# Numerical Modeling of Transcritical and Supercritical Fuel Injections Using a Multi-Component Two-Phase Flow Model

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

^{®}. To this end, the Peng-Robinson (PR) cubic equation of state (EOS) is considered and the solver is modified to account for the real-fluid thermodynamics. At high pressure conditions, the variable transport properties such as dynamic viscosity and thermal conductivity are accurately computed using the Chung transport model. To deal with the multicomponent species mixing, molar averaged homogeneous classical mixing rules are used. For the velocity-pressure coupling, a PIMPLE based compressible algorithm is employed. For both cryogenic and non-cryogenic fuel injections, qualitative and quantitative analyses are performed, and the results show significant effects of the chamber pressure on the mixing processes and entrainment rates. The capability of the proposed numerical model to handle multicomponent species mixing with real-fluid thermophysical properties is demonstrated, in both supercritical and transcritical regimes.

## 1. Introduction

_{pb}[3]. At subcritical conditions, the initial liquid jet, exhibiting surface tension, is broken-up through successive steps into liquid ligaments and fine atomized droplets, which then undergo evaporation and subsequent ignition. The jet disintegration behavior occurring under transcritical and supercritical conditions deviates from the subcritical conditions where the formation of liquid ligaments and droplets are replaced by turbulent gas jet mixing processes with widening of the diffusion layer of the liquid-gas interface [4,5].

## 2. Numerical Methodology

#### 2.1. Governing Equations and Solution Algorithm

#### 2.2. Real-Fluid EOS

**x**is the species mole fraction vector, and R is the universal gas constant. Coefficients a and b are model constants dependent on mixture composition, which account for the effects of intermolecular attractive forces and volume displacement, respectively. The parameter α is a function of both reduced temperature T

_{r}= T/T

_{c}and acentric factor ω, where the subscript ‘c’ indicates the critical condition.

#### 2.3. Real-Fluid Chung Transport Model

^{3}/mol, ${\rho}_{m}$ is the mixture density and ${\mathsf{\Omega}}^{*}$ is an empirical correlation for the reduced collision integral. The constants ${A}_{i}$ are linear functions of the acentric factor ${\omega}_{c,m}$, the reduced dipole moment ${\mu}_{r,m}$, and the association factor ${\kappa}_{m}$. The factor ${F}_{c}$ represents the empirical model function of the acentric factor, and it accounts for the shape and polarity of the molecules. A detailed discussion of the remaining coefficients is found in Chung et al. [47]. Similar molar weighted averages based mixing rules as aforementioned for the equation of state are applied to compute the mixture equivalent parameters, denoted with the subscript ‘m’.

#### 2.4. Model Implementation Remarks and Property Verifications

## 3. Results and Discussion

_{2}) and n-dodecane injected into a quiescent gaseous nitrogen (G-N

_{2}) environment are examined, under transcritical and supercritical conditions, for various chamber pressures and injection temperatures. For an engine, a higher efficiency is obtained at higher chamber pressures. This trend has motivated many researchers to investigate higher operating pressure conditions of the combustion chamber, as the thermal efficiency and power output in rockets, gas turbines or diesel engines is shown to increase. The injected fluid at sub/trans critical temperature into supercritical condition of the chamber typically heats up beyond its critical temperature as it mixes with the surrounding hot gas before it burns inside the combustion chamber and this process is referred to as “transcritical”. The liquid rocket engines operate at conditions above the thermodynamic critical point of the injected propellants. For example, the combustion chamber pressure with the L-H

_{2}/L-O

_{2}in Vulcain engine in the Ariane-5 can reach up to 11.5 MPa, Space Shuttle main engine was able to reach up to a maximum of 22.3 MPa [9] or more recent Merlin engines operate at 9.7 MPa.

