Ball Mill Calculations: Calculations & Excel Sheet

The ball mill calculations are the working numbers of the finish grinding plant: the critical speed of the mill, the operating speed, the ball charge weight and the composition, the degree of filling, the power consumption, the specific surface area of the product, the ventilation and the mill heat balance: the complete set of the calculations that the grinding engineer uses to design, to operate and to optimize the ball mill of the cement plant. The Excel workbook of this article is a dedicated ball mill calculation tool: the working file with the calculation sheets of the mill dimensions and the speeds, the ball charge and the grinding media, the power and the production estimates, and the optimization parameters that the grinding engineers of the industry use for the daily and the project work.

This article walks the reader through the complete set of the ball mill calculations: the mill geometry and the critical speed, the ball charge and the filling degree, the power consumption by the Bond method and the practical formulas, the specific energy and the Blaine, the ventilation and the cooling, the heat balance of the mill and the optimization practice with the worked numbers throughout. The Complete Cement Technical Package (931 files: the handbooks, the courses, the Excel tools and the presentations, $249.99 one-time, instant download through the secure PayPal payment) delivers the ball mill calculations workbook together with the complete library of the grinding and the process calculation tools.

1. The Ball Mill and Its Parameters

The ball mill is the classic grinding machine of the cement plant: the horizontal rotating cylinder, partially filled with the steel balls, in which the clinker and the gypsum are ground by the impact and the attrition of the cascading and the cataracting balls. The mill of the cement grinding is typically 3–5.2 meters in diameter and 10–16 meters in length, driven through the central or the girth gear at the 14–18 revolutions per minute, and filled with the 80–250 tonnes of the steel balls in the two or the three compartments:

• The compartments: the two-compartment mills of the open circuit and the closed circuit systems: the first compartment with the large balls (the 60–90 mm) for the coarse grinding, the second with the smaller balls (the 15–40 mm) for the fine grinding, and the three-compartment mills for the finer products: the compartment lengths and the ball sizes are the design variables of the mill;
• The grinding action: the balls lifted by the rotating shell to the top of the charge and cascading down on the material: the impact at the bottom of the charge crushes the coarse particles, and the attrition between the balls grinds the fines: the balance of the cascade and the cataract is set by the mill speed;
• The process variables: the mill speed (the percent of the critical), the ball charge weight and the composition, the liner profile, the feed rate and the fineness, the ventilation air and the cooling water: the calculation sheet of the package holds the formulas of all the variables;
• The cement grinding circuit: the closed circuit with the dynamic separator returns the oversize particles to the mill: the circulating load of the 150–300 percent: the calculation of the circuit (the feed, the product, the rejects) belongs to the same workbook family as the mill calculations;
• The purpose of the calculations: the design of the new mills, the verification of the operating mills, the optimization of the ball charge and the energy and the quality targets: the calculations of the sheet serve the grinding engineer in the daily and the project work.

The ball mill calculations of the package cover the complete parameter set of the grinding machine, and the article walks the reader through the formulas and the worked numbers one by one.

2. The Critical Speed and the Operating Speed

The first calculation of the ball mill sheet is the critical speed: the speed at which the balls at the mill shell are held against the lining by the centrifugal force and stop grinding, and the operating speed that follows the critical:

