Light-off and Cumulative Emissions in Catalytic Monoliths with

4668

Ind. Eng. Chem. Res. 2004, 43, 4668-4690

Light-off and Cumulative Emissions in Catalytic Monoliths with Nonuniform Catalyst Loading Karthik Ramanathan,† David H. West,‡ and Vemuri Balakotaiah*,† Department of Chemical Engineering, University of Houston, Houston, Texas 77204-4004, and The Dow Chemical Company, Freeport, Texas 77541

A 1-D two-phase model with nonuniform catalyst loading along the channel length is used to present a detailed analysis of light-off behavior and cumulative emissions in a catalytic monolith in which an exothermic reaction occurs. The influence of important design parameters such as the washcoat thickness, solid conductivity, channel geometry, and catalyst distribution along the channel on the transient time and cumulative emissions is analyzed. An analytical light-off criterion for the case of nonuniform catalyst loading is proposed, with the help of which we can determine the nature of ignition (front-end, middle, or back-end ignition). It is found that, for the wall reaction case, nonuniform catalyst loading with two zones of catalyst with more catalyst at the inlet favors front-end ignition and reduces the transient time as well as the cumulative emissions. Though high solid conduction is better for steady-state designs, better transient and equivalent steady-state performances can be obtained for finite solid conduction by distributing the catalyst appropriately. As can be expected, washcoat diffusion has a profound influence on light-off and defines a critical value of catalyst loading below which front-end light-off does not occur. For catalyst loading exceeding this critical value, there is a maximum value of washcoat thickness (corresponding to a washcoat Thiele modulus of 0.5 at inlet fluid temperature) below which light-off is not influenced by washcoat diffusion. Increasing the washcoat thickness beyond this critical value has no effect on the transient time or cumulative emissions. There is a range of catalyst loading in which uniform distribution leads to the optimal design, while outside this range, the optimal design corresponds to having more catalyst at the front. It is also found that parallel-plate or rectangular channel geometries with high aspect ratios lead to better transient and good steady-state performances. Finally, we show that an optimally designed monolith with proper catalyst distribution, solid conductivity, channel geometry and dimensions, and washcoat can reduce cold-start emissions substantially compared to the standard case. 1. Introduction Monolithic catalytic reactors are used in a variety of applications that include control of automobile emissions, oxidation of volatile organic carbons, and selective removal of NOx from exhaust gases. The monolithic reactor contains a large number of small, long channels (of diameter 0.5-2 mm and length 5-20 cm) in parallel through which the reacting fluid flows. The catalyst is deposited on the walls of the monolithic channels as a porous washcoat. The flow in the channels is laminar in most cases, with Reynolds numbers in the range of 100-1000. The reactant in the fluid phase is transported to the surface of the catalyst and within the catalyst by diffusion and is transported in the axial direction mainly by convection. Catalytic monoliths used in automobile emission control operate under highly transient conditions, and their performance during the warm-up period is an important factor in the design of catalytic converters. Because of the time-varying inlet conditions and the complex coupling of the physical and chemical processes in the monolith, improving the design requires a comprehensive understanding of the influence of various design and operating variables on the performance. Mathematical modeling and simula* To whom correspondence should be addressed. Tel.: (713) 743-4318. Fax: (713) 743-4323. E-mail: [email protected]. † University of Houston. ‡ The Dow Chemical Co.

tion may be used to identify the optimal designs and reduce the amount of experimentation needed. In this work, a 1-D two-phase transient model with position-dependent transfer coefficients is used to analyze in detail the influence of the various design variables on the transient time, cumulative emissions, and ignition behavior of the monolith. The design variables considered are the geometry of the channel, the total amount of catalyst loading, the catalyst distribution along the channel, the conductivity of the monolith walls, the washcoat thickness, and the channel dimensions. The operating variables studied are the inlet fluid temperature and the initial solid temperature. By analyzing the influence of these design and operating variables on cumulative emissions, we propose optimal monolithic designs that can reduce the cumulative emissions substantially. 2. Literature Review Previous studies of monoliths describing experimental, modeling, and simulation results are summarized in the books by Becker and Pereira1 and Hayes and Kolaczkowski2 and review articles by Cybulski and Moulijn3 and Groppi et al.4 Oh and Cavendish5 used a 1-D two-phase model with constant Nusselt and Sherwood numbers to simulate the transient behavior of the automobile catalytic converter with overall kinetics for all of the main reactions. Groppi et al.6 and Tronconi and Forzatti7 compared 1-D and 2-D model predictions

10.1021/ie034131v CCC: $27.50 © 2004 American Chemical Society Published on Web 03/17/2004

Ind. Eng. Chem. Res., Vol. 43, No. 16, 2004 4669

using position-dependent transfer coefficients for the 1-D model in the mass-transfer-controlled regime. Extensive studies also exist in the literature on the transient and light-off behavior of packed beds and monoliths. Most of these studies either used the dimensional form of the model equations or were confined to specific cases (reaction kinetics) and limited ranges of parameter values (sensitivity studies). Eigenberger,8 Oh and Cavendish,5 and Kirchner and Eigenberger9 presented transient simulations illustrating the light-off behavior and propagation of temperature fronts in packed beds and monoliths, respectively. Please et al.,10 Leighton and Chang,11 and Keith et al.12 analyzed the propagation of the ignition front in the monolith using a 1-D two-phase model with constant transfer coefficients (and with either zero or very small conduction in the solid). Kirchner and Eigenberger13 analyzed the cold-start behavior of automotive catalysts using an electrically heated precatalyst. The models considered by these authors do not include washcoat diffusion, which is known to have a profound influence on the light-off behavior of the converter.14 Cold-start (cumulative) emissions and design optimization to minimize cold-start emissions were not investigated in detail (in the global parameter space) in these previous studies. There have also been some earlier studies that discuss the influence of nonuniform catalyst loading (along the channel) on the transient and steady-state performance of the monolith. Oh and Cavendish5 studied the effect of nonuniform catalyst distribution and found that a linearly decreasing catalyst loading improves the transient performance of the monolith. Psyllos and Philippopoulos15 used a 1-D two-phase model to simulate the catalytic monolith and found that a parabolic activity distribution (decreasing from the entrance to the exit) has the shortest warm-up time. Cominos and Gavriilidis16 investigated the effect of axially nonuniform catalyst distribution on temperature and concentration gradients in monoliths using a 2-D model. They observe that exponentially decreasing catalyst distributions initiate light-off at the entrance of the monolith and also alleviate temperature gradients. Most of the earlier studies were confined to either circular channels or uniform catalyst loading or both. A recent study by Ramanathan et al.17 analyzed the steady-state behavior of catalytic monoliths with nonuniform catalyst loading. Tronci et al.18 presented transient simulations of catalytic monoliths using a 1-D two-phase model and found that the optimal catalyst distribution for reducing cumulative emissions is a monotonically decreasing or two-zone catalyst distribution with more catalyst near the inlet. The work of Tronci et al.18 concentrates on solving the optimization problem for a specific set of parameters and for the case of no diffusional limitations in the washcoat (wall reaction). A systematic analysis of the influence of the various design variables (geometry and dimensions of the channel, total amount of catalyst loading and its distribution along the channel, solid conductivity, and washcoat thickness) on the cumulative emissions, transient time, and light-off behavior is currently lacking. Such a systematic study is useful in obtaining an optimal monolith design. 3. Mathematical Model The mathematical model used is the same as that developed by Ramanathan et al.14 and is given by

u j

kc(x) ∂Cm )(C - Cs) ∂x RΩ m

u j Ffcpf

(1)

∂Tf h(x) (T - Ts) )∂x RΩ f

(2)

kc(x) (Cm - Cs) ) ac(x) Rv(Cs,Ts) δcη

(3)

∂Ts ∂2 Ts δwFwcpw ) δwkw 2 + ∂t ∂x (-∆HR)ac(x) Rv(Cs,Ts) δcη - h(x) (Ts - Tf) (4) De

∂2C ) ac(x) Rv(C,Ts); 0 < y < δc ∂y2

C ) Cs

at y ) 0;

∂C )0 ∂y

ηδcac(x) Rv(Cs,Ts) ) -De Cm ) Cin(t); Tf ) Tin(t) ∂Ts/∂x ) 0

(5)

at y ) δc (6)

|

∂C ∂y y)0

(7)

at x ) 0

(8)

at x ) 0, L

Ts(x,t)0) ) Ts0(x)

(9) (10)

Equations 1-4 describe the fluid- and solid-phase species and energy balances, respectively. Equations 5 and 6 describe the reactant concentration profile in the washcoat, while eq 7 gives the washcoat effectiveness factor (η). Equations 8-10 define the boundary and initial conditions. Here, x and t represent the axial coordinate along the channel and time, respectively. Tf and Cm represent the cup-mixing temperature and concentration in the fluid phase, while Ts and Cs denote the solid-phase temperature and the reactant concentration at the fluid-solid interface. ac(x) is the normalized activity (catalyst distribution) profile along the channel [(1/L)∫L0 ac(x) dx ) 1], and Rv(C,T) is the intrinsic reaction rate per unit volume of washcoat [Rv(C,T) ) Rs(C,T) Sv, where Rs(C,T) is the rate based on the unit internal surface area and Sv is the internal surface area per unit washcoat volume]. RΩ is the effective transverse (diffusion or conduction) length scale (RΩ ) AΩ/PΩ, where AΩ ) channel cross-sectional area open to flow and PΩ ) channel perimeter open to flow; RΩ ) Dh/4, where Dh ) channel hydraulic diameter), δc is the effective washcoat thickness (defined as the volume of the washcoat over the fluid-washcoat interfacial area), and δw is the effective wall thickness (defined as the sum δs + δc, where δs is the half-thickness of the solid wall without washcoat). The effective wall thickness in eq 4 may be replaced by the porosity of the monolith channel, given by

δw 1 - ) RΩ )

1 δw 1 δw 1+ + RΩ 4 RΩ

( )

2

(11a) ≈

1 δw 1+ RΩ

(11b)

The coefficients of the accumulation and conduction

4670 Ind. Eng. Chem. Res., Vol. 43, No. 16, 2004 Table 1. Effective Diffusion Length and Asymptotic Constants for Some Common Channel Geometries

terms in the solid-phase energy balance could be written more explicitly as

δwFwcpw ) δcFccpc + δsFscps

(12)

