Haber process and steam-coal gasification: Two standard thermodynamic problems elucidated using two...

Haber Process and Steam-CoalGasification: Two StandardThermodynamic ProblemsElucidated Using TwoDistinct ApproachesHOUSAM BINOUS,1 AHMED AHEED,1,2 MOHAMMAD M. HOSSAIN1,2

1Department of Chemical Engineering, King Fahd University of Petroleum & Minerals, Dhahran 31261, Kingdom of

Saudi Arabia

2KACST-TIC for CCS, King Fahd University of Petroleum & Minerals, Dhahran 31261, Kingdom of Saudi Arabia

Received 28 January 2015; accepted 26 May 2015

ABSTRACT: The present study shows how the direct Gibbs free energy minimization technique is sometimes

superior to the reaction coordinates—equilibrium constants method when the thermodynamic analysis of

complex systems is performed. In this respect, the above two methods are applied for two different processes to

determine their equilibrium compositions: (1) the Haber process: a simple problem consisting of a single gas-

phase reaction where both methods are expected to give the same result and (2) steam-coal gasification process

which ismore complex since it involves numerous reactions and solid-phase chemical species (i.e., coal or carbon).

Thus, in the second case the reaction coordinates—equilibrium constants method fails to provide correct

predictions of the equilibrium composition. In addition, the authors give the optimum conditions for the two

processes: high pressure for Haber process and high temperature for steam-coal gasification in agreement with

LeChatelier principle. In order to deal with deviation from the ideal-case assumption, the Peng–RobinsonEquation

of State (PR EOS) is implemented for both techniques and both case studies. All computations and figures are

generated using a single computer algebra, Mathematica©. © 2015 Wiley Periodicals, Inc. Comput Appl Eng Educ

9999:1–13, 2015; View this article online at wileyonlinelibrary.com/journal/cae; DOI 10.1002/cae.21672

Keywords: Gibbs free energy minimization; reaction coordinates; equilibrium constants; Haber process;

steam-coal gasification; mathematica

INTRODUCTION

Thermodynamic analysis calculates the equilibrium distributionof different chemical species involved in specific system undercertain conditions [1] (i.e., temperature, pressure and/or initialreactants’ mole fractions). Such system is subject to chemicalreactions occurring among the involved chemical species in

presence of a catalyst. Thus, it is very important to know theequilibrium conversions (or mole fractions), that can be achieved,in order to evaluate the catalyst efficiency and overallperformance.

To perform equilibrium conversions calculations in reactivesystems, the general idea is that the system attains the equilibriumwhen its total Gibbs free energy is at its global minimum value [2].To achieve this, two common methods are usually employed: (1)reaction coordinates—equilibrium constants method and (2)direct total Gibbs free energy minimization technique [1,2].

Generally, the direct Gibbs free energy minimizationtechnique is the preferred method if multiple reactions occursimultaneously. Indeed, this method does not require neither achemical reaction(s) set nor a knowledge of the equilibrium

Contract grant sponsor: King AbdulAziz City for Science andTechnology (KACST).

Correspondence to H. Binous (binoushousam@yahoo.com)

© 2015 Wiley Periodicals, Inc.

1

constants. The only thing that is needed is a list of all chemicalspecies involved in the system as well as a method of computationof their Gibbs free energy. Moreover, reaction coordinates—equilibrium constants method fails to handle solid reactants andproducts [1]. Here, we apply both methods for two different casestudies: (1) Haber process and (2) steam-coal gasification.However, instructors can choose other examples of reaction setssuch as biomass gasification, ethane steam cracking, catalyticcracking of crude oil, and methanol synthesis from syngas. Thepresent Gibbs free energy minimization approach has beenintroduced in two graduate level chemical engineering courses atKing Fahd University of Petroleum & Minerals (KFUPM): (1)CHE 530 or Advanced Reaction Engineering and (2) OGSF 508 orIndustrial Chemical Reactions—Catalysts—R&D Trends. Thisteaching helps students: (1) understand the thermodynamiccharacteristics of industrial chemical reactions, (2) find thefeasibility of a reaction, and (3) determine the displacement ofchemical equilibria as a function of operating conditions (i.e.,temperature and pressure) and reactants initial composition.Students appreciate especially the use of Mathematica© because itmake difficult mathematical problems easily tractable and allowthem to focus on the chemical engineering side of the problem athand.

Several pedagogical papers have dealt with thermodynam-ics [3–6] and chemical reaction engineering [7,8]. Most of thepedagogical articles considered reaction coordinates—equilibri-um constants methods and/or ideal gas-phase assumption (e.g.,Ref. 6) to estimate the equilibrium product compositions. Tothe best of our knowledge, only one article has applied theminimization of the Gibbs free energy technique to determine theequilibrium mole fractions of the coal methanation process [9].Although, this article have not compared the prediction betweenthe two after mentioned approaches. Each method has severaladvantages over the other. In some cases, the Gibbs free energyminimization approach shows clear superiority in accurateprediction. Especially, for the complex reaction system wheredefining the exact reaction sets is difficult (or in some cases almostimpossible). The thermodynamic equilibrium modeling usingGibbs free energy minimization technique has become a verypopular approach for studying new reactions [10–15].

