Uncertainty Analysis in the Determination of Fixed Carbon Content: A Comprehensive Technical Guide

Fixed carbon content is one of the most decisive quality indicators for coal, coke, graphite, and other carbonaceous materials, yet every determination carries a quantifiable measurement uncertainty that directly affects commercial settlement, process control, and conformity decisions. This article presents a complete uncertainty analysis of fixed carbon determination based on the ISO/IEC Guide 98-3 (GUM) framework and proximate analysis standards such as GB/T 212, ISO 17246, and ASTM D3172, covering the mathematical model, uncertainty sources, budget construction, and result reporting. It is written for laboratory engineers, quality managers, and technical staff in accredited third-party laboratories and industrial testing facilities.

Technical Overview

Fixed carbon (FC) is the non-volatile carbonaceous residue that remains after a sample has been heated under standardized proximate analysis conditions and moisture, ash, and volatile matter have been accounted for. Together with these three companion parameters, fixed carbon defines the proximate composition of coal, coke, and related carbonaceous materials, and it serves as a key input for pricing formulas, combustion and gasification calculations, and product specifications. In virtually all reference standards, fixed carbon is not measured directly but is obtained by difference from the other three proximate components.

The By-Difference Model

On an air-dried basis (ad), fixed carbon is calculated as:

FC(ad) = 100 − M(ad) − A(ad) − V(ad)

where M(ad), A(ad), and V(ad) are the moisture, ash, and volatile matter contents expressed in mass percent on the air-dried basis. Because the reported fixed carbon value inherits the measurement uncertainty of three separate determinations, its combined uncertainty is the quadrature sum of the individual component uncertainties. Any uncertainty evaluation that reports only a single component, or that treats fixed carbon as an independently measured quantity, is technically incomplete.

Why Uncertainty Analysis Is Required

• ISO/IEC 17025 requires laboratories to identify the contributions to measurement uncertainty in their testing and to evaluate the uncertainty where relevant for the validity of reported results.
• Coal and coke supply contracts frequently specify fixed carbon limits with bonus and penalty clauses, and the measurement uncertainty defines the defensible margin around these contractual limits.
• Statements of conformity, such as pass or fail decisions against a specification, must account for uncertainty through documented decision rules under ISO/IEC 17025.
• Method validation, interlaboratory comparison, and proficiency testing all rely on a properly constructed uncertainty budget for meaningful interpretation.

Testing Samples

Uncertainty evaluation must consider the physical and chemical characteristics of the material being tested, because sample heterogeneity and preparation quality frequently dominate the overall uncertainty. Typical sample categories include:

Coal and Anthracite

Bituminous coal, anthracite, and lignite prepared to the analytical fineness required by GB/T 212, ISO 17246, or ASTM D3173 through D3175. High-rank coals with high fixed carbon and low volatile matter demand particular attention to the volatile matter determination, because a small absolute error in volatile matter produces a proportionally larger relative effect on the calculated fixed carbon.

Coke and Semi-Coke

Metallurgical coke, foundry coke, and semi-coke (char) exhibit high fixed carbon contents, typically above 85 percent. For these materials, ash and volatile matter repeatability govern the fixed carbon uncertainty, and the coarser porosity of coke increases sub-sampling variance if sample division is not rigorously controlled.

Graphite and Graphite Products

Natural graphite, spherical graphite, and graphite electrode materials are characterized by fixed carbon or total carbon determination, often by combustion-infrared methods or by the indirect ash-plus-volatile approach. For high-purity graphite, uncertainty is dominated by calibration traceability to certified reference materials and by blank correction rather than by proximate repeatability.

Petroleum Coke and Carbon Black

Green and calcined petroleum coke, recarburizers, and carbon black are tested for fixed carbon in support of aluminum, steelmaking, and rubber industries. Volatile matter content and moisture sensitivity of these materials require strict air-drying control to prevent basis drift between gravimetric steps.

Biochar and Activated Carbon

Carbonized biomass and activated carbon products are increasingly specified by fixed carbon. Their high and variable ash contents, together with hygroscopic behavior, make moisture equilibration a significant Type B uncertainty contribution that must be included in the budget.

Testing Items

Primary Measurands

• Moisture content, M(ad), determined gravimetrically by oven drying at 105 to 110 degrees Celsius.
• Ash content, A(ad), determined gravimetrically by complete incineration at 815 degrees Celsius for coal and coke.
• Volatile matter content, V(ad), determined gravimetrically by heating in a covered crucible at 900 degrees Celsius for a fixed time, with a moisture correction applied.
• Fixed carbon content, FC(ad), calculated by difference and reported on the air-dried basis, with optional conversion to dry basis or dry, ash-free basis.

