## Abstract

An analytical transport/reaction model was developed to simulate the catalytic performance of ZnO nanowires as a catalyst support. ZnO nanowires were chosen because they have easily characterized, controllable features and a spatially uniform morphology. The analytical model couples convection in the catalyst flow channel with reaction and diffusion in the porous substrate material; it was developed to show that a simple analytical model with physics-based mass transport and empirical kinetics can be used to capture the essential physics involved in catalytic conversion of hydrocarbons. The model was effective at predicting species conversion efficiency over a range of temperature and flow rate. The model clarifies the relationship between advection, bulk diffusion, pore diffusion, and kinetics. The model was used to optimize the geometry of the experimental catalyst for which it predicted that maximum species conversion density for fixed catalyst surface occurred at a channel height of 520 *μ*m.

## Introduction

The cost of catalyst metals is continuously rising as demand increases due, in part, to tightening emissions regulations across the globe. Because of this, there is great incentive to use catalyst metals more effectively for the remediation of combustion emissions. This paper describes modeling work supported by experimental results investigating the performance of a platinum/palladium combustion emissions catalyst that uses ZnO nanowires as a support medium. The experimental results were modeled with an analytical species transport model that couples convection in the catalyst channel with diffusion and reaction in the catalyst substrate media.

While empirical models, semi-empirical models, numerical models, and commercial multiphysics packages have been used to predict catalyst behavior, the utility of two-dimensional analytical models that predict catalyst performance as a function of channel height and length and species concentration in both stream-wise and transverse dimensions has been sparsely addressed in the literature. Many previous studies have modeled catalyst performance using numerical and/or empirical models. A semi-empirical model for three-way catalysts with transient submodels and empirical mapping for species conservation was presented by Laing et al. at Ford Motor Company [1]. Numerous other models have accounted for heat transfer, transient behavior, oxygen storage, and other effects. Koltsakis et al. developed a model that considers transient heat transfer and transient temperature-dependent kinetics. The species concentration was spatially lumped in the transverse direction so only stream-wise transport was considered [2]. Lambert et al. at Ford Motor Company have done modeling and experimental investigations of NO_{x} catalysts in compression ignition engines. Their goal was to develop a model that can be integrated in a larger vehicle systems model. The model, known as SIMTWC, predicts catalyst performance as a function of volumetric throughput and temperature; the model accounts for heat transfer, transient species storage, and chemical kinetics using empirical submodels [3]. Heck and Farrauto have shown that a one-dimensional model that predicts stream-wise species concentration that is lumped in the transverse direction can provide relatively accurate results, but this model requires the Sherwood number, a dimensionless mass transfer coefficient that is equivalent to the Nusselt number, and other nondimensional numbers that have been obtained from empirical studies [4]. There has been some work in catalyst optimization using combined analytical and/or numerical models with empirical maps of operating conditions by Katare et al. [5,6]. These efforts are important for the design and manufacturing of industrial catalysts where rapid, accurate results are needed in order to meet tightening emissions standards and the rapid pace of production. This type of work would likely benefit from an accurate, fast, analytical species transport model because such a model would reduce the number of experiments needed to determine the optimal catalyst design.

Bhattacharya et al., as part of the catalyst group in Chemical Engineering at University of Houston, were the first to address the need for a two-dimensional model that coupled species convection in the stream-wise and transverse directions in the channel of a catalyst with species diffusion and reaction in the wash-coat [7,8]. They solved the partial differential equation (PDE) governing the physics of convection in the channel using separation of variables and a Fourier series expansion. This model also included the effects of various wash-coat shapes and channel shapes. Their model solved the PDE for discrete values of chemical kinetics, and they could model continuous kinetics for asymptotic conditions with extreme values of transverse Peclet number near zero or infinity.

Work done by Joshi et al. in the same group has demonstrated the potential usefulness of a low-dimensional model for real-time modeling and control of vehicle emissions systems [9–11]. This work presented a means of analytically determining Sherwood number based on duct geometry, flow rate, temperature, wash-coat geometry, wash-coat properties, and catalyst metal properties. This model was validated by comparison to the analytical model presented earlier by Bhattacharya et al. [7] and comparison to a COMSOL model. This work is valuable for several reasons: the model provides insight into which mechanism, out of kinetics, pore diffusion, or bulk diffusion, is the most dominant in limiting species conversion; parabolic flow is considered; and the 1D model can be used over a range of flow and temperature operating conditions.

