lable at ScienceDirect Energy 152 (2018) 371e382 Contents lists avai Energy journal homepage: www.elsevier .com/locate/energy Modeling of syngas composition obtained from fluidized bed gasifiers using KuhneTucker multipliers Jordan Amaro*, Andr�es Z. Mendiburu, Ivonete �Avila S~ao Paulo University (UNESP), School of Engineering, Energy Department, Guaratinguet�a, SP, CEP 12510-410, Brazil a r t i c l e i n f o Article history: Received 11 December 2017 Received in revised form 12 March 2018 Accepted 25 March 2018 Available online 26 March 2018 Keywords: Gasification Fluidized bed Modified chemical equilibrium model * Corresponding author. E-mail address: jordan.amaro.gutierrez7@gmail.co https://doi.org/10.1016/j.energy.2018.03.141 0360-5442/© 2018 Elsevier Ltd. All rights reserved. a b s t r a c t This work aims to develop a modified chemical equilibrium model to accurately determine the syngas (synthesis gas) composition obtained from fluidized bed gasifiers. In order to do so, an optimization method was applied to determine the correction factors which modify the chemical equilibrium con- stants, the carbon conversion efficiency and the enthalpy of reaction. The gasification agents considered for this study were: air, steam, airesteam, and airesteameoxygen. The optimization method used the KuhneTucker multipliers to obtain small RMS errors. A total of 76 experimental compositions of syngas were selected. Among these data 60 were used to obtain correlations for the correction factor, the carbon conversion efficiency and the enthalpy of reac- tion. Then, a modified chemical equilibrium model was formulated by a taking advantage of these correlations. The modified chemical equilibrium model was validated showing very good accuracy for the deter- mination of the syngas composition, the RMS error were found to be between 0.94 and 4.84. © 2018 Elsevier Ltd. All rights reserved. 1. Introduction Currently, gasification is one of the most widely used biomass applications [1e6]. Among the different gasifier types the fluidized bed gasifiers present several remarkable characteristics. Some of these characteristics are: good mixing between biomass and gasi- fying agents, low nitrogen content in syngas composition, high carbon conversion efficiency, moderate production of tars and elevated lower heating value of syngas [6]. There are two types of fluidized bed gasifiers, i.e. the bubbling and circulating fluidized bed gasifiers [4]. These gasifiers are schematically presented in Fig. 1. A detailed description of the characteristics and operation of fluidized bed gasifiers can be found in the literature [1e6]. The syngas (fuel gas obtained from gasification) composition can be basically obtained by two modelling approaches which are: chemical equilibrium modelling and kinetic modelling [4]. A chemical equilibrium model allows to calculate the concentrations of gaseous products of gasification at a given gasification temper- ature, these concentrations being invariable for a theoretically infinite reaction time [2]. Also, a chemical equilibrium model does not consider the geometrical characteristics of the gasifier and the m (J. Amaro). hydrodynamics of the mixing process between the biomass and the gasifying agents (air, steam or oxygen) [1]. On the other hand, a kinetic model is used to study the progress of chemical reactions that take place inside the gasifier, thus allowing to determine the concentrations of gaseous products at different positions along the gasifier evaluated in a given time. It takes into account the gasifier's geometry as well as its hydrodynamics [1]. In the works by Loha et al. [7] and by Karmakar et al. [8] a chemical equilibrium model (which does not consider char and tar as products) was developed. The chemical equilibrium model applied in both aforementioned works propose the solution of a system of equations. This system is constituted by the mass con- servation equations of carbon, hydrogen and oxygen, and the chemical equilibrium equations of the methane formation reaction and the homogenous water-gas reaction. Subsequently, by solving this system of equations, the number of moles of the gases present in the reaction are calculated. Finally, the adjustment of the chemical equilibrium model consists of multiplying correction factors (calculated by trial and error) to the chemical equilibrium constants to evaluate the approximation of the theoretical composition of the syngas to the experimental composition, thus originating a decrease of the RMS error [9]. A chemical equilibrium model that incorporates correction factors for chemical equilibrium constants is referred to in the present research as a modified chemical equilibrium model. A modified chemical equilibrium mailto:jordan.amaro.gutierrez7@gmail.com http://crossmark.crossref.org/dialog/?doi=10.1016/j.energy.2018.03.141&domain=pdf www.sciencedirect.com/science/journal/03605442 http://www.elsevier.com/locate/energy https://doi.org/10.1016/j.energy.2018.03.141 https://doi.org/10.1016/j.energy.2018.03.141 https://doi.org/10.1016/j.energy.2018.03.141 Nomenclature cop Specific heat capacity at constant pressure in molar basis, kJ/K-mol h o Specific enthalpy in molar basis, kJ/mol h o f�298 Specific formation enthalpy at normal conditions in molar basis, kJ/mol so Specific entropy in molar basis, kJ/K-mol go Gibbs free energy in molar basis, kJ/mol Ti Inlet temperature of the “ith” gasifying