Ball Mill Degree Of Filling Calculation: Complete Guide & Do

Ball mill degree of filling (also called the charge filling ratio or the degree of charge) is the single most influential operating parameter of a cement or raw grinding ball mill: it determines the mill power draw, the grinding efficiency, the product fineness and the wear of the grinding media. The degree of filling is defined as the percentage of the mill internal volume occupied by the ball charge (including the voids between the balls), and in operating practice it is measured by a simple and reliable geometric method: stopping the mill, measuring the distance from the mill shell top to the surface of the charge, and converting that measurement into a percentage with the help of the standard h/D relationship tables. This article is based on the widely used engineering workbook “Ball Mill Degree of Filling Calculation”, the Excel tool that performs the complete calculation chain: the h/D to filling degree conversion, the charge weight from the mill geometry and the bulk density, the critical speed of the mill, the operating speed, the mill power and the specific power consumption per tonne of product. The complete workbook is part of the Complete Cement Technical Package, the 931-file cement engineering library from cementequipment.org.

The degree of filling calculation is the foundation of every mill optimization: it tells the operator whether the mill is running too empty (wasting energy and wearing the lining through ball-to-ball impact without enough material), or too full (reducing the impact zone and risking media and diaphragm damage), and it is the first number checked when a mill’s production or fineness drifts. This guide explains the method, the formulas and the typical values, with the numerical example of the workbook reproduced throughout.

1. What the Degree of Filling Means in Mill Operation

The ball charge in a two-compartment cement mill typically occupies 26 to 32 percent of the mill internal volume, and this fraction is the degree of filling. The charge does the grinding: in the first (coarse) compartment, cascading and cataracting balls crush the coarse clinker; in the second (fine) compartment, smaller balls grind the material to cement fineness. The degree of filling sets the grinding action because it determines the number of balls, the active grinding surface and the energy that the mill can convert into grinding work: a higher filling means more balls, more surface and more power draw, up to the point where the charge becomes so deep that the balls cannot cascade properly and the power efficiency falls.

For this reason every mill has a designed degree of filling range. The design filling is fixed at the mill purchase from the grinding requirement (fineness, hardness and moisture of the feed, mill diameter and length ratio), and the operator maintains it by adding grinding media in a defined schedule: a new charge is installed at the design value, the charge slowly wears down (typically 40–120 g of media wear per tonne of material ground, depending on the material abrasiveness and the media quality), and the operator adds compensating balls periodically, checking the actual filling with the measurement method described below. The same measurement is used to audit the charge at mill stops, to calculate the exact charge weight for a complete re-ball, and to verify that the mill compartment filling distribution matches the design (a coarse compartment that has lost its large balls produces a coarse product no matter what the total filling shows).

2. The Measurement Method: H, De and the h/D Ratio

The degree of filling is measured when the mill is stopped, with the mill positioned so that the charge surface is horizontal and accessible through an inspection door or through the feed opening. The measurement procedure is: from the top inside of the mill shell (the point directly above the charge), measure the free distance down to the surface of the charge with a plumb line or a tape, at several points along the mill axis if the mill is long, and record the value H (in the workbook example, H = 2.86 m). The mill effective (inside lining) diameter De is taken from the mill drawings or measured (in the example, De = 4.2 m). The calculation then uses the height h, which is the free height measured from the centerline of the mill: h = H − De/2, i.e., the measured free height minus the mill radius (in the example: h = 2.86 − 2.10 = 0.76 m).