• The critical speed formula: n_c = 42.3 / √D, with the n_c in the revolutions per minute and the D the effective mill diameter (the inside diameter in the meters): the formula derived from the balance of the centrifugal force and the gravity at the top of the charge: for the mill of the 4.0 meters effective diameter: n_c = 42.3 / √4.0 = 42.3 / 2.0 = 21.15 rpm: the critical speed of the mill;
• The operating speed: the ball mills run at the 65–78 percent of the critical speed: the practical operating range of the industry: for the example mill at the 72 percent: n = 0.72 × 21.15 = 15.2 rpm: the operating speed of the mill: the typical values of the cement mills at the 14–17 rpm;
• The effect of the speed: below the 65 percent of the critical the balls cascade (the rolling down the charge surface, the attrition-dominated grinding), and above the 78 percent the balls cataract (the flight through the space, the impact-dominated grinding): the cement grinding favors the cascading regime with the high attrition, and the raw grinding with the coarse feed favors the cataracting: the operating speed of the cement mill is set to the lower part of the range;
• The speed and the power: the mill power rises with the speed to the optimum at about the 75–85 percent of the critical and falls beyond: the operating point of the mill balances the power, the grinding and the liner wear: the calculation sheet’s power rows follow the speed row;
• The wear consideration: the high speeds increase the ball and the liner wear: the operating speed of the industry practice at the 70–75 percent is the compromise of the grinding and the wear: the sheet’s recommended range guides the adjustments.

The critical speed calculation is the foundation of the mill sheet: the operating speed, the power and the ball behavior all follow from the diameter and the speed of the mill, and the formula of the example (the 21.15 rpm critical and the 15.2 rpm operating) is the first number of the design.

3. The Ball Charge: The Weight and the Filling Degree

The ball charge is the grinding tool of the mill, and its weight, its filling degree and its composition are the primary optimization variables of the grinding engineer:

• The charge weight: the weight of the balls in the mill = the mill volume × the filling degree × the bulk density of the charge: the internal volume of the example mill (the 4.0 m diameter and the 12 m length: V = π/4 × 4.0^2 × 12 = 150.8 m3), the filling degree of the 30 percent and the ball bulk density of the 4.65 t/m3: the charge weight = 150.8 × 0.30 × 4.65 = 210.4 tonnes of the balls: the charge weight of the cement mills at the 30–35 percent of the mill volume;
• The filling degree: the percent of the mill volume occupied by the static ball charge: the 28–35 percent of the cement mills: the filling measured by the free height of the charge (the standard method of the industry: the mill stopped and the surface height measured) or by the power draw correlations: the filling degree of the example at the 30 percent: the filling above the 35 percent reduces the impact space, and below the 28 percent the power and the grinding drop;
• The charge composition: the ball size distribution of the charge matches the grinding task: the first compartment with the 60–90 mm balls for the coarse clinker, the second with the 15–40 mm: the optimal composition follows the feed size distribution and the target fineness: the stepwise distribution of the practice (the 40 percent large, the 40 percent medium, the 20 percent small in the typical two-compartment charges):
• The media consumption: the ball wear of the 400–800 grams per tonne of the cement (the 0.04–0.08 percent of the charge): the media cost line of the cost sheet: the wear depends on the ball quality, the clinker grindability and the mill conditions: the sheet’s media rows track the consumption against the production;
• The top-up practice: the daily or the weekly top-up of the charge with the largest balls compensates the wear and maintains the charge level: the top-up calculation of the sheet: the required addition per period from the measured charge level.

The ball charge calculations give the grinding engineer the numbers of the daily operation: the charge weight to load, the filling to measure and the composition to maintain, and the workbook’s charge sheet organizes the three into the single calculation page.

4. The Power Consumption: The Bond Method

The power consumption of the ball mill is estimated by the Bond method, the classical energy method of the grinding industry, and the calculation sheet carries the standard formula:

• The Bond work index: the grindability of the material measured by the Bond test: the work index of the cement clinker at the 13–18 kWh/t, the gypsum at the 8–10, the raw materials at the 10–15: the work index is the material property that the Bond formula uses: the test of the package’s laboratory sheets determines the index;
• The Bond formula: the energy of the grinding from the feed and the product sizes: E = Wi × 10 × (1/√P80 − 1/√F80), with the E in the kWh per tonne, the Wi the work index in the kWh per tonne, and the P80 and the F80 the product and the feed 80-percent-passing sizes in the microns: the worked example: the clinker with the Wi of the 15.0 kWh/t, the feed F80 of the 25,000 microns and the product P80 of the 30 microns: E = 15.0 × 10 × (1/√30 − 1/√25,000) = 150 × (0.1826 − 0.0063) = 150 × 0.1763 = 26.4 kWh/t: the Bond energy of the grinding to the 30 microns;
• The practical power formula: the installed power of the mill estimated by the practice formula: P = 7.4 × D^2.5 × L × f, with the D the internal diameter (4.0 m), the L the length (12 m) and the f the filling factor (about the 0.95 at the 30 percent filling): P = 7.4 × 4.0^2.5 × 12 × 0.95 = 7.4 × 32.0 × 12 × 0.95 = 2,700 kW: the mill drive power of the example: the installed motor at the 3,000 kW with the drive losses;
• The specific power: the power per tonne of the product: the cement grinding at the 30–42 kWh/t of the cement (the open circuit) and the 26–36 (the closed circuit with the separator): the production of the example mill: the 2,700 kW / the 35 kWh/t = about the 77 t/h of the cement at the 3,000 Blaine: the production estimate of the sheet;
• The energy efficiency: the ball mill converts the 1–3 percent of the input energy into the new surface, the rest to the heat and the noise: the efficiency gap that the VRM and the high-pressure grinding rolls close: the energy context of the ball mill calculations.

The power calculations of the sheet estimate the mill demand, the production and the specific energy, and the worked numbers of the example (the 26.4 kWh/t by the Bond, the 2,700 kW power and the 77 t/h production) give the complete power picture of the mill.

5. The Fineness and the Blaine

The product fineness of the cement mill is the quality measure of the grinding, and the calculation sheet connects the fineness parameters to the mill operation:

• The Blaine specific surface: the surface area of the cement per unit of the mass (the cm2/g or the m2/kg), measured by the air permeability method of the standard (the Blaine apparatus): the typical cements at the 2,800–3,500 cm2/g (the 280–350 m2/kg), the high-strength grades above the 4,000: the Blaine is the primary fineness target of the mill;
• The residue on the sieves: the percent retained on the 45 micron sieve (the standard fineness measure): the 5–15 percent of the ordinary cements: the 90 micron residue of the coarser control: the sieve residues and the Blaine together describe the fineness distribution;
• The specific energy to the fineness: the energy per tonne rises with the target fineness: the rule of the practice: the 3–4 kWh/t per each 100 cm2/g of the Blaine increase in the normal ranges: the mill power and the production trade off against the Blaine: the sheet’s production rows show the trade-off;
• The separator and the circulating load: the closed circuit with the dynamic separator classifies the mill product: the circulating load of the 150–300 percent and the separator efficiency of the 60–80 percent (the efficiency = the fines recovered / the fines fed): the circuit calculations of the workbook family: the separator rejects return to the mill with the product
• The particle size distribution: the width of the distribution (the slope of the RRSB curve) decides the water demand and the strength development of the cement: the narrower distribution with the steeper slope gives the higher strength but the higher water demand: the fineness optimization of the plant balances the Blaine, the residue and the distribution.

The fineness section of the sheet connects the mill calculations to the cement quality: the Blaine and the residues are the targets that the power and the production numbers serve, and the grinding engineer operates the mill to the fineness specifications of the products.

6. The Ventilation and the Cooling of the Mill

The ball mill grinding generates the heat (the mill interior reaches the 100–120°C), and the ventilation of the mill carries away the heat and the water vapor, protecting the gypsum and the cement quality:

