CONCENTRATION SLIP AND ITS IMPACT ON HETEROGENEOUS COMBUSTION IN A MICRO SCALE CHEMICAL REACTOR. Bo Xu and Yiguang Ju

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1 Accepted for publication in Chemical Engineering Science, 005 CONCENTRATION SLIP AND ITS IMPACT ON HETEROGENEOUS COMBUSTION IN A MICRO SCALE CHEMICAL REACTOR Bo Xu and Yiguang Ju Department of Mechanical and Aeropace Engineering Princeton Univerity, Princeton, NJ 08544, USA Correponding author: Bo Xu Addre: Department of Mechanical and Aeropace Engineering Princeton Univerity E-Quad, Olden Street Princeton NJ, bxu@princeton.edu Tel: (609) Fax: (609) Short Title: Concentration Slip of Microcale Combution

2 Chemical engineering cience 005 CONCENTRATION SLIP AND ITS IMPACT ON HETEROGENEOUS COMBUSTION IN A MICRO SCALE CHEMICAL REACTOR Bo Xu and Yiguang Ju Department of Mechanical and Aeropace Engineering Princeton Univerity, Princeton, NJ 08544, USA ABSTRACT The rarefied ga effect on concentration lip and on heterogeneou combution in microcale chemical reactor wa invetigated. Firt, a concentration lip model to decribe the rarefied ga effect on the pecie tranport in microcale chemical reactor wa derived from the approximate olution of the Boltzmann equation. Second, the model wa verified uing the direct Monte-Carlo method for the pure diffuion problem at different Knuden number. The comparion howed that the preent analytical model for the concentration lip boundary condition reaonably predicted the rarefied ga effect in the lip regime. Finally, the impact of the concentration lip on the coupling between the urface catalytic reaction and the homogeneou ga phae reaction in a microcale chemical reactor wa examined uing the one-tep overall urface reaction model with a wide range of Knuden and Damköhler number. It wa hown that the rarefied ga effect ignificantly reduced the reaction rate of the urface catalytic oxidization for large Knuden number. Furthermore, it wa hown that the impact of lip effect on catalytic reaction trongly depend on the competition between the reaction rate and diffuion tranport. It wa found that the concentration lip caue a nonlinear reaction rate ditribution at large Damköhler number. The reult alo howed that an accurate prediction of the rarefied ga effect on catalytic reaction in microcale reactor ha to conider both the temperature lip and the concentration lip.

3 Chemical engineering cience 005 NOMENCLATURE a Thermal accommodation coefficient A Contant in Eq. 19 B c Denity-weighted colliion rate c p Specific heat at contant preure D Diffuion coefficient D a Damköhler number E g, E Activation energy of ga and urface reaction f -, f + Molecular velocity ditribution function H Channel height h k Enthalpy of the kth pecie K Equilibrium contant Kn Knuden number k Colliion frequency m Molecular ma NS Total pecie number n Molar concentration or number denity P Preure Pe Peclet number R, R 0 Ga or univeral ga contant T Temperature U, V Flow velocity in x and y direction v x, y, z Molecular thermal velocity υ Molecular mean thermal velocity W Molecular weight x, y, z Coordinate Y Ma fraction of pecie ψ Molecular tranport flux γ Ratio of the pecific heat λ Molecular mean free path κ Thermal conductivity µ, µ Fluid dynamic vicoity b Momentum accommodation coefficient σ Stoichiometric ma ratio ν, ν Molar concentration coefficient ρ Denity ω& Reaction rate Subcript f Fuel g Ga phae in Inlet boundary i Coordinate x,y, z k Specie o Oxygen Outer edge of the Knuden layer or urface w Wall ~ Nondimenional parameter * Nondimenional coordinate 3 3

4 Chemical engineering cience 005 INTRODUCTION The recent development in microfabrication technology ha led to great interet in microcale combution for power generation, chemical ening, and fuel converion. The reearch in microcale combution ha been motivated by the fact that hydrocarbon fuel ha an energy denity per unit ma (50 MJ/Kg) or volume that i 100 time more than the mot efficient lithium-ion batterie (0.5MJ/Kg). Furthermore, hydrocarbon fuel baed microcombutor alo have the advantage of lighter weight, longer life time, immediate recharge, and le environmental impact than batterie. Moreover, microcale combution alo ha higher thermal efficiency than macrocale combutor due to the increaed urface to volume ratio and heat recirculation. Depite the merit of microcale combution, combution in meo and micro cale yield challenge in undertanding the impact of heat lo, radical quenching, and nonequilibrium tranport on homogeneou and heterogeneou combution. It i well known that with the decreae of the combutor cale, the increae of larger urface to volume ratio dramatically increae the wall heat lo and lead to flame extinction. On the other hand, the reduction of thermal inertia at mall cale ignificantly reduce the repone time of the wall and lead to trong wall flame coupling. Thi flame-wall interaction can dramatically change the nature of flame propagation in mall cale device. In particular, at mall cale, the heat recirculation through the wall heat conduction yield new flame regime and intabilitie. Previou tudie have hown that the wall heat lo reult in the o called quenching ditance below which a flame cannot propagate [1]. However, Weinberg and co-worker [-4] tudied the heat recirculation effect by revering the burned ga to preheat the unburned mixture and their reult howed that the flammability limit can be extended and that there are two limit caued by heat lo and flow rate, repectively. Theoretical analye from Ronney [5] and Ju et al. 4 4

