## Abstract

The challenge in the design of oxy-combustors for direct-fired supercritical CO2 (sCO2) cycles is in addressing disparate performance metrics and objectives. Key design parameters to consider include, among others, injector design for mixing and flame stability, split of recycled CO2 diluent between injectors and cooling films, target flame temperatures to control noncondensable products, and strategies to inject the diluent CO2 for film cooling and thermal control. In order to support novel oxy-combustor designs, a high-fidelity yet numerically efficient modeling framework based on the CRUNCH CFD® flow solver is presented, featuring key physics-based submodels relevant in this regime. For computational efficiency in modeling large kinetic sets, a flamelet/progress variable (FPV) based tabulated-chemistry approach is utilized featuring a three-stream extension to allow for the simulation of the CO2 film cooling stream in addition to the fuel and oxidizer streams. Finite rate chemistry effects are modeled in terms of multiple progress variables for the primary flame as well as for slower-evolving chemical species such as NOx and SOx contaminants. Real fluid effects are modeled using advanced equations of states. The predictive capabilities of this computationally tractable design support tool are demonstrated on a conceptual injector design for an oxy-combustor operating near 30 MPa. Simulations results provide quantitative feedback on the effectiveness of the film cooling as well as the level of contaminants (CO, NO, and N) in the exhaust due to impurities entering from the injectors. These results indicate that this framework would be a useful tool for refining and optimizing the oxy-combustor designs as well as risk mitigation analyses.

## 1 Introduction

The challenge in the design of oxy-combustors for direct-fired supercritical CO2 (sCO2) cycles [1] is in addressing disparate performance metrics and objectives [2]. Typically, oxy-combustors feature a modest temperature rise in addition to the high combustor chamber pressure and near stoichiometric fuel/oxidizer ratio. To achieve a relatively low exit temperature from the combustor (operating at ∼1150 °C or ∼1423 K), a substantial fraction of the CO2 in the exhaust is recycled and injected back into the combustor as a diluent to modulate the temperature rise and control the turbine inlet temperature and combustion product species [3]. Impurities in the fuel (natural gas) or oxidizer streams can enter the combustor to generate additional noncondensable products such as NOx and SOx and reduce process efficiency. Contaminants can also impact chemical kinetic processes. Key design parameters to consider include, among others, injector design for mixing and flame stability, split of recycled CO2 diluent between injectors and cooling films, target flame temperatures to control noncondensable products, and strategies to inject the diluent CO2 for film cooling and thermal control. In order to support novel oxy-combustor designs, a high-fidelity yet numerically efficient modeling framework based on the CRUNCH CFD® flow solver is presented, featuring key physics-based submodels relevant in this regime. Previous attempts to perform CFD modeling of sCO2 oxy-combustion designs include the work of Strakey [3] and Adbul-Sater et al. [4].

For computational efficiency, a flamelet/progress variable (FPV)-based tabulated-chemistry approach is utilized to (a) accurately model the flame structure based on chemical kinetic models with a large number of species and (b) capture turbulence–chemistry interactions. The FPV formulation has been upgraded to a three-stream framework to allow for the simulation of the CO2 film cooling stream in addition to the fuel and oxidizer streams; the third stream accounts for the CO2 dilution and thermal effects. With a multitime scale (MTS) approach, finite rate chemistry effects are modeled in terms of multiple progress variables for the primary flame as well as for slower-evolving chemical species such as NOx and SOx contaminants. Real fluid effects are modeled using advanced nonideal equations of states. Validation test cases for the three-stream FPV formulation and MTS approach are presented in the paper.

The predictive capabilities of this computationally tractable design support tool are demonstrated on a conceptual injector design for an oxy-combustor operating near 30 MPa. This conceptual design features multiple shear-coaxial injectors and operates with natural gas. Multiple simulations with increasing levels of fidelity are presented to identify the effects of various physics being modeled. Numerical results indicate the establishment of lifted flames. Finite rate chemistry effects and ignition delay play a key role in the flame liftoff for the mass flow rates modeled. Turbulence–chemistry interactions are shown to reduce the flame temperature but broaden the high temperature zone and increase the heat flux to the chamber wall. Simulations results provide quantitative feedback on the effectiveness of the film cooling as well as the level of contaminants (CO, NO, and N) in the exhaust due to impurities entering from the injectors.

This paper is structured as follows. First, the FPV modeling framework in CRUNCH CFD® and its MTS extension are summarized, showcasing the Sandia Flame D [5] as a validation case. Next, the three-stream extension to the MTS-FPV approach is presented, followed by a verification case. Based on the enhanced modeling approach, simulation results for a conceptual injector design for an oxy-combustor operating near 30 MPa are presented in conjunction with predictions of NOx. These results indicate that this framework would be a useful tool for refining and optimizing oxy-combustor designs.

## 2 Flamelet/Progress Variable Modeling Framework in CRUNCH CFD®

A brief overview of the CRUNCH CFD® code and the FPV tabulated chemistry approach is presented. CRUNCH CFD® is an unstructured/multi-element flow solver based on a cell-vertex method and consists of a number of multiphysics modules as summarized in Fig. 1.

