Flotation Kinetics Rate

Fit log residual recovery against time with caller ultimate flotation recovery; intercept is estimated and no asymptote is invented.

Description

First-order flotation kinetics models recovery approaching an ultimate fraction.

First-order flotation kinetics models recovery approaching an ultimate fraction. Taking the logarithm of residual recovery turns that assumed model into a straight-line fit.^[1]

The tool returns fittedRatePerMinute, logResidualIntercept, fitRSquared. The computation uses supplied observations and parameters rather than acquiring measurements.

When to use Flotation Kinetics Rate

Use this for the following task: Fit log residual recovery against time with caller ultimate flotation recovery; intercept is estimated and no asymptote is invented. Prepare observations and calibration parameters on the same unit and reporting basis.

How to use Flotation Kinetics Rate and understand its results

Prepare the measured quantities and supplied model parameters. Match the units and align paired lists before calculating.
Enter timesMinutes, recoveryFractions, ultimateRecovery according to the field table, then run the tool.
Read fittedRatePerMinute, logResidualIntercept, fitRSquared. Use the calculation below to interpret the selected mean or the model’s signs, units and comparison flags.
Review the supplied assumptions and calibration before comparing or reporting the result.

Input field	Meaning	Value type and bounds	Requirement
timesMinutes	Times minutes	list of number	Required
recoveryFractions	Recovery fractions	list of number	Required
ultimateRecovery	Ultimate recovery	number	Required

Provide increasing nonnegative minutes and aligned nonnegative recoveries strictly below the positive supplied ultimate fraction, which is at most one. Fit slope and intercept and require a negative slope.

log(1−recovery/ultimate) = intercept−rate×time

The symbols and boundary conditions refer to the supplied fields and the implementation described above.

Output field	Meaning	Value type
fittedRatePerMinute	Fitted rate per minute	number
logResidualIntercept	Log residual intercept	number
fitRSquared	Fit r squared	number

json

{
  "input": {
    "timesMinutes": [
      0,
      1,
      2
    ],
    "recoveryFractions": [
      0,
      0.31477547222989327,
      0.5056964470628461
    ],
    "ultimateRecovery": 0.8
  },
  "output": {
    "fittedRatePerMinute": 0.49999999999999983,
    "logResidualIntercept": 0,
    "fitRSquared": 1
  }
}

Limitations and assumptions

The ultimate recovery is fixed by the caller rather than fitted. A fitted intercept need not be zero; the model does not prove first-order kinetics or account for competing mineral populations.

Numerical inputs must be finite. Input-shape and calculation checks do not establish measurement quality or the validity of a supplied calibration.

Alternative or Complementary methods

Choose Grade Recovery Curve when you need its operation: Build observed grade/recovery operating points from matched concentrate masses and assay fractions; points are separate trials, not additive products or a fitted optimization curve. Choose Leaching Kinetics for the complementary operation: Evaluate ideal reaction-controlled shrinking-core leaching 1−(1−X)^(1/3)=kt capped at complete conversion; excludes diffusion control and reagent depletion.

Background reference: Introduction to Mineral Processing.

References

Introduction to Mineral Processing — NPTEL, Indian Institute of Technology Kharagpur

Similar or alternative tools

Grade Recovery Curve
Build observed grade/recovery operating points from matched concentrate masses and assay fractions; points are separate trials, not additive products or a fitted optimization curve.
Leaching Kinetics
Evaluate ideal reaction-controlled shrinking-core leaching 1−(1−X)^(1/3)=kt capped at complete conversion; excludes diffusion control and reagent depletion.