study on the dry mechanism as well as the as the catalytic surface site density. Case, which involves a commercial monolith, uses the parameter a* which is the ratio of apparent catalytic area to geometric 7 The homogeneous reaction mechanism came from the wor of Davis et al.[84]. The surface CO mechanism came from the CH 4 /O on platinum proposed by Deutschmann et al.[, 6, 64]. Both the homogeneous and heterogeneous mechanisms are presented in their entirety in Appendix E.

47 area. Finally, case loos at the effect of external heat transfer on the predicted catalytic reaction front propagation as well as surface temperature. # # Ref Reaction Homogeneous Reactions from Davis et al. [84] A i (cm, mol, s) β i E i (Joules/mole) a O + O + M O + M.E a,b 7 CO + O (+M) CO (+M).6E+ 84 CO + O CO + O.9E+ 477 Heterogeneous Reations from Deutschmann et al. [] 4 4 O + Pt(s) O(s).8E c 5 O + Pt(s) O(s).E O(s) O + Pt(s).7E+ 6 Z O(s) 45 c 7 O + Pt(s) O(s) E+ 5 d 5 CO + Pt(s) CO(s).68E CO(s) CO + Pt(s) E CO (s) CO + Pt(s) E CO(s) + O(s) Pt(s) + CO (s).7e C(s) + O(s) Pt(s) + CO(s).7E Pt(s) + CO(s) C(s) + O(s) E+8 84 a Denotes the use of third body efficiencies which can be found in the source. b Denotes pressure dependent reactions. c Denotes sticing coefficient listed in column A i. d The order of CO adsorption is with regard to Pt(s) Table. Dry CO / O sub-mechanism on platinum from the wor of Deutschmann et al. [, 6, 64]

48 Case: teady / Transient teady-tate teady-tate Transient Experiment Reference: [54] [9] [44, 45, 85] Configuration Horizontal Tube Monolith Horizontal Tube Catalyst Pt Pt Washcoat Pt Parameter(s) Compared Fuel Conversion Fuel Conversion Rxn. Front Propagation; Temperature ubstrate N/A Cordierite N/A Cell Density (cells/cm ) N/A 6 N/A External Tube Radiative and Isothermal Adiabatic Conditions Convective Loss Joule heating: Ignition Method Hot Isothermal Channel Hot Inlet Gas.5W for seconds over last % of channel Channel Inlet Conditions Fuel CO CO CO Oxidizer / Diluent Air aturated at 8 K O / N Dry & aturated O Mass % Diluent 74.6% 99.% % Equivalence Ratio.746. Velocity (m/s) to Reynolds number, Re D 6 (see Table 5) 6 95 Temperature (K) 57 to 87 6 Pressure (atm) * * ** Channel Geometry Length (cm) 5.5 to 5.5 Cross-ection Circular Circular Circular Inner Hydraulic Diameter (mm).74 Outer Diameter (mm) N/A.4.95 Cross-ectional Open Area, A (mm ) Cross-ectional olid Area, A (mm ) N/A.8.79 A / (A+ A ) N/A Table. ummary of experimental configurations modeled in this wor. Values denoted with an asteris (*) were not explicitly stated in the reference but were assumed. The ambient pressure surrounding the channel for case (denoted by ** ) was slightly less-than (.97 atm)

49 5. Case : teady-tate Comparisons Isothermal Platinum Tube 5.. Experiment The first comparison of the model results is to the steady-state results from the experiment of Khitrin and olovyeva [54]. They performed a series of catalytic combustion experiments in a platinum tube using a simple isothermal configuration. The experiment involved a premixed gas flowing through a 5 mm long platinum cylindrical channel with a mm inner diameter. An electric heater ept the channel isothermal at temperatures ranging between 57K and 95K. The velocity in the channel varied between and 7 m/s with discrete values compared in this wor (, 4, and 7 m/s). The fuel was carbon monoxide in air tested at two equivalence ratios:.746 (% CO by volume) and.69 (5% CO by volume). The comparisons only use the % by volume cases since there were only slight differences between the % and 5% tests. The air had a constant humidity value by first saturating the air at room temperature and then passing it through condensing coils at 8K before entering a quartz heating chamber. The quartz heating tube, which transitioned directly into the platinum channel, brought the mixture to the desired test temperature. Thermocouple measurements verified that the channel maintained near isothermal conditions. Gas samples both at the entrance and exit of the channel provided conversion data although the authors did not report the details of the measurement technique. The conversion data as a function of channel temperature came directly from a magnified reproduction of figure in the reference. 5.. Model Parameters The model parameters matched the experimental conditions as closely as possible. The inlet gas temperature, T, as well as the temperature of the gas surrounding the tube, T, corresponded to the final steady-state temperature of the entire channel. The transient 4

50 model ran from a cold initial condition (T = K) to steady-state conditions. The surface initial condition was either Z CO(s) = or Z O(s) =. Despite the exothermic catalytic reactions, the channel temperature did approach isothermal conditions by setting the external heat-transfer coefficient to a very high value (h ext =, to 5, W/m /K). The external heat transfer coefficient was adjusted to yield a maximum temperature difference along the channel of less than K for the gas and 4K for the solid when compared to the inlet gas-temperature. The curves shown in Figure through Figure were generated by running the model to steady-state temperatures from 55K to 95K in K increments and plotting the fuel conversion at the exit of the channel. While not specifically reported in the experiment, the outer diameter of the tube was. mm for the computations although this dimension was of no consequence for the steady-state solution. The computations assumed that the catalytic active area equaled the geometric surface, that is a* =. since we are not dealing with a washcoat in these tests. For the majority of the comparisons, the computations used dry CO chemistry with the inlet mass (mole) fractions being.94% (%) carbon monoxide,.4% (.%) oxygen, and 74.6% (76.8%) nitrogen at atm pressure. 5.. Model vs. Experiment The primary parameter compared for this case is the steady-state exit conversion of fuel versus channel temperature for inlet velocities of, 4, and 7 m/s (Figure ). This figure presents three conversion versus temperature graphs which are staced vertically corresponding to, 4, and 7 m/s from top to bottom. The conversion is defined as the mass of gas-phase fuel reacted to the mass of gas-phase fuel fed into the reactor. Each of the graphs includes the residence time, τ L, in the channel (based on inlet 5