Fig. 1
Fig. 1

To define a robust system for stiff real fluid systems, a preconditioning approach has been implemented [6] following the work of Merkle, Venkateswaran, and coworkers [7]. In addition to treating numerical stiffness, the preconditioning formulation also permits specification of different algebraic equation of state (EoS) models [8,9] as well as a modular, tabular lookup procedure [10]. The baseline procedure for thermodynamic properties and their derivatives are defined using a cubic Soave-Redlich-Kwong EoS and transport properties using corresponding state principle proposed by Ely and Hanley [8,9]. Two equation turbulence models based on k$ϵ,$ as well as k$ω$ SST, are available. Unsteady hybrid Reynolds-averaged Navier–Stokes (RANS)-large eddy simulation models [11], which have been extensively validated for real fluid flows, are also featured. Generalized kinetics models can be specified through a Cantera-based chemistry manager. The formulation allows both detailed species as well as an FPV tabulated chemistry approach. In-built utilities to seamlessly switch from the FPV formulation to a detailed species formulation are also operational for production calculations.

The FPV tabulated chemistry framework [1214] is integrated with real fluid EoS properties and is a function of only a small number of scalars (typically three) regardless of the complexity of the chemical system. Since the computational cost due to chemistry increases quadratically with the number of transport equations, the FPV framework offers dramatic savings (about one order of magnitude) for detailed kinetic models with a large number of species. Accordingly, the detailed chemical composition of the fluid is replaced by solving equations for mixture fraction, its variance and progress variable. In the context of a RANS approach, the governing equations solved are: (a) continuity, (b) momentum, (c) energy, (d) mean mixture fraction transport, (e) mixture fraction variance transport, and (f) chemical progress variable transport. The progress variable is defined by the sum of selected chemical species mass fractions (typically major products). The transport equations for the continuity and momentum are in the standard form and hence not reported here. The Favre-averaged RANS transport equations of the mean mixture fraction (Z), mixture fraction variance (Vz), and the chemical progress variable (Yp) are given by
$∂ρ¯⟨Z⟩∂t+∂ρ¯⟨uj⟩⟨Z⟩∂xj−∂∂xj(ρ¯(DT+D)∂⟨Z⟩∂xj )=0$
(1)
$∂ρ¯Vz∂t+∂ρ¯⟨uj⟩Vz∂xj−∂∂xj(ρ¯(DT+D)∂Vz∂xj )=2ρ¯DT∂⟨Z⟩∂xj∂⟨Z⟩∂xj−2ρ¯ϵVzk$
(2)
$∂ρ¯⟨Yp⟩∂t+∂ρ¯⟨uj⟩⟨Yp⟩∂xj−∂∂xj(ρ¯(DT+D)∂⟨Yp⟩∂xj )=SP˙¯$
(3)
where $ρ, uj, D$, and $DT$ are the density, velocity components, molecular diffusivity, and turbulent diffusivity, respectively. The species composition and the progress variable source terms are precomputed and tabulated from solving the flamelet equations in the FPV approach [13]. A table lookup procedure is given by
$ϕ={⟨Yk⟩,⟨T⟩, ωP˙¯}=Φ(⟨Z⟩, Vz,⟨Yp⟩)$
(4)
and is used to retrieve the species composition, $Yk$, progress variable source term, $ωYp˙$ and other thermodynamic state variables in terms of the solved transported scalars ($Z, Vz, Yp$). For the tabulation, the solution to flamelet equations is convolved over a probability distribution function (PDF). For any thermo-chemical quantity, $ϕ$, the mean value is computed as
$⟨ϕ⟩(⟨Z⟩, Vz,⟨Yp⟩)=∫01ϕ(Z, Yp)P(Z;⟨Z⟩, Vz)P(Yp)dZ dYp$
(5)

where $P(Z;⟨Z⟩,Vz)$ is a beta PDF function. The other PDF distribution $P(Yp)$ is taken as a delta function in this work. The table lookup approach employed in CRUNCH CFD® is illustrated in the schematic of Fig. 2.

Fig. 2
Fig. 2

As a validation of the FPV approach for nonpremixed/partially premixed combustion, a well-known flame configuration for which extensive validation data are available was selected, namely the Sandia Flame D configuration [5]. The configuration involves a partially premixed central fuel jet surrounded by a pilot jet with a coflowing ambient air. The fuel is a 25/75% methane/air mixture, and the pilot flame which burns a premixture of C2H2, H2, air, CO2, and NO2 with an equivalence ratio of $ϕ$ = 0.77, and nominally has the same equilibrium composition and enthalpy as a CH4/air mixture. The pilot flame product composition can be characterized by a mixture fraction of Z = 0.27. The Reynolds number for the flame D configuration is $ReD$ = 22,400 based on the fuel stream.

The central jet bulk velocity is 49.6 m/s and the pilot velocity are 11.4 m/s. At these velocities, the flame is attached to the burner and local extinction/blowout effects are minimal. The computational grid used for the study is an axisymmetric mesh of domain size 100D × 50D, where D is the diameter of the central jet. The inlet guide walls of length 8D for both the fuel and the pilot are resolved by the computational grid. A two-dimensional (2D) axisymmetric mesh is used for this problem with a minimum mesh resolution of close to 30 μm. A 12-step, 16-species reduced methane oxidation mechanism developed by Sung et al. [15] is used for the FPV table generation, and the progress variable is set as sum of H2O and CO2 mass fractions.