The dimensionless ratio h/De is the parameter that carries the geometry (in the example: h/De = 0.181), and the degree of filling is read from the standard conversion table of the method, which relates the ratio of the free height to the mill diameter to the percentage of the mill volume filled by the charge. The relationship is purely geometric: a charge filling q percent of the mill volume leaves a free segment at the top whose chord depth relative to the diameter is exactly h/De, and the table tabulates the conversion for every ratio from 0 to about 0.42. For the example, the ratio h/De = 0.181 corresponds to a degree of filling q = 27.4 percent, a typical design value for a cement mill first compartment. The table also lists, alongside the degree of filling q, a companion coefficient a (rising from 0.425 at empty to 0.647 at h/De = 0.2) which the method uses in the power calculation block to account for the shape and position of the charge.

h/De	Degree of filling q (%)	h/De	Degree of filling q (%)
0.10	37.4	0.20	25.2
0.12	34.9	0.25	19.6
0.14	32.4	0.30	14.5
0.16	30.0	0.35	10.0
0.18	27.6	0.40	5.9

The method’s accuracy depends on the measurement discipline: the mill must be completely stopped and the charge surface must be at rest (a freshly stopped mill may have a slightly raised surface from the mill’s rotation), the measurement point must be the highest point of the mill interior, and the measurement must be repeated at several positions if the charge surface is irregular. In practice, plant engineers measure the free height through the inspection doors and through the empty feed and discharge openings, and the average of the measurements is used; the method gives the filling to within a few tenths of a percent when executed carefully, which is entirely sufficient for charge management.

3. The Charge Weight: From Filling to Tonnes

The degree of filling is a percentage, but the mill operator needs tonnes: the charge weight is the quantity that is installed, audited and re-added, and it is the direct result of the filling calculation. The charge weight follows from the mill geometry and the bulk density of the charge:

Charge (t) = q% × π/4 × De² × L × ρbulk

where q is the degree of filling in percent, De the effective (inside lining) mill diameter in meters, L the compartment (or mill) length in meters, and ρbulk the bulk density of the ball charge in t/m³. The bulk density of a compact ball charge is not the density of steel (7.85 t/m³) but the apparent density of the bed including the voids between the balls, which for loose balls is about 4.5 t/m³ — the value used in the workbook example — and for a compacted, interlocked charge rises to about 4.9–5.0 t/m³. The workbook example applies the calculation to a first compartment with De = 4.2 m, length L = 5 m and a measured filling of 27.4 percent: the charge volume is 0.274 × π/4 × 4.2² × 5 = 18.99 m³, and with the bulk density of 4.5 t/m³ the charge weight is 18.99 × 4.5 = 85.5 tonnes, exactly the result of the workbook.

The same formula serves every charge management task: the initial charge is calculated from the design filling, the audit charge at each mill stop is measured and compared with the design, and the quantity of balls to be added to restore the design filling follows from the difference. The compartment-by-compartment application of the formula (each compartment has its own filling, length and ball size distribution) gives the complete charge balance of the mill, and the sum of the compartments is the total media inventory that the plant keeps on its spare parts record.

4. The Critical Speed and the Mill Operating Speed

The grinding action of the mill depends on the mill speed relative to the critical speed. The critical speed nc is the rotational speed at which the centrifugal force on a ball at the top of the mill just balances gravity, so that the ball would start to stick to the shell instead of cascading; it depends only on the mill diameter, by the classic formula:

nc = 42.3 / √De   (rpm, with De in meters)

For the example mill, nc = 42.3 / √4.2 = 42.3 / 2.049 = 20.64 rpm, exactly the workbook’s value. No mill runs at the critical speed: the operating speed is expressed as a percentage of the critical speed, and for cement and raw mills the standard range is 65 to 78 percent of nc, with most mills running at 70–75 percent. The workbook example uses a speed ratio of 0.70, giving an operating speed of n = 0.70 × 20.64 = 14.45 rpm, again exactly the workbook’s value.

The speed ratio is the operator’s second lever after the filling: at speeds near the low end of the range the charge cascades gently (finer grinding by attrition, used for fine compartments), while at speeds near the high end the charge cataracts with higher impact energy (used where coarse clinker must be broken). The speed ratio and the degree of filling interact: a mill that runs too slow with a high filling loses its cataracting action and grinds poorly, while a mill that runs too fast throws the balls against the lining instead of the material, wasting energy and wearing the lining. The combination of the measured filling and the selected speed ratio is what the mill operator records, and it defines the grinding regime of the mill.

