Open Access Article

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Adam
Obrusník
*^{a},
Jiří
Dědina
^{b} and
Pavel
Dvořák
^{a}
^{a}Department of Physical Electronics at the Faculty of Science, Masaryk University, Kotlářská 2, CZ-61137 Brno, Czech Republic. E-mail: adam.obrusnik@gmail.com
^{b}Institute of Analytical Chemistry of the Czech Academy of Sciences, Veveří 97, 602 00 Brno, Czech Republic

Received
12th March 2020
, Accepted 29th May 2020

First published on 1st June 2020

This work presents a computational model of gas flow and hydrogen combustion in a diffusion flame. We validate the model using a simple miniature diffusion flame hydride atomizer, in which the H radical distribution was both simulated and determined by two-photon absorption laser-induced fluorescence. Good agreement of the simulations with the experiments was found. Since the performance of a hydride atomizer can be linked directly to the H radical distribution in the atomizer the presented model is a reliable predictive tool for atomizer optimization and for design of new atomizer geometries. The model is based on the open source laminarSMOKE framework and is itself open-source; it can be therefore leveraged by the scientific community for further theoretical studies on various hydride atomizers.

The aim of this was to work was to prove that a predictive computational model of the processes in a MDF can facilitate much easier optimization of the atomizer. To make this effort easier for the community, we do not only describe the physics and chemistry model of the atomizer but we also provide executable model files under an open-source academic license, so that they can be further utilized by the scientific community. Since the model is rather general it is also applicable to the optimization of the design and operating parameters of other hydride atomizers^{4} employed for AF and AA spectrometers, such as flames in gas shield atomizers, conventional externally heated quartz tube atomizers or multiatomizers.

Furthermore, we assess the sensitivity of the model results to uncertain experimental parameters (laboratory conditions, tube diameter and air contamination). This analysis provides insight into the uncertainty of both the model and the experiment and reveals factors to which the distribution of H radicals is most sensitive.

Finally, this work confirms that the GRI-Mech 3.0 reaction system,^{13} which was originally developed for natural gas combustion, can also be used for reliable predictive modeling of the combustion kinetics in hydrogen non-premixed flames. As such, this publication extends the list of studies that have experimentally validated the GRI-Mech 3.0 reaction system for various types of flames – see ref. 13 and references therein.

Numerical simulations similar to this work have been previously presented in the literature,^{14} where the fundamentals of hydrogen-air coflow flames were studied. Compared to ref. 14, this work offers model validation which is immediately relevant for analytical chemistry applications and, above all, it provides the model itself as a scientific instrument. To our knowledge, this is the first openly available numerical model of diffusion co-flow flames.

The model was not made from scratch but rather implemented using the open-source computational framework called laminarSMOKE+.^{15} Unlike the freely available reactingFoam solver and the commercially available ANSYS Fluent solvers,^{16} which have been designed for the more common case of turbulent combustion, laminarSMOKE+ is much more suited for building models of diffusion flames because it uses binary coefficients and mixture averaging for calculating the diffusion coefficients.

Firstly, the model solves the continuity equation for the mass of the gaseous mixture in the form

The momentum equation for the mixture velocity then takes the form:

Again, since the equation above is formulated for the entire gas mixture, the stress tensor T depends on the local gas composition and mixture temperature. The vector g is the acceleration vector due to gravity.

The mixture temperature is obtained from the energy equation of the mixture in the form:

q = −λ∇T |

Finally, a diffusion equation is solved for each species in the form

Unlike the abovementioned reactingFoam solver, the diffusion coefficients in the laminarSMOKE solver are calculated using mixture-rules^{17,18} from binary diffusion coefficients Γ_{jk}, which is currently the most accurate approach for diffusion-driven combustion. In the mixture-averaged approach, the binary diffusion coefficients are converted to Fick diffusion coefficients by calculating a weighted average according to the formula

As mentioned in the Introduction, the reaction set in this work is the subset of the GRI-MECH 3.0 reaction mechanism,^{13} from which we selected only the reactions which include the species mentioned above. The sub-set of reactions is provided in the ESI† in the CHEMKIN format and is also available in the open source model repository (see Section 2.4 below).

We did not list the reactions and their rates in this paper. Instead, we included both the reaction mechanisms in a CHEMKIN-compatible format as the ESI† to this publication.

Fig. 1 Computational geometry of the numerical model with dimensions and annotated boundaries (a) and the mesh used for discretization of the partial differential equations (b). |

There is a different boundary condition imposed for each boundary. For the inlet boundary, we prescribe the mean velocity of the flow, calculated from the flow rate and tube diameter. We also impose the Dirichlet boundary condition for the mass fractions of all the species – in the default case without impurities, all mass fractions are set to zero, except for argon and hydrogen.

For the wall boundary, we prescribe a zero-gradient boundary condition for the species' number densities and a no-slip boundary condition for mixture velocity. By assuming zero-gradient at the walls, we effectively neglect quenching of reactive species by surface reactions. This is a reasonable assumption because the area of the wall which is in contact with the flame is small.

The air intake boundary condition serves as a source of fresh air (oxidizer) in the model. Therefore, we set the mass fractions of all the species except for O_{2}, N_{2} and H_{2}O to zero. Regarding velocity, we impose a constant flow velocity of 0.05 m s^{−1} on the air intake, to provide a steady supply of air. In the experimental system, there is no forced flow outside the MDF tube, but the air circulates around it due to the buoyancy force. It is, however, not practical to assume a zero-gradient boundary condition for velocity on the intake because the system of equations would not have a unique solution.

Finally, the outflow boundary condition assumes a zero-gradient for both the species mole fractions and for the gas mixture velocity, acting as a free outflow for both these quantities.

