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Modelling the Complex Interactions Between Reformer and Reduction Furnace in a Midrex-Based Iron Plant Khalid Alhumaizi, AbdelHamid Ajbar* and Mustafa Soliman Department of Chemical Engineering, King Saud University, P.O. Box 800, Riyadh 11421, Saudi Arabia This article studies the complex mass and energy interactions between the reformer and the reduction furnace in an iron plant based on Midrex technology. The methodology consists in the development of rigorous first principle models for the reformer and the reduction furnace, in addition to models for auxiliary units such as heat recuperator, scrubber and compressor. In this regard, a one-dimensional heterogeneous model for the catalyst tubes which takes into account the intraparticle mass transfer resistance was developed for the reformer unit, while the furnace was modelled with bottom-firing configuration. As for the reduction furnace, the mathematical model was based on the concept of shrinking core model. The furnace was modelled as a moving bed reactor taking into consideration the effects of water gas shift reaction, steam reforming of methane and carburisation reactions. The model was first validated using data from a local iron/steel plant and was then simulated to determine key output variables such as bustle gas temperature, degree of metalisation, carbon content, ratio of hydrogen to carbon monoxide, reductants to oxidants ratio and required compression energy. The effects of key input parameters on the performance of the plant were studied. These parameters included recycle ratio, scrubber exit temperature, injected oxygen flow rate, flow rate of natural gas after reformer, to transition zone, to reformer and to cooling zone. Useful profiles were compiled to illustrate the results of the sensitivity analysis. These results may serve as guidelines for a further optimisation of the plant. Keywords: direct reduced iron, Midrex plant, reformer, shrinking core model, modelling, simulations INTRODUCTION T he direct reduction process has gained growing impor- tance in the last decades as a source of metallic units for electric arc furnaces used for the production of steel. The Midrex technology is the most important gas-based direct reduced iron (DRI) process and consists of around 58% of the world DRI production (Midrex, 2011). Moreover, in today environ- mentally focused industry, the gas-based direct reduction process offers a clear advantage over the conventional BF(blast fur- nace)/BOF(basis oxygen furnace) technology that has been the backbone of iron/steel production. The Midrex process produces only one-third of the carbon dioxide generated by a traditional BF process (Midrex, 2011). The issue of greenhouse gas emissions is of great importance given that the iron/steel industry is known to be a large polluter that emits around 15% of carbon diox- ide emissions within the industrial sector (Fujita et al., 2010). The study on ways to make the iron/steel industry more prof- itable by optimising the energy use is a subject that has received increasing attention in recent years, since the iron/steel indus- try is known to be one of the most energy-intensive industries (Larsson and Dahl, 2003). The analysis of energy use in this industry is, however, a complex task because there are a num- ber of material and energy flows that interact in sometimes unpredictable ways. Accordingly, different approaches have been followed in the literature. Such methodologies consist of thermo- dynamic analysis, economic models, methods based on process integration (PI) and model-based optimisation. Thermodynamic- based concept of exergy has been used to analyse the energy efficiency in many industrial sectors and in particular in the iron and steel industry (Fraser et al., 2006). However, exergy account- ing may not provide an answer to the analysis of all aspects of material and energy flows since, being a tool for analysis, it can not provide a framework for design or optimisation. Author to whom correspondence may be addressed. E-mail address: [email protected] Can. J. Chem. Eng. 9999:1–22, 2011 Copyright © 2011 Canadian Society for Chemical Engineering DOI 10.1002/cjce.20596 Published online in Wiley Online Library (wileyonlinelibrary.com). | VOLUME 9999, 2011 | | THE CANADIAN JOURNAL OF CHEMICAL ENGINEERING | 1 |
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Modelling the Complex Interactions BetweenReformer and Reduction Furnace in aMidrex-Based Iron PlantKhalid Alhumaizi, AbdelHamid Ajbar* and Mustafa Soliman

Department of Chemical Engineering, King Saud University, P.O. Box 800, Riyadh 11421, Saudi Arabia

This article studies the complex mass and energy interactions between the reformer and the reduction furnace in an iron plant based on Midrextechnology. The methodology consists in the development of rigorous first principle models for the reformer and the reduction furnace, in additionto models for auxiliary units such as heat recuperator, scrubber and compressor. In this regard, a one-dimensional heterogeneous model for thecatalyst tubes which takes into account the intraparticle mass transfer resistance was developed for the reformer unit, while the furnace wasmodelled with bottom-firing configuration. As for the reduction furnace, the mathematical model was based on the concept of shrinking coremodel. The furnace was modelled as a moving bed reactor taking into consideration the effects of water gas shift reaction, steam reforming ofmethane and carburisation reactions. The model was first validated using data from a local iron/steel plant and was then simulated to determinekey output variables such as bustle gas temperature, degree of metalisation, carbon content, ratio of hydrogen to carbon monoxide, reductantsto oxidants ratio and required compression energy. The effects of key input parameters on the performance of the plant were studied. Theseparameters included recycle ratio, scrubber exit temperature, injected oxygen flow rate, flow rate of natural gas after reformer, to transitionzone, to reformer and to cooling zone. Useful profiles were compiled to illustrate the results of the sensitivity analysis. These results may serve asguidelines for a further optimisation of the plant.

Keywords: direct reduced iron, Midrex plant, reformer, shrinking core model, modelling, simulations

INTRODUCTION

The direct reduction process has gained growing impor-tance in the last decades as a source of metallic unitsfor electric arc furnaces used for the production of steel.

The Midrex technology is the most important gas-based directreduced iron (DRI) process and consists of around 58% of theworld DRI production (Midrex, 2011). Moreover, in today environ-mentally focused industry, the gas-based direct reduction processoffers a clear advantage over the conventional BF(blast fur-nace)/BOF(basis oxygen furnace) technology that has been thebackbone of iron/steel production. The Midrex process producesonly one-third of the carbon dioxide generated by a traditional BFprocess (Midrex, 2011). The issue of greenhouse gas emissions isof great importance given that the iron/steel industry is knownto be a large polluter that emits around 15% of carbon diox-ide emissions within the industrial sector (Fujita et al., 2010).The study on ways to make the iron/steel industry more prof-itable by optimising the energy use is a subject that has receivedincreasing attention in recent years, since the iron/steel indus-try is known to be one of the most energy-intensive industries

(Larsson and Dahl, 2003). The analysis of energy use in thisindustry is, however, a complex task because there are a num-ber of material and energy flows that interact in sometimesunpredictable ways. Accordingly, different approaches have beenfollowed in the literature. Such methodologies consist of thermo-dynamic analysis, economic models, methods based on processintegration (PI) and model-based optimisation. Thermodynamic-based concept of exergy has been used to analyse the energyefficiency in many industrial sectors and in particular in the ironand steel industry (Fraser et al., 2006). However, exergy account-ing may not provide an answer to the analysis of all aspects ofmaterial and energy flows since, being a tool for analysis, it cannot provide a framework for design or optimisation.

∗Author to whom correspondence may be addressed.E-mail address: [email protected]. J. Chem. Eng. 9999:1–22, 2011Copyright © 2011 Canadian Society for Chemical EngineeringDOI 10.1002/cjce.20596Published online in Wiley Online Library(wileyonlinelibrary.com).

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PI tools, on the other hand, have been used successfully in theliterature for the analysis of interactions between mass and energyaspects. PI refers to the application of systematic methodologiesthat facilitate the selection and/or modification of processing stepsand the analysis of interconnections within the process, with thegoal of minimising the use of resources. Schultmann et al. (2004),for instance, presented a PI method for a basic oxygen furnace.