5 Chemical engineering cience 005 [6, 7] repredicted thi oberved phenomenon and demontrated the exitence of different flame regime in meocale channel caued by thi wall-flame thermal interaction [7]. Numerical imulation from Norton et al. [8] and Timothy et al. [9] howed that it wa poible to build microburner below the quenching ditance with a proper choice of wall material and reduced heat lo. Experimentally, a number of innovative way have been devied to reduce thermal loe and build up micro combutor near or below the quenching ditance, including the ingle pa [10], the Y-haped diffuion burner [11] and the three-dimenional Swi Roll [1, 13] counter flow heat-exchanger/combutor. A microburner with a cale of le than 1 mm ha alo been developed by Mael, Shannon and their co-worker [14] to allow elf-utained homogeneou combution, which i achieved by uing certain inulated alumina to reduce radical adorption and heat lo. Indutry and military application require that the micro power generator be portable a well a efficient. However, with the channel width further decreaed, it become very difficult to utain a table propagating flame due to the quenching effect of the wall and the requirement of high operating and ignition temperature of hydrocarbon fuel. On the other hand, catalytic combution can take advantage of the high urface to volume ratio and high tranport rate in microcale and generate enough radical to utain the reaction. Thi provide an opportunity for fater catalytic reaction than could be achieved with macrocale combutor. Many numerical imulation and experimental work have been conducted to tudy the ignition, tability and converion efficiency of catalytic microburner [13, 15, 16]. Some catalyzed microburner have been built to achieve utained catalytic microcombution over a wide range of compoition and at ubtantially lower temperature and preure than macro combutor where traditional homogeneou combution may not urvive. All of thee tudie were focued on combution in the continuum regime. The burner cale were o large (>100 µm) that the rarefied ga effect wa 5 5

6 Chemical engineering cience 005 ignored. However, when the length cale of the flow approache the mean free path of the combution mixture, there are no longer ufficient colliion between ga molecule to achieve thermodynamic equilibrium. For example, the mean free path of the partially burned methane-air mixture at 1000 K and one atmophere i around 0.1 µm. The reulting Knuden number for microcombution i between and 1. Norton et al. [17] have developed catalyzed microburner that allow elf-utained combution in channel with a gap of 50 µm and peak preure of 5000 Pa within a temperature range of K, which yielded a Knuden number of 0.0. A uch, in microcale combution, the reulting non-continuum and non-equilibrium tranport procee cannot be well predicted without reaonably conidering the rarefied ga effect. Slip model accounting for velocity and temperature in micro fluidic have been developed and employed in the numerically efficient continuum method to correct the non-equilibrium procee near olid boundarie [18]. The baic idea i to relax the traditional no-lip boundary to allow for the preence of lip on the urface while the equation applicable to the continuum regime are retained. It i well known that the temperature and velocity lip on the wall can greatly affect the energy exchange between the ga and the wall [19]. In a microcale combutor, thee lip will alo affect the catalytic urface reaction due to the trongly temperature dependent Arrheniu law and tranport propertie. A recent tudy of catalytic converion in micro-channel ha hown that lip flow ignificantly affect the converion efficiency [0]. Shankar and Glumac [1] tudied the temperature lip effect in a low preure catalytic combution ytem and they oberved a lip of 34K in the hydrogen/oxygen reaction ytem. Similar to the velocity and temperature, there i a poibility that the pecie concentration near the boundary may alo be very different from that at the boundary [], which ignificantly influence the urface reaction. 6 6

7 Chemical engineering cience 005 Raimondeau et al. [3] and Aghalayam et al. [4] numerically tudied the role of radical wall quenching in flame tability and wall heat flux and their reult howed that quenching of any of the important radical had a ignificant influence on the tability of the ytem due to the chemical and thermal coupling between the ga and the wall. It ha been hown that while the pecie concentration could be very uniform in a micro channel when the wall wa inert, certain radical could alo have a large gradient of concentration near the reactive urface. Thi could have been the cae epecially if the reaction peed wa larger than the tranport peed [4], which render a poible large radical dicontinuity between the ga and the wall. The rarefied ga urface reaction i alo an important iue in chemical vapor depoition becaue many depoition are conducted at very low preure environment. In pite of it great practical importance and the increaing interet in tudying the velocity and temperature lip in microfluidic, the radical or pecie concentration lip and it impact on catalytic reaction in micro and nano cale have not been well invetigated. Thi tudy wa motivated by the above dicuion and wa aimed to develop a lip condition of pecie concentration for the purpoe of numerical modeling of micro and nanocale chemical reactor. Firt, a concentration lip model wa developed from the approximate olution of the Boltzmann equation. Second, a pure diffuion problem wa olved uing the newly developed lip model and the reult were compared with thoe obtained by direct imulation of the Monte-Carlo (DSMC) method [5]. Third, the impact of lip boundary condition on the catalytic reaction wa invetigated uing the catalytic reaction in a low preure, two-dimenional micro channel. Finally, dicuion and concluion were made. A SLIP MODEL FOR SPECIES CONCENTRATION Slip model 7 7

