Tromp Curve for Cement Separators: Efficiency Guide

The Tromp curve (la courbe de Tromp) describes the probability, by particle-size class, of reporting to one selected separator stream—commonly the coarse/reject stream. From that curve, engineers derive the cut size, bypass behavior and separation sharpness. These parameters help explain separator performance, but circulating load, mill output and product particle-size distribution are determined by the complete mill–separator circuit, not by the curve alone.

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A Tromp curve is useful because it shows how each particle-size class is split between product and rejects. If the reject-stream convention is used, the curve rises from a low probability for very fine particles toward a high probability for coarse particles. The curve convention must always be stated, because some software and publications plot the complementary fine-stream probability instead.

1. The Separator and Its Place in the Closed Grinding Circuit

The closed circuit is the loop of the mill and the separator, and the separator is the quality gate of the loop:

• The circuit: the mill discharges the ground material to the bucket elevator, the elevator feeds the separator, the separator splits the feed into the fines (the finished cement, 1 to 5% residue on 45 µm) and the rejects (the coarse material, returned to the mill inlet): the rejects travel the loop again;
• The circulating load: commonly defined as reject flow divided by fresh feed flow. A separate circulation factor may be defined as separator feed divided by fresh feed. The terminology must be stated clearly because these two ratios differ by 1.0 on a mass basis. Both values are circuit-specific and should be calculated from measured streams.
• The separator types: the first-generation static conical separators (the “turbo”), the second-generation with the whizzer blades and the fans, the third-generation high-efficiency dynamic separators with the cage rotors and the air circuit: the curve quality improves with each generation;
• The efficiency meaning: the ideal separator sends every particle below the cut size to the fines and every particle above it to the rejects: the real separator misroutes a fraction of each: the misrouted fractions cost the mill the re-grinding work and the energy;

At steady state, fresh feed equals net product, while separator feed equals product plus rejects. The Tromp curve describes how particle sizes are split between those streams; it does not by itself determine mill capacity. Circuit performance must be evaluated with the material balance, separator behavior, mill discharge PSD, power and product quality together.

2. The Definitions of the Curve: The Fines, the Rejects and the Feed

The Tromp curve is built from three mass streams, and the definitions must be exact before any sample is taken:

• The separator feed (F): the mill discharge entering the separator: the mass flow in t/h and the complete particle size distribution of the stream;
• The fines (the product, Fines): the stream leaving as the finished cement: the fine fraction of the feed, classified;
• The rejects (the coarse, R): the stream returning to the mill: the coarse fraction;
• The mass balance: F = Fines + Rejects, and for every size class i: F·fi = Fines·pi + Rejects·ri, where fi, pi and ri are the mass fractions of the size class in the three streams;
• The classification probability: the fraction of the size class i that follows the rejects: Ci = Rejects·ri / (F·fi): the collection probability of the size class in the coarse stream: this Ci plotted against the particle size is the Tromp curve;

The balance equation is central to the method. In some plants separator feed is measured directly; in others it is reconstructed from product and reject streams. Either approach is valid if the flows and particle-size distributions are sampled over the same stable period and the resulting mass balance closes within an acceptance tolerance appropriate to the measurement method.

3. The Sampling Campaign: The Discipline of the Plant

The curve is only as good as the samples, and the file is emphatic about the sampling protocol:

• The sampling points: the fines stream at the separator outlet or the filter hopper, the rejects at the screw or the chute: the points must sample the full cross section of the moving stream, not the corner of the chute;
• The simultaneous sampling: fines, rejects and separator-feed samples—when a feed sampling point is available—should represent the same stable operating period. Rotor speed, airflow, mill feed and other relevant conditions must remain sufficiently stable for the test to be representative.
• The sample size: choose a representative mass and number of increments appropriate to the particle size, sampling point and laboratory method. Repeatability is more important than one universal kilogram or run-count rule.
• The condition log: the separator speed (rpm), the air flow (m³/h), the damper positions, the mill power, the feed rate and the fineness samples of the finished product: the curve is meaningless without the operating conditions that produced it;
• The sieve analysis: the dry sieving on the 45, 63, 90, 125, 200 and 315 µm sieves with the 1 to 2% residue check on the 90 µm: the fines stream needs the air jet sieving at the 45 µm because the fine cement agglomerates on the dry sieves: the residue on each sieve recorded to the 0.1%;

The campaign should be scheduled during stable, representative operation. Repeated curves are valuable for trending wear and operating drift, but the test interval should reflect process stability, separator wear, maintenance history and the cost of sampling rather than a fixed calendar rule.

