Coupled modeling and numerical simulation of gas flows laden with solid particles in de Laval nozzles

Abstract

Supersonic gas–solid separation is a new dedusting concept, in which a de Laval nozzle is employed to accelerate a gas laden with solid particles to supersonic speeds. In the present work, a correction for the curved nature of the nozzle walls is implemented in the turbulent boundary layer model. A gas-particle coupled quasi-one-dimensional flow model is established based on an Eulerian–Lagrangian method. Combining the flux vector splitting of gas-phase equations, fifth-order weighted essentially non-oscillatory spatial discretization, and three-step third-order total variation diminishing Runge–Kutta time marching, we create a set of uniformly high-order accurate, stable, and efficient numerical methods. The validation against previous experiments demonstrates the validity and accuracy of the numerical model. The dependence of the flow fields on parameters including particle size and mass loading, nozzle inlet stagnation temperature and pressure, and nozzle expansion angle is numerically computed and analyzed. The results show that the decrease in particle size or mass loading and the increase in inlet stagnation temperature, pressure, or nozzle expansion angle facilitate an increase in exit particle velocity. The increase in particle size, inlet stagnation pressure, or nozzle expansion angle and the decrease in particle mass loading or inlet stagnation temperature increase the exit gas Mach number. In the current parametric study, particle size and mass loading have the largest effects on exit particle velocity and exit gas Mach number, respectively.

Appendices

Appendix: Numerical methodology

Discretization methods for governing equations

In consideration of the prominently hyperbolic feature of the gas-phase conservation equations, a fifth-order WENO scheme [21] is employed to discretize the spatial derivative term of the flux vector (or convective term) on the left side of (1). However, it is necessary to conduct a characteristic splitting for the flux vector in advance. A modified Steger–Warming splitting method [18] is adopted in the present numerical model.

In order to make sure the spatial discretization for the gas-phase properties is at least third-order accurate, special treatments are required for grid nodes close to the nozzle inlet and exit boundaries. We combine a third-order upwind difference with a third-order upwind compact difference [18] to create the difference expressions for these nodes. Additionally, we adopt a simple third-order upwind difference approximation for the spatial derivatives such as the one of particle volume fraction in (19).

Shu and Osher’s [27] three-step third-order total variation diminishing Runge–Kutta (R–K) scheme is used for the difference approximation of the time derivatives.

Interpolation of the gas properties at particle locations

When we perform the calculations for the particle phase, the information of the gas phase should be provided in advance. However, the solution of the gas phase only provides the information at grid nodes. Therefore, we introduce a cubic polynomial interpolation for the properties of the gas phase at the particle positions.

Fig. 14

Algorithmic flowchart

Computational procedure of flow fields

The procedure for the present computations of the gas-particle two-phase flow is as shown in Fig. 14. The convergence criterion $E_\mathrm{r}<$ $\varepsilon _\mathrm{r}$ is adopted, where the residual of mass flow rate of the gas is defined by

$$\begin{aligned} E_{\mathrm{r}} =\max \left\{ {\left| {q_\mathrm{mg}^{\left( n \right) } -q_\mathrm{mg}^{\left( {n-{1}} \right) } } \right| /q_\mathrm{mg}^{\left( {n} \right) } } \right\} \end{aligned}$$

(31)

and the threshold value $\varepsilon _\mathrm{r} =$ 10$^{-5}$. For clarity, we also list the bullet points as follows:

• 1. Initialize the calculation with an isentropic gas flow.

• 2. Calculate $f_\mathrm{wf}$, $Q_\mathrm{w}$, and $G_\mathrm{w}$ at each node.

• 3. Calculate $v_\mathrm{g}$, p, $T_\mathrm{g}$, and $\rho _\mathrm{g}$ at the nozzle inlet.

• 4. Calculate $\rho _\mathrm{g}$, $v_\mathrm{g}$, $e^{*}$, e, $T_\mathrm{g}$, and p at each node.

• 5. Repeat (2)–(4) until the three-step R–K calculation is finished at this time step.

• 6. Calculate $E_\mathrm{r}$. If $E_\mathrm{r} \ge \varepsilon _\mathrm{r}$, turn back to (5). Otherwise (i.e., $E_\mathrm{r}<$ $\varepsilon _\mathrm{r})$, if the current calculation is for gas-particle two-phase flow, terminate the calculation; if the current calculation is just for pure gas flow, proceed to the following steps.

• 7. Inject particles by allocating $x_\mathrm{p}$, $v_\mathrm{p}$, and $T_\mathrm{p}$.

• 8. Reorder the new numbers for the remaining particles.

• 9. Calculate $N_{JI}$, $\varphi$,and $\alpha$ in each cell.

• 10. Calculate $\rho _\mathrm{g}$, p, and $T_\mathrm{g}$ at particle locations.

• 11. Calculate $f^\mathrm{gp}$, $f^\mathrm{pp}$, and $Q^\mathrm{gp}$ for each particle.

• 12. Calculate $v_\mathrm{p}$, $x_\mathrm{p}$, and $T_\mathrm{p}$ for each particle.

• 13. Calculate $G^\mathrm{pg}$ and $Q^\mathrm{pg}$ in each cell, and turn back to (5).