Keeping the above in mind, the present paper reports acomparison of the two approaches by considering two casestudies.

After a short description of the PR EOS, we provide insightsabout both resolution approaches. Then, the two case studies arepresented. Finally, we conclude the article by giving details aboutthe pedagogical value of the presented work.

THE PENG–ROBINSON EQUATION OF STATE

Even though the well-known Peng–Robinson EOS is especiallysuitable for hydrocarbon systems, it can also handle other commonsystems (except in case of sulfur present) containing water, carbondioxide and monoxide, nitrogen, and oxygen among otherchemical species [14]. This EOS can describe both liquid-phaseand vapor-phase behavior. It relates molar volume to pressure andtemperature as follows [15]:

P ¼ RT

V � b� aaðTÞV V þ bð Þ þ bðV � bÞ ð1Þ

where:

a ¼ 0:45724R2T2

c

Pc; b ¼ 0:0778

RTc

Pcð2Þ

a T�¼ 1þ 0:37464þ1:54226v�0:26992v2Þ 1�T

12r

� �� i2

h�ð3Þ

and

Tr ¼ TTc

ð4Þ

Note that the above equation is valid only for pure component. Inthe case of mixtures, the following mixing and combining rulesshould be used [1]:

P ¼ RTVmix � bmix

� amixaðTÞVmix Vmix þ bmixð Þ þ bmix Vmix � bmixð Þ ð5Þ

where amix and bmix are related to individual components ones by:

bmix¼Xni¼1

yibi; amix¼Xni¼1

Xnj¼1

yiyjaij; aij¼ffiffiffiffiffiffiffiffiaiaj

p1�kijÞ� ð6Þ

ai ¼ 0:45724aiðTÞR2Tc

Pci; bi ¼ 0:0778RTci

Pcið7Þ

ai Tð Þ ¼ 1þ ki 1�ffiffiffiffiffiffiffiTr i

p� �h i2;

ki ¼ 0:37464þ 1:54226vi � 0:26992v2i ð8Þ

and

Tri¼ TTc i

ð9Þ

Here, kij, the binary interaction parameters, can be set equal to zeroif the molecules are almost same in size and charge, or if thesystem approaches ideality, otherwise they have to be determinedeither experimentally or by using approximation models [2].

REACTION COORDINATES

Reaction coordinate—equilibrium constant method considers thatat equilibrium (minimum Gibbs free energy point) the differential(or change) of total Gibbs free energy with respect to the numberof moles (of each chemical species involved in the system) is zero.However, this is true only for constant temperature and pressuresystems. This method can also be applied for flow systems sinceeach point (specific temperature and pressure) can be dealt with asclosed system and because the Gibbs free energy is a state functionwhich means it does not depend on the path of the process [1].

The steps of this method are as follow:

a) Selection of complete and independent chemical reactionsset. Completeness means that the chemical reactions' setshould include all chemical compounds involved in thesystem. On the other hand, independence requires that eachchemical reaction has its own direction (combinations ofreactants and products) and is written in its overall form.To reduce complexity of this step, very common and goodsimplification can bemade by ignoring a reactionwith eithera relatively high or a relatively small equilibrium constant.

2 BINOUS ET AL.

Any mistake made in the reactions set selection will resultin very different results from what must be. Completenessproblems result from the absence of some chemicalspecies from the calculations. While independency prob-lems leads to either no solution or an infinite number ofsolutions for the studied system.

b) Relating mole fraction of different species to the reactioncoordinates of the selected chemical reactions.

yi ¼ni0 þ

Xjni jej

n0 þX

jnjej

ð10Þ

c) Calculating equilibrium constants for those reactions

Kj ¼ exp �DGjo

RT

� �ð11Þ

d) Relating equilibrium constants to mole fractions, then toreaction coordinates.Y

i

yifi

� �ni;j ¼ PPo

� ��nj

Kj ð12Þ

e) Solving the resulting system of non-linear algebraicequations to evaluate the values of reaction coordinates.Generally, this step done by using iterative method due topresence of fugacity coefficients (calculation of fugacitycoefficients needs knowing of the mole fractions and viceversa as stated by the above equation, so trial and errorprocedure is needed). A good initial guess for this trial anderror method considers all fugacity coefficients to be equalto unity (i.e., the ideal gas mixture assumption). In thepresent work, we make use of the built-in command ofMathematica© called FindRoot to solve simultaneously thecorresponding system of nonlinear algebraic equations. Inaddition, since we seek solution for various values of para-meters such as temperature, pressure and/or initial reactantmole fractions, the problem is readily tractable using arc-length continuation. A short description of this method isgiven in the following section of the present article.

f ) The final step is the calculation of the involved chemicalspecies mole fractions from the found reaction coordinates(back calculations). Note that in several instances there aremore than one solution of such system of nonlinearalgebraic equations. The constraints below allow thedetermination of the appropriate solution:X

i

yi ¼ 1; 0 � yi � 18i ð13Þ

The advantages of the reaction coordinates—equilibrium con-stants method resides in constructing a solvable algebraic systemby shifting the system’s degree of freedom to a zero value eventhough it may not seem to be the case when the set of molefractions relations used as is.