51 conditions). Figure shows three separate calculations for each velocity. These include calculations using different surface site distributions (Z O(s) or Z CO(s) = ) as well as from the plug-flow model. The computations in Figure use dry CO chemistry and both the Nusselt and herwood numbers are constant with Nu = 4.64 and h based on the heat and mass-transfer analogy (Table 4). CO Conversion (%) U = mps τ L = 5 ms U = 4 mps τ L = 4.4 ms U = 7 mps τ L =. ms 6 K 6 K 6 K Exp. Model -Z O(s) = Model-Z CO(s) = Plug 5 Dry Mechanism Nu = 4.6, a*= Channel Temperature (K) Figure. Comparison of steady-state model results (including a plug-flow model) to the experiment of Khitrin and olvyeva [54] for channel velocities. The inlet gas consisted of % CO (by volume) with the balance being air. 9 6

52 The experimental data in Figure show little conversion at low temperatures. ignificant conversion (i.e. light-off) begins at a temperature of approximately 6 K, 68 K, and 7K, for, 4, and 7 m/s, respectively. After light-off, there is a small temperature range where larger increases in conversion occur with further increases in temperature. This range is only K to K wide and is most obvious for the 4 m/s data. Further increases in temperature produce only modest increases in conversion. As velocity is increased, the total conversion decreases at a given temperature. Figure includes computations from a plug-flow model which neglects all diffusive terms. The plug-flow computations presented herein are from the PLUG code [] which is part of the CHEMKIN software pacage. The PLUG computations, which used isothermal channel conditions, automatically calculated the surface site fractions at the inlet (based on an initial guess) which evaluated to Z CO(s).97 for 5 K and Z CO(s).7 for 87 K for all velocities. The PLUG solution and the present model show similar characteristics to each other and the experimental data. Beginning at low temperature, the conversion increases only a small amount as temperature increases. Then, at a particular temperature, the light-off temperature, there is an abrupt increase in conversion. The abrupt increase in conversion continues as the temperature further increases K to K. At this point, the conversion increase with temperature drops off dramatically and higher temperatures yield only smaller increases in conversion. All the model computations tend to under predict the experimental light-off by approximately 5 K to 8 K. PLUG significantly over predicted the exit conversion at higher temperatures especially for the faster velocities. The result is not surprising because the surface 7

53 reactions are not mass-transfer limited in PLUG which subsequently allows significantly more conversion to occur []. The lower conversions at higher temperatures in the present model are due to mass-transport resistance limiting the conversion along the length of the channel. A surprising result is that PLUG predicted the onset of light-off at slightly higher temperatures than the present model with lateral mass-transfer. This effect is liely occurring because the finite rate mass-transfer in the latter case limits the transport of CO to the surface. This allows O to reach the surface more readily because it is present in excess (and hence is less influenced by mass-transfer) thus lighting-off at a slightly lower temperature. An important observation in Figure is the similarity of the PLUG solution to the model with mass-transfer for the m/s case. This result is liely due to the significantly longer residence time in the channel for the m/s case compared with the faster velocities. For the 4 and 7 m/s cases, the residence times are of the same order as the lateral (radial) diffusive mass-transport time scales. Figure also shows the effect of the initial surface-site occupancy on the solution for two initial surface conditions: Z CO(s) = and Z O(s) =. For the case of Z CO(s) =, the light-off temperature is approximately K higher than the Z O(s) = case. This effect is liely occurring because of the significantly higher sticing coefficient of CO relative to O (despite the latter being present in excess) and thereby requires a higher temperature to allow enough oxygen to reach the surface and begin reacting. Examination of the species profiles helps to better understand this observation. Figure 4 shows the species profiles at 4 separate temperatures near the onset of significant CO conversion for the 4 m/s case with O(s) initially occupying all surface 8

54 sites. The species presented are the bul and wall mass fractions of CO as well as the surface site fractions of CO(s), O(s), and Pt(s) or vacant platinum sites. The lowest temperature of 6K corresponds to just prior to light-off (and significant fuel conversion). From the upper most graphs of Figure 4, the data for 6K show that the wall mass-fraction is only slightly smaller (~.96 times) than the bul value along the majority of the channel length except for very near the outlet (x=5 cm). Here there is an abrupt decrease in the wall mass-fraction of CO (~.4 of the bul) it is at this point that the catalytic reaction rate becomes large (i.e. light-off occurs) causing a transition from a inetically controlled to a mass-transfer controlled reaction rate. As temperature increases, the abrupt decrease in wall mass-fraction shifts upstream and, at 7 K, the wall mass-fractions is entirely the lower value. The data in Figure 4 show that the bul mass-fraction decreases at a much larger rate after light-off (i.e. the reaction rates are much higher). When light-off occurs within the channel, the total fuel conversion is a combination of the slower conversion prior to light-off and the higher, mass-transfer limited, conversion after light-off. 9

55 . Z CO(s) = Bul Wall 6K CO(s) O(s) Pt(s) Z O(s) =.4 Z CO(s) = CO Mass Fraction K Z O(s) =.68 Z O(s) =.54 Z CO(s) = urface ite Fraction K Z O(s) = K Z CO(s) = Distance down the channel (cm) Figure 4. Computed steady-state profiles of CO mass fraction and select surface species along the length of the platinum channel for the 4 m/s case (Nu = 4.6) at 4 temperatures. The initial condition for the surface is Z O(s) =. The transition from inetic to mass-transfer limited reaction rates is accompanied by significant changes in surface species coverage (right hand side of Figure 4). Before light-off, the surface sites are primarily occupied by CO(s). This is despite the fact that the computation began with the entire surface initially occupied by O(s). The transient simulation shows that the surface is covered predominantly by O(s) during heat-up of the 4

56 channel until approximately 6K. At this temperature, light-off occurs at the channel inlet and subsequently propagates downstream towards the final steady-state. For the final steady-state, CO(s) remains the predominant surface species before the light-off point and the reactions appear to be limited by O(s). After light-off, the predominant surface species is O(s) and the reactions appear to be limited by CO(s). With Z CO(s) = initially (not shown), the transient simulations show that light-off begins at the channel exit and propagates upstream to a qualitatively similar (but quantitatively different) steady-state. The different time histories of the solution caused by the different initial conditions appear to lead to the slightly different steady-state profiles in Figure. After light-off, while in the mass-transfer controlled regime, the surface coverage is mostly O(s) owing to the sudden absence of CO in the adjacent gas-phase. In Figure 4, the light-off point shifts further upstream with increasing temperature and above roughly 64K, the entire channel is operating in a mass-transfer controlled mode. It is in this region where mass-transport coefficients (particularly for the limited species) are expected to play a significant role. Further down the channel, the reactions may again become inetically controlled as fewer reactants remain and the bul concentrations approach the wall values. The results in Figure, consistent with literature, show that catalytic reactions can be mass-transfer limited. To this point, the calculations used a single Nusselt number (Nu=4.64) while the herwood numbers, which vary with species (see Table 4), came from the analogy of heat and mass transfer (equation in the Bacground). The property values used in the calculation of Nu and h are based on the inlet mixture composition and do not vary significantly within the temperature range of the experiment. The value 4

