Herbst-Fuerstenau Ball Mill

Summary

The Herbst-Fuerstenau Ball Mill model represents continuous ball mill grinding using the population balance scale-up procedure proposed by Herbst and Fuerstenau in “Scale-up procedure for continuous grinding mill design using population balance models”, International Journal of Mineral Processing, 1980. The model is based on a linear population balance formulation in which breakage kinetics, material transport and energy input are represented explicitly.

The model should be used when ball mill product size distribution must be estimated from feed size distribution, mill geometry, ball filling, pulp loading, mill speed, discharge mode, residence-time-distribution approximation and component-specific breakage and selection parameters.

In this DPSIM implementation, the mill power, ball load, solids holdup and mean residence time are calculated internally from the mill geometry and operating parameters. The grinding response is then calculated for each component using a population balance solution represented by a series of perfectly mixed stages.

DPSIM model key: DPSIM.Comminution.HerbstFuerstenauBallMill
Category: Comminution
Subcategory: Mills
Display name: Herbst-Fuerstenau Ball Mill

Parameters

Derived parameters

Model Description

The Herbst-Fuerstenau Ball Mill model receives one feed stream and generates one product stream. The product stream preserves the feed solids flowrate and water flowrate. The product size distribution and component-by-size matrix are recalculated from the population balance model.

The effective mill length is calculated as:

$$L = R_{LD}D$$

The mill volume is:

$$V_{m} = \frac{\pi D^{2L}}{4}$$

Where:

The ball load per mill is calculated as:

$$M_{B} = (1 - \varepsilon)\rho_{B}J_{B}V_{m}$$

The solids holdup per mill is calculated as:

$$H = \rho_{p}J_{P}V_{m}X_{S}$$

Where:

The apparent charge density used in the power equation is calculated as:

$$\rho_{app} = \frac{M_{B} + M_{P}}{J_{B}V_{m}}$$

where:

$$M_{P} = \rho_{p}J_{P}\varepsilon\ J_{B}V_{m}$$

The net power per mill is calculated using a simplified tumbling mill power expression:

$$P_{net} = 0.238D_{ft}^{3.5}\left( \frac{L_{ft}}{D_{ft}} \right)N_{c}\rho_{app}\left( J_{B} - 1.065J_{B}^{2} \right)\sin\theta$$

The calculated power per mill is then corrected as:

$$P_{mill} = P_{net}f_{B}f_{P}$$

with:

$$f_{B} = 0.90928 + 0.7741J_{B}$$

Where:

The total calculated power is:

$$P_{total} = N_{parallel}P_{mill}$$

The power intensity used in the selection function is:

$$P_{H} = \frac{P_{mill}}{H}$$

The mean residence time is:

$$\tau = \frac{60H}{M_{S}},mill^{F}$$

with:

$$M_{S},mill^{F} = \frac{M_{S}^{F}}{N_{parallel}}$$

Where:

For each component c and active size class i, the selection rate is calculated as:

$$S_{c,i} = \frac{a_{c}\left( \frac{P_{H}}{60} \right)\left( \frac{x_{i}}{x_{crit,c}} \right)_{c}^{\alpha}}{1 + \left( \frac{x_{i}}{\mu_{c}} \right)_{c}^{\lambda}}$$

Where:

The last active size class is treated as terminal:

$$S_{c,N} = 0$$

The discharge correction is applied to the selection rates. For grate discharge, the rates are unchanged:

$$S_{c,i}^{*} = S_{c,i}$$

For overflow discharge, the current implementation applies:

$$S_{c,i}^{*} = 1.365\ S_{c,i}$$

Where:

For each component, the breakage distribution is calculated from a cumulative breakage function. The effective PHI value for a parent size class j is:

$$\varphi_{c,j}^{*} = \min\left( 1,\max\left( 0,\varphi_{c}\left( \frac{y_{j}}{y_{ref}},c \right)^{- \delta_{c}} \right) \right)$$

The cumulative breakage function is:

$$B_{c\left( x;y_{j} \right)} = \varphi_{c,j}^{*}\left( \frac{x}{y_{j}} \right)_{c}^{\gamma} + \left( 1 - \varphi_{c,j}^{*} \right)\left( \frac{x}{y_{j}} \right)_{c}^{\beta}$$

Where:

The differential breakage distribution for active classes is calculated as:

$$b_{ij,c} = 0,\ i \leq j$$

$$b_{ij,c} = B_{c\left( D_{i - 1};y_{j} \right)} - B_{c\left( D_{i};y_{j} \right)},\ i > j$$

Where:

The mill residence time distribution is approximated by N_m perfectly mixed stages in series. For each mixer stage and each active size class i, the product of the mixer is calculated recursively as:

$$m_{i}^{out} = \frac{m_{i}^{in} + \tau_{m}sum_{j = 1}^{i - 1}b_{ij,c}S_{c,j}^{*}m_{j}^{out}}{1 + \tau_{m}S_{c,i}^{*}}$$

with:

$$\tau_{m} = \frac{\tau}{N_{m}}$$

Where:

The model solves the population balance independently for each component. The calculated component retained masses are recombined into the total product retained distribution:

$$m_{i}^{P} = sum_{c}m_{c,i}^{P}$$

The product retained fraction is:

$$p_{i}^{P} = \frac{m_{i}^{P}}{M_{S}^{F}}$$

The product component fraction in each size interval is:

$$z_{c,i}^{P} = \frac{m_{c,i}^{P}}{m_{i}^{P}}$$

Where:

The parameters “Reserved - largest ball size (mm)” and “Reserved - ball size factor (0-disabled)” are retained for compatibility and future development, but they do not affect the current calculation.

The power equation is a simplified tumbling mill power estimate and should be calibrated before use for design conclusions.

The discharge correction is simplified. In the current implementation, overflow discharge applies a constant kinetic multiplier, while grate discharge leaves the selection rates unchanged.

The model requires calibrated component-specific selection and breakage parameters. The component-specific breakage and selection parameters allow different components to have different grinding behavior, but they should be fitted against laboratory, pilot or plant data whenever possible.

References

Herbst, J. A.; Fuerstenau, D. W. (1980). Scale-up procedure for continuous grinding mill design using population balance models. International Journal of Mineral Processing, 7, 1–31.

Herbst, J. A.; Fuerstenau, D. W. (1973). Mathematical simulation of dry ball milling using specific power information. Transactions AIME, 254, 343–348.

Austin, L. G. (1971–1972). A review introduction to the mathematical description of grinding as a rate process. Powder Technology, 5, 1–17.

Reid, K. J. (1965). A solution to the batch grinding equation. Chemical Engineering Science, 20, 953–963.

Rose, H. E.; Sullivan, R. M. E. (1958). A Treatise on the Internal Mechanics of Ball, Tube and Rod Mills. Chemical Publishing Company.