δwkw ) δckc + δsks

(13)

where the subscript s (c) refers to the support (catalyst or washcoat). De and Dm are the effective diffusivities of the reactant in the washcoat and fluid phase, respectively. y is the local distance (from the fluidwashcoat interface) coordinate in the washcoat, and η is the internal effectiveness factor in the washcoat. The other symbols have their usual meanings and are explained in the Notation section. The following expressions describe the local heatand mass-transfer coefficients in the monolith channel when the flow is laminar and fully developed at the entrance:

ShΩ(x) )

kc(x) RΩ ) Dm

{

( )

0.82

RΩ2u j xDm

ShH1,∞ 4

1/3

0 1, or Lef < 1). The transverse Peclet number (P), which is the ratio of the transverse diffusion time to the convection time, is one of the most important dimensionless groups that describes the monolith behavior. Note that P ) 1/16R(Re)(Sc), where R is the aspect ratio (hydraulic diameter to length), Re is the Reynolds number, and Sc is the Schmidt number. In this work, we present calculations only for the special case of a first-order reaction. This special case, while not too restrictive in applications, allows us to obtain some general results (valid for monotone kinetics) and qualitative trends describing the influence of various parameters on the transient behavior of the monolith. Specifically, the results presented here may be applied to determine transient emissions of hydrocarbons from catalytic converters. For the special case of a first-order reaction, we have

Rv(c,Ts) ) kv(Ts) c which in dimensionless form is given by

r(c,θs) ) c exp

[ ] θs

1+

θs γ

(26)

and the washcoat diffusion problem can be solved analytically to obtain

η)

tanh[Φ(z)] ; Φ(z) ) φxa(z) X ) φsxa(z) ΛX; Φ(z) θs X ) exp (27) θs 1+ γ

[ ]

Thus, the dimensionless effective reaction rate may be determined explicitly as

r(cs,θs) η ) cs

xX tanh(φxa(z) X) φxa(z)

Note that Φ(z) is the washcoat Thiele modulus at the local temperature and position z. We can further reduce the four model equations to three by eliminating the solid-phase concentration:

cs )

cm

(28)

a(z) φs xX tanh(φxa(z) X) 1+ ShΩ(z) φxa(z) 2

Thus, for the special case of linear kinetics considered here, the two-phase model equations (with washcoat diffusion) are given by

∂cm ) -a(z) Dare(cm,θs) ∂z

(29)

Lef ∂θm ) NuΩ(z) (θ - θm) ∂z P s

(30)

2 ∂θs 1 ∂ θs LefNuΩ(z) ) (θs - θm) + ∂τ Peh ∂z2 P a(z) BDare(cm,θs) (31)

with boundary and initial conditions given by eqs 23ad. Here, re(cm,θs) is the effective dimensionless reaction rate given by

xX tanh(φxa(z) X) re(cm,θs) ) cm

φxa(z)

a(z) φs xX tanh(φxa(z) X) 1+ ShΩ(z) φxa(z) 2

)

cmX Φ(z) Φ(z)2 + tanh Φ(z) ΛShΩ(z)

(32)

Equations 28 and 32 show the influence of external mass transfer as well as the washcoat diffusional effect on the reaction rate. If the second term in the denominator of eq 28 is much larger compared to unity, then cs , cm and the monolith is locally in the mass-transfercontrolled regime, whereas if the magnitude of the second term is negligible compared to unity, then cs ≈ cm and the monolith is locally in the kinetically controlled regime. The previous model equations (eqs 2931) reduce to the pseudohomogeneous model for the case of very small transverse Peclet number (P , 1) and local Damko¨hler number (φs2 , 1). Physically, this corresponds to long channel and/or small channel diameter


and low catalyst loading (P f 0 and φs2 f 0, but the ratio φs2/P ) Da is finite). In a pseudohomogeneous model, the transverse gradients are negligible and the solid- and fluid-phase concentrations and temperatures are almost equal at each axial position. It should be pointed out that in a pseudohomogeneous model the entire monolith is in the kinetically controlled regime (before or after ignition). The pseudohomogeneous model equations are given by

∂θ 1 ∂2θ ∂θ ) + a(z) BDar(c,θ) ∂τ Peh ∂z2 ∂z

(33a)

∂c ) -a(z) Dar(c,θ) ∂z

(33b)

r(c,θ) ) c

xX tanh(φxa(z) X) φxa(z)

(33c)

with the boundary and initial conditions given by

1 ∂θ - θ ) θm,in Peh ∂z ∂θ )0 ∂z

at z ) 0

at z ) 1

c ) cm,in

(34a) (34b)

at z ) 0

(34c)

The previous pseudohomogeneous model reduces further to the homogeneous model when washcoat diffusional limitations are negligible (φ f 0). The homogeneous model is given by eqs 33a,b and 34 with r(c,θ) ) c exp[θ/(1 + θ/γ)]. An analytical criterion for the validity of the pseudohomogeneous models has been given by Dommeti et al.22 and for the present case (with Lef ) 1 and no washcoat diffusion) may be expressed as

max [a(z)]φs2 exp ShΩ

[ ] B

B 1+ γ

< e-2

(35)

For typical values of the parameters (γ ) 25, B ) 10, Lef ) 1, ShΩ ) 1, and max [a(z)] ) 1), eq 35 is satisfied only if the catalyst loading is very low; i.e., φs2 e 10-4. With such low catalyst loading, an ignited state can be attained only if P values are very small (e10-3). As we shall show later, the design and operation of the monolith in the homogeneous or pseudohomogeneous regime leads to high-pressure drop and longer transient time. However, it may lead to a better steady-state design compared to cases in which the operation is in the heterogeneous regime. 4. Light-off and Cumulative Emissions In this section, we discuss in brief, the definition of light-off and cumulative emissions. By ignition or lightoff, we imply that the steady-state bifurcation diagram of the solid exit temperature θs(1) (or fluid-phase exit conversion) versus the inlet fluid temperature θf,in (or residence time or temperature) is an S-shaped diagram with the ignition point in the feasible region. According to our definition, the boundary between light-off and no light-off is determined by the hysteresis locus (the

Figure 1. Typical plots of the transient and cumulative emissions in the monolith: (a) dimensionless exit concentration; (b) cumulative concentration. Parameter values: Peh ) 500, γ ) 25, Lef ) 1, P ) 0.25, B ) 10, Λ ) 0.01, φs2 ) 0.025, θm,in ) 0, and θs0 ) -10.

appearance of ignition and extinction points in the bifurcation diagram of θs(1) versus θf,in). Once the monolith is ignited, the ignited length of the monolith is in the mass-transfer-controlled regime; i.e., cs , cm (except when eq 35 is satisfied), and before ignition, the monolith is in the kinetically controlled regime; i.e., cs ≈ cm. The calculation of the hysteresis locus (or the boundary between light-off and no light-off) with nonuniform catalyst loading was illustrated by Ramanathan et al.17 Hereafter, we shall assume that the inlet concentration (B), catalyst loading (φs2), reaction, and other parameters (γ, Λ, and Lef) are such that the monolith operates in the light-off region. For example, with uniform catalyst loading and B ) 10, γ ) 25, Lef ) 1, Λ ) 0.004, P ) 0.25, and Peh ) 333.34, the monolith operates in the light-off region if the catalyst loading exceeds the value given by φs2 ) 7 × 10-4. For nonuniform catalyst distribution [such as a7(z)], this value is reduced by another order of magnitude.17 [Note that there is a typographical error in ref 17. While the calculation of the hysteresis locus is done correctly, the dependence of Φ on a(z) is not shown explicitly.] Catalytic converters used in automobiles are aimed at reducing the cumulative emissions. The cumulative emissions (in moles) may be expressed as

Ecum ) VoCin

∫0τcm(τ′,z)1) dτ′

where Vo is the gas volumetric flow rate and Cin is the inlet molar concentration (assumed to be constant). In dimensionless form,

Ecum ) ccum(τ) ) VoCin

∫0τcm(τ′,z)1) dτ′

Figure 1 shows a plot of the exit concentration and the cumulative emissions for a typical set of parameters. The plot shows two clearly defined regions. The first

4674 Ind. Eng. Chem. Res., Vol. 43, No. 16, 2004

region is the initial transient period, where the monolith is in the kinetically controlled regime. In this region, the surface concentration and the fluid-phase concentrations are approximately the same and the reactant conversion is low (or emissions from the monolith are high). The second region is where the monolith is in the mass-transfer-controlled regime (which is also the steadystate asymptote), where the surface concentration (over the ignited length) is much less than the fluid-phase concentration and the exit reactant conversion is high (or low emissions). (Here, the second term in the denominator of eq 28 is much larger than unity.) It should be pointed out that when there is no ignition in the monolith, these two regions cannot be differentiated. For the cumulative emissions to be lower, the transient time should be lower and the steady-state conversion should be higher (or the slope of the steady-state asymptote should be very small). The transient time required to reach the steady-state asymptote may be defined as the time at which the mass-transfer and kinetic resistances are comparable or, in mathematical terms, when the two terms in the denominator of eq 28 are of equal magnitude. This transient time consists of the heat-up time and the ignition time. The heat-up time is the time required for the monolith to reach the inlet fluid temperature from its initial temperature. This process is mainly controlled by heat transfer from the fluid to the solid, and the heat generated by the reaction is negligible. This heat-up time depends mainly on the difference between the inlet fluid temperature and the initial solid temperature. The ignition time is the time required for the monolith to reach the ignition temperature from the inlet fluid temperature. [Remark: During this time period and in steady state, the direction of heat transfer is reversed (from solid to fluid) compared to that of the heat-up time (from fluid to solid).] The ignition time of the monolith depends more on the nature of the ignition (front-end, middle, or back-end), which depends mainly on the catalyst loading and the inlet fluid temperature. Frontend ignition leads to the shortest transient time, while back-end ignition has the longest transient time.14 In the transient asymptote, the entire monolith is in the kinetically controlled regime and the exit conversion is very small; hence, the exit concentration can be approximated by cm(τ,z)1) ≈ cm,in, and the cumulative emissions in the transient asymptote (τ < τig) may be expressed as

ccum(τ) ≈ cm,inτ ) τ

(cm,in ) 1)

(36)

The transverse Peclet number (P) is the main parameter in the system that determines the steadystate conversion/concentration. It has been shown by Balakotaiah and West20 that if the entire monolith is in the mass-transfer-controlled regime, then for values of P smaller than unity, the exit concentration is given by

cm,exit ) R1 exp(-ShT,∞/4P)