57 of Nu=4.64 corresponds to the asymptotic value (fully-developed) for heat-transfer in a circular tube with a constant heat flux and should better estimate the light-off position in the channel [7, 8]. As discussed in these references, the channel boundary condition resembles a constant flux boundary condition (both for temperature and species) up to the light-off position. pecies-> H H O CO N O CO herwood Number Table 4. Calculated herwood numbers for each species using the analogy of heat and mass transfer with Nu=4.64. The property values are based on the inlet mixture (% CO by volume in air with saturated water vapor) at 85K. Figure 5 shows how the mass-transfer coefficient (via the herwood number) affects the solution of the model. In this figure and the subsequent discussion, a single Nusselt number gauges the effect of the herwood number through the analogy of heat and mass-transfer. The specification of Nu for heat-transfer is less relevant for the near isothermal conditions in the channel. The advantage, however, is that a single number (i.e. Nu) can represent a series of herwood numbers similar to those presented in Table 4 for Nu=4.64. In this way, there is some distinction between the relative effectiveness of diffusion based on the species Lewis number. 4

58 U= mps Nu = 4.6 Nu = Nu = Plug CO Conversion (%) U=4 mps Nu = 8 Nu = 4.6 Nu = Nu = U=7mps Nu = 6 Nu = 8 Nu = 4.6 Nu = Nu = Nu = 46 Plug Nu = 46 Plug Channel Temperature (K) Figure 5. Computed results showing the effect of varying the mass-transfer effectiveness. The herwood number is related to the Nusselt number via the heat and mass-transfer analogy. The surface initial condition for these calculations is Z O(s) =. 8 9 Figure 5 shows that, at lower temperatures, the exit conversion is not affected significantly by variations in the mass transfer coefficient. As the temperature increases, the influence of mass-transfer becomes more important beginning near the region of light-off (or sudden increase in conversion). Well above the light-off temperature, where reactions are completely mass-transfer limited, the exit conversion decreases with Nu as expected. In Figure 5, the solution approaches the plug-flow solution as Nu increase. At Nu = 46, the current model results are almost identical to those of the plug flow model. 4

59 Figure 5 also shows that the light-off temperature decreases with decreasing Nu. Furthermore, there exists an optimum Nu (not necessarily ) which maximizes conversion. Looing at the 4 m/s data, the conversion is maximized as Nu at temperatures above 67K. Below this temperature, the conversion is maximized at a finite Nu. For example, there is only ~% conversion at 6K with Nu / h =. As Nu decreases down to 4.6, there is very little change in conversion. Then at Nu =, the conversion suddenly increases to approximately 5%. Further decreasing Nu to decreases the total conversion to ~5%. Figures 6 and 7 show both the species mass-fraction at the wall and surface site fraction as a function of Nu for CO and O (4 m/s data). The species data correspond to the values at the outlet of the channel this is done indicate whether or not light-off has occurred within the channel (refer bac to Figure 4 to see examples of the surface species profiles along the length of the channel). If light-off occurred then the wall mass-fraction of CO decreases significantly. Because O is in excess due to the lean conditions, its mass-fraction reduces only slightly. Figure 6 corresponds to 56K channel data. At this temperature, light-off has not been achieved for any Nu. The Nu = data, however, corresponds to conditions just prior to light-off which would occur with any further temperature increase or Nu decrease (see Figure 5). Figure 7 corresponds to 6K data and light-off has occurred for Nu (but not at Nu 4.64). 44

60 . Y W.5 CO (x) O.. Data corresponds to channel outlet Z CO(s) Z.95.9 T = 56K 4 m/s Z O(s) (x ) Z CO(s) * Z O(s) (x ) Nu Figure 6. Wall mass fraction, Y W (for CO and O ), and surface site fractions, Z (for CO(s) and O(s)), at the channel outlet versus Nu for 56K. Also shown is the product Z CO(s) Z O(s), which is proportional to the fuel conversion rate.. CO (x) Y W. O.. Data corresponds to channel outlet Z CO(s) Z.95.9 T = 6K 4 m/s Z O(s) (x 75) Z CO(s) * Z O(s) (x 75) Nu Figure 7. Wall mass fraction, Y W (for CO and O ), and surface site fractions, Z (for CO(s) and O(s)), at the channel outlet versus Nu for 6K. Also shown is the product Z CO(s) Z O(s), which is proportional to the fuel conversion rate. The sudden increase in conversion at Nu = for the 6 K case is accompanied by a dramatic change in surface species. Figures 6 and 7 show that CO(s) is the predominant surface species prior to light-off and that its site fraction, Z CO(s), remains 45

61 roughly uniform (.98 and.99 for 6K and 56K, respectively). The large amount of CO(s) on the surface inhibits O adsorption, thus limiting the reaction. The surface site fraction Z O(s) is small prior to light-off, however, it does increase as Nu decreases. The product Z CO(s) Z O(s), also shown in Figures 6 and 7, is proportional to the surface reaction rate (reaction 56 in Table ) which is the sole pathway for fuel conversion (i.e. CO production) via catalytic reactions. These figures show that fuel conversion increases with decreasing Nu and, in the case of 6 K, light-off occurs for values of Nu. Figures 6 and 7 show that a decreasing Nu effectively increases the reactivity of the surface mixture (hence the results of Figure 5). To understand why surface reactivity (i.e. the product Z CO(s) Z O(s) ) increase with decreasing Nu, it is necessary to examine the competition for free surface sites via the net adsorption of surface species. Looing bac at equation 7, species adsorption is influenced by: () species masstransfer coefficients, () species mass-fraction or concentration at the wall, () inetic parameters, and (4) surface temperature. In Figure 5, the mass-transfer coefficient and surface temperature are the independent parameters thus leaving the wall mass-fractions and inetic parameters to explain the variations in light-off temperature with Nu. Looing first at the inetic parameters may help to explain the temperature dependence of light-off with Nu. Table shows that O adsorption (reaction 4) decreases with increasing temperature while CO adsorption (reaction 5) increases with increasing temperature. This means that CO adsorption increases relative to O as temperature increases. With regard to the wall mass-fraction, decreasing Nu slows both the CO and O transport to the surface. Looing bac at Figures 6 and 7, this affects the CO wall mass- 46

