Mathematical Modeling of Biochar Pore Descriptors from Pyrolysis Temperature: Semi-Empirical Correlations for BET Surface Area, Total Pore Volume, and Mean Pore Diameter of Lignocellulosic Feedstocks

Abstract

Predicting the pore structure of lignocellulosic biochar from pyrolysis conditions without exhaustive experimental characterization remains an open challenge. We fit semi-empirical Arrhenius-type and power-law correlations linking pyrolysis temperature to $S_{\mathrm{BET}}$, $V_{\mathrm{T}}$, and ${\overline{d}}_p$ by nonlinear least squares, using 45 literature records from seven open access studies (12 feedstocks, 300–800 °C). The central finding is that feedstock category, not temperature alone, dominates variance in $S_{\mathrm{BET}}$: pooled calibration explains only $R^2=0.199$ (RMSE = 183 m² g⁻¹), whereas feedstock-stratified fitting recovers accuracy (e.g., RMSE = 43.6 m² g⁻¹ for grasses, a within-study estimate from a single source). The Arrhenius and power-law forms are statistically indistinguishable ($\Delta \mathrm{AIC}<2$); the Arrhenius form is adopted for physical interpretability. Pooled fits reach $R^2=0.691$ ($V_{\mathrm{T}}$) and $0.563$ (${\overline{d}}_p$). Leave-one-study-out cross-validation (RMSE = 184 m² g⁻¹) confirms that reliable prediction requires calibration data within the target feedstock category. The correlations are descriptive tools valid within their calibration envelope, not general predictive models. Estimation uses no machine learning; a benchmark against OLS and random forest models confirms that greater flexibility improves in-sample fit but not out-of-sample generalization.

1. Introduction

Biochar is the solid carbonaceous product of lignocellulosic biomass heated under limited oxygen supply, a process known as slow pyrolysis. Its principal uses span soil amendment (carbon sequestration, water retention, and cation exchange capacity), the adsorption of heavy metals and organic pollutants from aqueous media, and acting as a structural support for catalysts and microbial communities [1,2,3]. Across all three application domains, effectiveness depends on pore structure: surface area and micropore volume set adsorption capacity, while the meso- and macropore network governs mass transport resistance during both uptake and regeneration. Translating pyrolysis operating conditions into predictable pore structure is therefore a prerequisite for rational process design, yet no general predictive framework exists for slow pyrolysis biochar.

The pore structure of biochar is conventionally described by four scalar quantities determined from $N_2$ physisorption at 77 K, BET surface area ($S_{\mathrm{BET}}$, m²/g), total pore volume ($V_{\mathrm{T}}$, cm³/g), micropore volume ($V_{\mathrm{mic}}$, cm³/g), and mean pore diameter (${\overline{d}}_p$, nm), computed by the BET and BJH methods [4,5,6]. These scalar descriptors summarize a continuous pore size distribution (PSD) spanning micropores (<2 nm), mesopores (2–50 nm), and macropores (>50 nm). The shape of the PSD (how pore volume is partitioned across size classes) determines not only total capacity but also the selectivity and kinetics of mass transfer, so predicting it from operating conditions has direct engineering value. It should be stated at the outset that “porosity” in this work is operationalized through the three scalar descriptors $S_{\mathrm{BET}}$, $V_{\mathrm{T}}$, and ${\overline{d}}_p$; the full differential PSD curve and the surface fractal dimension are not modeled, because the compiled literature reports only these summary statistics. The correlations developed here therefore describe the magnitude of pore development, not the detailed evolution of pore size distribution, which remains future work.

Final pyrolysis temperature T exerts strong, systematic control on pore development within a given feedstock. Below approximately 400 °C, the devolatilization of hemicellulose and cellulose produces an amorphous char with $S_{\mathrm{BET}}$ typically below 50 m²g⁻¹; above 400 °C, progressive graphitization and tar evolution open a micropore network, and $S_{\mathrm{BET}}$ rises sharply, reaching 200 m²g⁻¹ to 500 m²g⁻¹ at 700 °C to 800 °C for wood-derived biochars [7,8]. Heating rate $\beta$ and residence time modulate the temperature response curve by controlling exposure duration at intermediate temperatures during the charring sequence [9]. Feedstock composition (cellulose, hemicellulose, and lignin fractions and mineral content) sets the amplitude of that response at any given temperature [2]. A central result of the present work is that this compositional amplitude, rather than temperature itself, accounts for the larger share of the variance in $S_{\mathrm{BET}}$ across the lignocellulosic feedstock space; temperature-only correlations, although mechanistically well-grounded, are therefore inherently limited as cross-feedstock predictors.

Current practice characterizes biochar pore structure on a case-by-case basis: for each feedstock–temperature combination, $N_2$ adsorption isotherms are collected, $S_{\mathrm{BET}}$ and $V_{\mathrm{T}}$ are extracted, and BJH analysis yields the PSD. This approach is resource-intensive, and the results are protocol-sensitive: Sigmund et al. [10] report that degassing temperature alone can shift the measured $S_{\mathrm{BET}}$ by up to $300\%$, compounding the inter-study variability that any predictive model must accommodate [11]. Machine learning models have addressed this gap: Li et al. [12] apply gradient boosting to predict $S_{\mathrm{BET}}$ and $V_{\mathrm{T}}$ from 27 input features with high in-sample accuracy, but the approach requires large training datasets, lacks physical interpretability, and cannot extrapolate reliably beyond the feature range of the training corpus. Semi-empirical correlations with explicit functional forms linking T and $\beta$ to pore structure descriptors (the standard approach in activated carbon science [13,14]) have not been systematically developed for slow pyrolysis biochar across multiple feedstock categories.