The results of the simulation are presented in Fig. 3. The contours of mean mixture fraction, temperature, progress variable and its source term shown in the figures are smooth and continuous. From the results, the flame seems to be attached to the burner walls as observed in the experiment. It must be noted that a nonzero progress variable source term is observed close to the fuel injection. This indicates that there is some progressive burning of the partially premixed fuel and the combustion is not completely mixing controlled in this region.

Fig. 3
Fig. 3

Based on the extensive experimental data available for the Sandia Flame D configuration, the experimental comparison of the simulation results along the central axis is reported in Fig. 4. From the line plot comparisons, a good agreement in results with the experimental data is seen for mixture fraction (Z), temperature (T), and for the selected species (O2 and OH). It is noted that the intermediate species OH, among others, is expected to be more sensitive to the selection of kinetic model used. In this case, a reduced chemistry is deployed, where the quasi-steady-state approximation [16] is introduced for selected species.

Fig. 4
Fig. 4

## 3 Multitime Scale Extensions to Flamelet/Progress Variable Framework

While the baseline FPV framework described above is accurate at modeling the evolution of the primary flame, it is unable to accurately model species that evolve over a much slower time scale, such as NOx or SOx contaminant levels, for instance, [17]. This is related to the fact that the production of contaminants can evolve through several different chemical pathways characterized by disparate time scales. For example, prompt NO production is relatively fast and has a similar time scale as the primary flame. However, NO continues to evolve on a much slower time scale and the predominant NO production is in the postflame zone. In Fig. 5, contour lines of mixture fraction variance are overlaid on top of the predicted NO distribution for same Sandia Flame D case [5] presented above. It is noted that the NO peaks downstream of the flame region and this postflame production of NO constitutes the major portion (more than 90%) of the overall NO emission. Furthermore, the NOx chemical source term for the postflame production is expected to be weakly dependent on turbulence–chemistry interactions unlike the primary flame region where turbulent fluctuations in mixture fraction are large. Thus, the key observation is that the predictions for contaminants such as NO cannot be accurately represented as a function of the primary flame progress variable [17].

Fig. 5
Fig. 5

In order to improve the solution accuracy of slower-evolving species, the solution of additional transport equations with appropriate source terms is required. These additional transport equations will solve for the species associated with the kinetics that are specific to the slower evolving contaminant chemistry mechanism and are not part of the primary flame chemistry set. The finite rate source terms to these additional transport equations are based on the reaction set and may include species from the primary flame set. The resulting framework is referred to as the MTS-FPV formulation, and a schematic of the approach is presented in Fig. 6. It is noted that while the discussion below refers to NOx for convenience, the approach is general and equally applicable to other slow evolving contaminants such as SOx.

Fig. 6
Fig. 6

In order to support the evaluation of the NOx chemistry within the MTS-FPV formulation, an enhanced implementation of the NOx chemical source term is required. Specifically, the net production rate of each NOx species (solved for as secondary progress variables) needs to be modeled as a function of the primary flame table lookup parameters (mixture fraction, Z, variance, Vz, and primary progress variable, Yp) as well as a function of pressure, P, temperature, T, and secondary progress variables themselves, e.g., YNO and YN. For example, as illustrated in Fig. 6, the chemical source terms for the secondary progress variables, i.e., NO and N, are dependent on N2, O, OH, and H (i.e., primary flame composition), in addition to the secondary progress variables themselves. These species concentrations need to be extracted and passed to the NOx source term evaluation.

The NOx chemical source term is implemented as a user defined function (UDF) for seamless integration with the MTS-FPV table lookup application programing interface (API) or using Cantera. The UDF can been generated using advanced automated reaction mechanism editing tools [16] to include the required NOx trace species reaction rate information. These automated editing tools are routinely used to perform the systematic reduction of reaction mechanisms and encapsulate all the details of the reduced kinetic model within a self-contained subroutine implementing the chemical source term.

The MTS-FPV framework has been demonstrated on the Sandia Flame D configuration using the simplified NOx chemistry kinetic set illustrated in Fig. 6. In Fig. 7, experimental data for NO concentrations are compared with numerical results from the MTS-FPV simulation along the central axial line and at a radial location of x/d = 3. From the line plot comparisons, a reasonable comparison is noted for the NO concentrations; the NO is overall underpredicted by the simulation, although it is of the same order of magnitude and qualitative trends are reasonably captured. The NO predictions compare very well with the experiments in the near field of the burner but start deviating further downstream. Noteworthy is that both MTS-FPV predictions and measured data indicate that NOx species are present in small amounts (of the order of 10−9 to 10−5). Since one of the assumptions of the MTS-FPV approach is that NOx chemistry is decoupled from the primary flame solution, the fact that NOx species are indeed trace species confirms that they do not significantly affect the transport/thermodynamic properties and the overall flowfield.

Fig. 7
Fig. 7

The MTS-FPV framework described above is applicable only to a “two-stream” configuration for fuel and oxidizer injection. However, sCO2 combustors feature a third stream where a substantial fraction of the CO2 in the exhaust is recycled and injected back into the combustor as a diluent to modulate the temperature rise and control the turbine inlet temperature.