_{2}) and n-dodecane (C

_{12}H

_{26}) used as working fluids in the present numerical study are tabulated in Table 1. In this numerical study, the injection of a single-component mixture, L-N

_{2}/G-N

_{2}and a binary (multicomponent) mixture, C

_{12}H

_{26}/N

_{2}are considered. In the case of cryogenic nitrogen into nitrogen, two species with identical properties are considered in the simulation, in order to track the injected species separately from the ambient one.

#### 3.1. Validation on Cryogenic Nitrogen Jet Data

_{2}) is not used as a propellant, but due to its easy handling in experimental measurements, it is widely used as a working fluid to investigate the turbulent jet mixing characteristics of the cryogenic liquid propellants along with a high-pressure gaseous nitrogen (G-N

_{2}) as chamber quiescent environment [8,37,55]. In this study, simulation characteristics similar to Mayer’s experimental near-critical “case-3” and supercritical “case-4” are adopted where the cold cryogenic L-N

_{2}jet is ejected from a circular nozzle of diameter d = 2.2 mm with constant injection temperature and uniform velocity into a chamber with supercritical conditions filled with a quiescent G-N

_{2}at pressure p

_{ch}= 4 MPa and temperature T

_{ch}= 298 K, respectively. The summary of these working parameters is tabulated in Table 2. Underlined conditions are referred to as the baseline conditions for the parametric study.

_{2}injected at transcritical condition of T

_{inj}= 126.9 K (${\rho}_{inj}$ = 439.8 kg/m

^{3}) and u

_{inj}= 4.9 m/s into G-N

_{2}chamber at p

_{ch}= 4 MPa and T

_{ch}= 298 K, compared with the experimental data (measured data extracted, ${\rho}_{inj}$ = 425.5 kg/m

^{3}) of Mayer et al. [55]. The result shows the presence of an intact core region in the initial jet up to the axial length x/d ≤ 8.5. The numerical density ratio is nearly uniform and slightly over-predicted with respect to the experimental data. This can depend not only on the PR EOS inaccuracies, but also on the experimental uncertainties as documented in the original reference [55]. Moving along the jet axis, when mechanical instabilities reach the jet centerline, the dense core disappears, and the centerline density profile starts to decline. However, in the transition to fully developed turbulent jet flow region at x/d > 8.5, the numerical density profile shows a trend similar to Mayer et al. experiments [8,55] for the test case-3. Figure 4b indicates the centerline non-dimensional time-averaged density profiles along the axial distance x/d for the validation case-4, i.e., L-N

_{2}injected at supercritical condition of T

_{inj}= 131 K and u

_{inj}= 5.4 m/s into G-N

_{2}chamber at p

_{ch}= 4 MPa and T

_{ch}= 298 K, compared with the experimental data of Mayer et al. [55]. The result shows again that the trend is similar to experimental data, but with slight underestimation in the non-dimensional centerline density for x/d > 10. In this supercritical case-4, however, experimental density data drops after a short distance from the nozzle (from x/d > 2) and the simulation correspondingly predicts a short intact core length up to about x/d = 5, less than the transcritical case-3.

_{2}conditions at different injection temperatures and chamber pressure conditions.

#### 3.2. Simulations of Cryogenic L-N_{2} Injections into G-N_{2}

_{2}injection under transcritical and supercritical conditions. The entire set of injection temperatures and ambient pressures tabulated in Table 2 is considered. As before, the rectangular domain is discretized using a uniform grid size with Δx = Δy = d/64 = 0.034375 mm, resulting into a total number of 5.12 million cells. The flow dependent time step Δt is such that a maximum convective CFL equal to 0.2 is maintained. The chamber is set at room temperature T

_{ch}= 298 K. At the jet inlet, a uniform flat velocity profile with u

_{inj}= 4.9 m/s is imposed.

_{inj}= 126.9 K, corresponding to the N

_{2}critical temperature, T

_{inj}= 128.5 K, slightly below the pseudo-boiling value at the 4 MPa, and lastly T

_{inj}= 131 K which is at supercritical.

