ie101558d

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Published: January 26, 2011

r 2011 American Chemical Society 2850 dx.doi.org/10.1021/ie101558d| Ind. Eng. Chem. Res. 2011, 50, 2850–

2864

ARTICLE

pubs.acs.org/IECR

Kinetic Parameter Estimation of a Commercial Fe-Zeolite SCR

Tae Joong Wang,* ,† ,‡ Seung Wook Baek,† Hyuk Jae Kwon,§ ,3 Young Jin Kim,§ In-Sik Nam,* ,§

Moon-Soon Cha,^

and Gwon Koo Yeo||

†Propulsion and Combustion Laboratory, School of Mechanical, Aerospace and Systems Engineering, Korea Advanced Institute of Science and Technology (KAIST), 373-1 Guseong-dong, Yuseong-gu, Daejeon 305-701, Korea§Environmental Catalysis Laboratory, Department of Chemical Engineering, Pohang University of Science and Technology (POSTECH), San 31, Hyoja-Dong, Pohang 790-784, Korea^Technology Center, ORDEG Corporation, 404 Mognae-dong, Danwon-gu, Ansan-si, Gyeonggi-do 425-100, Korea )Emission Research Team, Hyundai-Kia Motors, 772-1 Jangduk-dong, Hwaseong-si, Gyeonggi-do 445-706, Korea

ABSTRACT: In this work, an in-house computational code capable of simulating highly coupled physicochemical phenomenaoccurring in ammonia/urea SCR (selective catalytic reduction) was developed. On the basis of this computational code, the kineticparameters of catalytic reactions were newly calibrated using the experimental results obtained over a commercial ammonia/ureaSCR washcoated Fe-ion-exchanged zeolite-based catalyst. Powder-phase NH3 TPD (temperature-programmed desorption)experiments were performed to calibrate the kinetic parameters of NH3 adsorption and desorption, and core-out monolithexperiments were conducted to estimate the kinetic parameters of various deNOx reactions as well as NH3 oxidation. The currently established SCR model and kinetic parameters gave a good prediction for both steady-state and transient experimental results for a wide range of operating conditions. The main objectives of this study were to develop numerical tools and their implementationmethodologies that can be cost-eff ectively applied to the designand development of real-world ammonia/urea SCR systems. Detailsof the procedures and techniques in numerical modeling and kinetic parameter calibration are described step-by-step in this article.

1. INTRODUCTION

Ammonia/urea selective catalytic reduction (SCR) removes

NOx

emissions through catalytic reactions using ammonia orurea as a reducing agent. This SCR technology has been wellproven in a number of industrial stationary applications since the1970s.1 Although there are still several problems that need to beresolved or improved, ammonia/urea SCR is thought to be themost promising technology capable of lowering diesel NOx

emissions to levels required by increasingly stringent emissionregulations over the world.

For mobile SCR applications, aqueous urea is utilized inpractice as a reductant because of thetoxicity andsafety problemsinvolved in handling or transporting pure ammonia. In urea SCR,ammonia is generated in situ through thermal decomposition of urea and participates predominantly in deNOx reactions.2,3

However, there have been several reports that direct removal

of NO by urea itself and its decomposition byproduct, isocyanicacid, might play an important role in overall deNOx processes.4,5

Compared with other deNOx aftertreatments such as hydro-carbon SCR or LNT (lean NOx trap), the advantages of ureaSCR include higher conversion efficiencies over a wider tem-perature window, reduced fuel penalty, greater durability, andcost savings due to the lack of precious metals. On the contrary,potential limitations of urea SCR include system complexity,costs associated with urea dosing, the absence of an infrastructurefor urea, and ammonia slip.6

Selective catalytic reduction of NOx with ammonia was firstdiscovered over a platinum catalyst. However, platinum technol-ogy can be used only at temperatures below 250°C because of its

poor selectivity for NOx reduction at higher temperature.1 Inrecent years, three major catalysts have been widely employed forurea SCR: vanadium, Cu-zeolite, and Fe-zeolite. Surely, each one

shows diff erent performance characteristics in view of variousaspects such as operating temperature window, conversion level,human health eff ects, supply costs, and so on.

Vanadium is relatively cheaper and more resistant to sulfurpoisoning,7 but it is easily deactivated when exposed to the hightemperatures required for active regeneration of soot withoxygen in diesel particulate filters (DPFs).8 It is known thatlimited temperature for the use of vanadium-based SCR catalystsranges from about 600 to 650 °C,9,10 whereas some V 2O5/TiO2

catalysts are reported to be thermally stable up to 700 °C.11

Transition-metal-promoted zeolites are able to endure hightemperatures and achieve high NOx conversions. Two com-monly available zeolite SCR catalysts are based on iron andcopper.10 Cu-zeolite catalysts exhibit efficient NOx conversionsat relatively low temperatures with little or no NO2 , but display poor NOx conversions at elevated temperatures. On the otherhand, Fe-zeolite catalysts show better NOx conversions attemperatures as high as 600 °C or higher. However, they arenotasefficient as Cu-zeolite at lower temperatures in the absenceof NO2. Currently, Cu-zeolite formulations are favored when theexhaust gastemperature is lower than 450°C during the majority

Received: July 21, 2010 Accepted: December 17, 2010Revised: November 17, 2010

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2851 dx.doi.org/10.1021/ie101558d |Ind. Eng. Chem. Res. 2011, 50, 2850–2864

Industrial & Engineering Chemistry Research ARTICLE

of operation, whereas Fe-zeolite is preferred when the requiredtemperature for NOx conversion exceeds 450 °C.12 It is reportedthat the absolute upper limit on temperature for Fe-zeolite is925 °C, whereas that for Cu-zeolite is 775 °C.8 Note that the lateststate-of-the-art Cu-zeolite SCR catalyst shows a remarkable high-temperature hydrothermal stability up to 950 °C while main-taining stable low-temperature activity in NOx conversions.13

For the past several years, modeling and simulation have beenextensively employed for the design and development of mobileSCR systems. Numerical techniques are very helpful in establish-ing control technology of urea dosing, as well as in sustainingsynchronized operation with vehicles or other aftertreatmentdevices such as diesel oxidation catalysts (DOCs) and DPFs. A number of studies have progressively focused on developingaccurate numerical tools and their cost-eff ective implementation.