ARC-LENGTH CONTINUATION

Suppose one has to solve a system of nonlinear algebraic equationswhich solution depends on a parameter called a in the textthereafter. This parameter must be set first in order to get thesystem of equations’ solution. Generally getting the solution ofsuchlike systems as a continuous function of this parameter is noteasy task because of nonlinearity and solutions’multiplicity. Thus,

it is advisable to use a parametric method to see how the solutionchanges with a (especially if one solution is correct and the othersare rejected). An excellent approach consists in the use of theconcept of arc-length continuation by letting all of the system’sunknowns and the parameter functions in the arc-length [16],labeled s in the text thereafter. In such case, one has to introduce anadditional differential equation called the arc-length auxiliaryequation. Note that the resulting differential-algebraic systemneeds an initial condition to be solved. The true solution obtainedat a specific value of the parameter, a, should be used for s¼ 0. Allof the above computations are readily achieved using the built-inMathematica© commands called FindRoot and NDSolve.

In vector form the system of nonlinear algebraic equation iswritten as follows [16]:

f u;að Þ ¼ 0 ð14Þwhere the components of the vectors u and f are:

f ¼ f1;f2;f3; . . . ;fNf g; u ¼ fu1;u2; . . . ;uNg ð15Þ

Suppose the system has a solution u0 when a¼a0, orf u0;a0ð Þ ¼ 0, now by letting u ¼ u sð Þ and a¼a(s) and takingtotal derivative of f:

df ¼ fu u;að Þ duds

þ fa u;að Þ dads

¼ 0 ð16Þ

the unknowns number becomes Nþ1 as follow:du1ds ;

du2ds ; . . . ;

duNds ;

dads

, but the total differential has just N

independent equation, so the Nþ 1st equation is:

duds

� duds

þ da

ds

� �2

¼ 1 ð17Þ

In the present study, in order to obtain the equilibrium molefraction distribution using the reaction coordinates—equilibriumconstants method, one has to solve a system of nonlinear algebraicequations where either the temperature, the pressure and/or theinitial reactants number of moles is a regarded as the parameter, a.This makes the arc-length continuation technique a method ofchoice when one tackles such problems.

GIBBS-FREE ENERGY MINIMIZATION

Direct total Gibbs free energy minimization can be performed byusing any normal optimization method [2]. Here, the objectivefunction that should be minimized is the total Gibbs free energy,and since it depends on each chemical species number of moles,the number of moles must be taken as decision variables. It shouldbe noted that the number of moles are subject to constraintssteaming from the various atomic mass balance equations. It is inthese above constraints where the initial reactants mole fractionscome into play.

So the general form of our optimization problem is [1]:

minGtotalðfnigÞSubjected to: X

i

niaik ¼ Ak ð18Þ

So the optimum (minimum) point can be obtained analyti-cally (by using Lagrangian multipliers since all of constraintsare of equality ones), or numerically by using any suitable

HABER PROCESS AND STEAM-COAL GASIFICATION 3

algorithm [17] or the Mathematica© built-in command calledFindMinimum.

In case of using the Lagrangian multipliers, the equations tobe solved are [2]:X

i

niaik ¼ Akðk ¼ 1; 2; 3; . . . ;wÞ ð19Þ

Goi þ RTln

yif iPPo

� �þXk

lkaik ¼ 0ði ¼ 1; 2; 3; . . . ;NÞ ð20Þ

where lk are the Lagrangian multipliers. Note also that there is aneed of an iterativemethod (due to fugacity coefficients/number ofmoles dependency) when using this method. Moreover, thismethod gives all of the critical points (maxima, minima, andsaddle points) which adds another step to the calculations (i.e., thedetermination the global minimum). Because of this, using oneof the numerical minimization algorithms designed to converge tothe minimum point directly is the preferred approach. Regardingnumerical algorithms, any interior or exterior point algorithm canbe applied. However, in case of using interior point algorithm, onemust insure that the initial guess is feasible. Thus, to overcome thisdifficulty, it is suitable to use exterior point algorithm.

Total Gibbs free energy for all system is mainly thesummation of each single-phase total Gibbs free energy existing inthat system [2]. In this respect, it can be divided into two mainparts:

i. Gibbs free energy for gas and liquid phases (because twoof them calculated by applying the PR EOS and takingdifferent roots for the molar volume), and

ii. Gibbs free energy of solid phase.

