IMPLEMENTATION OF REDUCED MECHANISM IN COMPLEX CHEMICALLY REACTING FLOWS JATHAVEDA MAKTAL. Submitted in partial fulfillment of the requirements

Transcription

1 IMPLEMENTATION OF REDUCED MECHANISM IN COMPLEX CHEMICALLY REACTING FLOWS by JATHAVEDA MAKTAL Submitted in partial fulfillment of the requirements For the degree of Master of Science in Aerospace Engineering Advisor: Prof. Chih-Jen Sung Department of Mechanical and Aerospace Engineering CASE WESTERN RESERVE UNIVERSITY May 2009

2 CASE WESTERN RESERVE UNIVERSITY SCHOOL OF GRADUATE STUDIES We hereby approve the thesis/dissertation of JATHAVEDA MAKTAL candidate for the MASTER OF SCIENCE IN AEROSPACE ENGINEERING degree *. (signed) Dr Chih Jen Sung (chair of the committee) Dr James S T ien Dr Yasuhiro Kamotani (date) 3/30/09 *We also certify that written approval has been obtained for any proprietary material contained therein.

3 TABLE OF CONTENTS TABLE OF CONTENTS... i LIST OF FIGURES... iii LIST OF TABLES... vi ABSTRACT... vii CHAPTER-1:INTRODUCTION SCIENTIFIC BACKGROUND SKELETAL MECHANISM DIRECTED RELATIONAL GRAPH (DRG) DRG WITH ERROR PROPAGATION (DRGEP) DRG-AIDED SENSITIVITY ANALYSIS (DRGASA) REDUCED MECHANISM PARTIAL EQUILIBRIUM APPROXIMATION QUASI STEADY STATE ASSUMPTION (QSSA) COMPUTATIONAL SINGULAR PERTURBATION (CSP) BASIC FORMULATION AND ALGORITHM IDENTIFICATION OF QSS SPECIES OBJECTIVES THESIS ORGANIZATION CHAPTER-2: METHODOLOGY SCIENTIFIC BACKGROUND METHODOLOGY OF REDUCTION DIRECTED RELATIONAL GRAPH WITH ERROR PROPAGATION (DRGEPSA) SKELETAL MECHANISM REDUCED MECHANISM VALIDATION OF MECHANISMS i

4 2.3.1 SENKIN PSR SENKIN RESULTS CHAPTER-3: IMPLEMENTATION IN CFD SCIENTIFIC BACKGROUND PHYSICAL NUMERICAL PROCEDURE TURBULENT MODEL CHEMISTRY MODEL TRANSPORT MODEL PHYSICAL-NUMERICAL PROCEDURE GAS PHASE NUMERICAL MODEL THERMODYNAMIC AND TRANSPORT PROPERTIES COMBUSTION MODEL GEOMETRY AND MESH BOUNDARY CONDITIONS RESULTS VALIDATION OF FLUENT CHEMISTRY MODEL THREE DIMENSIONAL MODEL RESULTS CHAPTER-4: CONCLUSIONS AND FUTURE WORK CONCLUSIONS FUTURE WORK REFERENCES APPENDIX APPENDIX-1: SKELETAL MECHANISM APPENDIX-2: REDUCED MECHANISM ii

5 LIST OF FIGURES 2.1. Ignition delay as a function of the inverse of initial temperature for methane/air mixture of equivalence ratio of Ignition delay as a function of the inverse of initial temperature for methane/air mixture of equivalence ratio of Ignition delay as a function of the inverse of initial temperature for methane/air mixture of equivalence ratio of Temperature profile as a function of time for the methane/air mixture of equivalence ratio of 0.5, and P=20atm NO mass fraction profile as a function of time for the methane/air mixture of equivalence ratio of 1.0, and P=1atm CO mass fraction profile as a function of time for the methane/air mixture of equivalence ratio of 1.0, and P=1atm Comparison of adiabatic PSR among detailed, skeletal and reduced mechanisms Comparison of adiabatic PSR among detailed, skeletal and reduced mechanisms Comparison of adiabatic PSR among detailed, skeletal and reduced mechanisms.. 38 iii

6 2.10. Comparison of adiabatic PSR among detailed, skeletal and reduced mechanisms Comparison of adiabatic PSR among detailed, skeletal and reduced mechanisms Geometry and Grid of the model for CFD simulations Geometry and Grid of the model Comparision of temperature profiles for constant pressure ignition for the methane/air mixture of equivalence ratio of 1.0 at pressure of 1 atm Plot of temperature (K) profile for the methane/air mixture of equivalence ratio of 0.5 at pressure of 1 atm Plot of CH 4 mass fraction for the methane/air mixture of equivalence ratio of 0.5 at pressure of 1 atm Plot of O 2 mass fraction for the methane/air mixture of equivalence ratio of 0.5 at pressure of 1 atm Plot of H 2 O mass fraction for the methane/air mixture of equivalence ratio of 0.5 at pressure of 1 atm Plot of CO 2 mass fraction for the methane/air mixture of equivalence ratio of 0.5 at pressure of 1 atm Plot of velocity vectors in m/s for the methane/air mixture of equivalence ratio of 0.5 at pressure of 1 atm iv

7 3.10 Plot of temperature (K) for the methane/air mixture of equivalence ratio of 1.0 at pressure of 1 atm Plot of CH 4 mass fraction for the methane/air mixture of equivalence ratio of 1.0 at pressure of 1 atm Plot of O 2 mass fraction for the methane/air mixture of equivalence ratio of 1.0 at pressure of 1 atm Plot of CO 2 mass fraction for the methane/air mixture of equivalence ratio of 1.0 at pressure of 1 atm Plot of H 2 O mass fraction for the methane/air mixture of equivalence ratio of 1.0 at pressure of 1 atm Plot of Velocity vectors for the methane/air mixture of equivalence ratio of 1.0 at pressure of 1 atm Plot of Velocity vectors for the methane/air mixture of equivalence ratio of 1.0 at pressure of 1 atm Plot of CO mass fraction for the methane/air mixture of equivalence ratio of 1.0 at pressure of 1 atm Plot of NO mass fraction for the methane/air mixture of equivalence ratio of 1.0 at pressure of 1 atm...71 v

8 LIST OF TABLES 2.1 Speedup factors using the 19 species reduced mechanism compared with the detailed mechanism for φ= Speedup factors using the 19 species reduced mechanism compared with the detailed mechanism for φ= Speedup factors using the 19 species reduced mechanism compared with the detailed mechanism for φ= Speedup factors using the skeletal mechanism compared with the detailed mechanism for φ= Speedup factors using the skeletal mechanism compared with the detailed mechanism for φ= Speedup factors using the Skeletal Mechanism compared with the detailed mechanism for φ= Boundary conditions for the CFD simulations Boundary conditions for the CFD simulations Speed up factors for the methane/air mixture for CFD simulations..71 vi

9 Implementation of Reduced Mechanism in Complex Chemically Reacting Flows Abstract by JATHAVEDA MAKTAL Numerical investigation is conducted to study the combustion performance of Jet Stirred Reactor (JSR) by burning mixtures of CH 4 and air. Based on the detailed mechanism of GRI-Mech 3.0, a skeletal mechanism of 29 species and 150 elementary reaction steps and a reduced mechanism of 19 species and 15 lumped reactions are generated and computed to compare their predictive capabilities with the detailed mechanism. The skeletal mechanism is generated using the method of directed relational graph with error propagation and sensitivity analysis (DRGEPSA) and reduced mechanisms are generated using the method of Quasi Steady State (QSS) assumptions. The mechanisms are validated in homogeneous applications like perfectly stirred reactor (PSR) and auto-ignition. These mechanisms along with a similar 19 species ARM, a reduced mechanism from a previous study, are implemented into a commercial CFD package. A three dimensional modeling on a JSR is considered. The three dimensional turbulent flow inside the reactor is modeled using the Realizable k-ε model and enhanced wall functions are considered for near wall treatment. The equivalence ratios of 1 and 0.5 with the inlet temperature of 600K are considered for the boundary vii