5. The Mill Power Calculation

The power absorbed by a ball mill is the torque on the mill shell times the angular velocity, and the torque is created by the charge: as the mill rotates, the charge is lifted until it reaches its angle of repose, then it cascades down, and the displaced center of gravity of the charge produces a torque that the motor must overcome. The workbook’s power block computes the mill power from the charge weight, the mill diameter, the operating speed and the charge geometry coefficients of the method (the coefficient a from the h/De table and a torque coefficient, 0.73 in the example):

N = (2π n / 60) × T

where N is the absorbed power in kW, n the operating speed in rpm and T the charge torque in kNm, computed from the charge weight and the charge geometry. Applying the method to the example mill (charge 85.5 t, n = 14.45 rpm, torque coefficient 0.73, charge geometry coefficient a = 0.626 for h/De = 0.181) gives a charge torque of the order of 800 kNm and an absorbed power N = 1217.8 kW, the exact value reported by the workbook. This absorbed power is the power that actually grinds; the motor installed at the mill is larger by the mechanical losses of the gearbox, the pinion and girth gear, and the motor’s own efficiency, so a mill absorbing 1217.8 kW at the shell typically has an installed motor of 1400–1600 kW at the terminals.

The power formula shows the two variables that the operator controls: the power grows with the charge weight (the filling) and with the operating speed, and both are set by the grinding regime selected for the mill. The same formula is used in reverse for design: given a required mill power for a production target, the designer solves for the mill diameter, the charge weight and the speed, which is why the filling, the critical speed and the power are calculated together in the one workbook.

6. Specific Power Consumption: kWh per Tonne of Product

The final block of the workbook converts the mill power into the specific power consumption, the parameter that ties the mill directly to the plant economics: the specific power is the mill power divided by the production rate:

Specific power (kWh/t) = N / Production rate (t/h)

In the example, the mill produces 100 t/h and absorbs 1217.8 kW, giving a specific power of 12.18 kWh/t. This value is typical for a modern closed-circuit cement mill (9–15 kWh/t depending on the fineness, the clinker grindability and the circuit), while open-circuit mills run somewhat higher, and the same mill grinding raw meal with a high moisture and hardness requirement will show different values again. The specific power is the central mill performance figure because it merges the mill’s technical state (filling, speed, ball charge composition, liner condition, diaphragm and separator settings) into one number: a rising specific power at constant product quality signals a grinding problem (worn balls, wrong ball size distribution, liner profile wear, feed moisture), and a falling specific power at constant fineness usually signals an improvement (new media, better separator adjustment) or an upcoming problem (coarse product slipping through a worn diaphragm). The plant tracks the specific power daily, together with the mill throughput and the fineness, and the three numbers together describe the grinding circuit state completely.

7. Compartment-by-Compartment Filling and the Complete Mill Balance

In a two-compartment cement mill the two compartments have different jobs and therefore different fillings: the first (coarse) compartment breaks the clinker from 20–30 mm down to about 1–2 mm with large balls (60–90 mm) and runs at the higher filling (typically 29–32 percent), because a deep, energetic charge maximizes the impact energy that the coarse material needs; the second (fine) compartment grinds from 1–2 mm to cement fineness with small balls (17–40 mm) and runs at a slightly lower filling (typically 27–30 percent), because fine grinding is done mainly by surface attrition on a large number of small balls. The compartments are separated by the diaphragm, which keeps the coarse balls in the first compartment and lets the material pass, and the compartment lengths are designed from the power split: a typical cement mill gives 25–35 percent of the length and 30–40 percent of the power to the first compartment.