_{ch}, from 4 to 6 MPa, both above the nitrogen critical pressure, at fixed injected fluid temperature T

_{inj}= 126.9 K. These parametric studies are presented in the following subsections.

#### 3.2.1. Effect of Injection Temperature

_{inj}, of 126.9 K (column-1), 128.5 K (column-2), and 131 K (column-3), respectively. The time shown is t = 60 ms, corresponding to about 2.5 flow-through times (FTT) of the entire domain length. These qualitatively results indicate that an increase of the injection temperature from trans to supercritical conditions leads to improved mixing processes. The velocity gradients at the jet surface generate vortex rollups and which finally break up into ligament shaped structures, similar to other numerical studies [13]. In the supercritical case (at T

_{inj}= 131 K), no small structures are formed, while at the lower temperature (T

_{inj}= 126.9 K) tiny structures are visible. Kelvin-Helmholtz (K-H) instabilities can be observed in a shear layer, but with different scales. As the injected fluid temperature increases, the jet is more quickly heated up to a gas-like supercritical state and mixing appears more diffusive in nature.

_{2}injection are considered through the centerline non-dimensional profiles of density, temperature, velocity and through the calculation of the normalized mass flow rate across the transverse direction at varying axial distance. Results are shown in Figure 6a–d, respectively. In the initial laminar jets, up to axial distance x/d ≤ 5, any noticeable effect is observed on the density, velocity, and temperature ratios that follows the variation of injected fluid temperature. Similar behavior is even kept for further axial distance up to x/d = 7 for the two lowest injection temperatures, which are, to recall, lower than the pseudo-boiling temperature. The transition region between laminar to turbulent situated at the axial distance x/d ≤ 15 shows slight deviations at various injections temperature with axial distance. Along the axial length of the chamber, as the centerline jet temperature increases the density ratio decreases, so does the corresponding velocity profile with slight fluctuations.

_{2}and the G-N

_{2}environment.

_{inj}= 131 K, which is higher than the nitrogen pseudo-boiling temperature, so fully supercritical. Furthermore, the velocity profiles do not show any significant variation in the jet core width, but it slightly reduces at the jet center for the higher degrees of supercriticality.

#### 3.2.2. Effects of Chamber Pressure

_{ch}levels, 4 MPa and 6 MPa, are examined, while the remaining simulation parameters are maintained constant, as specified in Table 2. Figure 8 shows the instantaneous contour plots of temperature, density, pressure, viscosity, specific heat at constant pressure and velocity obtained from the aforementioned simulations, at time t = 60 ms, corresponding, as before, to 2.5 flow-through times (FTT) of the domain length. These results show that the increase in the chamber pressure from 4 MPa to 6 MPa leads to the transition of jet behaviors from the formation of liquid-like structure to gas-like phenomena, even though the injected cryogenic nitrogen temperature is transcritical, as 126.9 K is below the corresponding PB values. For example, the density field is clearly more diffuse at high pressure than at low pressure. More quantitative analyses are needed to assess the differences, and this is addressed in the following.

#### 3.2.3. Discussion

^{3}at 126.9 K, to 247 kg/m

^{3}at 131 K. Despite this large density change in the injected fluid, at constant injected mass flow rate (obtained through normalization), mixing does not change significantly, meaning that the injected fluid properties have minor effects in supercritical conditions.

#### 3.3. Simulations of N-Dodecane Injection into Hot Nitrogen Environment

_{2}environment, similarly to previous numerical studies [13,16]. The parameters used in the study of n-dodecane jets are tabulated in Table 3. As in the previous cryogenic application study, the effects of varying the inlet temperature and chamber pressure are discussed.

_{ch}= 11.1 MPa, and chamber temperature, T

_{ch}= 972.9 K, and with uniform injection velocity, u

_{inj}= 200 m/s.