agent into the gasifier, K aiði ¼ 1; :::; 7Þ NASA-Glenn coefficients biði ¼ 1; 2Þ NASA-Glenn integration constants R Universal gas constant, kJ/K-mol DBOA Dry basis without biomass ash DBWA Dry basis with biomass ash WBWA Wet basis with biomass ash DBWN Dry basis with nitrogen for the syngas composition DBON Dry basis without nitrogen for the syngas composition LHVBio Lower heating value of biomass, MJ/kg HHVBio Higher heating value of biomass, MJ/kg f i Number of moles for the “ith” substance per mole of carbon in the biomass, mol PMi Molar mass for the “ith” substance, g/mol ðPiÞDBWA Percentage of the “ith” substance in DBWA, % ðPAshÞWBWA Percentage of ash in WBWA, % ðPMoistÞWBWA Percentage of moisture in WBWA, % TMDBWA Total mass in DBWA Ho Total enthalpy, kJ xAir Air coefficient as a gasifying agent xi Number of moles for the “ith” substance, mol xT Total number of moles for gases present in the global gasification reaction, mol xSyn Number of moles of the syngas, mol nCC Carbon conversion efficiency, % AFExp Experimental air-fuel ratio, kg-air/kg-fuel AFStq Stoichiometric air-fuel ratio, kg-air/kg-fuel ER Equivalent ratio for gasification S=B Mass ratio of steam and biomass, kg-steam/kg- biomass O=B Mass ratio of oxygen and biomass, kg-oxygen/kg- biomass TGas Gasification temperature, K PGas Gasification pressure, atm Po Normal pressure, atm MFR Methane formation reaction WGHR Water-gas homogeneous reaction MRR Methane reforming reaction Ki Equilibrium constant for the “ith” reaction EPi Experimental percent amount of substance “ith”, % RMS Root mean square error FObj Objective function fMFR Correction factor for the MFR fWGHR Correction factor for the WGHR fMRR Correction factor for the MRR Chemical symbols C Carbon H Hydrogen O Oxygen N Nitrogen S Sulfur SiO2 Silicon dioxide H2O Water O2 Oxygen gas N2 Nitrogen gas H2 Hydrogen gas CO Carbon monoxide CO2 Carbon dioxide CH4 Methane SO2 Sulfur dioxide Greek symbols ratm Experimental percentage ratio between nitrogen and oxygen in the air Dh o React Enthalpy of global gasification reaction, kJ/mol miði ¼ 1; :::;6Þ KuhneTucker multiplier aiði ¼ 1;2;3Þ Variable used for notation change of nCC, xCO2 and xCH4 Superscripts o Magnitude evaluated at normal pressure Subscripts Ash Ash Moist Moisture l Liquid s Solid P Products R Reagents Steam Steam Air Air Oxy Oxygen Bio Biomass J. Amaro et al. / Energy 152 (2018) 371e382372 model for modeling the syngas composition is also used in other studies [10e13]. In the work by Radmanesh et al. [14] a kinetic model was developed. This model uses the hydrodynamics of the solid and gaseous phases, as well as diverse heterogeneous and homoge- neous reactions. The process of pyrolysis was considered very important and two kinetic models were used for this process. These models proved to be good to estimate the composition of the syngas and its LHV. In a similar way, in the work by Zheng and Morey [15], a biphasic kinetic model was developed, that model includes the kinetics of reaction and fluid dynamics for the gasifi- cation process of corn stover. The model predicts the compositions of syngas along the gasifier and the evolution of the particles over time, under different gasification conditions. From the obtained results, it is observed that the homogenous water-gas reaction and the residence time are very influential factors in the composition of the syngas. Thus, it was decided to use a modified chemical equilibrium model due to the simplicity of the model with respect to a kinetic model, in relation to the objective proposed in this research. In view of the actuality of modified chemical equilibrium models, one of the main objectives of this study is to calculate in an analytical and simple way the correction factors for the chemical equilibrium constants. Another objective of this research is to calculate a theoretical syngas composition very close to the experimental one to obtain a very low value of the RMS error. In Fig. 1. Schematic representation of: a) bubbling fluidized bed gasifier and b) circulating fluidized bed gasifier. J. Amaro et al. / Energy 152 (2018) 371e382 373 addition, it is intended to consider the char, represented by CðsÞ, as a product of the gasification process. Thus, establishing a more real modified chemical equilibrium model for the gasification. On the other hand, the present investigation is relevant for the following future studies: a) Study of the economic viability of the use of syngas in power generation systems (internal combustion engines, gas tur- bines or steam turbines). b) Thermodynamic study of syngas combustion in power gen- eration systems. c) Study of the energy viability for the selection of a biomass based on the LHV of the syngas obtained from a fluidized bed gasifier. d) Study of optimal gasification conditions to obtain the highest concentration of syngas components obtained from a fluid- ized bed gasifier. Each of the aforementioned future studies can be carried on by using the modified chemical equilibrium model proposed for: a) Gasificationwith air, b) Gasificationwith steam, c) Gasificationwith air and steam, and d) Gasification with air, steam and oxygen. 