(37)

where the constant R1 ) 0.819 for a fully developed laminar flow in a circular duct and ShT,∞ is the asymptotic Sherwood number for the constant wall concentration case (see Table 1 for values of R1 and ShT,∞ for different duct shapes). If the entire monolith is not in the mass-transfer-controlled regime, then the transverse

Peclet number (P) in the previous expression is calculated using the ignited length of the monolith. The steady-state conversion also depends on the channel geometry (through the asymptotic Sherwood number, ShT,∞). A higher asymptotic Sherwood number and a smaller value of P is required for higher conversions. Note that the slope of the steady-state asymptote given by eq 37 for a circular channel is 0.021 (conversion ) 0.979) and 0.000 088 (conversion ) 0.999 912) for P values of 0.25 and 0.1, respectively. (Remark: Equation 37 defines the minimal channel length required for a given level of steady-state conversion. This minimal length is determined only by the channel geometry and the asymptotic constants listed in Table 1.) 5. Light-off and Cumulative Emissions with Uniform Catalyst Loading In this section, we review and extend our previous work14,17 on light-off and cumulative emissions for the case of uniform catalyst loading. The analytical lightoff criterion for uniform catalyst loading may be expressed as14

(

)

4eBφs2 g(Peh) Bφs2 1 + >1 P LefNuH1,∞ f(φsxΛ)

(38)

where

f(φ) )

{

1 φ < 0.5 2φ φ < 0.5

(39)

In the previous expression, the function f(φ) describes the influence of washcoat diffusion while the function g(Peh) given by

g(Peh) )

1 2 σθ2/(1-σθ2)

(σθ )

, σθ2 )

2 2 (1 - e-Peh) Peh Pe 2 h (40)

describes the influence of solid conduction. Ignition occurs in the monolith if the previous criterion (eq 38) is satisfied. The first term in the criterion represents the ignition locus of the pseudohomogeneous model, and the heterogeneous contribution (or local ignition) appears through the second term. If the first term exceeds unity and the second term is negligible (compared to unity), we have a back-end ignition. If the second term exceeds unity, we have a front-end ignition irrespective of the magnitude of the first term. For the case where the sum exceeds unity and the second term less than unity and not negligible compared to the first term, then we have a middle ignition. In this case, the location in the reactor where ignition first occurs depends on the magnitude of the first term. As the first term continues to increase, the location moves from back to front. When the first term is much larger than unity (>10) and the second term is much smaller than unity, ignition occurs closer to the inlet. When the ignition criterion is barely satisfied and the magnitudes of both of the terms are less than unity, then it is always a back-end ignition. As can be seen from the criterion, the most important process influencing light-off is washcoat diffusion. It follows from eq 39 that washcoat diffusional limitation does not exist until ignition provided φ e 1/2. Thus, the minimum catalyst loading needed to obtain front-end


ignition with no washcoat diffusional limitation is given by

(41a)

4eBφs )1 LefNuH1,∞

(41b)

Solution of the previous simultaneous equations gives the critical catalyst loading and washcoat thickness needed to obtain front-end light-off without washcoat diffusional resistance:

Dm δc eB ) De RΩ LefNuH1,∞

(42a)

RΩδckvc(Tin) LefNuH1,∞ ) Dm 4eB

(42b)

[For typical values of B ) 10, Lef ) 1, and NuH1,∞ ) 4.364 (circular channel), we get Λc ) 6.23 and φsc2 ) 0.04.] For φs2 < φsc2, it is not possible to have front-end ignition but back-end/middle ignition can be obtained by choosing sufficiently small P values. However, this leads to a design that is close to the pseudohomogeneous regime and hence leads to longer channels or high-pressure drop, longer transient time, etc. For φs2 > φsc2, the washcoat thickness can be reduced so that the product φsxΛ is less than 0.5. Having a higher washcoat thickness will have no influence on the light-off behavior but can influence the steady-state performance if the monolith is not in the mass-transfer-controlled regime. In the mass-transfer-controlled regime, the concentration at the wall (cs) is much smaller compared to the fluid-phase concentration (cm), i.e., cs , cm, and the second term in the denominator of eq 28 is much larger than unity, i.e.

φs2a(z) xX tanh(φxa(z) X) φxa(z)

ShΩ(z)

.1

(43)

The previous equation should be satisfied for the entire channel to be in the mass-transfer-controlled regime. Equivalently, eq 43 defines the minimum catalyst activity needed so that the monolith is in the masstransfer-controlled regime (after ignition). We shall discuss in detail in section 8 how to choose the catalyst loading and the washcoat thickness so that the previous three conditions are satisfied. When there is front-end ignition but the monolith is not in the mass-transfer-controlled regime, the exit concentration may be approximated by

[(

cm,exit ) Rˆ 1 exp

4P 1 +

(44b)

For the wall reaction case (Λ f 0), eq 44b reduces to

2

φsc2 )

tanhΦ [-Φ ΛP ]

cm,exit ) exp

1 φ ) φsxΛ ) 2

Λc )

second term, and eq 44a further simplifies to

-Sh ˆ ˆ ΛSh 4Φ tanh Φ

]

)

(44a)

where Φ given by eq 27 is evaluated at the solid temperature after ignition with a(z) ) 1. The parameter Rˆ 1 varies between 1 and R1, while Sh ˆ varies between ShH1,∞ and ShT,∞ as Φ2 varies from 0 to ∞. In the pseudohomogeneous limit, the first term in the denominator of the exponential is negligible compared to the

[ ]

cm,exit ) exp

-φs2 X P

(44c)

It follows from eq 44 that the steady-state conversion can be much higher when the monolith is operated close to the pseudohomogeneous/homogeneous regime compared to the mass-transfer-controlled regime. (However, as stated earlier, the length of the monolith may be 1 or 2 orders of magnitude larger.) The other important parameters in the criterion are the channel geometry (appears through the asymptotic Nusselt number NuH1,∞), solid conduction [through the function g(Peh), which varies between 1 and 2.718], and transverse Peclet number (P). Channels with lower asymptotic Nusselt number (e.g., triangular) favor frontend ignition. As the solid conduction increases, the ignition behavior becomes more uniform along the channel length, and for infinite solid conduction, the entire channel ignites simultaneously. As long as the second term in the ignition criterion is greater than unity, the transverse Peclet number (P) does not affect the ignition behavior. However, if the second term is less than unity, then decreasing the transverse Peclet number changes the ignition behavior in the monolith from back-end to front-end/middle ignition depending on the magnitude of the second term. The transient time and cumulative emissions also depend on the same parameters (φs2, Λ, NuH1,∞, Peh, and P). It should be noted that higher catalyst loading, inlet temperature, adiabatic temperature rise, and activation energy always lead to better transient behavior (lower emissions). However, solid conductivity, channel geometry, and transverse Peclet numbers have contrasting effects on transient and steady-state operation. For a good steady-state design, the monolith should have high solid conduction, very small values of the transverse Peclet number (P < 0.1), and a parallel-plate channel or a channel with higher asymptotic transfer coefficients (ShT,∞). (Remark: If the catalyst loading is also low but φs2/P ) Da is finite, then the optimum steady-state design corresponds to operating the monolith close to the pseudohomogeneous/homogeneous regime.) Interestingly, with uniform catalyst loading, all of the above may lead to higher transient time and cumulative emissions (from τ ) 0 to some critical value). The heatup time is high for the case of very high or infinite conduction because the complete monolith has to be heated to the ignition temperature by only heat transfer from fluid to solid and the heat generated by the reaction is negligible during this period, whereas for the case of low solid conduction, the monolith can ignite locally (and only a small piece of the monolith needs to be heated). The effect of channel geometry on the performance of the catalytic monolith is much more complicated than the solid conductivity. The channel geometry appears in the 1-D two-phase model through the heat- and mass-transfer coefficients. Triangular and square channels that have low asymptotic transfer coefficients yield a larger region of multiplicity and hence require lower catalyst loading (or lower inlet fluid temperature) to ignite. Also, lower asymptotic transfer


coefficients are required for front-end ignition (from the analytical ignition criterion), and hence it lowers the transient time. It should be noted that the entire monolith will be in the mass-transfer-controlled regime in the case of a front-end ignition. However, lower transfer coefficients lead to lower steady-state conversions (eq 37) in the mass-transfer-controlled regime. On the other hand, higher asymptotic transfer coefficients will reduce the heat-up time and yield better steadystate conversions but lead to longer ignition time (because back-end ignition may be favored). The influence of the channel geometry on cumulative emissions and transient time is analyzed in more detail in the following sections. 6. Light-off with Nonuniform Catalyst Loading As stated earlier, it is important to understand the influence of the various design and operating variables to obtain a better design of catalytic monoliths. The results in the previous section are limited to uniform catalyst loading [a(z) ) 1]. We now extend these results to the case of nonuniform catalyst loading. The catalyst distribution function a(z) is chosen such that the catalyst activity at any axial position z is given by φs2(z) ) φs2a(z) and

Figure 2. Various nonuniform catalyst activity distributions.

catalyst distribution with more catalyst in the front is given by

∫01a(z) dz ) 1 so that the total amount of catalyst remains the same. The different continuous catalyst distributions studied by Ramanathan et al.17 are given by

a1(z) ) 2z a2(z) ) 3z2

a7(z) )

{

100/19 z < 0.1 10/19 z > 0.1

}

where the value of z* was taken to be 0.1 and the value of R* was taken to be 0.1. A general three-zone catalyst distribution is given by

a(z) )

1 + (R* - 1)H(z - z1)[1 - H(z - z2)] 1 + (R* - 1)(z2 - z1)

3

a3(z) ) 4z

2

a6(z) ) 1.5(1 - z ) When the catalyst distribution is divided into two zones and the catalyst activity in the first zone is 1/R* times that in the second zone, the catalyst distribution function a(z) is given by

a(z) )

1 + (R* - 1)H(z - z*) z* + R*(1 - z*)

