Tromp Curve I: Separator Efficiency Guide

The Tromp curve (la courbe de Tromp) is the identity card of the air separator: the curve that shows the probability of every particle size to end in the coarse rejects instead of the finished product: the separator of the closed grinding circuit classifies the mill discharge, and the quality of that classification is written in the Tromp curve: the cut size, the bypass and the imperfection: the three numbers that decide the circulating load, the mill output and the cement particle size distribution: this first part of the course teaches what the curve is, how it is measured and how it is drawn, with the complete worked example.

The Complete Cement Technical Package (931 files including this Tromp curve file, the separator handbooks, the Excel calculators and the training courses: $249.99 one-time: instant download via the PayPal payment) hosts the two Tromp parts with the calculation spreadsheets and the case studies: this article walks the first part: the definitions, the sampling campaign, the mass balance of the separator, the step-by-step curve computation and the reading of the results: the second part (the Tromp II file) develops the interpretation and the optimization.

The Tromp curve is the most useful curve of the grinding department, and the reason is the logic of its construction: every particle of every size either follows the fines or the rejects: the curve plots the probability of the “wrong” direction, the coarse direction, for every size class: the perfect separator would send everything below the cut size to the fines: the real separator loses the fine particles to the rejects (the bypass) and smears the cut: the curve shows both sins at a glance: this article makes the reader able to build the curve from the plant’s own samples within a day.

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: the ratio of the separator feed to the fresh feed: the typical modern circuits run at 150 to 300%: the separator sees 2.5 to 4 times the tons the mill receives fresh: the capacity of the separator is the hidden constraint of the circuit;
• 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;

The circuit equation that starts the course: Mill output = (the separator feed) × (the fines fraction of the feed): the separator feed is the mill capacity, and the fines fraction is the classification: the better the classification, the finer the product at the equal feed, or the more product at the equal fineness: the Tromp curve measures the “better”.

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 reader must internalize the balance equation because the whole method hangs on it: the feed is never measured directly in most plants (the separator feed is the elevator discharge), so the balance is solved from the two measured streams with the unknown feed eliminated: the elegant algebra of the method: the curve is computed from the fines and the rejects alone, and the feed follows from the balance: the next sections do the algebra with the real numbers.

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: the fines and the rejects samples taken at the same moment (within minutes) because the circuit drifts: the separator speed, the mill feed and the air flows must be stable for the hour before and during the sampling;
• The sample size: 1 to 2 kg per stream per run, three runs repeated: the composite of the runs for the sieve analysis: the two hours of the stable operation produce the reliable curve;
• 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 practical schedule: the samples taken on the Wednesday shift of the stable week, the sieving in the afternoon, the curve computed by the evening: the plant that samples monthly builds the history of its separator and catches the drift of the louvres and the wear before the product quality complains: the discipline of the campaign is half of the method.

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 %:

Size class (µm)	Fines % (p)	Rejects % (r)	Fines mass (kg)	Rejects mass (kg)
0 – 45	72.0	12.0	72.0	12.0
45 – 63	12.0	10.0	12.0	10.0
63 – 90	9.0	16.0	9.0	16.0
90 – 125	4.5	22.0	4.5	22.0
125 – 200	2.0	26.0	2.0	26.0
200 – 315	0.5	14.0	0.5	14.0

The balances: the mass split of the separator is unknown: the plant measures the rejects mass flow at 55.0 t/h against the 100 t/h of the separator feed (the elevator load): the mass balance check: Fines = 100 − 55 = 45 t/h: the balance of the total per size class: Feed·f = 45·p + 55·r: the class 45-63: 45×0.12 + 55×0.10 = 5.4 + 5.5 = 10.9 kg per 100 kg: the feed fraction of that class: 10.9%: the class check sums the feed at 100% ± the sampling tolerance: the audit of the balance is the first verification of the campaign.

The collection probability of each class: C = 55·r / (45·p + 55·r): the class 45-63: 5.5/10.9 = 0.505: the class 0-45: 6.6/38.4 = 0.172: the class 63-90: 8.8/12.9 = 0.682: the class 90-125: 12.1/14.1 = 0.858: the class 125-200: 14.3/15.2 = 0.941: the class 200-315: 7.7/7.95 = 0.969: the points plotted at the mid sizes of the classes produce the S-shaped curve: the shape of the classification.