## 4 Three-Stream Extension to Multitime Scale-Flamelet/Progress Variable Formulation

The extension of the MTS-FPV framework to a “three stream” configuration featuring diluent injection is discussed here. The diluent stream, i.e., the third stream, is modeled with a dedicated transport equation featuring no chemical source term. For this purpose, the third stream mixture fraction, ⟨Zd⟩ is tracked. This was accomplished by leveraging on the existing MTS-FPV framework that already features support for secondary progress variables. Specifically, the third stream was assigned to a secondary progress variable equation without chemical source term. The mass fraction composition of the third stream, ⟨Yd⟩, is defined for a given problem. For oxy-combustion applications, ⟨Yd⟩ is typically set to 100% CO2. The inflow temperature, ⟨Td⟩ is also defined.

A dedicated extension to the MTS-FPV API was added to support the implementation of the correction to the primary flame table lookup, $ϕ$, resulting in a corrected $ϕ3$
$ϕ3={⟨Yk,3⟩,⟨T3⟩,⟨ω˙p,3⟩}=Γ(⟨Z⟩,⟨Zd⟩,Vz,⟨Yp⟩)$
(6)

As discussed next, a mixing rule is typically applied for this correction. However, certain chemical source terms are inherently nonlinear and a special treatment is needed.

### 4.1 Diluted Chemical Species Composition.

With regard to chemical composition, the API call for the third stream introduces a correction to the primary flame species mass fraction array, ⟨Yk⟩. Specifically, ⟨Yk⟩ is scaled based on the relative level of dilution from the third stream represented by ⟨Zd
$⟨Yk,3⟩=⟨Yk⟩(1−⟨Zd⟩)+⟨Yd⟩⟨Zd⟩$
(7)

For oxy-combustion applications with 100% CO2 in the diluent stream, as a result of this correction, all chemical species mass fractions are reduced in magnitude with the exception of CO2, which now becomes the sum of the CO2 in the primary flame and of the CO2 in the diluent third stream. It is noted that CO2 may be present in the inflows for the fuel and/or oxidizer streams and that CO2 is generated as a result of the combustion processes modeled by the MTS-FPV table lookup. The thermal and chemical kinetic effects of the CO2 already present in the fuel and oxidizer streams are accurately modeled and captured during the generation of the lookup table. The inherent assumption is that CO2 dilution coming from other streams (e.g., third stream) uniformly affects the fuel and oxidizer streams and that the postcombustion products are mixed with and cooled by the CO2 diluent stream after combustion has taken place. To relax this restriction, further upgrades are planned as part of future efforts to more accurately capture the effects of nonuniformities in the CO2 mixing and the impact of cooling and finite rate chemical kinetics due to prompt CO2 dilution, i.e., very rapid mixing of CO2 that would directly affect the flame structure.

### 4.2 Diluted Progress Variable Source Terms.

Following the procedure for correction to the species composition from the diluent concentration, a similar correction is introduced for the primary flame progress variable source term, which is scaled based on the relative level of dilution from the third stream
$⟨ω˙p,3⟩=⟨ω˙p⟩(1−⟨Zd⟩)$
(8)

Here, there is no source term contribution coming from the nonreacting third stream. The limitations of this assumption are discussed in Sec. 4.4, and upgrades to relax this restriction will be undertaken as part of future work.

Unlike the primary progress variable source term, the chemical source terms for the secondary progress variables that model the slower-evolving NOx and SOx chemistries need a different correction. In order to ensure generality and portability of the secondary NOx/SOx chemistry in the MTS-FPV API, a self-contained chemistry manager is available based on the open-source freely redistributable Cantera modeling framework. An additional advantage of Cantera is that the kinetic model features supported encompass the vast majority of modern reaction mechanisms and include (i) arbitrary three-body collision efficiencies, (ii) reversible/irreversible and duplicate reactions in generalized Arrhenius form, (iii) arbitrary reaction orders for global reactions, (iv) pressure-dependent fall-off reactions with Lindemann, Troe (3- and 4-parameter) and Stanford Research Institute parameterization and beyond. Specifically, a minimal installation of the required Cantera libraries was extracted from a Cantera distribution so that a light-weight implementation of the secondary NOx/SOx chemistry can be attained. The API includes support for Cantera-based source term evaluations.

Under dilution, these chemical source terms are a function of (i) the three primary flame scalars (⟨Z⟩, Vz, and ⟨Yp⟩), (ii) the secondary progress variables ⟨YNOx,i⟩ (or ⟨YSOx,i⟩), and (iii) the diluent third stream mixture fraction ⟨Zd
$⟨ω˙NOx,i,3⟩=f(⟨P⟩,⟨T⟩,⟨Z⟩,Vz,⟨Yp⟩,⟨YNOx⟩,⟨Zd⟩)$
(9)
For $⟨ω˙NOx,i,3⟩$, the diluent correction to the primary species mass fractions is first applied before evaluating the source term
$⟨ω˙NOx,i,3⟩=f(⟨P⟩,⟨T⟩,⟨Yk,3⟩,⟨YNOx⟩)$
(10)

where $⟨Yk,3⟩$ was defined in Eq. (7).

### 4.3 Diluted Temperature and Derivatives.

While the tabulated temperature is available in the MTS-FPV table, this value is typically only retrieved for comparison against the temperature computed by the CRUNCH CFD® solver by solving the energy equation, which in its full form already accounts for the thermal effects of CO2 dilution. The tabulated value represents the primary flame temperature in the absence of third stream dilution. An estimation of the thermal cooling effect of dilution was implemented using a mixing rule, again, for comparison purposes only. It is noted that if an extension to the MTS-FPV approach that includes heat loss effects were to be used, an accurate estimation of the reference (i.e., tabulated) temperature would be required.

