Luận án tiến sĩ: Modeling of catalytic channels and monolith reactors

Modeling of Catalytic Channels and Monolith Reactors Abstract by PETER M STRUK This dissertation presents a “lumped” two-phase solid and gas model of transient catalytic combustion sui

Channel Geometry

Typical geometries in which surface reactions occur is that of a packed-bed or monolith reactor The catalytic packed-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 work 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 alike There are, of course, exceptions Researchers have explored heat loss effects near the periphery [13, 36] and 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 work looks 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 2.5.

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 [1] 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 [33] compares both lumped and distributed models of monolith catalytic combustors Examples of the lumped category include radial-averaged models [34-36] and even axially averaged models [37] The popular plug-flow model, PLUG [30], is an example of a lumped model There are many distributed models in the literature and many of these include calculated bulk heat and mass transfer coefficients suitable for use with lumped models [32, 43, 47-51] Raja et al [32] evaluated the various flow modeling assumptions including Navier-Stokes, boundary-layer, and plug- flow 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- transfer resistance between the bulk gas and reacting surface – this is the approach taken in this work

Heat and mass transport coefficients are typically expressed in dimensionless form in terms of the Nusselt (equation 1) and Sherwood (equation 2) numbers, respectively Many engineering applications use a constant value of Nu and Sh to estimate the heat and mass transfer between a bulk 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 = 3.655 for a constant temperature wall boundary condition For a constant heat flux wall boundary condition, the same analysis yields Nu = 4.364 k d

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 Shk numbers correlations have been routinely used in the past and have provided reasonable agreement between model and experiment Hayes and Kolaczkowski [1] as well as others [22, 33, 37-41, 52] discuss the selection of heat and mass-transfer coefficients for catalytic monoliths

The majority of computations in this work 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 [33] The value of the Sherwood number comes from the heat and mass transfer analogy shown in equation 3 [53] In this way, there is some distinction between the relative effectiveness of diffusion based on the species Lewis number A major focus of this work was to explore the sensitivity of the model results to variations in Nu and Sh

Gas-Phase Quasi-Steadiness

While steady-state models date back to the work of Khitrin and Solovyeva [54], transient models began appearing almost 20 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 works was a quasi- steady 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 S C S A S is typically very small so that neglecting the accumulation term in the overall energy balance on system causes minimal error [55].

Chemistry Modeling

Up until the early 1990’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- 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 1980 through 1990 saw the development of the CHEMKIN [60, 61] and Surface CHEMKIN [62] 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 [19], Deutschmann et al [21, 63, 64], Bond et al [20], 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 taking place

Heterogeneous reactions occur on a solid surface and exchange molecules with an adjacent fluid In this work, 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 back into the gas leaving a vacant surface site Furthermore, unreacted surface species may also desorb back into the gas-phase Kee et al [31] provide an excellent description of this process from a combustion perspective The textbook uses the framework set forth by Surface CHEMKIN [62] and is the convention adopted in this work The specific chemical mechanisms used in this study are presented and discussed in Appendix E

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, cracks 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 [19] and Deutschmann et al [63], estimated a value of Γ = 2.7066 x 10 -9 mol / cm 2 (1.63 x 10 15 sites / cm 2 ) from the atomic surface density of polycrystalline platinum and this value has been used subsequently by numerous studies [13, 20, 32, 64, 69] Hwang et al [70], however, claimed that this value has not been verified for accuracy and goes on to use a value of Γ = 1.757 x 10 -10 mol / cm 2 in their modeling work 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 work uses the CO sub-mechanism from Deutschmann’s CH4 and O2 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

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 Light- off is usually the onset of significant reactant conversion and is often (but not always) associated with a rapid rises in temperature [1] Furthermore, light-off is frequently characterized by the (typically abrupt) transition from kinetic to mass-transfer controlled surface reactions [40] Exceptions to latter are cases where a plug flow model is applicable and the reactions remain kinetically controlled Hayes and Kolaczkowski [1] provide some rough guidelines to help identify kinetic 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 bulk concentration then the reaction is kinetically controlled Similarly, if the surface concentration of the limiting reactant is less than 5% of the bulk then the reaction is mass-transfer controlled

In between, both mass-transport and kinetics are important The present work explores regimes which are kinetically or mass-transfer limited as well as cases which likely are affected by both kinetics and mass-transfer.

Solid-Phase Heat Transfer

Solid-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 [34, 38, 42, 43] Recently, a catalytic flame has been observed to propagate along a platinum channel due to solid-phase axial conduction [44, 45] Ramanathan et al

[38] defined an axial heat conduction Peclet number, Peh, and showed its effect on the light-off behavior of a monolith including both transient and steady-state results Similar to a traditional gas-phase Peclet number, the importance of axial heat conduction in the solid increases as Peh 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 [43, 71-73] The general consensus from the literature is that both solid-phase heat conduction and radiation exchange tend to affect the solid temperature by lowering peak values and broadening profiles This in turn can affect the performance of the monolith Hayes et al [43] 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.

Solution Methods

Typically, transient models (both lumped and distributed) require the solution of a system of partial differential equations (PDE) which can be computationally expensive

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 LSODE [76], LIMEX [77], DASSL [78] and DASPK [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.

Summary

The goal of this work is to develop a transient model which is applicable to both single catalytic channels as well as monolith reactors Based on the background presented, there are basic elements required to adequately model the relevant physics of a general catalytic channel These include (1) mass-transfer effects from the gas to the surface, (2) a solid phase which includes axial conduction, and (3) 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 compared to global chemistry The model can include the following simplifying assumptions 3 and still adequately capture the relevant physics: (1) lumped one- dimensional model with transverse heat and mass transfer coefficients and (2) gas-phase quasi-steadiness relative to a transient solid Such a model with detailed chemistry is notably lacking 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)

3 Appendix B provides further assumptions based on a detailed time-scale analysis of the physical processes in a catalytic channel

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 Bulk 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 bulk temperature and mass fraction at each axial location The model uses detailed kinetic 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 bulk material properties There are KS 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.

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 uA m& =ρ = 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 1 st term corresponds to energy advection down the channel while the second term is due to heat release from homogeneous reactions The 3 rd term of Equation 5 represents the sensible enthalpy change between adsorbing and desorbing species and involves the reaction rate, , of gas-phase species by surface reactions A positive represents the production (desorption) of gas-phase species due to surface reactions – these species enter the control volume at the surface temperature Thus, h s& k s& k k is evaluated at the surface temperature (T k ′ = T S ) if > 0 A value of < 0 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 bulk gas temperature ( = T) The parameter a* accounts for the potential surface area s& k s& k

T k ′ 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

Equation 6 shows the conservation of mass species This equation states that the change in mass of gas-phase species k along the length of the channel (term 1) comes from two sources: surface reactions, via lateral/radial mass-diffusion into or out of the control volume (Equation 7) and gas-phase reactions, ω s& k k

In equation 6, there are two values of gas-phase species: a bulk flow value, Y k , and a value adjacent to the catalytic surface, Y kW , 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 bulk mass properties of the control volume The flux of each species k 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, (adsorption or desorption), of gas-phase species are evaluated using the wall concentrations and surface temperatures as opposed to the bulk 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 657-661 [31] s& k

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 2) is equal in magnitude to term 4 of Equation 5 using appropriate heat transfer correlations External heat loss due to natural convection (term 3) 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 3 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 3 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 S , heat capacity,