62 fraction more drastically (on a percent basis) since O is present in excess due to the lean gas mixture. It is this relative decrease of wall CO compared to O with decreasing Nu, coupled with the temperature dependent inetics, that allow just enough O to adsorb to the surface (increasing Z O(s) ) thus enhancing surface conversion. This is further explained in Figure 8. Figure 8 shows the variation of wall equivalence ratio, φ W, at the outlet of the channel versus Nu for 4 temperatures: 56 K, 58 K, 6 K, and 6 K. This figure shows that φ W decreases with decreasing Nu for a given temperature. This occurs because decreasing Nu also decreases CO near the wall compared to O as just explained. For 56K, light-off did not occur but, for Nu =, the φ W at the outlet corresponded to the approximate φ W just prior to light-off (which would occur with any further temperature increase or Nu decrease). For the higher temperatures, the minimum Nu for light-off to occur increased. For instance, light-off occurred at Nu for 58 K while for 6K, light-off occurred at Nu. Furthermore, the corresponding light-off φ W becomes leaner (i.e. you need less CO in the gas) with increasing Nu and temperature for light-off to occur (as seen in Figure 5). This is a consequent of the CO adsorption rate increasing relative to O at higher temperatures. In summary, Figure 8 shows the unique combination wall equivalence ratio, mass-transfer rates, and temperature (for the given inetics) required for light-off. This combination then produces the results seen in Figure 5; that is, at lower rates of mass-transport (which reduces the amount of CO at the wall more so than O ), light-off is achieved at lower temperatures. 47

63 φ W at End of Channel K 58K Nu 6K Approx. φ W just prior to light-off 6K u = 4 m/s Figure 8. Wall equivalence ratio, φ W, at the channel outlet versus Nu for four temperatures: 56K, 58K, 6K, and 6K. The dashed line corresponds to the approximate φ W just prior to light-off (which would occur with any further temperature increase or Nu decrease). When specifying the mass-transfer coefficient, it is important to understand whether the flow is laminar, transitional, or turbulent. Table 5 shows the inlet Reynolds number based on channel diameter, Re D, for these particular tests at three select temperatures of the experiment. For the and 4 m/s cases, the flow in the channel is laminar while the 7 m/s cases may be transitional (for Re > ) especially at the lower temperatures (i.e. less viscous conditions). ince the model / experiment show that the light-off temperature is roughly 6K to 7K for the 7 m/s case (refer bac to Figure ), the impact of the enhanced mass transfer associated with transition is expected to be minimal because the channel is in a inetically limited regime. Inlet Velocity (m/s) Re (55K) Re (7K) Re (85K) Table 5. Calculated Reynolds numbers evaluated at the temperature e xtremes for the catalytic channel experiments of Khitrin and olovyeva. 48

64 To this point, the model has under predicted the light-off temperature seen in the experiment of Khitrin and olovyeva. Furthermore, variations in mass-transfer coefficients have not produced better agreement with experiment. A logical next step is to examine the various parameters of the inetic mechanism. This is accomplished first by varying the parameter Γ and examining its effect on the solution. The number of active catalytic sites per unit area, Γ, on the surface is difficult to measure. Because the model predicts higher conversion at lower temperature compared to experiment, a lower value of Γ (i.e. less active sites or catalyst deactivation) should improve agreement with the experiment. In Figure 9, Γ varies from 5% to % of the baseline value, Γ =.7 x -9 mol/cm, the predominant value in the literature [9, 6]. According to these references, the value of Γ was estimated from the density of platinum. The data in Figure 9 do not show a value of Γ which improves agreement with the experiment. omewhat surprisingly, reducing Γ does not shift light-off to the higher temperatures seen in the experiment. For.5Γ and.5γ, the resulting conversion profiles gradually increase with temperature and do not exhibit an abrupt increase in conversion at any temperature. For.75Γ and.γ, light-off occurs between 58K and 6K (depending on inlet velocity) with the specific temperature being insensitive to variations of Γ in this range. At higher values of Γ, the abrupt increase in conversion shifts to slightly cooler temperatures. The results in Figure 9 use an initial surface condition of Z O(s) =. With Z CO(s) = initially (Figure ), the results are similar for Γ < Γ. For Γ > Γ, there is virtually no shift of the conversion versus temperature profiles as seen with Z O(s) =. Rather, slightly more conversion is observed at the light-off point and at greater temperatures. This behavior is liely due, in part, to the significant 49

65 differences in sticing probability between CO and O as well as the species desorption characteristics. Furthermore, the species concentrations (via φ W as seen previously in Figure B) can potentially affect these results. 75 U= mps.5 Γ.75 Γ.5 Γ 5 Γ Γ.5 Γ 5 CO Conversion (%) U=4 mps Γ U=7mps.5 Γ Γ.75 Γ.5 Γ.5 Γ 5.5 Γ.75 Γ 5 Γ.5 Γ Γ.5 Γ Channel Temperature (K) Figure 9. Comparison of different values of Γ (surface site density) on the steady-state model results. The value of Γ =.7 x -9 mol/cm was estimated from the density of platinum [9, 6]. The initial surface condition was Z O(s) =

66 75 U= mps.5 Γ Γ.75 Γ.5 Γ 5 5 G.5 Γ CO Conversion (%) U=4 mps U=7mps G.5 Γ Γ.75 Γ.5 Γ.5 Γ 5.5 Γ.75 Γ 5.5 Γ G Γ.5 Γ Channel Temperature (K) 8 9 Figure. Comparison of different values of Γ (surface site density) on the steady-state model results. The value of Γ =.7 x -9 mol/cm was estimated from the density of platinum [9, 6]. The initial surface condition was Z CO(s) =. To this point, adjusting both h and Γ have not produced an adequate match of the light-off temperature to the experimental data. The comparisons between model and experiment suggest that the inetics must slow down in order to match the experiment. Varying the pre-exponential factor or sticing coefficient from 5% to 75% of the published value allows examination of the sensitivity of each individual surface reaction. ticing coefficients are limited to a maximum value of (or % probability that a wall collision will produce an adsorbed species) in the analysis. 5

67 Figures and show the effect of changing the pre-exponential factor or sticing coefficient by 5% and 75% for initial surface conditions of Z CO(s) = and Z O(s) =, respectively. In these figures, the blac line corresponds to the unmodified dry CO chemistry. Reactions 45, 55, 57, and 58 show virtually no effect on the computed solution when varying the pre-exponential factor or sticing coefficient by the values shown. This is not surprising for reactions 45 and 55 since there is very little O in the gas-phase present and CO desorption from the surface has a very low-activation energy and is not a rate-limiting step. Reaction 57 and 58 describe CO(s) surface dissociation into C(s) and O(s). The surface temperature is sufficiently low and does not favor the presence of significant C(s), hence the reactions are unimportant in these calculations. Figures and show that the most sensitive, and thus rate limiting, reactions are those dealing with major reactant species adsorption and desorption from the surface, steps 4-44 and Improvement of the model predictions with the experimental data occur when reducing the rate of either O adsorption to the surface (reactions 4 and 4) or CO desorption from the surface (reaction 54). In fact, reducing the CO desorption rate in reaction 54 to 5% (of the unmodified inetics) causes the model to match the experimental light-off temperature almost exactly. This is true, however, for the case of Z CO(s) = (Figure ), only. For cases started with Z O(s) = (Figure ), reaction 54 had almost no effect on the light-off temperature. This last point further demonstrates that the final steady-state solution can depend on the initial surface species distribution. Figures and suggest that most sensitive step in the dry CO mechanism is the adsorption of CO (reaction 5). imilar to the above trend, increasing the adsorption of CO relative to O has the effect of increasing the light-off temperature which improves 5