Reaction kinetics over a catalyst is one of the most significantfactors influencing the accuracy of mathematical models of catalytic reactor systems. However, because catalytic reactionsare aff ected by numerous chemicophysical factors, chemicalkinetics has a case-by-case nature depending on individualsystem configuration. This implies that the direct reuse of kinetic

parameters taken from other sources cannot be justified and,therefore, that calibration is required by all means. Despite theimportance of kinetic parameter calibration, it is hard to find afundamental work that addresses calibration processes for am-monia/urea SCR in detail. Therefore, this study is primarily intended to provide a detailed procedure and methodology fortuning kinetic parameters of various catalytic reactions occurringin ammonia/urea SCR.

In this study, selected model reactions and species masstransport equations were first mathematically described. Then,the partial diff erential forms of governing equations were nu-merically solved using an in-house Fortran 90 computationalcode that was developed through this work. With this numericaltool, kinetic parameters of ammonia adsorption/desorption were

newly calibrated on the basis of a powder-phase TPD experimentover a commercial Fe-zeolite catalyst. Also, kinetic parameters of various deNOx reactions and ammonia oxidation were newly estimated on the basis of core-out monolith SCR experimentsperformed at steady-state conditions. Finally, the simulationresults produced using the current model and kinetic parameters were validated with both steady-state and transient experimentalresults.

2. EXPERIMENTAL SECTION

2.1. NH3 TPD Experiments. An NH3 TPD analysis wascarried out through a powder-phase microreactor experimentover a commercial Fe-ion-exchanged zeolite-based catalyst. For

the catalyst preparation, the monolith form of a commercial SCR at 400/6.5 [cell density (cells/in.2)/wall thickness (m in.)] wascrushed and ground, and then a 0.1-g sample of catalyst powder was obtained. Note that this catalyst sample contained cordieritesubstrate as well as catalyst itself. After the powder-phase catalystsample was charged into a quartz tube microreactor, its pretreat-ment was conducted in situ at 500 °C for 2 h with flowing Ar gas.Thediameter of the microreactorwas 10.1 mm,and thethicknessof the catalyst sample layer was 2 mm. In this experiment, thereactor temperature was electrically controlled.

During the whole TPD test period, the volumetric flow rate of Ar feed gas was regulated to 50 cm3/min. At 654 s, a step input of 500 ppm NH3 was admitted into the reactor, so that the inlet

concentration of NH3 increased sharply from 0 to 500 ppm.From 654 to 5478 s, a 500 ppm NH3 feed was continuously supplied at a constant reactor temperature of 250 °C. This NH3

supply was shut down at 5478 s, so that the NH3 inletconcentration suddenly dropped from 500 to 0 ppm. From5478 to 9534 s, the reactor was flushed with Ar gas at 250 °C toremove the physisorbed species. From 9534 to 12702 s, the TPD

experiment was performed from 250 to 800 °C at a heating rateof 10 °C/min with continuous monitoring of the desorbedspecies including NH3 (m/e = 17) and its fragment, NH2

þ

(m/e = 16) by online mass spectrometer (Pfieff er/BalzersQuadstar, QMI422, QME125). Operating conditions for thisNH3 TPD analysis are summarized in Table 1.

2.2. Steady-State SCR Experiments. The catalytic activitiesof the same commercial SCR catalyst as used in the NH3 TPDexperiment were examined using an integral flow reactor systemunder steady-state conditions as shown in Figure 1. For thesetests, a core part was taken from the original SCR catalyst so thata small (diameter Â length = 20 mm Â 40 mm) core-outmonolith SCR catalyst was prepared. The following list providesthe feed gas compositions (commonly containing 5% O2 , 10%

H2O, and balance N2) and reactor space velocities for eachreaction test: (1) NH3 oxidation, 500 ppm NH3 at 10000 h-1;(2) NO SCR reaction, 500 ppm NH3 and 500 ppm NO at 10000and 15000 h-1; (3) NO2 SCR reaction, 500 ppm NH3 and500ppmNO2 at 30000-50000 h-1;and(4)NOx SCR reaction,500 ppm NH3 , 250 ppm NO, and 250 ppm NO2 at 30000-50000 h-1. Note that reactor space velocity is defined as the ratioof the total volumetric flow rate to the volume occupied by themonolith reactor.

In these core-out monolith SCR experiments, all measure-ment data were obtained after pretreatment of the catalystsunder air atmosphere at 500 °C for 2 h. Concentrations of NO, NO2 , and NH3 were measured by online chemilumines-cence NO-NOx analyzer (Thermo Electron Corporation,

model 42H), NO2 analyzer equipped with an electrochemicalcell (Testo, model 350M), and a nondispersive-infrared-(NDIR-) type NH3 analyzer (Rosemount Analytical, model880A), respectively.

2.3. Transient SCR Experiments. The transient catalyticactivities of the same size of core-out monolith SCR as used inthe previous steady-state experiments were also evaluated usingthe reactor system illustrated in Figure 1. In these transient tests,the inlet concentrations of NO and NO2 were changed with timeat constant reactor temperatures of 230 and 300 °C. Here, theNH3 inlet concentration was constantly set to 600 ppm. The sumof the NO and NO2 inlet concentrations was also regulated to600 ppm, while their ratio was changed with time as follows:

Table 1. Operating Conditions for NH3 TPD Analysis

parameter value and units

total experimental duration 12702 s

NH3 injection time 654 s

NH3 shut-off time 5478 s

beginning time of heating for TPD 9534 s

temperature limit for TPD 800 °C

temperature increase rate for TPD 10 °C/min

initial catalyst temperature 250 °C

volumetric flow rate of feed gas 50 cm3/min

NH3 inlet concentration 500 ppm

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NO/NO2 = (1) 600 ppm/0 ppmf (2) 500 ppm/100 ppm f(3) 400 ppm/200 ppm f (4) 300 ppm/300 ppm f (5) 200ppm/400 ppm f (6) 100 ppm/500 ppm f (7) 0 ppm/600ppm. The uniform feed of NO and NO2 at each step wasmaintained for 3 min except for step 1, which was continuedfor 13 min because the transient test was started with themonolith SCR being fresh and, therefore, sufficient time wasrequired for the catalyst surface to reach its steady state. Also, thetime interval between each step from 1 to 7 was regulated asfollows: step 1f 2 = 3 min, step 2f 3 = 2 min, step 3f 4 = 1min,step4f 5=0.5min,step5f 6=1min,step6f 7=2min.For all test periods, the feed gas composition was set to contain5% O2 , 10% H2O, and balance N2 , and the reactor space velocity was uniformly maintained at 40000 h-1.