C S , and thermal conductivity, k S, are all constant in the model

The effective catalytic surface area (often due to a high-surface area washcoat) can be different from the geometric surface area S∆x (see exploded view in Figure 34 of Appendix A) This can be achieved, for example, by a thin deposition of catalytic washcoat on a catalytically inert substrate material For thicker depositions, pore diffusion effects (e.g see [22, 50, 80]) need to be considered but are not accounted for in 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*>1) each kinetic pre-exponential factor by the factor a*

In this study, a* is constant Oh and Cavendish [34] 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 work, 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 1 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 11, 12 and 15 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 k (ratio of adsorbed species k to the total number of active sites at a specific axial location) Surface 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 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 1 during the entire integration provided that the initial condition sums to 1 and the reaction mechanism is balanced (hence 0)

Equations 10 and 11 show the molar rates of production of species due to gas- phase and surface reactions, respectively The rate of production of species k comes from summing the rate of progress of each reaction (forward and reverse) involving species k (Equation 12) For the gas-phase, the reaction rate constants, k fi and k ri , are evaluated at the bulk temperature while for surface reactions, k fi and k ri are evaluated at the surface temperature The various expressions for k fi and k ri , which typically take on an Arrhenius form, can be found in Chapters 9.3 and 11.6 of Kee et al [31] 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 sticking coefficients to the Arrhenius form

= ; for homogeneous reactions using bulk species (13)

= ; for surface reactions using gas species adjacent to surface, (14)

=Γ ; for surface reactions involving surface species (15)

For gas-phase reactions, the concentrations in Equation 12 are computed using the bulk mixture temperature and mass-fractions (equation 13) For surface adsorption and desorption reactions, the gaseous and surface concentrations are evaluated using the surface temperature and wall mass-fractions (equations 14 and 15, 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 bulk 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

Overview

In general, the dependent variables (T, T S , Y k , Y kW , and Z k ) 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 (solid- phase energy balance) are directly coupled to the gas-phase variables Y k 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 2) thus forming a large system of ordinary differential-algebraic equations (DAE) for the solid

Since 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 kinetically 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, took 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 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 (TS, Z k , Y kW , and ) were fixed at the particular instant in time The integration, which proceeds across each gas control volume beginning at the inlet, takes the necessary spatial step to handle stiff regions (i.e ignition) Such 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 s& k

The inclusion of detailed gas-phase and surface chemistry introduces numerical difficulty via mathematical stiffness Stiffness 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 package DASSL [78, 81] For larger scale problems (i.e those involving many equations, such as this case), the DASPK code [79], which was derived from DASSL, is more appropriate Because of the potential for a large number of equations arising from the solid phase discretization, the model used DASPK as the solver DASPK discretizes the differential variables of the ODEs using up to a 5 th order backward Euler method DASPK then solves the resulting system of nonlinear algebraic equations using a preconditioned General Minimal Residual iterative method [79]

Integrate Gas Phase Equations (From x=0 to L)

Figure 1 Flowchart of basic solution algorithm

Figure 1 shows the basic solution algorithm for the computer program After specifying the surface / solid initial condition, DASPK integrates the gas-phase equations (5-6) along the channel from x=0 to L (solid/surface-phase constant) Except after the very first spatial integration (t’ = 0), the program checks whether or not the gas-phase has converged (compared to the previous values checked 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 DASPK then performs an integration of the transient surface / solid equations (7-9) from t’=t to t + ∆t (gas-phase constant) Subsequently, 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

Integrate Solid Phase Equations (From t’= t to t+∆t)

Revert solid to values at t’ = t yes no

Store data at t’ transient integration interval ∆t The interval solution’s dependence, or more specifically independence, on ∆t is checked for calculations which require time accuracy For the calculations presented in this dissertation, a ∆t = 0.01ãτS provided a time-accurate (i.e independent of ∆t) solution The parameter τS is a characteristic time-scale of the solid based on internal heat transfer and is typically on the order of milliseconds (Appendix B).

Spatial Integration in x

The catalytic tube is discretized spatially as shown in Figure 2 There are a total of Kg + 1 gas-phase ODE equations That is one gas-phase energy equation (5) and Kg species equations (6) The gas-phase integration begins at the inlet (x=0 or iG=1) with specified boundary conditions for T and Y k The DASPK 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 k /dx, at the inlet which can be non-zero, particularly if reactions are taking place near the inlet DASPK computes these automatically using special algorithms and is discussed below

The spatial integration proceeds from x=0 (iG=1) to x=∆x2 (iG=2) using the spatial distribution of wall parameters T S , Y kW , Z k (at iS = 2)as well as , hs& k T , and h Dk 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 DASPK chooses the spatial step until reaching x=∆x2 (iG=2), where the solution components for T and Y k are output to a file The x-integration proceeds in this fashion to the next node at x=∆x2+∆x3

(iG=3) and repeats until reaching the end of the channel x=L (iG=n) For all computations presented, tests with various ∆x verified that the solution was grid independent

Figure 2 Discretization of catalytic channel into finite volumes Solid phase nodes (shown in black) represent the entire cross-sectional volume The inlet is at x=0.

Integration in Time

The solid / surface equations form a large matrix, n-1 times the sum Kg + KS +1, of ordinary differential-algebraic equations DASPK integrates these equations forward

C L bulk gas gas at the wall i G = i - 1 solid phase

… … … x = 0 =∆x 2 L in time by an interval ∆t assuming that the bulk gas-phase values remain constant during that interval For the time integration, the gas-phase values Y k , and T are evaluated at the center of the adjoining control volume (midpoint between iG = i and iG-1 = i -1) as shown in Figure 2 The user specifies the time interval ∆t while the solver DASPK 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 S = 300K) This procedure was necessary because the surface site fraction distribution (and corresponding wall mass-fraction) was not known 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 making it difficult to know the distribution a priori At low temperatures, the integration by DASPK was largely insensitive to the specification of initial surface site fraction unless a large fraction of unoccupied sites was specified (i.e ZPt(s) >> 0, which is physically unrealistic at low temperatures) causing integration errors to occur In this work, calculations started with either complete surface coverage by O(s) or CO(s).

Solver DASPK

The DASPK 5 code solves both the gas-phase system of ODEs and the solid / surface system of DAEs using the Krylov iterative method [79] DASPK is designed for general index-1 (or index-0) 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 unknowns [82] In the present system of equations, the gas-phase equations (5-6) are all ODEs, hence index-0 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-1)

DASPK 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 backward difference) at each time step in an iterative fashion using a preconditioner 6 In this work, the program used a general- purpose preconditioner (subroutines DBANJA and DBANPS) supplied with the DASPK 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 (iS+1and iS-1) influence any given solid control volume, iS (see Figure 2) The solver DASPK takes 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 worked for the solution

5 The DASPK code (this work used version 2) can be found on a web site sponsored by the Computational Science and Engineering department of the University of California, Santa Barbara[92]

6 A preconditioner is an approximation to the iteration matrix, used in Newton’s method, which may lead to a computationally less-expensive solution 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 16 where f j = T, T S , Y k , Y kW , or Z k DASPK 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 errj 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 5.3 of Brennan et al [81] recommends that the value of ATOL j be the value where the solution component is essentially insignificant and the value of