8 Chemical engineering cience 005 For a rarified gaeou flow, the colliion frequency between molecule and the wall are comparable with that between molecule. A a reult, flow cannot be conidered a a local thermodynamic equilibrium (LTE) any more. The degree that a ga deviate from the LTE can be meaured by the Knuden number (Kn), which i defined a Kn=λ/H, where λ i the molecular mean free path of the ga mixture and H i the characteritic length of the phyical problem. A flow with a higher Knuden number i aid to be more rarefied becaue of the relatively le inter molecule colliion. For medium Knuden number, that i, when a gaeou flow i within the o called Slip Boundary Regime (10-3 < Kn < 10-1 ), the effect of the thermodynamic non-equilibrium are limited only to a region of a few mean free path near the wall urface. Thi thin region i called the Knuden layer. Within the Knuden layer, the gradient of velocity, temperature, and pecie concentration are o teep that thee value at the edge of the Knuden layer are ignificantly different from thoe at the boundary (wall urface). Therefore, for imulation uing the continuum equation, lip model are needed to correct the non-equilibrium effect at the boundary. Different model [18-1] have been developed for velocity and temperature lip and been widely ued in micro fluidic and hyperonic boundary layer problem. However, the concentration lip model ha not been well etablihed. In the following, the lip boundary condition for pecie concentration from the ga kinetic theory wa derived. Figure 1 how the chematic of the Knuden layer. In Fig.1, variable with the ubcript w and repreent the propertie on the urface of the wall and at the outer edge of the Knuden layer, repectively. According to the lip model, if n denote the number denity of the tagged molecule per unit volume, in numerical imulation baed on the continuum model for the internal flow, the pecie number denity for the tagged molecule at the boundary hould be n intead of n w. Since the number denity of the tagged molecule at the wall concentration i 8 8

9 Chemical engineering cience much greater than that at the outer edge of the Knuden layer, the number of tagged molecule croing the Knuden layer via molecule colliion from the wall to the outer edge of the Knuden layer will exceed the number croing the Knuden layer in the oppoite direction. For the two-dimenional, weakly non-equilibrium gaeou flow with a zero velocity gradient, the thermal velocity ditribution function outide the Knuden layer and at the boundary can be given repectively a = z T v y T V v x T U v m kt v V v U v m kt m kt v V v U v m kt n f z y x z y x z y x ) ( ) ( / 5 ) ( ) ( 1 ) / ( 1 / ) ( ) ( exp ) / ( 3 ρ κ π (1) = + z T v y T V v x T U v m kt v V v U v m kt m kt v V v U v m kt n f z w y w x w z w y w x w w z w y w x w w ) ( ) ( / 5 ) ( ) ( 1 ) / ( 1 / ) ( ) ( exp ) / ( 3 ρ κ π () The net molecule number croing the Knuden layer from the left to the right in the poitive y- direction per unit area and per unit time i repreented by z y x y z y x y dv dv dv f v dv dv dv f v + + = 0 0 ψ (3) Thi flux hould be equal to the flux of the Stefan flow at the outer edge of the Knuden layer, y n D V n = ψ (4) where D=λυ/3 i the diffuion coefficient. From Eq.3 and 4, we can obtain the following lip boundary condition for pecie concentration

10 Chemical engineering cience 005 n Tw Vw 4 n nw + nw + λ T υ 3 y = (5) V 1+ υ If the macro velocity i negligible compared with the thermal velocity, which i alway true in microchannel flow, Eq. 5 reduce to n T 4 w = nw + λ T 3 n y (6) Thi form i imilar to the lip boundary condition of temperature and velocity. The firt term on the right hand ide of Eq.6 repreent the molecular flux produced through molecular thermal colliion and the econd term denote the correction of the molecular flux due to the concentration gradient. Therefore, the concentration lip depend both on the temperature lip and the concentration gradient on the outer edge of the Knuden layer. Validation of the lip model In order to validate the lip boundary condition of pecie concentration in Eq. 6, a pure diffuion proce wa modeled uing the DSMC method. The problem wa firt implified by neglecting the convective tranport and only retaining the diffuion proce. A uch, in the pure diffuion model, we conider the binary molecular diffuion of tagged molecule into the other molecule of the ame pecie (o-called elf-diffuion) [7], and aume uniform preure and temperature ditribution. If all the molecule of a ytem were exactly alike, there would be no experimental method by which one group of diffuing molecule could be ditinguihed from the other. However, in numerical imulation, it i poible to tag certain molecule o that they can be ditinguihed from the other and to treat the diffuion problem a if the two group of molecule were different

11 Chemical engineering cience 005 Figure how the chematic of the pure diffuion problem. The concentration of the tagged (untagged) molecule are, repectively, maintained at 0 (n w =n 0 ) on the boundary A and n w (0) on the boundary B. Employing one-dimenional geometry and the iothermal approximation with the lip boundary condition, a linear ditribution of the tagged molecule concentration can be obtained a n n y 4 8 = + Kn / 1 + Kn H 3 w 3 (7) The DSMC method treat the tagged and untagged molecule a different kind of molecule except that all of the propertie are the ame, which mean that phyically they are the ame pecie. The comparion between the analytic olution uing the lip model and the DSMC reult are hown in Fig. 3 and 4 for different Knuden number (Kn=0.01 and Kn=0.1). Parameter ued in the DSMC imulation are lited in Table 1. It i hown that both method predict the exitence of the concentration lip (jump) at the boundarie and that the magnitude of the concentration lip (jump) grow quickly with the increae of the Knuden number. Figure 3 how that the two method agree fairly well when the Knuden number i mall. It i alo een in Fig.4 that even when the Knuden number increae up to 0.1, the preent lip model i alo acceptable. However, a further increae of the Knuden number will lead to a ignificant deviation between the two reult, which may be becaue of the breakdown of the continuum aumption. Fortunately, mot of the problem intereted in microcale reactor fall into the range of Kn <0.1. NUMERICAL SIMULATIONS OF CATALYTIC REACTION IN A MICRO CHANNEL 11 11