(45)

where z* ( 0.9 10/24 z < 0.9 150/24 z > 0.9

where the value of z* was taken to be 0.9 and the value of R* was taken to be 10 and 15 for the two different distributions a4(z) and a5(z), respectively. The

where the three zones are from 0 to z1, z1 to z2, and z2 to 1. The catalyst distribution in the second zone is R* times the catalyst distribution in the first/third zone. The values of z1, z2, and R* for a8(z) are 0.3, 0.4, and 10, respectively, and the corresponding values for a9(z) are 0.1, 0.2, and 6, respectively. The values of z1, z2, and R* for the catalyst distribution a10(z) are 0.1, 0.5, and 6, respectively. Because the continuous catalyst distribution functions (such as linear, quadratic, and cubic functions) do not yield interesting results,17 the two- and three-zone catalyst distribution functions a(z) ) 1, a4(z), a5(z), a7(z), a8(z), a9(z), and a10(z) are used for simulations in this work. A plot of some of these distributions is given in Figure 2. We now extend the light-off criterion to monoliths with nonuniform catalyst loading [a(z) * 1] and also analyze the influence of the catalyst distribution on the light-off behavior. The ignition criterion for uniform loading (eq 38) contains two terms. As explained in detail elsewhere,14 the first term gives the ignition locus of the pseudohomogeneous model while the second term represents the contribution from the heterogeneous part of the model or local ignition (due to interphase gradients). Local ignition occurs whenever the solid-phase energy balance (eq 31 at steady state) starts having multiple solutions. For the case of zero solid conduction (Peh f ∞), negligible washcoat diffusional resistance, and nonuniform catalyst loading, the solid-phase bal-


ance starts to have multiple solutions when

4eBφs2a(z) LefNu(z)

>1

(46)

The local ignition now occurs at a location z, where the previous criterion is satisfied, unlike the case of uniform catalyst loading [a(z) ) 1], where the local ignition occurs at the front-end if the previous criterion is satisfied. Also, it can be seen easily that the first term (or the contribution from the pseudohomogeneous model) is not affected by the nonuniform catalyst loading for the case of negligible washcoat diffusional resistance. For this case, we find that it is the integral of the catalyst distribution function a(z) that appears in the first term, unlike the function a(z) itself as in the case of local ignition. Because the integral of the catalyst distribution function over the channel length is unity, the catalyst distribution function does not affect the first term of the ignition criterion. However, for the case of strong washcoat diffusional resistance (over the entire channel), the catalyst distribution function a(z) appears in the first term as an integral of the square root of a(z). Because the second term depends on the catalyst distribution, it depends on the dimensionless axial coordinate z. An approximate ignition criterion for monoliths with nonuniform catalyst loading may be expressed as

a(z) φs2 xX tanh(φxa(z) X) ShΩ(z)

g(Peh) Bφs2 w(φ) + f(φsxΛ) P 4eBφs2 max [a(z)] 1 > 1 (47) LefNuH1,∞ f(φsxmax [a(z)] Λ) where the functions g(Peh) and f(φ) accounting for solid conduction and washcoat diffusion are as given earlier in eqs 39 and 40. The function w(φ), which takes into account the effect of nonuniform catalyst loading on the washcoat diffusion for the pseudohomogeneous term, is given by

w(φ) )

As discussed in the previous section, front-end ignition or ignition occurring close to the inlet is preferred because the transient time is much lower (convection in the fluid phase and conduction in the solid phase cause the temperature front to travel in the flow direction faster as opposed to back-end ignition, where the temperature front travels against the flow direction only as a result of conduction in the solid phase). On the basis of the previous ignition criterion, we could distribute the catalyst [for a given catalyst loading (φs2)] such that a(z) is high near the inlet or close to the inlet so that the second term in the previous ignition criterion is greater than unity for z close to the inlet. Also, it should be noted that we could find values of φs2 for which the ignition criterion is not satisfied for uniform catalyst loading but is satisfied for nonuniform catalyst loading because of the catalyst distribution function a(z). Again, it should be pointed out that, in the case of frontend ignition, the entire monolith is in the mass-transfercontrolled regime, and hence the conversions will be higher when compared to a back-end or middle ignition (where the entire monolith need not be in the masstransfer-controlled regime). In the mass-transfercontrolled regime, the concentration at the wall (cs) is much smaller compared to the fluid phase concentration (cm), i.e., cs , cm, and the second term in the denominator of eq 28 is much larger than unity, i.e.

{∫

φ < 0.5

1

1

0

xa(z) dz φ > 0.5

(48)

In the previous criterion, we use the maximum of the function a(z) along the channel for the second term. If the previous criterion is satisfied, then there is ignition in the monolith. As before, if the first term exceeds unity and the second term is negligible (compared to unity), we have a back-end ignition. If the second term exceeds unity, then ignition occurs (irrespective of the magnitude of the first term) at the location where a(z) is a maximum. For the case where the sum exceeds unity and the second term is less than unity and not negligible compared to the first term, then we have a middle ignition. When the ignition criterion is barely satisfied and the magnitudes of both of the terms are less than unity, it is always a back-end ignition. In the case of uniform catalyst loading, if the second term exceeds unity, it means a front-end ignition, but with nonuniform loading, local ignition occurs where the catalyst loading is the maximum. [If catalyst loading is uniform over a finite length, say, 0 < z < 0.1 as in a7(z), then ignition occurs at the front end of this section.]

φxa(z)

.1

(49)

The previous equation should be satisfied for all values of z in the monolith for the entire channel to be in the mass-transfer-controlled regime. Equivalently, eq 49 defines the minimum catalyst activity needed at any position so that the monolith is in the mass-transfercontrolled regime at that position (after ignition). The optimal catalyst loading (φs2), distribution function [a(z)], and washcoat thickness (Λ) for the case of nonuniform catalyst loading will be discussed in detail in section 8. We show in Figure 3 the light-off behavior of monoliths with different catalyst distributions. The solid temperature profiles are shown as a function of the channel length and the dimensionless time. (Remark: One unit of dimensionless time τ corresponds to 10-15 s.) As can be seen, for the parameters used, back-end ignition is favored for channels with uniform distribution and front-end ignition is favored for channels having a7(z) distribution. It is important to note that the transient time taken for the monolith with uniform catalyst distribution is very high compared to other catalyst distributions. As previously explained, for backend ignition, the transient time is high because of slow propagation of the temperature front opposite to the flow direction. For the case of a three-zone catalyst distribution, the function a9(z) is a maximum for z values between 0.1 and 0.2, and this maximum value is 4. Because the second term in the criterion exceeds unity, ignition occurs at z ≈ 0.1. For the case of the three-zone catalyst distribution a9(z), the second term in the ignition criterion is 4 times higher than the value corresponding to the case of uniform catalyst loading. This effectively means that, instead of increasing the catalyst loading four times (with uniform catalyst loading), we could redistribute the catalyst to obtain the same performance (transient time and steady-state


Figure 3. Typical transient simulation showing the solid temperature profile in a monolith with different catalyst distribution functions. Parameter values: Peh ) 500, γ ) 25, Lef ) 1, P ) 0.25, B ) 10, Λ ) 1, φs2 ) 0.0175, θm,in ) 0, and θs0 ) -7.

conversion). For the case of a7(z) catalyst distribution, this factor is 5.26. Also, it is interesting to note that the catalyst distributions a4(z) and a5(z) ignite at z ) 0.9 and only 10% of the channel length is ignited and only 40% of the monolith is ignited for the case of uniform catalyst loading. 7. Influence of Design and Operating Variables on Cumulative Emissions In this section, we analyze the influence of the various design and operating variables on the cumulative emissions and the transient time. These variables include the total catalyst loading (φs2) and distribution [a(z)], washcoat thickness (Λ), conduction in the solid (Peh), channel geometry, channel dimensions (transverse Peclet number, P), initial solid temperature (θs0), and the inlet fluid temperature (θm,in). 7.1. Catalyst Loading and Distribution [Os2 and a(z)]. As shown in Figure 3, the catalyst distribution a(z) affects the ignition behavior strongly. Figure 4 shows the cumulative emissions plot in a circular channel for different catalyst distributions for the same set of parameters as those used in Figure 3. The distributions a4(z) and a5(z) ignite at z ≈ 0.9, and only 10% of the channel length is ignited; hence, the steadystate conversion is low. There is not much difference between the transient and steady-state asymptotes for these two catalyst distributions because the exit conversion is very small. Uniform catalyst loading favors a back-end ignition, and hence the transient time is high and it takes a long time to reach the steady-state

Figure 4. Cumulative emissions for different catalyst distributions in a circular geometry. Parameter values: Peh ) 500, γ ) 25, Lef ) 1, P ) 0.25, B ) 10, Λ ) 1, φs2 ) 0.0175, θm,in ) 0, and θs0 ) -7.

asymptote. The steady-state conversion is low (because less than 40% of the monolith is ignited) and also the time it takes for 40% of the monolith to ignite is very


Figure 5. Cumulative emissions for different catalyst loading (or local Damko¨hler number) in a circular geometry for the catalyst distribution function a7(z). Parameter values: Peh ) 500, γ ) 25, Lef ) 1, P ) 0.25, B ) 10, Λ ) 1, θm,in ) 0, and θs0 ) -7.

high because of back-end ignition. Catalyst distribution a7(z) favors a front-end ignition, and hence the transient time is small; however, because the entire monolith is ignited, the steady-state conversion is also high. The distributions a9(z) and a8(z) ignite at z ) 0.1 and 0.3, respectively. The distribution a10(z) ignites somewhere in the middle of the channel, and hence the transient time is slightly higher, but the steady-state conversion is higher than that obtained with a8(z) because of a longer ignited length of the monolith. For the parameter values chosen in Figure 4, the cumulative emissions (at τ ≈ 35) can be decreased by more than 80% by changing the catalyst distribution from uniform to a7(z). Hence, the catalyst distribution function has a profound influence on the transient time and cumulative emissions. The influence of the catalyst loading or the local Damko¨hler number (φs2) on the transient time is illustrated in Figure 5 for the catalyst distribution a7(z). For very low catalyst loading, there is no ignition in the monolith, and as the catalyst loading increases, the transient time keeps decreasing and the cumulative emissions become lower. As the catalyst loading is increased, the monolith can ignite at a lower temperature and hence the transient time decreases. The exit conversion at steady state does not depend on the catalyst loading as long as the entire monolith is in the mass-transfer-controlled regime, and hence the steady-state conversion remains the same for all local Damko¨hler numbers (φs2) exceeding some minimal value. 7.2. Washcoat Thickness (Λ). The washcoat thickness influences the transient and steady-state operation considerably. When the washcoat thickness is small (wall reaction case), there are no washcoat diffusional limitations and all of the catalyst present is accessible for the reactants. However, as the washcoat thickness continues to increase, washcoat diffusional limitations prevail, and after a critical washcoat thickness, strong