RTOL j=1.0x10 -(m+1) 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 1 The algebraic variables, Y kW , used more significant digits than the differential variables which seemed to facilitate the calculation of consistent initial conditions as discussed below j j j j RTOL abs f ATOL err = * ( )+ (16)

Table 1 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 S , Y k , Y kW , and Z k 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 bulk gas are specified (e.g T and Y k ) The gradients at the inlet, however, are not known (and not-necessarily zero) Similarly, the initial conditions are specified for the differential variables of the solid (T S and Z k ) Their initial derivatives as well as the algebraic variable, Y kW , also are not known The wall mass-fractions cannot be the same as the bulk when surface reactions are taking place ( ≠ 0, see Equation 7) DASPK automatically calculates consistent initial conditions (i.e dT/dx, dY s& k k /dx, dTS/dt, dZ k /dt, and Y kW ) given the initial differential variables [83] Early results showed that DASPK requires good initial guesses for the automatic routine to calculate consistent initial conditions, particularly for the algebraic quantities Y kW In the solution procedure, the initial “guessed” values for Y kW come from equation 7 where the inlet bulk mass-fractions, Y k , are used to compute For non-reacting conditions, the values of Y s& k kW ≅ Y k.

The numerical model presented herein predicts both steady-state and transient phenomena from 3 different experimental configurations all using CO as fuel Table 3 presents a summary of the modeled cases Cases 1 and 2 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 1 and 2 is the experimental configuration: Case 1 is an isothermal horizontal channel while Case 2 is a monolith reactor Case 3, 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 1 and 3, 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 2 Case 1 further presents a global sensitivity study on the dry mechanism as well as the as the catalytic surface site density Case 2, 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 work of Davis et al.[84] The surface CO mechanism came from the CH 4 /O 2 on platinum proposed by Deutschmann et al.[21, 63, 64] Both the homogeneous and heterogeneous mechanisms are presented in their entirety in Appendix E area Finally, case 3 looks at the effect of external heat transfer on the predicted catalytic reaction front propagation as well as surface temperature

Homogeneous Reactions from Davis et al [84]

Heterogeneous Reations from Deutschmann et al [21]

58 24 Pt(s) + CO(s) → C(s) + O(s) 1E+18 0 184000 a Denotes the use of third body efficiencies which can be found in the source b Denotes pressure dependent reactions c Denotes sticking coefficient listed in column A i d The order of CO adsorption is 2 with regard to Pt(s)

Table 2 Dry CO / O 2 sub-mechanism on platinum from the work of Deutschmann et al [21, 63, 64]

Steady / Transient Steady-State Steady-State Transient

Configuration Horizontal Tube Monolith Horizontal Tube

Catalyst Pt Pt Washcoat Pt

Parameter(s) Compared Fuel Conversion Fuel Conversion Rxn Front Propagation;

Conditions Isothermal Adiabatic Radiative and

Convective Loss Ignition Method Hot Isothermal Channel Hot Inlet Gas Joule heating:

2.5W for 3 seconds over last 10% of channel

Oxidizer / Diluent Air Saturated at 283 K O 2 / N 2 Dry & Saturated O 2

Reynolds’ number, Re D 60 – 3100 (see Table 5) 116 95

Cross-Section Circular Circular Circular

Table 3 Summary of experimental configurations modeled in this work Values denoted with an asterisk (*) were not explicitly stated in the reference but were assumed The ambient pressure surrounding the channel for case 3 (denoted by ** ) was slightly less-than 1 (≈0.97 atm)

Case 1: Steady-State Comparisons – Isothermal Platinum Tube

Experiment

The first comparison of the model results is to the steady-state results from the experiment of Khitrin and Solovyeva [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 150 mm long platinum cylindrical channel with a 2 mm inner diameter An electric heater kept the channel isothermal at temperatures ranging between 573K and 950K The velocity in the channel varied between 3 and 70 m/s with 3 discrete values compared in this work (3, 34, and 70 m/s)

The fuel was carbon monoxide in air tested at two equivalence ratios: 0.0746 (3%

CO by volume) and 0.1269 (5% CO by volume) The comparisons only use the 3% by volume cases since there were only slight differences between the 3% 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 283K 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 3 in the reference.

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 model ran from a cold initial condition (T S = 300K) to steady-state conditions The surface initial condition was either ZCO(s) = 1 or ZO(s) = 1 Despite the exothermic catalytic reactions, the channel temperature did approach isothermal conditions by setting the external heat-transfer coefficient to a very high value (hext= 10,000 to 500,000 W/m 2 /K) The external heat transfer coefficient was adjusted to yield a maximum temperature difference along the channel of less than 2K for the gas and 4K for the solid when compared to the inlet gas-temperature The curves shown in Figure 3 through Figure 13 were generated by running the model to steady-state temperatures from 550K to 950K in 10K 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 2.1 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* = 1.0 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 2.94% (3%) carbon monoxide, 22.43% (20.2%) oxygen, and 74.63% (76.8%) nitrogen at 1 atm pressure.

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 3, 34, and 70 m/s (Figure 3) This figure presents three conversion versus temperature graphs which are stacked vertically corresponding to 3, 34, and 70 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, τL, in the channel (based on inlet conditions) Figure 3 shows three separate calculations for each velocity These include calculations using 2 different surface site distributions (ZO(s) or ZCO(s) = 1) as well as from the plug-flow model The computations in Figure 3 use dry CO chemistry and both the Nusselt and Sherwood numbers are constant with Nu = 4.364 and Shk based on the heat and mass-transfer analogy (Table 4)

Figure 3 Comparison of steady-state model results (including a plug-flow model) to the experiment of Khitrin and Solvyeva [54] for 3 channel velocities The inlet gas consisted of 3% CO (by volume) with the balance being air

The experimental data in Figure 3 show little conversion at low temperatures Significant conversion (i.e light-off) begins at a temperature of approximately 610 K,

680 K, and 700K, for 3, 34, and 70 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 10K to 20K wide and is most obvious for the 34 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 3 includes computations from a plug-flow model which neglects all diffusive terms The plug-flow computations presented herein are from the PLUG code [30] which is part of the CHEMKIN software package 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 ZCO(s) ≅ 0.97 for 500 K and ZCO(s)

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 10 K to 30 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 50 K to 80 K

PLUG significantly over predicted the exit conversion at higher temperatures especially for the faster velocities The result is not surprising because the surface reactions are not mass-transfer limited in PLUG which subsequently allows significantly more conversion to occur [32] 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 likely occurring because the finite rate mass-transfer in the latter case limits the transport of CO to the surface This allows O2 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 3 is the similarity of the PLUG solution to the model with mass-transfer for the 3 m/s case This result is likely due to the significantly longer residence time in the channel for the 3 m/s case compared with the faster velocities For the 34 and 70 m/s cases, the residence times are of the same order as the lateral (radial) diffusive mass-transport time scales

Figure 3 also shows the effect of the initial surface-site occupancy on the solution for two initial surface conditions: ZCO(s) = 1 and ZO(s) = 1 For the case of ZCO(s) = 1, the light-off temperature is approximately 20K higher than the ZO(s) = 1 case This effect is likely occurring because of the significantly higher sticking coefficient of CO relative to

