Two-phase flow calculations are performed by solving mass, momentum, and energy conservation equations under a Lagrangian coordinate system. The conservation equations are expressed as follows:

Mass conservation equation
$\frac{D\mathrm{\rho}}{Dt}+\mathrm{\rho}\mathrm{\nabla}\cdot \stackrel{\u20d7}{V}=0$(1)

Momentum conservation equation
$\mathrm{\rho}\frac{d\stackrel{\u20d7}{u}}{dt}=F{\mathrm{\nu}}_{c}-\mathrm{\nabla}p-\frac{2}{3}\mathrm{\nabla}\cdot (\mathrm{\mu}\mathrm{\nabla}\stackrel{\u20d7}{u})+\mathrm{\nabla}2\mathrm{\mu}[\dot{\mathrm{\epsilon}}]$(2)

Energy conservation equation
$\mathrm{\rho}\frac{d}{dt}\left(\mathrm{\epsilon}+\frac{{\stackrel{\u20d7}{u}}^{2}}{2}\right)=\mathrm{\nabla}\cdot (\stackrel{\u20d7}{P}\cdot \stackrel{\u20d7}{u})\phantom{\rule{thinmathspace}{0ex}}+\phantom{\rule{thinmathspace}{0ex}}\mathrm{\rho}\stackrel{\u20d7}{g}\cdot \stackrel{\u20d7}{u}+\stackrel{\u20d7}{E}\cdot \stackrel{\u20d7}{J}-\mathrm{\nabla}\stackrel{\u20d7}{q}$(3)

The calculations in this article assume the following:

(1)

Particle is isothermal sphere with constant diameter

(2)

Average intensity of plasma fluctuation velocity is constant under turbulent conditions

(3)

The influences of boundary layer and flow source term are neglected

(4)

The evaporation of particles is neglected

The flow drag force and gravity are applied on particles in the movement process and can be represented as follows:
${m}_{d}\frac{d\mathrm{\upsilon}}{dt}={C}_{\mathrm{D}}\mathrm{\rho}\left(U-\mathrm{\upsilon}\right)|\frac{{A}_{d}}{2}+{m}_{d}g+Sm$(4)where ${m}_{d}$ is the particle mass, $\mathrm{\upsilon}$ is the particle velocity vector ($\mathrm{\upsilon}=ui+\mathrm{\nu}j+wk$), ${C}_{\mathrm{D}}$ is the drag coefficient, $\mathrm{\rho}$, $U$, and $p$ are density, velocity, and gas pressure, respectively, and ${A}_{d}$ is the particle windward area.

The drag coefficient ${C}_{\mathrm{D}}$ is a function of the Reynolds number [12]:
${C}_{\mathrm{D}}=\frac{24}{Re}+\frac{6}{1+\sqrt{Re}}+0.4$(5)The pressure-gradient force of the particle is written as [13]
$Fp=-\frac{\mathrm{\pi}}{6}{d}_{p}^{3}\mathrm{\nabla}P$(6)The thermophoretic force of the particle is written as
${F}_{\mathrm{T}}=-6\mathrm{\pi}\mathrm{\mu}\mathrm{\nu}{d}_{p}{C}_{s}\frac{1}{1+3{C}_{m}Kn}\frac{{K}_{g}/{K}_{p}+{C}_{t}{K}_{n}}{1+2{K}_{g}/{K}_{g}+2{C}_{t}Kn}\frac{\mathrm{\nabla}T}{T}$(7)where ${K}_{n}$ is the Knudsen number, ${K}_{g}$ is the gas thermal conductivity/W m^{−1} K^{−1}, ${K}_{p}$ is the particle thermal conductivity/W m^{−1} K^{−1}, ${C}_{s}$ is a constant (1.17), ${C}_{m}$ is a constant (1.14), ${C}_{t}$ is a constant (2.18) [14].

Heat transfer inside the particles can be written as follows:
$\frac{\mathrm{\partial}H}{\mathrm{\partial}t}=\frac{1}{{r}^{2}}\frac{\mathrm{\partial}}{\mathrm{\partial}r}\left({k}_{p}{r}^{2}\frac{\mathrm{\partial}T}{\mathrm{\partial}r}\right)$(8)where $r$ is the radial distance of particle/m, *H* is the enthalpy/J kg^{−1}, and ${k}_{p}$ is the thermal conductivity/W m^{−1} K^{−1}.

Heat balance item of the particle surface can be written as [15]:
$Q={k}_{p}{\left(\frac{\mathrm{\partial}T}{\mathrm{\partial}r}\right)}_{r={r}_{0}}=h({T}_{f}-{T}_{s})-\mathrm{\sigma}\mathrm{\epsilon}{T}_{s}^{4}+{Q}_{\mathrm{v}\mathrm{a}\mathrm{p}}$(9)where *h* is the coefficient of heat conduction, $\mathrm{\sigma}$ is the Boltzmann constant, $\mathrm{\epsilon}$ is the emission coefficient, ${T}_{s}$ is the particle surface temperature/K, ${T}_{f}$ is the plasma temperature/K, and ${Q}_{\mathrm{v}\mathrm{a}\mathrm{p}}$ is the energy taken away by particle evaporation/J.

Convective heat transfer coefficient can be written as follows:
${h}_{c}=\frac{{K}_{g}{N}_{u}}{d}$(10)where ${N}_{u}$ is the Nusselt number.

The mass loss ratio $\dot{m}$ caused by particle evaporation can be written as follows [16]:
$\dot{m}={K}_{\mathrm{M}\mathrm{I}\mathrm{X}}({C}_{s}^{\ast}-{C}_{f})\approx {K}_{\mathrm{M}\mathrm{I}\mathrm{X}}{C}_{s}^{\ast}$(11)where ${K}_{\mathrm{M}\mathrm{I}\mathrm{X}}$ is the total mass conductivity coefficient, ${C}_{s}^{\ast}$ is the concentration of particle surface evaporative substance/g · m^{−3}, and ${C}_{f}$ is the vapor concentration/g · m^{−3}.

The concentration of particle surface evaporative substance can be written as
${C}_{s}^{\ast}=\frac{M{P}_{v}}{R{T}_{s}}$(12)where *M* is the molar mass of steam/g · mol^{−1}, ${P}_{v}$ is the steam pressure/Pa.

The total mass conductivity coefficient can be written as follows:
${K}_{\mathrm{M}\mathrm{I}\mathrm{X}}={\left(\frac{1}{{h}_{m}}+\frac{1}{{h}_{e}}\right)}^{-1}$(13)where ${h}_{e}$ is the Langmuir evaporation rate coefficient and ${h}_{m}$ is the mass conduction coefficient.

The heat conduction caused by evaporation can be written as follows [17]:
${Q}_{\mathrm{v}\mathrm{a}\mathrm{p}}=\dot{m}\left[{L}_{e}+\left({H}_{v(Tf)}-{H}_{v(Ts)}\right)+\left({H}_{l(Ts)}-{H}_{v(To)}\right)\right]$(14)where ${L}_{e}$ is the enthalpy of steam/J · kg^{−1}.

Subscripts *v*, *l*, and *s* represent steam, liquid, and solid states, respectively.

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