Mathematical programming methods are also tools that belongto PI, and are increasingly being investigated in the iron/steelindustry (Larsson and Dahl, 2003; Larsson et al., 2006). Larssonet al. (2006), for instance, presented a PI-based optimisationapproach to study material and energy systems in an iron/steelplant. Due to the high temperatures and the different characteris-tics of the different flows in the iron/steel industry, the applicationof different PI methods is still an active area of research.

Model-based approach, followed in this article, is also a usefultool for analysis. In process modelling, mathematical equationsare used to describe the mechanism of processing operations.These equations are usually material, heat balances and designequations. The task of developing a rigorous model for the plantis, however, quite difficult. The plant consists essentially of thereformer and the reduction unit. These units are quite complexwithin themselves and are also connected to each other by recycleloops making the modelling and the simulation challenging tasks,which are exactly the objectives of the article, and represent themain novelty of this work.

Before the models for the individual reactor units are described,we present in the following a short overview of the modellingapproaches followed in the literature for the steam reformingand reduction reactors. We also outline the main assumptionsmade in our modelling work. The mathematical modelling of con-ventional steam reformers used in chemical and petrochemicalindustries (e.g. ammonia) has been well studied in the litera-ture (Soliman et al., 1988; Elnashaie et al., 1992; Pedernera etal., 2003). Various models with different degrees of complexitywere used for the design and simulation of these units. The mod-elling of reformers used in DRI plants did not receive similarattention in the literature. Murty and Murthy (1988), for instance,used the flux method for radiative transfer to develop models forreformer furnaces. Of notable contribution is the work of Farhadiet al. (2003) who carried out a detailed one-dimensional heteroge-neous model for the Midrex reformer. Pedernera et al. (2003), onthe other hand, studied the steady-state operation of a large-scaleprimary reformer by means of a heterogeneous two-dimensionalmodel, which accounts for the diffusion limitations in the cat-alyst particle. The authors uncovered strong radial temperaturegradients in the reformer tube, particularly close to the reactorentrance. In a recent study, Shayegan et al. (2008) developed atwo-dimensional heterogeneous model for a Midrex reformer. Theauthors presented arguments for the need to account for temper-ature gradients in radial dimension for their studied unit. Theseincluded low Reynolds number and low tube length to diameterratio.

In this work, a one-dimensional heterogeneous model for thecatalyst tubes which takes into account the intraparticle masstransfer resistance is developed for the reformer. The furnace ismodelled with bottom-firing configuration. Besides its relativesimplicity, the choice of a one-dimensional model is based onthe recent work of Shayegan et al. (2008) who concluded thatthe quality of predictions of the two-dimensional model dependedstrongly on fitted correlations for wall to fluid heat transfer. Theauthors also reported that one-dimensional models had better pre-dictions for flue gas temperatures. Moreover, the lack of any radial

measurements for the studied plant limits the usefulness of anytwo-dimensional model. Another worthy difference in this workis the choice of kinetics. Previous studies (Farhadi et al., 2003;Shayegan et al., 2008) used the simple first order kinetic model ofAkers and Camp (1955) for which an analytical solution is avail-able for the effectiveness factor, whereas this work uses a morerigorous Langmuir–Hinshwold type of kinetics (Xu and Froment,1989) for which a numerical solution for the effectiveness factorsis needed.

As for the reduction unit, a variety of studies on movinggas–solid bed reactors have been available in the literature sincethe early 1950s. These studies used a number of approaches,including simple linear models (Amundson, 1956), the shrink-ing core model for solid particles with different temperatures insolid and gas phases (Amundson and Arri, 1978), one and three-interface pellet models (Negri et al., 1987; Rao and Pichestapong,1994; Valipour et al., 2006) as well as the grain model with prod-uct layer resistance (Nouri et al., 2011).

In this article, the concept of the shrinking core model (Rao andPichestapong, 1994; Valipour et al., 2006) is adopted, and a one-dimensional non-isothermal moving bed model is derived for thefurnace. Unlike previous works (Parisi and Laborde, 2004; Nouriet al., 2011), both the kinetics of steam reforming of methaneand carburisation are included in the model. Moreover, all thereduction reactions are assumed to be reversible, which is morereminiscent of the actual behaviour in the reactor.

Plant CharacteristicsThe major components of a Midrex plant are the reduction fur-nace and the natural gas reformer with associated heat exchangeequipment (Figure 1). The furnace is fed at the top with a mixtureof lump ore and ore pellets. Reduction takes place as the chargedescends in counter-current contact with hot reducing gas. Thisgas is generated by catalytic reforming of natural gas together witha portion of the reducer offgas. The reformed gas temperature canbe raised to about 980◦C by injecting oxygen in its stream. The hotDRI is cooled to near ambient temperatures in the lower sectionof the furnace by means of inert cooling gas which circulates in aclosed loop. Around two-thirds of this gas are mixed with naturalgas as feed to the reformer, and the remainder is burned as a fuelto produce heat in the reformer.

The iron/steel plant, under study, is located in Saudi Arabia.Unlike most plants in the world where the scarp constitutes themajor part of the feed, the plant under study uses a feed whichconsists of large proportion of DRI, that is in the order of 80%,with the rest being scarp. Slag, oxygen and carbon injection prac-tices are different when melting DRI as compared to melting 100%scarp. DRI and scrap have different metallic content, with scrapinherently composed of iron. Intrinsic pieces of steel scrap usu-ally have about 98–99% metallic iron content. DRI, on the otherhand, has a 79–89% metallic iron content. The rest is composedof FeO, carbon, phosphorous, sulfur and gangue. Tables 1–3 sum-marise the characteristics of the plant. These include the designparameters of the reformer (Table 1), those of the reduction fur-nace (Table 2), as well as the nominal operating parameters of theplant (Table 3).

Models DevelopmentIn the following section, we present the derivation of mathemat-ical models for the reformer unit, the reduction unit and thevarious auxiliary units (heat recuperator, scrubber and compres-sor). For the sake of clarity, only model assumptions and kinetics

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Figure 1. Schematic simplified diagram for the Midrex process.

Table 1. Design parameters of reformer

Parameter Value

Number of tubes 521Inside tube diameter (m) 0.2Tube thickness (m) 0.024Tube length (m) 7.9Catalyst particles characteristic length (m) 0.0038Catalyst pellets bulk density (kg/m3) 250Catalyst pore radius (A) 80Tortosity 2.74

Table 2. Design parameters of reduction furnace

Parameter Value

Diameter (m) 5.5Height (m) 9.1Iron ore pellets radius (m) 0.012Porosity 0.2

are shown in the text, while the detailed model equations for thedifferent units are presented in the appendix.

Model for the ReformerAs was mentioned in the Introduction section, a one-dimensionalheterogeneous model for the catalyst tubes is derived for thereformer while the furnace is modelled with bottom-firing con-figuration.

Table 3. Nominal values of operating parameters for combinedreformer-reduction furnace plant

Parameter Value

Flowrate of hematite (Fe2O3) (ton/h) 130.0Flow rate of gangue (SiO2) (ton/h) 8.0Natural gas flow rate to reformer (103 Nm3/h) 22.4Natural gas after the reformer flow rate (103 Nm3/h) 2.9Oxygen after the reformer (103 Nm3/h) 1.1Natural gas flow rate to transition zone (103 Nm3/h) 5.0Natural gas flow rate to cooling zone (103 Nm3/h) 3.0Fraction of scrubber exit not sent to the burner 0.68Scrubber exit temperature (◦C) 53% Methane in natural gas 0.95

Reaction KineticsThe kinetic rate expressions considered in this work arethose developed by Xu and Froment (1989), based onLangmuir–Hinshelwood (Hougen–Watson) approach. The fol-lowing reactions are considered:

CH4 + H2O ↔ CO + 3H2 (1)

CO + H2O ↔ CO2 + H2 (2)

CH4 + 2H2O ↔ CO2 + 4H2 (3)

The kinetics of the dry reforming reaction:

CH4 + CO2 ↔ 2CO + 2H2 (4)

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Table 4. Kinetic rates for reforming (Xu and Froment, 1989).