Two methodological gaps stand out. First, no prior work has fitted parametric pore size distribution functions to BJH data from slow pyrolysis biochar and then correlated the PSD shape parameters to pyrolysis temperature, a connection that would enable PSD prediction from operating conditions alone. Second, while the qualitative relationship between T and $S_{\mathrm{BET}}$ is well documented [7,8], explicit functional forms with fitted parameters and uncertainty quantification have not been established for slow pyrolysis biochar across the five major lignocellulosic feedstock categories (wood, herbaceous, grass, shell, mixed). Without such correlations, practitioners cannot predict $S_{\mathrm{BET}}$ or $V_{\mathrm{T}}$ for a new (feedstock, T) combination without conducting new experiments.

This paper directly addresses the second gap through a systematic literature data approach. We compile a calibration dataset of 45 records from seven open access studies covering 12 distinct feedstocks over 300 °C to 800 °C and fit Arrhenius-type (phenomenological, not kinetic) and power-law semi-empirical correlations to $S_{\mathrm{BET}}\left(T\right)$, $V_{\mathrm{T}}\left(T\right)$, and ${\overline{d}}_p\left(T\right)$ by nonlinear least squares. Full parametric pore size distribution modeling is beyond the present scope, since the compiled sources report only scalar descriptors rather than differential volume curves, and is identified as future work. The estimation requires only the summary pore structure statistics available in the published literature. The scope is explicitly restricted to the solid carbonaceous product; volatile-phase yields (syngas composition, bio-oil fractions, and global mass balances) are outside the present framework.

The main contributions of this work are:

(i)

A compiled and quality-screened calibration dataset of 45 literature records from seven open access studies spanning 12 lignocellulosic feedstocks and 300 °C to 800 °C, with documented inclusion and exclusion criteria and per-record data quality flags.

(ii)

Arrhenius-type correlations for $S_{\mathrm{BET}}\left(T\right)$ (pooled and stratified by feedstock category, where identifiable), $V_{\mathrm{T}}\left(T\right)$, and ${\overline{d}}_p\left(T\right)$ (pooled), with fitted parameters, $95\%$ confidence intervals, and AIC/BIC comparison against a power-law alternative.

(iii)

The quantitative finding that feedstock category dominates variance in $S_{\mathrm{BET}}$: pooled calibration achieves $R^2=0.199$ (MAPE = $712\%$), while stratification by feedstock reduces error to MAPE = $15.1\%$ for the grass category, which directly bounds the scope of temperature-only predictive models.

(iv)

A leave-one-study-out cross-validation protocol that provides a study-level generalization test of the fitted correlations, independent of in-sample fit metrics.

Unlike machine learning surrogates that optimize predictive accuracy given large training sets [12], the semi-empirical framework presented here is designed for settings where data are sparse and interpretability matters: (i) the fitted parameters ($E_S$, ${\beta }_S$) carry a physically meaningful interpretation as apparent structural activation energies or scaling exponents; (ii) calibration requires only the summary pore structure statistics already available in the published literature; and (iii) the calibration envelope is explicitly defined, which makes extrapolation boundaries transparent. To test whether these design choices come at the cost of predictive accuracy, we benchmark the correlations for all three descriptors against ordinary least squares and random forest surrogates trained on the same data (Section 5.6); the data-driven models fit the training data better yet do not generalize better under study-level cross-validation, empirically supporting the parsimonious form for a dataset of this size.

The specific objectives of this work are: (1) to compile and quality-screen a calibration dataset of literature records spanning multiple lignocellulosic feedstock categories and the 300–800 °C temperature range; (2) to fit and compare Arrhenius-type and power-law semi-empirical correlations for $S_{\mathrm{BET}}\left(T\right)$, $V_{\mathrm{T}}\left(T\right)$, and ${\overline{d}}_p\left(T\right)$ by nonlinear least squares, both pooled and stratified by feedstock category; (3) to quantify prediction uncertainty through bootstrapped confidence intervals and leave-one-study-out cross-validation; and (4) to delineate the calibration envelope within which the proposed correlations provide reliable estimates and beyond which extrapolation is unsupported.

This paper proceeds as follows. Section 2 defines the pore structure descriptors and derives the semi-empirical correlation framework applied in this study. Section 3 describes the literature compilation protocol, inclusion criteria, and summary statistics of the calibration dataset. Section 4 details the nonlinear least squares fitting procedure, functional form selection, and model evaluation. Section 5 presents the calibration results, feedstock-stratified fits, and the leave-one-study-out validation. Section 6 interprets the results in their thermochemical and methodological context. Section 7 states the main conclusions and the conditions under which the fitted correlations are applicable.