_{ch}values are selected: 6 MPa (corresponding to the standard ECN Spray-A level), 11.1 MPa (high) and 30 MPa (ultra-high). Other operating parameters are kept constant, like injection temperature, T

_{inj}= 658.2 K, inlet velocity, u

_{inj}= 200 m/s and chamber temperature, T

_{ch}= 972.9 K. As for the previous section, both qualitative and quantitative analysis of the effects of different simulations conditions for diesel engines operating at high-load conditions are given hereafter.

#### 3.3.1. Effect of Injection Temperature

#### 3.3.2. Effect of Chamber Pressure

_{ch}of 6 MPa, 11.1 MPa, and 30 MPa, with constant chamber temperature, T

_{ch}= 972.9 K, and using fixed n-dodecane inlet temperature, T

_{inj}= 658.2 K. Figure 14 shows the comparison for the instantaneous contours of various flow variables for these cases. Results indicate that injecting n-dodecane fuel into a supercritical chamber leads to the formation of liquid-like jet structures of varying sizes. At low chamber pressure small scales are visible as in a pure atomization regime, while at high chamber pressure a transition towards gas-like phenomena is observed with larger scales and increased diffusive transport. In addition, an increased jet transverse spreading is also visible when moving from low to high chamber pressures.

#### 3.3.3. Discussion

^{3}to 412.2 kg/m

^{3}), explored through temperature changes (from 600.0 K to 736.8 K, respectively, at 11.1 MPa), causes minor effects on the injected mass based normalized mixing rate.

## 4. Conclusions

_{2}and n-dodecane into a chamber filled with quiescent G-N

_{2}environment. Qualitative analysis is performed for instantaneous solution fields, whereas quantitative aspects are discussed by examining non-dimensional averaged quantities, such as density, temperature, and velocity along axial or radial profiles, and by comparing normalized transverse mass flow rates along the axial distance. As a main finding of the study, for both cryogenic and diesel-like fuel injections, different chamber pressure conditions are found to affect the supercritical mixing processes more significantly than the injection temperatures. The present numerical study provides an insight of the jet behavior and the mixing processes at both trans- and super-critical conditions. Furthermore, the present numerical models demonstrated the capability to handle multicomponent species turbulent mixing processes with real-fluid thermophysical properties.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Pressure-temperature plots of density and specific heat at constant pressure, for pure nitrogen (

**left**), nitrogen/n-dodecane mixture—50% by volume (

**middle**), pure n-dodecane (

**right**), calculated with Peng-Robinson equation of state (PR-EOS). Red stars identify the pure and pseudo-fluid critical points. Yellow star identifies the real mixture critical point, evaluated by means of the NIST database [54].

**Figure 2.**Verification of CFD results (scatter points) against NIST data (line) vs. temperature for specific heat at constant pressure, density, and dynamic viscosity of (

**a**) nitrogen at 3, 4, and 5 MPa, and (

**b**) n-dodecane at 6, 11.1, and 30 MPa.

**Figure 4.**Centerline time-averaged non-dimensional density ratio profile along the axial distance obtained from the present numerical study of injection of cryogenic L-N

_{2}into supercritical chamber at p

_{ch}= 4 MPa and T

_{ch}= 298 K (

**a**) Case-3: at T

_{inj}= 126.9 K, u

_{inj}= 4.9 m/s and (

**b**) Case-4: at T

_{inj}= 131 K, u

_{inj}= 5.4 m/s with uniform grid resolution per jet diameter, d/64 compared with experimental test case-3 and case-4 of Mayer et al. [55].

**Figure 5.**Instantaneous contour plots obtained from simulations of cryogenic L-N

_{2}injected at 4.9 m/s into G-N

_{2}at supercritical chamber conditions, 4 MPa and 298 K, for various injection temperatures: first column 126.9 K, second column 128.5 K, third column 131 K. Rows, from top to bottom, show instantaneous fields of temperature T [K], density $\rho \text{}[\mathrm{kg}/{\mathrm{m}}^{3}$], pressure p [Pa], dynamic viscosity $\mu \text{}[\mathrm{Pa}\xb7\mathrm{s}$], specific heat at constant pressure c

_{p}[J/(kg K)], and velocity u [m/s].