3. MODELING

3.1. Reaction Kinetics and Mass Balances. NH3 Adsorption/ Desorption. In ammonia/urea SCR, NH3 molecules are chemi-

cally adsorbed both on active metal sites and on Br€onstedacid sites.14 Because NH3 plays a primary role in deNOx

processes, modeling of NH3 adsorption/desorption over thecatalyst surface is an essential part of the overall SCR model.Hence, NH3 adsorption/desorption should be realistically modeled for an accurate prediction of the performance of ammonia/urea SCR.

It is known that several NH3 molecules can be bound to oneactive metal site. Komatsu et al.15 reported that each copper ioncancoordinateuptofourNH3 molecules. Recently, based on thisreport, Olsson et al.16 presented an SCR model with detailedsurface descriptions. However, the current model assumes thatthere exists only one kindof surface site (denoted byS in eq1) on

which gaseous NH3 molecules are adsorbed or desorbed with 1:1adsorption stoichiometry. This assumption has been widely adopted in the literature for model simplicity.17-20

The adsorption and desorption of NH3 on an active site aredescribed by the following forward and backward reactions

NH3 þ S T NH3 3 S ð1Þ

Here, the symbol S denotes an active surface site, and the NH3

adsorption and desorption rates are modeled as

R a ¼ k aC NH3 ð1-θNH3 Þ ð2Þ

R d ¼ k dθNH3 ð3Þ

respectively, where each rate constant is formulated by an Arrhenius form such that

k i ¼ k oi exp - Ei

R uT s

, i ¼ a, d ð4Þ

To determine the adsorption/desorption activation energiesin eq 4, we have referred to other works conducted over vanadium-based catalysts.21,22 Therefore, in this model, NH3

adsorption is assumed to occur through a nonactivated processso that the adsorption activation energy, Ea , is set to zero. On theother hand, the NH3 desorption process is assumed to bestrongly dependent on surface conditions so that the desorptionactivation energy, Ed is represented by a Tempkin-type expres-sion in which Ed decreases linearly with increasing NH3 surfacecoverage, θNH3

, as

Ed ¼ Eodð1-RθNH3

Þ ð5Þ

Figure 1. Schematic flow diagram of the NH3 SCR reactor system.

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increasing temperature at constant space velocities of 10000 and15000 h-1. In these experiments, the temperature and space velocity levels were actually selected such that the measuredconversion rates varied over a broad range, thereby ensuringreliability and applicability of the newly calibrated kineticparameters.

The experimental result are shown in Figure 3, where it isobserved that the conversion rates of NO and NH3 are almostthe same. This indicates that the consumption ratio of NO/NH3 is nearly 1:1, which strongly supports the NO/NH3

stoichiometry given in reaction 14. The light-off temperature

of NO is about 210 °C at 10000 h-1 and about 227 °C at 15000h-1 , and the measured temperature at which 100% NO conversion first appeared was 276.5 °C at 10000 h-1 and 316.5 °Cat15000h-1. Complete conversion of NH3 was observed over a wide temperature range, and the NH3 conversion rate neverdecreased once it reached 100%. A similar behavior was alsoobserved for NO; hence, the NO conversion rate neverdecreased once it reached 100%, although it is expected thatNH3 oxidation occurs at temperatures over around 300 °C sothat the amount of NH3 required for complete NO consump-tion was not sufficient at this high temperature region con-sidering the NO/NH3 stoichiometry of 1:1. This can beattributed to the fact that, when the feed gas contains equalamounts of NH3 and NO with abundant O2 , almost all of theNH3 is consumed through the SCR reaction with NO and O2

than through NH3 oxidation with O2 because the NO SCR reaction is much faster than NH3 oxidation. Also, from theexperimental results, it can be inferred that the current Fe-zeolite catalyst has a good capability in adsorbing NH3 mole-cules because NO conversion does not decrease even at hightemperatures near 500 °C. For a vanadium-based catalyst, it has been reported that the NO removal activity decreases above380 °CbecauseofitsweakNH3 adsorption performance at hightemperatures.23

The model reaction of NO SCR is given by

4NH3 3 S þ 4NO þ O2 f 4N2 þ 6H2O þ 4S ð14Þ

This is the so-called standard SCR reaction, and its rate expres-sion is described as

R NO ¼ k NOC NOθNH3 ð15Þ

where the rate constant is modeled as the following Arrheniusform

k NO ¼ k o

NO exp -

ENO

R uT s

ð16Þ

To simulate the NO SCR process through the reactor, themass balances of gas- and solid-phase NH3 , gas-phase O2 , andgas-phase NO should be solved simultaneously.

First, the gas-phase NH3 mass balance is described by eq 6 because the NH3 molecule participates in various SCR reactionsas its adsorbed phase, not as its gas phase.

Second, the solid-phase NH3 mass balance is expresseddiff erently from either eq 7 or eq 12 because the consumptionof adsorbed NH3 through the NO SCR reaction should befurther taken into account. Therefore, it is represented by

DθNH3

Dt

¼ R a -R d -R ox -R NO ð17Þ

Third, the gas-phase O2 mass balance is given by

εmDC g , O2

Dt ¼ - uD , m

DC g , O2

Dx- ac , m

3

4R ox þ

1

4R NO

ð18Þ

where the coefficients 3/4 multiplying R ox and 1/4 multiplyingR NO were obtained by normalizing reactions 9 and 14, respec-tively, with respect to adsorbed-phase NH3.

Fourth, the gas-phase NO mass balance is expressed as

εmDC g , NO

Dt ¼ - uD , m

DC g , NO

Dx- ac , mR NO ð19Þ

where the coefficient 1 multiplying R NO results from normalizing

reaction 14 with respect to adsorbed-phase NH3.NO2 SCR Reaction. In general, for common ammonia/urea

SCR catalysts based on vanadium or zeolite, the deNOx perfor-mance is enhanced as the NO2 concentration increases becausethe reaction pathway consuming NO and NO2 simultaneously is very fast. However, as the ratio of NO2 toNOx becomes too high,the overall deNOx activity of the SCR catalyst is lowered again because the reaction pathway removing NO2 only is slow. Koebelet al.24 reported that, for a vanadium-based catalyst, the NO2/NOx ratio should not exceed 0.5 to maximize the deNOx

performance, whereas Baik et al.25 reported that the optimumNO2/NOx feed ratio for the best deNOx activity is 0.75 for bothFe-ZSM5 and Cu-ZSM5 catalysts and 0.5-0.75 for V 2O5/TiO2

catalyst. Also, Baik et al.25 concluded that the optimum NO2/

NOx feed ratio depends on the catalyst and its temperature.Therefore, for an accurate prediction of SCR performance at various NO/NO2 compositions and operating conditions, theSCR of NO2 by NH3 should be included in the overall SCR model.

