Population Balance Vertical Mill

The Population Balance Vertical Mill model represents vertical stirred 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 vertical mill product size distribution must be estimated from feed size distribution, mill geometry, screw speed, media filling, selection-function parameters and component-specific breakage parameters.

DPSIM model key: DPSIM.Comminution.SimplifiedVerticalMill
Category: Comminution
Subcategory: Mills
Display name: Population Balance Vertical Mill

Parameters

Derived parameters

Model Description

The Population Balance Vertical 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 calculates the apparent media charge volume:

$$V_{ch} = \pi\frac{D_{c}^{2}H_{c}J_{B}}{4}$$

The grinding media charge mass is calculated as:

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

The interstitial slurry mass is calculated as:

$$M_{IS} = \rho_{p}\varepsilon\ V_{ch}$$

The apparent charge density is:

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

Where:

The wall/screw gap is calculated as:

$$g_{ws} = \max\left( 0,\frac{D_{c} - D_{s}}{2} \right)$$

Where:

The net power per mill is calculated using the empirical vertical mill power equation:

$$P_{net} = K_{P}\left( \frac{H_{c}\left( J_{B},\% - 5 \right)}{100} \right)^{0.884}D_{s}^{2.232}g_{ws}\left( \frac{N_{s}}{60} \right)^{1.232}$$

Where:

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 wall/screw gap is a derived parameter. It is recalculated from chamber diameter and screw diameter during the model calculation and is not directly fitted.

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 stirred mill model.

The power calculation is an empirical vertical mill power estimate. The power constant should be calibrated for the mill type, geometry, media charge and operating range being represented.

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 and power calculation are common to all components.