12 Chemical engineering cience 005 Phyical model Here a implified geometry wa ued to demontrate how the concentration lip affect the catalytic reaction in microcale reactor. The chematic of a micro channel reactor i hown in Fig. 5. The width of the channel varie from 0.5 mm to 1 mm and the length from 1 to 5 mm. The top and bottom wall are catalytic. Since mot chemical reactor involve hydrocarbon fuel, we ue methane a an example fuel. The toichiometric methane and air premixture flow into the channel at a uniform velocity. The Reynold number i very mall o that the flow i aumed to be laminar. The wall temperature of the reactor i fixed at a contant value T w becaue the total chemical heat releae from the ga mixture i negligible compared with the thermal capacity of the wall. Fuel and air react with each other both in the ga phae and at the catalytic urface. To model the rarefied ga effect, the channel preure i reduced to 100 Pa to increae the Knuden number. The reulting Knuden number are between and 0.1. Governing equation and boundary condition The teady-tate Navier-Stoke equation, the energy equation, and the pecie conervation equation for the reactive internal flow are given a ρu x i i = 0 (8) U ( ) i P U r U i j ρu j = µ µ ' U µ (9) x j xi x j x j xi x j xi x i ρ x i x & (10) i k = 1 NS T ( c pu it ) = κ ω k hk x i x k ( ρ U iyk ) = ρd & ω k i Y x i (11) 1 1

13 Chemical engineering cience 005 The ideal ga tate equation i alo included to cloe the problem. Since our invetigation i focued on the lip effect, we only conider the one-tep overall reaction mechanim in both ga phae and urface reaction and aume contant thermal propertie. In the modeling of the lip phenomena on the wall, the following three kind of lip boundarie are conidered: (i) The velocity lip, which ha been proved valid in correcting the rarefied effect, can be written a: U b U = λ b y (1) (ii) The temperature jump can alo be derived from ga dynamic theory and written a: T T w = ( a γ 4κ )( )( a γ + 1 ρ υc p T )( ) y γ U + ( ) γ + 1 c p γ µ ( )( 3 γ + 1 ρ c p U ) x (13) (iii) The pecie boundary condition on the olid urface depend on the reactivity a well a the tranport proce between the wall and the ga, which i controlled by the Kn number and wa written in Eq.6. By ubtituting n k =ρy k /W k into Eq.6, we can get the lip boundary condition in term of pecie ma fraction. ( ρy ) = ( ρy ) k w T T w k ( ρy ) 4 λ 3 y k (14) The ga number denity at the catalytic urface i needed in order to calculate Y k,w. At firt ight, it eem reaonable to obtain ρ w from the ga equation of tate ince the wall temperature and preure i known. However, the ga equation of tate doe not apply within the Knuden layer due to the nonequilibrium effect. Since the ma fraction of all the pecie add up to unity, we add up Eq.14 for all pecie to yield a lip boundary condition for the ga denity, which i written a following 13 13

14 Chemical engineering cience 005 ρ w = T T w 4 ρ ρ λ 3 y (15) Reaction Model We here emphaize the effect of lip boundary condition on the ma tranfer between the ga and the wall. A uch, it i alo convenient to implify the chemitry by conidering the onetep model. The time cale of the catalytic reaction can be modeled by introducing the Damköhler number. The one-tep overall reaction i written a CH 4 +(O +4.36N ) = CO +H O+8.7N (16) The reaction rate for the ga phae can alo be written a: E g RTg ω& = k n n e (17) g g f o where the reaction coefficient k g i equal to m 3 /(mol*ec), the activation energy E g i 48.4 kcal/mole, and n f and n o are the mole concentration of fuel and oxygen. For the urface reaction, the Hougen-Waton model for methane oxidization over a palladium alumina catalyt ha been developed [8, 9] and the reaction rate can be expreed a O 4 ( + K P + K P + K P ) 3 1 O O CO CO H O H O E RT kp PCH e ω& = (18) where the K are either interpreted a adorption equilibrium contant for the active ite or a empirical contant with a unit in atm -1. Different method uch a Linear and Nonlinear Leat Square Method have been developed to determine the K baed on the reaction rate from the experimental data at certain temperature and preure and the detailed reaction mechanim. It wa found that the K are in an order of unity for many hydrocarbon catalytic reaction [8-31]