In the present steady-state NO2 SCR experiments on a core-out monolith SCR catalyst, the concentrations of NO2 and NH3

were measured at both the inlet and outlet of the catalyst withincreasing temperature at constant space velocities of 30000,40000, and 50000 h-1. The experimental result is illustrated inFigure 4, where the lowest temperature and space velocity levels were higher than those employed in the previous NO SCR experiment. This is because, as the temperature or space velocity

Figure 3. Measured conversions of (a) NO and(b) NH3 versus catalyst

temperature for the steady-state NO SCR reaction. Feed gas composi-tion: 500 ppm NH3 , 500 ppm NO, 5% O2 , 10% H2O, and balance N2.

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becomes lower than the currently adopted level, the formation of ammonium nitrite (NH4NO2)

26 through the gas-phase homo-geneous reaction begins to influence the experimental results. InFigure 4, the comparison between NO2 and NH3 conversionsreveals that the reaction stoichiometry of NO2/NH3 is almost1:1. Also, it is observed that both the NO2 and NH3 conversionrates decrease with increasing space velocity, whereas they increase with increasing temperature. Note that, unlike theprevious NO conversion, the NO2 conversion is lowered againat temperatures above around 350°C. Perhaps, this is mainly due

to NH3 oxidation in the corresponding temperature zone.The model reaction of the NO2 SCR reaction is

4NH3 3 S þ 3NO2 f7

2N2 þ 6H2O þ 4S ð20Þ

This reaction indicates that the stoichiometric ratio of NH3 toNO2 is 4:3. However, the current experimental result does notfollow this ratio; rather, it shows a 1:1 stoichiometry. This can beattributed to the fact that reactions 21 and 22 to produce N2Ooccur within SCR in addition to reaction 2026

3NH3 3 S þ 4NO2 f7

2N2O þ

9

2H2O þ 3S ð21Þ

4NH3 3 S þ 4NO2 þ O2 f 4N2O þ 6H2O þ 4S ð22Þ

The sum of reactions 20-22 leads to a 1:1 reaction stoichi-ometry for NH3/NO2. Therefore, the rate of the NO2 SCR reaction is expressed as

R NO2¼ k NO2

C NO2θNH3

ð23Þ


k NO2 ¼ k oNO2exp -

ENO2

R uT s

ð24Þ

To simulate the NO2 SCR process through the reactor, themass balances of gas- and solid-phase NH3 , gas-phase O2 , andgas-phase NO2 should be solved simultaneously.

First, the gas-phase NH3 mass balance is described by eq 6 because the reactions between all the gas-phase molecules arenegligible.

Second, the solid-phase NH3 mass balance should consider

the consumption of adsorbed NH3 by NO2 SCR reaction andtherefore it is represented asDθNH3

Dt ¼ R a -R d -R ox -R NO2 ð25Þ

Third, gas-phase O2 mass balance is also the same as eq 13 because gas-phase O2 does not participate in the NO2 SCR reaction.

Fourth, on the basis of the 1:1 stoichiometry for NH3/NO2 ,the gas-phase NO2 mass balance is expressed as

εmDC g , NO2

Dt ¼ - uD , m

DC g , NO2

Dx- ac , mR NO2 ð26Þ

where the coefficient 1 multiplying R NO2results from normal-

izing reactions 20-22 with respect to adsorbed-phase NH3 andintegrating them.NO x SCR Reaction. Rapid NOx conversion through the reac-

tion consuming equal amounts of NO and NO2 with NH3 haslong beenknown,27,28 which is considered as a practical means toincrease the performance of ammonia/urea SCR. Therefore, thefast SCR reaction consuming equimolar NO and NO2 (referredto as the NOx SCR reaction in this study) is finally modeled here.

In the current steady-state NOx SCR experiments on a core-out monolith SCR catalyst, the concentrations of NO, NO2 , andNH3 were measured at both the inlet and outlet of the catalyst with increasing temperature at constant space velocities of 30000, 40000, and 50000 h-1. The experimental results aredisplayed in Figure 5, where a comparison of the NOx and NH3

conversions reveals that the NOx/NH3 consumption ratio isnearly 1:1. This strongly supports the (NO þ NO2)/NH3

stoichiometry given in reaction 27. For a more detailed examina-tion of the measured NOx conversion results, each conversionrate of NO and NO2 is separately displayed in Figure 5c, wherethe NO2 conversion performance exceeds that for NO: NO2

shows 100% conversion over the entire temperature range, whereas NO does not. This indicates that the consumption rateof NO2 is higher than that of NO.29 This is because NO2 plays animportantrole in reoxidizing theactive site on thecatalystsurfacereduced in the activation (N2-) of NH3. Note that NO2 is astronger oxidant than O2.

28,30

The model reaction of the NOx SCR reaction is

4NH3 3 S þ 2NO þ 2NO2 f 4N2 þ 6H2O þ 4S ð27Þ

The rate expression of reaction 27 is written as

R NOx ¼ k NOx C NOC NO2θNH3 ð28Þ


k NOx ¼ k oNOxexp -

ENOx

R uT s

ð29Þ

To simulate the NOx SCR process through the reactor,the mass balances of gas- and solid-phase NH3 , gas-phaseO2 , gas-phase NO, and gas-phase NO2 should be solvedsimultaneously.

Figure 4. Measured conversions of (a) NO2

and (b) NH3

versus catalysttemperature for the steady-state NO2 SCR reaction.Feedgas composition:500 ppm NH3 , 500 ppm NO2 , 5% O2 , 10% H2O, and balance N2.


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First, the gas-phase NH3 mass balance is the same as in eq 6 because the reactions between all the gas-phase molecules arenegligible.

Second, the solid-phase NH3 mass balance is represented as

DθNH3

Dt ¼ R a -R d - R ox -R NO -R NO2 -R NOx ð30Þ

Third, the gas-phase O2 mass balance is the same as in eq 18, which is the governing equation of the NO SCR reaction becausegas-phase O2 does not participate in both the NO2 andNOx SCR reactions.

