Mathematical Model of a Two-Temperature Medium of Gas–Solid Nanoparticles with Laser Methane Pyrolysis

Abstract

A mathematical model of a two-phase chemically active medium of gas and solid ultrafine particles in the field of laser radiation with detailed heat transfer processes between the gas and particles is created. The mathematical model is a system of Navier–Stokes equations in the approximation of small Mach numbers and several temperatures, which describes the dynamics of a viscous multicomponent heat-conducting medium with diffusion, chemical reactions, and energy supply through laser radiation. A computational algorithm is developed for studying chemical processes in a gas–dust medium with the single-velocity dynamics of a multicomponent gas under the laser radiation. This mathematical model multiscale, i.e., is characterized by the presence of several very different temporal and spatial scales. The computational algorithm is based on the scheme of splitting by physical processes. For a two-phase medium of a multicomponent gas and nanodispersed solid particles, theoretical studies of multidirectional processes of thermal relaxation and specific heating-cooling of the components of a two-phase medium by laser radiation, thermal effects of chemical reactions, and intrinsic radiation particles are carried out. It is shown that laser radiation can form a significant gap between the particle temperature and the gas temperature and provide the activation of methane with conversion to ethylene and hydrogen. The developed numerical model will find its application in the creation of new technologies of laser thermochemistry.

Appendices

APPENDIX A

1.1 Estimation of the Velocity and Thermal Relaxation Time

We estimate the time of the velocity and thermal relaxation of particles in a gas as a function of the particle size. The thermal relaxation time of a particle ${{\zeta }_{i}}$ in a medium without evaporation is determined from Eq. (10) and the relation

$${{d{{T}_{i}}} \mathord{\left/ {\vphantom {{d{{T}_{i}}} {dt}}} \right. \kern-0em} {dt}} = - {{{{q}_{{{\text{conv}}}}}} \mathord{\left/ {\vphantom {{{{q}_{{{\text{conv}}}}}} {({{m}_{i}}{{C}_{{DV}}})}}} \right. \kern-0em} {({{m}_{i}}{{C}_{{DV}}})}} = ({{T}_{g}} - {{T}_{i}}){\text{/}}{{\zeta }_{i}},$$

and we obtain

$${{\zeta }_{i}} = \frac{{{{m}_{i}}{{C}_{{DV}}}({{T}_{i}} - {{T}_{g}})}}{{{{q}_{{{\text{conv}}}}}}} = \frac{{2{{m}_{i}}{{C}_{{DV}}}(\gamma - 1){{T}_{g}}}}{{\alpha \pi s_{i}^{2}p{{c}_{t}}(\gamma + 1)}}.$$

(A.1)

The velocity relaxation time of a particle t_i is determined [9, 21] from the relation:

$${{t}_{i}} = {{{{s}_{i}}{{\rho }_{s}}} \mathord{\left/ {\vphantom {{{{s}_{i}}{{\rho }_{s}}} {({{c}_{s}}{{\rho }_{g}})}}} \right. \kern-0em} {({{c}_{s}}{{\rho }_{g}})}} = {{s}_{i}}{{\rho }_{s}}{{c}_{s}}{\text{/}}p.$$

(A.2)

The results are shown in Fig. 5. Note that the found relaxation times are much shorter than the characteristic time of a gas mixture’s motion over the calculated length. This means that the problem of modeling the motion of a gas-dispersed medium is multiscale in time or rigid. When using explicit integration methods, a requirement arises for the time step: it must be less than the minimum of all the relaxation times.

Fig. 5.

Velocity (A.2) (left) and thermal (A.1) (right) relaxation times of spherical aluminum oxide particles in methane at atmospheric pressure. Rate relaxation times are given for methane temperatures of 850 and 1500 K. The thermal relaxation times are given for a methane temperature of 1500 K at different thermal accommodation coefficients. The thermal relaxation times for a methane temperature of 850 K at the same thermal accommodation coefficients practically do not differ from the values given on the right.

For such problems, approaches to the numerical solution based on the asymptotic approximation have been developed. The idea of such approaches is that it is possible to abandon the search for a numerical solution of the original problem and replace it with a solution of a simplified problem obtained from the original problem by expansion in a small parameter (the general idea is presented in [22]).¹ We illustrate this idea with an example of calculating the gas velocities u and dispersed phase ${v}$ satisfying the system

$$\left\{ \begin{gathered} d{v}{\text{/}}dt = {{g}_{{v}}} + (u - {v}){\text{/}}{{t}_{{{\text{rel}}}}} \hfill \\ du{\text{/}}dt = {{g}_{u}}, \hfill \\ \end{gathered} \right.$$

(A.3)

where ${{g}_{{v}}}$ and ${{g}_{u}}$ are accelerations acting on the dispersed phase and gas, respectively, as a result of the sum of all forces except friction, and ${{t}_{{{\text{rel}}}}}$ is the speed of the relaxation time. Note that system (A.3) corresponds to the case when the gas affects the dust, but the dust, due to its low concentration, does not change the gas velocity. At ${{t}_{{{\text{rel}}}}} \ll \tau$, where $\tau$ is the time step, the solution of the first equation at time $\tau$ (with an accuracy up to a second-order value from ${{t}_{{{\text{rel}}}}}$) has the form

$${v} = {{g}_{{{\text{rel}}}}}{{t}_{{{\text{rel}}}}} + u,$$

(A.4)

where ${{g}_{{{\text{rel}}}}} = {{g}_{{v}}} - {{g}_{u}}$. Thus, to find the velocity of the dispersed phase, instead of numerically solving the first equation from (A.3), which contains a small parameter, we can solve a much simpler algebraic equation (A.4). In addition, relation (A.4) makes it possible to evaluate the need for a single-velocity or two-velocity approximation. The gas velocity changes under the action of the viscosity, pressure gradient, and friction, and the modulus of the pressure force significantly exceeds the modulus of viscous forces. Then ${{g}_{{{\text{rel}}}}}$ for small concentrations of the dispersed phase will be proportional to the pressure gradient divided by the gas density. The maximum of this value can be estimated from the numerical data of our calculations with the gas flow without an admixture of dispersed particles ${{g}_{{{\text{rel}}}}}$ = $\max (\Delta p{\text{/}}\rho )$ ≈ 0.02 m s⁻². From this we obtain that the velocities of the gas and particles will differ by less than a thousandth of a percent. This makes it possible to neglect the difference in modeling the dynamics of the gas and dust medium.

In turn, the difference between the temperatures of the gas and particles depends on the intensity of the laser radiation absorbed by the particles, which is set externally and can vary over a wide range of values. From similar reasoning for temperature relaxation in a medium of a gas and dispersed phase, it follows that, despite the short thermal relaxation times relative to the time of the studied dynamics of the gas–dust mixture, the a priori transition to a one-temperature model for the gas and particles is unjustified. This also means that the estimated thermal relaxation times characterize the time intervals during which the particle temperature will reach a stationary value relative to the gas temperature, while the particle temperature will not necessarily be close to the gas temperature. In addition, the values of the particle and gas temperatures can be affected by chemical reactions, exothermic and endothermic, whose flow rate directly depends, in turn, on these temperatures.

APPENDIX B

1.1 Data for Determining the Heat Capacity, Viscosity, and Thermal Conductivity of the Components of a Mixture

Tables 2−Tables 4 show the coefficients for the polynomials of heat capacity, viscosity, and thermal conductivity for each of the mixture’s components taken from [25].