2. Determination of the correction factors 2.1. Biomass representation and properties The concentration of carbon in the biomass is important for the production of CO, CO2 and CH4, since carbon is one of the reactants of the exothermic reactions (fundamental in gasification) such as the carbon combustion reaction and the methane formation reac- tion [6]. In addition, these reactions provide the necessary heat for endothermic reactions such as the Boudouard reaction and the heterogeneous water-gas reaction, in which carbon is also one of the reactants [4]. The content of hydrogen in the biomass is very influential in the production of H2, CH4 and H2O present in the syngas (on a wet basis). This is because hydrogen helps the exothermic reactions of methane formation and methane refor- mation [6]. The oxygen content in the biomass promotes the combustion reactions thus helping the production of CO and CO2. A high concentration of oxygen in the biomass helps to reduce a little the mass flow of the oxidizing agent [16]. There are different values of admissible upper limits of tar concentration for syngas applications [1]. The tars are constituted by diverse compounds among which H2S, SO2, COS, NH3, HCN, ni- trides (NOx) are included [4]. Due to these upper limits, it is desirable that the biomasses have low concentration of sulfur and nitrogen in them. The ash and the moisture contents of the biomass have a great influence on the production of H2 and CO. This is because high concentrations of ash and moisture produce a great absorption of the energy supplied for gasification, as is the case of the gasification of rice husk [7,8,17e19] and of sugarcane bagasse [16] which pre- sent high concentration of ash and moisture, respectively. This energy absorption does not allow many endothermic reactions (such as the Boudouard reaction and the heterogeneous water-gas reaction) to occur inside the gasifier, thus causing the lack of H2 and CO production, important syngas components for use in systems of power generation [6]. In this research, the thermodynamic properties were evaluated by using the NASA-Glenn coefficients provided by McBride et al. [20]. These properties were the specific molar heat capacity at constant pressure ðcopÞ, the specific molar enthalpy ðhoÞ and the specific molar entropy � so � of chemical substances. The afore- mentioned thermodynamic properties are calculating according to equations (1) e (3). On the other hand, the Gibbs free energy � go � is obtained from the relation: go ¼ h o � Tso. cop ¼ � a1T �2 þ a2T �1 þ a3 þ a4Tþ a5T 2 þ a6T 3 þ a7T 4 � R (1) h o ¼ � �a1T �1þa2 lnTþa3Tþ a4 2 T2þa5 3 T3þa6 4 T4þa7 5 T5þb1 � R (2) so¼ � �a1 2 T�2�a2T �1þa3 lnTþa4Tþ a5 2 T2þa6 3 T3þa7 4 T4þb2 � R (3) The ultimate and proximate analysis of biomass are necessary in J. Amaro et al. / Energy 152 (2018) 371e382374 order to represent the biomass as a chemical formula. The ultimate analysis provides the mass concentrations of C, H, O, N and S in the biomass, and it can be expressed in dry basis without ash (DBOA) or dry basis with ash (DBWA). The proximate analysis provides the mass concentrations of fixed carbon, volatiles, moisture and ash in the biomass, and it is generally expressed in wet basis with ash (WBWA). There are several inorganic compounds contained in the biomass ash, because of this it is not possible to write a general chemical formula for the biomass ash. Therefore, in this work, biomass ash was assumed as being composed solely of silicon di- oxide ðSiO2Þ, as suggested by Souza-Santos [2] and applied in previous works [13,21]. In this work, the WBWA was used to express the number of moles of chemical elements in the biomass. Therefore, it was necessary to convert from DBOA or DBWA to WBWA. The expres- sions used to do so are presented in the supplementary material. A chemical representation of the biomass is shown in equation (4). C HfHOfONfNSfS þ fAshSiO2 þ fMoistH2OðlÞ (4) When the higher heating value ðHHVBioÞ and the lower heating value ðLHVBioÞ of the biomass were not available, the correlation provided by Channiwala and Parikh [22] was used to determine the HHVBio. That correlation is presented in equation (5) and provides the HHVBio in MJ/kg. Notice that in order to use this correlation it is � C HfHOfONfNSfS þ fAshSiO2 þ fMoistH2OðlÞ � þ xSteamH2Oþ xAirðO2 þ ratmN2Þ þ xOxyO2 �����! ð1� nCCÞCðsÞ þ xH2 H2 þ xCOCOþ xCO2 CO2 þ xCH4 CH4 þ � fN 2 þ ratmxAir � N2 þxH2OH2Oþ fSSO2 þ fAshSiO2 (10) necessary to express the biomass in DBWA. HHVBio¼0:3491ðPCÞDBWAþ1:1783ðPHÞDBWAþ0:1005ðPSÞDBWA� 0:1034ðPOÞDBWA�0:0151ðPNÞDBWA�0:0211ðPAshÞDBWA (5) for: 0:00% � ðPCÞDBWA% � 92:25%; 0:43% � ðPHÞDBWA% � 25:15%; 0:00% � ðPOÞDBWA% � 50:00%; 0:00% � ðPNÞDBWA% � 5:60%; 0:00% � ðPSÞDBWA% � 94:08%; 0:00% � ðPAshÞDBWA% � 71:40%; 4:745 MJ=kg � HHVBio � 55:345 MJ=kg (6) In order to apply the First Law of Thermodynamics, the enthalpy of formation of the biomass must be known, it can be determined by using equation (7). � h o f�298 � Bio ¼ HHVBioPMBio þ � 1molCO2 1molBio �� h o f�298 � CO2 þ � fH 2molBio � þ � fS 1molBio �� h o f�298 � SO2 where the molecular mass of the biomass is given by equation (8). PMBio ¼ PMC þ fHPMH þ fOPMO þ fNPMN þ fSPMS (8) This expression for the molecular mass of the biomass has also been adopted in other gasification studies [7,23]. The total enthalpy of biomass is presented in equation (9). Ho Bio¼1molBio � h o f�298 � Bio þfAsh � h o f�298 � SiO2 þfMoist � h o f�298 � H2OðlÞ (9) 2.2. Equilibrium modeling of the gasification process The gasification agents generally employed in fluidized bed gasification are steam [7,24,25], air [8,26,27], oxygen and their mixtures [17,23,28,29]. Consequently, the global