**Figure 6.**Centerline axial profiles of non-dimensional density, temperature, velocity and normalized transverse mass flow rate for different injection temperatures: 126.9 K, 128.5 K, and 131 K. Simulations of cryogenic L-N

_{2}injected at 4.9 m/s into G-N

_{2}at 4 MPa and 298 K.

**Figure 7.**Radial profiles of non-dimensional density, temperature, and velocity ratios at fixed axial distance x/d = 20, for different injection temperatures: 126.9 K, 128.5 K, and 131 K. Simulations of cryogenic L-N

_{2}injected at 4.9 m/s into G-N

_{2}at 4 MPa and 298 K.

**Figure 8.**Instantaneous contour plots obtained from simulations of cryogenic L-N

_{2}injected at 126.9 K and 4.9 m/s into G-N

_{2}at 298 K, for two different chamber pressures: first column 4 MPa, second column 6 MPa. Rows, from top to bottom, show instantaneous fields of temperature T [K], density $\rho \text{}[\mathrm{kg}/{\mathrm{m}}^{3}$], pressure p [Pa], dynamic viscosity $\mu \text{}[\mathrm{Pa}\xb7\mathrm{s}$], specific heat at constant pressure c

_{p}[J/(kg K)] and velocity u [m/s].

**Figure 9.**Centerline axial profiles of non-dimensional density, temperature, velocity, and normalized transverse mass flow rate for different chamber pressures: 4 MPa and 6 MPa. Simulations of cryogenic L-N

_{2}injected at 126.9 K and 4.9 m/s into G-N

_{2}at 298 K.

**Figure 10.**Radial profiles of non-dimensional density, temperature, and velocity at fixed axial distance x/d = 20, for different chamber pressures: 4 MPa and 6 MPa. Simulations of cryogenic L-N

_{2}injected at 126.9 K and 4.9 m/s into G-N

_{2}at 298 K.

**Figure 11.**Instantaneous contour plots obtained from simulations of n-dodecane injected at 200 m/s into a supercritical hot nitrogen, 972.9 K and 11.1 MPa, for various injection temperatures: first column 600 K, second column 658.2 K, third column 736.9 K. Rows, from top to bottom, show instantaneous fields of temperature T [K], density $\rho \text{}[\mathrm{kg}/{\mathrm{m}}^{3}$], pressure p [Pa], dynamic viscosity $\mu \text{}[\mathrm{Pa}\xb7\mathrm{s}$], specific heat at constant pressure c

_{p}[J/(kg K)], mole fraction of nitrogen x

_{N}

_{2}[-], and velocity u [m/s].

**Figure 12.**Centerline axial profiles of non-dimensional density, temperature, velocity, and normalized transverse mass flow rate for different n-dodecane injection temperatures: 600 K, 658.2 K and 736.8 K. Simulations of n-dodecane injected at 200 m/s into a supercritical hot nitrogen, 972.9 K and 11.1 MPa.

**Figure 13.**Radial profiles of non-dimensional density, temperature, and velocity ratios at fixed axial distance x/d = 20, for different n-dodecane injection temperatures: 600 K, 658.2 K, and 736.8 K. Simulations of n-dodecane injected at 200 m/s into a supercritical hot nitrogen, 972.9 K and 11.1 MPa.

**Figure 14.**Instantaneous contour plots obtained from simulations of n-dodecane injected at 658.2 K and 200 m/s into nitrogen at 972.9 K, for three different chamber pressures: first column 6 MPa, second column 11.1 MPa, third column 30 MPa. Rows, from top to bottom, show temperature T [K], density $\rho \text{}[\mathrm{kg}/{\mathrm{m}}^{3}$], pressure p [Pa], dynamic viscosity $\mu \text{}[\mathrm{Pa}\xb7\mathrm{s}$], specific heat at constant pressure c

_{p}[J/(kg K)], mole fraction of nitrogen x

_{N}

_{2}[-], and velocity u [m/s].