O2 (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 34 m/s case with O(s) initially occupying all surface sites The species presented are the bulk 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 610K corresponds to just prior to light-off (and significant fuel conversion) From the upper most graphs of Figure 4, the data for 610K show that the wall mass-fraction is only slightly smaller (~ 0.96 times) than the bulk value along the majority of the channel length except for very near the outlet (x cm) Here there is an abrupt decrease in the wall mass-fraction of CO (~0.42 of the bulk) – it is at this point that the catalytic reaction rate becomes large (i.e light-off occurs) causing a transition from a kinetically controlled to a mass-transfer controlled reaction rate As temperature increases, the abrupt decrease in wall mass-fraction shifts upstream and, at 700 K, the wall mass-fractions is entirely the lower value The data in Figure 4 show that the bulk 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

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 34 m/s case (Nu = 4.36) at 4 temperatures The initial condition for the surface is Z O(s) = 1

The transition from kinetic 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 channel until approximately 600K 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 ZCO(s) = 1 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 3

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 640K, 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 kinetically controlled as fewer reactants remain and the bulk concentrations approach the wall values

The results in Figure 3, consistent with literature, show that catalytic reactions can be mass-transfer limited To this point, the calculations used a single Nusselt number (Nu=4.364) while the Sherwood numbers, which vary with species (see Table 4), came from the analogy of heat and mass transfer (equation 3 in the Background) The property values used in the calculation of Nu and Sh are based on the inlet mixture composition and do not vary significantly within the temperature range of the experiment The value of Nu=4.364 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 [37, 38] 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

Table 4 Calculated Sherwood numbers for each species using the analogy of heat and mass transfer with Nu=4.364 The property values are based on the inlet mixture (3% CO by volume in air with saturated water vapor) at 850K

Discussion of Case 1

Case 1 compared the present model to the experimental results of Khitrin and Solovyeva 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 key 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 3 m/s data, both models predict nearly identical results but diverge for the 34 and 70 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, PeT 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 [38] For these computations, the PeT = 0.05 at 3 m/s and increases to PeT = 0.6 and 1.0 at 34 and 70 m/s, respectively (see Appendix B) Thus, this likely 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 10K to 30K wide) 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 Since both the present model and PLUG model predict this behavior, this reduction in the conversion increase with temperature is likely caused by the kinetics 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 / Shk combination which maximizes conversion At higher temperatures, Nu / Shk → ∞, maximizes the conversion, while at lower temperatures, a finite Nu / Shk produces a maximum conversion This latter effect is caused by subtle changes in the net adsorption of gas-species (owing to the temperature dependent kinetics as well as mass-transfer effects) which produce a concentration of surface species more favorable for light-off at lower Nu / Shk

The initial surface species distribution influenced the steady-state catalytic light- off 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 2 steady-states

Using the CO sub-mechanism from Deutschmann et al.[21, 63, 64], the model generally predicted a lower light-off temperature than seen in the experiment Calculations varying the mass-transfer coefficients via the Sherwood 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 kinetic constants Reducing the number of active surface sites per area, Γ, did slow the kinetics but produced results which were qualitatively different than seen in the experiment Thus, some adjustment of the kinetic 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 O2 determines the light-off temperature The reactions most sensitive to perturbations were the CO adsorption step followed by O2 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 O2 delayed ignition to higher temperatures and improved model agreement with experiment These results suggest that some adjustment of the kinetic constants is warranted.

Case 2: Steady-State Comparisons – Monolith

Experiment

The second test case for the model comes from the work of Ullah et al [39] 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 14 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

Figure 14 Schematic 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 [31]

The primary purpose of the experiment was to determine an appropriate mass- transfer 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 623K, a fuel

WashcoatSubstrate concentration of 097 mol/m 3 (balance air), and a volumetric flow rate of 183.3 cm 3 /s at STP The monolith reactor started at 15 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 12 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 0.5% (0.5%) CO and 0.29% (0.25%) O2 with the balance of 99.21% (99.25%) being N2 Figure 15 shows a summary of the experimental results

Figure 15 Measured CO concentration at the outlet of a commercial monolith from the work of Ullah et al [39] The inlet velocity for an individual channel was 6.23 m/s at a temperature of 623K The right axis shows the conversion calculated from the concentration measurements.

Model Parameters

The model parameters matched the experimental configuration as closely as possible Similar to the previous case, the model started from a cold initial condition for the solid (T S = 300K with either ZCO(s) = 1 or ZO(s) = 1) and ran to steady-state conditions

In some calculations, the monolith was heated only by the incoming hot (623K) gas In other calculations, the internal heat generation term provided additional heating for a few seconds to raise the monolith temperature to above 700K In this situation, the monolith was subsequently ‘cooled’ by the 623K inlet gas and exhibited a different steady-state under certain conditions

The single channel model represents an interior channel of the monolith (Figure 14) and thus uses an adiabatic boundary condition along the outside length of the channel (hext= 0 W/m 2 /K and ε =0) 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 62 cells / cm 2 , the corresponding individual cross-sectional cell area is 1.613 x 10 -2 cm 2 Ullah et al states that the nominally 1.08 mm 2 square monolith channels become approximately circular with a diameter of about 1.0 mm after the application of a “high surface area” washcoat (see Figure 14) Thus, the model calculations used a circular channel with an inner diameter of 1 mm Table 3 shows a summary of the geometric parameters The model computations used different channel lengths up of 12 cm, similar to the experiment For channel lengths smaller than 4 cm, the axial grid resolution was 0.1 mm For the longest channel length of 12 cm, the grid size increased to 1.2 mm The solution was grid independent even at the coarser grid spacing since there were no steep gradients along the length of the channel

The flow rate for the entire reactor (183.3 cm 3 /s) was given at STP conditions The entire reactor was 1.3 cm in diameter (1.327 cm 2 in cross-sectional area) and included approximately 82 total channels The individual channel flow rate was therefore 2.24 cm 3 /s yielding a channel average velocity of 2.84 m/s at STP At the gas inlet temperature of 623K, the channel velocity was 6.47 m/s which was the inlet value used in the calculations.

Model vs Experiment

Figure 16 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 takes place in this region (refer back to Figure 15) The experimental conversion data are shown in black solid circles There are no experimental temperature measurements except for the mention that gas temperatures typically rose 33K Figure 16 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 12 cm long monolith and show the axial variation along the reactor length (as opposed to exit variables as measured in the experiment) Symbols denote variable monolith lengths (e.g 0.125 cm, 0.25 cm, 0.5 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)

For cases with internal heat generation, two solutions occurred depending on the value of a* With a* < 2.3, the overall reactivity was too slow and only a low-conversion solution occurred (e.g see the green dotted line corresponding to a* = 1 in Figure 16) For a* > 2.3, a high-conversion or “ignited” solution occurred (see the line denoted

“Ignited or Mass Transfer Limited Branch” in Figure 16)

For cases where the reactor was heated only by the incoming 623K gas (dashed lines), Figure 16 shows two possible steady-state solutions for a* ranging between 2.3 to roughly 20 With a* < 2.3, 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 100% conversion between 3 and 4 cm of reactor length For a* = 15, the reactor achieved near 100% conversion by roughly 2 cm With a* ≥ 20, only 1 solution was again possible which matched the

“high-conversion” branch results seen with internal heat generation

The bottom graph of Figure 16 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 (~670K) 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 623K gas) For the cases heated by only the incoming flow (with a*> 2.3 but less than ~20), a gas-temperature of 623 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

Figure 16 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 [39] The top graph shows the conversion while the bottom graph shows the corresponding solid temperatures

Experiment a* = 1 - Both Gas Heating and Internal Heat Generation a* = 10 a* = 15

Ignited or Mass- (a* > 2.3) Transfer Limited Branch

Model Gas Heat Int Gen.