The complete mill balance therefore audits each compartment separately: the filling of each compartment is measured (through the feed opening for the first compartment and the discharge opening for the second, or during an internal inspection), the charge weight of each compartment is calculated with its own length and its own bulk density, and the total charge is the sum. The balance is closed by the media wear model: the plant records the ball additions per compartment over the operating period, converts them into tonnes added per 1000 tonnes of material ground, and compares the value with the known wear rate of the media type in that material; any unexplained difference between the calculated inventory and the actual additions is a red flag (media theft or loss through a broken diaphragm grate, ball classification changes, or an audit measurement error). The degree of filling is thus not a one-off number but the anchor of a continuous inventory system, exactly as the workbook’s charge and power blocks are designed to be re-run at every mill audit with the fresh measurement values.

8. Using the Degree of Filling in Daily Mill Operation

The degree of filling measurement is executed on every major mill stop and on a defined schedule (monthly or per operating hours), and its results feed three operational decisions. The first decision is the media addition: the measured filling of each compartment is compared with the design filling, and the missing tonnes are added as compensating balls, sized according to the ball charge composition rules of the companion tool in this package (the ball charge composition calculation). The second decision is the mill power management: the operator verifies that the mill draws the expected power at the measured filling and speed, because a mill that draws less than the charge predicts is losing its grinding action (charge slippage, liner wear, media breakdown), while a mill that draws more without producing more is wasting energy (over-filling, or a charge that has become too fine and compacted). The third decision is the production planning: the filling and the resulting power set the maximum production rate of the mill at the target fineness, and any production increase proposal starts from the filling calculation to check that the mill has the charge capacity to grind it.

The operating consequences of operating outside the design filling range are well documented in plant practice. An under-filled mill (filling below about 22 percent in a two-compartment cement mill) grinds with excessive ball-to-ball impact: the energy goes into the media and the lining instead of the material, the wear rate rises, the product overheats, and the free ball impacts crack the lining bolts. An over-filled mill (above about 33–35 percent) starves the cascade: the balls are cushioned in a deep bed, the coarse compartment cannot lift the material to the diaphragm, the mill “drowns”, the mill current rises toward the overload point, and the grinding collapses. The measured filling, checked against the mill’s known design range, is therefore the first troubleshooting step for every abnormal mill current, production or fineness pattern.

In the troubleshooting sequence, the filling check comes before any other mill investigation: the operator first verifies the mill current against the power predicted for the measured filling, then checks the media additions log, then the ball charge composition (with the companion composition calculation of this package), then the liner and diaphragm condition, and only then the process side (feed moisture, clinker grindability, separator settings). This ordering is deliberate, because the charge state explains the majority of mill performance problems: a mill whose specific power has crept upward while the filling dropped has lost charge and must be reballed before any separator adjustment will help; a mill whose current has risen while the filling rose above design is drowning and needs the charge reduced or the feed cut. The workbook’s numbers, kept in the daily mill log, turn each such diagnosis into a quantitative check instead of an opinion, which is why the filling measurement discipline is one of the strongest habits of the best grinding plants.

9. Accuracy, Repeatability and the Limits of the Method

The geometric measurement method has well-defined limits, and the competent engineer knows them. The method assumes that the charge surface is a horizontal plane, which is true only after the charge has settled with the mill stopped; the measurement through a single inspection door samples one point of the surface, so the practice of taking several readings (through the doors, the feed and discharge openings) and averaging is essential for a long mill. The method measures the total charge in the mill but cannot distinguish compartments: for a two-compartment mill with different fillings, the measurement gives a weighted result, and the compartment distribution must be estimated from the power balance or measured by internal inspection during a long stop.

The conversion table is computed for the ideal case of a spherical-charge surface with no liner effect; in practice the surface is disturbed by the liner profile, by the charge depth near the diaphragm and by the material filling the voids between the balls (the voids contain the material being ground, which makes the measured surface geometry slightly different from a pure ball charge). The error is small for charge management purposes, but it explains why the filling measurement and the power draw are always checked together: if the measured filling and the absorbed power disagree significantly with the method’s own power calculation, the engineer investigates (charge slippage, liner wear, diaphragm blockage, feed moisture) rather than trusting either number alone. Finally, the bulk density of the charge is not constant: a charge of worn, rounded balls has a different void fraction and apparent density than a fresh charge of sharp new balls, and the method’s standard value of 4.5 t/m³ should be adjusted from the actual measured density when the plant audits its true media inventory.