10 conditions. The dependence of turbulence on finite rate chemistry is considered by invoking turbulence-chemistry models. Comparing the results obtained from the skeletal and reduced mechanisms with the detailed mechanism it is found that there is a good agreement in the predictions like temperature and major species such as CH 4, O 2, H 2 O and CO 2. Among the pollutant species of CO and NO similar comparison shows that there is good agreement of CO between the mechanisms but the same cannot be said about NO. There is a considerable computational time savings in using the resulting skeletal and reduced mechanisms for the simulations, but the computational time saving achieved by the 19 species reduced mechanism is comparable to that of the 29 species skeletal mechanism. The time savings obtained by the reduced mechanism is not large as compared to the skeletal mechanism because the time consumed to solve the QSS relations is high. viii

11 CHAPTER-1:INTRODUCTION 1.1 SCIENTIFIC BACKGROUND In the last few decades a great progress has been made in the use of computation using detailed chemistry from zero dimensional homogeneous reactors to quasi-onedimensional laminar flames. A good agreement between the computational simulations and the experimental results show the extent of accuracy of both the computational codes and the detailed mechanisms [1]. It is quite routine now to use detailed mechanism to describe laminar flames in simple configurations. But for more complex flows involving turbulence, where it is necessary to model the turbulent flames or in engine simulations, mechanisms with few-global steps are often used. In such cases involving turbulence-chemistry interactions and multi dimensions the numerical solution of the governing equations with detailed chemistry is demanding in both time and memory requirements because of the large number of species and reactions in the detailed mechanisms. Along with the increase in the use of multi dimensional simulations for practical systems and the growth in the methods to provide accurate results for the same, there is a growing awareness in the importance of chemical kinetics in the combustion studies. Thus it becomes imperative to make them less demanding in terms of time and memory requirements. A step taken in this direction was to deduce reduced mechanisms from detailed and skeletal mechanisms, and typically consisting of four to five steps. The use of single global mechanism or simplified four to five step mechanism achieves the 1

12 reduction in the time and memory but compromises on the efficacy of the study of the phenomenon of combustion in detail. Also the applicability could be quite limited in terms of parametric range of simulations in which it can be used. Thus it is highly essential that the mechanism be simplified without losing the accuracy for a wider parametric range [2].Various mathematical methods have been proposed and used to achieve the reduction. Before moving on to the actual process of developing the reduced mechanism it is important to know and understand the comprehensiveness, i.e. what constitutes a satisfactory mechanism for a fuel/oxidizer mixture. In the absence of a unified theory that would ascertain the completeness of a mechanism in all the classes of the combustion phenomenon over all thermodynamic and system parametric variations. The more phenomena it can describe over as wide a range of parameter as possible the more comprehensive the mechanism would be considered. But this process would increase the size of the mechanism and the number of species. In numerical calculations due to the need to perform matrix inverse manipulations for describing the species concentration the computation time varies quadratically with the number of species. Thus it is favorable to reduce the number of species whose concentration needs to be solved by the differential equations [3]. The elementary reactions at the molecular level depend upon the energies and the frequency of collision between the molecules. Thus the effect of these parameters in the macro scale can be seen as the dependence on global temperature, molar 2

13 concentrations of the fuel and oxidizer and the dependence of concentrations on the system pressure. This results in the inclusion of extensive and independent variations of the system pressure, temperature and the concentration of the mixture through equivalence ratio. Along with the comprehensiveness based on the global parameters, hierarchical nature of the fuel oxidization process must also be considered. For example since H 2 is an intermediate in the oxidation of hydrocarbons the H 2 oxidation must be a part of the mechanism of hydrocarbon combustion. If all the reactions not pertaining to the H 2 oxidation are stripped off then the mechanism should result to that of H 2. Thus it could be said that for a mechanism to be comprehensive it must include the comprehensive sub- mechanisms of the oxidation of intermediaries. Now that the framework for the comprehensiveness has been laid it is important to know as to how to investigate the same. The comprehensiveness of the mechanism can be investigated at different levels of complexity for combustion phenomena. Usually the studies of kinetics are performed in homogeneous systems like shock tube, rapid compression machines, perfectly stirred reactors. The assumption being that the transport and flow effects are not present in them. Thus studies must be conducted in non-homogeneous systems like flames to complement the homogeneous studies. The performance of the mechanism must be judged also on its ability to describe different combustion phenomena which involve non-homogeneity like the phenomena involving flames. 3

14 Kinetics studies which are flame based are affected by the flow and diffusion effects. The kinetic information extracted is indirect in such studies and is through the assessment of the adequacy and validity of proposed mechanisms in describing the flame phenomenon. Since the flame structure spans extensive ranges of variation in terms of system pressure, reactant concentration and flame temperature, studies involving flame impose stringent and extensive constraints on the extraction and validation of the kinetic information. Validation of the proposed mechanism can be done at different levels. At the global level comparison can be done for example, the laminar flame speeds of premixed flames, the ignition delays of homogeneous mixtures obtained from shock tube and flow reactor studies. At the detailed level, one can compare the evolution of the temperature and concentrations in a system [3]. There are different levels of reduced chemistry that are derived from the detailed mechanisms. The first level of reduction is from detailed to skeletal mechanism. Further reduction i.e. the second level of reduction is based on two assumptions namely, quasi steady state assumption for species and partial equilibrium for elementary reactions. By these assumptions a reduced mechanism is derived from either detailed mechanism itself or from the skeletal mechanism. 4

15 1.2 SKELETAL MECHANISM The procedure that is widely used for generating a reduced mechanism usually involves an intermediate step. The detailed mechanism is assessed based on the importance of species and unimportant species and the reactions associated with them are eliminated. The mechanism thus generated is called the skeletal mechanism. As the nature of the reactions in this is elementary like the detailed mechanism itself, one can readily apply them and obtain a substantial savings in the computation time as compared to the detailed mechanism. Further reduction techniques can still be applied to the skeletal mechanism to result in the generation of the reduced mechanism. Several techniques can be used for generation of skeletal mechanism. Few of the examples are reaction rate analysis [4], detailed reduction [5], and computational singular perturbation [6]. The method of reaction rate analysis assumes the species to be redundant if by eliminating all its consuming reactions it induces no significant error to the remaining species. This method though simple to use is time consuming as validation for each eliminated species must be done. The importance of the time taken for the reduction will be discussed later. The method of detailed reduction identifies the unimportant reactions by comparing its reaction rate with that of pre-selected controlling reaction. The identification of the controlling reaction is not straight-forward for large mechanisms due to lack of rigorous definition of controlling behavior and the change in the controlling process under different conditions [7]. Moreover a slow reaction need not necessarily mean that it is unimportant. Computational Singular Perturbation method can be used to identify and eliminate the elementary reactions that are not 5

16 important for any species by comparing their importance with the help of importance index. The potential extent of reduction by using the methods where analysis is being done on the reactions and there by eliminating the reactions is small, this is because the number of governing equations are not reduced as no species are being eliminated. Hence to achieve a considerable reduction in the mechanism and also keeping in mind the time taken for the process of reduction a better technique could be the use of Directed Relation Graph. 1.3 DIRECTED RELATIONAL GRAPH (DRG) It is straight-forward to identify and eliminate the unimportant reactions which contribute negligibly to the production rate of every species, it is more complicated to identify and eliminate the unimportant species due to the coupling of the species. For example a species A can be strongly coupled to a species B directly if they appear together in a reaction, or indirectly if each of them is strongly coupled to another species C even if they themselves do not appear together in any reaction. Due to presence of such a coupling removal of a species from the detailed mechanism may require removal of all the species strongly coupled to it. By similar reasoning if a species has to be retained in the mechanism then all the species strongly coupled with it must be retained. To quantify the direct influence of one species on another a normalized contribution of species B to the production rate of species A, namely r AB can be defined as [7,8] 6

