**Greg Mehos**, director | Greg Mehos & Associates

Agglomeration is the process of converting fine powder particles into larger ones. This is often desirable to increase a material’s bulk density, reduce dust, and improve flowability. In wet agglomeration, the fine particles are wetted with a liquid, preferably water, which acts as a binder between the particles. Wet agglomeration equipment includes pin mixers, plough mixers, disk pelletizers, fluidized beds, and other technologies.

The optimal amount of liquid to add to a powder — the amount that gives green (or new) agglomerates their greatest strength — is typically 40 to 90 percent of the powder’s full saturation. The saturation level is the fraction of the total void space in the powder that can be filled with a liquid. When the liquid is added to a dry powder, liquid bridges will begin to form at contact points between particles. This is known as the pendular state of saturation. Moisture is attracted to the interfaces between powder particles by capillary forces. As the saturation level increases, the powder approaches the funicular state, where all internal solid surfaces become surrounded by liquid. At this point, the mixture becomes more paste-like, the tensional forces disappear, and the agglomerates become significantly weaker. When the powder becomes fully saturated, it reaches the capillary state, and at higher moisture levels, the mixture begins to behave as a slurry. The powder saturation states are shown in **Figure 1**.^{1}

Producing an agglomerate with the optimal saturation state is critical and requires careful control of the liquid and solids feedrates. Providing a steady liquid stream is usually not a problem, but providing a stable powder feedrate or ensuring a constant solids-to-liquid ratio may be a challenge with some powders. Mass-flow hoppers, in which the hopper walls are steep enough and low enough to allow powder to flow along the walls, are more likely to provide steady solids discharge rates. In addition, the size of the hopper outlet must be large enough to prevent a cohesive arch from developing and stopping flow. Methods for designing mass-flow hoppers are discussed in the article “Hopper Design Principles” by G. Mehos and D. Morgan.^{2}

While an outlet diameter greater than the minimum will prevent cohesive arching, it may not necessarily be large enough to allow the desired steady discharge rate, specifically if the powder is fine. A coarse powder’s steady-state discharge rate from a hopper can be determined from a force balance.

Consider a hopper with the geometry shown in **Figure 2**.

An equilibrium force balance on the powder inside the hopper is given by

**Equation 1**

where *a* is the powder’s acceleration, *ρ** _{b}* is its bulk density,

*g*is the acceleration due to gravity,

*P*is the interstitial gas pressure,

*z*is the axial coordinate (the origin is the hopper’s apex), and

*dP/dz*is the gas pressure gradient. To be able to calculate a solids discharge rate, we’ll need to derive an equation that provides a velocity, so we employ the following trick

**Equation 2**

where *v* is the powder’s velocity, and *t* denotes time. Hence, **Equation 2** can be rewritten as

**Equation 3**

If the bulk density is assumed constant

**Equation 4**

where *A* is the cross-sectional area of the hopper outlet. Therefore

**Equation 5**

For round outlets

**Equation 6**

**Equation 7**

At the hopper outlet

**Equation 8**

where *B* is the outlet diameter and *ϴ*′ is the hopper angle (from vertical). Therefore

**Equation 9**

and

**Equation 10**

where the subscript *o* denotes the hopper outlet.

A similar analysis for planar hoppers with flat walls and slotted outlets yields

**Equation 11**

where *B* is the outlet width.

The discharge rate, *ṁ** _{s}*, is the product of the powder’s velocity, its bulk density at the outlet,

*ρ*

*, and the cross-sectional area of the outlet,*

_{bo}*A*. Thus

_{o}**Equation 12**

where *m* is equal to 1 for conical hoppers and equal to 0 for hoppers with straight walls and slotted outlets.

Because the gas pressure gradient at the outlet, *dP/dz*|* _{o}*, is less than zero,

**Equation 12**shows that fine powders can have dramatically lower discharge rates than coarse powders. A fine powder’s maximum flowrate can be several orders of magnitude lower than that of coarser materials. Two-phase flow effects are significant due to the movement of interstitial gas as the powder compresses and expands during flow. Solids and gas pressure profiles in bins for coarse (high permeability) and fine (low permeability) powders are shown in

**Figure 3**.

In a bin’s straight-walled section, the stress level increases with depth, causing the material’s bulk density to increase and its void fraction to decrease, squeezing out a portion of the interstitial gas. This gas leaves the bulk material through its top, free surface. To flow in a bin’s hopper section, the consolidated material expands as it moves toward the outlet, reducing its bulk density and increasing its void fraction. This expansion results in a reduction of the interstitial gas pressure to below atmospheric (that is, a vacuum), causing gas counterflow through the outlet if the pressure below the outlet is atmospheric. A vacuum will develop even if the bin’s top is vented. At a critical solids discharge rate, the solids contact pressure reduces to zero, and efforts to exceed this limiting discharge rate will result in erratic flow.

The pressure gradient is related to the material’s permeability and the gas slip velocity, *u _{g}*, by Darcy’s law

**Equation 13**

Using Darcy’s law and applying continuity to the gas phase, the relationship between the air and solids flowrates can be derived

**Equation 14**

where *K _{o}* is the powder’s permeability at the hopper outlet,

*ρ*

*is its bulk density at the outlet, and*

_{bo}*ρ*

*is its bulk density at a location inside the hopper where the pressure gradient is equal to zero (that is, the gas pressure is at a minimum). A value of*

_{bmp}*ρ*

*equal to the powder’s bulk density at the solids stress at the cylinder-hopper junction is often used for design purposes,*

_{bmp}^{3}as

**Figure 3**shows that the maximum solids stress in the cylinder is approximately equal to the stress where the gas pressure gradient is zero.

^{4}The solids stress,

*σ*

*, at the cylinder-hopper junction can be calculated using the Janssen equation*

_{1max}**Equation 15**

where *R _{H}* is the hydraulic radius of the cylinder,

*k*is the Janssen coefficient (typically 0.4 to 0.6),

*ϕ*′ is the wall friction angle, and

*h*is the height of bulk material inside the cylinder.

Combining** Equations 13** and **14** yields a quadratic

**Equation 16**

The maximum steady solids discharge rate is the product of the powder’s velocity, its bulk density at the outlet, and the cross-sectional area of the outlet

**Equation 17**

To ensure a steady solids discharge rate and allow agglomerates with maximum strength and consistent properties, the hopper that feeds the agglomeration equipment should have an outlet large enough to prevent an arch and allow the required throughput.

**PBE**

# References

- G. Mehos and C. Kozicki, “Consider Wet Agglomeration to Improve Powder Flow,”
*Chemical Engineering*, Vol. 121, No. 1 (January 2011). - G. Mehos and D. Morgan, “Hopper Design Principles,”
*Chemical Engineering*, Vol. 126, No. 1 (January 2016). - K. Johanson, “Successfully Dealing with Erratic Flow Rates,”
*Powder Pointers*, Vol. 3, No. A (2009). - T. A. Royal, and J. W. Carson, “Fine Powder Flow Phenomena in Bins, Hoppers, and Processing Vessels,” presented at Bulk 2000: Bulk Handling Towards the Year 2000, London, 1991.

**For further reading**

Find more information on this topic in articles listed under “Agglomeration” in our article archive.

**Greg Mehos**, PE (978-799-7311) is a chemical engineering consultant specializing in bulk solids handling, storage, and processing and an adjunct professor at the University of Rhode Island. He received his BS and PhD in chemical engineering from the University of Colorado and his master’s from the University of Delaware. He’s a Fellow of the American Institute of Chemical Engineers.

**Greg Mehos & Associates • Westford, MA978-799-7311 • www.mehos.net**

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