The modeling portion of this work solves the species transport PDE in the catalyst channel and couples it to the reaction/diffusion ordinary differential equation (ODE) in the nanowire (porous media) region. The PDE is solved using separation of variables and Fourier series expansion. The solution technique includes a systematic procedure to rapidly solve for the eigenvalues that allows for nondiscrete, continuous values of Thiele modulus, i.e., a dimensionless ratio of reaction rate to diffusion rate within the substrate media, over a continuous range. The procedure to solve the PDE for continuous values of kinetics and Peclet number is a key contribution of this work. This will be discussed in Sec. 4.2. The objective of the modeling work is to develop and demonstrate a simple, yet powerful, analytical design tool for optimizing catalytic reactors.

The motivation for the experimental part of this research was to use a material system with a relatively spatially uniform distribution of features such as pore size, support layer thickness, nanowire diameter, and nanowire length; and a low tortuosity. In ZnO nanowires, these properties can be precisely controlled so that the catalyst support material can be characterized well enough to provide inputs for the analytical model to fit the data. Procedures for growing ZnO nanowires on a wide variety of substrates are available in the literature [12–16]. Typical nanowire diameters produced by these methods range from 20 nm to 100 nm. In addition, ZnO has excellent thermal stability [17,12], making it appropriate for the high temperature engine exhaust environment.

The concept of using substrates with nanostructures, including nanowires, as combustion catalyst supports has been studied previously with some success by Guo et al. [18]. In their novel investigation, Guo et al. synthesized SiC nanowires to serve as supports for Pd to be used in methane oxidation. They discovered that, due the presence of periodic stacking faults along the length of the nanowires, small circumferential ditches, or grooves, can be etched in the nanowires. These grooves serve as a geometrical barrier to Pd agglomeration, thus preserving uniform Pd distribution within the nanowire support. The work presented here did not explore the use of crystallographic flaws in the nanowires as a means of inhibiting catalyst metal agglomeration, but this work did provide a useful means of experimentally validating an analytical model that was developed concurrently with the experimental work.

Nanowires will likely have different pore-region diffusion of reactants in the catalyst media because they can have greater effective porosity and average diffusion length scale than traditional γ-alumina. Numerous studies by Coppens's group have demonstrated the importance of optimizing pore morphology. These studies have used analytical and numerical models to explore the effects of fractal pore geometries, bimodal pore size distributions, continuous pore size distributions, and other pore morphology parameters on diffusion in porous substrate media [19–22]. Coppens et al. sometimes refer to certain nanoscopic fractal structures as fjords. In light of the fact that Coppens in most likely Scandinavian, this is somewhat humorous, and probably intentional.

One complication associated with the use of zinc oxide is the possibility of strong metal support interaction (SMSI) between the nanowires and catalyst metal affecting hydrocarbon oxidation and/or sintering performance [23]. SMSI can explain a range of effects including hindering the rate of catalyst metal agglomeration, increasing the rate of catalyst metal agglomeration, hindering the rate of species conversion, or increasing the rate of species conversion. The affect of SMSI on hydrocarbon oxidation was not directly addressed in this research because this type of SMSI effect was expected to be small since the fraction of catalyst sites on each Pt/Pd particle that were in contact with the nanowire surface was small. Also, this research was not intended to explore the industrial usefulness of ZnO nanowires as a catalyst support so agglomeration, i.e., sintering, performance was not studied.

The experiments were used to empirically validate the model. In all experiments and modeling, the hydrocarbon under consideration was propane, and we assumed first order kinetics [24].

## Experimental

### Nanowire Growth.

Zinc oxide nanowires were grown on flat Si wafers using an aqueous solution technique from Greene et al. [13–15]. First, growth sites for the nanowires had to be created. An ethanol solution with 25 mM of zinc acetate dihydrate was prepared. Next, the zinc acetate solution was dropped onto a 20 mm × 76.2 mm Si wafer and allowed to remain on the wafer for 10 s. The wafer was then rinsed with pure ethanol and blown dry with air. These steps were repeated four times for a total of five drop, rinse, dry cycles. After the zinc acetate washing process was complete, the wafer was placed into a furnace at 300 °C for 30 min. This created ZnO seeds that would subsequently serve as growth sites.

The actual growth process occurred in an aqueous solution of 500 ml de-ionized water that was prepared with 25 mM zinc nitrate hexahydrate, 25 mM hexmethylenetetraammine, and 5 mM polyethyleneamine. The solution was placed in a beaker on top of a hot plate, and the wafer was placed facing slightly downward at a ∼85 deg angle from the bottom of the container. Placing the wafer facing slightly downward at an angle prevented bubbles from forming on the downward facing growth surface while also preventing precipitate from accumulating on the growth face. The hot plate was set to maintain the bath at 90 °C for 3 h and this was repeated two times with fresh solution to ensure long nanowires were grown. After the bathing process, the surface was rinsed using tap water to remove any precipitate from the nanowires.