4. The Worked Example: The Numbers of the Curve Construction

The file works a complete example, and this article reproduces it: the separator of the 100 t/h finish circuit, the samples of the fines and the rejects sieved into the six classes, all numbers in %:

For the worked example, separator feed is 100 t/h and rejects are 55 t/h, so fines are 45 t/h. For each size class, the feed fraction is calculated from Feed·f = 45·p + 55·r. For the 45–63 µm class: 45×0.12 + 55×0.10 = 10.9 t/h-equivalent per 100 t/h feed, so that class represents 10.9% of the separator feed. Repeating the calculation for every class gives feed fractions that sum to 100%, which is the first balance check of the example.

Using the reject-stream convention, the collection probability for each class is C = 55·r / (45·p + 55·r). The resulting values are approximately: 0–45 µm = 0.169; 45–63 µm = 0.505; 63–90 µm = 0.685; 90–125 µm = 0.857; 125–200 µm = 0.941; and 200–315 µm = 0.972. These points are plotted against a stated representative particle size for each class to build the Tromp curve.

5. The Cut Size: The d50 of the Curve

The cut size (the d50, the separazione size) is the size at which the collection probability equals 50%: half of the particles of that size go to the fines, half to the rejects: the size where the classifier “cuts” the distribution:

• The reading: in this example, the 45–63 µm class already has a reject probability of about 0.505. Therefore d50 lies close to the representative size assigned to that class—roughly the low-to-mid 50 µm range if a conventional class midpoint is used. The exact reported d50 depends on the chosen class-size convention and interpolation method, which should be stated in the calculation sheet.
• The control lever: d50 moves with rotor speed, airflow, guide-vane setting, feed loading and material properties. Higher rotor speed commonly shifts the cut finer, but the magnitude of the response is machine-specific and should be established from the separator’s actual operating curve.
• The target: d50 should be interpreted together with bypass, sharpness and the separator-feed PSD. Two separators can produce similar cement residue with different Tromp curves, so the appropriate cut size should be established from the actual product specification and circuit performance rather than a universal d50 range.
• d50 and product fineness: changing the separator cut can influence product Blaine, residue and PSD, but the result also depends on mill discharge PSD, feed rate and circulating load. Rotor speed is therefore one control variable in a coupled circuit, not a direct one-to-one Blaine control.

d50 should be taken from the Tromp curve or from a validated calculation method. Product residue alone is not sufficient to determine separator cut size reliably because residue depends on both the separator curve and the particle-size distribution entering the separator.

6. The Bypass: The Fine Particles Lost to the Rejects

The left end of the Tromp curve carries the second vital number: the bypass, the percentage of the fine particles that follows the rejects as if no classification existed:

• The definition: bypass is inferred from the fine-size end of the Tromp curve using the stated curve convention. It should not be equated automatically to the probability of one broad class such as 0–45 µm, because that class contains particles across a wide size range. Reliable bypass estimation requires sufficiently fine size classes or an appropriate fitted/extrapolated curve.
• The meaning: fines misplaced into the reject stream are unnecessarily recirculated and may increase grinding work, separator load and media wear. The effect on output and specific energy is circuit-specific and should be quantified from the plant material balance and before/after performance data.
• The causes: the mechanical leaks (the fine material short-circuiting through the seals and the louvres without the classification), the air currents carrying the fines with the coarse stream, the overload of the separator, the coarse particle turbulence dragging the fines;
• The benchmark: compare bypass with the separator’s clean/design performance measured using the same sampling and curve convention. A sustained increase from that baseline can indicate wear, leakage, build-up, poor dispersion or a changed operating point.