Figure 16 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 12 cm reactor At the same axial location, both the conversion and temperature were lower for the short reactors compared with the 12 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 16 correspond to regimes of kinetically and mass-transfer limited surface reactions, respectively With a*< 2.3, only a kinetically limited solution occurred With a* > 2.3, the monolith reactor was in a mass- transfer controlled regime for all cases beyond about 4 cm For cases with a* > 2.3 but less than about 20, the reactor could be kinetically 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* > 20, the kinetics were sufficiently fast and only a mass-transfer limited branch solution occurred irrespective of the heating history

Figure 17 shows the calculated temperature and select species profiles along the channel length up to 4 cm (from the L = 12 cm calculation) The model parameters are a* and Nu=4.364 with the corresponding Sh The experiment reported a maximum gas-temperature rise of approximately 33K With complete CO conversion, the model predicted a gas temperature rise across the monolith of 48K (see green line in Figure 17) 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

Transitioning from Kinetic to Mass-Transfer Control t = 23.3 sec t = 32.2 sec t = 46.6 sec.

Figure 17 Temperature and select species mass and site fraction as a function of monolith length up to 4cm (the calculation used 12 cm) The model parameters are a* and Nu=4.364

The data in Figure 17 show that the CO mass fraction at the wall is approximately 90% of the bulk value at the inlet and decreases to roughly 0.1% at a channel length of 1.5 cm (similarly for the O2 mass-fraction) Per the definitions of Hayes and Kolaczkowski [1] presented earlier, the monolith reactor is operating in a kinetically controlled regime at the inlet but becomes completely mass-transfer controlled by 1.5 cm Although difficult to see in the figure, the monolith remains mass-transfer controlled with the wall concentration remaining below 1% of the bulk value along its remaining length

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 kinetic 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 kinetically 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 steady- state solutions were accompanied by different light-off positions in the channel For the case shown Figure 17, light-off occurred just downstream of the inlet (at about 3 cm) when the surface reached about 650K and then propagated upstream due to solid heat conduction (see temperature history of solid in Figure 17) Similar transient behavior occurred for all the computations heated by the inlet gas irrespective of the surface initial condition

Figure 18 shows the computed temperature and species profiles using the parameters a*0 and Nu=4.364 (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 ~ 0.5 cm) and only needed a short distance to propagate to exactly the inlet The final steady-state surface concentration at the inlet is approximately 1% of the bulk 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

Z CO(s) Z O(s) Z Pt(s) a*0 Nu=4.364 t = 18.6 sec. t = 27.9 sec.

Figure 18 Temperature and select species mass and site fraction as a function of monolith length up to 4cm (the calculation used 12 cm) The model parameters are a*0 and Nu=4.364

In the previous cases of this section, the model generally over-predicted the exit conversion after roughly 0.25 cm under mass-transfer limited conditions (see Figure 16)

A single Nu and Sh 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 19 shows the effect of changing the Nu / Sh on the steady- state conversion using a* = 30 along a 12 cm reactor The profiles do not match precisely for any Nu / Sh combination, although there is better agreement with a Nu < 4.364 This is in agreement with the correlation suggested by Ullah et al, which estimates a global Sherwood number for CO to be 1.75 [39]

Figure 19 Effect of Nu / Sh on the steady-state conversion of CO as a function of reactor length The Sherwood number comes from the analogy of heat and mass transfer The computations use a*0 and correspond to the mass-transfer limited or high-conversion solution.

Discussion of Case 2

The parameter a* can be interpreted as either a surface area enhancement as discussed previously or simply a kinetic adjustment factor which accelerates the kinetics 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 20 to 30 times the geometric surface area is not an unreasonable number In fact, numbers as large as

1500 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 20 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 20 times the geometric surface area This assumes that the monolith is, in fact, operating in a mass-transferred limited branch Figure 16 shows solutions with a* less than 20 that agree reasonably with the experiment These are, in turn, kinetically limited near the reactor inlet

The parameter a* may also be a simple kinetic adjustment factor The authors provided little information about the catalyst and only allude to “platinum” group metals in their work Therefore, the surface kinetic mechanism, explicitly formulated for platinum, may not be appropriate A value of a* = 2.4 can provide reasonable agreement with experiment if the surface is initially heated to an elevated temperature Multiplying the surface reaction rates by 2.4 is not an unreasonable adjustment given the unknown catalyst.

Case 3: Transient Propagation – Single Horizontal Platinum Tube

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 20) 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 4.0 cm in total length (3.5 cm outside a support fitting), with inner and outer diameters of 0.74 and 0.95 mm, respectively The inlet velocity varied from 1 to 3 m/s The fuel was carbon monoxide in pure oxygen with an equivalence ratio ranging from 0.1 to 2 The gasses were 99.5% pure CO (Matheson Co., CP Purity with typically less than 15 ppm water but can be as high as 50 ppm) and 99.5% pure O2 (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 broke 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 20 shows images of the ignition and propagation sequence for an inlet flow condition of φ = 1 and 2 m/s Figure 20a shows the illuminated platinum tube attached to the flow system using a support fitting A Kanthal igniter (hotwire), dissipating approximately 12.4 watts (9.9 V across the wire drawing 1.25 amps), externally heated the last 10% of the tube near the outlet for 3 seconds (Figure 20b) – an indicator light visible in the images shows when the igniter is powered After 3 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 20c shows the propagation of the catalytic flame midway along the tube while Figure 20d 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

Figure 20 Ignition and propagation sequence of a CO / O 2 (φ = 1) catalytic reaction along the inside of a platinum tube (0.74 mm ID, 0.95 mm OD) The inlet gas velocity is 2 m/s a b c d

The experiment diagnostics included both standard video and a 12-bit infrared camera taking images of the platinum tube From the video, which produced the images seen in Figure 20, computer analysis tracked the position of the leading edge of the catalytic flame zone at 30 fps The analysis used the NASA Spotlight software [86] In some cases, the analysis used the IR images (see Figure 40 discussed below) instead of the video particularly when the glowing region was very dim (usually at low and high equivalence ratios) Only 2.6 cm of the entire 3.5 cm tube was in the field of view of both cameras (the images in Figure 20 were made specifically for presentation purposes and sacrificed some resolution) The analysis of the IR images also provided temperature measurements of the exterior surface of the platinum

Figures 21 through 24 show the position of the leading edge of the glowing region as a function of time for equivalence ratios ranging from φ = 0.1 to φ = 2.0 Each figure shows 10 different equivalence ratios with up to 3 separate experimental tests for each condition In these figures, the tube outlet corresponds to a position of 3.5 cm Figures

21 and 22 correspond to lean (φ = 0.1 to 1.0) and rich (φ = 1.1 to 2.0) mixtures, respectively In these cases, the gasses came directly from the gas-bottles through the mixing system Although there is likely some amount of water vapor present in these gasses (perhaps 10 to 50 ppm), these tests are referred to as the “dry CO” tests in subsequent discussion Figures 23 and 24 show data with water vapor intentionally added to the inlet mixture using the bubbler These tests are referred to as the “wet CO” tests in the subsequent discussion