Figure 6. Cumulative emission plot for different washcoat thicknesses for two catalyst distributions [uniform and a7(z)] for a circular geometry. Parameter values: γ ) 25, Lef ) 1, P ) 0.25, B ) 10, φs2/Λ ) 0.2, θm,in ) 0, Peh ) 500, and θs0 ) -7.

diffusional limitations exist. When strong diffusional limitations exist, though the catalyst loading will be high, the fraction of available catalyst sites for the reactants is reduced. Also, as we continue to increase the washcoat thickness, the inlet fluid temperature for ignition continues to decrease until the critical washcoat thickness is reached, after which increasing the washcoat thickness any further does not decrease the inlet fluid temperature required for ignition.14 It should be noted here that both the parameters φs2 and Λ have washcoat thickness in their definitions and hence increasing the washcoat thickness increases both φs2 and Λ but the ratio φs2/Λ remains constant. As we increase the washcoat thickness (δc) for a fixed value of the metal loading (kv), both of the terms in the ignition criterion continue to increase until the critical washcoat thickness is reached, after which there is no influence of the washcoat thickness on the ignition criterion. However, it is important to note the fact that, as both φs2 and Λ conitue to increase, the ignition behavior changes from back-end to front-end ignition. Hence, we expect the transient time also to decrease as we increase the washcoat thickness until the critical value of the washcoat thickness is reached, after which it does not decrease any further. Here, we present transient simulations for the case of thick and thin washcoat for different catalyst distributions and for finite solid conduction. Figure 6 shows a cumulative emissions plot for uniform catalyst distribution and the two-zone catalyst distribution a7(z) for different values of washcoat thickness (keeping the catalyst loading fixed). The ratio φs2/Λ in these plots is fixed at 0.2. As we increase the washcoat thickness, the transient time continues to decrease until washcoat diffusion becomes strong, after which the transient time does not decrease any further. For very low washcoat thickness (Λ ) 0.01), there is no ignition in the monolith, and as we increase the washcoat thickness, the ignition behavior changes from back-end ignition to front-end ignition (because increasing the washcoat


thickness also increases φs2) and hence the transient time and cumulative emissions are lower. Once strong diffusional limitations start to exist in the monolith, increasing the washcoat thickness does not decrease the cumulative emissions any further. For uniform catalyst distribution, the cumulative emissions plot remains the same for any values of Λ > 10 and does not affect the transient time and steady-state conversion. For the twozone catalyst distribution a7(z), the cumulative emissions plot remains the same for any values of Λ > 5, as opposed to the value of 10 in the case of uniform catalyst distribution. Because operating the monolith with strong washcoat diffusional limitations is not preferable, the optimum value of the washcoat thickness would be the one corresponding to φ ) 1/2, for which the ignition temperature is the lowest and washcoat diffusional limitations do not exist. [Note: When a nonuniform catalyst distribution is used, φ depends on the axial coordinate, and in this case, we would like to have the maximum value of φ be equal to 0.5, i.e., max (φxa(z)) ) 1/2.] It should be noted that as the washcoat thickness increases the solid conduction also increases. Because the washcoat thickness is usually much smaller than the support thickness, we assumed that changing the washcoat thickness has little influence on the solid conduction or the heat Peclet number (Peh); i.e., kcδc , ksδs. Hence, for the calculations presented in Figure 6, we have assumed that the heat Peclet number is constant. 7.3. Solid Conduction. The influence of solid conduction on transient and steady-state behavior in catalytic monoliths with uniform catalyst distribution has been analyzed by Ramanathan et al.14,17 It has been found that high solid conduction yields better conversions during steady-state operations. The reason for this behavior is that, for higher solid conduction, the solid temperature is nearly uniform in the entire channel and, once ignited, the entire channel will be in the masstransfer-controlled regime, leading to better conversion. It was also shown14,17 that a higher value of solid conductivity (lower Peh value) increases the region of multiple solutions (in the sense that the metal loading needed to obtain an ignition point in the bifurcation diagram of exit cup-mixing conversion versus inlet fluid temperature decreases by a factor of 30-80 as Peh decreases from infinity to zero). This also means that we could choose the catalyst loading such that ignition occurs in channels with high solid conduction and there is no ignition in channels with low solid conduction. The influence of solid conduction on the transient time is more complicated than its influence on steady-state operation. Depending on the catalyst loading and other design and operating variables, high or low solid conduction may be preferred for a better transient performance. As mentioned earlier, the transient time depends on the initial solid temperature, the difference between the initial solid temperature and the ignition temperature, and also the nature of the ignition. This transient time is made up of the heat-up time and ignition time. If solid conduction is high, the entire solid has to be heated to the ignition temperature by heat transfer from the fluid phase before ignition occurs, whereas for the case of low solid conduction, ignition occurs locally (as opposed to high solid conduction, where the ignition is uniform) and hence the entire monolith need not be heated to the ignition temperature. For low solid conduction, once the ignition temperature

Figure 7. Cumulative emission plot for two different φs2 values (0.0175 and 0.05) at different heat Peclet numbers for a circular geometry. Parameter values: γ ) 25, Lef ) 1, P ) 0.25, B ) 10, Λ ) 0.004, θm,in ) 0, and θs0 ) -10.

is reached locally, the monolith ignites and the heat produced by the reaction helps to heat the rest of the monolith. Figure 7 shows the cumulative emissions plot for two different values of the catalyst loading (Figure 7a with φs2 ) 0.0175 and Figure 7b with φs2 ) 0.05) for a uniform catalyst distribution for different values of the heat Peclet number (Peh). (Remark: Typical Peh values are 300-1000 for ceramic monoliths and 10-50 for metallic monoliths.) For φs2 ) 0.0175, back-end ignition is favored in the monolith for finite solid conduction. As can be seen, infinite conduction in the solid gives the shortest transient time and lower cumulative emissions. The steady-state conversion is higher for infinite conduction because the entire monolith is ignited, and for finite values of the solid conduction, the entire monolith is not ignited; hence, the conversions are lower. In this case, though the monolith ignites locally for finite conduction, the ignition first occurs near the back end and hence the transient time is higher and the steady-state conversion is lower. Hence, the nature of ignition also plays an important role in determining the transient time and cumulative emissions for finite solid conduction. For φs2 ) 0.05, front-end ignition is favored in the monolith. Because it is a front-end ignition, for finite (and low) solid conduction, the monolith ignites locally near the inlet and the heat produced by ignition (and convection in the fluid phase) helps to heat the rest of the monolith to the ignition temperature and hence the transient time is lower. Because of front-end ignition, the entire monolith is ignited even for the finite conduction values, the steadystate conversion is high, and its value is the same for all values of the heat Peclet number. Any value of the heat Peclet number (Peh) greater than 100 is considered to be low solid conduction because it allows the monolith to ignite locally. (Note: The slope of the steady-state asymptote in the cumulative emissions plot gives the steady-state conversion.) In Figure 7b, the catalyst loading was approximately 3 times higher to favor front-end ignition. Instead of increasing the catalyst loading by this factor, the same


Figure 8. Cumulative emission plot for different heat Peclet numbers and the catalyst distributions a7(z) and a9(z) for a circular channel. Parameter values: γ ) 25, Lef ) 1, P ) 0.25, B ) 10, φs2 ) 0.0175, Λ ) 0.004, θm,in ) 0, and θs0 ) -10.

loading as that used in Figure 7a but with different catalyst distributions a7(z) and a9(z) is used to favor front-end ignition for any finite conduction in the monolith. Figure 8 shows the plot of cumulative emissions for different heat Peclet numbers (Peh) and for two different catalyst distributions a7(z) and a9(z). There is not much difference in the cumulative emissions plot for the case of infinite solid conduction, the reason being that the monolith ignites uniformly and hence the catalyst distribution does not affect the ignition behavior. However, there is a considerable difference in the cumulative emissions plot for finite solid conduction between uniform a7(z) and a9(z) distributions. The two catalyst distributions a7(z) and a9(z) favor front-end ignition at z ) 0 and 0.1, respectively. Hence, the transient time and cumulative emissions are considerably lower for both of these distributions. 7.4. Channel Geometry. The geometry of the channel is one of the design parameters that could influence the steady-state and transient behavior of catalytic monoliths. When a 3-D model is written for describing transport within each channel, the effect of channel geometry is apparent in the model equations. However, when the model is averaged out in the transverse directions, the effect of channel geometry appears only through the heat- and mass-transfer coefficients. In this section, we discuss briefly the influence of the channel geometry on the cumulative emissions. Steady-state bifurcation analysis shows17 that triangular channels have a larger region of multiple solutions followed by square, circular, and parallel-plate channels. Geometries with lower asymptotic Nusselt numbers show a larger region of multiplicity. Also, as is evident from the ignition criterion (eq 47), channel geometries with lower asymptotic Nusselt numbers (NuH1) (like triangular channels) satisfy the ignition criterion (and favor front-end ignition because it appears in the denominator of the second term) easily compared to channels with high asymptotic Nusselt numbers (like

parallel-plate channels). For lower cumulative emissions, the transient time (which is made up of the heatup time and ignition time) should be lower and the steady-state conversion should be higher. As mentioned earlier, the heat-up time depends on the difference between the initial solid temperature and the inlet fluid temperature. The heat-up time will be lower if the heattransfer coefficient is higher. However, ignition requires a lower heat-transfer coefficient (so that a higher fraction of the heat generated is trapped in the solid). The other important part of the cumulative emissions is the steady-state conversion, which depends on the asymptotic mass-transfer coefficient (Sherwood number). The exit fluid conversion is given by eq 37, and higher conversions are obtained for channels with higher asymptotic Sherwood numbers (ShT). Hence, channels with lower transfer coefficients will favor frontend ignition and the ignition time will be less (but the heat-up time will be high and steady-state conversions will be low), whereas channels with higher transfer coefficients will have a good steady-state conversion and the heat-up time will be less but the ignition time will be high. It is possible to choose parameters such that channels with lower transfer coefficients ignite, but channels with higher transfer coefficients do not ignite. This was already shown in Ramanathan et al.14 It should be pointed out here that if all of the geometries favor front-end ignition, then it is always better to choose the one which has the highest asymptotic transfer coefficient because that would give better conversions and the heat-up time would also be small. As mentioned earlier, with the help of nonuniform catalyst loading, we could favor front-end ignition in the monolith by distributing the catalyst accordingly. For example, ignition would not occur in a parallel-plate channel (high transfer coefficient) with uniform catalyst loading (and will occur in channels with low transfer coefficients such as triangular and square channels), but it is possible to redistribute the catalyst in such a way as to favor ignition. This is shown in Figure 9. For the set of parameters in Figure 8, a parallel-plate channel would not ignite with a uniform catalyst distribution. However, with catalyst distributions a7(z) and a9(z), there is ignition in the monolith. As can be seen from the figure, for the catalyst distribution a7(z), all of the channel geometries have front-end ignition and hence the ignition time is almost the same. The heat-up time is less for parallel-plate channels because of the higher heat-transfer coefficient. (Note: In this figure, the initial solid temperature θs0 ) -10 and hence the heat-up time is considerable.) Because the heat-up time is less for parallel-plate channels and the ignition time is almost the same for all channel geometries, parallel-plate channels have the lowest transient time and also better steady-state conversions (because of the higher asymptotic Sherwood number). As can be expected, the conversion is higher for parallel-plate channels followed by circular, rectangular, square, and triangular channels. Simulation with catalyst distribution a9(z) yields interesting results. In this case, all of the geometries have a front-end ignition except parallel-plate channels, in which middle ignition occurs. Because it is a front-end ignition for other channel geometries, the transient time is almost the same for these channels but parallel-plate channels have a higher transient time because of the middle ignition. However, it should be noted that the conversion is still higher in parallel-plate channels