Fourth, the gas-phase NO mass balance is expressed as

εm DC g , NODt

¼ - uD , m DC g , NODx

- ac , m R NO þ 12

R NOx

ð31Þ

where the coefficients 1 multiplying R NO and 1/2 multiplyingR NO

x were obtained by normalizing reactions 14 and 27, respec-

tively, with respect to adsorbed-phase NH3.Fifth, the gas-phase NO2 mass balance is given by

εmDC g , NO2

Dt ¼ - uD , m

DC g , NO2

Dx- ac , m R NO2 þ

1

2R NOx

ð32Þ

where the coefficients 1 multiplying R NO2and 1/2 multiplying

R NOx were obtained by normalizing reactions 20-22 and 27,

respectively, with respect to adsorbed-phase NH3.Boundary and Initial Conditions. Adequate boundary and

initial conditions are required to obtain a complete set of solutions for the above governing equations. Gas-phase mass balances commonly require two boundary conditions (i.e., the

Dirichlet condition at inlet and the Neumann condition atoutlet) and one initial condition as follows

C g , iðt , x ¼ 1Þ ¼ given at inlet,

i ¼ NH3 , O2 , NO, NO2 ð33Þ

DC g , iðt , x ¼ I max Þ

Dx¼ 0, i ¼ NH3 , O2 , NO, NO2 ð34Þ

C g , iðt ¼ 0, xÞ ¼ 0, i ¼ NH3 , O2 , NO, NO2 ð35Þ

The solid-phase NH3 mass balance requires no boundary

condition but only one initial condition as follows

θNH3ðt ¼ 0, xÞ ¼ 0 ð36Þ

3.2. Numerical Solutions. Discretization of Governing Equa-tions. As a first step to obtaining the numerical solutions of thegoverning equations that were defined above, the SCR reactor was represented by a one-dimensional computational domaindivided into 200 equally spaced small cells. In this study, a cell-centered coordinate system was adopted so that the total numberof nodal points was 202 (i.e., I max = 202); two nodes coincided with both the inlet and outlet cell faces, and the other 200 nodes were placed on the center of each cell.

Then, partial diff erential forms of the governing mass balances were discretized on each computational node to yield theirlinearized forms. In this study, the gas-phase mass balances werediscretized using the Euler implicit method,31 which was un-conditionally stable for all time steps and second-order-accuratefor space and first-order-accurate for time, by employing centralspace-diff erencing and forward time-diff erencing schemes. Thesolid-phase mass balances were discretized using a simpleforward time-diff erencing scheme, which resulted in first-order-accurate solutions for time.

Numerical Techniques. With the boundary and initial condi-tions, the linearizedgoverning equationswere numerically solvedregarding the kinetics of each reaction by developing an in-housecomputational code written in Fortran 90.

In obtaining the numerical solutions, an iterative calculationtechnique was implemented because the gas and solid phases arecoupled each other. As a convergence criterion, the iterativecalculation was continued until the error defined in eq 37 reachedas small as 10-3 at every time step

δn ¼

X I max - 1

x ¼ 2

XY

Y n - Y n- 1

Y n

ð37Þ

where the superscript n indicates current iteration step and n - 1denotes previous iteration step at the same time step. Althoughthis error calculation formula was identical for all reactions,

Figure 5. Measured conversions of (a) NOx , (b) NH3 , and (c) NO andNO2 versus catalyst temperature for the steady-state NOx SCR reaction.Feed gas composition: 500 ppm NH3 , 250 ppm NO, 250 ppm NO2 , 5%O2 , 10% H2O, and balance N2.


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the variable Y diff ered for each reaction, as summarized inTable 2.

To prevent solution divergence, this study employed anunder-relaxation technique in which an under-relaxation factorof 0.2 was used in calculating each reaction rate except for theNOx SCR reaction (0.15 was employed for this reaction). Actually, these values were determined after many preliminary

calculations based on the tradeoff

between speed and stability insolution convergence; a larger under-relaxation factor causesgreater speed but poorer stability, and vice versa.

Setting an incremental time for obtaining simulation resultsalso required trial and error because there is a tradeoff betweensolution accuracy and computing duration. In simulating the

NH3 TPD analysis, a time increment of 1 s was used. In obtainingthe steady-state simulation results for NH3 oxidation, NO SCR,NO2 SCR, and NOx SCR, the transient calculations werecontinued until steady state was reached for each operatingcondition. Here, the time increment of transient simulation was set to 10 s. Also, in obtaining transient simulation resultsfortheNOx SCR reaction, a time incrementof 1 s was employed.

4. KINETIC PARAMETER ESTIMATION

NH3 Adsorption/Desorption. Using the currently devel-oped computational code, kinetic parameter calibrations forNH3 adsorption and desorption were carried out on the basisof the experimental results. Figure 6 displays a comparison of thecalculated and measured results on NH3 TPD analysis, whichshows a fairly good agreement. Here, the simulation results wereobtained using a trial-and-error method in which the kineticparameters of NH3 adsorption and desorption were altered untilthe deviation between simulation and experiment was mini-mized. Both the initial and newly calibrated kinetic parametersare summarized in Table 3.

In Figure 6, the initial stage reveals that the NH3 exitconcentration is equal to the inlet concentration (500 ppm)after some duration due to the adsorption of NH3 onto thecatalyst surface. A similar behavior was detected after the shut-off of NH3 feed. From about 5500 s, the NH3 exit concentrationslowly decreased with time because the adsorbed NH3 desorbed

Table 2. Convergence Criterion Variables for Each Reaction

reaction convergence criterion variable

NH3 adsorption/desorption Y = θNH3 , C g,NH3

, R a , R d

NH3 oxidation Y = θNH3 , C g,NH3

, C g,O2 ,

R a , R d , R ox

NO SCR Y = θNH3 , C g,NH3

, C g,O2 ,

C g,NO , R a , R d , R ox , R NO

NO2 SCR Y = θNH3 , C g,NH3

, C g,O2 ,

C g,NO2 , R a , R d , R ox , R NO2

NOx SCR Y = θNH3 , C g,NH3

, C g,O2 , C g,NO ,

C g,NO2 , R a , R d , R ox , R NO , R NO2

, R NOx

Figure 6. Measured and simulated NH3 exit concentrations together with NH3 inlet concentration and reactor temperature in NH3 TPD

experiments.