Gas and Liquid Phase Mixture Gibbs Free Energy

Firstly, the liquid-phase mixture can be dealt with as if it is gas-phase with change of the value of molar volume only. This thegreat advantage of the cubic equations of state such as the PREOS.It should be pointed out that separate calculations of each species’fugacity can be avoided by using the concept of mixing andcombining rules (amix and bmix) and using the residual propertyrelation [14]:

Ggas;mix ¼ Gigmix þ GR

mix ð21ÞThe ideal gas mixture Gibbs free energy is given by:

Gigmix ¼

Xni¼1

yiGigi þ RT

Xni¼1

yilnyi ð22Þ

The ideal gas Gibbs free energy of pure substance is given by:

Gigi ¼

ZT298:15

Cp;idT � TZT

298:15

Cp;i

TdT þ RTln

PP1

�Si 298:15 T � 298:15ð Þ þ Gi 298:15 ð23Þwhere 298.15K is the reference temperature while P1 is thereference pressure.

One can notice that, all of the terms on the right hand side ofthe equation (23) are functions in temperature except one, whichhas also pressure dependency.

Regarding residual Gibbs free energy, it is the very sameformula for pure substances that is used but with somemodifications made in the EOS constants by using mixing andcombining rules shown below.

For pure components and by using Peng–RobinsonEOS [14]:

GR

RT¼ Z � 1� ln Z � Pb

RT

� �� a

2ffiffiffi2

pbRT

lnV þ 1þ ffiffiffi

2p� �

b

V þ 1� ffiffiffi2

p� �b

!ð24Þ

Thus, for mixtures, we find what follows:

GRmix

RT¼ Zmix � 1� ln Zmix � Pbmix

RT

� �

� amix

2ffiffiffi2

pbmixRT

lnVmix þ 1þ ffiffiffi

2p� �

bmix

Vmix þ 1� ffiffiffi2

p� �bmix

!ð25Þ

Zmix ¼ PVmix

RTð26Þ

where Vmix is determined from the roots of the cubic PR EOS. It isworth mentioning at this point that the relation between Gibbsfree energy and number of moles clearly comes from the molefractions, which appear in molar Gibbs free energy:

yi ¼ninT

ð27Þ

On the other hand, in the reaction coordinates—equilibriumconstants method, one needs to calculate fugacity coefficientsfrom the PR EOS as follows [14,18]:

f i ¼ exp

z� 1ð ÞBi

B� ln z� Bð Þ

� A

2ffiffiffi2

pB

2Xnc

j¼1yiAij

A� Bi

B

0@ 1Alnzþ 1þ ffiffiffi

2p� �

B

zþ 1� ffiffiffi2

p� �B

!! ð28Þ

Ai ¼ aiP

RT2 ; Bi ¼ biPRT

ð29Þ

Aij ¼ffiffiffiffiffiffiffiffiffiAiAj

p ð1� kijÞ ð30Þ

It should be clear that there are three values for z or v, whichresult from solving the PR EOS at specific temperatures andpressures. Either two of them have complex conjugate values(because the polynomial coefficients are real ones), in such case,there is only one phase existing in the system which is either thevapor or liquid phase. The other case where all values are real, insuch case the largest value corresponds to the vapor-phase molarvolume while the smallest one is its liquid-phase counterpart andthis can be proved by stability tests [2].

Solid Phase Gibbs Free Energy

Fortunately, total Gibbs free energy of solid phase can be setequal to the sum of each total Gibbs free energies for allcomponents existing in such phase due to negligible solid-solidinteractions.

ntotal; solid Gsolid; mix ¼Xi

ni; solidGi; solid ð31Þ

In this respect, Gibbs free energies of solid species (reactants andproducts such as carbon or coal in the second example) arecalculated and modeled as a function of temperature only (sincesolid phase is not affected by pressure remarkably).

4 BINOUS ET AL.

Thus the total Gibbs free energy of the system is:

Gtotal ¼ ntotal; gasGgas; mix þ ntotal; solidGsolid; mix ð32Þ

THE TWO CASE STUDIES

Haber Process

The Haber process produces ammonia gas by using hydrogen andnitrogen gases as main reactants and iron as catalyst according tothe following reaction:

3H2 þ N2?2NH3 ðr:1Þ

This chemical process is easy enough (i.e., a single reaction in thegas-phase) to be completely studied and understood thermody-namically. A single gas-phase reaction implies neither dependencynor completeness problems are expected when the reactioncoordinates—equilibrium constants method is utilized. In addi-tion, no solid material is involved in such system, so resultsobtained by performing the reaction coordinates—equilibriumconstants method should coincide with those obtained by usingdirect Gibbs free energy minimization technique.

