Understanding the relationship between velocity and flow rate in the Discrete Phase Model (DPM) has been a daunting challenge for ANSYS Fluent users.
This article aims to clarify the common confusion surrounding these terms in DPM injection panel, shown in Fig. 1.
Figure 1: Velocity Magnitude and Total Flow Rate in DPM Injection window
DPM Velocity and Mass Flow
Basically, in fluid mechanics fundamentals, the flow rate linked through velocity by the following equation:
Flow rate=Density×Velocity×Cross-sectional Area
![\[ \dot{ m } = \rho . V . A \]](https://cfdland.com/wp-content/ql-cache/quicklatex.com-f64ccabfdade60fe73df752f20ed6ac5_l3.png "Rendered by QuickLaTeX.com")
So whenever the cross-sectional area and density are known, setting one parameter (either velocity or flow rate) allows the automatic calculation of the other. So why does ANSYS explicitly ask for the definition of both velocity and flow rate?
Dealing with continuous phase, the above equation must hold true as the fluid occupies the entire cross-sectional area of the flow path. However, in discrete phase cases, the scenario changes. For instance, imagine a spray nozzle. Here, the liquid is atomized into droplets, creating a discontinuous phase where droplets do not occupy the entire cross-section. As a result, while the mass flow rate remains consistent with the continuous phase, the velocity of the droplets differs from the velocity calculated using the above equation.
Figure 2: Discrete Phase of liquid droplets in spray nozzle
Understanding Velocity & Flow Rate in DPM
To illustrate the practical application, consider a spray nozzle atomizer. Suppose you supply a specific flow rate to the atomizer and calculate the velocity based on the cross-sectional area. In that case, the resulting velocity will often differ significantly from the real velocity observed in experiments. As mentioned in the previous section, this is happening because the droplets formed do not occupy the entire cross-sectional area, leading to variations in velocity.
In such cases, the velocity should be set based on experimental data. This ensures that the simulation accurately reflects real-world conditions.
The equation of motion for a particle in DPM is given by:
![\[ \frac{d u_p}{d t} = \frac{18 \mu}{\rho_p d_p^2} C_D (\mathbf{u} - \mathbf{u}_p \]](https://cfdland.com/wp-content/ql-cache/quicklatex.com-64db73289d3854e7265aec3f19b5790b_l3.png "Rendered by QuickLaTeX.com")
where:
- up is the particle velocity,
- μ is the dynamic viscosity,
- ρp is the particle density,
- dp is the particle diameter,
- CD is the drag coefficient,
- u and up are the fluid and particle velocities, respectively.
The momentum equation for the continuous phase is:
![\[ \rho \frac{\partial \mathbf{u}}{\partial t} + \rho (\mathbf{u} \cdot \nabla) \mathbf{u} = -\nabla p + \mu \nabla^2 \mathbf{u} + \mathbf{F}_D \]](https://cfdland.com/wp-content/ql-cache/quicklatex.com-b052398c50f3a26d0e6f72a69c8011c8_l3.png "Rendered by QuickLaTeX.com")
where FD is the drag force exerted by the particles on the fluid, which is a function of both the particle velocity and the flow rate Q.
You can get access to dpm cfd
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Discrete Phase Model Physical Model