### Catalyst Preparation.

The first step in preparing for catalyst testing was sputter coating the nanowires using an 80%Pt/20%Pd sputter target in a sputter coater with a quartz crystal deposition monitor. The quartz crystal deposition monitor is a disc that is coated during the sputter process, and as more material is deposited on the quartz surface, the frequency at which the quartz vibrates is reduced. The mass of material that is deposited on the surface can be determined based on the change in frequency, and the film thickness can be determined if the material density is known. This thickness is based on an assumed uniform film of sputtered material. In the context of sputter coating nanowires, the sputter thickness is the amount of material that would form a film of the desired thickness if it were uniformly deposited on a flat surface without any nano-scale texture. This method for coating the catalysts with Pt/Pd was chosen because it provided an easy way to determine the loading of catalyst metal. Scanning electron microscopy (SEM) and transmission electron microscopy (TEM) were used to qualitatively determine nanowire morphology, catalyst particle size, and catalyst particle distribution.

### Catalyst Testing.

The hydrocarbon conversion efficiency was tested using a temperature controlled tube furnace and a Horiba Mexa-584 L exhaust analyzer that measured hydrocarbon, carbon monoxide, carbon dioxide, and oxygen concentrations. A system schematic of the catalyst test rig and a detailed view of the furnace are shown in Figs. 1(a) and 1(b), respectively. The test rig consists of propane and air flowing through a mass flow controller and a rotameter, respectively. Then the propane and air are mixed, and the combined fuel/air mixture is allowed to flow either through or around the furnace to a hydrocarbon analyzer. The inlet gas was a mixture of propane and air with a fuel-air equivalence ratio of approximately 0.1, or a concentration of approximately 4200 ppm. This fuel/air mixture was used in all experiments and simulations in this work.

Conversion efficiency was determined from the measured hydrocarbon concentrations upstream and downstream of the furnace. The hydrocarbon analyzer manufacturer's stated measurement accuracy was ±5%, but the reproducibility of the measurements showed a precision uncertainty of less than 1%. The hydrocarbon conversion efficiency was based solely on the difference in concentrations so the uncertainty of the conversion efficiency was a function of the precision uncertainty only. The air flow was controlled using a manually calibrated rotameter with an uncertainty of ±2% at 500 sccm which was the estimated uncertainty of the total flow rate since the contribution of the propane flow rate uncertainty was negligible. The calculated uncertainty in the conversion efficiency was less than ±2%. The temperature variation immediately upstream and immediately downstream of the reactor was measured using thermocouples; the temperature rise was less than 2 °C with a fuel/air mixture flowing, a furnace set point of 450 °C, and approximately 25% hydrocarbon conversion.

Two wafers were placed in series on the bottom surface of the alumina sample holder used for Si wafers. The reactor length was 152.4 mm or twice the length of a single Si wafer. The holder consisted of an alumina rod with a 3 mm flow slot machined into the side. The flow slot was a narrow slot machined along the length of the alumina insert. It allowed the nanowire coated Si wafers to be inserted into the alumina insert; the width of the slot determined the height between the surface of the Si wafer and the wall above it. The alumina insert was placed in the quartz tube. The flow entered and exited the alumina insert via a round hole cut in each end of the alumina insert. The flow passed through the hole on the inlet side to the slot, and from the slot on the outlet side through the hole on the exit side. A photo and schematic of the alumina sample holder are shown in Figs. 2(a) and 2(b). The flow entered and exited the alumina sample holder via 5 mm holes that were drilled in both ends of the alumina sample holder. After entering the alumina holder, the fuel/air mixture then flowed over the two Si wafers in the 2.5 mm gap. The gap for the flow was the gap height of the alumina sample holder minus the thickness of the Si wafer.

## Results and Discussion

### Substrate Characterization.

SEM was used to determine the nanowire areal density, length, diameter, and an estimate of the porosity of the nanowire material; an SEM image of nanowires grown on a silicon wafer is shown in Fig. 3. These nanowires were typically 2 *μ*m in length with an average diameter of approximately 100 nm. This results in a calculated specific surface area of $7(\u2003m2/g)$. The nanowire coverage was quite dense over the entire substrate, with approximately 10 nanowires per 1 *μ*m^{2}.