The Reaction Coordinates—Equilibrium Constants MethodApplied to the Haber Process. Firstly, by simple considerations,all of mole fractions are related to the extent of the reaction asfollow:

yH2¼ nH2 0 � 3e

n0 � 2eð33Þ

yN2¼ nN2 0 � e

n0 � 2eð34Þ

yNH3¼ nNH3 0 þ 2e

n0 � 2eð35Þ

Since hydrogen and nitrogen are elements, their Gibbs offormation energies are zeros [1], so Gibbs energy change for thisreaction is simply that one for ammonia formation (under sameconditions). This allows us to write an expression for the singleequilibrium constant involved in the process:

K ¼ exp �DGNH3 f

RT

� �ð36Þ

Now, by substituting mole fractions in terms of reaction extent orcoordinate:

nNH30þ2e

n0�2e fNH3

� �2nH20

�3en0�2e fH2

� �3 nN20�en0�2e fN2

� �1 ¼ PPo

� �2

exp �DGNH3 f

RT

� �ð37Þ

This single equation is easily tractable using a trial and errormethod due to fugacity coefficients dependency onmole fractions.In the present work, we choose to solve simultaneously the systemnonlinear algebraic of equations consisting of the above equation,the three definitions of fugacity coefficients and the cubic PR EOS.Moreover, we apply arc-length continuation in order to determinethe solutions at various pressures and temperatures (see Fig. 1with a block diagram explaining the solution steps usingMathematica©).

Direct Gibbs Free Energy Minimization Applied to the HaberProcess. The problem statement when this method is used is thefollowing:

minGtotalðnH2 ; nN2 ; nNH3 Þ

Figure 1 Algorithm for the solution of the Haber process using the arc-length continuation technique and the reactioncoordinate method. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

HABER PROCESS AND STEAM-COAL GASIFICATION 5

subjected to:

2 0 3

0 2 1

" # nH2

nN2

nNH3

264375 ¼ AH

AN

" #ð38Þ

nH2 ; nN2 and nNH3 � 0

where:

AH

AN

" #¼ 2nH2 0 þ 3nNH3 0

2nN2 0 þ nNH3 0

" #ð39Þ

Since the reaction is taking place in the gas-phase, we have:

Gtotal ¼ ntotal;gasGgas; mix ð40ÞFigure 2 gives the block diagram for the solution steps involved inthis calculation using Mathematica©. Figure 3 shows the molefractions of hydrogen, nitrogen, and ammonia. The solid linesrepresents the mole fractions calculated by equilibrium constants—reaction coordinates method using arc-length continuationwhile the rhombus shape points refers to the results obtained byapplying Gibbs free energy minimization. As expected andexplained previously, it is clear that both approaches give thesame results. Using the built-in Mathematica© command, Manipu-late, it is possible for user of the program tovary the temperature andobserve its effect instantaneously on the graph. As expected, it isfound that ammonia synthesis is favoredbyhighpressures (i.e., uptoa few hundred bars) as well as low temperatures.

Steam-Coal Gasification

Steam-coal gasification process is much more complicated thanthe Haber process. All limitations of the reaction coordinates—equilibrium constants method apply. Indeed, there is a plethora ofreactions and a solid-phase component is involved in the reactionscheme.

Table 1 shows the chemical reactions taking place in thesteam-coal gasification process [19].

The Reaction Coordinates—Equilibrium Constants MethodApplied to the Steam-Coal Gasification. If all reactions aretaken into account, a dependency problem will arise as explainedbelow. If pressure is atmospheric, one can naively consider thesystem of equations given below:

yCOfCOyH2fH2

yH2OfH2O

¼ K2 ð41Þ

y2COf2CO

yCO2fCO2

¼ K3 ð42Þ

yCH4fCH4

y2H2f2H2

¼ K4 ð43Þ

y2H2f2H2y2COf

2CO

yCO2fCO2

yCH4fCH4

¼ K5 ð44Þ

y3H2f3H2yCOfCO

yH2OfH2OyCH4fCH4

¼ K6 ð45Þ

yCO2fCO2

yH2fH2

yCOfCOyH2OfH2O

¼ K7 ð46Þ

However, this set of equations has no solution as a result of theconflict created by substituting equilibrium relations for chemicalreactions (r.2) and (r.4) in the relation for reaction (r.6), which doesnot lead to K2

K4¼ K6 as expected. In addition, one can note that this

system consists of five unknowns (i.e., the five equilibrium molefractions) while there are six equations. Thus, the problem is over-determined (i.e., we have a negative degree of freedom) and not allthe equations are independent from each other. Thus, a carefullyselected chemical reaction set must be used, one can propose thefollowing set:

Cþ 2H2O?CO2 þ 2H2 ðr:8Þ

Figure 2 Algorithm for the solution of the Haber process using the Gibbs free energy minimization technique. [Colorfigure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

6 BINOUS ET AL.

Table 1 List of the Reactions Involved in the Steam-Coal Gasification Process

Chemical Reaction DG0f ð298KÞ½kj=mole� DH0

f ð298KÞ½kj=mole� K (8008C)

Cþ H2O?H2 þ CO(r.2) 89.82 130.41 7.04

Cþ CO2?2CO(r.3) 118.36 172.62 6.50

Cþ 2H2?CH4(r.4) �50.27 �74.90 0.050

CH4 þ CO2?2COþ 2H2(r.5) 168.63 123.76 132.01

CH4 þ H2O?COþ 3H2(r.6) 140.10 205.31 169.18

COþ H2O?H2 þ CO2(r.7) �28.54 �42.20 1.00

Figure 3 Comparison of hydrogen, nitrogen and ammonia mole fractions calculated by Gibbs free energy minimizationtechnique and the reaction coordinates—equilibrium constants method. [Color figure can be viewed in the online issue,which is available at wileyonlinelibrary.com.]