gasification reac- tion proposed herein has been determined with the aim of including all commonly used gasifying agents. In the adopted global reaction, unconverted carbon ðCðsÞÞ can be found in the products. Thus, a carbon conversion efficiency was included and it is represented by nCC [18,21]. The global gasification reaction adopted in this study is presented in equation (10). The parameter ratm represents the number of moles of nitrogen per mole of oxygen on the air. In this research, nitrogen and oxygen percentages were assumed to be 79% and 21%, respectively, this assumption implies that ratm ¼ 3:76. When air is the gasification agent, the gasification equivalence ratio (ER) is used. The ER is defined as the ratio of the experimental air-fuel ratio to the stoichiometric air-fuel ratio, ER ¼ AFExp=AFStq. Therefore, the coefficient of the air used in the gasification is defined in the equation (11). xAir ¼ ER � 1þ fS þ fH 4 � fO 2 � (11) When steam is the gasification agent, the steam to biomass ratio (S/B) is used. Therefore, the number of moles of steam can be ob- tained from equation (12). S=B ¼ xSteamPMH2O PMBio (12) � h o f�298 � H2OðlÞ (7) J. Amaro et al. / Energy 152 (2018) 371e382 375 When oxygen is the gasification agent, the ratio of oxygen mass flow rate to biomass mass flow rate (O/B) is used. This ratio is defined in equation (13). O=B ¼ PMO2 � _VOxy . 22:4 � _mBio (13) Where _VOxy is the volumetric flow rate of oxygen in Nm3/h, _mBio is the mass flow rate of biomass in kg/h, and 22.4m3 is the volume occupied by 1 kmol of an ideal gas at normal pressure and tem- perature conditions. Therefore, the number of moles of oxygen can be determined by equation (14). xOxy ¼ ðO=BÞPMBio PMO2 (14) Once the global gasification reaction has been adopted and the number of moles of the reactants have been determined, the Law of Mass Conservation was applied to each chemical element in the reaction. After some algebraic manipulations, the expressions shown in equations (15) e (18) were obtained. The total number of moles of the gaseous products ðxTÞ is given by equation (18). xCO ¼ nCC � xCO2 � xCH4 (15) xH2 ¼ CH2 þ nCC þ xCO2 � 3xCH4 (16) xH2O ¼ CH2O � nCC � xCO2 þ xCH4 (17) xT ¼ CT þ nCC � 2xCH4 (18) where CH2 , CH2O and CT are input parameters that depend only on the reactants: CH2 ¼ fH 2 � fO � 2xOxy � 2xAir þ 2fS (19) CH2O ¼ fO þ fMoist þ 2xOxy þ xSteam þ 2xAir � 2fS (20) CT ¼ fH 2 þ fMoist þ xSteam þ fN 2 þ ratmxAir þ fS (21) It should be noticed that equations (15)e (18) have beenwritten in such a way that nCC, xCO2 and xCH4 are independent variables, while, xCO, xH2 , xH2O and xT are dependent variables. The First Law of Thermodynamics was applied to the global gasification reaction, and the resulting expression is presented in equation (22). There is no work crossing the boundaries of the system, and changes in kinetic and potential energy are negligible. 1molBioDh o React ¼ Ho P � Ho R (22) where: Ho P ¼ ð1� nCCÞh o C þ xH2 h o H2 þ xCOh o CO þ xCO2 h o CO2 þ xCH4 h o CH4 þ xH2Oh o H2O þ � fN 2 þ ratmxAir � h o N2 þ fSh o SO2 þ fAshh o SiO2 (23) Ho R ¼ Ho Bio þ xSteamh o H2O þ xAirh o O2 þ xAirratmh o N2 þ xOxyh o O2 (24) TSteam, TAir and TOxy are the inlet temperatures of steam, air and oxygen, respectively, which are used to evaluate their enthalpies. The gasification temperatures ðTGasÞ are provided in the experi- mental works and they are used to determine the enthalpies of the products. In order to complete the system of equation for determining the number of moles of the products at the gasification temperature TGas, the chemical equilibrium condition was applied. Three re- actions have been commonly adopted to model the equilibrium composition of the gasification in fluidized bed gasifiers. These reactions are the methane formation reaction (MFR) [18,19,30e32], wateregas homogeneous reaction (WGHR) [7,8,19,33,34] and methane reforming reaction (MRR) [19,32,33], which are shown in equations (25) e (27), respectively. Cþ 2H2⇔CH4 � Dh o React ¼ �74:8 kJ=mol � (25) COþH2O⇔H2 þ CO2 � Dh o React ¼ �41:1 kJ=mol � (26) CH4 þH2O⇔3H2 þ CO � Dh o React ¼ 206:1 kJ=mol � (27) The chemical equilibrium constants obtained from the reactions presented above are shown in equations (28) e (30), respectively [1,35,36]. KMFR ¼ xCH4 xT� xH2 �2 � PGas Po ��1 (28) KWGHR ¼ xCO2 xH2 xCOxH2O (29) KMRR ¼ xCO � xH2 �3 xCH4 xH2OðxTÞ2 � PGas Po �2 (30) Generally, the gasification pressure ðPGasÞ in fluidized bed gas- ifiers is equal to the normal pressure ðPoÞ, thus the ratio of gasifi- cation pressure to normal pressure might have no effect on equations (28) and (30). 