Bypass is one indicator of non-ideal classification, but it should be interpreted together with cut size, sharpness, feed PSD and operating conditions. The objective is not to chase one isolated number; it is to understand why material is being misplaced and whether that misplacement is affecting product quality or circuit efficiency.

7. The Imperfection and the Sharpness of the Classification

The steepness of the central part of the curve is the sharpness of the classification, expressed by the imperfection (the I value):

One common imperfection definition is I = (d75 − d25) / (2 × d50), where d25, d50 and d75 are read from the same stated Tromp-curve convention. Lower values indicate a steeper transition around the cut, but meaningful comparison requires the same calculation method, sampling procedure and particle-size analysis.

• The worked example: d25, d50 and d75 should be obtained from the plotted or fitted curve using a stated representative size for each class and a consistent interpolation method. Because the original table contains broad size classes, reporting highly precise d25/d50/d75 values would imply more resolution than the data support.
• Interpretation: a lower imperfection value generally indicates a sharper cut, but there is no single universal good/bad threshold for all separator types and duties. Compare against the OEM design, the clean-machine baseline and comparable tests performed with the same method.
• The effect: a sharper cut can reduce misplacement around the cut size and unnecessary recirculation. Its effect on product PSD, strength, circulating load and output depends on the mill discharge PSD and the selected operating point and must be confirmed from plant and laboratory data.
• The limit: real separators cannot produce an ideal step-function cut because of turbulence, particle interactions, feed dispersion and measurement uncertainty. Optimization should focus on verified circuit performance rather than pursuing an arbitrary imperfection target.

The imperfection completes the triple reading of the curve: the d50 tells where the cut is, the I tells how sharp it is, the bypass tells how much leaks: the three numbers of the separator on one piece of paper: the file’s summary sheet carries the three boxes and the target ranges for the separator generation.

8. The Presentation of the Curve: The Cumulative and the Probability Plots

The raw points of the example need the professional presentation, and the file teaches the two standard forms:

• The linear plot: the collection probability C against the particle size on the linear axes: the S-curve with the floor at the bypass: the daily working form that the operators read directly;
• The probability scale: the C plotted on the probability (the probit) axis against the log of the size: the normal distribution of the classification errors becomes the straight line in the middle region: the straight-line fit gives the d50 and the standard deviation of the cut: the slope of the line is the sharpness;
• The log-normal reading: the log-normal separation theory: the classification of the real separators approximates the log-normal: the d50 and the sigma extracted from the probit plot feed the efficiency models of the circuit;
• The Excel practice: the calculation sheets of the package: the input of the sieve tables, the automatic balance check, the plot and the three parameters: the plant replicates the curves monthly and archives the trend;

The presentation discipline: every curve carries the conditions block (the date, the separator speed, the air, the feed rate, the mill power) and the three parameters box: the curves of the different weeks compare only with the equal conditions: the file’s template prints the conditions on the chart corner so the comparisons stay honest.

9. The Typical Curves: The Separators of the Three Generations

The curve is the fingerprint of the separator type, and the file tabulates the typical fingerprints:

The generation comparison is useful for understanding design evolution, but retrofit value cannot be inferred from separator generation alone. The correct modernization case compares the existing Tromp curve, product quality, circulating load and circuit energy with a supplier-backed post-upgrade target measured under comparable operating conditions.

10. The Validation of the Curve: The Checks of the Calculation

Not every curve is trustworthy, and the file teaches the validation checks before the curve enters the decision-making:

• The mass-balance check: calculated stream fractions should close consistently with the measured flows and PSDs. Set the acceptance tolerance from the sampling method, flow-meter uncertainty and laboratory repeatability rather than one universal percentage.
• The shape check: a reject-probability Tromp curve will usually rise with particle size, but local non-monotonic behavior can result from sampling error, agglomeration, particle-shape effects or a real fish-hook response. Investigate the cause rather than automatically deleting the point.
• The split check: compare the calculated fines/reject split with independently measured stream flows where available. Judge the difference against the known uncertainty of the flow measurements and sampling system.
• The repeatability check: compare repeated runs made under the same operating point. Define acceptance from the observed repeatability of the plant sampling and laboratory method rather than a fixed universal difference in collection probability.
• The common sense check: the d50 against the product residue and the Blaine: the curve that contradicts the laboratory of the day is suspect before any statistics: the data must fit the plant before it fits the theory;

The validation section is short but decisive: the file’s position is that one verified curve beats ten pretty curves: the plant that applies the checks automatically in the spreadsheet (the package’s calculator does the checks in the background) builds the trustworthy database that the Tromp II part uses for the optimization.