Given that the CRUNCH CFD® solver uses a preconditioned formulation, it becomes necessary to implement also the corrections to the derivatives of species mass fractions with respect to the MTS-FPV scalars. For this purpose, the chain rule is applied to $⟨Yk,3⟩$. A numerical derivation is applied to $⟨ω˙NOx,i,3⟩$.

### 4.4 Limitations of Current Three-Stream Approach.

The upgraded implementation of the MTS-FPV modeling framework in CRUNCH CFD® allows for a computationally efficient engineering-level representation of CO2 dilution. The key assumption currently is that the CO2 not already present in the fuel and oxidizer streams (e.g., from film cooling) mixes only downstream of the flame with the postcombustion products, i.e., after combustion has taken place. Therefore, a limitation of this approach is that thermal and kinetic effects due to prompt mixing of the CO2 not already present in the fuel and oxidizer streams are not captured. As part of future work, an extension to the tabulated chemistry parameterization is envisioned that introduces the extent of CO2 dilution as an additional parameter. As a result, the parameterized lookup table will be four-dimensional requiring a more sophisticated framework for data extraction.

## 5 Verification of Three-Stream Implementation

In this section, the CFD results are presented as a verification of the upgraded implementation of the three-stream multitime scale approach. The framework is applied to a gas-phase lean direct injection (LDI) geometry with injection of diluent. Specifically, the case selected for verification of the upgraded MTS-FPV modeling framework entails a modified 2D representation of a NASA-relevant swirl-venturi LDI (SV-LDI) design featuring a single injector element [18]. This combustor configuration was selected for its simplicity and for being amenable to modifications. The schematic of this simplified setup is shown in Fig. 8. A third stream representing film cooling is represented by the red arrow.

Fig. 8
Fig. 8

By producing a swirl-stabilized flame, significant fuel–air mixing occurs prior to ignition, resulting in partially premixed conditions. The combustor consists of a convergent-divergent injector section opening into a cylindrical combustion chamber with a diameter of 1 in. Case setup conditions are summarized in Table 1.

Table 1

Simulation conditions for 2D LDI configuration

Operating conditions
Operating pressure∼0.5 MPa
Equivalence ratio0.4
Inlet fuel temperature400 K
Inlet air temperature400 K
Fuel mass flow rate3.6 × 10−4  kg/s
Air mass flow rate1.54 × 10−2  kg/s
Operating conditions
Operating pressure∼0.5 MPa
Equivalence ratio0.4
Inlet fuel temperature400 K
Inlet air temperature400 K
Fuel mass flow rate3.6 × 10−4  kg/s
Air mass flow rate1.54 × 10−2  kg/s

Specifically, gaseous methane fuel and air are injected at 400 K and the chamber pressure is maintained at ∼0.5 MPa. The air flow comes out of a swirler in the actual rig and is therefore provided with a swirl component. The diluent is injected with a normal velocity of 2 m/s also at 400 K. As noted from experiments and numerical studies of similar configurations, the flame in the combustion is swirl-stabilized. In the actual rig, a fuel spray rather than gaseous fuel is injected and therefore the velocity of the gaseous fuel is adjusted to impose the same global equivalence ratio of ϕ = 0.4 as generally used in the experiments. A subsonic characteristic based outflow condition is imposed downstream of the combustion chamber. A steady-state kϵ based RANS formulation is used for the simulation.

A unique feature of the swirl stabilized combustors is the presence of a central recirculation bubble immediately downstream of the fuel injection system. The fuel and the oxidizer become partially premixed due to the mixing induced by the swirl and turbulence and the flame is anchored around the central vortex bubble. The verification case presented next was successfully run to steady-state convergence with the same large Courant-Friedrichs–Lewy number used in the original case that did not introduce dilution.

### 5.1 Verification Case.

To test the three-stream MTS-FPV model, the LDI configuration was modeled with air in the third stream. The advantage of using air is that a direct comparison is possible by assuming the air as inert and the air as reacting with the fuel stream. This case is referred to as case 2 here. Another case (case 1) was also run to exercise the MTS capabilities, but is not shown here due to space constrains. The comparison is accomplished by running:

• Case 2A: inert air diluent: three-stream problem with inert air as the third stream, and

• Case 2B: reacting air diluent: two stream problem with the oxidizer stream also used for the third stream of case A.

Results of this comparison are presented in Figs. 911. We note that the flowfields for cases 2A and 2B are expected to be similar if the air injected as a third stream from the top boundary does not participate in the combustion. Figure 9 shows the distribution of the mass fraction of air diluent in terms of the secondary progress variable (case 2A only). Figures 10 and 11 compare case 2A (bottom) against case 2B (top) in terms of O2 mass fraction and temperature distributions, respectively. It can be observed that distributions are overall qualitatively quite similar. However, for the reacting air diluent case (case 2B), less O2 remains compared to case 2A due the partial consumption of the O2 entrained in the primary combustion region (recirculation bubble), a fraction of which is coming from the air injected at the top boundary. Similarly, a slightly broader high temperature combustion region is attained for the reacting air diluent case (case 2B), as seen in Fig. 11. However, importantly we note that the flowfield (and in particular the temperature) at the top wall where the diluent air in introduced is very similar between the two-stream reacting air configuration and three-stream inert air configuration, providing confidence that the modifications for the three-stream MTS-FPV framework are operating correctly.