Figure 21 Leading edge of catalytic reaction versus time in a platinum tube for φ = 0.1 to 1 with a dry CO / O 2 mixture flowing at 2 m/s A hot-wire heats the outlet end of the tube for 3 seconds (denoted by the red vertical line) There are up to 3 tests plotted for each φ The tube outlet is at 3.5 cm

Figure 22 Leading edge of catalytic reaction versus time in a platinum tube for φ = 1.1 to 2 with a dry CO / O 2 mixture flowing at 2 m/s A hot-wire heats the outlet end of the tube for 3 seconds (denoted by the red vertical line) There are up to 3 tests plotted for each φ The tube outlet is at 3.5 cm

Figure 23 Leading edge of catalytic reaction versus time in a platinum tube for φ = 0.1 to 1 with a wet CO / O 2 mixture flowing at 2 m/s A hot-wire heats the outlet end of the tube for 3 seconds (denoted by the red vertical line) There are up to 3 tests plotted for each φ The tube outlet is at 3.5 cm

Catalytic flame remained near tube outlet with small movement up- stream and then back downstream.

Catalytic flame remained near tube outlet with small movement up- stream and then back downstream.

Figure 24 Leading edge of catalytic reaction versus time in a platinum tube for φ = 1.1 to 2 with a wet CO / O 2 mixture flowing at 2 m/s A hot-wire heats the outlet end of the tube for 3 seconds (denoted by the red vertical line) There are up to 3 tests plotted for each φ The tube outlet is at 3.5 cm

The mixtures with φ = 0.1 and 0.2 (dry) and φ = 0.1, 0.2, and 0.3 (wet) did not propagate after the igniter was turned off and removed For the dry CO tests, all of the remaining equivalence ratios propagated The wet CO tests propagated uniformly up to φ=1.6 For φ = 1.7 and 1.8, the catalytic flame front propagated both upstream and then back downstream before ultimately propagating again upstream and anchoring itself near the fitting For φ > 1.8, the catalytic flame remained near the tube outlet with only small amounts of propagation upstream and then back to the outlet For the leaner tests (φ = 0.3 and 0.4), the dry CO tests exhibited a plateau region (i.e the catalytic reaction zone stabilized briefly at the outlet) after the igniter was removed – this phenomena was more pronounced for the wet CO tests which exhibited a clear plateau region up to φ = 0.9

For most of the tests, the propagation was linear with time The notable exception was the richest wet CO cases which were excluded from the subsequent analysis The propagation velocity came from a linear curve fit of the data ranging from 25% to 75% of the total propagation distance Figure 25 shows the propagation velocity versus equivalence ratio for both dry and wet CO tests The velocity was insensitive to the actual pixel intensity tracked by the computer (< 0.3%.change in velocity with a 20 count change in intensity) Furthermore, a linear regression analysis showed that the 95% confidence intervals on the slope ranged between roughly +/-0.7% to +/-1.5%

The data in figure 25 show that the propagation velocity increases with increasing equivalence ratio, reaches a maximum near or slightly below φ = 1, and then decreases for richer mixtures Between roughly φ = 0.6 and 1.3, the propagation speed did not vary significantly These trends were the same for both dry and wet CO Furthermore, only subtle differences in propagation velocity occurred between these conditions For leaner cases, the wet CO tests propagated slightly faster while, for richer cases, the dry CO tests propagated somewhat faster The minimum equivalence ratio at which propagation occurred was φ = 0.4 for the wet CO cases which was slightly richer than the dry cases, which propagates at φ = 0.3 Furthermore, the wet CO cases did not propagate uniformly for φ > 1.6 and were not included in the graph The model calculations shown in the figures are discussed below

Wet Dry Dry a Dry b a 2x Ext Nat Convection b Modified Reaction Step 54 No light-off for > φ with 2.5W after 3 sec.

Figure 25 Measurements and model predictions of the propagation velocity for the catalytic reaction front versus φ for both dry and wet CO tests.

Model Parameters

The model conditions matched the experiment as closely as possible For the gas- phase, the inlet boundary conditions were a temperature of 300K, a velocity of 2 m/s, a pressure of 1 atm, and a gas-mixture of CO with pure O2 ranging from φ=0.1 to 2.0 Water vapor saturated the inlet feed at 300K for some calculations The model used wet

CO chemistry for the cases that include water vapor otherwise the model used dry CO chemistry (Appendix E)

The solid energy equation (equation 8) requires 2 boundary conditions at each axial face due to the conduction term At the inlet face, the model maintains a temperature of 300K to simulate the large heat sink due to the metal fitting present in the experiment (see Figure 20a) The outlet face of the solid channel is adiabatic for the computations The exterior of the channel has heat loss from convection and radiation The external heat transfer coefficient, h0, comes from the natural convection correlations of Churchill and Chu [74] The radiation losses use a gray body approximation with a constant emissivity ε = 0.15 [87]

The solid / surface requires initial conditions for surface temperature as well as the gas and surface species distribution along the inner length of the channel The initial surface temperature was 300K for all cases Calculations used one of 2 different initial surface species distributions (ZCO(S) =1 or ZO(S) =1) The computations assumed that the catalytic active area equaled the geometric surface, that is a* = 1, since the tests used a pure platinum tube As in the previous cases, the initial condition for the remaining unknown, Y kW , came from the surface flux-balance (equation 7) For non-reacting conditions, it is approximately equal to the bulk value, Y k

The ignition scheme uses internal heat generation along the last (downstream) 10% of the tube for 3 seconds and attempts to mimic the hotwire in the experiment The majority of the calculations used a heat generation term, , of 2.5W which was sufficient to achieve light-off and propagation for most of the tests While the power dissipated by the hotwire is approximately 12W in the experiment, a significant fraction of that energy goes initially into heating the wire and later is lost to the surroundings via radiation and convection Thus, it is not unreasonable that only ~20% of the power dissipated in the igniter wire actually is transferred to the platinum tube Table 3 summarizes the parameters used in this case q& gen

For the calculations presented in this section, a spatial step of 0.2 mm was sufficient for grid independence while the time step ∆t = 0.01ãτS or roughly 22 milliseconds achieved temporal independence Recall that ∆t is not the timestep used by the transient solver but rather is the time between successive spatial integrations of the quasi-steady gas-phase The transient solution was independent of the parameter ∆t at this value (and lower) For the calculations presented in this section, each second of physical time (during propagation) took roughly 4 minutes of CPU time using the dry CO tests and about 160 minutes of CPU time for the wet CO tests 8 With internal heat generation on, the computations took significantly longer (roughly 25 and 460 minutes for dry and wet CO, respectively) for 1 second of physical time.