Parameter Expression

k1 [kmol/bar/kgcat h] 9.49 × 1015 exp(−240.1/RT)k2 [kmol/kgcat h bar] 4.39 × 106 exp(−67.13/RT)k3 [kmol/kgcat h bar] 2.29 × 1015 exp(−243.9/RT)kCO [bar−1] 8.23 × 10−5 exp(−70.65/RT)kCH4 [bar−1] 6.65 × 10−4 exp(38.28/RT)kH2O [−] 1.77 × 105 exp(−88.68/RT)kH2 [bar−1] 6.12 × 10−9 exp(82.9/RT)k1 [bar2] exp(−26830.0/T + 30.114)k2 [bar2 exp(4400/T − 4.036)

are negligible with respect to the steam reforming reactions (reac-tions 1 and 3). CO2 mainly hinders the forward reactions (2) and(3). The authors (Xu and Froment, 1989; Froment and Bischoff,2010) used an adsorption–desorption mechanism model consist-ing of 13 steps and reached the following rates equations:

r1 = k1

(PCH4PH2O

P2.5H2

−P0.5H2PCO

K1

)/DEN2 (5)

r2 = k2

(PCOPH2O

PH2

−PCO

K2

)/DEN2 (6)

r3 = k3

(PCH4PH2O

P3.5H2

−P0.5H2PCO2

K1K2

)/DEN2 (7)

With

DEN = 1 + kCOPCO + kH2PH2 + kCH4PCH4 + kH2OPH2O

PH2

(8)

where Pi is the partial pressure of component i. The expressionsfor reaction rate constants k1, k2 and k3, absorption constants kCO,kH2O, kCH4 and kH2 and the equilibrium constants K1 and K2 aresummarised in Table 4.

Reformer Model AssumptionsThe following are the main assumptions used for the developmentof the reformer model:

• The reactor is at steady state.• Radial distribution of the temperature and the concentrations

of the different components inside the reactor are uniform (i.e.one-dimensional model).

• Heat and mass diffusions in the longitudinal direction are negli-gible considering the very high gas velocity at which the reactoris operated (i.e. axial dispersion is negligible).

• Ideal behaviour is assumed for the gases.• Mass and heat transfer resistances between the fluid and the

particle surface are negligible.

The detailed model equations for the reformer unit are given inthe appendix.

Model for Reduction FurnaceIn this section, we present the derivation of a model that describesthe direct reduction of iron oxides in the furnace. As was men-tioned in the introduction, the concept of the shrinking core model

is adopted. A mathematical formulation is obtained that definesthe mass flux of the reactant gas in terms of bulk gas phaseand pellet properties. A complete pellet model is developed forthe reduction furnace using three consecutive reaction schemes.Afterwards, a one-dimensional non-isothermal moving bed modelis derived for the furnace.

Iron Oxides ReductionThree consecutive reactions are considered for the reduction ofiron oxide ore (hematite). The reduction reactions by hydrogenare represented by:

• Hematite reduction to magnetite:

3Fe2O3 + H2 ↔ 2Fe3O4 + H2O (9)

• Magnetite reduction to wustite:

Fe3O4 + 4x−3x

H2 ↔ 3x

FexO + 4x−3x

H2O (10)

• Wustite reduction to iron:

FexO + H2 ↔ xFe + H2O (11)

In the case of reduction by CO, the three reactions are:

• Hematite reduction to magnetite:

3Fe2O3 + CO ↔ 2Fe3O4 + CO2 (12)

• Magnetite reduction to wustite:

Fe3O4 + 4x−3x

CO ↔ 3x

FexO + 4x−3x

CO2 (13)

• Wustite reduction to iron:

FexO + CO ↔ xFe + CO2 (14)

Unlike previous works (Parisi and Laborde, 2004; Nouri et al.,2011), all the reactions are assumed to be reversible. This is specif-ically the case for the most difficult reduction step from wustiteto iron which is highly reversible (Negri et al., 1991).

Pellet ModelThe three consecutive reactions are used to develop a pellet modelbased on the shrinking core scheme. The following assumptionsare made:

• Quasi steady-state assumption.• The reduction reactions are assumed to take place on three

sharp interfaces; one between iron and wustite, the secondbetween wustite and magnetite and the third between mag-netite and hematite.

• The gas diffusivities between the different layers are the samefor the reducing gases and are calculated based on bulk gasconditions. (A more rigorous model that assumes that gas dif-fusivities change with composition from one layer to anotherwould complicate unnecessarily the model.)

• The water gas shift reaction takes place only close to the topgas exit, and is modelled as suggested by Negri et al. (1991).

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Figure 2. Schematic representation of the three-interface pellet model.

• There is no volume change during the reduction. The reductionprocess occurs at three advancing interfaces inside the hematiteparticle (Figure 2).

The rate of moving for each interface is determined in terms ofrate constants for surface reactions and bulk gas conditions. Thekey assumption in the model is the quasi steady-state approxi-mation, that is the mass flux of the reactant diffusing towardsan interface is equal to the amount of the gas reacted plus theremaining amount which diffuses to the following interface. Thediffusion of the gaseous product is related to the reactant bythe stoichiometric ratio which is 1/1 for both hydrogen and car-bon monoxide. In our derivation, the subscripts 1–5 are used toindicate the hematite/magnetite, magnetite/wustite, wustite/Fe,Fe/boundary layer, and boundary layer/bulk interfaces, as shownin Figure 2. The reaction rate �j of the reducing gas at the pelletinterface j is expressed in the following form:

�j = kj(4�r2

j

)(cj− cj,p

Kj

)(15)

where kj is the rate constant at interface j, rj the radius, cj theconcentration of the reducing gas, cj,p the concentration of theproduct gas and Kj is the equilibrium constant of the reductionreaction. For the two reducing gases (H2 and CO) we have a totalof six rate equations.

Let x, y and z be the reaction rates of one reducing gas(e.g. hydrogen) with hematite (layer 1), magnetite (layer 2) and

wustite (layer 3) respectively. The rates are defined by:

x = c1−(c1,p

/K1

)R1

, y = c2−(c2,p

/K2

)R2

, z = c3−(c3,p

/K3

)R3

(16)

where Rj is the resistance of the interfacial reaction and is equalto 1/(kjr2

j ). The rates x, y and z are related to �j simply by:

�1 = 4�x, �2 = 4�y, �3 = 4�z (17)

The reduction rate in the first layer (magnetite/hematite) canalso be expressed in terms of the resistances of the intraparticle dif-fusion of the reactant (Rm) and the product (Rm

p ) in the magnetiteregion:

x = c2−c1Rm

= c1−(c1,p

/K1

)R1

= c1,p−c2,pRm

p(18)

where

Rm = r2−r1Dm

e r1r2, Rm

p = r2−r1Dm

epr1r2(19)

Dme and Dm

ep are the effective diffusion coefficients of the reduc-ing gas and the product gas respectively. They are computed usingthe turtosity and porosity of the layer. Similar equations can bewritten for the other layers.

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Reduction Furnace ModelA moving bed model is developed for the DRI furnace. The fol-lowing assumptions are used:

• Steady-state operation.• Both gas and solid streams move in plug flow with no radial

temperature or concentration gradients, that is one-dimensionalmodel.