Fig. 9
Fig. 9
Fig. 10
Fig. 10
Fig. 11
Fig. 11

## 6 Simulations of Baseline Supercritical CO2 Oxy-Combustor Design

In this section, we apply the “three-stream” MTS-FPV framework described above toward simulations of a conceptual injector design for an oxy-combustor operating near 30 MPa and at a peak temperature of ∼1800 °C (or ∼2073 K). The goal is to demonstrate the ability to provide design support by characterizing the flame structure, flame stability, and emissions (excess O2, CO, unreacted hydrocarbons) of this preliminary baseline design under various conditions. A systematic study identifying the impact of various physics (e.g., ignition delay, turbulent chemistry interactions, etc.) was conducted.

### 6.1 Combustor Configuration Overview and Flow Conditions.

Gas Research Institute (GTI) has been developing a novel oxy-combustor concept for direct-fired supercritical CO2 power cycles. This combustor design is based on multi-element shear coaxial injectors with a chamber pressure of about 30 MPa. Each element includes a central fuel (natural gas) nozzle, surrounded by an annulus which directs flow of oxidant (O2/sCO2 mixture) around the fuel. The injector elements are housed in a faceplate with back-side cooling provided by the O2 oxidant. Additional sCO2 is injected at the base of the chamber wall to provide film cooling, while the balance is injected further downstream in the chamber.

The simulations have been performed for a 45 deg wedge with periodic conditions on the side boundaries. The computational grid had approximately 2 × 106 cells, which provided sufficient accuracy for steady-state performance calculations with wall function procedure at the walls. While the computational cost for the GTI combustor precluded a detailed grid resolution study, we have previously looked at grid resolution effects for the Sandia Flame D calculation (refer to Ref. [13]) as well as high pressure sCO2 flow in de-Laval nozzles with condensation [19]. The design objective was to select velocity ratios in the injector to ensure adequate mixing of fuel and oxidizer, while the bluff body of the fuel nozzle serves to stabilize the flame. The O2:CO2 ratio of the oxidizer in the injector was chosen to attain a flame temperature of ∼1800 °C (or ∼2073 K).

The kinetic mechanism that was used to generate the parameterized FPV table is based on work by Vasu and coworkers at UCF [20] and has been validated for methane combustion at high pressures. The details of the kinetic mechanism are given in Table 2; the reaction set has 25 species and 142 reactions. We note that, by employing an FPV table lookup procedure, the approximate reduction in computational cost relative to a finite rate simulation is an order of magnitude (∼10), which makes the FPV procedure very attractive as a high-fidelity design support tool. As the number of species in the kinetic mechanism increase, the savings in computational cost will be even higher. Real fluid models based on cubic equations of state and validated for liquid rocket combustors are used for the calculations presented here.

Table 2

Kinetic mechanism of Manikantachari et al. [20]

List of chemical species
1H214CH3O2
2H15CH3OH
3O216CH3O
4O17CH2OH
5H2O18CH2O
6OH19HCO
7H2O220C2H6
8HO221C2H5
9CO22C2H4
10CO223C2H3
11CH424Ar
12CH325N2
13CH3O2H
List of chemical species
1H214CH3O2
2H15CH3OH
3O216CH3O
4O17CH2OH
5H2O18CH2O
6OH19HCO
7H2O220C2H6
8HO221C2H5
9CO22C2H4
10CO223C2H3
11CH424Ar
12CH325N2
13CH3O2H

The natural gas composition from the NETL report [21] used for the fuel had to be modified since the kinetic mechanism used in the study did not account for all the hydrocarbons. The modified composition relative to the NETL pipeline composition is shown in Table 3. We note that the methane content is increased from 93.10% to 94.10% and a small amount of Argon is added to keep the heat content of the revised composition close to the original composition. We note that we do not expect the changes in the composition to significantly affect the solutions computed.

Table 3

Fuel stream composition

Chemical speciesNatural gas composition (by volume)Adjusted natural gas composition (by volume) for CFD modeling
CH493.10%94.14%
C2H63.20%3.20%
C3H80.70%0.00%
n-C4H100.40%0.00%
CO21.00%1.00%
N21.60%1.60%
Ar0.00%0.06%
Chemical speciesNatural gas composition (by volume)Adjusted natural gas composition (by volume) for CFD modeling
CH493.10%94.14%
C2H63.20%3.20%
C3H80.70%0.00%
n-C4H100.40%0.00%
CO21.00%1.00%
N21.60%1.60%
Ar0.00%0.06%

### 6.2 Baseline Simulation Results.

Simulations have been performed on the GTI combustor using the “three-stream” MTS-FPV framework where the FPV table was generated around 30 MPa and the prescribed inlet temperatures for the kinetic mechanism. To better understand the impact of different physics, a series of calculations were carried out where the physics models were systematically turned on (as input flags using the same FPV table) to better understand their relative effects. Three model variants were computed:

• Case 1: basic flamelet model; no progress variable, no turbulence–chemistry interactions. This model gives a mixing-controlled solution where the flame is attached.

• Case 2: flamelet model with progress variable, but no turbulence chemistry interactions. This model identifies the effect of ignition delay and finite rate kinetics on the flame structure relative to case 1.