Model vs Experiment

Figure 26 compares the model prediction to the experimentally measured propagation of the catalytic reaction in a platinum tube for φ ranging from 0.2 to 2 with a dry CO / O2 mixture These calculations used ZCO(s) = 1 initially, however, propagation was not sensitive to whether ZCO(s) = 1 or ZO(s) = 1, initially For space savings, only the even equivalence ratios were compared to the model In this figure, there are 2 curves for each φ corresponding to the location of maximum catalytic heat release (shown with hatched lines in the figure) and the location of the first occurrence of 650K from the inlet (shown with a solid black line) In tracking the maximum heat release, the heat release needed to satisfy the criterion of being greater than 0.01 W Tracking the 650K temperature best matched the anchor (or upstream most) position of the catalytic reaction in the experiment This technique better represented the experimental measurements

8 The computations used a variety of processors including an Intel P4 (2.0 to 2.6 GHz) and Macintosh Power PC G5 (2.0 to 2.5 GHz) processors The reported CPU times are typical values which used the leading edge of the glowing region – the edge likely corresponds to a specific temperature – and is the technique used in all subsequent analyses

The data in Figure 26 show reasonable agreement between the model and experiment especially in the richer cases (φ > 1) In fact, for φ = 1.2, 1.4, and 1.6, the model almost matches the experiment exactly (when comparing the catalytic front defined by the position of the 650K isotherm) For the richer cases, the model begins to exhibit curvature (i.e deceleration) in the position versus time plot The model predictions of the catalytic anchor point also diverge for the richer cases with the experiment propagating closer towards the fitting In the model, the upstream 300K boundary condition may not faithfully represent the experiment The experiment uses a non-metallic ferrule to seal the Pt tube in the fitting and likely limits the thermal contact between the tube and the fitting As such, the slower propagation speeds of the rich cases may allow the tube within the fitting to warm which subsequently permits the reaction to propagate further upstream

With φ < 1, the model predicts a significantly faster propagation velocity than observed experimentally The model does capture the qualitative feature of a decelerating propagation speed with decreasing φ but the magnitude is significantly different between model and experiment Also, the model predicted light-off and propagation at φ = 0.2, while the experiment did not propagate at these values At φ = 0.1 (not shown), the model showed light-off but the reaction remained near the outlet of the channel

The model propagation velocity shown back in Figure 25 (black lines) corresponds to the model data from Figure 26 Similar to the experiment, the velocity comes from a curve fit to the region from 25% to 75% of the propagation distance (Figure 26 shows the curve fits) For the model, the propagation distance was defined beginning at location of the catalytic front at 3 seconds (heat-generation turned off) and ending at the anchor point Clearly, the model over-predicts the propagation velocity for φ < 1 For richer cases, there is good agreement between the model and experiment

Dry CO Experiment vs Model Model: 2.5W, 3sec, Nu=4.364

Figure 26 Comparison of model and experiment looking at the propagation of a catalytic reaction front along a Pt tube for φ ranging from 0.2 to 2 with a dry CO / O 2 mixture flowing at 2 m/s (Z CO(s)

=1) The model predictions show both the position of the 650K isotherm and location of maximum catalytic heat release versus time The position 3.5 cm corresponds to the tube outlet

Figure 27 shows the effect of varying the input power via the internal heat generation term, , from 1.5 to 3W in 0.25 W increments on the transient propagation of the catalytic reaction front The results in Figure 27 are for a single equivalence ratio of φ = 1 (with Z q& gen

CO(s) = 1 initially) Similar trends, however, were present for both lean and rich conditions The red vertical hatched line denotes hotwire shut-off in the experiment and removal of internal heat generation in the model The catalytic reaction did not propagate for = 1.5 W and 1.75 W after 3 seconds For 2 W, the reaction does propagate and the solution matches the position of the experimental reaction front initially The reaction front then propagates at a rate of 5.4 mm/s and diverges from the experiment As the input power increases, propagation occurs earlier even prior to the removal of heat generation In all these cases, however, the propagation velocity is largely insensitive to input energy (only 2 percent change from 2W to 3W) q& gen

Figure 27 Effect of varying the input power on the transient evolution of the catalytic flame The test condition was φ = 1 using dry CO with Z CO(s) = 1 initially

The calculations presented thus far used internal heat and mass transfer coefficients that corresponded to the constant flux asymptote This boundary condition, up to catalytic light-off, is the best choice for a reasonable prediction of light-off according to most previous research [37, 38] Figure 28 shows how varying Nu (and correspondingly Sh) from 3.66 to 6 affects the propagation of the catalytic reaction front

In this figure, Nu =3.66 corresponds to a constant surface temperature while Nu = 6 is the largest value reported in the literature [32] Only slight differences in the propagation occurred with these variations of Nu and Sh

Figure 29 compares model predictions to the measured position of the catalytic reaction for the wet CO data As in the previous cases, the figure shows data for φ from 0.2 to 2 The most striking difference between the wet and dry CO data takes place just after ignition With the wet CO, the catalytic reaction front lingers near the tube outlet for φ between 0.4 and 1.0 This phenomena is more prevalent in the wet CO cases but did occur for φ=0.3 and 0.4 with dry CO (refer back to Figure 21) The model does not capture this difference between wet and dry CO but does show a delay in propagation The dry and wet computations, however, are nearly identical as is seen by the small difference in propagation velocity shown previously in figure 25 (black lines)

Dry CO Model: 2.5W, 3sec, Nu / Sh Comparison

Figure 28 Model results with varying Nu (and Sh via the heat and mass-transfer analogy) that compare the propagation of a catalytic reaction in a Pt tube for φ ranging from 0.2 to 2 with a dry

CO / O 2 mixture flowing at 2 m/s (Z CO(s) =1) The position 3.5 cm corresponds to the tube outlet

Catalytic flame remained near tube outlet with small movement up- stream and then back downstream.

Wet CO Experiment vs Model Model: 2.5W, 3 sec., Nu = 4.364

Figure 29 Comparison of model and experiment looking at the propagation of a catalytic reaction front inside a Pt tube for φ=0.2 to 2 with a wet CO / O 2 mixture flowing at 2 m/s (Z CO(s) =1) The model predictions show the position of the 650K isotherm versus time The position 3.5 cm corresponds to the tube outlet

Other experiments showed good model agreement using the current kinetic mechanism under lean conditions [21] Thus, the present results which show good agreement under rich but not lean conditions are surprising In the experiment, there were no special measures to prevent room currents across the tube Furthermore, an exhaust fan was near the experiment to prevent the poisonous CO from accumulating Thus, it is possible that some forced convection may have affected the experimental results To better understand the role of external heat transfer, the model explores the sensitivity of the external heat transfer coefficient and emissivity on the solution

Figure 30 shows the effect of varying the external heat transfer coefficient on the model results This figure presents 2 calculations: one using a constant h0 and the other using twice (2x) the natural convection correlation The constant h0 correspond to a value of 100 W/ m 2 ãK which allowed the model to match the experimental propagation velocity at φ = 1.0 Subsequent calculations using this h0 yielded better agreement with the experiment for φ < 1 but produced worse agreement for φ > 1 (see hatched green lines back in Figure 25) Specifically, good quantitative agreement with the experiment occurred in the range of φ = 0.6 to 1.0 Below this φ, the model showed better qualitative agreement with experiment in terms of slowing but the propagation velocities did not match for φ = 0.4 For φ = 0.1, the catalytic reaction did not propagate in both the model and experiment For the rich cases, the model and experiment do not agree as the external heat loss slows the propagation significantly for φ = 1.2 and extinguishes the flame at higher φ

Figure 30 also shows calculations using 2 times the natural convection coefficient This had the effect of increasing the overall heat transfer in a fashion that varied along the tube proportional to natural convection As with the constant h0 = 100 W/ m 2 ãK cases, the propagation velocity better matched the experiment for φ 0) A value of < 0 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 bulk gas temperature (T′ = T if < 0) s& k s& k s& k