2. Theoretical Framework

This section establishes the mathematical basis for describing pore structure in biochar. We define the key structural descriptors (Section 2.1) and derive the semi-empirical relationships that link pyrolysis operating conditions to the structural descriptors (Section 2.2).

2.1. Pore Structure Descriptors

The pore structure of biochar is characterized by the following descriptors, consistent with IUPAC nomenclature [6]:

$S_{\mathrm{BET}}$:

BET specific surface area (m²/g), determined from the nitrogen adsorption isotherm in the relative pressure range $p/p_0\in [0.05,0.30]$.

$V_{\mathrm{T}}$:

Total pore volume (cm³/g), estimated from the amount of $N_2$ adsorbed at $p/p_0=0.99$.

$V_{\mathrm{mic}}$:

Micropore volume (cm³/g), pores with width $w<2$ nm, obtained by the t-plot or Dubinin–Radushkevich method.

${\overline{d}}_p$:

Average pore diameter (nm), approximated as ${\overline{d}}_p=4V_{\mathrm{T}}/S_{\mathrm{BET}}$ assuming cylindrical geometry. This relation yields an effective mean pore size and is adequate for meso- and macropore-dominated biochars. Its validity decreases as micropore fraction increases: in highly microporous biochars ($f_{\mathrm{mic}}>0.7$, typically at $T>650$ °C), slit-shaped or ink-bottle geometries deviate substantially from the cylindrical assumption [6], and ${\overline{d}}_p$ should be interpreted as an apparent descriptor rather than a true geometric mean.

$f_{\mathrm{mic}}$:

Micropore fraction, $f_{\mathrm{mic}}=V_{\mathrm{mic}}/V_{\mathrm{T}}$.

$D_{\mathrm{f}}$:

Surface fractal dimension (dimensionless, $2\le D_{\mathrm{f}}\le 3$), estimable from a Frenkel–Halsey–Hill analysis of the full nitrogen adsorption isotherm; it is not calibrated here, since the compiled dataset provides only scalar descriptors, and is identified as future work (Section 6.6).

2.2. Semi-Empirical Correlations with Pyrolysis Conditions

Pyrolysis temperature $T\left(K\right)$ is the primary driver of pore development within a given feedstock [7,8]; across feedstocks, composition sets the amplitude of that response (Section 5.2). Based on the mechanistic sequence of devolatilization, polycondensation, and graphitization, we postulate below the functional forms for $S_{\mathrm{BET}}\left(T\right)$; the same two forms are applied to $V_{\mathrm{T}}\left(T\right)$ and ${\overline{d}}_p\left(T\right)$ in Section 3.3.

2.2.1. The Phenomenological Nature of the Arrhenius Form

The correlations below adopt an Arrhenius-type functional form as a phenomenological descriptor, not as a representation of any specific chemical reaction. The exponential temperature dependence reflects the aggregate effect of overlapping decomposition reactions whose individual rate constants follow Arrhenius-type dependencies over the 300 °C to 800 °C window. The fitted parameter $E_S$ is therefore an apparent, or structural, activation energy: a curve-fitting parameter that captures the composite temperature sensitivity of pore opening, not the activation energy of an individual reaction step. This usage follows the convention established in the activated carbon literature [13,14], where Arrhenius-type structural correlations have long been distinguished from kinetic activation energies derived from TGA/DTG measurements of individual reaction channels [15].

2.2.2. BET Surface Area

$$S_{\mathrm{BET}}\left(T\right)=S_0\exp \left({\displaystyle \frac{-E_S}{RT}}\right),$$

(1)

where $S_0$ (m²/g) and $E_S$ (kJ/mol) are empirical constants, and $R=8.314$ J mol⁻¹K⁻¹ is the gas constant.

Alternatively, if a simpler power-law relationship is preferred,

$$S_{\mathrm{BET}}\left(T\right)={\alpha }_S{\left({\displaystyle \frac{T}{T_{\mathrm{ref}}}}\right)}^{{\beta }_S},$$

(2)

with $T_{\mathrm{ref}}$ = $773$ K (500 °C) as the reference temperature. Model selection between Equations (1) and (2) follows the criterion in Section 4.2.

3. Materials and Methods

3.1. Literature Dataset for Model Calibration

The calibration dataset was assembled from a systematic search of Web of Science, Scopus, and Google Scholar using the query terms biochar, BET surface area, pore size distribution, pyrolysis temperature, and $N_2$ adsorption. The search was restricted to peer-reviewed articles published between 2000 and 2024. A record was included only if it satisfied all six criteria simultaneously: (i) the final pyrolysis temperature T fell within 300 °C to 800 °C; (ii) the thermal regime corresponded to slow pyrolysis, or the heating rate was not specified and could reasonably be classified as slow; (iii) at least one of the descriptors $\{S_{\mathrm{BET}},V_{\mathrm{T}},V_{\mathrm{mic}}\}$ was explicitly reported; (iv) $N_2$ physisorption at $77$ K was the primary characterization technique; (v) the feedstock was lignocellulosic, belonging to one of the wood, herbaceous, grass, shell, or mixed categories; and (vi) pore size distribution analysis was performed using BJH applied to the $N_2$ desorption branch, following the standard convention for mesopore characterization [5,6].