Unlike many studies focusing on combustion simulation, which chooses the simulated and experimental flame radius or length and flame velocities as the main observable,^{19–21} this work goes into greater detail by using the H radical 2D maps as the observable. The rationale behind this is that the distribution of H radicals has a determining influence on an atomizer's performance, as detailed in the Introduction.

In order to obtain the absolute concentration of atomic hydrogen, the sensitivity of the detection was calibrated by measurement of Kr TALIF.^{25,26} Kr atoms with a known concentration (1.5–8%) in Ar gas were excited by using a laser with a wavelength of 204 nm and the generated Kr fluorescence was detected. Then, the atomic hydrogen concentration was calculated by means of the formula

N_{H} = N_{Kr}(S_{H}/S_{Kr})(E_{Kr}/E_{H})^{2}(ν_{H}/ν_{Kr})^{2}(σ^{(2)}_{Kr}/σ^{(2)}_{H})(q_{Kr}/q_{H})(K_{Kr}/K_{H}), |

The 1D radially resolved data allow us to quantitatively compare the model predictions with the experiment. The data are plotted in Fig. 3 for 2 mm above the top of the atomizer tube. It is apparent that, in terms of absolute values, the number density of H is within the same range. The experimental data are, however, more diffuse compared to the simulation data. This is most likely caused by the jittering of the flame, which effectively changes the position over the 10 s integration time of the ICCD camera.

Fig. 3 Spatial distribution of H radicals along a horizontal line, 2 mm above the top of the quartz tube. |

The more diffuse nature of the experimentally measured data is also apparent in the 2D spatially resolved data, shown in Fig. 4. However, the model tends to capture very accurately the length of the flame and the overall shape of the H-rich regions in several ratios of argon/hydrogen supplied to the atomizer. Apparently, H radicals are present especially in the interfacial mixing region, where the argon/hydrogen mixture supplied to the atomizer mixes with ambient air.

The model and experiment consistently show that the length of the flame can be controlled by the argon flow rate – compare Fig. 4(b) and (c). This is because increasing the argon flow rate effectively increases the flow velocity and the combustion of hydrogen occurs over a greater distance. At the same time, however, increasing the argon flow rate decreases the absolute density of H radicals. On the other hand, increasing the fraction of H_{2} (under the same flow rate of the Ar/H_{2} mixture) does increase the flame length and H radical density – compare Fig. 4(a) and (c).

Simulation label | Description |
---|---|

Dry air | Humidity of laboratory air set to 0% |

Humid air | Humidity of laboratory air set to 100% |

Smaller tube | Tube i.d. decreased to 5.6 mm |

Larger tube | Tube i.d. increased to 6.4 mm |

Impurity 0.1% | 0.1 molar% of air added to the gas inlet |

Two observables were chosen for quantifying the uncertainty. One is the absolute yield of H radicals produced by the atomizer – i.e. the number density of hydrogen radicals integrated over the simulation domain. The second observable is the length of the flame L_{f} which is defined as the position on the z-axis, where the mass fraction of H_{2} drops below 0.1% of the maximum value. In both cases, we plot the change in these observables with respect to the default simulation – the value of ΔL_{f} = +10% corresponds to a 10% increase of the flame length w.r.t the default simulation.

Firstly, in Fig. 5, we plot the sensitivity of the total hydrogen yield. Apparently, the main sources of uncertainty in the presented setup are the tube diameter. Having a tube with a diameter increased by 0.4 mm reduces the total H yield by approximately 10% while decreasing the tube diameter increases the H yield by approximately 6%. The humidity of ambient air plays only a small role in the total H yield, with the H yield decreasing with increasing humidity. This result is reasonable, considering that H_{2}O molecules will deplete some of the H radicals, ultimately converting them to H_{2}O_{2}. Finally, the 0.1% air impurity in the gas inlet has only minor impact on the total H yield.

In Fig. 6, we plot the sensitivity of the simulated flame length to experimental uncertainties. In this case, only the tube diameter has non-negligible influence within the range of experimental uncertainties specified in Table 1. The result is to be expected as a larger tube diameter decreases the initial velocity of the gas flow while a smaller diameter increases it, thereby stretching the flame longer.

The model developed has been used for simulating H radical production in a simple MDF atomizer. By comparing the simulated H density profile with TALIF measurements of the same quantity, we confirmed that the predictions of the model are correct, both in terms of the shape of the flame and in terms of the absolute values of H radical concentrations. The main difference between the simulation and the experimental data lies in the fact that the experiment shows H density profiles which are more diffuse. However, this observation may be attributed both to the imperfections of the model (underestimated diffusion) and experimental imperfections (moving or oscillatory flame position).

We have also used the model to assess the sensitivity of the MDF atomizer to uncertain laboratory conditions and geometrical parameters. The ability to quantify how laboratory conditions and other experimental parameters influence the performance of an atomizer is another example of how the computational model can be utilized.

Since the H radical distribution controls what happens in atomizers, it has become obvious that the presented model is a powerful predictive tool that can be utilized for the design and optimization of not only MDF atomizers but other atomizer configurations as well. The model supports both 2D axially symmetrical setups, e.g. flames in gas shield atomizers, and setups which require full 3D geometry, e.g. conventional externally heated quartz tube atomizers or multiatomizers.^{3}

By providing the model simulation files under an open-source license, we allow and invite other members of the analytical chemistry community to modify the geometry or operating conditions and use the tool for their research.

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## Footnote |

† Electronic supplementary information (ESI) available: kinetic_scheme.pdf: the reaction mechanism used in this work; mdfatomizer-laminarsmoke-20200129.zip: a ZIP archive with geometry, kinetics and model configuration files for the default configuration. Includes a README file with execution instructions and all the kinetic- and thermo-chemistry input data in CHEMKIN format. See DOI: 10.1039/d0ja00099j |

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