Table 3. Kinetic Parameters of NH3 Adsorption and Desorption

parameter initiala new b

pre-exponential factor of the NH3 adsorption rate, k ao 0.93 m3/(mol s) 0.6 m3/(mol s)

pre-exponential factor of the NH3 desorption rate, k do 1.0 Â 1011 s-1 2.0 Â 1010 s-1

activation energy of the NH3 desorption rate at zero coverage, Edo 181.5 kJ/mol 180.0 kJ/mol

parameter for NH3 surface coverage dependence, R 0.98 0.7

active-site density based on the solid part of monolith, ac,m/εm 200.0 mol/m3 200.9 mol/m3

a Initial valuesadopted from Olssonet al.19 b Newvalues obtained through thebestfit of the simulationto NH3 TPDexperimentsover a commercialFe-zeolite catalyst.

Figure 7. Arrhenius plot with a least-squares regression curve to newly fitted pre-exponential factors of the NH3 oxidation rate. Curve informa-

tion: slope = 4459.07, intercept = 13.6364.



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from the catalyst surface. From about 9500 s, NH3 desorption was enhanced by the heating of the catalyst (TPD experiment),and finally, complete desorption of NH3 was achieved.

In Table 3, it should be noted that the value of the catalyticactive-site density is given based on a monolith because themodeling of other reactions was conducted for a monolith

configuration. Actually, the active site was first obtained basedon a packed bed and was then converted to a monolith-based value. The conversion was performed using the following simplerelation between monolith and packed-bed properties

ac , p ¼ ac , m

F b , p

F b , m

ð38Þ

The current calibration first gives the active-site density basedon the packed-bed bulk volume, ac,p = 148.4 mol/m3. Then,eq 38 yields the active-site density based on the monolith bulk volume, ac,m = 110.0 mol/m3. In the literature, the active-sitedensity is often defined based on only the solid part of catalyst,not the bulk.19,22 Therefore, for a convenient comparison, theactive-site density in Table 3 is given basedon thesolidpart of the

monolith SCR catalyst. Note that a solid-part-basis property can be simply obtained from a bulk-basis property by dividing by the void fraction.

NH3 Oxidation. For a realistic simulation of NH3 oxidation,NH3 adsorption and desorption should be considered together.This is because the oxidation of NH3 takes place in its adsorbedstate onto active sites, so that the NH3 oxidizing capability isdependent on its adsorption/desorption characteristics. There-fore, in this study, kinetic parameter tuning for NH3 oxidation was performed using the already-estimated kinetics of NH3

adsorption/desorption.Using the currently developed computational code, kinetic

parameter calibration for NH3 oxidation was carried out on the

basis of the steady-state experimental results given in Figure 2.In this calibration, only the measurement data obtained above345.5 °C (i.e., four high-temperature data points in Figure 2) were employed because other data measured below 285.5 °Cexhibited zero NH3 conversion.

An Arrhenius plot for the pre-exponential factors of the NH3

oxidation rate that were newly fitted is illustrated in Figure 7. Toobtain these numerical fit results, the iterative calculation wascontinued until the deviation of the calculated NH3 exit con-centration from the measured value became as small as 0.1 ppm.The initial values of the kinetic parameters employed here weretaken from Olsson et al.19 and are listed in Table 4. From the Arrhenius plot shown in Figure 7, both the pre-exponential factor

and activation energy of NH3 oxidation rate were newly deter-mined, as also summarized in Table 4. A detailed procedure toderive those newkinetic parameters canbe found in Wang et al.32

In Figure 8, a comparison of the simulation results obtainedusing the newly calibrated kinetic parameters with the experi-mental results is exhibited, and the agreement between them isquite good.

NO SCR Reaction. For an accurate simulation of theNO SCR reaction, NH3 adsorption/desorption and NH3 oxidation should be considered together. Therefore, in this study, kinetic param-eter tuning for the NO SCR reaction was performed using thealready-estimated kinetics of NH3 adsorption/desorption andNH3 oxidation.

Table 4. Kinetic Parameters for NH3 Oxidation and Various deNOx

Reactions

pre-exponential factor [m3/(mol s)] activation energy (kJ/mol)

reaction initiala new b initiala new b

NH3 oxidation 1.2 Â 1011 8.36 Â 105 162.4 125.3

NO SCR reaction 2.3 Â 108 7.48 Â 106 84.9 77.2

NO2 SCR reaction 1.1 Â 107

2.55 Â 104

72.3 49.3NOx SCR reaction 1.9 Â 1012 2.50 Â 109 85.1 67.1

a Initial values adopted from Olsson et al.19 b New values obtained through the best fit of simulation to steady-state SCR experiment over a commercialFe-zeolite catalyst.

Figure 8. Comparison of calculated and measured NH3 exit concentra-tions for steady-state NH3 oxidation.

Figure 9. Arrhenius plot with a least-squares regression curve to newly fitted pre-exponential factors of the NO SCR reaction rate. Curveinformation: slope = 931.536, intercept = 15.8271.



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On the basis of the currently developed numerical tools, thekinetic parameters of the NO SCR reaction were calibrated using

the steady-state experimental results given in Figure 3. In thiscalibration, both the data sets measured at 10000 and 15000 h-1

were simultaneously utilized. However, several high-temperaturedata were excluded because of their complete NO conversion.The exact magnitude of the reaction rate cannot be estimatedfrom the data with zero exit concentration because reaction rateslarger than a certain threshold value, corresponding to completeNO consumption exactly at the SCR exit, also yield zero NOemissions. In addition, some measurement data located far fromthe regression curve on the Arrhenius plot were also kept out of the calibration. In the end, four data points at 10000 h-1 andthree data points at 15000 h-1 were actually employed for thiscalibration.

Figure 9 presents an Arrhenius plot for the pre-exponentialfactors of the NO SCR reaction rate that were newly fitted. Toobtain each data point in Figure 9, an iterative calculation was

performed while updating the pre-exponential factor. The stop-ping criterion was that the deviation between the calculated andmeasured species concentrations at the SCR exit became as smallas 0.1 ppm. In this study, because NO and NH3 were bothemployed for stopping criteria, two fitted data sets were pro-duced at one catalyst temperature. Analyzing the Arrhenius plot with reference to Wang et al.32 gave a new pre-exponential factorand activation energy of the NO SCR reaction rate, as listed inTable 4. The initial values of the kinetic parameters used in thiscalibration were taken from Olsson et al.19 and are also summar-ized in Table 4.