68 model predictions with experiment (although in this instance the CO conversion just after light-off is significantly higher than experiment). Reducing the CO adsorption rate, however, has drastic effects on the exit conversion. When reducing the pre-exponential factor by 5% or greater for reaction 5, the qualitative shape of the computed exit conversion versus temperature changes significantly. This may be due in part to representing the CO adsorption in Arrhenius form as opposed to representing this reaction in terms of a pure sticing coefficient (see Appendix E). Clearly then the competition between adsorbing and desorbing CO and O determines the light-off temperature. Decreasing the relative rate of O adsorption (reactions 4 and 4) has the effect of delaying light-off to higher temperatures (i.e. more surface sites remain occupied by CO requiring higher temperatures for light-off). A similar affect is seen when the rate of CO adsorption (reaction 5) is increased. In these cases, the adsorption steps show very little dependence on the surface initial condition. The desorption steps, however, are affected by the initial species coverages on the surface. For surfaces covered initially by CO(s), the CO desorption step can influence the steady-state solution significantly (reaction 54 in Figure ). For the case of initial O(s) coverage, the dependence is much less pronounced. These calculations suggest that some adjustment of the inetic constants is warranted. Case presents calculations which further support adjustment of the mechanism. 5

69 Reaction 4: O + Pt(s) >O(s) Reaction 54: CO(s) >CO+Pt(s) 5 5 I.C. CO(s) Reaction 4: O + Pt(s) >O(s) Reaction 55 :CO (s) >CO +Pt(s) 5 5 CO Conversion (%) 5 Reaction 44: O(s) >O +Pt(s) Reaction 45: O+Pt(s) >O(s) 5 Reaction 56: CO(s)+O(s) >CO (s)+pt(s) Reaction 57: C(s)+O(s) >CO(s)+Pt(s) 5 5 Reaction 5: CO+Pt(s) >CO(s) Reaction 58: CO(s)+Pt(s) >C(s)+O(s) 5.75 A.5 A 5 Exp..5 A/γ. A/γ.75 A/γ Channel Temperature (K) Figure. Global sensitivity study of the dry CO mechanism conducted by changing the preexponential constant or sticing coefficient by 5% and 75% relative to the published value. The data corresponds to 4 m/s and the blac lines are the unmodified inetics. In these calculations, the surface is initially covered by CO(s). 54

70 Reaction 4: O + Pt(s) >O(s) Reaction 54: CO(s) >CO+Pt(s) 5 5 I.C. O(s) Reaction 4: O + Pt(s) >O(s) Reaction 55 :CO (s) >CO +Pt(s) 5 5 CO Conversion (%) 5 Reaction 44: O(s) >O +Pt(s) Reaction 45: O+Pt(s) >O(s) 5 Reaction 56: CO(s)+O(s) >CO (s)+pt(s) Reaction 57: C(s)+O(s) >CO(s)+Pt(s) 5 5 Reaction 5: CO+Pt(s) >CO(s) Reaction 58: CO(s)+Pt(s) >C(s)+O(s) 5 5 Exp..5 A/γ. A/γ.75 A/γ Channel Temperature (K) Figure. Global sensitivity study of the dry CO mechanism conducted by changing the preexponential constant or sticing coefficient by 5% and 75% relative to the published value. The data corresponds to 4 m/s and the blac lines are the unmodified inetics. In these calculations, the surface is initially covered by O(s). 55

71 The final comparison using the data of Khitrin and olovyeva loos at the influence of water vapor on the predicted exit conversion versus channel temperature. The experimenters explicitly mentioned that the inlet feed was saturated with water vapor at 8K prior to entering a quartz heating channel. To gauge the effect of water vapor, the model compares computations utilizing both the wet and dry CO mechanisms outlined in Appendix E. All cases presented prior to this section used the dry CO mechanism. For the wet CO mechanism, the inlet mass (mole) fractions were.9% (%) carbon monoxide,.4% (.%) oxygen, 7.88% (75.67%) nitrogen, with.76% (. %) water vapor. The partial pressure of saturated air at 8K (to match the experiment) approximated the amount of water vapor in the inlet gas mixture. For this comparison, the inlet pressure of the dry CO tests was reduced in the computations by the partial pressure of water vapor to.9878 atm. This preserved the mass flow rates (and the partial pressures) of the common species in both the wet and dry cases. Figure shows the effect of adding water vapor to the inlet feed by comparing the computed conversions using the wet versus dry CO mechanism for two different initial surface conditions (Z CO(s) = and Z O(s) = ). For these computations, Nu = 4.64 and the herwood numbers are those in Table 5. With the surface initially covered by CO(s), the water vapor had no effect on the steady-state solution (see green solid lines in Figure ). With O(s) initially occupying the surface, the results (green hatched lines) showed a small difference when compared with the dry CO calculations for the higher velocities. pecifically, the saturated water vapor caused the light-off temperature to shift approximately K lower for 4 and 7 m/s. For the m/s cases, the computations showed that there were no distinguishable differences with or without the presence of 56

72 water vapor in the inlet feed. The presence of water in the feed stream may be an additional source of O(s) to the surface (through reactions 45, 5, and 49). These results are not surprising given that H O has a large sticing coefficient comparable with, but not as large as, CO. While computations show it to be a minor influence in this experiment, water vapor, if present in sufficient concentrations, can compete with CO for free surface sites. 75 U = m/s τ L = 5 ms Exp. 5 5 I.C. O(s) CO(s) Dry Wet CO Conversion (%) U = 4 m/s τ L = 4.4 ms I.C. O(s) CO(s) Dry Wet 75 5 U = 7 m/s τ L =. ms 5 I.C. O(s) CO(s) Dry Wet Channel Temperature (K) 9 Figure. Computed results showing the effect of saturated water vapor (using a wet CO mechanism) in the inlet feed on the steady-state conversion. 57