2.3. Optimization method to obtain correction factors There are different studies which present advances regarding the calculation of the correction factors. These studies are briefly described below. In the work by Loha et al. [7] the adjustment of the chemical equilibrium model was made by multiplying the chemical equi- librium constants of the methane formation reaction and the ho- mogenous water-gas reaction by the correction factors of 0.93 and 0.71, respectively. In this way, average RMS error over six samples of syngas composition was decreased from an initial value of approximately 3.34 to a final value of 2.62. In the work by Jarungthammachote and Dutta [10] correction factors with values 11.28 and 0.91 were multiplied to the chemical equilibrium constants of the methane formation and homogeneous water-gas reactions, respectively. This adjustment was made to approximate the modified chemical equilibrium model to the experimental data, being the factor 11.28 and 0.91 necessary to adjust the concentration of CH4 and CO, respectively. In the work by Barman et al. [12] the correction factor of the chemical equilibrium constant of the methane formation reaction was calculated by increasing in steps of 0.5 a correction factor that was initially 1. At the end of this process, the correction factor selected was 3.5. For the homogeneous water-gas reaction, a J. Amaro et al. / Energy 152 (2018) 371e382376 correction factor was not included due to the high gasification temperature, instead it was considered that this reaction reaches chemical equilibrium. In the work by Mendiburu et al. [13] the adjustment of the chemical equilibrium constants of the methane formation and homogenous water-gas reactions was performed. The correction factor of the methane formation reactionwas obtained by adjusting the model to the experimental data of the selected works. A similar process determined the correction factor of the homogeneous water-gas reaction. In the work by Lim and Lee [37], empirical relations (based on Fig. 2. Optimization meth the ER parameter) were used to calculate the correction factors for the methane formation and homogeneous water-gas formation reactions. 2.3.1. The objective functions As aforementioned, the correction factors are used to better approximate the experimental syngas composition using chemical equilibrium models. In other works, these correction factors have been obtained by trial and error without relying in any analytical method for their calculation [7,10e13]. In the present work, the correction factors have been determined through an optimization od solution scheme. J. Amaro et al. / Energy 152 (2018) 371e382 377 method which uses the KuhneTucker multipliers [38e42]. The procedure followed in this section is depicted in Fig. 2. The proposed method consists in the minimization of an objective function that is derived from the definition of RMS error [9]. The objective function depends on the number of moles of syngas ðxSynÞ derived from the experimental composition reported in scientific articles. The syngas composition is generally given in dry basis with nitrogen (DBWN) or dry basis without nitrogen (DBON). DBWN is mostly used when air is the gasifying agent, while DBON is generally used when steam or oxygen are the gasifying agents. Therefore, the number of moles of syngas ðxSynÞ is different from the total number of moles of gases ðxTÞ, because the later considers the number of moles of steam ðxH2OÞ and sulfur dioxide ðfSÞ present in the products while the former does not. The experimental con- centration of the species “i” (H2, CO, CO2, CH4 or N2) is defined by equation (31). EPi% ¼ 100xi xSyn % (31) The number of moles of syngas, xSyn, in DBWN is given by equation (32). On the other hand, the value of xSyn in DBON is given in equation (33). xSyn ¼ 100 � fN 2 þ ratmxAir � EPN2 (32) xSyn ¼ CH2 þ 2nCC þ xCO2 � 3xCH4 (33) The RMS error obtained is presented in equation (34) being “n” the number of species of the syngas. Hence, the objective functions obtained for DBWN and DBON are given by equations (35) and (36), respectively. RMS ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPn i¼1 � EPixSyn � 100xi �2x�2 Syn n vuuut (34) Objective function for DBWN FObj � nCC; xCO2 ; xCH4 � ¼: FObj ¼ X i¼H2;CO;CO2;CH4;N2 � EPixSyn � 100xi �2 (35) Objective function for DBON FObj � nCC; xCO2 ; xCH4 � ¼: FObj ¼ X i¼H2;CO;CO2;CH4 � EPixSyn � 100xi �2x�2 Syn (36) The independent variables of the objective functions have their domains defined in [0,1]. On the other hand, the co-domains of the objective functions are defined in R (set of real numbers). It is important to mention that the objective function for DBWN and DBON are a strictly convex and a convex function, respectively. 2.3.2. Optimization using KuhneTucker multipliers In the present section the variables nCC, xCO2 and xCH4 have been substituted by a1, a2 and a3, respectively. It is important tomention that the optimization that uses the KuhneTucker multipliers is the generalization of the Lagrange multipliers optimization method. Thus, in order to perform an optimization using the KuhneTucker multipliers ðmÞ it is essential to define the Lagrangian expression (L) with inequality constraints obtained from the domains of in- dependent variables (nCC, xCO2 and xCH4 ). The Lagrangian expres- sion is shown in equation (37). While, the independent variables have to satisfy the inequality constraints, obtained from their do- mains, presented in equation (38). L ¼ FObj þ X3 i¼1 miðai � 1Þ � X3 i¼1 miþ3ai (37) 0 � ai; ai � 1 and mj � 0 for i ¼ 1 to 3 and j ¼ 1 to 6 (38) The KarusheKuhneTucker conditions [38e42] were applied to equation (37) obtaining six equations (Complementarity con- straints) which are the following: miðai � 1Þ ¼ 0 and miþ3ai ¼ 0 for i¼ 1 to 3. Notice that the conditions stablish that 0 � ai and ai � 1. From the experimental results [7,8,14,16,17,19,23,25,28,29,43] it is known that the number of moles of CO, CO2 and CH4 must be greater than zero. Among the different possibilities that can satisfy the equations, some were discarded, for instance, it was discarded that: a) a2 ¼ 1 or a3 ¼ 1. b) a2 ¼ 0 or a3 ¼ 0. c) a1 ¼ 0. Therefore, some of the KuhneTucker multipliers could be