TEM was used to qualitatively determine nanowire size, Pt/Pd particle size, and Pt/Pd particle distribution. A TEM image of a nanowire sputter coated with 1 nm Pt/Pd is shown in Fig. 4. In Fig. 4, the average particle size is about 3 nm, and the average particle spacing is also about 3 nm. Sputter coating is a process that utilizes ballistic and diffusional transport of Pt/Pd particles from the sputter target (the source material) to the sample on which deposition is occurring. The result of the ballistic effects is that nanowire surfaces facing the sputter target are more densely coated than surfaces facing away from the sputter target, causing a shadow effect. This shadow effect can be seen on the nanowires in Fig. 4.

### Catalyst Performance.

The hydrocarbon analyzer was used to determine hydrocarbon conversion efficiency by measuring the percentage decrease in hydrocarbon concentration downstream versus upstream of the catalyst. Conversion efficiency was found to be a function of the sputter thickness of the Pt/Pd layer. As an example, a plot of hydrocarbon conversion efficiency versus Pt/Pd sputter thickness at 450 °C and 500 sccm of fuel/air flow rate is shown in Fig. 5. Figure 5 shows that increasing Pt/Pd loading initially increases conversion efficiency until 20 nm of Pt/Pd loading; then at some point between 20 nm and 50 nm, the conversion efficiency decreases with further Pt/Pd deposition. A likely cause of this precipitous decrease in conversion efficiency is agglomeration of Pt/Pd particles during the deposition process, resulting in larger particles, which reduces the total amount of exposed area of the particles. The larger particles also might have reduced surface kinetic rates [25]. SEM images showing the increase in particle agglomeration associated with increasing Pt/Pd sputter amounts are shown in Figs. 6(a) and 6(b). As observed in Figs. 6(a) and 6(b), for the case of higher metal loading, the Pt/Pd particles begin to agglomerate and lose available surface area.

The effect of temperature and flow rate on conversion efficiency is shown in Fig. 7. Shown are three flow rates for Si wafers with nanowires sputter coated with 10 nm Pt/Pd.

In Fig. 7, flow rates correspond to space velocities of 1640 h^{–1}, 3281 h^{–1}, and 4921 h^{–1}. With increasing flow rate, the conversion efficiency is decreased because the residence time in the reactor is decreased.

In order to understand the significance of transport considerations, an analytical model that couples convective species transport in the channel with diffusive species transport and simplified chemical reaction in the porous media was developed.

## Model

The species concentration within the catalyst channel and porous media of a catalyst consisting of two parallel plates was modeled to understand the underlying transport physics governing the performance of the nanowire catalysts. A schematic of the system that was modeled is shown in Fig. 8. The purpose of this modeling effort was to develop a means of comparing different catalyst designs and to create a design tool for optimizing catalyst performance as a function of flow rate, temperature, channel geometry, wash-coat thickness, substrate morphology, catalyst metal loading, and catalyst metal activity. A benefit of this would be the ability to compare the performance of different real catalysts without relying on space velocity. The parallel plate model was used to predict the results for a flat catalyst plate in internal flow, as used in the experiments.

### Governing Equations.

In Eq. (1), *ρ* is the gas density, *u* is the stream-wise bulk velocity, *x* is the stream-wise coordinate, $D$ is the bulk diffusivity of the species of interest into air, *Y* is the mass fraction of the species of interest, and *y* is the perpendicular coordinate. The mass fraction, *Y* (*x*, *y*), was resolved in both the stream-wise (*x*) and transverse (*y*) directions.

*y*,

*h*is the height of the flow channel [

*m*], $x\u02dc$ is the dimensionless form of the stream-wise coordinate

*x*, $Y\u02dc$ is the scaled form of the species mass fraction

*Y*, and

*Y*

_{0}is the inlet mass fraction of the reactant species. The result of applying these scaling variables is a simplified equation for species concentration in the channel

where $Peh=(uh/D)$ is the transverse Peclet number with respect to duct height.

*y** is a coordinate system defined as zero at the edge of the porous media and increasing in the direction of the center-line of the flow channel, and $kpore[1/s]$ is the effective volumetric reaction rate coefficient in the porous media. Equation (6) is a variation of the equation for mass balance in porous media given by Morbidelli et al. [27]. Equation (6) can be nondimensionalized by defining the following scaling variable:

where *h*_{pore} [m] is the height of the porous media.

where $\phi \u2261(kporehpore2/Dpore)$ is the Thiele modulus within the porous media [27].