Records were excluded if the pyrolysis reactor was a fluidized bed (fast pyrolysis), if microwave pyrolysis was used, if pore structure was characterized exclusively by CO₂ adsorption or mercury porosimetry, or if T fell outside the 300 °C to 800 °C range. Studies relying on CO₂ adsorption alone were excluded because this technique probes a different pore size window than $N_2$ adsorption at $77$ K and would introduce systematic incompatibility into the fitted correlations [11].

After screening 93 records from 11 independent studies, 45 records from 7 studies were retained (48 excluded). The retained sources are Ronsse et al. [8], Askeland et al. [16], Chatterjee et al. [17], Herrera et al. [18], Muzyka et al. [9], and Novak et al. [19]. The 45 records cover 12 distinct feedstocks: 2 wood species, 5 herbaceous materials, 2 grasses, 2 shells, and 1 mixed feedstock. Temperature coverage is 5 records at 300 °C to 400 °C, 6 at 400 °C to 500 °C, 9 at 500 °C to 600 °C, 10 at 600 °C to 700 °C, and 10 at 700 °C to 800 °C. In terms of variable completeness, $S_{\mathrm{BET}}$ is available for all 45 records; $V_{\mathrm{T}}$ for 22; $V_{\mathrm{mic}}$ for 26; and ${\overline{d}}_p$ for 27, either reported directly or computed as ${\overline{d}}_p=4V_{\mathrm{T}}/S_{\mathrm{BET}}$ where $V_{\mathrm{T}}$ and $S_{\mathrm{BET}}$ were both available.

Table 1. A summary of the seven studies retained in the calibration dataset.

Study	Feedstock Category	Feedstocks (Representative)	N	T Range (°C)	BJH Branch	Degassing T
Chatterjee et al. [17]	Grass	Miscanthus, switchgrass	16	300–700	Desorption	NR
Herrera et al. [18]	Shell	Rice husk	6	400–700	Desorption	NR
Ronsse et al. [8]	Wood, Herbaceous	Pine wood, wheat straw	7	300–750	Desorption	NR
Askeland et al. [16]	Wood	Pine sawdust	5	300–750	Desorption	NR
Muzyka et al. [9]	Mixed	Mixed biomass	3	500–800	Desorption	NR
Novak et al. [19]	Shell	Pecan shell	2	300–700	Desorption	NR

Notes: NR = not reported. Degassing temperature before $N_2$ adsorption was not consistently reported in any of the seven source studies. All studies used the BJH desorption branch; the two candidate records that applied BJH to the adsorption branch were excluded (see inclusion criterion vi above). Data from Novak et al. [19] were obtained via secondary citation, treated with additional caution. Rice husk [18] is classified as shell because this study explicitly characterizes the material as a silica-rich agricultural hull; see Section 3.2 for classification rules.

summarizes the seven retained studies with their feedstock categories, record counts, temperature ranges, BJH branch, and degassing status. Figure 1 summarizes the compiled dataset by plotting $S_{\mathrm{BET}}$ against pyrolysis temperature for all 45 included records, colored by feedstock category.

Four data quality issues warrant acknowledgment. First, the degassing temperature applied before $N_2$ adsorption was not reported in any of the seven retained studies (Table 1, column “Degassing $T$”). Sigmund et al. [10] report that degassing temperature alone can shift the measured $S_{\mathrm{BET}}$ by up to $300\%$; this unmeasured source of variability cannot be controlled as a covariate in the present calibration and contributes to the residual variance across studies. Second, the single-study origin of the grass group—all 16 grass records derive from Chatterjee et al. [17]—means that the grass group MAPE of $15.1\%$ reflects within-study prediction error for a common measurement protocol, not between-study generalization; it should not be interpreted as the expected accuracy for a new study using different equipment or degassing conditions. Third, two records from Novak et al. [19] (pecan shell feedstock) were obtained via secondary citation and were treated with additional caution during fitting. Fourth, the BJH method was applied to the desorption branch in all included studies [5]; the two candidate records that applied BJH to the adsorption branch were excluded (criterion vi above) to maintain methodological consistency across the dataset.

3.2. Feedstock Classification Rules

Each record was assigned to one of five feedstock categories (wood, herbaceous, grass, shell, or mixed) using the following hierarchy: (1) the source study’s own terminology was adopted where unambiguous; (2) when the study provided no category, IPCC biomass classification conventions were followed; and (3) where botanical and compositional criteria conflicted, silica content was the tiebreaker.

Under this hierarchy, rice husk was classified as shell for records from Herrera et al. [18], because their study explicitly characterizes rice husk as a high-silica agricultural hull ($15\%$ to $20\%$ SiO₂) and treats it as a structural shell material. Rice husk from studies that classify it as an herbaceous straw residue (a convention found in some agronomic databases) would instead be assigned to the herbaceous category; no such records were retained in this dataset after applying the inclusion criteria above. Consequently, the herbaceous group in this dataset comprises maize straw, wheat straw, and analogous crop residues; it does not include rice husk. Practitioners applying the correlations to rice husk should use the shell category parameters while acknowledging that the shell group is not well-identified at the available sample size (see Section 6.7).

3.3. Model Calibration Procedure

All semi-empirical correlations are estimated by nonlinear least squares (NLLS) using scipy.optimize.curve_fit in Python 3.x. The objective function minimized in each case is the sum of squared residuals between the observed and predicted values of the target descriptor.