15 Chemical engineering cience 005 In thi work, we aume the preure i very low (0.001~0.01 atm) o that the rarefied ga effect i ignificant even at meocale. Since the goal of thi work i to tudy the lip effect intead of the reaction mechanim, we can implify the problem by auming that the product of K and P i much maller than unity within the preure range being conidered. Furthermore, the reaction contant k i varied and choen in uch a way that the reulting Damköhler number cover the whole range of time cale of the practical catalytic reaction. The denominator of Eq.18 can be evaluated with the inlet condition and written a A. A uch, Eq.18 reduce to = AkP PCH 4 O E RT ω& e (19) By further normalizing the pecie concentration uing the inlet fuel concentration, the fuel reaction rate i rewritten a the following ' '' ' ν k E E WF ( ν f ν f ) PW = ~ ' ' ν ~ k RT ν k RT A σ k, f Y f, in Y e = BcA k Y e 0 k Y f, in R TWk ω f (0) where ubcript f repreent the fuel and in repreent the inlet boundary; σ k,f i the toichiometric ma ratio of pecie k to the fuel; W i the molecular weight; ν k and ν k are the molar concentration coefficient of the reactant and product. By nondimenionalizing the reaction rate with κ/(c p H), the Damköhler number can be defined a D a E RTmax HBc = Ae (1) κ / c p The Damköhler number defined in Eq. 1 repreent the ratio of the characteritic time of diffuion to the reaction. A unity Damköhler number implie equal rate of urface reaction and fuel ma tranport due to diffuion. The activation energy E i 4.6 kcal/mole. To invetigate the impact of concentration lip on the coupling between tranport and the catalytic reaction in a wide range of time cale of interet, Da i varied from 0.1 to 5 in thi tudy

16 Chemical engineering cience 005 At the inlet, the velocity, temperature and pecie concentration are aumed to be uniform. At the outlet, zero gradient boundary condition are employed for all the variable. In our imulation, the length of the channel i at leat five time longer than the width. In typical catalytic urface reaction, the reaction temperature i uually le than 1000 K. A a reult, the urface reaction dominate over the ga phae reaction due to it low activation energy. When the preure i very low or the channel i very narrow, the wall temperature will not increae too much becaue the chemical heat releae from the urface reaction i very mall compared with the heat capacity of the wall. Therefore, the wall temperature can be aumed to be contant. At the olid-ga interface, the lip boundary concentration for the velocity, temperature and pecie concentration are applied. Thi i neceary a long a the Kn number i large enough (Kn >0.01). For the preent channel width (0.5~1 mm) and preure (100 Pa), the Kn number range i from 0.03~0.15. The computation domain i the upper half channel and the problem i olved with the SIMPLE method. Non-uniform meh grid are ued and the grid width near the inlet and the wall i much maller than that near the outlet and the center line in order to capture the teep gradient of temperature and concentration. The effect of meh ize i tudied by decreaing the cell width. The meh ize i choen in uch a way that a further decreae in the cell width doe not change the numerical reult. In our problem, the wall temperature T w i fixed at a typical catalytic reaction temperature (700 K). In our figure, the pecie concentration i nondimenionalized by the inlet parameter and the length i nondimenionalized by the channel width. The N-S equation a well a the ma and pecie conervation equation are olved uing the SIMPLE method. Reult and dicuion 16 16

17 Chemical engineering cience 005 In our imulation, the inlet velocity i fixed at 0.35 m/ and the preure i varied from 00 to 1000 Pa. The reulting Peclet number of the flow, which i a meaure of the relative rate of convective to diffuive tranport, varie from to At firt, the velocity lip effect i numerically invetigated. Our reult how that although the velocity lip will ignificantly affect the flow field, it ha very little influence on the pecie concentration and temperature ditribution becaue the convective tranport i much lower than the diffuive tranport for mall Peclet number. Therefore, in the following tudy even though all the three lip condition (velocity, temperature, and concentration) are conidered, focu will be placed on the impact of temperature and concentration lip on the rate of catalytic reaction. Figure 6 how the temperature contour near the inlet of the channel with and without the lip effect (velocity, temperature, and concentration) at a Kn of It i hown that the lip effect dramatically decreae the heat tranfer between the ga and wall. It i clearly een that in the lip cae, becaue of the temperature lip, the increae of ga temperature near the wall become much lower (Fig. 6b) than in the non-lip cae (Fig. 6a). Note that in order to iolate the lip effect on the catalytic reaction from the effect of wall heat tranfer, in our imulation, the temperature of the catalytic urface i fixed at 700 K. Therefore, the temperature jump will not affect the urface reaction rate through the Arrheniu law. However, in the cae when the wall temperature i not fixed and the thermal diffuivity and thermal inertia of the wall are mall, thi temperature jump will lead to a very large departure of the reaction rate from the non-lip model. In order to undertand the impact of temperature and pecie concentration lip on the urface reaction rate, numerical imulation are conducted at variou inlet ga temperature. Figure 7, 8 and 9 how the ditribution of the urface reaction rate along the channel, repectively, at three typical ga inlet temperature (300 K, 600 K, and 1000 K). The dah-dot line repreent the reult without any lip effect; the doted line are the reult with only 17 17

18 Chemical engineering cience 005 temperature lip; and the olid line are thoe with both temperature and pecie concentration lip. It i een that in all three cae the lip effect caue a decreae in reaction rate along the flow direction. Moreover, the reduction of the reaction rate and the effect of temperature lip on the reaction rate are trongly dependent on the initial ga temperature. For inlet ga temperature of 300 K and 600 K, which are below the wall temperature, Fig. 7 and 8 how that the temperature lip model yield a higher reaction rate than the non-lip model. Thi i becaue the temperature lip low down the increae of the ga temperature near the wall and thu increae the diffuion flux of the reactant to the catalytic urface via the increae of molecular number denity. On the other hand, when the inlet ga temperature i higher than the wall temperature, Fig.9 how that the temperature lip model yield a decreae of the reaction rate compared to Fig 7 and 8. Thi i becaue the temperature lip low down the cool down of the ga mixture near the wall and thu reduce the diffuion flux of reactant to the catalytic urface due to the decreae of the molecular number denity. Therefore, it can be concluded that the temperature lip effect on catalytic reaction i highly dependent on the temperature difference between the mixture and wall temperature. However, it i intereting to note that when both the temperature and concentration lip are conidered, independent of the temperature difference between the wall and the mixture, the predicted reaction rate i alway maller than that predicted by the non-lip model. Thi i becaue the concentration lip alway tend to decreae the number denitie of the reactant on the catalytic wall (ee Eq.6) and the role of temperature lip i only to modify the magnitude of the concentration lip effect. Therefore, it hould be pointed out that numerical imulation with only temperature lip effect may lead to mileading concluion in the rarefied ga effect on catalytic reaction in micro chemical reactor. Furthermore, at a low ga temperature (300 K), Fig