Uncertainty Deliverables

• A documented uncertainty budget listing every standard uncertainty component, its source, its distribution, and its sensitivity coefficient.
• The combined standard uncertainty u_c of the fixed carbon result.
• The expanded uncertainty U = k × u_c with the stated coverage factor, typically k = 2 for a coverage probability of approximately 95 percent.
• The reported result in the complete format, for example FC(ad) = (80.00 ± 0.32) percent, together with the decision rule applied when conformity to a specification is stated.

Testing Methods

Reference Proximate Analysis Procedures

The component determinations are performed in accordance with recognized reference standards, including GB/T 212 for the proximate analysis of coal, ISO 17246 together with ISO 11722, ISO 1171, and ISO 562 for hard coal, and ASTM D3172 with ASTM D3173, D3174, and D3175 for coal and coke. Automated thermogravimetric procedures validated against these reference methods, such as those described in ASTM D7582, may be used provided their equivalence is demonstrated over the applicable concentration range. All gravimetric steps rely on analytical balances calibrated against traceable mass standards, and furnaces are characterized for temperature accuracy and uniformity.

Mathematical Model and Sensitivity Coefficients

From the by-difference model FC = 100 − M − A − V, the sensitivity coefficients are the partial derivatives with respect to each input quantity, each equal to −1 in absolute value. Assuming the three component determinations are uncorrelated, the law of propagation of uncertainty from ISO/IEC Guide 98-3 (GUM) yields:

u_c(FC) = √( u²(M) + u²(A) + u²(V) )

When results are converted to the dry basis using FC(d) = FC(ad) × 100 / (100 − M(ad)), the moisture term appears in both the numerator and the denominator, creating a correlation that must be handled with the appropriate covariance term or, preferably, with a numerical propagation method such as the Monte Carlo approach described in ISO/IEC Guide 98-3 Supplement 1.

Type A Evaluation

Type A components are quantified statistically from replicate observations. The preferred sources are pooled standard deviations from in-house repeatability studies and interlaboratory reproducibility data. Where published repeatability limits r are available from the standards themselves or from ISO 5725 validation studies, the repeatability standard deviation is obtained as s_r = r / 2.83, reflecting the relationship r = 2√2 × s_r for a normal distribution. For example, a repeatability limit of 0.30 percent corresponds to s_r ≈ 0.106 percent for a single determination, which reduces to approximately 0.075 percent when duplicate determinations are averaged.

Type B Evaluation

Type B components are estimated from calibration certificates, equipment specifications, and physical judgment, using appropriate probability distributions:

• Analytical balance resolution: a rectangular distribution with half-width equal to half the readability gives u = a / √3. For a 0.1 mg readability balance and a 1 g test portion, this contribution is below 0.01 percent absolute and is usually negligible but is retained for budget completeness.
• Balance calibration uncertainty: taken from the calibration certificate as u = U / k and combined over the number of weighings involved in each gravimetric difference.
• Furnace temperature tolerance: the effect of the permitted temperature deviation on mass loss is quantified from validation data and treated as rectangular; it is most significant for the volatile matter determination at 900 degrees Celsius.
• Timing and atmosphere control: deviations in heating duration or purge gas flow during volatile matter testing, bounded by instrument specifications.
• Sample homogeneity and sub-sampling variance: estimated from replicate test portions of the same laboratory sample; for coarse or porous materials such as coke this is frequently among the dominant components.
• Basis stability: residual moisture drift between individual determinations caused by incomplete air equilibration of the sample.

Combined and Expanded Uncertainty

The combined standard uncertainty is calculated by quadrature summation of all components weighted by their sensitivity coefficients. The effective degrees of freedom may be verified with the Welch–Satterthwaite formula; in practice, when repeatability data are extensive and Type B components dominate, a coverage factor of k = 2 provides a coverage probability of approximately 95 percent. The expanded uncertainty is rounded to two significant figures, and the reported value is rounded to the same decimal place, in accordance with GUM rounding conventions.