73 5..4 Discussion of Case Case compared the present model to the experimental results of Khitrin and olovyeva. The experimental data showed clear trends that were generally reproduced by the model. Computations showed the sensitivity of the solution to a variety of parameters within the model. This section summarizes the ey findings and provides some interpretations of the observations. The present model (which includes mass-transfer) and the plug-flow model showed similar trends although it is clear that mass-transfer effects become more important (after light-off) at low residence-times. For the m/s data, both models predict nearly identical results but diverge for the 4 and 7 m/s cases at higher temperature the PLUG model drastically over predicted the exit conversion for the higher velocities. Mass-transfer effects in the present model limit the conversion at higher temperature for the larger velocity cases. A parameter which can gauge the importance of mass-transfer effects is the transverse Peclet number, Pe T. This parameter, which is the ratio of radial (or transverse) diffusion time to residence time, is an important parameter governing the behavior of catalytic monoliths [8]. For these computations, the Pe T =.5 at m/s and increases to Pe T =.6 and. at 4 and 7 m/s, respectively (see Appendix B). Thus, this liely represents a transition between conditions when mass-transfer effects are important. Both the present model and the PLUG model showed qualitatively similar characteristics with respect to the rate of conversion increase with temperature. Beginning at low temperature, the conversion increases only a small amount as temperature increases. Then, at the light-off temperature, there is an abrupt increase in conversion. After light-off, there is a small temperature range (about K to K wide) 58

74 where larger increases in conversion occur with further increases in temperature. After this range, there is a significant drop-off in the rate of conversion increase with temperature which causes a noticeable inflection point in the data. ince both the present model and PLUG model predict this behavior, this reduction in the conversion increase with temperature is liely caused by the inetics which become limited as a reactant (fuel in this case due to the lean conditions) is depleted. Mass-transfer, however, does influence at what temperature light-off occurs as well as the subsequent inflection point (i.e. decrease in conversion rate increase with temperature) just discussed. Furthermore, there exists an optimum Nu / h combination which maximizes conversion. At higher temperatures, Nu / h, maximizes the conversion, while at lower temperatures, a finite Nu / h produces a maximum conversion. This latter effect is caused by subtle changes in the net adsorption of gas-species (owing to the temperature dependent inetics as well as mass-transfer effects) which produce a concentration of surface species more favorable for light-off at lower Nu / h. The initial surface species distribution influenced the steady-state catalytic lightoff temperature. These different initial conditions, either complete O(s) or CO(s) coverage, led to different time evolutions which ultimately led to different steady-states. The differences were most pronounced near catalytic light-off. For the case of initial O(s) coverage, light-off occurred at the channel inlet and propagated downstream to the final steady-state. With initial CO(s) coverage, the transient simulations showed that light-off began at the channel exit and propagated upstream to its final steady-state. Because of this hysteresis effect, the transient model proved valuable in determining these steady-states. 59

75 Using the CO sub-mechanism from Deutschmann et al.[, 6, 64], the model generally predicted a lower light-off temperature than seen in the experiment. Calculations varying the mass-transfer coefficients via the herwood number did not match the experimental light-off temperature. Also, calculations suggested that the small amount of water vapor present in the experiment did not significantly affect the light-off temperature. Owing to the simplicity of the isothermal channel, the only other obvious parameters which can influence light-off are the surface inetic constants. Reducing the number of active surface sites per area, Γ, did slow the inetics but produced results which were qualitatively different than seen in the experiment. Thus, some adjustment of the inetic parameters in the mechanism is warranted. A sensitivity study on the individual reactions of the dry CO mechanism showed that competition between the net adsorption of CO and O determines the light-off temperature. The reactions most sensitive to perturbations were the CO adsorption step followed by O adsorption. The desorption steps, particularly CO(s), showed sensitivity to the surface initial condition. The calculation showed that increasing the net rate of CO adsorption compared to O delayed ignition to higher temperatures and improved model agreement with experiment. These results suggest that some adjustment of the inetic constants is warranted. 6

76 5. Case : teady-tate Comparisons Monolith 5.. Experiment The second test case for the model comes from the wor of Ullah et al. [9] who conducted a set of experiments using a lean mixture of CO and air in a monolith reactor. This data is useful for examining the parameter a* (apparent catalytic area to geometric surface area) which may be important when modeling a catalytic washcoat. Figure 4 shows a schematic of the monolith configuration which represents the configuration tested in the experiment. The monolith used in this study came from a commercial vendor and very little data (other than geometry) regarding the catalyst material and preparation was available. The paper implies that the substrate was cordierite and that the washcoat consisted of platinum group metals (platinum, rhodium, and palladium) impregnated in alumina. ubstrate mm Washcoat Figure 4. chematic representation of monolith configuration tested in this section. The model represents the center channel of the monolith and is used to characterize the entire monolith performance. The left image is from Kee et al. [] The primary purpose of the experiment was to determine an appropriate masstransfer correlation for use in monolith reactors. The authors reported steady-state CO exit concentration as a function of reactor length for an inlet condition of 6K, a fuel 6

77 concentration of.97 mol/m (balance air), and a volumetric flow rate of 8. cm /s at TP. The monolith reactor started at 5 cm and was subsequently shortened by cutting a portion of the reactor away after each successive test thus achieving a variable reactor length. Data was reported up to a reactor length of cm. For each reactor length, an infra-red gas analyzer measured the gas-composition at the exit of the reactor. In terms of mass (mole) fractions, the reactor feed stream consisted of.5% (.5%) CO and.9% (.5%) O with the balance of 99.% (99.5%) being N. Figure 5 shows a summary of the experimental results. Conc. (mol/m ) x Conversion (%) Reactor Length (cm) Figure 5. Measured CO concentration at the outlet of a commercial monolith from the wor of Ullah et al. [9]. The inlet velocity for an individual channel was 6. m/s at a temperature of 6K. The right axis shows the conversion calculated from the concentration measurements. 5.. Model Parameters The model parameters matched the experimental configuration as closely as possible. imilar to the previous case, the model started from a cold initial condition for the solid (T = K with either Z CO(s) = or Z O(s) = ) and ran to steady-state conditions. In some calculations, the monolith was heated only by the incoming hot (6K) gas. In 6