3.3.1. BET Surface Area

Two functional forms are fitted to $S_{\mathrm{BET}}\left(T\right)$: the Arrhenius-type expression given by Equation (1) and the power-law expression given by Equation (2). In the power-law form, the reference temperature is $T_{\mathrm{ref}}$ = $773.15$K (≈500 °C), chosen as the approximate midpoint of the 300 °C to 800 °C fitting window and the threshold above which micropore development accelerates; centering the normalization here improves the numerical stability of the NLLS fit. Each form is fitted separately for each feedstock category with at least 5 records (wood, herbaceous, grass, shell). Goodness of fit is assessed by the root mean square error (RMSE), the coefficient of determination $R^2$, and three percentage-based metrics. The mean absolute percentage error,

$$\mathrm{MAPE}={\displaystyle \frac{100}{n}}{\displaystyle \sum _{i=1}^n}\left|{\displaystyle \frac{y_i-{\widehat{y}}_i}{y_i}}\right|,$$

(3)

is undefined for $y_i=0$ and is inflated severely as $y_i\to 0$, which motivates the stratified reporting described in Section 5.4. The mean absolute error,

$$\mathrm{MAE}={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n}|y_i-{\widehat{y}}_i|,$$

(4)

reports prediction error in the same physical units as the target variable and is not distorted by near-zero observations. The symmetric mean absolute percentage error,

$$\mathrm{sMAPE}={\displaystyle \frac{100}{n}}{\displaystyle \sum _{i=1}^n}{\displaystyle \frac{|y_i-{\widehat{y}}_i|}{\left(\right|y_i|+|{\widehat{y}}_i\left|\right)/2}},$$

(5)

is bounded between $0\%$ and $200\%$; it is less sensitive than the MAPE to near-zero observations because the denominator incorporates both observed and predicted values [20]. Here, $y_i$ represents observed values, and ${\widehat{y}}_i$ represents model predictions. Reporting several complementary metrics is warranted precisely because the dataset is small and spans a 250-fold range in $S_{\mathrm{BET}}$, over which no single metric is robust: the MAPE is destabilized by near-zero low-temperature records, RMSE is dominated by high-$S_{\mathrm{BET}}$ outliers, and MAE and sMAPE provide bounded, unit-consistent complements. The metrics are therefore interpreted jointly rather than individually. Model selection between the two forms follows the Akaike and Bayesian information criteria [21], as described in Section 4.2.

3.3.2. Total Pore Volume and Average Pore Diameter

The same two functional forms are applied to $V_{\mathrm{T}}\left(T\right)$ and ${\overline{d}}_p\left(T\right)$ for feedstock groups where the available $N$ is at least 5. Where ${\overline{d}}_p$ was not reported directly, it was calculated as ${\overline{d}}_p=4V_{\mathrm{T}}/S_{\mathrm{BET}}$ using the cylindrical pore approximation established in Section 2.1.

3.3.3. Uncertainty Quantification

Parameter uncertainty is quantified by two methods depending on group sample size.

Asymptotic covariance matrix CI (groups with $N>10$). For groups with sufficient data, the $95\%$ confidence interval for each parameter ${\widehat{\theta }}_i$ is ${\widehat{\theta }}_i\pm 1.96\sqrt{C_{\mathit{ii}}}$, where $C_{\mathit{ii}}$ is the diagonal entry of the estimated covariance matrix returned by the NLLS solver [22]. A convergence flag (✓or †) records whether the optimizer returned a valid (positive semi-definite) covariance matrix; † indicates an ill-conditioned matrix whose CIs are unreliable.

Parametric residual bootstrap CI (groups with $N\le 10$).

For small-group fits, the asymptotic approximation can be unreliable because the covariance matrix estimate is based on few observations. In these cases, $95\%$ CIs are computed by parametric residual bootstrap ($B=500$ replications) following Efron and Tibshirani [23]: (i) fit the model to the original $n$ observations and record residuals $e_i=y_i-{\widehat{y}}_i$; (ii) form a bootstrap sample by drawing $n$ residuals with replacement from $\left\{e_i\right\}$, adding them to the fitted values ${\widehat{y}}_i$, and clipping the result at zero to preserve non-negativity; (iii) refit the model to the bootstrap response vector and record the parameter estimates; (iv) repeat steps (ii)–(iii) $B=500$ times; and (v) report the $95\%$ CI as the interval from the 2.5th to the 97.5th percentile of the bootstrap parameter distribution. Residual resampling (rather than case resampling) is used because the temperature grid $T_i$ is fixed by the experimental design, and only the response values carry sampling uncertainty. The choice of $B=500$ was confirmed to be adequate by inspecting CI width stability: increasing it to $B=1000$ changed reported CI half-widths by less than $2\%$ in all fitted groups. Bootstrap CIs are identified in the parameter tables with the superscript “$b$”.

3.4. Cross-Validation

No independent experimental dataset is available for external validation. Cross-validation on the literature dataset therefore serves as the primary internal consistency check, assessing whether the fitted correlations generalize across studies rather than merely fitting to the pooled sample.