A more informative form of the energy equation (Equation 25) can be formulated by substituting species conservation (Equation 21) into Equation 24 Upon examination of Equation 25, the 5th and 6th terms appear to cancel leaving the equation independent of surface reactions In essence, the difference in enthalpy due to species change is already accounted for (term 2 in Equation 22 or terms 2 & 4 in Equation 24) irrespective of whether they change due to gas or surface reactions These terms, however, do not precisely cancel as can be seen by careful examination of the control volume in Figure 4 The 5th term in Equation 25 represents the change in enthalpy from the left to right face of the control volume and thus is evaluated at the bulk temperature The 6th term comes from the enthalpy exchange at the surface of the control volume Mass exiting the control volume ( < 0) is at the bulk temperature Mass entering the control volume ( > 0) is at the surface temperature and thus either releases or requires energy to change its temperature to the bulk The final form of the gas-phase energy equation is shown in Equation 26 and uses a temperature T′ (Equation 27) that is equal to the bulk value, T, for < 0 or is equal to the surface temperature, T s& k s& k s& k S , if s& k > 0

Side View Front View Exploded View

Figure 38 Solid phase control volume (including catalytic surface shown in red)

The solid-phase energy equation is derived based on the control volume (AS⋅dx) shown in Figure 5 and includes heat transfer to and from the gas inside the tube, external heat transfer to the surroundings, heat generation terms due to catalytic reactions and

Joule heating, and axial heat conduction (Equation 28) Internal radiation exchange has been neglected The external heat loss due to natural convection (term 3) and radiation

(term 4) allows modeling of a single catalytic channel The external heat loss from natural convection comes from correlations of horizontal tube by Churchill and Chu [74]

For a single (central) channel of a monolith reactor, the external heat loss terms are set to zero The enthalpy of absorbing and desorbing gas species (term 5) are evaluated just as described previously for the gas-phase (i.e Equations 26 and 27) Again, if s& k > 0 (gas- species k is desorbed from the surface) then h k is evaluated at the surface temperature (T′

= T S ) For values of s& k < 0, or gas-species k adsorbing to the surface, h k is evaluated at the bulk gas temperature (T′ = T) Because the solid is thermally thin, the surface catalytic reactions (those involving surface species only) are modeled as surface heat generation (term 6) in the solid A prescribed volumetric heat generation (term 7) simulates Joule or external heating of the solid Term 8 represents the axial heat transfer due to conduction The solid density, ρ S ; heat capacity, C S ; and thermal conductivity, k S ; are all assumed constant in the model

The molar rates of production of species due to gas-phase reactions (per volume) and surface reactions (per area) are shown in Equations 29 and 30, respectively The rate of production of species k comes from summing the rate of progress of each reaction (forward and reverse) involving species k, as shown in Equation 31

pixels of resolution across the diameter of the tube

The camera measures photons that are incident on the detector (in a given integration period) and converts this value into a voltage This voltage is subsequently digitized into a ‘counts’ value between roughly 8000 and 13000 – it is not clear why the manufacturer chose the large offset from zero, however, it is of no consequence in the analysis The number of photons leaving the surface of the platinum tube as a function of wavelength and surface temperature is given by equation 56 where all of the symbols are

10 The section was prepared with the assistance of Dr Daniel L Dietrich of NASA Glenn Research Center and Benjamin P Mellish of the National Center for Space Exploration Research Furthermore, Mr Mellish conducted all of the experiments and analysis of the catalytic propagation tests

3.5 cm defined in the nomenclature section Only a fraction of the photons reach the detector surface (due to attenuation by various factors) and only a fraction of those produce a signal (due to the quantum efficiency effect) Equation 57 is an expression which represents a theoretical number, KD, that is proportional to the detected photons by the camera per unit area per unit time In this equation, QE(λ) is the detector quantum efficiency while ƒF(λ) is the transmisive factor which accounts for attenuation of photons as they pass through a medium to the detector This analysis consider only the attenuation of light through the narrow band-pass filter shown in Figure 41 and neglect all other potential sources of attenuation (such as air)

Figure 41 Transmission characteristics of the filter used in IR imaging of platinum tube

The analysis assumes a constant value for the quantum efficiency and emissivity across the narrow wavelength range of the filter This key assumption allows the QE and ε to be moved outside of the integral in equation 57 thereby simplifying the calculation of

KD Equation 58 now defines a new non-dimensional quantity, κ, which is a ratio of camera counts of an arbitrary object at a given temperature, T, to the counts of a blackbody at a specific temperature, T0 If the arbitrary object is a blackbody source, then κ is denoted as κBB A blackbody calibration source provided data to develop a functional relationship between temperature (i.e number of photons) and camera counts (Figure 42) Additionally, this figure shows the function κBB at an arbitrary reference temperature, T0 = 650K, which came from integrating equation 57 at many temperatures across the range shown in Figure 42

Blackbody Temperature Measurement measured counts corresponding platinum surface temperature

Figure 42 Relationship between Blackbody temperature to the camera counts and the parameter κ from equation 58

Equation 59 shows an expression for κ of the platinum surface, called κPt In this expression, the relationship between κBB and κPt differs by only a factor of the (constant) emissivity of the platinum surface Thus, the temperature at which this equality holds true is also the surface temperature of the platinum tube

As an example of the previous discussion, assume that a camera pixel produces a counts value of 8500 while imaging the surface of the platinum Figure 42 shows that a counts values of 8500 corresponds to a κ =3.57 which basically states that there is 3.57 times the number of photons being detected compared to those detected when viewing a blackbody at our reference temperature T0 = 650K Figure 42 further shows that the corresponding blackbody temperature which produces this signal is roughly 835K However, the signal is not from a blackbody but from a platinum surface with an unknown surface temperature, T From the relationship in equation 59 and assuming ε 0.15, the corresponding κBB(T) = 3.57 / 0.15 or 23.8 Using the functional relationship in Figure 42, κBB = 23.8 which corresponds to a temperature of 1424K, the temperature of the platinum surface Thought of in another way, the platinum surface must be at a significantly higher temperature (~1424K), because of its relatively low emissivity, compared to a blackbody (~835 K) to produce the detected photons

Custom software analyzed the IR images using the above process with curve fits of the data shown in Figure 42 The analysis was automatic and conducted along the length of the platinum tube using a 5-pixel average centered on the tube diameter The image had roughly 20 pixels of resolution across the tube diameter reducing the likelihood of angular dependent effects Due to optical limitations, the camera viewed only about 2.6 cm of the overall 3.5 cm length tube nearest the fitting

The calculations presented in the text assumed that the emissivity of the platinum tube, ε = 0.15 This value varies somewhat in the literature and does have a slight temperature dependence for the total normal emissivity [91] At 800K, the reference reports ε ≅ 0.1 which increases to a value of ε ≅ 0.2 at about 1600K For the example given above (which is near the maximum temperatures observed in the experiment), the resulting temperatures are 1660K, 1424K, and 1291K for ε = 0.1, 0.15, and 0.2, respectively These types of error bars are included on the peak temperatures shown in the figures of the text A preliminary comparison (not presented) using a platinum tube with small thermocouples spot welded at various axial distances shows good agreement with the IR measurements using an ε =0.15 for the temperature range of this experiment (700K – 1300K)

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