11. The Use of the Curve in the Daily Operation

Beyond the monthly campaigns, the curve principles serve the daily rounds of the grinding department:

• The residue interpretation: the product residue on 45 µm is the single-point shadow of the curve: the residue up with the stable Blaine points the d50 up (the separator speed down or the air up);
• The circulating-load reading: a change in circulating load at similar product quality can come from separator performance, mill discharge PSD, feed grindability, material balance error or an internal restriction. Confirm the cause with stream data before attributing it to the separator.
• The energy reading: the kWh/t trend against the curve history: the circuit that loses the output efficiency shows the curve decay weeks before the cost reports: the quarterly curve is the early warning of the louvre wear and the seal deterioration;
• The maintenance trigger: a sustained deterioration in bypass, cut sharpness or airflow/pressure behavior relative to the clean baseline can justify inspection. Trigger levels should be defined from the separator’s normal variability, wear history and production impact.

The practical value of the Tromp curve is in trend comparison under known operating conditions. Routine product-quality data can indicate when performance is drifting, while a repeat Tromp test provides a structured way to determine whether the cause lies in classification, mill discharge, or the measurement system.

12. The Historical Trend of the Curves: The Aging of the Separator

The single curve is the photograph, the series of the curves is the movie, and the file insists that the plants build the historical archive: the aging of the separator and its ancillaries writes itself in the drift of the curve parameters:

• Guide-vane wear: erosion can disturb airflow distribution and classification selectivity. A repeated Tromp test may show a change in cut, bypass or sharpness, but the size and rate of that change depend on the separator design, material abrasiveness and operating conditions.
• Seal deterioration: worn seals or airlocks can create leakage and material short-circuiting that changes the Tromp curve. Confirm the effect from measured performance before assigning the cause to a seal alone.
• The fan and the rotor balance: the dynamic balance of the cage rotor decays with the uneven wear, the air distribution across the annulus becomes asymmetric, and the curve develops the two-step shape with the plateau: the vibration survey and the re-balancing restore the curve;
• The air circuit fouling: the dust deposits in the separator body and the ducting narrow the flow passages: the effective air velocity rises, the d50 falls without the operator touching the speed: the unexpected fineness rise is the symptom, the inspection of the internals the cure;
• The baseline discipline: archive the curve of a clean, stable or newly overhauled separator and compare later tests against it. Define maintenance triggers from normal test repeatability, wear history, product impact and OEM limits rather than universal parameter deltas.

A historical series of Tromp curves is most useful when it is tied to maintenance records, product quality, throughput and specific energy. If performance drifts and a maintenance action restores the curve and circuit KPIs toward baseline, the plant gains evidence that the intervention addressed a real classification loss. The economic value should be calculated from the site’s measured production and energy data.

13. The Curve and the Cement Quality: The PSD Bridge

The Tromp curve ends its first part with the bridge to the product quality, because the separator parameters become the concrete properties of the cement:

• The particle-size distribution of the product: product PSD results from the mill discharge PSD combined with separator classification. Changes in the distribution can affect hydration, packing, water demand and strength development, but the relationship is cement-specific and should be confirmed by laboratory testing rather than converted directly into a fixed MPa gain.
• The water demand: PSD shape can influence packing and concrete water demand, but the direction and magnitude also depend on cement chemistry, ultrafine content, particle shape and admixture interaction. Compare cements using mortar/concrete tests rather than assuming that a sharper separator cut automatically lowers water demand.
• The strength optimization: separator settings should be optimized together with Blaine, residue, full PSD and strength results. A lower coarse tail may improve performance in some cases, but no fixed Blaine/residue pair guarantees higher strength across different clinkers and cement formulations.
• The additive and the limestone interplay: the limestone and the slag additions shift the product distribution: the separator settings follow the recipe changes: the curve campaigns after every recipe change document the new envelope and the Blaine-residue pair the laboratory targets;
• The customer-facing evidence: the cement sales engineers quote the PSD and the residue data to the ready-mix customers: the plant that holds the curve archive answers the “why does the cement behave differently this month” with the numbers: the curve is the quality documentation of the grinding;