Worked Example of an Uncertainty Budget

For an anthracite laboratory sample measured on the air-dried basis with M(ad) = 1.50 percent, A(ad) = 10.20 percent, and V(ad) = 8.30 percent, the fixed carbon content is FC(ad) = 100 − 1.50 − 10.20 − 8.30 = 80.00 percent. A representative uncertainty budget is:

• Moisture: u(M) = 0.05 percent, dominated by duplicate repeatability with a minor balance contribution.
• Ash: u(A) = 0.08 percent, combining repeatability, balance calibration, and residual carbonization effects.
• Volatile matter: u(V) = 0.13 percent, combining repeatability, furnace temperature tolerance, timing, and the moisture correction embedded in the volatile matter calculation.
• Combined standard uncertainty: u_c(FC) = √(0.05² + 0.08² + 0.13²) = √0.0258 ≈ 0.161 percent.
• Expanded uncertainty: U = 2 × 0.161 ≈ 0.32 percent (k = 2, coverage probability approximately 95 percent).
• Reported result: FC(ad) = (80.00 ± 0.32) percent.

Direct Instrumental Methods and Their Uncertainty Profiles

For graphite, recarburizers, and high-purity carbons, total carbon may be determined directly by high-temperature combustion with infrared detection, and fixed carbon derived by combining total carbon with ash and volatile matter data. The uncertainty profile of these methods differs from gravimetric proximate analysis: the dominant contributions are the calibration uncertainty transferred from certified reference materials, sample mass weighing, blank stability, and material homogeneity. Laboratories must construct a budget appropriate to the actual method chain used rather than assuming the classical proximate profile applies.

Applications

• Coal trade and commercial settlement: defining the defensible margin around contractually specified fixed carbon limits and supporting dispute resolution between trading partners.
• Coke production and blast furnace operation: controlling fixed carbon consistency as an input to coke quality indices and furnace thermal management.
• Foundry and steelmaking recarburizers: verifying carbon yield of graphite and petroleum coke additives used for carbon adjustment in molten iron and steel.
• Combustion and gasification engineering: supplying fixed carbon values with realistic uncertainties for carbon conversion efficiency models and mass balance calculations.
• Graphite electrodes, battery anode materials, and activated carbon: demonstrating compliance of high-fixed-carbon products with tight customer specifications.
• Accreditation and conformity assessment: supporting the scope of accreditation of laboratories under ISO/IEC 17025 and underpinning documented decision rules for pass or fail statements.

FAQ

What is fixed carbon and how is it determined?

Fixed carbon is the calculated non-volatile carbonaceous residue in a carbonaceous material. In standard proximate analysis it is not measured directly; it is obtained by subtracting moisture, ash, and volatile matter from 100 percent on the air-dried basis, following GB/T 212, ISO 17246, or ASTM D3172.

Why is the uncertainty of fixed carbon larger than that of any single component?

Because fixed carbon is calculated by difference, its combined standard uncertainty is the quadrature sum of the moisture, ash, and volatile matter uncertainties. In addition, since fixed carbon is typically a large number close to 100 percent, the volatile matter and ash determinations, which have the largest absolute uncertainties, contribute most strongly to the fixed carbon uncertainty.

What coverage factor should be used when reporting the expanded uncertainty?

A coverage factor of k = 2 is standard practice, corresponding to a coverage probability of approximately 95 percent under the assumption of a normal distribution. The choice should be justified through the effective degrees of freedom, and both k and the resulting expanded uncertainty must be stated in the report.

How should a fixed carbon result with uncertainty be reported?

The recommended format is FC(ad) = (80.00 ± 0.32) percent, where 0.32 percent is the expanded uncertainty obtained using a coverage factor k = 2 corresponding to a coverage probability of approximately 95 percent. The basis of the result, air-dried, dry, or dry ash-free, must always be stated.

What are the dominant uncertainty sources in practice?

In routine coal and coke testing, the dominant contributions are usually the repeatability of the volatile matter determination, furnace temperature control, and sample heterogeneity. Balance resolution and calibration are rarely dominant for 1 g test portions but are included for completeness.

Can the uncertainty of fixed carbon be reduced?

Yes. Averaging duplicate determinations reduces Type A components by a factor of the square root of the number of replicates, rigorous air-drying stabilizes the basis, improved sample division lowers sub-sampling variance, and tighter furnace calibration reduces the temperature-related Type B component. The achievable reduction is limited by the intrinsic repeatability of the standardized methods.

Is fixed carbon the same as total carbon?

No. Fixed carbon is an empirical construct of proximate analysis, whereas total carbon is the elemental carbon content measured by combustion techniques. The two values differ because ash contains carbonates, volatile matter contains some carbon, and fixed carbon contains residual inorganic residues; they should never be interchanged in specifications or uncertainty budgets.