• The ventilation flow: the air flow through the mill at the 0.8–1.2 m/s of the gas velocity in the mill shell: the ventilation volume of the example mill: the mill cross-section of the 12.57 m2 × the velocity of the 1.0 m/s = 12.6 m3/s = 45,300 m3/h: the ventilation air of the mill;
• The heat removal: the mill heat balance: the heat of the grinding (the 26–36 kWh/t of the specific power, nearly all converted to the heat) removed by the material, the ventilation air and the mill shell: the temperature of the mill discharge cement at the 90–120°C: the gypsum dehydration threshold at the 100–110°C (the gypsum converts to the hemihydrate, the loss of the setting control): the ventilation and the water injection protect the gypsum;
• The mill temperature calculation: the discharge temperature estimated from the balance of the grinding heat and the cooling: T_out ≈ T_in + (the grinding heat − the shell and the air cooling) / (the material × the specific heat): the sheet’s heat rows compute the expected temperature and flag the exceedances;
• The water injection: the water injected into the mill at the 0.5–1.5 percent of the feed evaporates and cools the interior: the water injection row of the sheet: the water of the cooling must evaporate fully (the residual moisture of the cement below the 0.5–1 percent):
• The mill heat balance: the complete heat balance of the mill: the input (the feed heat, the grinding heat) against the output (the product heat, the air heat, the water evaporation, the shell losses): the mill balance of the package’s sheet mirrors the kiln balance methodology at the smaller scale.

The ventilation and the cooling calculations protect the cement quality of the mill, and the sheet’s heat rows give the engineer the control of the mill temperature: the gypsum protection, the coating and the quality consistency of the grinding.

7. The Liner and the Diaphragm Considerations

The internal parts of the ball mill (the liners and the diaphragms) shape the grinding action, and the calculation sheet includes the considerations of the mill internals:

• The liner profiles: the first compartment with the lifting liners (the wave or the step liners) that lift the large balls for the impact grinding, the second with the classifying liners that direct the balls to the finer sizes along the mill: the liner selection affects the power draw (the wave liners at the higher power) and the grinding: the sheet’s liner rows record the profile types and their effects;
• The liner wear: the lifter height of the liners wears with the service: the worn liners reduce the lifting and the power: the power draw of the mill is the indirect measure of the liner condition: the sheet’s power trend rows flag the liner replacement points;
• The diaphragms: the compartment diaphragms retain the balls and pass the material: the slot sizes and the open areas set the material flow: the blocked or the worn diaphragm slots restrict the mill: the mill audit of the package checks the diaphragm condition with the power and the fineness trends;
• The charge and the liner interaction: the effective volume of the mill changes with the liner wear (the worn liners increase the internal diameter and the charge volume): the recalculations of the charge weight and the power follow the liner age: the sheet’s recalculation rows handle the corrections;
• The maintenance planning: the liner life of the 4–8 years for the inlet and the first compartment, the 2–4 for the second: the maintenance schedule of the mill aligned with the liner wear data of the sheet: the reliability practice of the grinding plant.

The internals chapter completes the mechanical picture of the mill calculations: the liners and the diaphragms are the wearing parts that change the mill behavior, and the sheet’s records and the recalculation rows keep the operating numbers current with the mechanical condition.

8. The Optimization Practice with the Calculations

The ball mill calculations of the package support the daily optimization of the grinding plant, and the practice of the well-run plants follows the standard cycle. The worked numbers of the article’s example mill assemble into the complete parameter table of the workbook:

Parameter	Formula / basis	Example value
Effective mill diameter	inside diameter	4.0 m
Critical speed	n_c = 42.3 / √D	21.15 rpm
Operating speed (72%)	0.72 × n_c	15.2 rpm
Mill volume (12 m length)	π/4 × D2 × L	150.8 m3
Ball charge (30% filling)	V × f × 4.65 t/m3	210.4 t
Bond grinding energy	Wi × 10 × (1/√P80 − 1/√F80)	26.4 kWh/t
Mill power	P = 7.4 × D^2.5 × L × f	2,700 kW
Production estimate	power / specific energy	~77 t/h