17 (a) 1-2(b) where ν Ai is the stoichiometric coefficient of species A in the i th reaction, subscript i designates i th elementary reaction and ω i is the net production rate of the reaction i. It is seen that if the normalized contribution of r AB is sufficiently large, the removal of species B from the mechanism will induce a significant error on the production rate of species A and vice versa. In such a case we say that the A strongly depends on species B. In order to quantify the dependence of species a threshold value ε is defined such that, for r AB < ε, the dependence can be considered negligible. Thus DRG can be constructed by the following rules: 1) Each node in DRG is uniquely mapped to a species in the detailed mechanism 2) There exists a directed edge from A to B if and only if r AB ε For each species there exists a group of species which are reachable from A, and this set of species is defined as the dependent set of A denoted as S A. If a species A must be kept in the mechanism then its dependent set S A should be kept as well. Procedurally, by obtaining the rates of elementary reactions, the DRG can be constructed by evaluating the contribution of each elementary reaction, ω i, to the edges affected by it. 7

18 After the construction of the DRG a set of species which have to be kept called starting set is selected.the starting set can simply consist of a single species namely fuel or a pollutant species such as NO, because through it the oxidizer as well as radicals that are products of reaction involving them is coupled. A deep first search is applied to the graph for each starting species to identify the dependent set. By identifying the dependent set for each starting species, the species constituting the skeletal mechanism is the union of all the dependent sets. The elementary reaction set of the skeletal mechanism is obtained by retaining all the elementary reactions of the detailed mechanism which contain the species of the skeletal mechanism and eliminating the rest of the elementary reactions. The skeletal mechanism obtained has errors bounded by the user specified value of the threshold value ε under which it is developed. The skeletal mechanism is obtained by sampling a group of points in the parametric space in typical applications. While these typical applications can be homogeneous systems of Perfectly Stirred Reactor (PSR) and auto-ignition, and diffusive systems of planar laminar flame propagation, it is more appropriate and expedient to use homogeneous systems because they are chemistry controlled phenomenon and because of the significantly reduced calculation time. For a sufficiently wide range of pressures, equivalence ratios and initial temperatures, the sampled data points should cover most of the typical conditions under which the mechanism is to be applied. Skeletal mechanisms can be generated with different levels of accuracy by assigning different threshold values. The smaller the ε the larger is the 8

19 skeletal mechanism obtained, converging to detailed mechanism as ε approaches zero. It would be reasonable to expect that strongly coupled groups exist in large mechanisms, and intra-group couplings are strong while inter-group couplings are relatively weak. Thus the number of species in the skeletal mechanism could vary abruptly as the threshold value is varied. The existence of such jumps facilitates in the selection of the threshold value. The reduction is most efficient in the neighborhood of such jumps. 1.4 DRG WITH ERROR PROPAGATION (DRGEP) DRGEP extends DRG by considering error damping along the graph searching path. For example consider the path A B C. If an error is made in C then it is propagated to A by B. Thus the longer the path in the graph it is expected that the error induced to the starting set would be damped. The major difference of this new selection method as compared to DRG is that it allows a finer selection of the species as it looks at the error made on the species will propagate to the starting set of species. The benefit of using error propagation along with DRG is to eliminate more species. In this method it is important to choose the starting species carefully because only the starting species would be described accurately and the other species would be adjusted [9]. 1.5 DRG-AIDED SENSITIVITY ANALYSIS (DRGASA) Among the species retained in the skeletal mechanism after DRG not every species in the skeletal mechanism is of equal importance to the target species and global parameters. Thus it is possible to eliminate those species which have minimal effect on 9

20 the target species or global parameters even though they may have some effect on other species which are not of significant interest. Identifying such set of species which can be eliminated is not straightforward and the error induced by such elimination also would not be small. The way these species would be coupled is highly non-linear. A simple and reliable way to eliminate such species is by computing the worst case is by eliminating each such species one at a time. This approach is similar to that of a brute force sensitivity analysis instead of analysis by means of perturbation of species concentration. This process of reduction is more CPU-time demanding than simple DRG because the auto ignition or PSR simulations have to be carried out for multiple cases. The time required for this process can be reduced by a certain amount by using the information obtained in the DRG. From this information the number of species that need to be tested can be reduced. For example consider a large mechanism, for this mechanism it is reasonable to expect that species obtained when ε > 0.4 would be strongly and directly coupled to target species and elimination of such species would induce a large error to the system. These species can therefore be eliminated from the elimination test. Also species with critical values of ε < 0.2 can be safely removed from the mechanism. Thus only species within a small majority of ε have to be considered. This results in reduction in the time taken for the reduction. Due to the non-linearity of the reaction mechanism the error caused by removal of a set of species is not just the sum of the errors induced by the elimination of each species. Thus an iterative procedure is applied to arrive at a skeletal mechanism with the desired 10

21 level of accuracy. Procedurally for each iteration using the DRGASA method, DRG analysis is first applied to obtain the critical value of ε for each species. Species with ε smaller than a user specified value ε 1 can be readily eliminated. Also all the species with ε larger than ε 2 are retained. Secondly the sensitivities E i, defined as the worst case relative error induced to the parameters of interest due to elimination of each species i, are computed for species between the user specified values of ε 1 ε ε 2. Third the species are sorted by the calculated values of E i in ascending order. Species starting with the smallest E i are eliminated until the worst case error in the parameter of interest is larger than the user specified error level. This procedure is repeated until no further species can be eliminated. Thus the computation time of the reduction process can be minimized. The skeletal mechanism obtained by the above DRGASA method is therefore minimal and that any further removal species would cause a larger than acceptable error [10]. 1.6 REDUCED MECHANISM The objective of the whole of the reduction process is to obtain a mechanism called reduced mechanism. For this the detailed or skeletal mechanism is assessed based on the nature of species or the reactions. The quasi steady state (QSS) approximation to certain species and partial equilibrium assumptions to certain reactions are applied to obtain the reduced mechanism with smaller number of rate equations to solve. The QSS species are excluded from the reaction and are solved using algebraic equations. Thus 11

22 the total no of differential equations is lessened. The nature of the reactions in this is not elementary like the detailed mechanism itself, but is lumped PARTIAL EQUILIBRIUM APPROXIMATION Partial equilibrium approximation assumes that the forward and backward rates of reaction are much larger than the net reaction rate such that we can approximate the net reaction rate to zero. This method is not a preferred option as it not straightforward to implement as compared to quasi steady state assumption QUASI STEADY STATE ASSUMPTION (QSSA) In the process of complex chemical reaction scheme leading to products from the reactants, lot of intermediate species are formed. Some of these play a vital role in the progression of the overall reaction scheme because they provide linkage between the individual reactions. The consumption and regeneration of these intermediates occur rapidly, but at approximately equal rates such that their concentrations can be considered to remain constant. One of the major issues involved in the reduction process based on QSSA is the identification of QSS species. One of the methods used for identification of QSS species is Computational Singular Perturbation (CSP) technique. 1.7 COMPUTATIONAL SINGULAR PERTURBATION (CSP) CSP is a systematic mathematical procedure to do boundary-layer type singular perturbation analysis. The basic idea of CSP is to separate the time scales and there by facilitate in further solving procedure. 12