10. Frequently Asked Questions

Q1. How is the degree of filling of a ball mill measured?

With the mill stopped, measure the free height H from the top inside of the shell to the charge surface, subtract the mill radius to get h, divide by the mill diameter De to get h/De, and read the filling percentage from the standard conversion table. The example gives H = 2.86 m, De = 4.2 m, h/De = 0.181 and a filling of 27.4 percent.

Q2. What is the normal degree of filling of a cement mill?

The ball charge normally occupies 26 to 32 percent of the mill internal volume; two-compartment cement mills typically run at 27–30 percent with the coarse compartment at the higher end of the range. Below about 22 percent the mill grinds wastefully, and above about 33–35 percent it drowns.

Q3. How is the charge weight calculated from the filling?

Charge (t) = filling% × π/4 × De² × L × bulk density. With a 4.2 m effective diameter, a 5 m compartment, 27.4 percent filling and a bulk density of 4.5 t/m³, the charge is 85.5 tonnes, the example of the workbook.

Q4. What is the critical speed of a ball mill and how is it used?

The critical speed is the speed at which the balls would stick to the shell: nc = 42.3/√De rpm (20.6 rpm for a 4.2 m mill). The mill runs at 65–78 percent of this value (70 percent, i.e., 14.45 rpm, in the example), because the cascading grinding action requires a speed below the critical point.

Q5. What is a normal specific power consumption of a cement mill?

About 9–15 kWh/t for modern closed-circuit cement mills (12.2 kWh/t in the workbook example at 100 t/h production and 1217.8 kW absorbed power), depending on the fineness target, the clinker grindability, the mill condition and the separator performance.

Q6. Why does the measured filling disagree with the mill power draw?

Because the filling measurement is geometric and assumes an ideal charge, while the power depends on the actual charge behaviour: slippage of the charge, liner profile wear, media size degradation, material in the voids and diaphragm condition all shift the real power away from the calculated value. The two numbers are checked together, and a significant disagreement is itself a diagnostic.

Q7. How often should the degree of filling be measured?

At every major mill stop and at the regular audit interval (typically monthly or every few thousand operating hours), and additionally whenever the mill current, the production rate or the fineness drifts without a clear cause. Each measurement updates the charge balance, the media addition schedule and the specific power monitoring, so the measurement is an operating routine, not a one-off engineering task.

11. Final Summary

The degree of filling is the master parameter of ball mill operation, and the calculation workbook turns the simple stop-and-measure procedure into the complete chain of mill numbers: the measured free height H and the ratio h/De give the filling percentage from the standard conversion table, the filling and the mill geometry with the charge bulk density give the charge weight in tonnes, the critical speed formula gives the mill speed base, the speed ratio and the charge geometry give the absorbed power, and the power with the production rate gives the specific power consumption in kWh/t. The example calculation (27.4 percent filling, 85.5 t charge, 20.6 rpm critical speed, 14.45 rpm operating speed, 1217.8 kW and 12.2 kWh/t) is a complete, typical mill audit in one page. Applied compartment by compartment and repeated on the plant’s audit schedule, the method gives the mill operator the full charge inventory, the media wear tracking and the grinding regime verification that keep the mill inside its design window.

The complete workbook is part of the 931-file Complete Cement Technical Package, together with the ball charge composition and piece weight calculator, the kiln loading tool, the bag filter calculations and the full engineering library of books, manuals and courses covering the whole cement process. Get the entire package with one PayPal payment — instant download, lifetime access — and keep the complete cement engineering reference on your desk.

This file is part of the Complete Cement Technical Package (931 files) available from cementequipment.org. Respective rights holders; library copy for the licensed single user.