As stated in the Discrete Phase Model Physical Models article, ANSYS Fluent predicts the trajectory of particles by integrating the force balance on the particle. This force balance is written in a Lagrangian reference frame and has an independent term shown by **F****D **that **represents drag force per unit particle mass**.

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ToggleDiscrete Phase Model drag laws describe the forces exerted on discrete particles by the surrounding fluid, influencing particle trajectories and overall flow dynamics. ANSYS Fluent offers various drag law formulations, accessible from the following path (see Fig. 1):

Injection Window ==> Physical Models Tab

Understanding these drag laws and their appropriate application is essential for engineers who aim to optimize their DPM simulations.

Figure 1: Discrete Phase Model (DPM) drag laws in ANSYS Fluent

In some references, 14 draw laws are introduced for discrete phase model. However, there are **7** drag laws provided in ANSYS fluent for Discrete Phase Model. The rest of the drag laws are dedicated to the **Dense Discrete Phase Model**. This article concentrates on the theoretical and practical descriptions of DPM drag laws.

## 1) DPM-Spherical Drag Law

The **Spherical drag law** is the most commonly used drag model for the Discrete Phase Model (DPM) simulations. As its title suggests, it is designed for particles that are approximately **smooth** **spherical** in shape. Theoretically, the software calculates the drag coefficient taken from the following equation:

Where:

- CD represents the drag coefficient
- a1, a2, and a3 are constants
- Re represents the Reynolds number

It can be concluded that the drag coefficient completely __depends on the Reynolds number__. For better visualization, see Fig. 2. This plot illustrates a clear relationship between drag coefficient versus Reynolds number. The graph shows three distinct regions:

- A
**linear**decrease from**10^-3 <Re< 10^2** - A more gradual decrease from
**10^2<Re<10^4** **Re>10^4**

The overall trend shows a decreasing relationship, with the drag coefficent values ranging **from 10^5 to 10^-1**.

Figure 2: Drag coefficient Vs. Reynolds number for Discrete Phase Model spherical drag law

## 2) DPM Non-Spherical Drag Law

In most cases, it is an idealistic view to consider all particle spheres. Realistically, the particles are mostly like **distorted spheres**. This approach is taken by using the Nonspherecial drag law in ANSYS Fluent. Avoiding the theoretical complex formulation, there is an important term inside the formulation that plays a critical role in calculations, known as **shape factor(ϕ)**. Fig. 3 depicts the nonspherical drag law model in ANSYS Fluent. After selecting this model, the software asks for a definition of the shape factor. The shape factor is a dimensionless number that quantifies the deviation of a particle’s shape from a perfect sphere. __φ ranges from 0 to 1__, where **1 represents a perfect sphere**.

Where:

**Cd,sphere**is the drag coefficient for a sphere, calculated based on the particle Reynolds number**ϕ**is the shape factor

Figure 3: Nonspherical drag law in ANSYS Fluent

**Tip**: Complex shapes may be estimated through experiments or detailed particle analysis. However, these shape factor values can be used:

- Sphere: 1.0
- Cube: ~0.81
- Cylinder: ~0.70

**Tip 2**: ANSYS Fluent visualizes all the particles as spheres and does not distinguish them by shape factor.

### Effect of shape factor on drag coefficient in different Reynolds

Fig. 4 is the graph illustrating the relationship between drag coefficient and Reynolds number for different shape factors. Reynolds number from 0.1 to 10^6, while the drag coefficient from 0.1 to 1000. Multiple curves are plotted, each corresponding to a different shape factor ranging from 0.1 to 1. All curves show a general decreasing trend as the Reynolds number increases, with the drag coefficient dropping at lower Reynolds numbers and leveling off at higher Reynolds numbers.

The curves for higher shape factors (φ = 0.8, 0.9, 1) closely overlap, especially at lower Reynolds numbers, while the curves for lower volume fractions (φ = 0.1, 0.2, 0.5) remain distinctly separated throughout the Reynolds number range.

Figure 4: Drag coefficient Verses Reynolds Number for nonspherical drag law

## 3) DPM Stokes-Cunningham Drag Law

This model modifies the standard Stokes drag law to account for __non-continuum effects__. According to fluid mechanics fundamentals, it becomes significant for particles with small diameters, in __sub-micron____ scale__. In other words, when the particle size approaches the mean free path of the fluid molecules.

The Stokes-Cunningham drag formulation is not our concern, but Fluent asks for the **Cunningham correction factor** after the selection of the Stokes-Cunningham drag law in the DPM panel (see Fig. 5). Here is the Cunningham factor formulation:

where:

- λ is the
**mean free path**of the fluid, - dp is the
**particle diameter**, - A1, A2, and A3 are constants.

The values of the constants A1, A2, and A3 can vary depending on the conditions and the specific fluid, but a __commonly used__ set of values for air at standard conditions is:

**A1=1.257 , A2=0.4 , A3=1.1**

So, the formulation simplify to this form:

For example, the mean free path of air at standard atmospheric pressure and temperature (20°C) is approximately 0.066 µm. let’s say we have a particle with a diameter of 0.1 µm. Given the λ=0.066 µm ,dp=0.1 µm: Cc≈2.889.

Figure 5: Stokes-cunningham drag law panel in ANSYS Fluent

## 4) DPM High-Mach-Number Drag Law

The **High-Mach-Number Drag Law** in ANSYS Fluent is specifically designed to handle the drag force calculations on particles moving at high velocities, particularly when:

**Mach Number > 0.4****Particle Reynolds > 20**

**Tip:** Particle Reynolds number is computed using the following equation:

Where:

- ρf is the fluid density (kg/m³).
- u is the relative velocity between the particle and the fluid (m/s).
- dp is the diameter of the particle (m).
- μ is the dynamic viscosity of the fluid (Pa·s or kg/(m·s)).

## 5) DPM Dynamic Drag Law

Initially, the dynamic drag law is only accessible when the droplet Break-__up__ model is enabled. This model is applicable in **almost any circumstance** because, naturally, the droplets don`t remain spherical after injection. Nevertheless, the drag coefficient drastically increases as the droplets move through the air, especially when the Weber number is large.

Drag coefficient of a distorted spherical droplet is always higher compared to the drag exerted on a sphere based on the following equation:

where **y** is the droplet distortion.

Y=0 ==> No distortion (**sphere**)

Y=1 ==> Maximum distortion (**disk**)

Figure 6: Dynamic drag law in DPM – Droplet distortion after spray

## 6-7) DPM Ishii-Zuber & Grace Drag Laws

**Ishii-zuber** and **Grace** drag laws are very alike. They are also called **Bubbly flow drag laws**. These models are well-suited for predicting the drag force in bubbly flows where the liquid is the continuous phase and the gas forms dispersed bubbles. However, apart from the basic formulation, each has its own application:

**Ishii-zuber**is the best option in cases where the gas phase is dispersed in the form of bubbles within a liquid. Common applications include**boiling flows, bubble columns, and other gas-liquid reactors**.- The
**Grace**drag model is an ideal choice for**granular****flows**, such as**fluidized****beds**, where**particle concentration is high**and interactions between particles is expected. It is also useful in**sedimentation**processes and other applications involving dense suspensions of solid particles.

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