• Ideal gas behaviour is assumed for the reducing gases.• Spherical pellets are assumed with equal size and with constant

voidage of packing.• Heat transfer from the reactor wall to the surrounding is

assumed to occur through an effective convective heat transfercoefficient.

The detailed model equations for the reduction furnace aregiven in the appendix. In addition to individual models forreformer and reduction units, simple models were also developedfor the recuperator, scrubber and compressors, and are given inthe appendix.

Assumptions for Transition and Cooling ZonesThe main role of the transition zone in the reduction furnace is tocarburise the product iron which saves energy in the subsequentrefining in the electrical arc furnace. As to cooling zone, its mainrole is to cool the product iron. The gas temperature profiles inthe transition and cooling zones are very steep because the flowrate of gas is small while that of the solid is large, resulting inlarge changes in temperature. The solution of models for bothtransition and cooling zones lead to difficulties in convergencebecause of the severe steepness of the profiles. For this purpose,we made a simple analysis to approximate the events taking placein the transition zone. In the present study, we assume that 95%of the methane in the transition zone decomposes into carbonand hydrogen, and that 5% of the hydrogen is used in the fur-ther reduction of iron oxides. As to cooling zone, the methanedecomposition is very minor and could be neglected.

Solution StrategyThe developed computer code starts by reading the design andoperating data for reduction furnace unit, reformer, oxygen flowrate, enrichment natural gas flow rate and composition. The pro-gram then carries out calculations for reduction furnace materialand heat balances equations, calculate process gas compositionafter scrubber, compressor energy consumption, and carries outmaterial balances on the scrubber, compressor and recycle loop.The scrubbed gas after compression is divided into two streams:one goes to the reformer furnace to be burned while the other goesto the reformer where it is mixed with fresh natural gas. For thereformer calculations, we have two sides: the furnace side calcu-lations, where the adiabatic flame temperature and radiation heattransfer equations are calculated. These calculations are based onguessed values for the reformer tubes outside temperature. On thereformer tube side, differential material and heat balances requirethe calculations of the catalyst effectiveness factors. The programthus calculates the effectiveness factors for CO2 and CH4 from thecatalyst particle module at each differential step length, and thencalculates bulk gas concentrations, temperature and pressure ateach step. It then calculates new tube outer wall temperature sothat the transferred heat from the furnace is equal to the heat trans-ferred from the tube to the reacting mixture. Iterations are carriedout until the outer wall temperature becomes constant within a

predetermined tolerance. The program then carries out materialand heat balance calculations for oxygen injection, adds enrich-ment natural gas and adds natural gas from cooling and transitionzones to calculate bustle gas mix flow rate and composition.

The method of spline orthogonal collocation (Soliman, 1992)is used to solve the resulting boundary value problems. The mainadvantage of this technique is its adequate representation of themodel with minimum number of variables and its ability to solvemoderately steep profiles.

In order to validate the model against the industrial data, a num-ber of parameters were fitted. These include the wall to fluid heattransfer coefficient in the reformer unit. The fitting is justified bythe recent work of Shayegan et al. (2008) who showed that thequality of predictions of their two-dimensional model dependedstrongly on the fitted correlations for heat transfer coefficient. Theother parameters are the effectiveness factors for methane andcarbon dioxide in the reformer. As for the reduction reactor, thefitting parameters are kinetic parameters for the side reactions.These include methane decomposition, carbon monoxide dispro-portionation and carbon monoxide reduction by hydrogenation,as explained in the appendix. The fitting is justified by the fact

Table 5. Validation of the model

Process Reformed Bustle OverallParameter gas gas gas plant

T (◦C)Plant NA 948.0 980.0Model 53 947.0 976.0

CO (dry) %Plant 24.5 36.5 36.0Model 26.4 34.1 34.0

CO2 (dry) %Plant 22.5 3.5 3.6Model 14.8 2.6 2.5

H2 (dry) %Plant 48.0 57.5 56.0Model 56.5 60.8 58.7

CH4 (dry) %Plant 3.0 1.2 2.8Model 0.9 1.4 3.6

N2 (dry) %Plant 2.4 1.7 1.7Model 1.4 1.1 1.2

Metalisation %Plant 94.0Model 95.1

CO (wet) %Plant 31.6

CO2 (wet) %Plant 2.3

H2 (wet) %Plant 54.5

CH4 (wet) %Plant 3.4

H2O (wet) %Plant 7.1

N2 (wet) %Plant 1.1

Carbon content %Plant 2.5Model 2.4

Compression energy (kwh)Model 9068.0

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that the experiments used to get these kinetics are at conditionsdifferent from dry reforming conditions.

RESULTS AND DISCUSSIONThe simulation results for the plant at the base case conditions areshown in Table 5. Also shown are the plant data which consistof dry gas composition, metalisation degree and the carbon con-tent. The results compare well except for the process gas wherethe simulated results show low mole fractions of methane and car-bon dioxide as well as a large mole fraction of hydrogen. It shouldbe noted that since the model predicts that too much methane isbeing steam reformed then it is natural that the hydrogen molefraction is large. Another reason for these discrepancies couldbe due to the fact that the chosen kinetics predict the reductionto occur more by carbon monoxide than by hydrogen. Measure-ments errors in the plant data could also explain these model-plantmismatches.

Key performance indicators are the degree of metalisationwhich is 94.0% from the plant data and 95.1% from simulations.The discrepancy is most likely due to the assumptions made inthe transition zone, as explained in the previous section. The ratioH2/CO (on a dry basis) predicted by the model is 1.72 and isclose to the value of 1.56 obtained from the plant. The Midrex

Figure 3. Profiles of the effectiveness factors along the steam reformer.

Figure 4. Profile of the gas temperature in the reduction reactor.

Figure 5. Profile of the solid temperature in the reduction reactor.

Figure 6. Profiles of mole fractions in the reduction reactor.

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Table 6. Qualitative effect of the different operating parameters on the performance of the plant: (+) increases, (−) decreases, (Max) exhibits amaximum, (Min) exhibits a minimum

An increase in Bustle Reformer inlet Metallisation Carbon (H2 + CO)/ Compressionthe variable temp. temp. degree content H2/CO (H2O + CO2) power

NG to reformer Max Max Max Max + Max +NG after reformer Max Max Max Max + Max +Oxygen after reformer + − + + − − −Recycle ratio − − Max − Min − +Scrubber exit temp. − − − − + − +NG flow rate to cooling zone + + + + + + +NG flow rate to transition zone + + + + + + +Ratio of all flow rates − Max − − + − +

Figure 7. Effects of recycle ratio on the performance of the plant.

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technology recommends that the ratio (H2/CO) should be about1.5 and not less than 1.4 to avoid carbon deposition in thereformer. A larger value would mean lower reduction furnacetemperature but this can be dealt by, for instance, increas-ing oxygen injection. The reductants to oxidants ratio, that is(H2 + CO)/(H2O + CO2) obtained from simulations is 9.16 whilethese data are not available from the plant since the sampling isdone on a dry basis. Too small values of this ratio would reducemetalisation while larger values would lead to carbon deposi-tion on the reformer catalyst. The percentage of carbon is around2.4% from simulations and 2.5% from the plant data. This value,together with the mole fraction of CO2 of 3.6% suggests that the

plant is operating at reasonable conditions. Very low values ofCO2 can lead to carbon deposition in the reformer.

Before the results of the parametric study of the overall plantare shown, it may be useful to discuss some results pertinentto the individual performance of steam reforming and reductionreactors. Figure 3a,b shows the profiles of the effectiveness fac-tors for methane and carbon dioxide along the steam reformerlength. The values of the effectiveness factors are very small,indicating that the reactions are very fast and take place ina very small layer at the surface of the catalyst. This alsoindicates diffusional limitations which is consistent with the con-clusions reached in previous studies (Elnashaie and Abashar,

Figure 8. Effects of scrubber exit temperature on the performance of the plant.