78 other calculations, the internal heat generation term provided additional heating for a few seconds to raise the monolith temperature to above 7K. In this situation, the monolith was subsequently cooled by the 6K inlet gas and exhibited a different steady-state under certain conditions. The single channel model represents an interior channel of the monolith (Figure 4) and thus uses an adiabatic boundary condition along the outside length of the channel (h ext = W/m /K and ε =). There is heat transfer along the front face of the monolith the heat transfer coefficient (Nu ~ 8) comes from stagnation point theory (Appendix D). The downstream face of the monolith is adiabatic. The channel thermophysical properties used the values of cordierite (neglecting the contribution of the washcoat). The model assumes a thin washcoat (i.e. no washcoat diffusion effects). The parameter a* (apparent to geometric surface area) was a variable adjusted to match the experimental results. Using a cell density of 6 cells / cm, the corresponding individual cross-sectional cell area is.6 x - cm. Ullah et al. states that the nominally.8 mm square monolith channels become approximately circular with a diameter of about. mm after the application of a high surface area washcoat (see Figure 4). Thus, the model calculations used a circular channel with an inner diameter of mm. Table shows a summary of the geometric parameters. The model computations used different channel lengths up of cm, similar to the experiment. For channel lengths smaller than 4 cm, the axial grid resolution was. mm. For the longest channel length of cm, the grid size increased to. mm. The solution was grid independent even at the coarser grid spacing since there were no steep gradients along the length of the channel. 6

79 The flow rate for the entire reactor (8. cm /s) was given at TP conditions. The entire reactor was. cm in diameter (.7 cm in cross-sectional area) and included approximately 8 total channels. The individual channel flow rate was therefore.4 cm /s yielding a channel average velocity of.84 m/s at TP. At the gas inlet temperature of 6K, the channel velocity was 6.47 m/s which was the inlet value used in the calculations. 5.. Model vs. Experiment Figure 6 includes two graphs both as a function of reactor length: the top showing fuel conversion while the bottom shows reactor temperature. Only the data up to 4 cm in length is presented since the majority of the CO conversion taes place in this region (refer bac to Figure 5). The experimental conversion data are shown in blac solid circles. There are no experimental temperature measurements except for the mention that gas temperatures typically rose K. Figure 6 uses several line styles and symbols to represent the model calculations depending on the reactor length as well as heating profile. Lines represent computations using a cm long monolith and show the axial variation along the reactor length (as opposed to exit variables as measured in the experiment). ymbols denote variable monolith lengths (e.g..5 cm,.5 cm,.5 cm, up to 4 cm.) and represent the value at the exit of the reactor. Computations, in which the incoming gas heats the solid, use dashed lines and the plus symbol (+). Computations, using internal heat generation and incoming gas for heating, are shown with solid lines and the triangle symbol ( ). The different colors correspond to different values of a* (ratio of catalytic area to geometric surface area). 64

80 For cases with internal heat generation, two solutions occurred depending on the value of a*. With a* <., the overall reactivity was too slow and only a low-conversion solution occurred (e.g. see the green dotted line corresponding to a* = in Figure 6). For a* >., a high-conversion or ignited solution occurred (see the line denoted Ignited or Mass Transfer Limited Branch in Figure 6). For cases where the reactor was heated only by the incoming 6K gas (dashed lines), Figure 6 shows two possible steady-state solutions for a* ranging between. to roughly. With a* <., only a low-conversion solution occurred. As a* increased, a solution occurred which transitioned from low to high conversion across the length of the reactor. For example, computation with a*= reached near % conversion between and 4 cm of reactor length. For a* = 5, the reactor achieved near % conversion by roughly cm. With a*, only solution was again possible which matched the high-conversion branch results seen with internal heat generation. The bottom graph of Figure 6 shows the computed steady-state temperatures of the solid which correspond the conversion data above. This graph shows that the solid temperature near the inlet is the highest (~67K) for the ignited or mass-transfer limited solutions. Because of the high temperature, the reactor lights off near the inlet and releases sufficient heat to maintain the higher surface temperature (despite being cooled by the incoming 6K gas). For the cases heated by only the incoming flow (with a*>. but less than ~), a gas-temperature of 6 K is insufficient to allow the inlet region of the monolith to heat up to achieve light-off without external heating. Light-off does occur further downstream for these cases. Upstream axial heat conduction, however, is insufficient to warm the inlet region of the monolith. 65

81 8 Ignited or Mass- (a* >.) Transfer Limited Branch a* = a* = 5 Nu = 4.64 Experiment Conversion (%) 6 4 a* = Kinetic Limited Branch Model Gas Heat Int. Gen. L=cm L=Var. a* = - Both Gas Heating and Internal Heat Generation 4 68 a* = olid Temperature (K) a* = 5 a* = a* = 6 Channel Length (cm) Figure 6. Computed results showing the effect of varying a* on the conversion of CO as a function of reactor length. The experimental data is from Ullah et al. [9]. The top graph shows the conversion while the bottom graph shows the corresponding solid temperatures. 4 66

82 Figure 6 also shows calculations using a variable reactor length (symbols). These values correspond to the exit of the reactor. The variable reactor length calculations better represent the experiment, however, require separate calculations to generate each datum and, thus, only a few are computed. These calculations showed slightly different results compared with the profiles of a cm reactor. At the same axial location, both the conversion and temperature were lower for the short reactors compared with the cm reactor. This is due to solid phase axial conduction and upstream heat loss to the face of the monolith which lowers the temperature across the shorter monoliths. At axial locations near the inlet, the longer monolith receives heat from downstream catalytic reactions via solid axial conduction, whereas, the shorter monoliths are cut-off and do not achieve complete conversion and thus release less heat. The low and high conversion branches of Figure 6 correspond to regimes of inetically and mass-transfer limited surface reactions, respectively. With a*<., only a inetically limited solution occurred. With a* >., the monolith reactor was in a masstransfer controlled regime for all cases beyond about 4 cm. For cases with a* >. but less than about, the reactor could be inetically controlled at lengths less than 4 cm but transitioned to mass-transfer control further downstream as long as the inlet region did not receive any additional heating. For a* >, the inetics were sufficiently fast and only a mass-transfer limited branch solution occurred irrespective of the heating history. Figure 7 shows the calculated temperature and select species profiles along the channel length up to 4 cm (from the L = cm calculation). The model parameters are a*= and Nu=4.64 with the corresponding h. The experiment reported a maximum gas-temperature rise of approximately K. With complete CO conversion, the model 67

83 predicted a gas temperature rise across the monolith of 48K (see green line in Figure 7). This is not unexpected since the experiment did not achieve complete CO conversion and also that there could be some heat loss from the exterior channels of the monolith in the experiment. T (K) Y x a*= Nu=4.64 Transitioning from Kinetic to Mass-Transfer Control t =. sec. Mass-Transfer Control t =. sec. teady Trans. Gas olid Bul Wall t = 46.6 sec. CO O Z -4-7 Z CO(s) Z O(s) Z Pt(s) Channel Length (cm) Figure 7. Temperature and select species mass and site fraction as a function of monolith length up to 4cm (the calculation used cm). The model parameters are a*= and Nu= The data in Figure 7 show that the CO mass fraction at the wall is approximately 9% of the bul value at the inlet and decreases to roughly.% at a channel length of.5 cm (similarly for the O mass-fraction). Per the definitions of Hayes and Kolaczowsi [] presented earlier, the monolith reactor is operating in a inetically controlled regime at the inlet but becomes completely mass-transfer controlled by.5 cm. Although difficult to see in the figure, the monolith remains mass-transfer controlled with the wall concentration remaining below % of the bul value along its remaining length. 68