The cross-validation protocol is leave-one-study-out (LOSO-CV). In each of the 7 folds, one contributing study is held out; the model is fitted on the remaining six studies and used to predict the descriptors for the held-out study. This procedure deliberately tests generalization at the study level, which is the natural unit of systematic variation in the dataset: studies differ in instrument, analyst, degassing protocol, and feedstock provenance.

Performance is reported for each fold as the MAPE and RMSE on held-out predictions. These are then aggregated across folds (unweighted mean) to give the LOSO-CV MAPE and RMSE. The in-sample $R^2$ from the full-dataset fit is reported alongside the LOSO-CV MAPE to make explicit the gap between fitting and prediction accuracy. A large gap between in-sample $R^2$ and out-of-sample MAPE signals overfitting, which would be particularly consequential given the limited $N$ in each feedstock group.

4. Model Development and Parameter Estimation

4.1. Overview of Modeling Strategy

The modeling strategy proceeds in two stages:

(i)

Correlation fitting —Arrhenius-type and power-law semi-empirical correlations for $S_{\mathrm{BET}}\left(T\right)$, $V_{\mathrm{T}}\left(T\right)$, and ${\overline{d}}_p\left(T\right)$ are fitted by nonlinear least squares, pooled and stratified by feedstock category. Functional form selection uses AIC and BIC (Section 4.2 and Section 4.3).

(ii)

Cross-validation—A leave-one-study-out (LOSO-CV) protocol assesses generalization at the study level, which is the natural unit of systematic variation in the compiled dataset (Section 3.4).

The correlation framework itself employs no machine learning; all fitting is performed using classical nonlinear least squares estimation. Data-driven surrogates (OLS and random forest) are introduced solely as a benchmark to test whether added model flexibility improves genuine generalization (Section 5.6).

4.2. Model Selection

The two candidate functional forms for each pore structure descriptor—Arrhenius (Equation (1)) and power-law (Equation (2))—are compared using the Akaike information criterion (AIC) and the Bayesian information criterion (BIC):

$$\begin{array}{cc}\hfill \mathrm{AIC}& =2p-2\ln \widehat{\mathcal{L}},\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill \mathrm{BIC}& =p\ln n-2\ln \widehat{\mathcal{L}},\hfill \end{array}$$

(7)

where $p$ is the number of free parameters, $n$ is the number of data points, and $\widehat{\mathcal{L}}$ is the maximized log-likelihood under a Gaussian error assumption. When $|\Delta \mathrm{AIC}|<2$, both models are treated as empirically indistinguishable; in such cases, the model reported as the “preferred” form in Table 2 is the one with the marginally lower AIC value, but this designation carries no statistical weight, and either form is equally defensible. Note that when both models are fitted to the same dataset with the same number of parameters ($p=2$ in both cases), identical AIC values can arise if the residual sum of squares is identical to the precision reported—this is expected and reflects the near-equivalent fit documented throughout.

Table 2. Model selection statistics for all fitted target-by-group combinations.

Group	Target	AIC Arrhenius	AIC Power-Law	Winner	ΔAIC
pooled	$S_{\mathrm{BET}}$	601	601	Arrhenius	0.546
grass	$S_{\mathrm{BET}}$	87.1	88.1	Arrhenius	1.00
herbaceous	$S_{\mathrm{BET}}$	241	241	Arrhenius	0.500
shell	$S_{\mathrm{BET}}$	88.5	88.7	Arrhenius	0.169
wood	$S_{\mathrm{BET}}$	111	111	Arrhenius	0.232
pooled	$V_{\mathrm{T}}$	−79.7	−78.6	Arrhenius	1.04
grass	$V_{\mathrm{T}}$	−34.3	−33.2	Arrhenius	1.02
herbaceous	$V_{\mathrm{T}}$	−48.8	−48.5	Arrhenius	0.270
pooled	${\overline{d}}_p$	129	129	Arrhenius	0.474
grass	${\overline{d}}_p$	−15.4	−15.4	Power-law	0.0225
herbaceous	${\overline{d}}_p$	77.3	76.6	Power-law	0.693

Notes: $\Delta \mathrm{AIC}={\mathrm{AIC}}_{\mathrm{Power}\text{-}\mathrm{law}}-{\mathrm{AIC}}_{\mathrm{Arrhenius}}$; positive values indicate that the Arrhenius model achieves a lower (better) AIC. By convention, $|\Delta \mathrm{AIC}|<2$ implies that the two forms are empirically indistinguishable at the available sample size [21]. The AIC columns are rounded to three significant figures, so the two forms may display identical AIC values even where $\Delta \mathrm{AIC}$ is a small nonzero number; the reported $\Delta \mathrm{AIC}$ is computed from the full-precision AIC values, not from the rounded entries. In every case, $|\Delta \mathrm{AIC}|<2$, so the “Winner” column identifies only the marginally lower AIC and carries no statistical weight. The wood and shell $S_{\mathrm{BET}}$ rows are included for completeness; the functional form comparison rests on the residual sum of squares and remains valid even though the fitted parameters of these two groups are not identified (Section 5.2) and are not reported. Sample sizes per group are reported in Table 3 and Table 4.

Table 2. Model selection statistics for all fitted target-by-group combinations.