### Solution.

*λ*is the

_{n}*n*th eigenvalue and

*A*and

_{n}*B*are the

_{n}*n*th Fourier coefficients. The symmetry condition along the center-line of the channel, given by

where *η* is hydrocarbon conversion efficiency. A novel procedure was employed to solve for the eigenvalues as a function of chemical kinetics. This is explained in detail in Eq. (6), but a brief explanation will be given here. First, eigenvalues were found graphically and tabulated. These tabulated values were then used with a spline fit algorithm to determine the eigenvalues for arbitrary kinetics. Lastly, the spline fit was used to provide an initial guess for a numerical root finding solver which accurately solved for the exact eigenvalues beyond eight significant figures. This was the most important contribution of this work.

A detailed solution of the model, including closure for the transport and kinetics parameters, is presented in Appendix A.

### Meaning of Dimensionless Parameters.

Before proceeding, it is pedagogically important to discuss the significance of the various nondimensional parameters associated with this model. The nondimensional parameters are highly useful because they provide a means of discerning which physical mechanisms (chemical kinetics, pore diffusion, and/or bulk diffusion) are most important in limiting the performance of a catalyst. The relevant nondimensional parameters are the transverse Peclet number, Pe* _{n}*, the Dahmköhler number, Da, and the Thiele modulus,

*ϕ*. Pe

*is the ratio of stream-wise advection relative to transverse (bulk) diffusion, Da is the ratio of effective surface chemical kinetics at the wall relative to transverse (bulk) diffusion in the flow, and*

_{h}*ϕ*is the ratio of effective volumetric chemical kinetics in the porous media (nanowires or

*γ*-alumina wash-coat) relative to pore diffusion. Based on these definitions, Fig. 9 shows some useful observations that can be made from these nondimensional parameters.

### Model Results.

Experimental results for conversion efficiency versus temperature were used to calibrate the kinetic fit parameters in Eq. (A17), and the model was used to predict conversion efficiency versus temperature for various flow rates for which experimental data were available. For all results, a four term approximation was used in the Fourier series expansion. Values used for key parameters in the model are shown in Table 1, and parameters used for calculation of the diffusion coefficients are shown in Table 2. The results are shown in Fig. 10.

Diffusion | A | |||||
---|---|---|---|---|---|---|

Parameter | Coefficient, $D$ | Porosity, ϵ | Tortuosity, τ | Wash-Coat | Thickness, h_{pore} | T_{a} |

Value | 18.9 mm^{2}s^{–1}@300 °C | 0.97 | 1.03 | 5 μm | 201.5 × 10^{3}^{ }s^{–1} | 5,739 K |

26.7 mm^{2}s^{–1}@450 °C |

Diffusion | A | |||||
---|---|---|---|---|---|---|

Parameter | Coefficient, $D$ | Porosity, ϵ | Tortuosity, τ | Wash-Coat | Thickness, h_{pore} | T_{a} |

Value | 18.9 mm^{2}s^{–1}@300 °C | 0.97 | 1.03 | 5 μm | 201.5 × 10^{3}^{ }s^{–1} | 5,739 K |

26.7 mm^{2}s^{–1}@450 °C |

Species | Molecular weight (g/mol) | Effective molecular diameter (Å) |
---|---|---|

Air | 28.964 | 3.617 |

Propane | 44.10 | 4.934 |

Species | Molecular weight (g/mol) | Effective molecular diameter (Å) |
---|---|---|

Air | 28.964 | 3.617 |

Propane | 44.10 | 4.934 |

In Fig. 10, the purpose of the 1000 sccm control experiment was to observe HC conversion not due to catalytic reaction on the nanowire surface. The control experiment consisted of the same rig with the alumina holder, but no catalyst samples were placed inside of the alumina holder. The flow rate of 1000 sccm was chosen because this resulted in approximately the same velocity in the catalyst channel as was the case with catalyst samples in the holder. In the control experiment, all reactions were considered to be bulk reactions or catalytic reactions on the alumina holder.

As shown in Fig. 10, the fit of the model to the experimental data is quite good through a temperature of 425 ∘C. For higher temperatures the model was observed to underpredict the experimental hydrocarbon conversion efficiency. This may have been the result of homogeneous gas-phase reactions in the bulk flow; the control plot shows that these reactions begin to occur at around 500 ∘C.