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 the example curve, the C values pass the 0.50 line between the classes 45-63 and 63-90: the interpolation gives the d50 at about 62 µm: the separator at these conditions cuts at 62 µm:
• The control lever: the d50 moves with the separator rotor speed, the air flow and the damper: the higher the speed, the smaller the d50 (the finer the product): the relation is roughly proportional: +10% speed moves the d50 by about 8 to 15% down, depending on the separator geometry;
• The target: the d50 is set so that the fines stream meets the product residue: the OPC at 2% residue on 45 µm runs the d50 in the 25 to 45 µm range for the modern high-efficiency separators, against the 60 to 90 µm of the old first-generation machines at the same product;
• The d50 and the Blaine: the Blaine of the product is largely set by the d50 and the bypass: the lower the d50, the higher the Blaine: the plant tunes the Blaine by the separator speed at the constant mill conditions: the curve shows why the tuning works;

The d50 is the first number the operators quote, and the file teaches its estimation from the residue curve alone as the quick check: the d50 ≈ the sieve size at which the feed residue and the product residue cross in the plot of the cumulative distributions: the approximation serves the daily rounds, the full curve serves the decisions.

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: the asymptotic value of the curve at the smallest sizes: the C of the finest class: in the example, the C at 0-45 of 0.172 and the curve floor near 0.15 to 0.20: the bypass of the separator is about 15 to 20%:
• The meaning: the bypassed fine material recycles to the mill and is ground again: the fine re-grinding is wasted work: the energy, the ball wear and the separator load all pay for the bypass: each 10% of the bypass costs the circuit roughly 5 to 10% of the mill output;
• 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 targets: the modern high-efficiency separators achieve the bypass of 5 to 15%, the old units run at 25 to 40%: the reduction of the bypass from 30 to 15% is the classic modernization gain: the output +8 to 15% at the equal fineness;

The bypass is the “leak” of the classifier, and the Tromp curve quantifies it with one number: the second part of the course (the Tromp II) teaches the hunt for the bypass sources: the seals, the air flows and the loading: in this first part, the message is the measurement: the plant that cannot see its bypass cannot fight it.

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):

The definition: I = (d75 − d25) / (2 × d50): where the d75 and the d25 are the sizes at the collection probabilities of 75% and 25%: the perfect classification has the vertical curve: d75 = d25 = d50 and I = 0: the real separators: I = 0.20 to 0.45 for the air classifiers:

• The reading of the example: the curve crosses the 0.75 between the classes 63-90 and 90-125 (the d75 ≈ 82 µm) and the 0.25 between 45-63 and 63-90 (the d25 ≈ 50 µm): with the d50 = 62: I = (82 − 50)/(2 × 62) = 32/124 = 0.26: the decent classifier;
• The good and the bad: the high-efficiency separators at I = 0.20 to 0.30, the old whizzer units at 0.35 to 0.45, the static cones at 0.50 and above: the lower the I, the sharper the cut and the better the circuit:
• The effect: the sharp cut means the rejects contain only the true oversize and the fines contain no oversize: the mill stops the re-grinding of the near-cut fines, the product PSD narrows and the strength improves: the same circulating load produces more finished product;
• The limits: the I below 0.15 becomes physically difficult: the turbulence of the real air streams resists the ideal cut: the file warns against chasing the I below 0.20 in the normal cement separators: the money is in the bypass reduction first;

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:

Separator generation	d50 range (µm)	Bypass %	Imperfection I	Circuit output index
First generation (static cone)	60 – 90	25 – 40	0.45 – 0.60	0.85 – 0.90
Second generation (whizzer)	45 – 70	20 – 35	0.35 – 0.45	0.90 – 0.95
Third generation (dynamic cage)	25 – 45	5 – 15	0.20 – 0.30	1.00 (reference)

The table reads as the modernization roadmap: the plant that moves from the second to the third generation gains the output index of 5 to 10% at the equal fineness, plus the wider control range of the d50: the curves of the file show the evolution graphically: the bypass floor dropping and the central slope steepening: the separator replacement is one of the best-return projects of the grinding department, and the Tromp curves document the before and the after with the single sheet of paper.