Group	Target	AIC Arrhenius	AIC Power-Law	Winner	ΔAIC
pooled	$S_{\mathrm{BET}}$	601	601	Arrhenius	0.546
grass	$S_{\mathrm{BET}}$	87.1	88.1	Arrhenius	1.00
herbaceous	$S_{\mathrm{BET}}$	241	241	Arrhenius	0.500
shell	$S_{\mathrm{BET}}$	88.5	88.7	Arrhenius	0.169
wood	$S_{\mathrm{BET}}$	111	111	Arrhenius	0.232
pooled	$V_{\mathrm{T}}$	−79.7	−78.6	Arrhenius	1.04
grass	$V_{\mathrm{T}}$	−34.3	−33.2	Arrhenius	1.02
herbaceous	$V_{\mathrm{T}}$	−48.8	−48.5	Arrhenius	0.270
pooled	${\overline{d}}_p$	129	129	Arrhenius	0.474
grass	${\overline{d}}_p$	−15.4	−15.4	Power-law	0.0225
herbaceous	${\overline{d}}_p$	77.3	76.6	Power-law	0.693

Notes: $\Delta \mathrm{AIC}={\mathrm{AIC}}_{\mathrm{Power}\text{-}\mathrm{law}}-{\mathrm{AIC}}_{\mathrm{Arrhenius}}$; positive values indicate that the Arrhenius model achieves a lower (better) AIC. By convention, $|\Delta \mathrm{AIC}|<2$ implies that the two forms are empirically indistinguishable at the available sample size [21]. The AIC columns are rounded to three significant figures, so the two forms may display identical AIC values even where $\Delta \mathrm{AIC}$ is a small nonzero number; the reported $\Delta \mathrm{AIC}$ is computed from the full-precision AIC values, not from the rounded entries. In every case, $|\Delta \mathrm{AIC}|<2$, so the “Winner” column identifies only the marginally lower AIC and carries no statistical weight. The wood and shell $S_{\mathrm{BET}}$ rows are included for completeness; the functional form comparison rests on the residual sum of squares and remains valid even though the fitted parameters of these two groups are not identified (Section 5.2) and are not reported. Sample sizes per group are reported in Table 3 and Table 4.

Table 3. Fitted parameters for $S_{\mathrm{BET}}$—Arrhenius and power-law models, pooled and for the identifiable feedstock categories (grass and herbaceous). The wood and shell groups are not identified (bootstrap confidence intervals exceed the point estimates), and no fitted parameters are reported for them; see Section 5.2.

Group	Model	Param1	95% CI	Param2	95% CI	RMSE	MAPE (%)	AIC	BIC	N	Conv.
pooled	Arrhenius	2194	[3535]	16,010	[12,289]	183	712	601	604	45	✓
pooled	Power-law	179	[69.6]	2.22	[1.67]	185	759	601	605	45	✓
grass	Arrhenius	1922	[1798 b]	14,053	[6698 b]	43.6	15.1	87.1	87.3	8	✓
grass	Power-law	221	[49.0 b]	1.79	[0.913 b]	46.4	16.4	88.1	88.3	8	✓
herbaceous	Arrhenius	17,697	[36,769]	34,248	[17,180]	90.1	298	241	243	20	✓
herbaceous	Power-law	89.5	[53.3]	4.46	[2.16]	91.2	331	241	243	20	✓

Notes: Arrhenius model: $S_{\mathrm{BET}}\left(T\right)=S_0\exp (-E_S/\mathrm{RT})$; ${\mathrm{Param}}_1=S_0$ (m²/g), ${\mathrm{Param}}_2=E_S$ (J/mol). Power-law model: $S_{\mathrm{BET}}\left(T\right)=\alpha {(T/T_{\mathrm{ref}})}^{\widehat{\beta }}$, $T_{\mathrm{ref}}=773.15K$; ${\mathrm{Param}}_1=\alpha$ (m²/g), ${\mathrm{Param}}_2=\widehat{\beta }$ (dimensionless). Half-widths of 95% CIs: parametric bootstrap ($B=500$, residual resampling) for groups with $N\le 10$ (grass); asymptotic covariance matrix for $N>10$ (herbaceous, pooled) [22]. Convergence flag: ✓ = optimizer converged to valid covariance matrix; ^b = covariance matrix ill-conditioned (reported CI unreliable). RMSE in m²/g. Applicable range: $T=300$ °C to 800 °C; slow pyrolysis (undefined or low heating rate); lignocellulosic feedstocks belonging to categories listed; $N_2$ adsorption at 77 K with BJH desorption analysis.

Table 3. Fitted parameters for $S_{\mathrm{BET}}$—Arrhenius and power-law models, pooled and for the identifiable feedstock categories (grass and herbaceous). The wood and shell groups are not identified (bootstrap confidence intervals exceed the point estimates), and no fitted parameters are reported for them; see Section 5.2.