A parameterized plot of conversion efficiency versus temperature for four flow rates for experimental and model results is shown in Fig. 11. Note that the parameters were calibrated at only the 250 sccm condition. Figure 11 shows good trendwise correlation with the data, illustrating the model's ability to capture the integrated effects of transport even with the limitation of simple one-step kinetics. Its ability to fit the experimental data suggests that with further development for specific applications it may be useful for future catalyst design. For the experimental data, values of the nondimensional parameters are shown in Table 3. For each temperature, the minimum and maximum values of Pe* _{h}* are specified, and the values are proportional to the flow rates. Note that Pe

*is weakly dependent on temperature because specific volume and diffusion coefficient both increase with increasing temperature, though not with the same order.*

_{h}T | Pe (flow rate dependent)_{h} | Da | ϕ |
---|---|---|---|

300 | 11.0 to 42.4 | 2.99 × 10^{–3} | 14.9 × 10^{–6} |

375 | 10.3 to 41.5 | 7.92 × 10^{–3} | 42.4 × 10^{–6} |

450 | 9.79 to 39.2 | 16.8 × 10^{–3} | 96.4 × 10^{–6} |

T | Pe (flow rate dependent)_{h} | Da | ϕ |
---|---|---|---|

300 | 11.0 to 42.4 | 2.99 × 10^{–3} | 14.9 × 10^{–6} |

375 | 10.3 to 41.5 | 7.92 × 10^{–3} | 42.4 × 10^{–6} |

450 | 9.79 to 39.2 | 16.8 × 10^{–3} | 96.4 × 10^{–6} |

The model provides a convenient means of comparing the relative importance of bulk diffusion, pore diffusion, and chemical kinetics through the Dahmköhler number and the Thiele modulus as shown in Fig. 9.

For the experimental conditions, the Reynolds number was between 5.63 and 7.19. The values of Reynolds and Peclet numbers can be used to estimate the entrance lengths as 0.07% of the channel length for fully developed flow and 0.17% of channel length for fully developed species concentration. This means that entrance effects were generally negligible, and thus, only a few terms of the Fourier series expansion were required to satisfy the inlet boundary condition. Based on the values of Pe* _{h}*, Da, and

*ϕ*, all experiments were in the kinetics limited regime (lower right quadrant of Fig. 9). This regime resulted in a species concentration profile that was nearly laterally uniform within the channel. As reactants were depleted at the wall, lateral diffusion from the bulk flow replenished these reactants sufficiently fast that the lateral gradient in species concentration was small. Thus, the plug flow assumption was reasonable because the shape of the transverse species concentration profile was not dependent on the flow rate for the range of flow rates in the experiments. This was also supported by a numerical model that accounted for a fully developed parabolic flow profile that will be shown and discussed in Appendix B.2.

In the future, this model might be useful as a means of validating kinetic mechanisms that have been determined by vacuum chamber techniques using the equation relating volumetric reaction rate coefficient in the porous media with reaction rate coefficient on the particle surface (Eq. (A18)) to back out the local surface kinetics for a particular reactant species and catalyst metal [28].

### Optimization of Channel Height.

In the experiments the species conversion efficiencies were limited by kinetics, but nonetheless, there is some benefit in optimizing channel geometry. As an example of the model's potential usefulness for catalyst monolith optimization, it was used to predict how diffusion effects vary with channel geometry and to determine the channel geometry that maximizes species conversion per unit volume, i.e., species conversion density, for a catalytic reactor of fixed catalyst surface area. This optimization strategy minimizes package size of a catalytic reactor for a given amount of catalyst metal and support material.

Bejan [29], da Silva, and others [30–32] have developed models and optimization techniques that are highly effective for maximizing compact heat exchanger performance for a given package space and pumping power requirement. Given the analogy between heat and mass transfer, these optimization techniques are relevant for catalytic reactors. The heat transfer analysis of Bejan [29] was adapted to species transport and used here to optimize the species conversion density of the catalytic reactor.

*μ*m. The 16.7 cm/s velocity corresponds to 500 sccm for the 2.5 mm plate spacing. The velocity was calculated using

*t*is the plate thickness, and

*U*and

*h*have been previously defined. As

*h*approaches zero, velocity approaches infinity. The temperature was 400 °C. A plot of conversion efficiency per unit volume versus channel height between adjacent catalyst plates is shown in Fig. 12

This shows that the optimum conversion efficiency density occurs at a channel height of 520 *μ*m. For channel height less than 520 *μ*m, Pe* _{h}* increases asymptotically with decreasing channel height. This is because the flow volume approaches zero as channel height approaches zero, causing an asymptotic increase in velocity and space velocity (inverse of residence time). Consequently, the residence time becomes too short relative to the transverse diffusion time to achieve high conversion rates. For channel height greater than 520

*μ*m, Pe

*decreases with increasing channel height and Da increases linearly. As channel height increases beyond 520*

_{h}*μ*m, species conversion efficiency increases, but because the volume of the reactor increases faster, the species conversion density decreases. A more useful application of this optimization strategy should be employed to account for pressure drop effects, but this is beyond the scope of this work.