23 1.7.1 BASIC FORMULATION AND ALGORITHM A general chemical system can be defined as 1-3 Where y is the concentration vector of all the species, S the stoichiometric coefficient matrix, and F(y) the rate vector of elementary reactions. Differentiating Eq 1-3 with respect to time we get 1-4 where J=dg/dy is the time independent Jacobian matrix. By using Decomposition of J, Eq 1-4 can be written as =Λ.f, f=b.g, Λ=( 1-5 where A is a matrix of column basis vectors and B is the inverse matrix of A. In Eq1-5 if A matrix is built up by ideal basis vectors, then Λ reduces to diagonal matrix and the modes f are decoupled [6]. As the Jacobian matrix J is time independent, a refinement procedure is warranted for [6], but this may be difficult in many cases. If only the time scales of the species are of interest the Jacobian matrix can be approximated to be locally time independent and the refinement procedure can be replaced by an eigenvalue eigenvector decomposition of the Jacobian matrix. Since oscillatory modes may appear in the chemical systems, it is necessary to decompose J in complex space in order to diagonalize Λ. The definitions of 13

24 the modes, their timescales, the radical pointer, and the participation index needs to be defined in complex space. When oscillatory modes appear matrices A and B are complex in nature. Complex conjugate pairs of columns and rows in each of the matrix needs to be treated in the definitions of CSP data.[7] A i = A ir +ia ii 1-6 A i+1 = A ir - ia ii 1-7 B i = B i R + ib i I 1-8 B i+1 = B i R - ib i I 1-9 where the i th and (i+1) th columns of A and the i th and (i+1) th row of B are complex conjugate pairs, and the subscript R and I denote the real and imaginary parts, respectively. Each pair of complex conjugate modes in Eq 1-5 are converted into a pair of real modes by algebraic manipulation: f i = B i R. g 1-10 f i+1 = B i I. g 1-11 These two modes are coupled with each other through the complex conjugate eigenvalue pair. The corresponding projection matrices associated with the pair of oscillatory modes in Eqs 1-10 and 1-11 are 14

25 Q i = 2A ir. B i R 1-12 Q i+1 = 2A ii. B i I 1-13 The radical pointers R i,r and R i+1,r are the r th diagonal elements of Q i and Q i+1, respectively. The radical pointer indicates how parallel the axis of r th species is with respect to the i th mode, and is used to find the QSS species [7] IDENTIFICATION OF QSS SPECIES By the assumption that the modes in the fast subspace are exhausted, CSP can eliminate the short timescales related to QSS species. From Eq 1-5 we see that if the i th diagonal element of Λ is a large negative number, the i th mode will be quickly exhausted compared to slow modes. If the species is associated with only one exhausted mode that is if the radical pointer is almost unity, the species can be identified as a QSS species. However in most cases the species is projected onto several modes of different time scales. Thus further procedure is required for the identification of the QSS species. Keeping in mind that a species projected more onto the fast subspace than other species would be a better QSS candidate and that the QSS species generally have lower concentrations, the time scales of independent modes are averaged to an indicator weighted by sufficiently large radical pointers. That is a species averagely belonging to the faster subspace, which consists of a group of faster modes can be exhausted faster than a species belonging to a subspace that is less fast. 15

26 The CSP data above are only the local chemistry information and at certain time step under a specific condition. The QSS species identified in this manner is valid only locally. As the conditions change in the system the species identified as QSS species at one instance may not be such good QSS species later. Thus to obtain a global QSS species, the above analysis must be done at all the points in the range of interest. The QSS species thus obtained could be called a global QSS species for the range of conditions under which it has been identified [11]. 1.8 OBJECTIVES Combustion of fuels has provided a majority of energy needs. Combustion is a phenomenon through which the energy trapped in various fuels is converted from chemical form to heat. This energy is then further utilized for generating steam, electricity, space heating, transportation, cooking and many other applications. At the molecular level, the fuel and oxidizer can undergo a change in their electronic configuration to form or break bond. Therefore bringing the fuel and oxidizer in the close proximity of each other forms a challenging part of design of any combustion equipment. Experimental Investigation gives useful data that forms the basis for designing such equipment while theoretical developments attempt to explain the experimental results. With the advent of numerical simulations and the costs of performing experiments on massive systems it becomes cost effective in utilizing the numerical simulations for predicting the results [12]. 16

27 Most fuels in combustion involve a large number of species and reactions in their oxidation process. Knowledge of all such steps and intermediate species is necessary in understanding the combustion behavior of the fuels. Lot of research has resulted in the mathematical and numerical models to describe and handle the complex nature of chemical reactions. But the nature of fluid flows in the combustion systems are varied and range from laminar to turbulent. In majority of equipment the flow is turbulent and it is desired to be so to effect a better mixing of the reactants. The flow of fluid can have additional characteristics like swirl, compressibility and unsteady effects. Thus it is imperative to model the turbulence appropriately. With the presence of reacting species in such a simulation, it becomes more cumbersome as additional terms have to be modeled. This is due to the source terms of each species and the energy released therein. This energy released or absorbed has an effect on the flow field and vice versa. Such a coupling further complicates the process of obtaining a solution. This affects the mixing of reactants. Thus to successfully model such conditions it is not only important to understand the turbulence and complex chemistry but also to have a good descriptive means to model the interactions between the same. With the advent of commercially available CFD codes of the simulation of the combustion processes there is an increase in the use of such codes in the industry. To understand the phenomena in detail it is important to have detailed chemistry implemented in these codes. There are various limitations to implement the same, like the large amount of computational resources and memory requirement along with the 17

28 limitations placed in the number of species and reactions placed by these commercial codes. Thus to be able to simulate the combustion processes in detail without consuming a large amount of resources it is imperative to use reduced mechanisms. Also as we have seen in this chapter it is important to use the reduced mechanism in non-homogenous and flame conditions to test the comprehensiveness of the reduced mechanism. Some work has been done to implement the reduced mechanism in the commercial CFD codes, but most of the work is either in two-dimensional or compares the results with the homogeneous cases. Thus it is important to simulate the same phenomenon in three dimensions and compare the results between using the detailed mechanism and the reduced mechanism. This work is aimed at simulating the combustion process using automatically reduced mechanism in a three-dimensional CFD model. A commercially available CFD package, FLUENT is used to implement the reduced mechanism. A simplified model of Jet Stirred Reactor is taken. A certain amount of non-homogeneity is expected in this reactor. Simulations with the detailed and reduced chemistry of different levels are performed in order to assess the efficacy of implementation of reduced mechanisms. 1.9 THESIS ORGANIZATION In Chapter 2 of this thesis we shall look into the methodology used in generating the skeletal and reduced mechanism. We shall further validate the same by using PSR or auto-ignition codes. Chapter 3 will deal with the implementation of the detailed, skeletal and reduced mechanism in the commercial CFD codes. In Chapter 4 some 18

29 conclusions are drawn on the results presented in Chapter2 and 3 and some thoughts for future work is provided. 19

30 CHAPTER-2: METHODOLOGY 2.1. SCIENTIFIC BACKGROUND Methane is a major component in the natural gas. Owing to this the study of chemical kinetics of methane has received extensive interest in both practical applications and fundamental combustion research. Detailed mechanisms for methane have been developed and also various reduced mechanisms of various complexities have been developed. The reduced mechanism has been developed with various number of species. While the models with fewer global steps are good for qualitative analysis only, reduced mechanisms with larger number of species can mimic detailed mechanisms with a great accuracy. One can expect this accuracy for a wide range of parametric space even with inclusion of NO chemistry [13]. In the last chapter we have seen the modifications or additions done in the methods used for generation of reduced mechanism. We have seen the additions to the basic DRG method to obtain a more accurate reduced mechanism, i.e. use of DRGEP and DRGASA. Keeping in tune with the kind of developments in the methods of reduced mechanism, an algorithm was developed with combination of these methods to generate a reduced mechanism. The new algorithm combines aspects from both the DRGEP and DRGASA to result in a method DRGEPASA [14]. Further QSS Assumptions were made for species there by eliminating them from the mechanism and solving those using algebraic relations. The advantage of employing QSS assumption after the DRGEPA process is slated to be that there will be less number of algebraic relations to solve. In the last chapter we have also 20