Figures 10 and 11 compare the calculated and measuredresults for the NO SCR reaction at space velocities of 10000

Figure 10. Comparisons of calculated and measured exit concentra-tions and conversion rates of (a) NO and (b) NH3 for the steady-state

NO SCR reaction. Space velocity = 10000 h-1

.

Figure 11. Comparisons of calculated and measured exit concentra-tions and conversion rates of (a) NO and (b) NH3 for the steady-stateNO SCR reaction. Space velocity = 15000 h-1.

Figure 12. Arrhenius plot with a least-squaresregression curve to newly fi

tted pre-exponential factors of the NO2 SCR reaction rate. Curveinformation: slope = 2767.15, intercept = 10.1469.

Figure 13. Comparisons of calculated and measured exit concentra-tions of (a) NO2 and (b) NH3 for the steady-state NO2 SCR reaction.Space velocity = 30000 h-1.





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and 15000 h-1 , respectively. A good agreement is observed for both species and for both space velocities. The simulation results

show that the conversion rate of NO reaches 100% above around300 °C, and then, it becomes slightly lower with further increasein temperature, which was not observed in the experiments. Thisdiff erence can be attributed to the fact that the rate of NH3

oxidation with O2 becomes appreciable in the correspondinghigh-temperature region and, therefore, the amount of NH3

required to react with NO is not sufficient there.NO2 SCR Reaction. Fora realistic simulation of the NO2 SCR

reaction, NH3 adsorption/desorption and NH3 oxidation should be considered together. Therefore, in this study, kinetic param-eter tuning for the NO2 SCR reaction was conducted using thealready-estimated kinetics of NH3 adsorption/desorption andNH3 oxidation.

On the basis of the currently developed numerical tools,kinetic parameter calibration for the NO2 SCR reaction wascarried out using the steady-state experimental results given inFigure 4. In this calibration, the data measured at 30000, 40000,and 50000 h-1 were simultaneously employed. However, mea-surement data showing 100% conversion were excluded.



Figure 16. Arrhenius plot with a least-squaresregression curve to newly fi

tted pre-exponential factors of the NOx SCR reaction rate. Curveinformation: slope = 2165.30, intercept = 21.6393.

Figure 17. Comparisons of calculated and measured exit concentra-tions of (a) NO (b) NO2 , and (c) NH3 for the steady-state NOx SCR reaction. Space velocity = 30000 h-1.





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Figure 12 shows an Arrhenius plot for the newly fitted pre-exponential factors of theNO2 SCRreactionrate. To obtain eachfitted data point presented in Figure 12, an iterative calculation

was performed while updating the pre-exponential factor. Thestopping criterion was that the deviation between the calculatedand measured species concentrations at the SCR exit became assmall as 0.1 ppm. Here, both NO2 and NH3 were employed forstopping criteria, so that two sets of fitted data were produced atone catalyst temperature. Analyzing the Arrhenius plot withreference to Wang et al.32 gave a new pre-exponential factorand activation energy of the NO2 SCR reaction rate, as summar-ized in Table 4. The initial kinetic parameters employed for thiscalibration were taken from Olsson et al.19 and are also listed inTable 4.

Figures 13-15 show comparisons of the calculated andmeasured results for NO2 SCR reaction at space velocities of 30000, 40000, and 50000 h-1 , respectively. At 30000 h-1 , quite

good agreement is observed for both species. However, thedeviation between simulation and experiment becomes larger asspace velocity increases, especially at 50000 h-1.

NO x

SCR Reaction. For a realistic simulationof the NOx SCR reaction, NH3 adsorption/desorption, NH3 oxidation, NO SCR,and NO2 SCR should be considered together. Therefore, in thisstudy, kinetic parameter tuning for the NOx SCR reaction wasconducted using the already-estimated kinetics of NH3 adsorp-tion/desorption, NH3 oxidation, NO SCR, and NO2 SCR.

On the basis of the currently developed numerical tools,kinetic parameter calibration for the NOx SCR reaction wasperformed using the steady-state experimental results given inFigure 5. In this calibration, all of the data measured at 30000,

40000, and 50000 h-1 were simultaneously utilized. However,measurement data showing 100% conversion were excluded because they are not useful in estimating the kinetic parameters

as described previously. In the end, only the following low-temperature data were employed in the calibration: two datapoints below 269 °C at 30000 h-1 , three data points below 304 °C at 40000 h-1 , and three data points below 298 °C at50000 h-1.

Figure 16 displays an Arrhenius plot for the newly fitted pre-exponential factors of the NOx SCR reaction rate. To obtaineachfitted data point presented in Figure 16, an iterative calculation was performed while updating the pre-exponential factor. Thestopping criterion was that the deviation between the calculatedand measured exit concentrations became as small as 0.1 ppm.Here, both NO and NH3 were used for stopping criteria, so thattwo sets of fitted data were produced at one catalyst temperature.Note that NO2 was not selected for a stopping criterion because

all of the measurement data for NO2 showed 100% conversion. Analyzing the Arrhenius plot with reference to Wang et al.32

yielded a new pre-exponential factor and activation energy, assummarized in Table 4. The initial kinetic parameters used in thiscalibration were taken from Olsson et al.19 and are also listed inTable 4.

Figures 17-19 compare the calculated and measured resultsfor the NOx SCR reaction at space velocities of 30000, 40000,and 50000 h-1 , respectively. For NH3 , good agreement isobserved between the simulation and experiment at all space velocities. However, for NO and NO2 , the simulation resultsshow some deviations from the experimental results, which becomes larger as space velocity increases. Especially for NO2 ,

Figure 18. Comparisons of calculated and measured exit concentra-tions of (a) NO, (b) NO2 , and (c) NH3 for the steady-state NOx SCR reaction. Space velocity = 40000 h-1.

Figure 19. Comparisons of calculated and measured exit concentra-tions of (a) NO, (b) NO2 , and (c) NH3 for the steady-state NOx SCR reaction. Space velocity = 50000 h-1.



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the discrepancy becomes considerable in both the low- and high-temperature regions.

5. TRANSIENT VALIDATION OF THE MODEL

For the NOx SCR reaction, a validation of the transientsimulation results produced using the currently obtained kineticparameters and computational code was performed with thetransient experimental results.