19 Chemical engineering cience how that there exit a non-monotonic reaction rate ditribution near the channel entrance. Thi non-monotonic phenomenon i not predicted by the temperature lip model. It i a direct reult of the competition between the diffuion tranport and the reaction conumption of the reaction. A detailed explanation will be given later in the dicuion of the Damköhler number effect. The Knuden number effect on the pecie number denity lip i hown in Fig. 10. It i een that the mole concentration lip increae with the increae of Knuden number. At Kn number of 0.1, it i een that the mole concentration lip i very ignificant (up to 50%) near the entrance of the channel where the urface reaction rate i very large. A the reaction proceed, the reactant are conumed and their concentration decreae. A a reult, the reaction rate decreae ignificantly due to the depletion of the reactant and the diffuion rate become fater when compared to the reaction rate. The diffuion render more uniform concentration profile and the concentration lip dramatically decreae along the channel. Figure 11 how the lip effect on the urface reaction rate at different Damköhler number. The reult how that the lip effect become increaingly ignificant when Damköhler number i larger than unity. A i hown in Eq.15, the magnitude of the lip effect i proportional to the gradient of pecie concentration and temperature. When the Damköhler number i large, the reaction proce i diffuion controlled and a fater urface reaction will caue a larger concentration gradient and thu a larger lip effect. On the other hand, if the Damköhler number i mall, the problem become reaction controlled. A a reult, the relatively fater diffuion proce yield a more uniform ditribution of temperature and pecie near the reaction urface. Therefore, the lip effect become maller at mall Damköhler number. In practical application, depending on the reactor working condition, the ytem Damköhler number can be either greater or le than unity

20 Chemical engineering cience 005 Since the Damköhler number i a meaure of the reaction rate, the urface reaction rate will continue to increae if we increae the Damköhler number. A the Damköhler number i increaed from 0.1 to 5, Fig. 11 how that both the lip model and the non-lip model how a monotonic increae of the reaction rate. However, if we further increae the Damköhler number to 10, the reaction rate will be dominated by diffuion tranport. Fig.11 how that a nonmonotonic ditribution of the reaction rate appear near the entrance of the reaction channel. Thi phenomenon i imilar to the monotonic reaction rate ditribution in Fig. 7. Therefore, the nonmonotonic reaction rate ditribution hown in Fig. 7 i actually caued by the competition between chemical reaction and pecie diffuion repreented by the Damköhler number. A large Damköhler number yield a non-monotonic ditribution and a mall Damköhler number lead to a monotonic ditribution. It i clearly hown that, for the inlet ga temperature of 1000 K, a monotonic ditribution of the urface reaction rate in Fig. 9 will hift to a non-monotonic ditribution in Fig. 1 only by increaing the Damköhler number from to 3. Figure 1 how the reaction rate ditribution imilar to Fig.9 by only increaing the Damköhler number from to 3. It i een that the reaction proce can be divided into three regime. Regime I (A-B) how a rapid drop of the reaction rate; in regime II (B-C), the reaction rate increae lowly while in regime III (C-D) it continue to decreae lowly. The nondimentional oxygen mole concentration profile at different location are hown in Fig. 13. Fig. 13a correpond to location A on the ditribution of the urface reaction rate in Fig. 1. The olid line are the reult without any lip effect and dahed one are the reult of lip effect. It i een that the lip effect can dramatically change the pecie ditribution. Even though the ga phae concentration with lip effect i higher than that without lip effect, the concentration at the wall i lower when lip effect i conidered. Thi i caued by the lip of the concentration at the urface. Thi reult alo explain why the urface reaction rate i lower even when the ga phae 0 0

21 Chemical engineering cience 005 molar concentration of the reactant are higher in the lip cae. At the inlet of the channel, the pecie concentration are very uniform and there i no concentration lip caued by the concentration gradient. The number denitie of the reactant at the wall are large and the reaction i fater than the diffuion. The fat reaction quickly deplete the reactant near the wall but the reactant concentration far away from the wall remain almot unchanged. Thu, the concentration gradient build up near the wall (Fig. 13a). A a reult, the number denitie of the reactant decreae quickly due to the concentration lip until point B (Fig. 13b). At location B, both the reaction rate and the number denitie of the reactant reach their minimum and the diffuion reache it maximum due to the large gradient of the pecie. Now the diffuion i fater than the reaction. Thi mean that the local Damköhler number become mall. The diffuion can now bring more reactant to the wall than the reaction can conume, cauing a low increae of the number denitie of the reactant. At the ame time, the urface reaction rate increae and the gradient of the pecie a well a the diffuion decreae until at location C (Fig. 13c) where the urface reaction rate top increaing. After location C, the decreae of the concentration lip will not increae the number denitie at the wall becaue the overall pecie concentration in the reactor i decreaing due to the reaction. In the non-lip cae, the concentration profile are quite different from thoe of the lip cae. In Fig. 13a, the oxygen number denity i increaing toward the wall. Thi i actually caued by the cooling effect of the wall. The cooling effect tend to increae the number denity while the reaction intend to decreae it. In the non-lip cae, the temperature gradient i much larger near the inlet of the channel than in the lip cae, which i hown in Fig. 6. Therefore, initially the cooling effect of the wall dominate in the non-lip cae. After location B (Fig. 13b), the urface reaction dominate and the oxygen number denity begin to decreae toward the wall. At location D (Fig. 13d), we can ee that the overall concentration in the lip cae i higher than that 1 1