ob- tained which are shown in equation (39). miþ1 ¼ 0 for i ¼ 1 to 5 (39) Substituting the values of the KuhneTucker multipliers, shown in equation (39), into the system of equations obtained from the Lagrangian expression, the result is the system shown in equations (40) e (42). vFObj va1 þ m1 ¼ 0 (40) vFObj vaiþ1 ¼ 0 for i ¼ 1 to 2 (41) m1ða1 � 1Þ ¼ 0 (42) This system of equations is limited by the inequalities presented in equation (38), but knowing that equation (39) is also satisfied. Now the objective functions for DBWN [equation (35)] and for DBON [equation (36)] are replaced in equations (40) and (41) to then solve the system of equations (40) e (42). In the case of the objective function for DBWN there are two possible solutions: (a) for the case in which m1 ¼ 0 and nCCs100%, and, (b) for the case in which m1s0 and nCC ¼ 100%. Then, for the solution (a) the values of nCC, xCO2 and xCH4 are determined by the equations (43) e (45). On the other hand, for solution (b) the value of nCC ¼ 100% is replaced into equations (44) and (45). The number of moles of the other species can be obtained from themass balance given in equations (15) e (18). The solution (a) is preferred over the solution (b) because the nCCs100%. nCC ¼ � EPH2 þ 9EPCO þ 8EPCO2 þ 12EPCH4 ��fN 2 þ ratmxAir � EP�1 N2 � CH2 10 (43) xCO2 ¼ 8nCC � 5CH2 þ � 5EPH2 þ 11EPCO2 þ 2EPCH4 � 13EPCO ��fN 2 þ ratmxAir � EP�1 N2 29 (44) xCH4 ¼ 12nCC þ 7CH2 þ � 2EPCO2 þ 3EPCH4 � 7EPH2 � 5EPCO ��fN 2 þ ratmxAir � EP�1 N2 29 (45) J. Amaro et al. / Energy 152 (2018) 371e382378 In the case of the objective function for DBON the solution was obtained by using the fsolve solver of the Matlab software [44e46]. The values obtained for nCC, xCO2 and xCH4 were then replaced into equations (15) e (18) to determine the number of moles of the other species in the products. The correction factors for the chemical equilibrium constants of the three chemical reactions [8,19,33] presented in equations (25)e (27) are denoted by fMFR, fWGHR and fMRR, respectively. The ex- pressions for determination of the correction factors are presented in equations (46) e (48). The number of moles of the species in equations (46) e (48) are those determined in the solutions for DBWN and DBON. fMFR ¼ xCH4 xT � PGas Po ��1 KMFRx2H2 (46) fWGHR ¼ xCO2 xH2 KWGHRxCOxH2O (47) Fig. 3. Modified chemical equilibr fMRR ¼ xCO � xH2 �3�PGas Po �2 KMRRxCH4 xH2Ox 2 T (48) The correction factors were determined using the data taken from several experimental articles [7,8,14,16,17,19,23,25,28,29,43]. The fMFR was found to be on the interval ½2:4044; 1236:3325�, the fWGHR on the interval ½0:0990; 2:2639� and fMRR on the interval ½3:9033x10�07; 0:6015�. As can be observed the interval of fWGHR is not so small nor so large with respect to the intervals of fMFR and fMRR. Therefore, the fWGHR was selected for the modeling process which is going to be presented in the next section. It is also important to point out that it is not recommendable to simply use the experimental syngas composition and obtain the number of moles of the species in the products. That is because the value of nCC and the composition of the solid residue are not re- ported in most experimental articles. Finally, the enthalpy of reaction Dh o React is determined with equation (22) by considering in the products the number of moles of each component as determined by the optimization method. ium model solution scheme. Table 1 Distribution of selected experimental compositions of syngas. Gasifying agent References Number of data Number of data for validation Air [8,14,16,43] 28 5 Steam [7,19,25] 17 4 Air and steam [17,23,28,29] 20 5 Air, steam and oxygen [23] 11 2 J. Amaro et al. / Energy 152 (2018) 371e382 379 3. Modified chemical equilibrium model The proposed modified chemical equilibrium model consists of three correlations for the following parameters: carbon conversion efficiency ðnCCÞ, correction factor ðfWGHRÞ and enthalpy of the global gasification reaction ðDho ReactÞ. These magnitudes were replaced in a system of two equations that is constituted by: The First Law of Thermodynamics and the modified chemical equilib- rium equation for the homogeneous water-gas reaction [equation (53)]. The proposed correlations are related to the values obtained from the substitution of the optimal numbers of moles for the theoretical composition of the syngas. To calculate these optimal numbers of moles, the objective functions based on the definition of the RMS error were minimized. A disadvantage of the proposed modified chemical equilibriummodel is the assumption that the tar is not present in the products of the proposed global gasification reaction. This assumption was made due to the small production of tar which is generally obtained from fluidized bed gasifiers. These three correlations were obtained by applying linear re- gressions [47]. The solution scheme for the modified chemical equilibrium model is depicted in Fig. 3. Seventy-six samples of syngas compositions from different experimental articles were selected [7,8,14,16,17,19,23,25,28,29,43]. The gasification agents were air, steam, airesteam, and air- esteameoxygen. The set of 76 samples was divided into a set containing 60 samples and another set with 16 samples. The first one was used to obtain the correlations for nCC, fWGHR and Dh o React while the second one was used to validate the modified chemical equilibrium model. The samples