31 see that some of the zero-dimensional and one-dimensional codes have reached a point where the simulations can be trusted. In this chapter we shall make use of such codes to implement the reduced mechanism generated. 2.2 METHODOLOGY OF REDUCTION The DRGASA method is very effective in generating a optimal skeletal mechanism, but one of the main issues concerned with this methods is the time taken for the process of reduction as it involves sensitivity analysis. Since the DRG method itself is found to be inefficient to identify unimportant species, the number of species on whom the sensitivity analysis has to be performed is large. DRGEP on the other hand is found to be a superior method in finding the unimportant species from the detailed mechanism. In this method it considers the propagation of errors from the target species into account i.e. the contribution to the production / consumption of species far away in the path from the target species will be very little. As the computation of the normalized contribution is incorrect it could be misleading. Since DRGEP underestimates the interaction coefficients of some of the important species, sensitivity analysis is needed to clearly identify the importance of those species. Taking into account these factors sensitivity analysis is integrated into DRGEP for effective mechanism reduction resulting in DRGEPSA [14]. CSP analysis is performed on the skeletal mechanism thus generated and QSS species are found. From this a reduced mechanism is generated. 21

32 2.2.1 DIRECTED RELATIONAL GRAPH WITH ERROR PROPAGATION (DRGEPSA) DRGEPSA is carried out in a twostep process. First a preliminary skeletal mechanism is generated by using the DRGEP method. In this step the skeletal mechanism is generated by an iterative process such that the error is less than the user specified error percentage. For the second step this skeletal mechanism is used and sensitivity analysis is performed on the same. A threshold value is chosen and for all the species below this threshold value sensitivity analysis is performed. The induced error for each of these species is obtained and they are ranked in ascending order. The species are then eliminated one at a time in that order till the error for the skeletal mechanism exceeds the specified error tolerance. The skeletal mechanism thus obtained would be highly optimized as to goes though both DRGEP and sensitivity analysis. It has been seen that the computational effort involved for DRGEPSA is substantially less than DRGASA as the number of species whose sensitivity has to be performed is less [14] SKELETAL MECHANISM As a first step in the reduction in the detailed mechanism, a skeletal mechanism is generated. For this we consider GRI-Mech 3.0 [15] and eliminate the unimportant species and reactions. The reduction study in this study is based on data sampled from a homogeneous application namely auto-ignition which is a typical application for the study of ignition. The parameter range covers pressures from atm, equivalence ratio from and temperature ranges from K.The temperature range was 22

33 increased to upto 2000K compared to the 1800 K used for the previous studies of reduced mechanism [13].The temperature increase facilitated the inclusion of C 2 H 2 species in the mechanism. The method of DRGEPSA was applied to the starting mechanism of GRI-Mech 3.0 containing 53 species and 325 reactions. The error tolerance specified for the generation of the skeletal mechanism was set at 5%. The algorithm utilized an iterative process to obtain a threshold value for the method such that the final error was within 5%. The advantage of using the iterative method is that it optimizes the skeletal mechanism for the input parametric range. The skeletal mechanism thus generated contained 29 species and 150 reactions. The skeletal mechanism is listed in appendix. The target species were chosen such that the resultant skeletal mechanism contained the NO chemistry for the prediction of NO formation REDUCED MECHANISM The detailed mechanism of methane i.e GRI-Mech 3.0 is further used to perform analysis using the theory of CSP as explained in the previous chapter. By using the theory of CSP we separate the fast and slow time scales. For doing so a characteristic time scale τ c is required. For the auto-ignition cases the ignition delay is chosen. A safety factor α, say is applied to ensure that the time scales of the fast processes are sufficiently shorter than τ c [14]

34 where f fast is the time scale of the fast modes. The separated fast and slow subspaces are then used for determining whether species is a good QSS candidate. After this a reduced mechanism of 19 species is so generated that it is similar to the 19 species mechanism generated by [16]. The list of species in the reduced mechanism is listed in appendix. This is different from the mechanism of 12-step of ARM [1,2] in that the O radical is considered as a non QSS species. Also QSS assumption of radical O has found to increase in the reduction error [13]. 2.3 VALIDATION OF MECHANISMS The skeletal and reduced mechanisms thus generated are implemented and validated in homogeneous applications like PSR and auto-ignition SENKIN The most widely used tool for the computing auto ignition parameters is SENKIN package [17], a subset of CHEMKIN [18] originally created by Sandia National Laboratory. It is s a computer program that is used to predict time-dependent chemical behavior of a homogeneous gas mixture in a closed system. The set of equations are solved by a package called DASAC (Differential Algebraic Sensitivity Analysis Code). It performs the time integration using a Backward Differential Formula (BDF) [17] PSR The PSR [19] code created by Sandia National Laboratory is the steady state version of the Continuously Stirred Tank Reactor (CSTR) and is considered as the simple and useful tool for studying chemical kinetics. CSTRs have been used for many years to 24

35 study the chemistry of various batch chemical processes including combustion. Since these are relatively inexpensive to build and easy to operate there is a considerable value in using these reactors to help in the understanding of important problems such as pollutant formation. The stirred reactor consists of a small thermally insulated chamber with single inlet and outlet with constant mass flow through it. A steady flow of fuel and oxidizer are introduced into the reactor such that the high intensity turbulent mixing causes the contents of the rector to be spatially uniform. Thus the rate of conversion from reactants to products is controlled by chemical reaction rates and not by mixing process. In addition to the fast mixing the modeling of the well stirred reactor requires certain assumptions namely that the walls must be non-catalytic and that the flow through the reactor has to characterized by a nominal residence time [19]. This simplification leads to a unique time scale associated with the through flow of the reaction that is the residence time. At large residence times, the reactor is set to approach the chemical equilibrium limits. At very low residence time the flame extinction is set to occur because the chemical reaction cannot be sustained [20]. The description of the process occurring within the perfectly stirred rector is obtained by relating the conservation of mass and energy to the net generation of chemical species in the reactor volume. This mainly results in a set of algebraic equations. The equations are solved by a damped modified newton algorithm. But if during the course of the iteration of the newton algorithm fails to converge then the solution is estimated by solving the time-dependent transient equations by time 25

36 integration. The response to chemical kinetics with the residence times is a useful test for reduced chemistry SENKIN RESULTS The ignition delay in the auto-ignition cases is calculated as a function of initial temperature of the mixture for a range of pressures and equivalence ratios. These calculations were done for the detailed GRI Mech 3.0, the 29 species skeletal mechanism and the 19 species reduced mechanism which was generated inhouse Methane/Air Phi= Auto-Ignition Delay (Sec) Detailed Mech P=1atm Skeletal Mech P=1atm Reduced Mech P=1atm Detailed Mech P=10atm Skeletal Mech P=10atm Reduced Mech P=10atm Detailed Mech P=20atm Skeletal Mech P=20atm Reduced Mech P=20atm /T(1/K) Figure 2.1. Plot of ignition delay as a function of the inverse of initial temperature for constant pressure ignition for the methane/air mixture of equivalence ratio of

37 10 2 Methane/Air Phi= Auto-Ignition Delay (Sec) GRI 3.0 P=1atm Skeletal Mech P=1atm Reduced Mech P=1atm GRI 3.0 P=10atm Skeletal Mech P=10atm Reduced Mech P=10atm GRI 3.0 P=20atm Skeletal Mech P=20atm Reduced Mech P=20atm /T(1/K) Figure 2.2. Plot of ignition delay as a function of the inverse of initial temperature for constant pressure ignition for the methane/air mixture of equivalence ratio of