The quality message of the chapter: the laboratory measures the Blaine and the residues, the curve measures the cause: the two readings together give the grinding department the complete control picture: the second part of the course takes the optimization: the target curve shapes for the cement types and the step-by-step tuning that the operators apply to reach them.

14. Practical Tromp-Curve Test Checklist

1. Define the curve convention before sampling: probability to rejects or probability to fines.
2. Hold the grinding circuit at a stable, representative operating point and record rotor speed, airflow, feed rate, mill power and product quality.
3. Collect representative fines and rejects samples over the same period; sample separator feed as well when a reliable point is available.
4. Measure the stream mass flows or use a validated balance method to determine the fines/reject split.
5. Analyze all streams with the same particle-size method and class boundaries.
6. Calculate the feed PSD and collection probability for each size class, then verify the material balance and test repeatability.
7. Plot the curve using a stated representative size for each class and extract d25, d50, d75, bypass and imperfection using one consistent method.
8. Compare the result with the same separator’s clean/design baseline and correlate any change with throughput, energy, PSD and maintenance condition before taking action.

15. Frequently Asked Questions

How often should the Tromp curve be measured?

Measure often enough to establish a reliable baseline and detect meaningful drift. Repeat the test after major separator maintenance, retrofit, unexplained quality change or persistent circuit-performance deterioration. The routine interval should reflect wear rate, process stability and the effort required for representative sampling.

What samples are needed for the curve?

The fines stream and the rejects stream, taken simultaneously during the stable operation, 1 to 2 kg each, three runs: the mill feed rate, the separator speed and the air flows logged: the sieve analysis on the 45, 63, 90, 125, 200 and 315 µm sieves: the feed stream is not sampled: the mass balance reconstructs it.

What is a good bypass value?

There is no universal good bypass value because the reported number depends on separator design, operating duty, particle-size analysis and the Tromp-curve convention. Compare against the same separator’s clean/design baseline and investigate a sustained deterioration using airflow, feed distribution, wear, leakage and loading data.

What is the difference between the d50 and the residue on 45 µm?

The residue is the fraction of the product coarser than the sieve, the d50 is the separator cut size: the two connect through the curve and the product distribution: the same residue can come from the different curves, and the cement strength differs: the d50 describes the separator, the residue describes the product, the curve ties both.

Can the curve be computed in the Excel spreadsheet?

Yes, and the package includes the ready calculator: the sieve tables of the two streams are typed in, the balances and the checks run automatically, the curve plots and the three parameters (d50, bypass, imperfection) print: the plant’s engineers type, the sheet decides, the operator reads: the calculator is the tool of the file.

Why is the curve called the Tromp curve?

The method is named after the scientist who developed the classification analysis technique in the coal preparation of the early 20th century: the Tromp curve, also called the partition curve or the selectivity curve, migrated from the coal washing to the cement separators: the name honors the origin, the method serves the cement.

16. Conclusion

The Tromp curve is the measurement instrument of the separator: the d50, the bypass and the imperfection read the classification of the machine in three numbers: this first part of the course built the complete method: the sampling discipline, the mass balance, the worked example with the real numbers and the validation checks: the reader now builds the curve of any separator of his plant within a day: the second part of the course takes the numbers and converts them into the output, the energy and the quality of the circuit.

The Complete Cement Technical Package includes both Tromp-curve parts together with calculation sheets, separator references and case-study material. The complete 931-file library is offered for $249 as a one-time purchase with instant download access immediately after payment.

Related Separator and Grinding Guides

• High Efficiency Separator: Complete Technical Guide
• Separator Operation in Cement Grinding
• Classifiers and Separators: Complete Guide
• Cement Grinding Systems: Complete Technical Guide

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.