84 The computed steady-state results showed no sensitivity to the initial surface species distribution when comparing a surface completely covered by O(s) versus CO(s). This result was true for both the inetic and mass-transfer limited computations presented. Based on comparisons to the previous case, this result is not surprising when the reactor was in a mass-transfer controlled regime. For inetically controlled reactions, however, the reason for this was not immediately clear. Examination of the transient simulations leading to the steady-state, however, revealed that light-off occurred similarly for all cases heated solely by the incoming gas. In the previous case, the different steadystate solutions were accompanied by different light-off positions in the channel. For the case shown Figure 7, light-off occurred just downstream of the inlet (at about cm) when the surface reached about 65K and then propagated upstream due to solid heat conduction (see temperature history of solid in Figure 7). imilar transient behavior occurred for all the computations heated by the inlet gas irrespective of the surface initial condition. Figure 8 shows the computed temperature and species profiles using the parameters a*= and Nu=4.64 (i.e. completely mass-transfer limited). All of the cases on the mass-transfer limited branch showed results virtually identical to these. Light-off occurred nearer the inlet (in this case at L ~.5 cm) and only needed a short distance to propagate to exactly the inlet. The final steady-state surface concentration at the inlet is approximately % of the bul value. These results confirm that the entire monolith reactor is operating under mass-transfer control. This emphasizes the need to include mass-transfer in the computations. 69

85 T (K) Y x a*= Nu=4.64 t = 8.6 sec. t = 7.9 sec. Bul Wall teady Trans. Gas olid CO O - Z -4 Z CO(s) Z O(s) Z Pt(s) -7 Channel Length (cm) Figure 8. Temperature and select species mass and site fraction as a function of monolith length up to 4cm (the calculation used cm). The model parameters are a*= and Nu=4.64. In the previous cases of this section, the model generally over-predicted the exit conversion after roughly.5 cm under mass-transfer limited conditions (see Figure 6). A single Nu and h describes both the heat and mass transfer coefficient since, in these cases, the entry affects are limited to very short distances in the channel (~ fully developed by 5mm). Figure 9 shows the effect of changing the Nu / h on the steadystate conversion using a* = along a cm reactor. The profiles do not match precisely for any Nu / h combination, although there is better agreement with a Nu < This is in agreement with the correlation suggested by Ullah et al, which estimates a global herwood number for CO to be.75 [9]. 4 7

86 Conversion (%) Experiment Nu = 4.64 Nu = Nu = Nu = Channel Length (cm) Figure 9. Effect of Nu / h on the steady-state conversion of CO as a function of reactor length. The herwood number comes from the analogy of heat and mass transfer. The computations use a*= and correspond to the mass-transfer limited or high-conversion solution Discussion of Case The parameter a* can be interpreted as either a surface area enhancement as discussed previously or simply a inetic adjustment factor which accelerates the inetics by the factor a* (mathematically they are equivalent see equations 5, 7, and 8 in the Model Description). In terms of a surface area enhancement, a factor of to times the geometric surface area is not an unreasonable number. In fact, numbers as large as 5 have been reported in the literature [4]. The results in this section showed that, no matter the heating history of the monolith, values of a* greater than about ensured that the entire length of the reactor was mass-transfer limited at steady-state. If a* is interpreted as a surface area enhancement factor, the current analysis can only conclude that the surface area enhancement is greater than about times the geometric surface area. This assumes that the monolith is, in fact, operating in a mass-transferred limited 7

87 branch. Figure 6 shows solutions with a* less than that agree reasonably with the experiment. These are, in turn, inetically limited near the reactor inlet. The parameter a* may also be a simple inetic adjustment factor. The authors provided little information about the catalyst and only allude to platinum group metals in their wor. Therefore, the surface inetic mechanism, explicitly formulated for platinum, may not be appropriate. A value of a* =.4 can provide reasonable agreement with experiment if the surface is initially heated to an elevated temperature. Multiplying the surface reaction rates by.4 is not an unreasonable adjustment given the unnown catalyst. 7

88 5. Case : Transient Propagation ingle Horizontal Platinum Tube 5.. Experiment Miller et al. [44, 45] conducted several experiments that showed a catalytic reaction front propagating along the inside of a platinum tube (see Figure ). The reactions were catalytic since tests using stainless steel tubes with similar dimensions did not produce internal flames. The platinum tube (99.95% purity) was 4. cm in total length (.5 cm outside a support fitting), with inner and outer diameters of.74 and.95 mm, respectively. The inlet velocity varied from to m/s. The fuel was carbon monoxide in pure oxygen with an equivalence ratio ranging from. to. The gasses were 99.5% pure CO (Matheson Co., CP Purity with typically less than 5 ppm. water but can be as high as 5 ppm) and 99.5% pure O (Air Products Co. industrial grade with up to 7.8 ppm water vapor). In some cases, a bubbler deliberately saturated the inlet gas with water vapor. The bubbler was a porous sphere submerged in a sealed container of water. The flow system diverted each gas through a separate bubbler. The gasses formed small bubbles on the exterior of the porous sphere and subsequently broe away floating to the surface before continuing to the Pt channel. Calculations show that the gas saturates with water vapor during the bubble s ascent to the water s surface. The experiment began with a room temperature pre-mixed gas flowing through the Pt tube also at room temperature. Figure shows images of the ignition and propagation sequence for an inlet flow condition of φ = and m/s. Figure a shows the illuminated platinum tube attached to the flow system using a support fitting. A Kanthal igniter (hotwire), dissipating approximately.4 watts (9.9 V across the wire drawing.5 amps), externally heated the last % of the tube near the outlet for 7

89 seconds (Figure b) an indicator light visible in the images shows when the igniter is powered. After seconds, the test operator manually translates the hotwire away from the tube which was glowing dull orange. The glowing region then propagated upstream along the channel at a slow speed (~ mm/s). Figure c shows the propagation of the catalytic flame midway along the tube while Figure d shows the catalytic flame stabilized inside the channel near the inlet. The mechanism for flame stabilization is heat loss to the large fitting at the inlet. Fitting.5 cm a Kanthal igniter Igniter indicator light b Propagating Catalytic Flame c Anchored Catalytic Flame d Figure. Ignition and propagation sequence of a CO / O (φ = ) catalytic reaction along the inside of a platinum tube (.74 mm ID,.95 mm OD). The inlet gas velocity is m/s. 74