38 10 2 Methane/Air Phi= Auto-Ignition Delay(Sec) GRI 3.0 P=1atm Skeletal Mech P=1atm Reduced Mech P=1atm GRI 3.0 P=10atm Skeletal Mech P=10atm Reduced Mech P=10atm GRI 3.0 P=20atm Skeletal Mech P=20atm Reduced Mech P=20atm /T(1/K) Figure 2.3. Plot of ignition delay as a function of the inverse of initial temperature for constant pressure ignition for the methane/air mixture of equivalence ratio of 2.0 We can see from Figures that there is a excellent agreement between the detailed, skeletal and the reduced mechanisms. On examination of the percentage error caused by the skeletal and reduced mechanism, the calculations indicate that the error percentage for the skeletal mechanism is below 5% which was the error tolerance with which the mechanism was generated. For the reduced mechanism it was found that the 28

39 error percentage is below 10%. But as the reason for generating reduced or skeletal mechanisms is to reduce the computational time, comparison was performed between the times taken for the reduced or skeletal mechanism to the time taken for the detailed mechanism. For this purpose a speed up factor is defined as the ratio of time taken by the detailed mechanism to the time taken by the skeletal or reduced mechanism. On making this comparison it was observed that the skeletal mechanism gave an average of 3 as the speed up factor and the speed up factor was fairly consistent for the temperature and pressure range considered. But the reduced mechanism was not found to be showing similar characteristics. It was observed that in some of the low temperature and high temperature cases the reduced mechanism performed poorly in the aspect of computational time savings. The time consumed by the reduced mechanism was either comparable to the time taken by the detailed mechanism or more than that. Tables show the speed up factor for reduced and skeletal mechanism for the range of the cases. 29

40 Initial Temperature Speed Up Factor Pressure=1atm Speed Up Factor Pressure=10atm Speed Up Factor Pressure=20atm E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E+00 Table 2.1. Speedup factors using the 19 species reduced mechanism compared with the detailed mechanism for φ=0.5 Initial Temperature Speed Up Factor Pressure=1atm Speed Up Factor Pressure=10atm Speed Up Factor Pressure=20atm E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E+00 Table 2.2. Speedup factors using the 19 species reduced mechanism compared with the detailed mechanism for φ=1.0 30

41 Initial Temperature Speed Up Factor Pressure=1atm Speed Up Factor Pressure=10atm Speed Up Factor Pressure=20atm E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E+00 Table 2.3. Speedup factors using the 19 species reduced mechanism compared with the detailed mechanism for φ=2.0 Initial Temperature Speed Up Factor Pressure=1atm Speed Up Factor Pressure=10atm Speed Up Factor Pressure=20atm E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E+00 Table 2.4. Speedup factors using the skeletal mechanism compared with the detailed mechanism for φ=0.5 31

42 Initial Temperature Speed Up Factor Pressure=1atm Speed Up Factor Pressure=10atm Speed Up Factor Pressure=20atm E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E+00 Table 2.5. Speedup factors using the skeletal mechanism compared with the detailed mechanism for φ=1.0 Initial Temperature Speed Up Factor Pressure=1atm Speed Up Factor Pressure=10atm Speed Up Factor Pressure=20atm E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E+00 Table 2.6. Speedup factors using the Skeletal Mechanism compared with the Detailed Mechanism for φ=1.0 32

43 Till now we have seen the global response characteristics of the reduced mechanisms. For further validation it is necessary to compare the behavior of parameters like temperature and pollutant species i.e. CO and NO over the time integration range. Figure 2.4 compares the temperature profile of the skeletal mechanism and 19 species reduced mechanism with the detailed mechanism and we can see that there is an excellent agreement between the mechanisms Methane/Air Phi= Temperature(K) GRI Mech 3.0 Skeletal Mech Reduced Mech Time(Sec) Figure 2.4.Temperature profile as a function of time for the methane/air mixture of equivalence ratio of 0.5, and P=20atm 33

44 0.016 Methane/Air Phi= Mass Fraction of NO GRI Mech 3.0 Skeletal Mech Reduced Mech Time(Sec) Figure 2.5.NO mass fraction profile as a function of time for the methane/air mixture of equivalence ratio of 1.0, and P=1atm Figure 2.5 shows the comparison of the NO mass fractions of skeletal and reduced mechanism with the detailed mechanism. We can see from the Figure 2.5 that there is excellent agreement between the mechanisms. This condition of equivalence ratio 1.0 at pressure of 1 atm is considered because similar conditions are faced in the CFD simulations performed in the next chapter and thus would help in comparison of the 34

45 behaviors of the parameters in different computational codes. Thus for the pollutant species i.e. NO and CO this different condition is used. 0.1 Methane/Air Phi= Mass Fraction of CO GRI Mech 3.0 Skeletal Mechanism Reduced Mechanism Time(Sec) Figure 2.6.CO mass fraction profile as a function of time for the methane/air mixture of equivalence ratio of 1.0, and P=1atm We also can see from the Figure 2.6 that there is a good agreement between the mechanisms. 35

46 2.3.4 PSR RESULTS The temperature in a PSR as a function of residence time is computed for a range of pressures and equivalence ratios for a fixed inlet temperature of 600K. These calculations were done for the detailed GRI-Mech3.0, the 29 species skeletal mechanism and the 19 species reduced mechanism Methane/Air, Phi=0.5, T inlet =600K Temperature(K) Detailed for P=1atm Skeletal for P=1atm Reduced for P=1atm Detailed for P=10atm Skeletal for P=10atm Reduced for P=10atm Detailed for P=20atm Skeletal for P=20atm Reduced for P=20atm Residence Time (Sec) Figure 2.7. Comparison of adiabatic PSR among detailed, skeletal and reduced mechanisms 36

47 2500 Methane/Air, Phi=1.0,T inlet =600K 2400 Temperature(K) Detailed for P=1atm Skeletal for P=1atm Reduced for P=1atm Detailed for P=10atm Skeletal for P=10atm Reduced for P=10atm Detailed for P=20atm Skeletal for P=20atm Reduced for P=20atm Residence Time(Sec) Figure 2.8. Comparison of adiabatic PSR among detailed, skeletal and reduced mechanisms We can see from the Figures that there is a excellent agreement between the detailed, skeletal and the reduced mechanisms. On examination of the percentage error caused by the skeletal and reduced mechanism, the calculations indicate that the error percentage for the skeletal mechanism is below 5% which was the error tolerance with which the mechanism was generated. For the reduced mechanism it was found that the error percentage is still less than 5% unlike the higher error percentage seen in the SENKIN cases. 37

48 Methane/Air, Phi=2.0,T =600K inlet Detailed for P=1atm Skeletal for P=1atm Reduced for P=1atm Temperature(K) Residence Time (Sec) Figure 2.9. Comparison of adiabatic PSR among detailed, skeletal and reduced mechanisms 38

49 1900 Methane/Air, Phi=2.0,T inlet =600K 1880 Temperature(K) Detailed for P=10atm Skeletal for P=10atm Reduced for P=10atm Residence Time (Sec) Figure Comparison of adiabatic PSR among detailed, skeletal and reduced mechanisms 39

50 1900 Methane/Air, Phi=2.0,T inlet =600K 1880 Temperature(K) Detailed for P=20atm Skeletal for 20atm Reduced for 20atm Residence Time (Sec) Figure Comparison of adiabatic PSR among Detailed, Skeletal and Reduced mechanisms 40