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: the sum of the calculated feed fractions must equal 100% ± 1 to 2%: the deviation beyond the tolerance means the sampling missed a stream or the sieving drifted;
• The monotonicity check: the collection probability must rise monotonically with the size: the dips of the curve flag the sieve errors or the sampling bias: the class with the odd C is re-sieved or the campaign repeated;
• The split check: the calculated mass split (the fines fraction of the feed) is compared with the measured flows: the deviation beyond 5% flags the flow measurement error: the balance discipline closes the loop;
• The repeatability check: the two or three runs of the campaign plotted together: the curves within ±0.05 of the C values confirm the stable operation: the spread beyond that means the circuit was drifting and the campaign must be redone;
• 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: the rising circulating load at the constant fineness and the constant mill feed: the curve explanation: the bypass up or the cut size drifted: the operator checks the separator internals before touching the mill;
• 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: the bypass rising by 5 points or more from the baseline curve: the inspection trigger of the internal wear parts: the curve data justifies the maintenance budget with the numbers, not the opinions;

The daily message: the curve is not the annual ritual but the running gauge: the plant that plots the quarterly curves and the monthly residues against the conditions keeps the separator inside the envelope, and the mill output stays at the design: the first part of the course ends here, with the instrument in the hands: the second part teaches what to do with it.

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:

• The louvre wear: the guide louvres of the classification zone erode with the abrasive cement dust: the worn louvres lose their directing precision, the turbulence rises, the imperfection creeps up: the quarterly curves show the I value drifting from 0.25 toward 0.35 over the campaign of two years;
• The seal deterioration: the bypass rises as the internal seals and the air locks wear: the C of the finest class climbs from 8 to 20%: the bypass is the first parameter to age and the first to be fixed, because the seal replacement is the cheap maintenance of the shutdown;
• 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: the plant that archives the curve of the new or the overhauled separator as the baseline compares every later curve against it: the parameter deltas above the thresholds (bypass +5 points, I +0.05) trigger the planned maintenance: the curves make the maintenance plan for the separator;

The worked history of the file: the plant of the example ran its quarterly curves for three years: the bypass rose from 12 to 26%, the output fell 7% at the equal fineness, the energy rose 2 kWh/t: the analysis traced the drift to the worn seals and the eroded louvres: the 40,000 USD overhaul restored the curve to the baseline and the output with it: the archive paid for itself many times over in the three years of the avoided losses: the file’s message: the curve is not the diagnostic of the day, it is the chronicle of the machine.

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: the product PSD is the convolution of the mill discharge distribution and the separator curve: the sharp curve (the low I) produces the narrower PSD with the higher fraction in the 3 to 32 µm window: the window of the strength development: the 3 to 32 µm fraction correlates with the 28-day strength: the gain of 5% in that fraction typically shows as 2 to 4 MPa on the 28-day mortar strength;
• The water demand: the wide PSD with the high fines tail raises the water demand of the concrete: the sharp classification cuts the overground <3 µm fraction, lowers the water demand and improves the workability: the same Blaine with the different curve shapes gives the different concrete behavior: the curve is the hidden ingredient of the mix;
• The strength optimization: the modern cement recipe (the high Blaine, the low residue) needs the d50 and the I in the optimum window: the plants that tune the separator with the curve data alongside the laboratory data find the strength reserve in the PSD shape without the extra energy: the classic example: the 3,800 Blaine cement with the 2.5% residue at 45 µm outperforms the 4,000 Blaine cement with the 4% residue, and the curve explains why: the coarse tail dominates the weakening;
• 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. Frequently Asked Questions

How often should the Tromp curve be measured?

Quarterly in the well-run plants, and always after the separator overhauls, the louvre changes, the motor speed changes and the product type switches: the monthly campaigns are justified in the plants with the old separators or the frequent quality changes: the curve is cheap: two hours of sampling and sieving.

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?

5 to 15% for the modern high-efficiency dynamic separators, 20 to 35% for the second generation, 25 to 40% for the old static cones: the bypass above 20% in a modern separator demands the investigation of the seals, the louvres and the loading: each 10% of the bypass costs about 5 to 10% of the circuit output.

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.

15. 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 parts with the calculation sheets and the case studies: the one-time $249.99: the instant download: the 931 files of the cement library: the classification knowledge of the grinding plants, organized: the curve of the separator, drawn and read: the engineering of the closed circuit, mastered.

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.