**Figure 15.**Centerline axial profiles of non-dimensional density, temperature, velocity and normalized transverse mass flow rate for different chamber pressures: 6 MPa, 11.1 MPa, and 30 MPa. Simulations of n-dodecane injected at 658.2 K and 200 m/s into nitrogen at 972.9 K.

**Figure 16.**Radial profiles of non-dimensional density, temperature and velocity at fixed axial distance x/d = 20, for different chamber pressures: 6 MPa, 11.1 MPa, and 30 MPa. Simulations of n-dodecane injected at 658.2 K and 200 m/s into nitrogen at 972.9 K.

**Table 1.**Critical condition parameters for pure species [46].

Species | T_{c} [K] | p_{c} [MPa] | v_{c} [cm^{3}/mol] | Acentric Factor, ω |
---|---|---|---|---|

N_{2} | 126.192 | 3.3958 | 89.41 | 0.0377 |

C_{12}H_{26} | 658.2 | 1.82 | 22.064 | 0.5764 |

**Table 2.**Working parameters of the cryogenic nitrogen injection case, from Mayer et al. [55].

Input Parameters | Values |
---|---|

Injected Fluid | cryogenic nitrogen, N_{2} |

Ambient chamber gas | nitrogen, N_{2} |

Computational domain, L × H | 54 d × 27 d (~120 mm × 60 mm) |

Total cell count (uniform grid) | 5.12 × 10^{6} cells |

Computational cells per jet diameter | 64 |

Injection nozzle diameter, d | 2.2 mm |

Inlet velocity (uniform profile), u_{inj} | 4.9 m/s and 5.4 m/s |

Chamber temperature, T_{ch} | 298 K |

Chamber pressure, p_{ch} | 4 MPa and 6 MPa |

Injection temperature, T_{inj} | 126.9 K,128.5 K and 131 K |

Input Parameters | Values |
---|---|

Injected Fluid | n-dodecane, C_{12}H_{26} |

Ambient chamber gas | nitrogen, N_{2} |

Computational domain, L × H | 55 d × 28 d (~5 mm × 2.5 mm) |

Total cell count (uniform grid) | 5.12 × 10^{6} |

Computational cells per jet diameter | 64 |

Injection nozzle diameter, d | 90 μm (ECN Spray-A) |

Inlet velocity, u_{inj} | 200 m/s (uniform profile) |

Chamber temperature, T_{ch} | 972.9 K |

Chamber pressure, p_{ch} | 6 MPa, 11.1 MPa and 30 MPa |

Injection temperature, T_{inj} | 600 K, 658.2 K and 736.9 K |

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**MDPI and ACS Style**

Ningegowda, B.M.; Rahantamialisoa, F.N.Z.; Pandal, A.; Jasak, H.; Im, H.G.; Battistoni, M.
Numerical Modeling of Transcritical and Supercritical Fuel Injections Using a Multi-Component Two-Phase Flow Model. *Energies* **2020**, *13*, 5676.
https://doi.org/10.3390/en13215676

**AMA Style**

Ningegowda BM, Rahantamialisoa FNZ, Pandal A, Jasak H, Im HG, Battistoni M.
Numerical Modeling of Transcritical and Supercritical Fuel Injections Using a Multi-Component Two-Phase Flow Model. *Energies*. 2020; 13(21):5676.
https://doi.org/10.3390/en13215676

**Chicago/Turabian Style**

Ningegowda, Bittagowdanahalli Manjegowda, Faniry Nadia Zazaravaka Rahantamialisoa, Adrian Pandal, Hrvoje Jasak, Hong Geun Im, and Michele Battistoni.
2020. "Numerical Modeling of Transcritical and Supercritical Fuel Injections Using a Multi-Component Two-Phase Flow Model" *Energies* 13, no. 21: 5676.
https://doi.org/10.3390/en13215676