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12 August 2022 Inhalation Industry methods e industry-standard CFD method is the finite volume method (FVM), which solves the transient equations of fluid flow, treating the fluid as a contin- uum—that is, not attempting to model the motion of individual molecules. Finite element method (FEM) solvers also exist. Equations for conservation of mass, energy and momentum are solved, often called the Navier-Stokes equations. FVM operates by discretiz- ing the entire fluid region into a grid or mesh of small volumes, known as cells. e simulation proceeds from an initialized fluid state, with the Navier-Stokes equations solved in each cell at a series of small timesteps, each new fluid state dependent on the last. Steady simulations have artificial timesteps, allowing convergence towards a single solution. Simulations involving transient device actuation or patient inhalation are inherently transient, needing an unsteady simulation, where the timesteps represent the passing of time from a known initial state into a simulated future. Single-phase Eulerian simulation With this method, each cell is a fixed observation point for the fluid as it flows past—known mathe- matically as an Eulerian description (after Leonhard Euler). e output from a basic industrial CFD solver can provide the flow velocity and direction, pressure, temperature and fluid composition in each cell as a function of time, which, when viewed as a whole, provides an overall visualization of the predicted fluid flow. Figure 1 shows the mesh for simulation of air flow through a USP-IP throat, the simulated flow velocity at the 90° corner, and a smoothed version with cell interpolation to give a continuous visualiza- tion of flow velocity typical of that usually presented. Multi-phase simulation Of relevance for simulating sprays and aerosols is the ability of CFD solvers to track particles and inter- faces. is allows simpler simulation of multi-phase flow, e.g., bubbly, flash boiling flows within the expansion chamber of a pMDI, particle-laden flows of a DPI or liquid jets of a nasal spray. A standard method for simulating an aerosol is Lagrangian parti- cle tracking, also known as discrete droplet modeling [29]. A Lagrangian description of fluid flow (named for Joseph-Louis Lagrange) moves with the fluid, so each particle is modeled from an observation point that moves along with it. For aerosol simulations, the ambient gas is modeled in the previously-described Eulerian sense (the 'Eule- rian phase') and individual droplets or particles (the 'Lagrangian phase') are tracked from the time of their creation until they disappear. e fate of particles may be coalescence with another particle, break-up under aerodynamic forces into child particles, depo- sition on solid surfaces, evaporation or exit from the simulated region. Each droplet or particle is assumed Figure 1 (a) Structured mesh at the center lines of a USP-IP throat CFD model, showing location of results in (b) and (c); (b) CFD- simulated cell values of air velocity magnitude at throat corner; (c) Smoothed air velocity magnitudes at throat corner. (a) (b) (c) 3.8e+00 3 2 1 0.0e+00 U Magnitude

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