Study on the sound attenuation coefficient of O2 in power plant furnace based on LAMMPS

3.1. Heat capacity calculation

Heat capacity and isothermal compressibility are derived from the partial derivatives of volume, internal energy, and enthalpy with respect to temperature and pressure. These thermodynamic derivatives can be directly obtained from the fluctuations of the system at a fixed composition. In the canonical ensembles (NVT/NPT) adopted for heat capacity and compressibility calculations, the system energy is averaged over all accessible microscopic states. For a given microscopic state S with energy E_S, the mean squared deviation $⟨{\left(E-\stackrel{-}{E}\right)}^2⟩$ is defined as the energy fluctuation:

where ${\rho }_S$denotes the probability that the system is in the microstate $S$, for regular distribution:

where, the former term represents the sum of kinetic energy terms, including translational kinetic energy, rotational kinetic energy, and vibrational kinetic energy, and the latter term represents the sum of potential energy terms. According to the equipartition theorem, the kinetic energy term can be expressed in the form of Eq. (8), where $N_A$ is Avogadro’s constant, $i$ is the degree of freedom, and $k_B$ is Boltzmann’s constant.

So:

Fig. 1Heat capacity

The results (Fig. 1) show that in the low-temperature region, the $c_v$ of the non-rigid model is 846.45 J·kg^-1·K^-1, which is notably higher than the experimental value of 653.125 J·kg^-1·K^-1, with a relative error of 29 %. In comparison, the $c_v$ value of the rigid model is 644.63 J·kg^-1·K^-1, corresponding to a relative error of only 1.3 % relative to the experimental data. This discrepancy arises because molecular vibrational degrees of freedom remain unexcited at low temperatures, and the inclusion of vibrational degrees of freedom in the flexible model leads to an overestimation of heat capacity.

In the high-temperature region (Fig. 1), the $c_v$ value of the non-rigid model is 848.45 J·kg^-1·K^-1, which gradually approaches the experimental value of 868.65 J·kg^-1·K^-1, with a relative error below 2 %. Meanwhile, the $c_v$ value of the rigid model is 645.02 J·kg^-1·K^-1, and its relative error relative to the experimental data rises to 24 %. This is attributed to the excitation of molecular vibrational degrees of freedom at high temperatures, which cannot be captured by the rigid model.

3.2. Isothermal compressibility

${\kappa }_T$ is the isothermal compression coefficient, defined as the rate of change of gas volume with pressure under isothermal conditions. According to fluctuation theory:

As depicted in Fig. 2, temperature exerts a negligible effect on the ${\kappa }_T$ over 300-1200 K, where ${\kappa }_T$ fluctuates near 1×10⁻⁶Pa⁻¹. Specifically, ${\kappa }_T$ is 0.911×10^-6Pa^-1 at 300 K and 0.932×10^-6Pa^-1 at 1200 K, with a relative fluctuation below 2.3 % and a relative error within 9 % relative to the experimental value of 0.9996×10^-6Pa^-1. At 1200 K, increasing pressure from 81060 Pa to 111457.5 Pa reduces ${\kappa }_T$ from 1.4×10^-6Pa^-1 to 0.85×10^-6Pa^-1 (a drop of ~39 %), as elevated pressure suppresses gas compressibility. This behavior is critical for on-site calibration of furnace acoustic thermometry: pressure fluctuation-induced corrections to ${\kappa }_T$ are required to avoid large errors in sound velocity calculation and ensure accurate temperature inversion.

Fig. 2Isothermal compressibility

3.3. Thermal conductivity calculation

Thermal conductivity is calculated using Fourier’s law:

Fig. 3Temperature distribution

As obtained through simulation (Fig. 3), when the temperature rises while the pressure remains constant, and when the pressure varies within the range of 81060 Pa to 111457.5 Pa at a constant temperature, there is no significant difference in the temperature difference, and the thermal conductivity changes very little. This indicates that under furnace combustion conditions, the effect of temperature variations on thermal conductivity is much greater than that of pressure fluctuations. The finally calculated thermal conductivities are 0.042 W·m^-1K^-1 at 500 K, 0.075 W·m^-1K^-1 at 1000 K, and 0.089 W·m^-1K^-1 at 1200 K.

3.4. Viscosity calculation

In this study, the RNEMD method is used to calculate viscosity, which is computed using the following formula:

By comparing the simulated data images at different temperatures (Fig. 4), the increase in viscosity can be clearly observed. This is because as the temperature rises, gas molecules move more vigorously, making them more likely to collide with other molecules and exchange momentum, which intensifies the internal friction between fluid layers and increases viscosity. During combustion in the furnace, the pressure is at a slightly negative state below 101325 Pa. Simulation results also indicate that typical pressure fluctuations in the furnace (slightly below 101325 Pa negative pressure) have little effect on the viscosity of oxygen, with values remaining around 55×10⁻⁵ Pa·s with only slight fluctuations.

Fig. 4Viscosity simulation value

3.5. Sound attenuation coefficient

Since the attenuation coefficient is proportional to the square of the frequency, their ratio is constant. The calculated values are 1.25×10^-11 s²/m at 300 K, 1.89×10^-11 s²/m at 500 K, 2.0×10^-11 s²/m at 1000 K, and 2.3×10^-11 s²/m at 1200 K. respectively. These values are significantly larger than the room-temperature value, indicating that the increase in temperature has an important effect on the attenuation of sound waves. This is because sound waves are absorbed due to viscosity and heat conduction during propagation. Acoustic wave attenuation is mainly caused by viscous dissipation and heat conduction dissipation. At constant pressure, high temperatures lead to a decrease in gas density, which in turn increases the speed and wavelength of sound. The increase in viscosity and thermal conductivity further exacerbates attenuation.