Figure 9. Cumulative emission plot for different channel geometries and the catalyst distributions a7(z) and a9(z). Parameter values: Peh ) 500, γ ) 25, Lef ) 1, P ) 0.25, B ) 10, φs2 ) 0.0175, Λ ) 0.004, θm,in ) 0, and θs0 ) -10.

followed by circular, rectangular, square, and triangular channels. It is interesting to note that, for this catalyst distribution, triangular channels lead to lower cumulative emissions until a particular time (τ ≈ 20) and, similarly, square channels are better than parallel-plate channels until τ ≈ 26, after which parallel-plate channels have lower cumulative emissions. This is because the steady-state conversion in parallel-plate channels is higher. Eventually, parallel-plate channels will have better cumulative emissions than other geometries. As long as front-end (or close to front-end) ignition can be achieved by appropriately distributing the catalyst along the channel length, it is better to choose a channel geometry with a higher asymptotic transfer coefficient. However, choosing geometries (like circular or parallel-plate) that have a constant peripheral curvature may not be really advantageous. When the ignition behavior is simulated with a 2-D model, it was found6,23 that, for channels with sharp corners, ignition occurs first at the corners and then spreads along the channel periphery, and this behavior is due to the variation of the Nusselt/Sherwood number along the channel perimeter. For channels with constant peripheral curvature, ignition occurs uniformly along the periphery. At the corners, the heat-transfer coefficient is lower and hence ignition first occurs at the corner. This feature cannot be captured by the 1-D model because the transverse direction is averaged out, and by ignition in the 1-D model, it is assumed that the entire channel perimeter is ignited. [Note: For the simulations of the 1-D model, the Nu(Sh)H1 values have been used and hence the ignition predicted by using these transfer coefficients would be conservative, in the sense that it would actually predict a late ignition and the location where the entire channel perimeter is ignited, as corners would have ignited before.] It may not be practical to have parallel-plate channels for catalytic converters. Hence, rectangular channels

Figure 10. Cumulative emission plot for rectangular and circular geometries and the catalyst distributions a7(z) and a9(z). Parameter values: Peh ) 500, γ ) 25, Lef ) 1, P ) 0.25, B ) 10, φs2 ) 0.0175, Λ ) 0.004, θm,in ) 0, and θs0 ) -10.

with an aspect ratio of 1:2 [Nu(Sh)H1 ) 4.123, Nu(Sh)H2 ) 3.02, and Nu(Sh)T ) 3.391] or 1:3 [Nu(Sh)H1 ) 4.795, Nu(Sh)H2 ) 2.97, and Nu(Sh)T ) 3.956] would be better suited to reduce the cumulative emissions. In fact, a rectangular channel with a high aspect ratio of 1:3 would yield better steady-state conversions when compared to a circular channel because it has a higher value of ShT and also the corner effect of the geometry would reduce the ignition time. In Figure 10, we compare the cumulative emissions for circular and rectangular geometries for two different catalyst distributions a7(z) and a9(z). As expected, rectangular channels with an aspect ratio of 1:3 gives better cumulative emissions compared to the circular geometry. (The difference in the cumulative emissions is rather small at τ ) 30 but continues to increase with τ.) 7.5. Channel Dimensions/Transverse Peclet Number (P). The parameter that depends on the channel dimensions is the transverse Peclet number, P. As the length of the channel is increased, the transverse Peclet number decreases but the pressure drop in the channel increases. The dimensionless pressure drop (or the Euler number) is given by

Eu )

-∆P fRe ) 8P Ff u j2

(50)

where -∆P is the pressure drop in the channel, f is the friction factor, and Re is the Reynolds number. The factor fRe for different channel geometries is given in Table 1. One of the main constraints in increasing the length of the channel would be the pressure drop. The transverse Peclet number also affects the steady-state conversion as given by eq 37, and as P increases, the steady-state conversion decreases. (Note: As discussed in section 4, for very small values of P and φs2, the monolith operates in the pseudohomogeneous regime. In this regime, the length of the monolith required to


Figure 11. Cumulative emission plot for two catalyst distributions [uniform and a7(z)] for different values of the transverse Peclet numbers for a circular channel. Parameter values: Peh ) 500, γ ) 25, Lef ) 1, B ) 10, φs2 ) 0.0175, Λ ) 0.004, θm,in ) 0, and θs0 ) -10.

achieve the same conversion as that in the masstransfer-controlled regime may be 1-2 orders of magnitude higher.) Figure 11 shows a plot of the cumulative emissions for the case of uniform catalyst loading and catalyst distribution a7(z) for different values of the transverse Peclet number. For the parameter values used, the second term in the ignition criterion is less than unity ()0.436 for uniform catalyst loading and 2.2948 for a7(z) at z ) 1). As P is varied from 0.25 to 0.01, the first term changes from 0.716 to 17.892 and hence ignition in the monolith changes from back-end (for P ) 0.25) to close to front-end (for P ) 0.01) for the case of uniform catalyst loading. For P ) 0.25, uniform catalyst loading favors back-end ignition and hence the transient time is high. (Remark: The time t* is scaled using the transverse diffusion time in contrast to τ, which is scaled with respect to the residence time, L/u j .) For smaller values of P, the ignition is somewhere in the middle of the channel and the transient time is low. For small values of P, the steady-state exit conversion is very high, as given by eq 37. The same set of parameters when used with the catalyst distribution a7(z) favors frontend ignition for all values of P, and the value of P affects only the steady-state exit conversion and does not affect the transient time considerably. Hence, the P value does not affect the transient time if the monolith ignites near the inlet. It is interesting to note that the transient time for P ) 0.01 is slightly higher than that for P ) 0.05 or 0.1 and can be attributed to the fact that the monolith operates close to the pseudohomogeneous limit. It is also important to note that a front-end ignition obtained by local ignition at the inlet [like the a7(z) distribution, where the second term in the ignition criterion exceeds unity] leads to lower transient times and cumulative emissions. Reducing the P value beyond a certain value does not really help to reduce the transient time, and it also increases the pressure drop; hence, small values of P would lead to an overdesign (or a bad design) of

Figure 12. Cumulative emission plot for different initial solid temperatures and the catalyst distribution a7(z) for circular and parallel-plate channels. Parameter values: γ ) 25, Lef ) 1, P ) 0.25, B ) 10, φs2 ) 0.0175, Λ ) 0.004, θm,in ) 0, and Peh ) 500.

the monolith. The optimum P value for a monolith with front-end ignition would depend on the desired conversion and can be obtained using eq 37. 7.6. Initial Solid Temperature (θs0). The initial solid temperature also plays an important role in the transient time (particularly the heat-up time). Though the inlet solid temperature is an operating variable and not a design variable, it is worth understanding the effect of the inlet solid temperature on the cumulative emissions and transient time. As can be expected, when the initial solid temperature is very low (cold start), the transient time is higher and hence the cumulative emissions are also higher. The heat-up time (time required for the solid temperature to reach the ignition temperature) depends mostly on the heat-transfer coefficient and conduction in the solid. Channels with a high heat-transfer coefficient reduce the heat-up time (as explained earlier). When the solid conduction is very high, the entire solid has to be heated to the ignition temperature from the initial solid temperature and hence the heat-up time will be high, whereas for the case of low solid conduction, the monolith can ignite locally and the entire solid need not be heated to the ignition temperature; hence, the heat-up time will be low. This was explained earlier in Figure 9. This is true only for the case of front-end ignition, and we assume that with the help of nonuniform catalyst loading we could distribute the catalyst so as to favor front-end ignition. Figure 12 shows a plot of the cumulative emissions for a circular channel at different initial solid temperatures. (Note: One unit of the dimensionless temperature is equivalent to about 18 °C.) The steadystate conversions remain the same, and only the transient (heat-up) time increases as the initial solid temperature gets lower. We also show the cumulative emissions plot for the case of a parallel-plate channel for different initial solid temperatures in the same figure. As can be seen, the difference in transient time


tive emissions decrease. For the case of uniform catalyst distribution, the effect of time-varying inlet temperature is not as pronounced because the ignition time is larger (because of back-end ignition) than the heat-up time, but for the catalyst distribution a7(z), the ignition time is small (front-end ignition) compared to the heat-up time and hence the effect of time-varying inlet temperature is more pronounced. 8. Optimal Design for Minimizing Cumulative Emissions

Figure 13. Cumulative emissions plot for time-varying inlet temperatures for the uniform and catalyst distribution a7(z) for a circular channel. Parameter values: γ ) 25, Lef ) 1, P ) 0.25, B ) 10, φs2 ) 0.0175, Λ ) 0.004, Peh ) 500, and θs0 ) -10.