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1993). Intraparticle convection could also be a factor in theefficiency of the process, especially given the small size of thepores. Some investigators (Quinta Ferreira et al., 1992; Oliveira etal., 2010) showed that intraparticle convection can be promotedby using large-pore structured catalysts that reduce intraparticlegradients.

As for the reduction reactor, the profiles of gas and solid tem-perature are shown in Figures 4 and 5. It can be seen that sincethe gas is fed counter-currently to the solid, the gas temperaturedecreases starting from the dimensionless distance of 1 (bustlegas) to the dimensionless distance of 0 (top gas). This decrease isdue to the endothermic nature of the reaction and to the fact thatthe solid is fed at colder temperature. Figure 5 shows the profile

of solid temperature. The profile is quite close to that of the gastemperature.

Figure 6 shows the profiles for the mole fractions (wet gas basis)of hydrogen and carbon monoxide in the reduction reactor. Forhydrogen, starting from bustle gas inlet conditions (dimensionlessdistance of 1), both steam reforming and methane decomposi-tion produce hydrogen which leads to the increases in its molefraction. Subsequently, the reduction of the oxides dominate,causing a decrease in the hydrogen mole fraction. It should benoted that at top gas conditions (dimensionless distance of 0)there is the constraint that water gas shift reaction is at equi-librium which leads to an increase in the hydrogen molefraction.

Figure 9. Effects of injected oxygen flow rate on the performance of the plant.

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As for carbon monoxide, the reforming leads to the increase inits mole fraction at the bustle gas inlet conditions (dimensionlessdistance of 1) followed by a decrease in the mole fraction as resultof its use in the reduction of the iron oxide.

Parametric StudyIn the following, we present simulations showing the effects of anumber of operating parameters on the performance of the plant.The effects of the following parameters are studied:

• Recycle ratio.• Scrubber exit temperature.

• Injected oxygen flow rate.• Flow rate of natural gas to reformer.• Flow rate of natural gas after reformer.• Flow rate of natural gas to transition zone.• Flow rate of natural gas to cooling zone.

Besides these input parameters, we will be interested in know-ing the effect of increasing all the flow rates including the oreflow rate on the performance of the plant. Thus, we introduceas an other input variable an artificial parameter (rf) defined asthe ratio of actual flow rates to the flow rates in the nominalcase.

Figure 10. Effects of natural gas flow rate to reformer on the performance of the plant.

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As for plant performance, the following key output parametersare selected:

• Bustle gas temperature.• Reformer inlet temperature.• Degree of metalisation.• Carbon content.• Ratio H2/CO.• Reductants to oxidants ratio (H2 + CO)/(H2O + CO2).• Required compression energy.

The variations of each input parameter around the nominalvalue were made quite wide but keeping in perspective that somebounds are to be placed on some output parameters, for example

the metalisation degree should not go below 90%, the temper-ature of the reduction furnace should not exceed 1250 K whilethe ratios H2/CO and (H2 + CO)/(H2O + CO2) should remain inthe range mentioned in the previous section. Besides the plots,a summary of the sensitivity analysis is provided in Table 6where the qualitative effects of the different input parameters areillustrated.

Effect of Recycle RatioThe effects of recycle ratio on the plant performance are shownin Figure 7. The recycle ratio was varied from 0.65 to 0.75, thenominal value being 0.68. An increase in the recycle ratio leadsto a lower amount of fuel to the reformer furnace, which means

Figure 11. Effects of natural gas flow rate after reformer on the performance of the plant.

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less reforming, less reductants to oxidants ratio, lower bustlegas temperature, lower inlet temperature to the reformer, lessmetalisation, less carbon content and more compressor energy.However, we will have more dry reforming than steam reformingand thus lower H2/CO, which yields higher temperature insidethe reduction furnace, since more reduction is taking place by theexothermic CO reduction. Lower temperatures favour high H2/COratio. This is why we see a minimum at the profile of H2/CO.Higher reduction temperatures in the furnace, on the other hand,cause higher metalisation which is in conflict with lower met-alisation due to lower bustle gas temperatures. This is why the

metalisation profile exhibits a maximum with the increase in therecycle ratio.

Effect of Scrubber Exit TemperatureThe effect of scrubber exit temperature is shown in Figure 8.The values of this temperatures were varied from 49 to 56◦Caround the nominal value of 53◦C. An increase in the scrubberexit temperature means more water vapour in the process gas,more steam reforming, more flow rates, less bustle gas tempera-ture and less reduction of iron ore. This leads to less metalisation,

Figure 12. Effects of natural gas flow rate to transition zone on the performance of the plant.

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less carbon content, lower reductants to oxidants ratio and highercompression energy. As for the H2/CO ratio, it increases due tomore steam reforming of the excess water. Although a low scrub-ber temperature is desired, it should not decrease below certainlimits, since a minimum amount of water is needed to preventcarbon formation in the reformer tubes.

Effect of Injected Oxygen Flow RateThe effect of injecting oxygen is shown in Figure 9. The flowrate was varied in a wide range from 300 to 1700 Nm3/h aroundthe nominal value of 1100 Nm3/h. As the flow rate of oxygenincreases, some reformed hydrogen will burn. This leads to anincrease in the bustle gas temperature leading to more reduction

and carburisation which leads to an increase in metalisation andcarbon content. This also leads to a decrease in the ratio of reduc-tants to oxidants. Since we have less reductants in the process gas,there will be less H2 and CO to burn in the reformer furnace, lead-ing to lower flue gas exit temperature and lower inlet temperatureof the reformer.

Increasing the injected oxygen flow rate could also lead (as inthis case) to less H2/CO because of the combustion of hydrogenby oxygen. Lower reformer inlet temperatures mean less reform-ing, less overall flow rates and hence less compression energy.Although increasing the injected oxygen flow rate is desirable,there is a maximum bustle gas temperature for the safe operationof the reduction furnace.

Figure 13. Effects of natural gas flow rate to cooling zone on the performance of the plant.

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Effect of Natural Gas Flow Rate to Reformer and AfterReformer

As it can be seen in the summary Table 6, both these flow rates(NG to reformer and NG after reformer) have similar effects onthe plant performance. The natural gas flow rate to reformer wasvaried from 21,000 to 26,500 Nm3/h around the nominal valueof 22,400 Nm3/h. Figure 10 shows the results for the changes innatural gas flow rate to reformer. The flow rate after reformerwas, on other hand, varied from 1000 to 7100 Nm3/h around thenominal value of 2900 Nm3/h. The results of sensitivity analysisare shown in Figure 11.

An increase in natural gas flow rate will lead initially to morereforming, more reduction and more fuel to the reformer furnace.This means higher bustle gas temperature, higher reformer inlettemperature, higher H2/CO ratio, higher (H2 + CO)/(H2O + CO2)ratio, higher metalisation and higher carbon content. Compres-sion energy increases because of the increase in natural gas flowrate and process gas flow rate. However, as the flow rate of naturalgas increases, conditions causing less reforming in the reduc-tion furnace occur due to the limited capacity of the reformer.This leads to lower reductants to oxidants ratio. Also, because ofthe limited capacity of the recuperator, reformer inlet tempera-ture starts to drop at certain flow rate. Consequently, bustle gas

Figure 14. Effects of ratio of all flow rates on the performance of the plant. The enlargement in the upper left figure shows the maximum exhibited bythe profile of reformer inlet temperature.

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temperature starts to drop. Combination of lower reducants to oxi-dants ratio and lower bustle gas temperature leads to a decrease inmetalisation and carburisation. This explains the maxima in theprofiles of metalisation, carburisation and reductants to oxidantsratio.