22 Chemical engineering cience 005 in the non-lip cae, which i a direct reult of the decreaed reaction rate due to the lip effect. CONCLUSIONS A lip model for pecie concentration wa developed to conider the rarefied ga effect at the boundary in microcale reactor and nanocale tructure. The model howed that lip effect wa proportional to the Knuden number. The comparion with DSMC reult howed that the preent lip model could offer reaonable prediction of the lip effect within the range of the lip regime (Kn between 0.1 and 0.01). Numerical imulation of the two-dimenional micro channel catalytic urface reactor howed that the pecie concentration lip reduced the pecie tranport between the wall and the ga, and thu dramatically decreae the urface reaction rate. Thi reduction became more profound at large Knuden number. The reult alo demontrated that the impact of lip effect on catalytic reaction trongly depended on the Damköhler number of the reaction ytem, in particular at large Damköhler number. Moreover, a non-monotonic ditribution of the reaction rate at large Damköhler number wa found. Furthermore, the reult howed that the net effect of temperature lip on catalytic reaction depended on the temperature difference between the wall and the ga mixture. An accurate prediction of the rarefied ga effect on catalytic reaction in microcale reactor need to conider temperature lip together with concentration lip. Detailed catalytic reaction mechanim need to be emphaized in future tudie. ACKNOWLEDGEMENT Thi reearch i partially upported by the National Science Foundation reearch grant via contract CTS REFERENCE

23 Chemical engineering cience F. A William. Combution Theory, Benjamin/ Cumming, (1985).. F. J. Weiberg, Nature 33:39 (1971). 3. S. A. Lloyd, F. J. Weinberg, Nature 57: (1975). 4. S. A. Lloyd, F. J. Weinberg, A Recirculating Fluidized Bed Combutor for Extended Flow Range, Combut. Flame Vol. 7, pp (1976). 5. P. D. Ronney, Analyi of Non-Adiabatic Heat-Recirculating Combutor, Combut. Flame, Vol. 135, pp (003). 6. Y. Ju, W. Choi, An Analyi of Sub-Limit Flame Dynamic Uing Oppoite Propagating Flame in Meocale Channel, Combut. Flame, Vol. 133, pp (003). 7. Y. Ju, and B. Xu, Theoretical and Experimental Studie on Meocale Flame Propagation and Extinction, Proc. Combut. Int. Vol 30, to appear, D. G. Norton, D. G. Vlacho, Combution Characteritic and Flame Stability at the Microcale: A CFD tudy of Premixed Methane/Air Mixture, Chemical Engineering Science, Vol. 58, pp (003). 9. T. T. Leach, C. P. Cadou, Theo Role of Structural Heat Exchange and Heat Lo in the Deign of Efficient Silicon Micro-Combutor, Proc. Combut. Int. Vol 30, to appear, R. B. Peteron, J. M. Hatfield, MEMS 3, pp (001). 11. C. Miee, R. I. Mael, M. Short, M. A. Shannon, Diffuion Flame Intabilitie in A 0.75 mm Non-Premixed Microburner, 30th international ympoium on combution, Chicago (004). 1. L. Sitzki, K. Borer, S. Wuow, E. Schuter, K. Maruta, P. Ronney, A. Cohen. AIAA , 38th AIAA Space Science and Exhibit, Reno, NV. (001) 13. J. Ahn, C. Eatwood, L. Sitzki, P. D. Ronney, Ga-Phae and catalytic combution in Heat-Recirculating Burner, 30th international ympoium on combution, Chicago (004). 14. R. I. Mael, M. Shannon, Microcombutor Having Submilimeter Critical Dimenion, U. S. Patent 6,193,501, (001). 15. D.G. Norton, D.G. Vlacho, Hydrogen Aited Self-Ignition of Propane/Air Mixture in Catalytic Microburner, 30th international ympoium on combution, Chicago (004). 3 3

