Population Balance Ball Mill

Summary

The Population Balance Ball Mill model represents ball mill grinding using the population balance grinding framework developed by Austin, Klimpel and Luckie in Process Engineering of Size Reduction: Ball Milling, with the analytical batch-grinding solution originally derived by Reid in “A solution to the batch grinding equation”, Chemical Engineering Science, 1965. The model also follows the energy-based scale-up concept associated with Herbst and Fuerstenau, where the grinding response is related to the specific energy applied to the mill contents.

The model should be used when ball mill product size distribution must be estimated from feed size distribution, mill geometry, operating speed, charge filling, ball filling, selection-function parameters and component-specific breakage parameters.

In this DPSIM implementation, the model uses a simplified ball-charge power calculation to estimate the specific energy applied per mill. The population balance is solved using an analytical matrix solution with a three-stage residence-energy approximation.

DPSIM model key: DPSIM.Comminution.SimplifiedBallMill
Category: Comminution
Subcategory: Mills
Display name: Population Balance Ball Mill

Parameters

Derived parameters

Model Description

The Population Balance 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 model first estimates the mill charge volume:

$$V_{ch} = J\ \pi\frac{(0.305D)^{2}(0.305L)}{4}$$

The ball charge mass is calculated as:

$$M_{B} = (1 - \varepsilon)\rho_{B}J_{B}\pi\frac{(0.305D)^{2}(0.305L)}{4}$$

The interstitial slurry mass is calculated as:

$$M_{IS} = \rho_{p}U_{IS}\varepsilon\ J_{B}\pi\frac{(0.305D)^{2}(0.305L)}{4}$$

The apparent charge density is:

$$\rho_{app} = \frac{M_{B} + M_{IS}}{V_{ch}}$$

Where:

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

$$P_{net} = 0.238\frac{D^{3.5}L}{D}N_{c}\rho_{app}\left( J - 1.065J^{2} \right)\sin\theta$$

The ball-charge power used for the specific energy calculation is:

$$P_{B} = \frac{P_{net}M_{B}}{V_{ch}\rho_{app}}$$

The specific energy basis used in the population balance solution is:

$$E_{bar} = \frac{P_{B}}{M_{S}},mill^{F}$$

with:

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

Where:

The component breakage function is expressed in cumulative form as:

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

Where:

The differential breakage matrix is obtained from the cumulative breakage function:

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

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

$$b_{Nj,c} = B_{c\left( D_{N};y_{j} \right)}$$

Where:

The selection function is calculated as:

$S_{i} = a\frac{\left( \frac{x_{i}}{x_{\max}} \right)^{\alpha}1}{1 + \left( \frac{x_{i}}{\mu} \right)^{\lambda}}$ere:

The last size class is treated as an absorbing terminal class:

$$S_{N} = 0$$

The population balance solution is calculated using the analytical matrix form:

$$P_{c} = T\ J\ T^{- 1}F_{c}$$

Where:

The transfer matrix is calculated recursively:

$$T_{ij} = 0,\ i < j$$

$$T_{ii} = 1$$

$$T_{ij} = \frac{sum_{k = j}^{i - 1}b_{ik}S_{k}T_{kj}}{S_{i} - S_{j}},\ i > j$$

The diagonal residence-energy matrix uses a three-stage approximation:

$$J_{ii} = \frac{1}{\left( 1 + \frac{S_{i}E_{bar}}{N_{m}} \right)_{m}^{N}}$$

with:

$$N_{m} = 3$$

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 terminal size class is treated as an absorbing class with zero selection. Residual component mass is assigned to the terminal class to close the component mass balance.

The residence-energy term uses a three-stage approximation rather than a pure plug-flow exponential solution. The model should therefore be treated as a simplified population balance representation rather than a rigorous hydrodynamic ball mill model.

The power calculation is a simplified ball-charge power estimate. It should not be used as a detailed mill power design method without calibration.

The model requires calibrated selection and breakage parameters. The component-specific breakage parameters allow different components to have different breakage behavior, but the selection function is common to all components.