51 CHAPTER-3: IMPLEMENTATION IN CFD 3.1. SCIENTIFIC BACKGROUND Most combustion processes have a recirculating zone in some way or the other. This recirculation zone stabilizes the reaction zone and helps in better mixing of the reactants. The flow field in a large scale combustion process will probably be turbulent and therefore three dimensional [21]. The presence of turbulence in the flow field and because of the complex nature of the nature of turbulent flows it requires special treatment. Turbulent flows consist of length scales along which the energy is transferred. The kinetic energy enters in the large eddies and results in the break-up into smaller eddies which further results in the break-up into smaller eddies. The energy is dissipated in these smaller eddies. There are two characteristic phenomena that can be seen in the turbulent flows mainly coherent structures and intermittency. Coherent structures are a those portions of the turbulent flows that have large length scales and long residence times. There exists a part of the flow that switches between turbulent and non-turbulent in nature thus causing non-uniformity. This is called intermittency. Resolving all the length scales in turbulent flow by means of Direct Numerical Simulation (DNS) requires a very large amount of resources in terms of memory and computational time required. The other ways of handling the turbulent flow equations are by averaging and solving for the averaged quantities. But this averaging gives problems in terms of new averaged quantities for which closure must be attained. For this purpose lot of models both of the nature of two equation like K-epsilon, K-Omega 41

52 and others are devised. But the presence of the reactions in such a flow poses a certain amount of complexity. The presence of reactants can cause the breakup of eddies, mixing and subsequently steepen the gradient of fuel and oxidizer. Thus for the turbulent flows involving reactions the molecular properties like Prandtl number and Lewis Number plays a role [12]. 3.2 PHYSICAL NUMERICAL PROCEDURE As we have seen that there are various models to describe the turbulent flows we look at the models used in the simulation for the purpose of describing the flow and chemistry. As the objective was to implement the reduced mechanisms in a commercial CFD codes, FLUENT has been considered as the commercial CFD code. In this package there are different models to choose from. As we have mentioned in Chapter 1 of this thesis a simplified Jet Stirred Reactor (JSR) was considered. This particular geometry is considered because it involves complex and turbulent flow field and it is expected to possess non-homogeneity under certain conditions. We shall now delve into the models chosen and the treatment of the flow and chemistry by each of them TURBULENT MODEL As we have seen in the preceding section that turbulent flows are characterized by fluctuating flow fields and that it is too computationally expensive to simulate them directly time-averaging or ensemble averaging is performed on the exact governing equations to make them computationally less expensive. To solve for the unknown variables obtained by this averaging turbulence models are needed. As there is no single 42

53 model that is universally accepted as being superior to all class of problems, the choice of the turbulence model depends on the physics encompassed in the flow, established practice of the specific class of problem, the level of accuracy required, the available computational resources and the available time for the computation. The simplest complete models of turbulence are the two-equation models in which the two separate transport equations allows the turbulent velocity and length scales to be determined independently[22]. Among such two equation models the k-ε model is most widely used complete turbulence model and is incorporated in most commercial CFD codes [23]. As this models has become very popular in the simulating of practical engineering flow calculations FLUENT provides this model and its improvements. One such model obtained after the improvements is the realizable k-ε model. The benefit of this model is in its accurate prediction of the spreading rate of both planar and round jets and its expectation in providing superior performance for flows involving rotation, boundary layers under strong adverse pressure gradients, separation and recirculation. The transport equations of the realizable k-ε model are where C 1 = max [0.43,η/(η+5)], η =S(k/ε), S =, S ij = 43

54 In these equations, G k represents the generation of turbulent kinetic energy due to the mean velocity gradients, G b is the generation of turbulence kinetic energy due to buoyancy, Y m represents the contribution of the fluctuating dilatation in compressible turbulence to the overall dissipation rate. C 2 and C 1ε are constants. σ k and σ ε are turbulent Prandtl numbers for k and ε, respectively. S k and S ε are user-defined source terms. Presence of walls significantly affects the turbulent flows. No slip condition that has to be satisfied at the wall affects the mean velocity field, viscous damping reduces the tangential velocity fluctuations very close to the wall while kinematic blocking reduces the normal fluctuations. Therefore for successful predictions of wall bounded turbulent flows accurate representation of the flow in the near-wall region is necessary. The near-wall region is largely subdivided into three layers. The innermost layer, called the viscous sub layer the flow is almost laminar, and the molecular viscosity plays a dominant role in momentum and heat or mass transfer. In the outer layer, called the fully-turbulent layer, turbulence plays a major role. Last but not the least there is an interim region between the viscous sub layer and the fully turbulent layer where the effects of molecular viscosity and turbulence are equally important [24]. To deal with the presence of walls in the turbulent flows there are two approaches to modeling the near wall region. In one approach semi-empirical formulas called wall functions are used to bridge the viscosity-affected region between the wall and fully-turbulent region instead of resolving the same. In another approach the region 44

55 to be resolved is meshed all the way to the wall to include viscous sub-layer. This approach is usually called near-wall-modeling approach. The use of wall modeling approach saves a lot of computation time because the viscous sub-layer need not be resolved. One such wall modeling function used for the purpose of simulation is called Enhanced Wall Treatment in FLUENT. Before we move on to the enhanced wall functions it is important to know a parameter y +. It is the distance from the wall measured in viscous lengths and is denoted by y + = ρµ t y /µ where y is the distance from the wall [23]. As y + is similar to local Reynolds number, its magnitude is expected to determine the relative importance of viscous and turbulent process. For near wall approach modeling the y + has to be less in order to be able to resolve the viscous sublayer. With the use of enhanced wall function the unstructured mesh, coarse grids and a relatively larger y + can be had and one would expect to have a higher accuracy. In FLUENT a single wall law for the entire wall region is achieved by blending linear (laminar) and logarithmic (turbulent) laws of the wall by the use of a function [22] 3-3 where blending function is given by; 3-4 where α=0.01 and b=5. This formula guarantees the correct asymptotic behavior for large and small values of y +. 45

56 3.2.2 CHEMISTRY MODEL Similar to the modeling of the turbulent flow it is also important to model the chemistry involved in combustion simulations. FLUENT models mixing and transport of chemical species by solving conservation equations for each component. The conservation equation has the general form of 3-5 where R i is the net rate of production of species i by chemical reaction and S i is the rate of creation by addition from the dispersed phase and any user-defined sources and J i is the diffusion flux of species i. This equation is solved for N-1 species where N is the total number of fluid phase chemical species present in the system. Since the mass fraction of the species must sum to unity, the N th mass fraction is determined as one minus the N-1 solved mass fractions. The expression for R i that is the net rate of production of species is dependent on the model used for species. One of the models in FLUENT that deal with the chemistry is the Eddy Dissipation Concept (EDC) model [22]. In the finite rate chemistry methodology of solving for the species the fluctuations due to the presence of turbulence is not considered. In this model the effects of turbulence is also considered. From the Arrhenius rate expression for the reversible reaction, the molar rate of creation /destruction of species i in the reaction r is given by

57 Where is the molar rate of creation/destruction of species i in the reaction r, N is the number of species in the system, ν i,r is the stoichiometric coefficient for species i in the right hand side of reaction r, ν i,r is the stoichiometric coefficient for species i in the left hand side of the reaction r, k f,r is the forward rate constant for reaction r and k b,r is the backward rate constant for reaction r. Γ accounts for the net effect of third bodies on the reaction rate. The EDC model assumes that the reactions occur in small turbulent structures known as fine scales. The inert part is referred to as surrounding fluid. The length fraction of the fine scales in modeled as 3-7 where * denotes fine scale quantities and C ξ is a volume fraction constant and is equal to The volume fractions are calculated as ξ *3. The species are assumed to react in the fine structures over a time scale defined as 3-8 Where C τ is a time scale constant equal to The combustion at the fine scales is assumed to occur as a constant pressure reactor. Reactions proceed over the time scale τ *, governed by Arrhenius rate given by Eq 3-6. Finally the source term in the Eq 3-5 is modeled as 47

