Recovery yield is the result of a measurement model, not a number that stands on its own

In instant-coffee production, recovery yield is usually used to answer a very practical question: from a given amount of raw material or input liquid, how much valuable product is ultimately obtained. The formula can be simple, but the data behind it comes from several different measurements, each carrying its own error.

Because recovery yield is calculated from these quantities, the uncertainty of each measurement propagates into the final result. NIST emphasizes that a measurement result is only complete when accompanied by a quantitative statement of uncertainty, and when the quantity of interest is calculated from several input quantities, the combined uncertainty needs assessing according to a consistent procedure.

This has an important consequence: 17.0% and 17.0% ± a meaningful level of uncertainty don't carry the same meaning as 17.00% reported as if precision to the hundredths place had been demonstrated. The number of decimal places doesn't make a measurement more accurate.

So before calculating recovery yield, you need to define the measurand — the quantity you actually want to know. Recovery yield by wet mass, recovery yield by dry matter, and recovery yield by soluble solids are different indicators. If this isn't agreed upon in advance, comparing between batches or lines can become meaningless.

Where sources of uncertainty usually lie

The first source of uncertainty is mass measurement. Scales have their own resolution, repeatability, and calibration deviation. If mass is inferred from flow rate and time, the uncertainty of both quantities feeds into the result.

The second source is moisture. This variable is especially important for instant products because the same powder mass can represent a different actual dry-matter amount. If recovery yield is compared between batches without converting moisture to the same basis, part of the observed variation may just reflect differences in moisture rather than true process differences.

The third source is sampling and analysis of soluble-solids content. A small sample must represent an entire tank or batch. If the liquid isn't homogeneous or the powder isn't evenly distributed, sampling error can exceed the analytical instrument's own error.

The fourth source is time synchronization. An hourly average flow rate doesn't necessarily represent the exact amount of material in a given batch. If input and output are taken at different time windows, the recovery-yield calculation can produce a number that doesn't correspond to the same actual event.

Finally, there's the model itself. If part of the material held in piping or an intermediate tank isn't accounted for, the recovery-yield formula has already omitted a real variable. NIST also notes that a measurement model needs to include quantities capable of contributing significantly to the uncertainty of the final result.

From individual measurements to the uncertainty of recovery yield

Suppose an indicator is calculated as Y = f(X1, X2,..., Xn). When the inputs all carry uncertainty, the combined uncertainty of Y can be assessed using the law of propagation of uncertainty when the appropriate conditions apply. NIST describes how to calculate this based on sensitivity coefficients for each input variable.

In the simple case, if recovery yield is the ratio of product mass to input mass, the uncertainty of both numerator and denominator needs considering. If recovery yield is calculated on a dry-matter basis, the moisture of both sides becomes an additional variable needing propagation.

There are two useful approaches. The first is building bottom-up: identify each source of uncertainty from the scale, flow meter, analysis, and sampling, then propagate them into the result. The second is looking top-down by repeating the entire measurement under representative conditions and observing the actual spread of results.

When the model is nonlinear or the input distribution doesn't suit a simple approximation, Monte Carlo simulation or another numerical method can be used. But a sophisticated tool doesn't replace correctly identifying the input variables. A very refined model but missing an important variable still produces a misleading conclusion.

Reporting recovery yield so readers know how confident to be in the number

A valuable recovery-yield report should state at least the definition of recovery yield, the mass basis, the time interval or batch being calculated, the main inputs, and the accompanying uncertainty. With only a single percentage figure, the reader has no way to know how much difference between two values is actually meaningful versus just noise.

NIST recommends that when reporting results and uncertainty, you should describe the uncertainty components, how they were assessed, and how a coverage factor was chosen when using expanded uncertainty. This doesn't mean every operating report must be as long as a full measurement report, but the key components should still be available for whoever needs to verify the data.

In process management, uncertainty also helps set a reasonable action threshold. If two batches differ in recovery yield by very little compared with the measurement's own uncertainty, adjusting the process based on that difference alone can create a false sense of “optimization.” Conversely, once a difference clearly exceeds the uncertainty range, that's a much stronger signal worth investigating.

When recovery yield is placed within the same system as mass balance, equipment data, and batch history, it becomes an explainable indicator. Its value doesn't lie in having more decimal digits, but in knowing exactly how reliable that number is and what decision it should be used to support.

References

• NIST Technical Note 1900 (2015). Simple Guide for Evaluating and Expressing the Uncertainty of NIST Measurement Results. DOI: 10.6028/NIST.TN.1900.
• Possolo, A. (2013). Five Examples of Assessment and Expression of Measurement Uncertainty.
• NIST Technical Note 1297. Appendix A: Law of Propagation of Uncertainty; Section 7: Reporting Uncertainty.
• NIST Engineering Statistics Handbook. Propagation of error considerations.