= Blank-allocation design for carbon recovery
\b[Blank-allocation design for carbon recovery] chooses how many specimen and blank measurements to make under a stated cost and uncertainty model. Because a shared blank affects all corrected results, additional specimen readings alone can leave the dominant uncertainty unchanged. The design objective should explicitly include the blank estimate.
For independent specimen totals with individual standard deviation $\sigma_T$ and blanks with standard deviation $\sigma_B$, estimate a net mean by $\overline T-\overline B$. Its variance is
$$
V=\frac{\sigma_T^2}{n_T}+\frac{\sigma_B^2}{n_B}.
$$
Let each specimen measurement cost $c_T$ and each blank cost $c_B$, with budget $C=c_Tn_T+c_Bn_B$. An original continuous optimization using a Lagrange multiplier gives
$$
\frac{n_T}{n_B}=\frac{\sigma_T}{\sigma_B}\sqrt{\frac{c_B}{c_T}},\qquad
V_{\min}=\frac{(\sigma_T\sqrt{c_T}+\sigma_B\sqrt{c_B})^2}{C}.
$$
These formulas assume positive counts, independent observations and known planning variances. Integer counts and minimum replication constraints still require checking.
For stipulated $\sigma_T=1$ mg, $\sigma_B=2$ mg, specimen cost 3 minutes, blank cost 1 minute and budget 60 minutes, a feasible design uses nine specimen measurements and thirty-three blanks. It costs exactly 60 minutes and gives variance $1/9+4/33\approx0.2323\,\mathrm{mg^2}$. Eighteen specimen measurements and six blanks also fit the budget but give approximately $0.7222\,\mathrm{mg^2}$. The more specimen-heavy design is less precise in this model.
The optimization concerns precision, not proof that background subtraction is unbiased. OECD's carbon-dioxide evolution method uses inoculum blanks, while its general principles note that endogenous activity with the test chemical may differ from the blank control.<Test No. 301: Ready Biodegradability by OECD>[General principle §4; method 301 B §§1–3]{ref} More blanks estimate their own mean more precisely without establishing perfect additivity.
<Batch-specific blank allocation> is important when preparation day, medium or collection time changes the background. Pooling all blanks into one common estimate can be inappropriate if the specimen groups do not share that background. Conversely, treating one reused blank estimate as independent for every specimen discards the <shared-blank covariance in carbon recovery>.
A practical planning study therefore records the independent experimental unit, actual preparation costs, temporal grouping and pilot variance uncertainty. The GUM's treatment of correlated inputs supplies the framework for retaining common corrections.<Guide to the expression of uncertainty in measurement by JCGM>[§§5.2.2–5.2.4]{ref} Evaluate feasible designs under plausible alternative variance components rather than reporting a continuous optimum as a universal replication requirement.
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