used for the validation were pro- portionally selected from each type of gasifying agent or agents. Table 1 presents the distribution of the number of samples adopted to develop each modified chemical equilibrium model. The modi- fied chemical equilibriummodel introduces three correlated values through one of the equations (49), (50), (51) or (52), which should be used together with the coefficients provided in Table 2. The Table 2 Coefficients for correlations shown in equations (49) e (52). Coefficients Ga Air Steam Z¼ Eq. (49) Z¼ Eq. (50) nCC ð%Þ fWGHR Dh o React ðkJ=molÞ nCC ð%Þ fWGHR Dh o R ðkJ=m x fMoist fAsh fMoist e e e I �2979.4090 �522.3036 �2241.0450 100 7.3278 �143.3 C1 402.3987 �550.5976 84.4509 0 0.0104 �0.70 C2 �3391.6953 0.5249 �1903.5383 0 �20.7697 39.82 C3 3042.3198 519.8738 2139.1648 0 18.4715 �6.51 C4 �10.8580 �0.0124 �11.5510 0 �5.1779 0.00 C5 76.2600 14.7958 �303.1547 0 2.4434 0.00 C6 0.0000 �16.3996 0.0000 0 0.0000 0.34 C7 �0.0212 0.0004 0.1789 � e e R2 0.9662 0.9030 0.9877 1 0.9700 0.98 obtained correlations are shown below. Gasification with air Z ¼ Iþ C1fAsh þ C2fMoist þ C3e x þ C4 � fMoist fAsh � þ C5ER þ C6ER 2þC7TGas (49) Gasification with steam It was assumed that carbon conversion efficiency ðnCCÞ is 100% because this value has always been obtained for all the optimiza- tion method's calculations. Z ¼ Iþ C1 � fMoist fAsh � þ C2ðS=BÞ þ C3ðS=BÞ2 þ C4ðS=BÞ3 þ C5 � TSteam TGas � þ C6TGas (50) Gasification with airesteam Z ¼ Iþ C1fAsh þ C2fMoist þ C3ER þ C4ðS=BÞ þ C5 � TAir TGas � þ C6 � TSteam TGas � (51) Gasification with aireoxygenesteam In this case the carbon conversion efficiency ðnCCÞ was also assumed to be 100% because this value has always been obtained for all the optimization method's calculations. Z ¼ Iþ C1ER þ C2ERðS=BÞ þ C3½ERðS=BÞ�2 þ C4ðO=BÞ þ C5ðO=BÞ2 þ C6TGas (52) The value of 100% was adopted for carbon conversion efficiency when using the correlations for the carbon conversion efficiency presented in equations (49) and (51) exceeds 100%. fWGHRKWGHR ¼ xCO2 xH2 xCOxH2O (53) For the calculation of the composition of the syngas using the modified chemical equilibrium model, the values obtained by the model have to be substituted in the system of two equations con- formed by the application of the First Law of Thermodynamics [equation (22)] and themodified chemical equilibrium equation for the homogenous water-gas reaction [equation (53)]. The solution of this system of equations are the number of moles of methane and sifying agent Air and steam Air, steam and oxygen Z¼ Eq. (51) Z¼ Eq. (52) eact olÞ nCC ð%Þ fWGHR Dh o React ðkJ=molÞ nCC ð%Þ fWGHR Dh o React ðkJ=molÞ e e e e e e 802 111.4569 �6.7994 124.7102 100 0.6108 117.0359 79 13.2464 �21.4213 �1038.5164 0 0.0000 �468.2000 12 �537.7031 48.2313 765.7268 0 0.7189 0.0000 10 157.4910 1.4444 �277.8947 0 0.0530 0.0000 00 24.1496 0.1007 49.2069 0 �15.5142 �229.5642 00 �115.0108 8.5565 65.7674 0 37.9479 0.0000 16 44.0283 �3.6423 �354.1102 0 0.0016 0.0205 e e e e e e 62 0.9643 0.8843 0.9978 1 0.6676 0.9833 Table 3 Gasification conditions for the application of the optimization method. Exp. Reference ER S/B O/B TGas (�C) 1 Sarker et al. [43] 0.30 0.00 0.00 876 2 Arteaga-P�erez et al. [16] 0.34 0.00 0.00 803.5 3 Karmakar et al. [8] 0.45 0.00 0.00 600 4 Radmanesh et al. [14] 0.66 0.00 0.00 800 5 Karmakar and Datta [19] 0.00 1.70 0.00 750 6 Loha et al. [7] 0.00 1.32 0.00 690 7 Vecchione et al. [25] 0.00 1.00 0.00 830 8 Campoy et al. [28] 0.23 0.18 0.00 752 9 Loha et al. [17] 0.35 0.80 0.00 850 10 Sethupathy Subbaiah et al. [29] 0.18 0.30 0.00 650 11 Campoy et al. [23] 0.27 0.43 0.00 755 12 Campoy et al. [23] 0.36 0.32 0.1728a 808 a Value obtained by using the oxygen flow of 1.5 Nm3/h provided by the authors. J. Amaro et al. / Energy 152 (2018) 371e382380 carbon dioxide. Finally, we will proceed to calculate the other chemical substances using the equations of conservation of the mass presented in equations (15) e (17). 4. Results and discussions All the syngas experimental compositions selected for the application of the optimization method are presented in Tables S1 to S4 of the supplementary material. The number of moles ob- tained by applying the optimization method are presented in Tables S5 to S8. The correction factors, enthalpies of reaction and the KuhneTucker multipliers ðm1Þ are shown in Tables S9 to S12. Finally, the theoretical compositions of the syngas when the Table 4 Results obtained from the optimization using different gasifying agents. Exp. Reference H2 ð%Þ CO ð%Þ CO2 ð%Þ CH ð%Þ 1 Sarker et al. [43] 3.82 13.15 13.23 3.1 2 Arteaga-P�erez et al. [16] 5.85 15.23 14.14 5.1 3 Karmakar et al. [8] 9.56 11.08 21.66 3.5 4 Radmanesh et al. [14] 4.60 11.25 14.47 0.5 5 Karmakar and Datta [19] 51.15 18.23 26.57 4.0 6 Loha et al. [7] 49.43 15.53 29.14 5.9 7 Vecchione et al. [25] 49.17 19.17 22.13 9.5 8 Campoy et al. [28] 14.60 13.80 16.90 5.2 9 Loha et al. [17] 13.00 14.30 20.50 2.8 10 Sethupathy Subbaiah et al. [29] 21.22 15.78 15.96 6.2 11 Campoy et al. [23] 31.20 22.90 35.80 10.2 12 Campoy et al. [23] 27.40 34.20 34.60 3.8 Table 5 Correction factors and enthalpy of reaction obtained from the optimization method. Exp. ER S/B O/B Dh o React ðkJ=molÞ 1 0.30 0.00 0.00 �2.17 2 0.34 0.00 0.00 �3.87 3 0.45 0.00 0.00 �63.40 4 0.66 0.00 0.00 �153.00 5 0.00 1.70 0.00 159.90 6 0.00 1.32 0.00 134.01 7 0.00 1.00 0.00 124.12 8 0.23 0.18 0.00 �14.16 9 0.35 0.80 0.00 21.13 10 0.18 0.30 0.00 