• The mill audit: the periodic stop audit of the mill: the charge level and the composition measured (the charge sampling), the liner condition recorded, the diaphragm slots checked: the audit data entered into the sheet, the charge and the power recalculated: the audit findings drive the corrections;
• The charge optimization: the ball composition adjusted to the feed and the fineness: the larger balls for the harder clinker and the coarser feed, the smaller for the fine grinding: the charge optimization of the practice gains the 2–5 percent of the production or the energy;
• The speed and the filling: the speed and the filling checked against the optimum ranges: the speed adjustments for the cataracting regime, the filling maintained at the design: the sheet’s ranges guide the settings;
• The circuit optimization: the separator settings, the circulating load and the airflow tuned with the mill: the combined optimization of the mill and the separator gains more than the mill alone: the circuit calculations of the workbook family support the combined work;
• The monitoring: the daily log of the power, the production, the Blaine, the temperature and the media: the trends of the sheet: the deviations from the calculated values trigger the investigations: the monitoring cycle keeps the mill on the design numbers.

The optimization practice closes the loop of the calculations: the sheet gives the numbers, the audit verifies the numbers, the corrections restore the numbers, and the monitoring holds the numbers: the grinding plant that runs the discipline runs the ball mill on the calculated values of the workbook.

9. The Ball Mill vs. the Modern Grinding Systems

The ball mill remains the standard of the cement grinding, but the modern plants compare it with the newer systems, and the calculation sheets of the package provide the comparison basis:

• The ball mill strengths: the reliability, the flexibility of the products (the wide fineness and the distribution range), the simplicity of the operation and the low maintenance skills demand: the ball mill remains the workhorse of the cement grinding worldwide;
• The vertical roller mill: the VRM of the cement grinding at the 22–32 kWh/t against the 30–40 of the ball mill: the higher efficiency, the smaller footprint and the integrated drying: the limits of the VRM in the cement grinding (the product distribution narrower, the surface quality debates) addressed by the modern mill designs: the VRM calculation sheets of the package’s family serve the comparison;
• The HPGR (high-pressure grinding rolls): the pre-grinding of the clinker at the 60–80 percent of the energy saving before the ball mill: the hybrid circuits (the HPGR + the ball mill) reduce the specific energy to the 24–30 kWh/t: the hybrid circuit calculations of the package’s workbook family;
• The conversion economics: the energy savings of the modern systems against the capital of the conversions: the cost sheet of the package quantifies the paybacks: the grinding system decisions of the plants are the calculations plus the money;
• The future of the ball mill: the ball mill remains the standard of the flexible cement grinding, and the modern plants run the hybrid circuits with the ball mill at the core: the ball mill calculations of the package are the standing tool of the standard and the hybrid designs.

The comparison chapter places the ball mill calculations in the context of the industry: the ball mill is no longer the only answer, but it remains the reference, and the calculations of the sheet serve the evaluation of all the grinding systems against the ball mill baseline.

10. The Daily Log and the Trend Analysis

The calculation sheet of the package is the basis of the mill’s daily log, and the log is the memory of the grinding plant: the daily entries of the production, the power, the fineness, the temperatures and the media, compared with the calculated values and reviewed as the trends:

• The daily entries: the production in the tonnes per hour, the mill power and the specific power, the Blaine and the sieve residues of the product, the mill discharge temperature, the ventilation flow, the separator settings and the feed moisture: the log rows of the sheet next to the calculated reference values;
• The deviation review: the daily deviations of the measured values from the calculated references: the power above the calculated with the production below signals the charge or the liner issues; the Blaine drift with the constant settings signals the separator or the feed changes; the temperature rise signals the ventilation or the water injection problems: the deviation review of the log is the early warning system of the mill;
• The trend charts: the weekly and the monthly trends of the key numbers: the specific power trend shows the gradual changes (the liner wear, the media degradation, the clinker quality trends), the Blaine trend shows the product consistency, and the media consumption trend shows the charge condition: the trend review of the monthly meeting reads the mill’s health;
• The maintenance triggers: the log trends trigger the maintenance actions: the power drop of the 5–8 percent from the reference signals the liner wear and the charge inspection; the media consumption rise signals the ball quality or the charge condition problems; the temperature trend triggers the ventilation and the water checks: the predictive maintenance of the mill runs on the log;
• The optimization feedback: the log records the optimization actions and their effects: the charge adjustments, the speed changes, the separator settings: the before and after numbers of the log document the gains: the knowledge base of the plant’s grinding practice grows with the log, and the calculation sheet of the package is the format of the record.