Effect of Natural Gas Flow Rate to Transition Zoneand Cooling ZoneAs it can be seen in the summary Table 6, these two streams (NGto transition zone and NG to cooling zone) have the same effecton the process. The flow rate to transition zone was varied from3000 to 6700 Nm3/h around the nominal value of 5000 Nm3/h.The results of the sensitivity analysis are shown in Figure 12. Theflow rate to cooling zone was, on the other hand, varied from2000 to 4750 Nm3/h around the nominal value of 3000 Nm3/h.The sets of Figure 13 show the results of the sensitivity analysis forthis stream. Both flow rates to transition or cooling zone providehydrogen for the reduction zone due to methane decomposition,and thus increase the reductants to oxidants ratio. An increasein natural gas flow rate will lead to more reforming, more reduc-tion and more fuel to the reformer furnace. This means higherbustle gas temperature, higher reformer inlet temperature, higherH2/CO ratio, higher (H2 + CO)/(H2O + CO2) ratio, higher metali-sation and higher carbon content. Compression energy increasesbecause of the increase in natural gas flow rate and process gasflow rate. Therefore, although it is desirable to increase flow ratesof natural gas to the cooling and transition zones, increasing theseflow rates above certain limits could cause carbon formation inthe reformer tubes.

Effect of Increase in all Inlet Flow Rates Including OreIn this section, we present the simulation results for the increasein all inlet flows including the ore flow rate. The parameter (rf)is introduced to represent a ratio multiplying all the inlet flowsincluding ore flow rate. The value of (rf) was varied from 0.925 to1.072, the unity being the nominal value. The results of changes inrf are shown in Figure 14. For a fixed design, increasing the valueof rf means less metalisation, less reforming, less reductants tooxidants ratio and higher compressor energy. The bustle gas tem-perature decreases but the inlet temperature to the reformer showsa maximum (shown in the enlargement of the first figure). Thismaximum is due to the initial decrease in the amount of heat goingto the reformer which later increases. The ratio H2/CO increasesdue to low exit reformer temperature (lower bustle gas temper-ature), since at low temperatures the water gas shift reaction isfavoured, with CO going to CO2 and H2O going to H2.

CONCLUSIONSThis article presented the modelling and simulations of a DRIplant based on Midrex technology. The approach followed in thiswork was the development of rigorous models for the individualunits of the plant followed by validation and simulations stud-ies. Mathematical modelling, while sometimes difficult, is alwaysuseful when the model incorporates the essential elements of theprocess and can be validated against plant data. The level of rigorin the development of these models was constrained by the com-plexity of the numerical solution as well as the availability of plantdata to validate the resulting model. Given the high temperaturenature of the process involved, the only data available from theplant are some temperature measurements, dry gas compositionand metalisation degree.

In this regard, a detailed one-dimensional heterogeneous modelwas developed for the reformer. The need for a two-dimensionalmodel was deemed not useful given that practically no data areavailable for radial measurements.

A detailed model based on the well-known shrinking core pelletwas developed for the reduction furnace. The models for the tran-sition and cooling zones were approximated by simple algebraicequations since a detailed model in these zones would result in toosteep profiles (and stiff problems) while small extent of methanedecomposition takes place in the transition zone and even less inthe cooling zone. When these individual models were integratedtogether, the effect of a number of key operating parameters onthe performance of the plant was clearly identified in appropriateplots.

The sensitivity study could provide some practical guidelinesfor the optimisation of the plant, subject to the constraints onthe prevention of carbon deposition in the reformer tubes. In thisregard, the parametric study has revealed that the scrubber exittemperature could be lowered to the extent of providing enoughwater vapour to prevent carbon formation. Recycle ratio shouldbe optimised subject to the same constraint. There is an optimumnatural gas/iron ore ratio. If we would like to increase naturalgas flow rate, it is preferable that this addition to be done inthe transition zone where methane decomposes to carbon andhydrogen, and thus increases carburisation and metalisation. It isalso preferable that this natural gas is at elevated temperatures.Injected oxygen flow rate should be high enough to increase thebustle gas temperature to the level where no iron ore coagulationis possible.

ACKNOWLEDGEMENTSThis work was made possible by a generous grant from theNational Plan for Science and Technology (Project # 08-ENE337-2).

APPENDIX: MODEL EQUATIONS

Model Equations for ReformerA one-dimensional heterogeneous model is used to describe thecatalyst tube, which takes into account the intraparticle masstransfer resistance. One reformer tube performance is assumed tobe representative of any other tube in the furnace, by assuming thefeed gas to be distributed equally among the reformer tubes, andthat heat flux passing through each tube wall is the same regard-less of the location of the tube inside the furnace with respect tothe location of the burners.

Reactor EquationsThe rate of disappearance and formation of CH4 and CO2 (takenas key components) are given by:

RCH4 = r1 + r3, RCO2 = r2 + r3 (A.1)

where r1 and r2 are reaction rates for reactions (1) and (2) alreadypresented in the text. Let the conversions of methane and carbondioxide be defined by:

xCH4 =(nCH4,f−nCH4

)nCH4,f

(A.2)

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xCO2 =(nCO2−nCO2,f

)nCH4,f

(A.3)

The molar flow rate ni of the component i in the reacting mix-ture can be expressed, using stoichiometric relations, in terms ofthe molar feed rate ni,f and the conversions xCH4 and xCO2 . Stoi-chiometric relations in terms of xCH4 and xCO2 can be written asfollows:

nCH4 = nCH4,f−xCH4nCH4,f (A.4)

nH2O = nH2O,f−(xCH4 + xCO2

)nCH4,f (A.5)

nCO = nCO,f + (xCH4−xCO2

)nCH4,f (A.6)

nCO2 = nCO2,f + xCO2nCH4,f (A.7)

nH2 = nH2,f + (3xCH4 + xCO2

)nCH4,f (A.8)

The total molar flow rate nT is given by:

nT = nT,f + 2xCH4nCH4,f (A.9)

The material, energy and momentum balance equations can bewritten as follows:

dxCH4

dl= A���CH4

RCH4

nCH4,f(A.10)

dxCO2

dl= A���CO2

RCO2

nCH4,f(A.11)

dTdl

= 1CpUs

{���H1�CH4(r1 + r2) + ���H2�CO2(r2 + r3) + 4

U

dti(TS−T)

}(A.12)

dPTdl

= f

(G2

�gdP

)(A.13)

T is the temperature of the reacting mixture, Ts the temperatureof inner tube skin, PT the total pressure along the reformer, Athe reformer tube cross sectional area, �� the bulk density of thecatalyst, �CH4 and �CO2 are the effectiveness factors associated withthe reactions of CH4 and CO2, respectively, �Hi (i = 1,2) the heatsof reactions, us the superficial gas velocity, G the gas mass velocity,dp the catalyst particle diameter and dti the internal diameter ofthe reformer tubes.

The heat transfer coefficient U needed in the energy balanceequation (Eq. A.12) is calculated using the correlation developedby De Deken et al. (1982); and Froment and Bischoff (2010).

Ergun correlation is used to calculate the friction coefficient(f) in the momentum balance Equation (A.13) for large Reynoldsnumber:

f = 1.751−εε3

(A.14)

where ε is the bed void fraction.

Catalyst Particle EquationsThe catalyst pellet is assumed to be a slab with a characteristiclength lc. This is a simplification since the actual shape of thecatalyst is a cylinder with holes. However, the adopted assump-tion will simplify the modelling. Besides, the effectiveness factorscomputed based on the selected shape will be further tuned tovalidate the model.