for circular and parallel-plate channels is more pronounced for the case of θs0 ) -10. This is expected because the channel geometry or solid conduction does not affect the heat-up time considerably if the initial solid temperature is close to the ignition temperature. 7.7. Inlet Fluid Temperature [θm,in(τ)]. The inlet fluid temperature (θm,in) was assumed to be constant with time in all of the simulations earlier. However, for catalytic converters used in automobile emission control, the inlet temperature varies with time initially and reaches a steady value. (In practice, the inlet gas temperature varies from 300 K to around 450-500 K in about 30-60 s, after which it remains almost a constant.) We approximate the initial variation of the inlet fluid temperature by a ramp, and hence the inlet fluid temperature can be expressed as

θm,in(τ) )

{

θm,f - θm,i µ θm,f - θm,i τ> µ

θm,i + µτ τ
1 LefNuH1,∞

(52)

Note that the function f in the ignition criterion equals unity for the wall reaction case. Also, after ignition we want the entire monolith to be in the mass-transfercontrolled regime (for good steady-state conversions). Hence, eq 49 for wall reaction may be expressed as

4Xφs2a(z) > 10 ShH1,∞

(53)

(Remark: The factor 10 in eq 53 is arbitrary but gives cs < 0.1cm, which is close enough to mass-transfer control.) Note that X ) exp[θs/(1 + θs/γ)] in eq 53 is evaluated at the solid temperature after ignition. For front-end ignition, eq 52 should be satisfied at the inlet (z ) 0), and if the criterion for mass-transfer control (eq 53) is satisfied for min [a(z)], then it will be satisfied for all z. Hence, the two previous equations are satisfied if

4eBφs2amax >1 LefNuH1,∞ 4Xφs2amin > 10 ShH1,∞ where amax and amin are the maximum and minimum values of the distribution function. [Note amax ) max


Figure 14. Cumulative emissions for different catalyst distributions in a circular geometry for the wall reaction case. Parameter values: Peh ) 500, γ ) 25, Lef ) 1, P ) 0.15, B ) 10, φs2 ) 0.0175, θm,in ) 0, and θs0 ) -7.

[a(z)] ) a(0), and for the case of uniform catalyst loading, amax ) amin ) 1.] For typical values of B ) 10, γ ) 25, ShH1,∞ ) NuH1,∞ ) 4.364, and Lef ) 1, the previous conditions can be written as

φs2amax > 0.04

(54a)

φs2amin > 8.62 × 10-3

(54b)

It is easily seen that, for the case of uniform catalyst loading, the minimum catalyst loading (φs2) is 0.04 and, for values below this, it is not possible to have frontend ignition, but back-end/middle ignition can be obtained by choosing the sufficiently small values of P. However, for the case of nonuniform catalyst loading, the previous two conditions can be satisfied for any φs2 > 8.62 × 10-3 (by choosing amin and amax accordingly). Note that, for the case of uniform catalyst loading, the ignition criterion determines the minimum metal loading required whereas, for the case of nonuniform catalyst loading, the criterion for mass-transfer control determines the minimum metal loading. For a local Damko¨hler number of 0.0175 and other typical parameter values previously mentioned, we show the plots of cumulative emissions for different values of amax and compare them with the case of uniform catalyst loading. For φs2 ) 0.0175, uniform catalyst loading favors a (close to) back-end ignition, but by redistributing the catalyst, we could favor front-end ignition. We fix the value of amin so that the entire monolith is in the mass-transfer-controlled regime, yielding amin ) 0.493 (from eq 83), and change amax to find the optimum two-zone distribution. Now using eq 82 we find that amax g 2.29 for front-end ignition. If a two-zone catalyst distribution is chosen using eq 70, it is easily seen that z* e 0.282 and R* e 0.215. We choose different values of z* ( 100 is preferable. The best geometry for cold start (θs0 e -7) is the one which has a higher transfer coefficient because it decreases both the heat-up time and the exit concentration (eq 37). Hence, it is best to choose a parallel-plate channel or a rectangular channel with higher aspect ratios (if feasible). The transverse Peclet number (P) is chosen such that we obtain the desired steady-state conversion. From eq 37, the transverse Peclet number for an exit conversion of 0.995 can be obtained as

Pe

ShT 4 ln(200R1)

The previous expression is obtained assuming that the entire monolith is in the mass-transfer-controlled regime after ignition. For a rectangular geometry with an aspect ratio of 1:3, the optimum P value obtained using the previous expression is approximately 0.2, while for parallel plates, P ) 0.36. As mentioned in the previous section, using a lower value of P than that required would not only increase the pressure drop but may also increase the transient time. The previous criteria can be summarized as

4eBφs2amax >1 LefNuH1,∞

(55)

4Xφs2amin > 10 ShH1,∞

(56)

Peh > 100

(57)

Pe

ShT 4 ln(200R1)

(58)

Hence, for the case where there are no washcoat diffusional limitations (before and after ignition), the optimal monolith design parameters are chosen based


on the above criteria to reduce the transient time and cumulative emissions. It is important to note that the minimum metal loading with nonuniform catalyst distribution along the channel is lower than that for the case of uniform catalyst loading. Hence, for the wall reaction case, nonuniform catalyst loading (distributed such that the previous criteria are satisfied) always leads to lower transient time and lower cumulative emissions than uniform catalyst loading. As shown in Figure 14, for a given amount of catalyst loading and other design and operating variables, we could find an optimum two-zone distribution that minimizes the cumulative emissions. 8.2. Influence of Washcoat Diffusion on the Optimal Design. When washcoat diffusional limitations are not negligible, depending on the catalyst loading and the washcoat thickness, the monolith may be in the washcoat diffusion-limited regime. As before, we require front-end ignition for lower transient time and cumulative emissions. However, for a given catalyst loading, if front-end ignition is favored, then there is a critical washcoat thickness given by φ ) 1/2 up to which the transient performance can be improved, and for washcoat thickness greater than this critical value, the transient performance is not affected. In addition, for good steady-state performance, we want the entire monolith to be in the mass-transfer-controlled regime. The above constraints can be written in mathematical terms as

1 f(φsxamax

4eBφs2amax >1 Λ) LefNuH1,∞

(59a)

φ ) φsxamaxΛ < 0.5

(59b)

4φsxamin xX > 10 ShH1,∞ xΛ

(59c)

The first condition when satisfied favors front-end ignition. The second condition when satisfied means that there is no washcoat diffusional limitation until ignition. The third condition when satisfied forces the entire monolith to be in the mass-transfer-controlled regime after ignition. (Note: Because after ignition the monolith will be in the washcoat diffusion-limited regime, the “tanh” term in eq 49 is assumed to be unity.) For lower transient time and cumulative emissions, the previous three conditions should be satisfied. For typical values of B ) 10, γ ) 25, P ) 0.25, ShH1,∞ ) NuH1,∞ ) 4.364, and Lef ) 1, the previous conditions may be rewritten as

24.91φs2amax

>1

(60a)

φsxamaxΛ < 0.5

(60b)

f(φsxamaxΛ)

32.593φsxamin

xΛ

> 10

(60c)

It is interesting to note that the first two conditions contain both the reaction rate constant (kv), which depends on the metal loading, and the washcoat thickness (δc), whereas the third condition depends only on

the metal loading and not on the washcoat thickness (because φs/xΛ is independent of δc). Hence, the third equation determines the critical metal loading required for the monolith to be in the mass-transfer-controlled regime and is a more stringent condition on the metal loading than the first two conditions. This means that if the third condition is satisfied, we can always find a critical washcoat thickness above which (front-end) ignition will occur in the monolith. Equation 60c defining the minimum metal loading needed for the entire monolith to be in the masstransfer-controlled regime may be written as

φs

xΛ

>

0.3067

xamin

(61)

If the metal loading is taken to be the minimum required value defined previously, then the range of washcoat thickness for which there is no (front-end) ignition, ignition with no washcoat diffusional limitations, and ignition with washcoat diffusional limitations are given by

amin w no (front-end) ignition Λ < 0.426 amax

x

amin amin < Λ < 1.63 w 0.426 amax amax no washcoat diffusion limitation before ignition

x

amin w amax washcoat diffusion limitations before ignition

Λ > 1.63

In all of the previous cases, after ignition the entire monolith will be in the mass-transfer-controlled regime. Unlike the wall reaction case, the minimum metal loading required is given by the mass-transfer-controlled criterion for both uniform and nonuniform catalyst loading. The minimum metal loading required for the case of nonuniform catalyst loading will be higher than that required for uniform catalyst loading because amin is always less than unity. Hence, the maximum feasible metal loading (φs/xΛ) will place constraints on the optimum catalyst distribution. With front-end ignition, it is clear that having the maximum possible amount of catalyst near the inlet improves the transient performance. For any value of the metal loading (φs/xΛ > 0.3067), the value of amin should be chosen such that eq 61 is just satisfied. The value of amax can be chosen to be the maximum possible value (depending on practical restrictions), and this determines the range of values of the washcoat thickness. For example, for the case (φs/ xΛ) ) 0.3067, for the condition to be barely satisfied amin ) 1 and hence amax ) 1; hence, uniform catalyst loading is the best possible distribution. For values of metal loading (φs/xΛ < 0.3067), eq 61 cannot be satisfied for any values of amin e 1 and hence the entire monolith will not be in the mass-transfercontrolled regime. In this case, the value of amax is determined first by satisfying the ignition criterion equation (59a), and the criterion for no washcoat diffusional limitations and the maximum possible value of amin (such that ∫10a(z) dz ) 1) is then found. If the


Figure 15. Cumulative emissions for different catalyst distributions in a circular geometry. Parameter values: Peh ) 500, γ ) 25, Lef ) 1, P ) 0.15, B ) 10, φs2 ) 0.0274, Λ ) 9.1287, θm,in ) 0, and θs0 ) -7.

ignition criterion is already satisfied for amax ) 1 (uniform catalyst loading), then it is better to have uniform catalyst loading because amin will be equal to unity and the entire monolith will be as close to the mass-transfer-controlled regime as possible. Only if the ignition criterion is not satisfied for uniform catalyst loading is the value of amax determined by the ignition criterion. (The value of amax is determined in this case so that we could at least favor front-end ignition so that the transient time will be lower.) In this particular case, where the entire monolith will not be in the masstransfer-controlled regime, the only way to decrease the cumulative emissions is to decrease the transverse Peclet number so as to have better steady-state conversions. However, such a design will be approaching the pseudohomogeneous limit, leading to a higher pressure drop. The previous conditions and criteria can be summarized as