HABER PROCESS AND STEAM-COAL GASIFICATION 7

Cþ H2O?CO þ H2 ðr:9Þ

Cþ 2H2?CH4 ðr:10ÞAccording to this proposed system, we have what follows:

yCO2fCO2

y2H2f2H2

yH2OfH2O

� �2 ¼ K8 ð47Þ

yCOfCOyH2fH2

yH2OfH2O

¼ K9 ð48Þ

yCH4fCH4

y2H2f2H2

¼ K10 ð49Þ

The dependency versus temperature of K8, K9, and K10 are shownin Figure 4. It is clear from the trend of these curves that reactions(r.8) and (r.9) are endothermic but reaction (r.10) is exothermic(i.e., K10 for reaction [r.10] decreases as T increases). Also, theequilibrium constant, K9, for reaction (r.9) is larger than K8 forreaction (r.8) at higher temperatures. Thus, one expects methane,water, and carbon dioxide to be depleted at higher temperatures infavor of carbon monoxide and hydrogen. Also, a higher molar

Figure 4 Equilibrium constants for reactions r.8, r.9, and r.10. [Color figure can be viewed in the online issue, which isavailable at wileyonlinelibrary.com.]

8 BINOUS ET AL.

ratio r of water to carbon shifts reaction (r.8) to the product side,with more carbon dioxide being formed.

The relations among mole fractions and reactions’ extentsare:

yH2O ¼ nH2O0 � 2e8 � e9n0 þ e8 þ e9 � e10

ð50Þ

yH2¼ nH2 0 þ 2e8 þ e9 � 2e10

n0 þ e8 þ e9 � e10ð51Þ

yCO2¼ nCO2 0 þ e8

n0 þ e8 þ e9 � e10ð52Þ

yCO ¼ nCO0 þ e9n0 þ e8 þ e9 � e10

ð53Þ

yCH4¼ nCH4 0 þ e10

n0 þ e8 þ e9 � e10ð54Þ

Now, by substituting equations 50 through 54 into equations 47through 49, the system will be have zero degree of freedom

Figure 5 Equilibrium mole fractions of Methane, carbon monoxide, hydrogen, and carbon dioxide on a water-free basiscalculated by reaction coordinates—equilibrium constants method and arc-length continuation. [Color figure can beviewed in the online issue, which is available at wileyonlinelibrary.com.]

HABER PROCESS AND STEAM-COAL GASIFICATION 9

(three unknowns and three independent equations). Theunknowns are now the three extents of reactions. This maylead to more than one solution and one has to select the solutionthat provides mole fractions laying in the interval (0,1). Resultsobtained by applying such approach and arc-length continuationare displayed in Figure 5. Using the built-in Mathematica©

command, Manipulate, it is possible for user of the program tovary the pressure and observe it effect instantaneously on thegraph. As expected, it is found that syngas (i.e., hydrogen andcarbon monoxide) production is favored by low pressures (i.e.,

atmospheric pressure) as well as high temperatures. In Figure 6the extents of reaction for the three reactions (i.e. e8; e9 and e10)are plotted.

Direct Gibbs Free Energy Minimization Applied to the Steam-Coal Gasification. Based on the above description, one has tosolve the following optimization problem:

minGtotalðnH2O; nH2 ; nCO2 ; nCO; nCH4 ; nCÞ

Figure 6 Extent of reactions for reactions r.8, r.9, and r.10 calculated by reaction coordinates—equilibrium constantsmethod and arc-length continuation. [Color figure can be viewed in the online issue, which is available atwileyonlinelibrary.com.]

10 BINOUS ET AL.

Subjected to:

2 2 0 0 4 0

1 0 2 1 0 0

0 0 1 1 1 1

264375

nH2O

nH2

nCO2

nCOnCH4

nC

2666666664

3777777775¼

AH

AO

AC

264375 ð55Þ

nH2O; nH2 ; nCO2 ; nCO; nCH4 and nC � 0

where:

AH

AO

AC

264375 ¼

2nH2O0 þ 2nH2 0 þ 4nCH4 0

nH2O0 þ 2nCO2 0 þ nCO0

nCO2 0 þ nCO0 þ nCH4 0 þ nC0

264375 ð56Þ

It is obvious that there is only one component in this systemappears in solid state which is carbon namely. Thus, we have:

Gtotal ¼ ntotal;gasGgas;mix þ nCGC ð57Þ

Figure 7 Equilibrium mole fractions of methane, carbon monoxide, hydrogen, and carbon dioxide on a water-free basiscalculated by direct Gibbs free energy minimization. [Color figure can be viewed in the online issue, which is available atwileyonlinelibrary.com.]

