In our previous blog, we explained the 7 standard Discrete Phase Model (DPM) drag laws. Now, we look at the other seven choices in ANSYS Fluent. These are for the Dense Discrete Phase Model (DDPM). Many users find DDPM confusing. Because of this, we will briefly introduce it before we look at the formulas.
What is the Dense Discrete Phase Model (DDPM)?
Sometimes, we deal with flows that have many particles. These particles are packed tightly. They push each other and change the main fluid. When the particle volume fraction is high (typically between 10% and 50%), the DDPM approach is used. It is a much wiser choice than standard DPM.
DDPM extends the standard model. It accounts for particle-particle interactions. Also, it calculates the fluid volume displaced by the particles. You can find this option in the Hybrid Models group box under the Multiphase Model section (see Fig. 1). It is an Eulerian-Lagrangian approach, even though it sits next to Eulerian-Eulerian models. We use DDPM for applications like fluidized beds, pneumatic conveying systems, cyclone separators, and coal combustion furnaces.
Figure 1: Dense Discrete Phase Model in Multiphase model section of ANSYS Fluent
How DDPM Drag Laws Work
When DDPM is activated in ANSYS Fluent, the software makes a change. The ‘averaged-discrete-phase-drag’ option for the Drag Coefficient is automatically selected. This method is based on Equation 14-499 in the Fluent Theory Guide. It calculates the interaction between the discrete and fluid phases. It does this by using averaged values of fluid drag for all particles in a fluid cell.
The momentum exchange coefficient, K_{dpm}, is computed like: K_{dpm} = \frac{\sum F_{i,drag} \Delta t_i}{\sum \Delta t_i}
At each time step, Fluent evaluates the drag for individual particles. It uses the drag law you specify in the Set Injection Properties dialog box (Fig.2). Then, an average drag is computed. This average is weighted by each particle’s residence time in the cell.
Figure 2: DDPM drag laws in ANSYS Fluent
1) Wen-Yu DDPM Drag Law
Wen-Yu is the first model for computing drag force on particles in DDPM. The formula is a math correlation. It accounts for the particle Reynolds number.
The Wen-Yu formula for the drag coefficient (CD) is given by:
\[ C_D = \frac{24}{\alpha_f Re_s} \left[ 1 + 0.15 (\alpha_f Re_s)^{0.687} \right] \]
We usually use the Wen and Yu model specifically for dilute phase flows. In dilute flows, the volume fraction of the dispersed phase is very low. It is much lower than the primary fluid phase. Therefore, the dispersed phase has a negligible effect on the overall flow.
2) Gidaspow DDPM Drag Law
The Gidaspow drag model is very popular. It combines two well-known drag models: the Wen-Yu model and the Ergun equation. The software chooses between them based on the void fraction.
- For dilute flows (void fraction > 0.8): It uses the Wen-Yu model (shown above).
- For dense flows (void fraction ≤ 0.8): It uses the Ergun equation.
The Gidaspow formula for the dilute drag coefficient (CD) is given by:
c_D = \frac{24}{\alpha_f Re_s} [1 + 0.15(\alpha_f Re_s)^{0.687}]
When the flow is dense, the Ergun equation is used to find the fluid-solid exchange coefficient (Ksl):
K_{sl} = 150 \frac{\alpha_s^2 \mu_f}{\alpha_f^2 d_s^2} + 1.75 \frac{\rho_f \alpha_s |v_s - v_f|}{d_s}
Where:
- CD = drag coefficient
- αf = volume fraction of the fluid phase
- αs = volume fraction of the solid phase
- Res = particle Reynolds number
- ds = particle diameter
We highly recommend the Gidaspow model for dense fluidized beds.
3) Huilin-Gidaspow DDPM Drag Law
As the name says, this model is a refined version of the original Gidaspow model. It introduces improvements. These improvements fix some limitations of the standard Gidaspow law. Primarily, the enhancement is a better blending function. This new function provides a smoother transition between the dense packing limit and the dilute flow limit. Specifically, this smooth switch works when the solid volume fraction is less than 0.2.
The application is similar to the Gidaspow model. I use it for Fluidized Beds where the particle concentration may vary in different regions.
4) Syamlal-Obrien DDPM drag law
The Syamlal-O’Brien drag model works based on the terminal velocities of particles. The deep fundamentals are not our main concern here. However, you need to know the formula. The drag coefficient (CD) is given by:
\[ C_D = \left( 0.63 + \frac{4.8}{\sqrt{Re_s / v_{r,s}}} \right)^2 \]
Where:
- CD is the drag coefficient
- Res is the particle Reynolds number
- vr,s is the terminal velocity correlation for the solid phase
As you can see, CD is a function of the volume fraction and the relative Reynolds number. This allows the model to account for particle concentration and flow conditions.
This model is very useful in gas-solid fluidized beds and any gas-solid system.
Tip: We often find that the Syamlal-O’Brien model provides more precise results for gas-solid flows compared to the Gidaspow model. This is true especially in systems with varying particle concentrations.
5) Gibilaro DDPM drag law
The Gibilaro model is another DDPM option. L.G. Gibilaro developed it. It is based on a theoretical framework. This framework accounts for fluid-particle interactions more dynamically than simple models.
The ANSYS Guide recommends the Gibilaro drag law for the simulation of circulating fluidized beds.
6) EMMS DDPM drag law
The EMMS (Energy-Minimization Multi-Scale) drag law is mostly used for gas-solid flows. This model is designed for cases where meso-scale flow structures are dominant. Sometimes, these small structures cannot be resolved using fine grid resolution. In the EMMS method, the mesoscale structures are broken down into two parts: a cluster phase and a dilute phase. The drag coefficient is given by:
\[
C_D = \begin{cases}
0.44, & Re_s \geq 1000 \\
\frac{24}{Re_s} \left( 1 + 0.15 Re_s^{0.687} \right), & Re_s < 1000
\end{cases}
\]
The coefficients α, β, and γ are functions of the gas phase volume fraction. You can find the formulas for these in the tables within the Fluent Theory Guide.
Application: The EMMS drag model is beneficial for gas-solid flows and monodispersed granular flows. We use it when the Wen-Yu or Gidaspow models over-predict the solids flux. In conclusion, the EMMS law is very accurate compared to traditional models, despite the higher computational cost.
7) Filtered DDPM drag law
The Filtered drag law is an advanced approach. It is mostly used for gas-particle flows, particularly in large-scale fluidized bed systems.
When should you choose the Filtered drag law? In fact, this model is designed to solve a specific problem. It helps simulate monodispersed two-phase flows on coarse meshes. For example, it is the best choice for catalyst particles (ranging from 20 μm to 100 μm) in large-scale beds.
We have already discussed the importance of the grid in my Grid Independence CFD guide. In large-scale flows, structures range from microscale to macroscale. Traditional models require very fine mesh resolutions (often just a few particle diameters). The Filtered drag model does not need this. Therefore, it is a perfect alternative for simulating gas-solid flows with coarse meshes.
Conclusion
Choosing the correct drag law is the most important step in DDPM. If your flow is dilute, Wen-Yu is often enough. However, if you simulate a dense fluidized bed, models like Gidaspow or EMMS must be used. In fact, the wrong choice can completely ruin your results. In this guide, I explained all 7 options to help you make the right decision.
Sometimes, setting up these complex particle tracking models is very difficult. If you want to see our professional offerings, please visit our DPM CFD Simulation Services category. Also, if you need an expert to do the hard work for you, you can easily order your CFD project here. We are always ready to help you with your simulations.