• Case 3: complete FPV model with progress variable and turbulence chemistry interactions. The complete model will identify the effect of turbulence chemistry interactions relative to case 2.

The centerline temperature contours for cases 1–3 are shown in Fig. 12. For case 1, which is mixing controlled, we observe that the flame is attached to the injector post and vigorous combustion controlled by the mixing is taking place as expected. For case 2, where ignition delay and finite rate effects are accounted for, we observe that the flame lifts-off and is no longer attached to the post. The flame temperature is approximately the same as case 1, but the flame structure downstream is not as wide. Results for case 3, which is the complete model, show that the flame liftoff distance becomes slightly larger compared to case 2, but that the effect is not dramatic. However, the downstream structure of the flame is altered quite significantly. While the peak flame temperatures are lower, the combustion is more diffused and occurs over a wider region. Indeed, the exit temperature distributions for case 3 are closer to the mixing-controlled case 1 solution rather than case 2. As we shall discuss later below, this also results in increased heat flux to the chamber wall indicating the need for more CO2 for film cooling compared to the preliminary design here.

Fig. 12
Fig. 12

In Figs. 13 and 14, we plot the mixture fraction and oxygen mass fraction for cases 1–3. Both these plots are generally consistent with the temperature plot. Consistent with the decreased combustion in cases 2 and 3 relative to case 1, we observed that the unburned oxygen accumulates at the exit of the fuel injector since ignition is delayed till further downstream.

Fig. 13
Fig. 13
Fig. 14
Fig. 14

Isosurfaces of water concentrations colored by temperature are shown in Fig. 15 to highlight the three-dimensional structure of the flame. The effect of the ignition delay on the flame liftoff is clearly evident for cases 2 and 3. The broader, more diffuse flame front from turbulence chemistry interactions are striking for case 3 shown in Fig. 15. Despite the lower peak flame temperature, the broader flame zone results in the water isosurface contours having a higher temperature than case 2, where the flame does not spread as much.

Fig. 15
Fig. 15

The effectiveness of the diluent CO2 for film cooling as well as for reducing the exit temperature is evaluated here. Figure 16 shows mass fraction of diluent CO2 on the symmetry plane. The CO2 injected for film cooling appears to be (i) losing its effectiveness before the second injection port downstream where the CO2 is injected for mixing and (ii) reducing the exhaust temperature. With regard to the CO2 that is injected downstream, we note that it is very effective at cooling the wall, but the mixing with combustion products may need to be improved by changing the angle of injection. The temperature on the chamber wall is plotted in Fig. 17. As we expected, we note that the boundary layer film cooling is becoming less effective by the diluent injection port downstream with the wall temperature starting to rise quite sharply. Most interesting is the observation that the temperature rise is highest for case 3 despite the lower flame temperature and largest ignition delay. This clearly indicates that the turbulence effects result in a broader high temperature zone increasing the heat flux to the wall even though the peak flame temperatures are lower. The large mass of CO2 injected at the second port is very effective in cooling the chamber wall; however, the mixing with the core combustion flow could potentially be improved since this will lead to a stratified temperature field entering the turbine.

Fig. 16
Fig. 16
Fig. 17
Fig. 17

Finally, we plot the CO concentration for the case 3 flowfield in Fig. 18. We note peak mass fractions of 0.008 for CO, which is an order of magnitude higher than what an equilibrium calculation indicates. This may indicate that higher CO2 injector flow rates may be required to lower the level of CO production. However, for a given injector design, increasing CO2 flow rates increases injector pressure loss, which also decreases cycle efficiency. Therefore, any increase in CO2 flow rate through the injector would likely have to be coupled with larger element annuli and larger injector diameters. This example illustrates some of the complexity in optimizing the design of sCO2 combustors arising from many interconnecting factors.

Fig. 18
Fig. 18

## 7 Simulations of NOx Production in Oxy-Combustor Design

A MTS analysis is demonstrated on the sCO2 shear coaxial combustor described above. Here, a NOx chemistry subset is overlaid over the basic combusting flowfield to quantify the NOx noncondensable contaminant levels that are generated. The formulation is predicated on the assumption that the contaminant levels are very dilute and do not affect the flame structure. However, since the time scales for these reactions occur over much longer time scales than the primary flame, they need to be solved separately with a finite rate calculation for the specified kinetic subset.

The flowfield for the complete FPV model (case 3) with finite rate chemistry effects and turbulence–chemistry interactions is analyzed with the MTS formulation. Here, the simplified NOx chemistry kinetic subset shown in Table 4 from Ref. [18] is implemented in a finite rate integration procedure, where a transport equation for each species in this subset is solved for assuming the mean flowfield for velocity, density, temperature, and species concentrations from the FPV solution. Essentially, we obtain an overlaid solution for contaminants such as NOx (or SOx if the appropriate kinetic subset is solved) under the assumption that their concentrations are low enough not to alter the mean combusting flowfield or the primary flame structure.