The daily log closes the loop of the ball mill calculations: the sheet gives the references, the log records the measurements, the deviations trigger the actions and the trends guide the maintenance: the grinding plant that runs the discipline operates the mill on the numbers of the workbook day after day.

11. Frequently Asked Questions

What is the critical speed of the ball mill and why does it matter?

The critical speed is the rotation speed at which the balls are held against the mill shell by the centrifugal force: n_c = 42.3 / √D revolutions per minute (the D in the meters): the example mill of the 4.0 m diameter has the critical speed of the 21.15 rpm. The mills operate at the 65–78 percent of the critical: below the range the grinding is weak, and above the range the balls centrifuge and stop grinding: the operating speed of the cement mills at the 70–75 percent of the critical.

How much does the ball charge weigh?

The charge weight = the mill volume × the filling degree × the ball bulk density: the example mill (the 4.0 m by the 12 m, the 30 percent filling, the 4.65 t/m3) carries the 210 tonnes of the balls. The cement mills typically carry the 80–250 tonnes at the filling degrees of the 28–35 percent, and the filling is measured by the charge surface height after the stop or by the power draw.

How is the power of the ball mill calculated?

The practical power formula P = 7.4 × D^2.5 × L × f (the D and the L in the meters, the f the filling factor) gives the mill demand: the example at the 2,700 kW, with the Bond method E = Wi × 10 × (1/√P80 − 1/√F80) estimating the grinding energy to the target fineness (the 26.4 kWh/t of the example to the 30 microns). The production of the mill follows from the power and the specific energy.

What are the typical specific power consumptions of the cement grinding?

The cement grinding in the ball mills runs at the 30–42 kWh/t in the open circuit and the 26–36 in the closed circuit with the separator, the VRM at the 22–32 and the hybrid HPGR+ball mill circuits at the 24–30. The specific consumption depends on the fineness target (the 3–4 kWh/t per each 100 cm2/g of the Blaine in the normal ranges), the clinker quality and the circuit efficiency.

Why does the mill temperature matter?

The grinding converts the power into the heat, and the mill interior reaches the 100–120°C: the gypsum of the cement dehydrates above the 100–110°C and the setting control is lost, and the hot cement causes the quality and the handling problems. The ventilation and the water injection cool the mill, and the heat balance of the sheet predicts the discharge temperature and flags the exceedances.

Is the ball mill calculations workbook included in the package?

Yes: the Complete Cement Technical Package (931 files) includes the original ball mill calculations workbook with the speed, the charge, the power, the fineness and the ventilation sheets, together with the ball mill optimization courses, the grinding handbooks and the complete library of the cement engineer: the purchase below delivers the file and the package.

12. Conclusion

The ball mill calculations are the working numbers of the finish grinding plant: the critical speed and the operating speed (the 21.15 rpm and the 15.2 rpm of the example), the ball charge and the filling (the 210 tonnes at the 30 percent), the power by the Bond and the practice formulas (the 26.4 kWh/t and the 2,700 kW), the fineness and the Blaine targets, the ventilation and the mill heat balance, and the optimization practice that keeps the mill on the calculated values. The workbook of the package organizes the complete set into the calculation sheets, and the article walked the reader through the formulas and the worked numbers one by one.

The workbook, the grinding knowledge and the complete library of the process calculation tools are part of the Complete Cement Technical Package: the 931 files, the one-time payment of $249.99, the instant download and the lifetime access. The purchase button below delivers the ball mill calculations workbook and the whole package of the cement engineer in one download.

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.