Uniform catalyst loading best for LefNuH1,∞ 2eB

e

φs

xΛ

10ShH1,∞ e

4xX

Nonuniform catalyst loading best for φs

xΛ

10ShH1,∞ 4xX

Optimum washcoat thickness φ ) φsxΛamax ) 0.5 We show in Figure 15 plots of the cumulative emissions for different catalyst distributions for the case where the catalyst loading is less than the minimum required for front-end ignition; i.e., φs/xΛ < LefNuH1,∞/2eB. We

fix Λ ) 9.1287 and φs2 ) 0.0274 so that the washcoat thickness is optimum for the case of uniform catalyst loading. For other typical parameter values, we find amax such that the ignition criterion (eq 60a) is satisfied, and this yields amax > 2.15. Different two-zone catalyst distributions are chosen, the value of z* is fixed, and the corresponding value of amin is calculated such that ∫10a(z) dz ) 1. (Note: Sometimes barely satisfying the criterion for front-end ignition may not favor front-end ignition because of the conservativeness of the criterion.14 It is better if the term in eq 60a exceeds 1.25 or 1.5 so that the monolith will definitely ignite at the front end. Also, because position-dependent transfer coefficients are used in the model and the asymptotic transfer coefficients are used in the ignition criterion, using this factor of 1.25 or 1.5 will take into account this effect and the conservativeness of the criterion.) Figure 15 shows the cumulative emissions plot for different values of z* and for the uniform catalyst distribution. As expected, uniform catalyst distribution gives the highest cumulative emissions. However, among the different two-zone catalyst distributions, the optimum distribution depends on the time as the curves intersect each other. This is because when the monolith is not in the mass-transfer-controlled regime, the steadystate conversion is not very high and the slope of the steady-state asymptote in the cumulative emissions plot is not small. The steady-state conversions are different for the distributions. Hence, the optimum distribution will depend on the amount of time the monolith is in use. However, the important point is that nonuniform catalyst distributions give much lower cumulative emissions than the uniform catalyst distribution when φs/ xΛ < LefNuH1,∞/2eB. Though not shown here (in plots), we see that uniform catalyst loading is better when LefNuH1,∞/2eB e φs/xΛ e 10ShH1,∞/4xX because front-end ignition will be favored and the steady-state conversion will be higher than that obtained using a nonuniform catalyst distribution. Though nonuniform catalyst loading will reduce the ignition time by having more catalyst at the front, the steady-state conversion for nonuniform loading will be less because the term on the left-hand side of eq 60c will be much lower because of amin, whereas for the case of uniform catalyst loading, this term will be higher and the monolith will be as close to the masstransfer-controlled regime as possible. [As shown below, the ignition time does not reduce significantly when there is more catalyst at the front (than that required for front-end ignition).] When the catalyst loading φs/xΛ > 10ShH1,∞/4xX, then it is easily seen that nonuniform catalyst loading is always better. This is because both uniform and nonuniform catalyst loading will favor front-end ignition and the entire monolith will be in the mass-transfer-controlled regime. The only way to reduce the cumulative emissions is to reduce the ignition time, which can be done by distributing the excess catalyst loading (than that required for attaining the mass-transfer-controlled regime) near the inlet. As in the wall reaction case, the solid conduction and the transverse Peclet number are chosen such that eqs 57 and 58 are satisfied. Again, it should be noted that when washcoat diffusional limitations are negligible before and after ignition, then nonuniform catalyst loading is always better.


8.3. Transient Time. For the case of front-end ignition, we can evaluate the solid-phase energy balance at the inlet where the solid conduction term can be neglected and θm ≈ θm,in and cm ≈ cm,in, and this equation can be used to approximate the transient time. The transient time may be expressed as

τig )

∫θθ

dθs (62) LefNuH1,∞ Bcm,inΘ (θs - θm,in) 4P

s,trans

s0

where

Θ)

a(0) φs2 exp[θs/(1 + θs/γ)] P{Φ(0)/tanh[Φ(0)]} + [4Φ(0)2/ΛShH1,∞] Φ(0)2 ) φs2a(0) Λ exp

[

θs 1 + θs/γ

]

(63)

(64)

and θs,trans is the root of the equation (the temperature at which the two terms in the denominator of the Θ expression are of equal magnitude)

x [ ] ( x [ ])

φs

a(0) Λ exp

φs

θs

θs 1+ γ

Figure 16. Transient time as a function of the initial solid temperature and the amount of catalyst at the inlet [φs2a(0)] for a circular channel. Parameter values: γ ) 25, Lef ) 1, B ) 10, Λ ) 0.004, and θm,in ) 0.

tanh

a(0) Λ exp

θs

θs 1+ γ

)

ΛShH1,∞ 4

(65)

In eq 62, θs0 represents the initial solid temperature, a(0) represents the magnitude of the distribution function at the inlet, and θs,trans represents the transition temperature (transition from kinetically controlled to mass-transfer-controlled regime). Because the value of P would be chosen so as to have 99.5% conversion, the steady-state asymptote will essentially be flat (with slope 0.005) and hence the cumulative emissions will depend only on the transient time. In the transient asymptote, the cumulative emission is given by eq 36 and depends only on the transient time, which can be evaluated using eq 62. Figure 16 shows the influence of the initial solid temperature (θs0) and the amount of catalyst at the inlet [φs2a(0)] on the transient time. As the amount of catalyst at the inlet increases, the transient time continues to decrease. When the initial solid is very cold, the effect of the factor [φs2a(0)] is more pronounced. For higher initial solid temperatures, the amount of the catalyst at the inlet does not affect the transient time significantly. (Note that the transverse Peclet number P can be factored out in the previous expression for the transient time and the figure shows the scaled transient time tig ) τig/P.) 9. Conclusions and Discussion We have presented a detailed transient and cumulative emissions analysis of a catalytic monolith with uniform as well as nonuniform catalyst distribution for the case of a single exothermic reaction. The influence of design parameters such as the washcoat thickness, channel geometry, catalyst loading and distribution, and solid-phase conductivity on the cumulative emissions is

studied to arrive at some simple design criteria for catalytic monoliths that have good transient and steadystate performance. As can be seen intuitively, the optimum monolith design for reducing cumulative emissions depends on the maximum possible catalyst loading (and hence is linked to washcoat design and properties), the time horizon (or the time at which cumulative emissions are compared), and the maximum allowable pressure drop. If pressure drop is not a concern and the time horizon is long (τ . 1 or, in practical terms, cumulative emissions are compared at 30 min or larger), then optimal designs correspond to operation close to the pseudohomogeneous limit. For such designs, though the transient time is longer (and cumulative emissions are higher for small τ), the slope of the steady-state asymptote given by eq 44c or eq 44b can be very close to zero and any other design having a finite slope will lead to higher cumulative emissions (for τ f ∞). On the other hand, when we place constraints on the pressure drop and have a shorter time horizon (τ values around 10 or, in practical terms, a time limit of 15 min or less), then optimal designs correspond to minimizing the transient time and operating close to the mass-transfercontrolled regime. In this work, we have focused mostly on optimal designs with fixed pressure drop. Even with this constraint, the optimal design depends strongly on washcoat properties (φs2 and Λ or, equivalently, the maximum possible metal loading that depends on the washcoat internal area or pore size and effective diffusivity in the washcoat and the thickness). For the wall reaction case, we have shown that the optimal design always corresponds to front-end ignition with more catalyst at the inlet. For realistic cases of finite washcoat thickness and metal loading, the optimal design corresponds to front-end ignition but the catalyst distribution along the channel may or may not be uniform. The simple analytical criteria given here may be used for


evaluating and designing monoliths if the washcoat properties and the reaction parameters are known. The analysis presented here can be extended in two major directions. First, the single reaction linear kinetics used here can be replaced by multiple reactions with nonlinear kinetics (e.g., Langmuir-Hinshelwood type) or more detailed microkinetic models accounting for finite rates of adsorption, desorption, and interactions at the catalytic site level. Second, this work can be extended to cases of more complex washcoat compositions where the metal loading varies with the depth in the washcoat or there exist several different types of active sites. The resulting mathematical models are expected to be computationally demanding, but the insight obtained from the detailed study of the simpler case presented in this work can be useful in guiding the numerical simulations.

δc ) effective thickness of the washcoat δs ) effective thickness of the support δw ) effective wall thickness η ) effectiveness factor θ ) dimensionless temperature Λ ) dimensionless washcoat thickness µ ) ramp rate ξ ) dimensionless coordinate for washcoat diffusion τ ) dimensionless time φs ) Thiele modulus (local Damko¨hler number) Φ ) washcoat Thiele modulus at the local solid temperature φ ) washcoat diffusion Thiele modulus Subscripts and Superscripts f ) fluid phase m ) cup mixing s ) solid phase

Acknowledgment This paper is dedicated to Professor Gerhardt Eigenberger whose pioneering work on the transient behavior of catalytic reactors has been an inspiration to us. This work was supported by grants from the Robert A. Welch Foundation, Texas Advanced Technology Program, and The Dow Chemical Co. Notation Roman Letters a(z) ) catalyst activity distribution function B ) dimensionless adiabatic temperature rise c ) dimensionless concentration C ) concentration Cm,in ) inlet fluid concentration De ) effective diffusivity of the reactant in the washcoat Dm ) diffusion coefficient in the fluid phase Da ) monolith Damko¨hler number E ) activation energy Eu ) Euler number (or dimensionless pressure drop) h(x) ) position-dependent heat-transfer coefficient kc(x) ) position-dependent mass-transfer coefficient kf ) fluid thermal conductivity kw ) solid thermal conductivity kv ) first-order rate constant per unit washcoat volume L ) length of the monolith channel Lef ) fluid Lewis number Nu(z) ) Nusselt number P ) transverse Peclet number Peh ) heat Peclet number Pr ) Prandtl number Rg ) universal gas constant RΩ ) transverse diffusion length ()half the channel hydraulic radius) Re ) Reynolds number Sc ) Schmidt number Sh(z) ) Sherwood number T ) fluid temperature Tf,in ) inlet fluid temperature u j ) average fluid velocity in the channel Vo ) volumetric gas flow rate x ) coordinate along the length of the channel y ) distance coordinate in the washcoat z ) dimensionless coordinate along the length of the channel Greek Letters R ) monolith aspect ratio (4RΩ/L) R1 ) Fourier weight γ ) dimensionless activation energy

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Received for review September 16, 2003 Revised manuscript received January 6, 2004 Accepted January 13, 2004 IE034131V

Light-off and Cumulative Emissions in Catalytic Monoliths with

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