HABER PROCESS AND STEAM-COAL GASIFICATION 11

Figure 7 displays the results of such minimization procedure (i.e.,the water-free equilibrium mole fractions, for methane, carbonmonoxide, hydrogen and carbon dioxide, versus temperature).Again Using the built-in Mathematica© command, Manipulate, itis possible for user of the program to vary either the pressure or thesteam/coal initial ratio and observe their effect instantaneously onthe graph. This is a very useful pedagogical feature that instructorscan use in the classroom. Indeed, recalculation for differentparameters and display of the results takes only a few seconds. Asexpected, it is found that syngas (i.e., hydrogen and carbonmonoxide) production is favored by low pressures (i.e.,atmospheric pressure) as well as high temperatures. The resultusing this methodology is far more accurate than the previous one(i.e., using reaction coordinates—equilibrium constants method).Indeed, the calculation using the Gibbs free energy takes intoaccount more accurately the solid-phase reactant (i.e., coal).

CONCLUSION

From the present study the following list of conclusion is drawn:

1. The reaction coordinates—equilibrium constants methodrequires clearly defined reactions incorporating all thespecies involved in the reactions.

2. The Gibbs free energy minimization technique does notrequire the stoichiometric reactions. This approach relieson thermodynamic databases that contain the values of thestandard Gibbs energy of the components.

3. The reaction coordinates—equilibrium constants methodalways considers the solid phase reactant activity as unity.

4. The Gibbs free minimization approach can take intoaccount solid reactants such as coal. Therefore, this methodrepresents the system more accurately.

It isworth, at this point, stressing the importance of this presentpedagogical contribution to the teaching of chemical engineeringthermodynamics. Indeed, we find equilibrium mole fractions fortwo important processes that are often taught in senior undergradu-ate and/or first-year graduate level courses. We use a commontechnique that is ubiquitous in most chemical engineeringthermodynamic courses: the reaction coordinates—equilibriumconstants method. However, we go beyond what is usuallyconsidered by studying high-pressure effects through the incorpo-ration of the fugacity coefficients using the PREOS. In addition, weuse theGibbs freeminimization technique, which is less commonlytaught in undergraduate courses despite its versatility. Indeed, wesee in our second case study (i.e., the steam-coal gasificationprocess) that it is easily implemented in Mathematica© using thebuilt-in command called FindMinimum. In addition, this methodcan handle solid-phase reactants such as coal and requires no priorknowledge of the numerous chemical reactions taking place in theprocess. Finally, all the Mathematica© codes, used in the presentedtheoretical calculations, are available upon request from thecorresponding author or at the Wolfram Demonstration Project(http://demonstrations.wolfram.com/index.html).

ACKNOWLEDGMENTS

The research team acknowledges the financial support providedby King AbdulAziz City for Science and Technology (KACST) to

this research under KACST-TIC for CCS project no. 03. The teamalso acknowledges the facilities and support provided byKFUPM.

AbbreviationsEOS equation of statePR Peng–Robinson

NOMENCLATURE

aik number of atoms of kth element involved in ithcomponent

Ak total number of atomic masses of kth element in thesystem

Cp,i ith component constant pressure heat capacity (J/mol�K)

Gi298:15 Gibbs free energy of ith component at 298.15 K (J/mol)Go

i standard Gibbs free energy for ith component (J/mol)Gig

mix ideal gas Gibbs free energy of the mixture (J/mol)GR

mix residual Gibbs free energy of the mixture (J/mol)DGo

i jth chemical reaction Gibbs free energy (J/mol)kij binary interaction parameter between ith and jth

componentsKj jth chemical reaction equilibrium constantnT total number of moles in the systemni number of moles of ith componentn0 initial total number of molesni0 initial number of moles of ith componentP Pressure (bar)Pc critical pressure (bar)Po standard pressure (bar)R universal gas constant (J/mol�K)Si298:15 ith component entropy at 298.15 K (J/mol�K)T Temperature (K)Ti critical temperature (K)Tr reduced temperatureV molar volume (cm3/mol)yi mole fraction of component iz compressibility factor

GREEK LETTERS

ej extent of jth chemical reactionlk kth lagrange multipliervi,j stoichiometric coefficient of ith component in jth chemical

reactionbfi ith component fugacity coefficientw acentric factor

REFERENCES

[1] J. M. Smith, H. C. Van Ness, and M. M. Abbot, Introduction tochemical engineering thermodynamics, 7th ed., McGraw-Hill HigherEducation, New York, 2005.