58 ) 3-9 where Y * i is the fine-scale species mass fraction after reacting over the time τ *. As it has been mentioned that combustion in fine scales is assumed to occur as a constant pressure reactor, the reactions are represented by an isobaric, adiabatic, perfectly stirred reactor. The radiation from or to the fine structures is neglected. Y * i is obtained as a steady state solution of 3-10 where w * i is the reaction rate of i th species, ρ * is the density of the gas in the fine scales and Y 0 i is the mass fraction of the species in the inert part [25] TRANSPORT MODEL There are two ways to model the diffusion i.e. the term J i in the Eq 3-5. Once is the Fick s law approximation and the other is full multi component diffusion. In the Fick s law of diffusion it is assumed that the binary diffusion coefficients of all pairs of species are equal. This approximation may be sufficient for most of the flows where the mixture composition is not changing. One of the ways of handling the multi component diffusion is by a method of kinetic theory. This method allows for the changing diffusion coefficients. In these method Lennard-Jones parameters σ i and (ε/k B ) are defined for each species, where k B is the boltzmann constant. The solver uses a modified Chapman- Enskog formula to compute the diffusion coefficient 48

59 3-11 where p abs is the absolute pressure, and Ω D is the diffusion collision integral, which is a measure of the interaction of the molecules in the system. Ω D is a function of the quantity T D * where 3-12 and (ε/k B ) ij for the mixture is the geometric average PHYSICAL-NUMERICAL PROCEDURE In the last section we have seen the equations that handle the flow and the chemistry and also as said earlier the mechanisms generated and used in Chapter 2 are implemented in the three-dimensional CFD calculations. The following section deals with the specifics of the models and conditions used in the CFD computations GAS PHASE NUMERICAL MODEL The three-dimensional governing equations of continuity, momentum, energy, turbulence and species are solved using the pressure based solver in FLUENT. Turbulence is modeled using the Realizable k-ε equations. Enhanced wall functions are used to resolve the flow field near the wall. Heating and viscous effects were also taken into account. A segregated implicit solver with PISO (Pressure-Implicit Split-Operator) 49

60 algorithm for pressure velocity coupling, PRESTO scheme for pressure and second-order upwind scheme were used for the governing equations. The gradients and derivatives of the governing equations are computed using the Green-Gauss Node Based method which is second order spatially accurate [26] THERMODYNAMIC AND TRANSPORT PROPERTIES The thermodynamic and transport properties are temperature and species dependent. The mixture density is computed using the ideal gas law. The specific heat capacity of individual species is computed with piecewise polynomials. The thermal conductivity and viscosity of the individual species were based on the Chapman-Enskog theory. Lennard-Jones potentials are used to compute the binary diffusion coefficients COMBUSTION MODEL To have higher accuracy and to see the behavior of the chemistry in the threedimensional turbulent flows, finite rate chemistry is used. As the expectation was that the there would be some amount of non-homogenity in the system, the assumption that the overall reaction rate is governed only by chemistry was not considered. The fluctuations caused by the presence of turbulence were taken into account. Thus a turbulence-chemistry model which combined the concepts of finite rate chemistry while considering the fluctuations caused by the turbulence was taken. This resulted in an EDC model in the FLUENT. The detailed mechanism skeletal mechanism and reduced mechanism were read in the CHEMKIN format. A User Defined subroutine (UDF) was 50

61 written in C-language to link the subroutine of the reduced mechanism with the FLUENT GEOMETRY AND MESH A simple geometry of a sphere with four inlets and an outlet is considered. As the geometry is not symmetrical the entire geometry is modeled and meshed. The creation of the model and meshing was done in GAMBIT [27]. The volume of the geometry is meshed using unstructured T-Grid meshing scheme using only tetrahedral elements. A schematic of the geometry and the grid is shown in Fig 3.1. The surface in red color is the outlet and the other openings are inlets. Figure 3.1. Geometry and Grid of the model for CFD simulations 51

62 3.3.5 BOUNDARY CONDITIONS Boundary conditions have to be set for the inlets and outlets for simulations. There are two equivalence ratios for which the simulations are performed. One lean that is equivalence ratio of 0.5 and other stoichiometric condition are considered. The boundary conditions for different equivalence ratios are different as shown in Table 3.1. PSR calculations are performed because of the similarity of the geometry considered. From these two calculations two conditions are chosen such that the residence times are near the extinction point. Apart from this a no-slip and adiabatic conditions are applied at the wall and the operating pressure is 1 atm. φ=1.0 Surface Boundary type Parameter Value Velocity Temperature 1.8m/s 600K Inlet Outlet Velocity Inlet Outflow Turbulence Intensity 10% CH4 (Mass Fraction) O2 (Mass Fraction) 0.22 N2 (Mass Fraction) 0.73 Table3.1 Boundary conditions for the CFD simulations 52

63 φ=0.5 Surface Boundary type Parameter Value Velocity Temperature 0.229m/s 600K Inlet Outlet Velocity Inlet Outflow Turbulence Intensity 10% CH4 (Mass Fraction) 0.03 O2 (Mass Fraction) 0.23 N2 (Mass Fraction) 0.73 Table3.2 Boundary conditions for the CFD simulations 3.4 RESULTS VALIDATION OF FLUENT CHEMISTRY MODEL Before we take on to simulate the mechanisms in the three-dimensional model, tests were performed to check the chemistry calculations performed by FLUENT. For this purpose a simplified model is taken with just one node to simulate homogeneous conditions. Auto-ignition is simulated in this model. Detailed, skeletal and reduced mechanisms are run and the results are compared with the results obtained from SENKIN calculations. The grid is as shown in Fig 3.2, in which the red lines are the walls and the central node is the only node at which computations are performed. 53

64 Figure 3.2. Geometry and Grid of the model Comparision for Phi=1.0 SENKIN Time(Sec) Detailed Mechanism Skeletal Mechanism Reduced Mechanism SENKIN Temperature(K) Time(Sec) Figure 3.3. Comparison of temperature profiles for constant pressure ignition for the methane/air mixture of equivalence ratio of 1.0 at pressure of 1 atm 54

65 Figure 3.3 shows that the results of the simulations in FLUENT for detailed, skeletal and reduced mechanism match with the corresponding calculation performed in SENKIN with the detailed mechanism. Based on this result, it is concluded that the chemistry model in FLUENT is adequate THREE DIMENSIONAL MODEL RESULTS As mentioned earlier three-dimensional simulations were performed for two equivalence ratios. For the computations parallel processing was performed. CASE ITS cluster was used to perform the simulations. The grid was partitioned along the principle axis to run in the different nodes. Before we look at the results it is important to determine the parameters that are to be compared. For this purpose we shall consider temperature, mass fractions of CH 4, O 2, H 2 O, CO 2 and the velocity vectors. Along with this pollutant species like CO and NO are also considered. Apart from knowing what parameters to compare it is also important to know what values to compare. For this purpose we consider two values i.e. peak value and the mass averaged quantity denoted by the subscript mavg which is computed by dividing the summation of the product of density, cell volume and the selected field variable by the summation of the product of density and cell volume. We shall first look at the results from the equivalence ratio 0.5 and then for equivalence ratio of 1.0. All the plots are taken on a diametrical plane which bisects the two horizontal inlets and the outlet. 55

66 Figure 3.4. Plot of temperature (K) profile for the methane/air mixture of equivalence ratio of 0.5 at pressure of 1 atm with detailed mechanism, skeletal mechanism, reduced mechanism and ARM reduced mechanism in clockwise direction Figure 3.4 presents the temperature contours for the mechanisms used. We can see that the temperature has dropped in the skeletal and reduced mechanisms. The peak temperature i.e. T peak for the detailed mechanism is K where as it is K, 1792K and K for skeletal, reduced and ARM reduced mechanism respectively. Thus we can see that the results predicted lie within 10% error when compared with the detailed mechanism. Also the mass averaged temperature T mavg is K for the detailed. The T mavg values for the skeletal, reduced and ARM reduced mechanisms are 56