Table 4

NOx kinetic model

ReactionCollision frequency (mol cm s K)Temperature exponentActivation energy (cal/mol)
N + NO ↔ N2 + O3.0 × 10120.300
N + O2 ↔ NO + O6.4 × 1091.003170
N + OH ↔ NO + H6.3 × 10110.500
N + N + M ↔ N2 + M2.8 × 1017−0.750
ReactionCollision frequency (mol cm s K)Temperature exponentActivation energy (cal/mol)
N + NO ↔ N2 + O3.0 × 10120.300
N + O2 ↔ NO + O6.4 × 1091.003170
N + OH ↔ NO + H6.3 × 10110.500
N + N + M ↔ N2 + M2.8 × 1017−0.750

The mass fractions for NO and N for the case 3 flowfield shown earlier in Figs. 1216 are given in Fig. 19. We note that the NO production occurs downstream of the flame as expected. However, the mass fraction of the NO (order of 10−10) and N (order of 10−15) are extremely low indicating very little NOx production. In addition to the uncertainty in the chemical kinetics models deployed (as discussed next), the low levels of available N2 present in the fuel stream (1.6% mass) and the dilution effects of the CO2 contribute to the limited production of NOx species.

Fig. 19
Fig. 19

It is unclear if the levels of the NOx production are being affected by the selected baseline chemistry or by the simplified NOx kinetic set that is being used to compute their levels. In future efforts, a more comprehensive study will be undertaken where the sensitivity to more detailed kinetic sets both for the NOx production as well as the baseline flame chemistry will be evaluated to ensure that the levels being predicted are accurate. Most importantly, a NOx chemistry set valid for the extreme sCO2 oxy-combustion conditions is necessary, both in terms of pressure range and extent of CO2 dilution. This set will need to be compatible with the appropriate baseline chemistry for methane or natural gas.

## 8 Conclusions

The development of a high-fidelity computational framework within the CRUNCH CFD code has been presented. To reduce the computational cost of modeling large kinetic mechanisms, an MTS-FPV formulation was employed where the chemical kinetics is represented in a parametrized form allowing the local species composition to be obtained from a table lookup approach. For contaminants such as NOx or SOx, which evolve over a much slower time scale, an overlaid finite rate kinetic model has been developed that is applicable when the contaminants are dilute.

The MTS-FPV API was upgraded to support the introduction of a third stream that is nonreacting and that represents the CO2 diluent stream in oxy-combustion applications. The formulation requires the solution of an additional transport equation for a progress variable that represents the concentration of the diluent. Modifications to the thermodynamics decode from the FPV table are described. The upgraded MTS-FPV modeling framework was successfully evaluated using a 2D LDI configuration to showcase the implementation of the three-stream multitime scale approach. Simulations were performed by introducing a third stream from the top boundary similar to the diluent injection stream in sCO2 combustors. The three-stream MTS-FPV framework was demonstrated for a case where the diluent stream is air. Simulations for this case performed robustly and provided physical results without significant alteration of the flame structure.

Simulations of a preliminary sCO2 shear coaxial combustor designed by GTI have been demonstrated with the three-stream MTS-FPV framework developed in this effort. The parameterized FPV table lookup procedure provides significant savings in computational cost while retaining adequate fidelity to provide valuable design support. The simulations presented here indicate that finite rate ignition delay effects results in a lifted flame indicating that injector postdesign can be improved for better flame holding. Turbulent chemistry effects only marginally increase the flame liftoff height. However, turbulent chemistry effects have a substantial effect on the flame structure downstream; while the peak flame temperatures are lower, the flame zone is much broader and combustion more complete. The turbulence effects also increase the heat flux to the chamber wall. Analysis of the diluent CO2 injected both for film cooling and further downstream to condition the exit temperature indicated that the mass flow for the film cooling may need to be increased to prevent temperature rise. The design for the CO2 injection downstream may also need to be altered if increased mixing with the core combustor flow is desired.

A MTS analysis has been demonstrated on the sCO2 shear coaxial combustor. Here, a NOx chemistry subset is overlaid over the basic combusting flowfield to quantify the NOx noncondensable contaminant levels that are generated. Results with a simplified NOx kinetic mechanism indicate very low levels of NOx production in the sCO2 combustors. Sensitivity studies with more complex kinetic models, both for the primary flame chemistry as well as for the NOx chemistry are envisioned as part of future work.

## Acknowledgment

This work was primarily supported by the U.S. Department of Energy (DOE) through an STTR Phase I program (Contract No. DE-SC0019640) with Dr. Bhima Sastri as Program Manager. The authors would also like to acknowledge the helpful technical discussions with Prof. Subith Vasu and his students at the University of Central Florida and are very grateful for their support. This research used resources of the National Energy Research Scientific Computing Center; a DOE Office of Science User Facility supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231.

This paper was prepared as an account of work sponsored by an agency of the U.S. Government. Neither the U.S. Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the U.S. Government or any agency thereof. The views of and opinions of authors expressed herein do not necessarily state or reflect those of the U.S. Government or any agency thereof.

## Funding Data

• U.S. Department of Energy, Small Business Innovative Research and Small Business Technology Transfer (Grant No. DE-SC0019640; Funder ID: 10.13039/100007002).

## Nomenclature

• API =

application programing interface

•
• FPV =

flamelet progress variable

•
• MTS =

multitime scale

•
• NOx =

nitrogen oxides

•
• PDF =

probability density function

•
• RANS =

Reynolds-averaged Navier–Stokes

•
• sCO2 =

supercritical CO2

•
• SOx =

sulfur oxides

•
• UDF =

user defined function

•
• Vz =

mixture fraction variance

•
• Yk =

kth species mass fraction

•
• Yp =

primary progress variable

•
• Z =

mixture fraction

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