[2] J. W. Tester and M. Modell, Thermodynamics and its applications,3rd ed., Prentice Hall, Upper Saddle River, NJ, 1996.

[3] R. Baur, J. Bailey, B. Brol, A. Tatusko, and R. Taylor, Maple and theart of thermodynamics, Comput Appl Eng Educ 6 (1998), 223–234.

[4] Y. Liu, Development of instructional courseware in thermodynamicseducation, Comput Appl Eng Educ 19 (2011), 115–124.

12 BINOUS ET AL.

[5] F. Cruz-Perag�on, J. M. Palomar, E. Torres-Jimenez, and R. Dorado,Spreadsheet for teaching reciprocating engine cycles, Comput ApplEng Educ 20 (2012), 681–691.

[6] K. T. Klemola, Chemical reaction equilibrium calculation task forchemical engineering undergraduates—Simulating Fritz Haber’sammonia synthesis with thermodynamic software, Chem Eng Educ48 (2014), 115–120.

[7] L. M. Porto and R. H. Ogeda, Java applets for chemical reactionengineering, Comput Appl Eng Educ 6 (1998), 67–77.

[8] R. Rodrigues, R. P. Soares, and A. R. Secchi, Teaching chemicalreaction engineering using EMSO simulator, Comput Appl Eng Educ18 (2010), 607–618.

[9] A. L. Myers, Computation of Multiple Reaction Equilibria, ChemEng Educ 25 (1991), 112–116.

[10] J. Mazumder and H. I. de Lasa, Fluidizable La2O3 promotedNi/g-Al2O3 catalyst for steam gasification of biomass: Effect ofcatalyst preparation conditions, Appl Catal B Environ 168 (2015),250–265.

[11] Y. Cao and W-P. Pan, Investigation of chemical looping combustionby solid fuels. 1. process analysis, Energy Fuels 20 (2006), 1836–1844.

[12] M. Asadullah, Barriers of commercial power generation usingbiomass gasification gas: A review, Renew Sustain Energy Rev 29(2014), 201–215.

[13] H. I. de Lasa, E. Salaices, J. Mazumder, and R. A. Lucky, Catalyticateam gasification of biomass: Catalysts, thermodynamics andkinetics, Chem Rev 111 (2011), 5404–5433.

[14] S. I. Sandler, Chemical and engineering thermodynamics, 3rd ed.,Wiley, NY, 1999.

[15] D. Y. Peng and D. B. Robinson, A new two-constant equation of state,Ind Eng Chem Fundamen 15 (1976), 59–64.

[16] H. Binous and A. A. Shaikh, Introduction of the arc-lengthcontinuation technique in the chemical engineering graduate programat KFUPM, Comput Appl Eng Educ 23 (2015), 344–351.

[17] M. S. Bazaraa, H. D. Sherali, and C. M. Shetty, Nonlinear program-ming: Theory and algorithms, 3rd ed., JohnWiley & Sons, New York,2006.

[18] H. Binous, Applications of the Peng–Robinson equation of state usingmathematica, Chem Eng Educ 42 (2008), 47–51.

[19] E. Salaices, B. Serrano, and H. De Lasa, Biomass catalytic steamgasification thermodynamics analysis and reaction experiments ina CREC riser simulator, Ind Eng Chem Res 49 (2010), 6834–6844.

BIOGRAPHIES

Dr. Housam Binous a visiting AssociateProfessor at King Fahd University Petroleum&Minerals, has been a full time faculty memberat the National Institute of Applied Sciences andTechnology in Tunis for eleven years. He earneda Diplome d’ing�enieur in biotechnology fromthe Ecole des Mines de Paris and a PhD inChemical Engineering from the University ofCalifornia at Davis. His research interestsinclude the applications of computers in chemi-

cal engineering.

Mr. Ahmed Aheed was born in Khartoum(Sudan) in 1989. Mr. Aheed received a B.Sc.degree in chemical engineering from KhartoumUniversity. Since 2012, Mr. Aheed has beenaccepted as a full time graduate student in theChemical Engineering department at King FahdUniversity of Petroleum & Minerals (KFUPM).Currently, he is working on his M.S. thesis onthermodynamic modelling under the supervisionof Dr. HousamBinous and Dr.Mozahar Hossain.

Dr. Mozahar Hossain is an Associate Professorin Chemical Engineering at King Fahd Univer-sity of Petroleum & Minerals (KFUPM). M.Mozahar Hossain is also a Professional Engi-neer, currently registered with ProfessionalEngineers of Ontario, Canada. He is a memberof AIChE and ACS of Saudi Arabian Sections.Dr. Hossain serves as an Editorial Boardmemberof the International Journal of Chemical ReactorEngineering. His formal fields of research are

catalysis and chemical reaction engineering. His main focus is onthe development of next generation technologies in order to use valuableresources such as petroleum feed stocks and biomass resources moreefficiently with minimum environmental impacts.

HABER PROCESS AND STEAM-COAL GASIFICATION 13