95.75 11 0.27 0.43 0.00 �17.41 12 0.36 0.32 0.1728a �75.96 a Using the oxygen flow of 1.5 Nm3/h provided by the authors, this quantity is obtain optimization method is applied are presented in Tables S13 to S16. For some experimental articles [28,29,43] considered in the present study it was assumed that the percentage of nitrogen was the difference between 100% and the concentrations of H2, CO, CO2 and CH4 present in the products of the global gasification reaction [equation (10)]. The syngas composition was obtained by applying the input parameters shown in Table 3, which are the same input parameters reported in the experimental articles considered for the present study. Table 4 shows the syngas composition obtained from the opti- mization method application. These compositions are very close to experimental ones, as expected. It can be observed from Table 4 that the calculated syngas compositions are closer to the experi- mental compositions when air is used as gasifying agent. Other information regarding the optimization method is presented in Table 5, where the enthalpies of reaction, correction factors and KuhneTucker multipliers ðm1Þ are reported. In order to analyze the values of the enthalpies of reaction ðDho ReactÞ with respect to each gasifying agent presented in Table 5, it is important to notice that a negative value of the enthalpy of reaction means that the process has liberated heat (exothermic). In the case of gasification with air, it can be observed in Table 5 that the increase of ER produces a decrease on the value of Dh o React, which means that more exothermic reactions are taking place due to the increase of the available oxygen. In the case of the gasification with steam, it can be observed in Table 5 that an increase of S/B produces an increase of Dh o React, which means that more endothermic reactions are taking place. In the case of gasificationwith steam there is more hydrogen available 4 N2 ð%Þ RMS error from this work RMS error from the reference 4 66.66 0.88 e 3 59.65 1.08 e 5 54.15 0.90 1.21 0 69.19 0.33 e 4 0.00 1.41 3.62 0 0.00 2.02 2.12 3 0.00 1.19 e 0 49.50 0.00 e 0 49.40 0.09 e 6 40.78 0.95 e 0 0.00 0.00 e 0 0.00 3.70 e fMFR fWGHR fMRR m1 1236.33 0.1257 3.90� 10�07 0.00 507.15 0.0990 2.81� 10�06 0.00 9.24 0.9887 0.0643 0.00 59.18 0.3546 6.39� 10�05 0.00 3.81 0.6681 0.0042 22.07 2.71 0.8819 0.0206 32.93 16.64 1.2020 0.0007 7.18 43.39 0.4464 3.04� 10�04 0.00 85.94 0.3690 1.65� 10�05 0.00 8.09 0.2923 1.31� 10�02 0.00 37.83 0.5716 2.77� 10�04 0.00 30.59 0.4940 2.61� 10�04 339.93 ed. Table 6 Validation of modified chemical equilibrium model applied to fluidized bed gasification. Reference ER S/B Oxygen flow rate ðNm3=hÞ O/Ba TGas(�C) nCC ð%Þ H2 ð%Þ CO ð%Þ CO2 ð%Þ CH4 ð%Þ N2 ð%Þ RMS Karmakar et al. [8] 0.35 0.00 0.00 0.00 700 82.68 14.63 22.47 16.67 1.32 44.91 Predictive model 0.35 0.00 0.00 0.00 700 94.32 15.09 19.31 17.05 1.75 46.80 1.68 Radmanesh et al. [14] 0.32 0.00 0.00 0.00 805 e 9.20 16.20 12.70 2.50 59.60 Predictive model 0.32 0.00 0.00 0.00 805 71.58 13.56 14.01 15.00 2.40 55.03 3.16 Arteaga-P�erez et al. [16] 0.34 0.00 0.00 0.00 801.5 e 5.40 16.18 13.67 3.54 60.49 Predictive model 0.34 0.00 0.00 0.00 801.5 76.43 8.31 12.99 15.63 4.53 58.55 2.33 Sarker et al. [43] 0.35 0.00 0.00 0.00 874 e 3.84 14.31 14.98 2.67 64.20 Predictive model 0.35 0.00 0.00 0.00 874 63.11 5.69 11.26 15.01 2.99 65.04 1.65 Karmakar and Datta [19] 0.00 1.32 0.00 0.00 650 84.10 47.25 11.25 31.90 9.60 0.00 Predictive model 0.00 1.32 0.00 0.00 650 100.00 48.49 11.69 32.31 7.51 0.00 1.25 Loha et al. [7] 0.00 1.00 0.00 0.00 750 e 49.50 23.70 21.20 5.60 0.00 Predictive model 0.00 1.00 0.00 0.00 750 100.00 46.71 27.87 20.73 4.69 0.00 2.56 Vecchione et al. [25] 0.00 0.70 0.00 0.00 830 e 47.93 21.11 22.81 8.15 0.00 Predictive model 0.00 0.70 0.00 0.00 830 100.00 46.91 21.84 20.64 10.61 0.00 1.76 Campoy et al. [28] 0.19 0.28 0.00 0.00 727 89.00 16.20 11.50 18.60 5.90 47.80 Predictive model 0.19 0.28 0.00 0.00 727 60.57 16.60 11.00 19.60 6.60 46.20 0.94 Loha et al. [17] 0.35 0.50 0.00 0.00 750 71.90 9.20 12.80 20.80 2.10 55.10 Predictive model 0.35 0.50 0.00 0.00 750 79.61 10.90 14.30 20.40 1.70 52.70 1.50 Sethupathy Subbaiah et al. [29] 0.18 0.30 0.00 0.00 750 81.37 20.92 17.56 16.41 4.50 40.61 Predictive model 0.18 0.30 0.00 0.00 750 67.43 21.93 17.20 14.83 5.52 40.51 0.97 Campoy et al. [23] 0.25 0.31 1.40 0.1235 781 96.00 31.60 32.90 24.90 10.60 0.00 Predictive model 0.25 0.31 1.40 0.1235 781 100.00 34.70 36.30 24.60 4.50 0.00 3.82 Campoy et al. [23] 0.24 0.58 1.00 0.1190 765 96.00 34.30 23.50 31.30 10.90 0.00 Predictive model 0.24 0.58 1.00 0.1190 765 100.00 38.40 28.80 28.30 4.60 0.00 4.84 a This magnitude is calculated according to the theoretical development proposed in this work. J. Amaro et al. / Energy 152 (2018) 371e382 381 for the gasification reaction; however, it is necessary to provide more heat to the reaction by means of an electric furnace, for example. The gasification with airesteam mixtures show a more complicated behavior in Table 5. We can analyze this behavior by considering experiments 8, 9 and 10 in Table 5. It is observed that for the experiment 8 the exothermic reactions dominate since Dh o React is negative, then, for experiment 9 the values of ER and S/B are increased and the endothermic reactions dominate since Dh o React is positive. An interesting comparison appear when we consider experiment 8 and 10, in which ER8>ER10, S/B8