The material balance equations for the catalyst pellet take thefollowing form:

d2PCH4,p

dω2= RCH4RTl

2c

DCH4,e

(A.15)

d2PCO2,p

dω2= −RCO2RTl

2c

DCO2,e

(A.16)

with the boundary conditions,

PCH4,P = PCH4,s and PCO2,P = PCO2,s at ω = 1.0 (A.17)

dPCH4,P

dω= dPCO2,P

dω= 0 at ω = 0 (A.18)

Pi,p and Pi,s designate, respectively, the partial pressure of com-ponent i in catalyst particle and on catalyst particle surface, ω thedimensionless coordinate of the catalyst pellet and Di,e is the effec-tive molecular diffusitivity of component i. The slab is assumedto be isothermal while external mass and heat transfer resistancesare assumed negligible.

The physico-chemical parameters such as viscosity, thermalconductivity, molecular diffusitivities are obtained using standardcorrelations (Reid and Sherwood, 1958; Froment and Bischoff,2010).

It should be noted that in the model development, we assumedthat the flux of each species depends only on its concentration gra-dient, which is a simplification. A more rigorous model such as thedusty gas model would account for the contribution of the fluxesof other components. However, this will complicate the modelling.Moreover, some investigators (e.g. Elnashaie and Abashar, 1993)have compared the dusty gas model against simple Fickian typemodels for the steam reforming under industrial conditions, andconcluded that in the case of low steam to methane ratios thepredictions of the different models are in good agreement.

Model of the FurnaceOne type of firing will be considered which is the bottom-firedfurnace. Roesler (1967) introduced the two-flux model for one-dimensional furnaces. The model is capable of taking into accountthe radiative heat along and normal to the gas flow. We will adoptin this work the Roesler model that was modified by Filla (1984)to take the following form:

d2E1

dl2= ˛1

{ˇ(E1− T4

g

) + εtat

(E1− T4

t,0

) + εrar((1− )E1− E2)}

(A.19)

d2E2

dl2= ˛2

{εtat

(E2−(1− )T4

t,0

) + εrar( E2−(1− )E1)}

(A.20)

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with the following boundary conditions:At l = 0,

1˛1

dE1

dl= − 1

˛2

dE2

dl= εr

2−εr{(1− )E1− E2} (A.21)

At l = L,

− 1˛1

dE1

dl= 1˛2

dE2

dl= εr

2−εr{(1− )E1− E2} (A.22)

E1 and E2 are, respectively, the grey and clear gas component heatgas flux, Tg the temperature of furnace gas, Tt,0 the outer tubesurface temperature, the Stefan-Boltzman constant, the bandradiation fraction, ˇ the absorption coefficient and εt, are, respec-tively, the emissivities of refractory slab and tube wall. In thismodel, it is assumed that the gas emissivity can be approximatedby:

εg =

{1−exp

(−BL−

2

)}(A.23)

where L− is the beam length and

˛1 = ˇ + at + ar , ˛2 = at + ar (A.24)

where ar and at are, respectively, the refractory furnace area andthe tube side furnace area to furnace free volume half ratio. Therefractory temperature Tr is given by:

Tr =(E1 + E2

)1/4

(A.25)

The differential heat balance on the flue gas stream takes thefollowing form:

GgCgd[Tg−F

(T∗−Tg,0

)]dl

= 2ˇ[E1− Eg

](A.26)

where Gg is the process flue gas mass velocity, T* the adiabaticflame temperature, F the fraction of the fuel burned along thereformer and Eg is the gas emissive powers in side fired furnaces.

The flue gas inlet temperature Tg,0 is given by:

Tg,0 =[E1,0

]1/4

(A.27)

where E1,0 is the value of E1 at the gas inlet.The heat transfer Qr by radiation to the reformer tubes is given

by:

Qr = 2εtat(Er−Et)V (A.28)

where Et is the tube emissive powers in side fired furnaces, Er =T4

r and V is the free volume of the reformer.

Model Equations for Reduction Furnace

Mass balance-solid phaseIn the first step, mass balance equations are written for iron oxidesin the interface positions as particles move down through the reac-tor. The changes in iron oxides along the bed are represented interms of the variation of the radii of the shrinking interfaces for the

three layers. For the magnetite/hematite interface, the change inthe hematite concentration is defined as the oxygen fraction con-sumed during the reduction at this interface. Consider a volumeof element (�/d)d2

t Z, the change in the hematite concentration inthe pellets of this element, in terms of the interface radius, can beexpressed as follows:

�CFe2O3 =

solid volume in Fe2O3 layer︷ ︸︸ ︷(1−εh)�

(4/3�r3

1

)�h

(1−εb)�/4d2t �Z︸ ︷︷ ︸

volume of reactor solid phase

(A.29)

where Z is the axial coordinate, r1 the effective radius of interfacebetween hematite and magnetite in the pellet, dt the diameter ofthe furnace and εh and �h are, respectively, the porosity and molardensity of hematite.

The number of moles of hematite consumed in this element isrelated to the reaction rates by the following balance equation:

(�

4dt

2)

(1−εb)us�CFe2O3 = −4�(xH2 + xCO

)ϕh−m

(A.30)

In this equation, we use the stoichiometric ratio: one atom ofoxygen is removed from the pellet for each mole of hydrogenor carbon monoxide reacted. The parameter ϕh−m is simply therelation between the consumed oxygen atoms and the reducedhematite molecules. xH2 and xCO are reaction rates with respectto H2 and CO, respectively, us the solid velocity and εb is the bedbulk porosity.

Introducing the dimensionless reactor height:

� = Z

L(A.31)

the balance equation for the hematite concentration in terms ofthe interface radius can be rewritten as a differential equation:

ϕh−mus(1−εh)(1−εb)�h

(1−εb)L

d((4/3)�r3

1

)d�

= −4�(xH2 + xCO

)(A.32)

The molar flux rate for the solids is given by:

Gs = us�h(1−εb)(1−εh) (A.33)

where �h and εh are the density and porosity of the hematite.Thus, Equation (A.32) can be rewritten for the radius r1 of the

magnetite/hematite interface as follows:

dr1d�

= −L(1−εb)(xH2 + xCO

)r21Gsϕh−m

(A.34)

Similar equations can be written for the wustite/magnetite, andFe/wustite layers:

dr2d�

= −L(1−εb)(yH2 + yCO

)r22Gsϕm−w

(A.35)

dr3d�

= −L(1−εb)(zH2 + zCO

)r23Gsϕ

w−Fe(A.36)

where r2 and r3 are, respectively, the effective radius of interfacebetween magnetite and wustite and between wustite and iron.

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ϕm−w and ϕw−Fe denote, on the other hand, oxygen atoms react-ing/mole of magnetite converted to wustite, and oxygen atomsreacting/mole of wustite converted to iron.

For these equations, the initial conditions are given by:� = 0, r1 = r2 = r3 = r4 (r4 is the particle diameter) (A.37).