24 Chemical engineering cience G. A. Boyarko, C. J. Sung, S J. Schneider, Catalyzed Combution of Hydrogen-Oxygen in Platinum Tube for Micro-Propulion Application, Proc. Combut. Int. Vol 30, to appear, D. G. Norton, E. D. Wetzel, D. G. Vlacho, Fabrication of Single-Channel Catalytic Microburner: Effect of Confinement on the Oxidation of Hydrogen/Air Mixture, Ind. Eng. Chem. Re., Vol. 43, pp (004). 18. Matthew J. McNenly, Michael A. Galli, Iain D. Boyd, Slip Model Performance for Micro-Scale Ga Flow, AIAA, Microcale Heat Tranfer (003). 19. Kavehpour H. P., Faghri M, Aako Y., Effect of Compreibility and Rarefaction on Gaeou Flow in Microchannel, Numerical Heat Tranfer Part A, Vol. 3, pp (1997). 0. Sauro Succi, Meocopic Modeling of Slip Motion at Fluid-Solid Interface with Heterogeneou Catalyi, Phyical Review Letter, Vol. 89, No. 6, pp (00). 1. N. Shankar, N. Glumac, Experiental Invetigation into the Effect of Temperature Slip on Catalytic Combution, Eatern State Section Meeting of the Combution Intitute, Penn. State Univ. (003).. D. E. Roner, D. H. Papadopoulo, Jump, Slip, and Creep Boundary Condition at Nonequilibrium Ga/Solid Interface, Ind. Eng. Chem. Re., Vol. 35, No.9, pp (1996). 3. S. Raimondeau, D. Norton, D. G. Vlacho, R. I. Mael, Modeling of High-Temperature Microburner, Proceeding of the Combution Intitute, Vol. 9, pp (00). 4. P. Aghalayam, P. A. Bui, D. G. Vlacho, The Role of Radical Wall quenching in Flame Stability and Wall Heat Flux: Hydrogen-Air Mixture, Combut. Theory Modeling, Vol., pp (1998). 5. G. A. Bird, Molecular Ga Dynamic and the Direct Simulation of Ga Flow, Clarendon, Oxford, (1994). 6. J. Warnatz, U. Maa, R.W. Dibble, Combution, Springer, p.80 (1995). 7. F. W. Sear, An Introduction to Thermodynamic, the Kinetic Theory of Gae, and Statitical Mechanic, Addion-Weley publihing company, London, p.69 (1959). 8. W. G. Hunter, R. Mezaki, A Model Building Technique for Chemical Engineering Kinetic, A. I. Ch. E. J. Vol. 10, pp (1964). 9. J. R. Kittrell, W. G. Hunter, R. Mezaki, The Ue of Diagnotic Parameter for Kinetic 4 4

25 Chemical engineering cience 005 Model Building, A. I. Ch. E. J. Vol. 1, pp (1966). 30. J. R. Kittrell, R. Mezaki, Ind. Eng. Chem., Vol. 58, No. 5, p. 59 (1966). 31. J. R. Kittrell, R. Mezaki, Ind. Eng. Chem., Vol. 59, No., p. 8 (1967). 5 5

26

27 Chemical engineering cience 005 Figure Caption Table 1: Parameter ued in DSMC imulation. Fig. 1 Concentration profile within the Knuden layer Fig. Schematic of the pure diffuion problem Fig. 3 Specie concentration profile for Kn=0.01 Fig. 4 Specie concentration profile for Kn=0.1 Fig. 5 Schematic of the micro channel reactor Fig. 6 (a) Temperature contour without lip effect (Kn=0.14) Fig. 6 (b) Temperature contour with lip effect (Kn=0.14) Fig. 7 Surface reaction rate along the channel (T in =300K, Da=5, Kn=0.1) Fig. 8 Surface reaction rate along the channel (T in =600K, Da=5, Kn=0.1) Fig. 9 Surface reaction rate along the channel (T in =1000K, Da=, Kn=0.1) Fig. 10 Mole concentration lip of oxygen along the channel at Da=5, T in =600K Fig. 11 Surface reaction rate at different Damköhler number along the channel (T in =600K, Kn=0.1) Fig. 1 Surface reaction rate along the channel (T in =1000K, Da=3, Kn=0.1) Fig. 13 Ditribution of oxygen mole concentration at different poition at Kn=0.1, Da=3, T in =1000K 7 7

28 Chemical engineering cience 005 Table 1: Parameter ued in DSMC imulation Parameter Value Temperature 373K Molecular diameter 4E-10m Molecular Ma 6.64E-6 Vicoity power law 0.81 Number denity 1.4E0 Reciprocal of VSS Time Step.5E-6 Cell Size 100 Molecule Table 1: Parameter ued in DSMC imulation 8 8

29 Boundary Knuden Layer n w n y Fig.1

30 y H U x Fig.

31 n / n DSMC Slip Model y* Fig. 3

32 n / n DSMC Slip Model y* Fig. 4

33 U e, T e, P 0 CO +N +H O Catalyt Fig. 5

34 .0E E E E-05.0E E E E E E E E E E E E-03 a b Fig. 6

35 ω No lip Temperature lip Temperature and pecie lip T in =300K Da= x* Slip, w dot, title Fig. 7

36 ω No lip Temperature lip Temperature and pecie lip T in =600K Da= x* Fig. 8

37 ω No lip Temperature lip Temperature and pecie lip T in =1000K Da= x* Fig. 9

38 0.4 n* O Slip Kn=0.0 Kn=0.05 Kn= x* what doe it mean? Da=?, T=?, title Fig. 10

39 With lip Without lip ω Da= x* Fig. 11

40 Chemical engineering cience 005 ω A I B II C Without lip Temperature lip Temperature and pecie lip T in =1000K Da= x* III D Fig. 1 (ABCD too mall, w dot, title) 40 40

41 Chemical engineering cience 005 n* o a x*= b x*=0.07 n* o c x*= d x*= y* y* Fig. 13 (line i too thin, letter are too mall) 41 41