Figure 20 displays the transient SCR inlet conditions for bothreactor temperatures of 230 and 300 °C. The total measurementduration was 2430 s for both cases. In Figures 21 and 22, thetransient simulation results are compared with the experimental

results, and it can be seen that there is quite good agreement.However, for the NO exit concentration, some deviations can beobserved in the early part of the transient process, especially at300 °C. Also, for the NH3 exit concentration, the simulationresults exhibit a smooth nature in their transient variations, whereas the experimental results show a discrete characteristic.

In the transient simulations, the activation energy of eachreaction rate was changed slightly from the values presented inTable 4 for the best fit to the experimental results. This tuning was carried out on the basis of the steady-state zone of thetransient experiments. The currently calibrated activation en-ergies are given in Table 5. Note that a time increment of 1 s wasemployed in the current transient calculations.

Figure 21. Comparisons of calculated and measured exit concentra-tions of (a) NO, (b) NO2 , and (c) NH3 for the transient NOx SCR reaction at a constant reactor temperature of 230 °C. Space velocity =40000 h-1.

Figure 20. Variations in the SCR inlet concentrations of NO, NO2 , andNH3 with time for both reactor temperatures of 230 and 300 °C.

Figure 22. Comparisons of calculated and measured exit concentra-tions of (a) NO, (b) NO2 , and (c) NH3 for the transient NOx SCR reaction at a constant reactor temperature of 300 °C. Space velocity =40000 h-1.

Table 5. Activation Energy of Each Reaction Rate Yielding the Best Fit of the Simulation to Transient Experiments atConstant Reactor Temperature

activation energy (kJ/mol)

reaction 230 °C 300 °C

NH3 oxidation 125.3 125.3

NO SCR 81.4 80.9

NO2 SCR 48.3 50.8

NOx SCR 68.4 68.8




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6. CONCLUSIONS

In this study, an in-house computational code for simulatingthe performance of ammonia/urea SCR was developed. On the basis of this numerical tool, kinetic parameter calibrations for various catalytic reactions were successfully conducted using thesteady-state experimental results obtained for a commercial ammo-nia/urea SCR washcoated Fe-ion-exchanged zeolite-based catalyst.

Also, the transientsimulation results were validated with experimentalresults. A summary of the main results of this study is as follows:

1. The NH3 TPD experimental results are nicely described by the simulation results obtained using the newly calibratedkinetic parameters. Both adsorption and temperature-pro-grammed desorption of NH3 over SCR catalyst surface are well-predicted.

2. At a space velocity of 10000 h-1 , NH3 oxidation with O2

began at about 350 °C, and the conversion rate reachednearly 60% at 500 °C. For steady-state NH3 oxidation, thesimulation results obtained using the newly calibratedkinetic parameters showed quite good agreement withthe experimental results.

3. In the NO SCR experiments, complete NO conversion wasmaintained up to catalyst temperatures as high as 500 °C ata space velocity of 10000 h-1. For thesteady-state NO SCR reaction, the simulation results obtained using the newly calibrated kinetic parameters displayedexcellent agreement with the experimental results.

4. At space velocities of 30000 to 50000 h-1 , NO2 removalthrough the NO2 SCR reaction was found to be enhanced with increasing catalyst temperature up to around 350 °C. Above this temperature range, the NO2 conversion ratedecreased again because of the lack of NH3 by its oxidation with O2. For the steady-state NO2 SCR reaction, the simula-tion results produced using the newly estimated kineticparameterscloselyfollowedtheexperimental results; however,

the simulation error became larger as space velocity increased.5. From the steady-state NOx SCR experiments, the consump-

tionratio ofNOx , (NO þ NO2)/NH3 , appeared to be almost1:1. Also, it was observed that the conversion rate of NO2 washigher than that of NO. For the NH3 concentration, thesimulation results obtained using the newly estimated kineticparameters nicely predicted the experimental results. How-ever, for the NO and NO2 concentrations, some deviations were found between the simulations and the experiments.

6. Transient NOx SCR reaction processes at constant re-actor temperatures of 230 and 300 °C were well-predicted by the currently obtained kinetic parameters and numerical code.

’AUTHOR INFORMATION

Corresponding Author*Tel.: þ82-31-270-1378 (T.J.W.). Fax: þ82-31-270-1399(T.J.W.). E-mail: [email protected] (T.J.W.), [email protected] (I.-S.N.).

Present Addresses‡ Advanced Combustion & Engine Research Team, Institute of Technology, Doosan Infracore, 39-3 Sungbok-dong, Suji-gu, Yongin-si, Gyeonggi-do 448-795, Korea.

3Energy Laboratory, Samsung Advanced Institute of Technol-ogy (SAIT), Samsung Electronics Co., Ltd., San 14, Nongseo-dong, Giheung-gu, Yongin-si, Gyeonggi-do 446-712, Korea.

’ACKNOWLEDGMENT

This work was supported by a Korea Science and EngineeringFoundation (KOSEF) grant funded by the Korean government(MEST) (2010-0000353).

’NOMENCLATUREac = catalytic active-site density (mol/m3)

C i = molar concentration of species i (mol/m3) Ea = activation energy of NH3 adsorption (J/mol) Ed = activation energy of NH3 desorption ( J/mol) Ed

o = activation energy of NH3 desorption at zero coverage(J/mol)

k a = rate constant of NH3 adsorption [m3/(mol s)]k ao = pre-exponential factor of NH3 adsorption rate constant

[m3/(mol s)]k d = rate constant of NH3 desorption (s-1)k do = pre-exponential factor of NH3 desorption rate constant

(s-1)k i = rate constant of reaction i [m3/(mol s)]k io = pre-exponential factor of rate of reaction i [m3/(mol s)]

R a = NH3 adsorption rate (s-1)R d = NH3 desorption rate (s-1)R NO = NO SCR reaction rate (s-1)R ox = NH3 oxidation rate (s-1)R u = universal gas constant [J/(mol K)]t = time (s)T = temperature (K)uD = superficial velocity (m/s)x = axial position (m)Y = convergence criterion variable

Greek Symbols

R = parameter for the dependence of the ammonia surfacecoverage

δ = iterative error

ε = void fraction (macroscopic bulk porosity)θNH3

= ammonia surface coverageF = density (kg/m3)

Subscripts

a = NH3 adsorption b = bulk d = NH3 desorptiong = gas phasei = dummy index m = monolithp = pore, packed-beds = solid phase

Superscriptsn = iterative index

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