## Conclusions

An analytical model was developed to demonstrate that a physics-based, analytical transport submodel along with an empirical chemical kinetics submodel can predict catalyst performance remarkably well. ZnO nanowires were used as a support material in fundamental experiments that were used to calibrate and validate the model. The model showed that separation of variables with a spline fit and numerical solver to determine the eigenvalues for continuous values of kinetics can effectively be used to fit experimental data using empirically determined one step Arrhenius kinetics. After fitting the chemical kinetics parameters to a single set of experimental results at constant flow rate and varied temperature, the agreement between the model and several sets of experimental results at several different flow rates demonstrated that a simple analytical model can capture the essential physics of the catalyst system remarkably well. This was further validated by comparing the analytical model to a finite difference model with more detailed physics.

The model results implied that the hydrocarbon conversion efficiency in the experimental results was rate limited by chemical kinetics. The model provides a straightforward way of predicting conversion efficiency and species concentration profile as a function of channel height, channel length, and wash-coat thickness or nanowire length when catalyst particle size and catalyst particle surface kinetics are fixed for a catalytic reactor with known flow rate and temperature operating conditions. In addition, the model clarifies the relationship between important nondimensional parameters such as Peclet number, Dahmköhler number, and Thiele modulus. The model has shown that maximum species conversion density for fixed catalyst surface area in a reactor consisting of stacked parallel plates occurs at a channel height of 520 *μ*m.

## Acknowledgment

Support for this research was provided by the Texas Hazardous Waste Research Center, Lamar University, Beaumont, TX. The authors would like to acknowledge the helpful comments of Gordon Bartley at Southwest Research Institute and Kerrie Gath at Texas State University. The authors gratefully acknowledge the help from Bob McCabe and Hung-wen Jen at Ford Motor Company.

### Appendix A: Detailed Model Solution

##### Applying Boundary Conditions.

*λ*, and the Fourier coefficients,

_{n}*A*, the model for the channel must be coupled to the model for the porous media. To do this, two interface boundary conditions are needed. First, the species concentration must be the same for both regions at the interface

_{n}*C*

_{1}and

*C*

_{2}are integration constants that are not yet known. The outer edge of the porous media has an impermeable wall boundary condition

which provides the solution for the eigenvalues.

To rapidly solve the model for an arbitrary value of Da, eigenvalues were tabulated as a function of Da, and a spline interpolation function was used to estimate the eigenvalues, *λ _{n}*. The result from this spline interpolation was then used to provide a starting estimate for a numerical solver that determined the exact eigenvalues with a higher level of accuracy. The numerical solver could not be used directly because the eigenvalues have asymptotic behavior, and thus, the numerical solver needed accurate starting estimates for the eigenvalues. This technique enabled the model to run over a continuous range of varied Da without the need for user interaction which allowed the use of nondiscrete, continuous values for chemical kinetics.

##### Closure for Transport and Kinetics Parameters.

*ϵ*is the porosity of the substrate material and

*τ*is the tortuosity of the porous media. $DKn$ is the Knudsen diffusivity, defined as

which is the bulk diffusivity divided by the Knudsen number when the Knudsen number is greater than one. The Knudsen number is defined as $Kn\u2261(\lambda /L)$, where *λ* is the mean free path of the gas species and *L* is the characteristic length scale associated with the geometry of the catalyst. The tortuosity was unknown for the ZnO nanowire substrates so it was modeled as *τ* ≈ 1/*ϵ* as suggested by Wang et al. [19]. Porosity was estimated by counting nanowires in an SEM image and dividing the total cross-section area of the nanowires by the total area of the image. The porosity was 97% based on the SEM image in Fig. 3. The inter-nanowire spacing was used as the length scale for calculating the Knudsen number and this resulted in a Knudsen number that was always less than unity. Therefore, the Knudsen diffusivity was assumed to be equal to bulk diffusivity in Eq. (A15).

where *A* is a pre-exponential fit parameter, *T* is the gas temperature, and *T _{a}* is the activation temperature for the reaction. The pre-exponential fit parameter and activation temperature were determined by fitting the data to conversion efficiency over a range of temperatures for a fixed flow rate.

where $kpart[1/s]$ is the particle reaction rate coefficient, $np[#/m3]$ is the number of catalyst particles per unit volume, and *A _{p}*[

*m*

^{2}] is the exposed surface area per catalyst particle. In Eq. (A18), a catalyst particle is a typical Pt/Pd nano-particle as shown on the nanowire in Fig. 4. The dark dots are images of the particles. The Pt/Pd deposition thickness was used as a surrogate variable for catalyst metal particle loading. Equation (A18) is presented as means of showing how average kinetics at the individual particle level might be determined if particle density and surface area are known.