Mass balance-gas phaseTwo mass balance equations are written for the reducing gases,one for hydrogen and the other for carbon monoxide. The hydro-gen molar balance in the element (�/4)d2

t Z is given by:

gas volumetric flowrate︷ ︸︸ ︷�

4dt

2ugεb ×�CH2 =molar flux of H2︷︸︸︷jH2ap ×�

4d2t (1−εb)�Z (A.38)

The mass flux of hydrogen jH2 is related to the reduction ratesat all interfaces by the following relation:

JH2ap =

moles of H2 consumed in all interfaces per unit time︷ ︸︸ ︷4�

(xH2 + yH2 + zH2

)(4/3)�r3

4︸ ︷︷ ︸Pellet volume

(A.39)

The mass balance equation for hydrogen can be rewritten asfollows:

εbug�g

(4/3

)�r3

4

(1−εb)L�g

dCH2

d�= 4�

(xH2 + yH2 + zH2

)(A.40)

Simply, this equation relates the change in hydrogen concen-tration along the bed with the amount of hydrogen used to reducethe three types of iron oxides. The gas mass velocity is defined asfollows:

G = εbuG�G (A.41)

Thus, Equation (A.40) can be rewritten as follows:

dCH2

d�= 3(1−εb)L�g

r34G

(xH2 + yH2 + zH2

)(A.42)

Similar equation can be derived for CO:

dCCO

d�= 3(1−εb)L�G

r34G

(xCO + yCO + zCO) (A.43)

with the following boundary conditions:

� = 1, CH2 = CH2, inlet, CCO = CCO, inlert (A.44)

Energy balance-solid phaseThe differential heat balance for solid relates the change in solidtemperature along the bed with the heat transferred from the gasphase into the solid phase, and the heat generated or consumedby the reduction reactions.

(�/4)d2t

molar fux rate of solid︷ ︸︸ ︷us(1−εb)(1−εh)�h Cps�Ts =

number of pellets in unit volume︷ ︸︸ ︷(1−εb)(�/4)d2

t �Z

(4/3)�r24

× 4�r24h

(Tg−Ts

)︸ ︷︷ ︸heat transfer per pellet

(A.45)

Tg and Ts are, respectively, gas and solid temperature along thereactor, h the heat transfer coefficient between the gas and solidand Cps the solid heat capacity.

The solid energy balance equation can be rewritten as:

dTs

d�= 3hL(1−εb)

GsCpsr4

(Tg−Ts

) + 3hL(1−εb)GsCpsr

34HR (A.46)

with the boundary conditions:

� = 0, ∈Ts = Ts, inlet (A.47)

HR defines the heat generated or consumed by the reductionreactions, and it is given by:

HR = (−�H1)xH2 + (−�H2)yH2 + (−�H3)zH2

+ (−�H4)xCO + (−�H5)yCO + (−�H6)zCO (A.48)

with �Hi being the heat of reaction for reaction Ri, (i = 1–6).

Energy balance-gas phaseThe heat balance for the gas phase is written as:

gas molar flowrate︷ ︸︸ ︷G(�/4)d2

t Cpg�Tg = (4�r2

4

)total area ofpellets︷ ︸︸ ︷

(�/4)d2t (1−εb)�Z

(4/3)�r34

× h(Tg−Ts

)︸ ︷︷ ︸heat flux of solid

(A.49)

Model Modification to Account for Steam Reformingand CarburisationIn this part, we include the effect of other side reactions that couldoccur in the reduction furnace. All these reactions are assumed totake place on the iron layer. These reactions are:

Steam reforming reaction (Takahashi et al., 1986):

CH4 + H2O ↔ CO + 3H2 (A.50)

The rate equation for this reaction is given by:

rmf = A1e(6.77×103)/RT(PCH4PH2O−PCOP

3H2

KM

)VFe (A.51)

and in the reverse reaction:

rmb = A2e(4.18×103)/RT(PCOP

3H2

−PCH4PH2OKM

)VFe (A.52)

Methane decomposition reaction (Sawai et al., 1998):

CH4 ↔ C + 2H2 (A.53)

with the rate equation:

rd = A3e(−55×103)/RT

PO.5H2

(PCH4−

P2H2ac

Kd

)VFe (A.54)

where ac is the carbon activity.Carbon monoxide disproportionation (Grabke, 1965):

2CO ↔ C + CO2 (A.55)

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The reaction rate equation is:

rc =(A4e(−27.2×103)/RTP0.5

H2+ A5e(−8.8×103)/RT

)(P2

CO−PCO2ac

Kc

)

(A.56)

Carbon monoxide reduction by hydrogen (Grabke, 1965):

CO + H2 ↔ C + H2O (A.57)

rr = A6e(−16.48×103)/RT

(PCOPH2−

PH2Oac

Kr

)(A.58)

The subscripts m, d, c, r denote steam reforming, methanedecomposition, carbon monoxide disproportionation, and carbonmonoxide reduction by hydrogenation respectively. As was men-tioned in the text, the parameters Ai (i = 1,6) appearing in thereaction rates will be used as tuning parameters to fit industrialdata. Reactions involving carbon and iron to form carbides willnot be included in the model due to the scarcity of information.

The reactor model will include two additional mass differ-ential equations balances in the gas phase: one for methaneconsumption:

dCCH4

d�= rm + rd

G(4/3)�r34

(1−εb)L�g (A.59)

and the second one for carbon formation:

dCC

d�= rd + rc + rr

2Gs(4/3)�r34

(1−εb)L�G (A.60)

The gas balance equations (Eqs. A.42 and A.43) for hydrogenand carbon monoxide are modified to the following forms:

dCH2

d�= 3(1−εb)L�g

r34G

(xH2 + yH2 + zH2 + rr−2rd−3rm

4�

)(A.61)

dCCO

d�= 3(1−εb)L�G

r34G

(xCO + yCO + zCO + (rr + 2rc−rm)

4�

)(A.62)

The carbon activity ac is based on the formula given by Chip-man (1972):

logaC = 2300/T−0.92 + (3860/T)C + logC

1-C(A.63)

where C is atom C/atom Fe.The solid differential heat balance equation (A.46) is modified

to include the heat of all side reactions:

dTs

d�= 2hL(1−εb)

GsCpsrr

(Tg−Ts

) + 3hL(1−εb)GsCpsr

34HR + L(1−εb)

(r34−r3

3

)GsCpsr

34

× ((−�H1)rm + (−�Hd)rd + (−�Hc)rc + (−�Hr)rr)

(A.64)

KineticsThe kinetics used in this work are those developed by Tsay et al.(1976a, 1976b)

For hydrogen production, the kinetic rates constants are:

k1 = 160 exp

(−22, 000RT

)(A.65)

k2 = 23 exp

(−17, 000RT

)(A.66)

k3 = 20 exp

(−15, 200RT

)(A.67)

For the reduction of carbon monoxide, the kinetic rate constants(m/s) are:

k1 = 2700 exp

(−27, 200RT

)(A.68)

k2 = 25 exp

(−17, 600RT

)(A.69)

k3 = 17 exp

(−16, 600RT

)(A.70)

Model for the RecuperatorThe recuperator is a heat exchanger where a cold side of heatcapacity Cc is heated from tc to Tc with heat source of capacity Ch

from temperature th to Th. For counter-current flow, and assumingthat Ch <Cc, the following relation holds:

Th−tcth−TC

= e−N(1−Ch/Cc) = ˛ (A.71)

where

N = UA

Ch(A.72)

Therefore,

Th−tC = ˛(th−TC) (A.73)

TC = th−1˛

(Th−tC) (A.74)

The heat balance, on the other hand, yields:

CC(TC−tC) = Ch(th−Th) (A.75)

Eliminating Tc between these two equations yields:

Th = ˇth + (1−ˇ)tC (A.76)

with

ˇ = CC−Ch

(CC/˛)−Ch(A.77)

Equation for the CompressorThe compressor power is given by:

k

�(k−1)P1V

[(CR)

k−1k −1

](A.78)

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Page 21: Mid Rex 7

where CR is the compressor ratio (P2/P1), k = 1.35 the polytropiccoefficient and �= 0.7 the efficiency.

Equation for the ScrubberThe equation for the scrubber is provided by the vapor pressureof water (PV) given by:

log10 (760Pv) = 8.07131− 1730.63233.426 + t(◦C)

(A.79)

where PV is in atm.

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Manuscript received March 11, 2011; revised manuscriptreceived April 12, 2011; accepted for publication April 19, 2011

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