The model is now fully closed and can be used to calculate conversion efficiency or species concentration in two dimensions in the porous media and channels of a catalyst as a function of flow rate, temperature, channel height, channel length, wash-coat thickness or nanowire length, catalyst particle size, and catalyst particle surface kinetics.

### Appendix B Model Validation

##### Numerical Model.

A numerical finite difference model was also implemented to validate the assumption that four terms in a Fourier series expansion is sufficient to capture the inlet effects and the assumption of a uniform flow profile. The channel was modeled using 100 grid points in the transverse direction and grid points of variable spacing in the stream-wise direction. A numerical integrator determined the optimal variable spacing of stream-wise grids. To model each node, a version of Eq. (5) that was modified to account for a parabolic flow profile was discretized using a an upwind differencing scheme for the convection on the left-hand-side and a central differencing scheme for the diffusion on the right-hand-side. The interface boundary condition (Eq. (A10)) was satisfied using a finite difference equation in the transverse direction.

##### Validation Results.

Plots comparing the species concentration profiles for one term and four terms in the Fourier series expansion as well as the numerical model that accounts for a fully developed parabolic flow profile are shown in Figs. 13(a)–13(c). This shows that for the same flow rate and temperature, the species concentration profile predicted by the one-term and four-term Fourier series solutions is somewhat, albeit minimally, different. Visible changes in the species concentration profile cease to happen when at least four terms are used in the Fourier series expansion, and this is validated by the agreement between the numerical model (Fig. 13(c)) and the four term solution (Fig. 13(b)). For this reason, the results will be based on the more accurate four-term solution. A solution technique with more than four terms is unnecessary because after the fourth term, additional terms are nearly independent of surface Dahmköhler number, and the Fourier coefficients become vanishingly small. The fifth term varied less than one percent over a range of Da from 0.01 up to 0.25, and the fifth Fourier coefficient for this condition was 2.22 × 10^{–3}. Even the species concentration profile predicted by the single term model is almost indistinguishable from the four term model, and, additionally, the numerical model that accounted for fully developed parabolic flow gave results that are in good agreement with the four term model. This justifies the assumption of plug flow that was required for the analytical solution technique.

As yet another means of showing that the analytical model is effective for predicting conversion efficiency, a conversion efficiency was calculated for varied number of terms used in the Fourier series and compared to the numerical model. A plot showing this result for the Fourier series approximations varied from 1 to 10 terms and the numerical model is shown in Fig. 14. The results shown in Fig. 14 indicate that the predicted conversion efficiency is weakly dependent on the number of terms used in the Fourier series expansion. Also, there is good agreement between the analytical models and the numerical model that accounted for fully developed parabolic flow, providing even further justification for the assumption of plug flow used in the analytical model.

### Nomenclature

*A*=pre-exponential fit parameter $(1/s)$

*A*=_{n}Fourier coefficient

*A*=_{p}exposed surface area per particle (m

^{2})*B*=_{n}Fourier coefficient

*C*_{1}=unknown integration constant

*C*_{2}=unknown integration constant

- $D$ =
bulk diffusivity of dilute reactant species into air $(m2/s)$

- $DKn$ =
Knudsen diffusivity ($m2/s$)

- $Dpore$ =
effective diffusivity in the porous media $(m2/s)$

- Da =
surface Dahmköhler number

*h*=height of flow channel (m)

*h*_{pore}=height of porous media (m)

*k*_{part}=catalyst particle reaction rate coefficient

*k*_{pore}=effective volumetric reaction rate coefficient in the porous media $(1/s)$

- Ł =
characteristic length scale for Knudsen number (m)

*n*=_{p}number of catalyst particles per unit volume (m

^{–3})- Pe
=_{h}transverse Peclet number, $uh/D$

*t*=plate thickness

*T*=gas temperature (K)

- τ =
tortuosity

*T*=_{a}activation temperature (K)

*u*=velocity of plug flow $(m/s)$

*x*=stream-wise coordinate (m)

- $Y\u02dcint$ =
mass fraction at porous media/channel interface

*Y*=mass fraction of reactant species

*y*=transverse coordinate (m)

*Y*_{0}=mass fraction of reactant species at inlet

*ϵ*=porosity

- η =
species conversion efficiency

- λ =
mean free path (m)

*λ*=_{n}eigenvalue in Fourier series expansion

*ϕ*=thiele modulus

- ρ =
density of gas $(kg/m3)$

- ∼ =
dimensionless form of variable