Group	Model	Param1	95% CI	Param2	95% CI	RMSE	MAPE (%)	AIC	BIC	N	Conv.
pooled	Arrhenius	2194	[3535]	16,010	[12,289]	183	712	601	604	45	✓
pooled	Power-law	179	[69.6]	2.22	[1.67]	185	759	601	605	45	✓
grass	Arrhenius	1922	[1798 b]	14,053	[6698 b]	43.6	15.1	87.1	87.3	8	✓
grass	Power-law	221	[49.0 b]	1.79	[0.913 b]	46.4	16.4	88.1	88.3	8	✓
herbaceous	Arrhenius	17,697	[36,769]	34,248	[17,180]	90.1	298	241	243	20	✓
herbaceous	Power-law	89.5	[53.3]	4.46	[2.16]	91.2	331	241	243	20	✓

Notes: Arrhenius model: $S_{\mathrm{BET}}\left(T\right)=S_0\exp (-E_S/\mathrm{RT})$; ${\mathrm{Param}}_1=S_0$ (m²/g), ${\mathrm{Param}}_2=E_S$ (J/mol). Power-law model: $S_{\mathrm{BET}}\left(T\right)=\alpha {(T/T_{\mathrm{ref}})}^{\widehat{\beta }}$, $T_{\mathrm{ref}}=773.15K$; ${\mathrm{Param}}_1=\alpha$ (m²/g), ${\mathrm{Param}}_2=\widehat{\beta }$ (dimensionless). Half-widths of 95% CIs: parametric bootstrap ($B=500$, residual resampling) for groups with $N\le 10$ (grass); asymptotic covariance matrix for $N>10$ (herbaceous, pooled) [22]. Convergence flag: ✓ = optimizer converged to valid covariance matrix; ^b = covariance matrix ill-conditioned (reported CI unreliable). RMSE in m²/g. Applicable range: $T=300$ °C to 800 °C; slow pyrolysis (undefined or low heating rate); lignocellulosic feedstocks belonging to categories listed; $N_2$ adsorption at 77 K with BJH desorption analysis.

Table 4. Fitted parameters for $V_{\mathrm{T}}$ and ${\overline{d}}_p$—pooled Arrhenius and power-law models.

Target	Model	Param1	95% CI	Param2	95% CI	RMSE	MAPE (%)	AIC	BIC	N	Conv.
$V_{\mathrm{T}}$	Arrhenius	2.30	[2.39]	21910	[8381]	0.0361	52.5	−79.7	−77.5	22	✓
$V_{\mathrm{T}}$	Power-law	0.0776	[0.0225]	2.88	[1.09]	0.0370	56.4	−78.6	−76.4	22	✓
${\overline{d}}_p$	Arrhenius	0.119	[0.143]	−23352	[6663]	2.45	45.9	129	132	27	✓
${\overline{d}}_p$	Power-law	4.70	[1.21]	−4.01	[1.21]	2.47	50.8	129	132	27	✓

Notes: Arrhenius model: $Y\left(T\right)=Y_0\exp (-E_Y/\mathrm{RT})$; ${\mathrm{Param}}_1=Y_0$, ${\mathrm{Param}}_2=E_Y$ (J/mol). Power-law model: $Y\left(T\right)=\alpha {(T/T_{\mathrm{ref}})}^{\widehat{\beta }}$, $T_{\mathrm{ref}}$ = $773.15$ K; ${\mathrm{Param}}_1=\alpha$, ${\mathrm{Param}}_2=\widehat{\beta }$. Units: $V_{\mathrm{T}}$ in cm³/g; ${\overline{d}}_p$ in nm. RMSE in units of target variable. Half-widths of 95% CIs from NLLS covariance diagonal. Applicable range: $T=300$ °C to $T=800$ °C; slow pyrolysis; pooled across feedstock categories with available data; BJH-derived descriptors from $N_2$ adsorption at $77$ K. ${\overline{d}}_p$ values at $T>650$ °C should be treated as apparent means due to BJH limitations in micropore regime.

Table 4. Fitted parameters for $V_{\mathrm{T}}$ and ${\overline{d}}_p$—pooled Arrhenius and power-law models.

Target	Model	Param1	95% CI	Param2	95% CI	RMSE	MAPE (%)	AIC	BIC	N	Conv.
$V_{\mathrm{T}}$	Arrhenius	2.30	[2.39]	21910	[8381]	0.0361	52.5	−79.7	−77.5	22	✓
$V_{\mathrm{T}}$	Power-law	0.0776	[0.0225]	2.88	[1.09]	0.0370	56.4	−78.6	−76.4	22	✓
${\overline{d}}_p$	Arrhenius	0.119	[0.143]	−23352	[6663]	2.45	45.9	129	132	27	✓
${\overline{d}}_p$	Power-law	4.70	[1.21]	−4.01	[1.21]	2.47	50.8	129	132	27	✓

Notes: Arrhenius model: $Y\left(T\right)=Y_0\exp (-E_Y/\mathrm{RT})$; ${\mathrm{Param}}_1=Y_0$, ${\mathrm{Param}}_2=E_Y$ (J/mol). Power-law model: $Y\left(T\right)=\alpha {(T/T_{\mathrm{ref}})}^{\widehat{\beta }}$, $T_{\mathrm{ref}}$ = $773.15$ K; ${\mathrm{Param}}_1=\alpha$, ${\mathrm{Param}}_2=\widehat{\beta }$. Units: $V_{\mathrm{T}}$ in cm³/g; ${\overline{d}}_p$ in nm. RMSE in units of target variable. Half-widths of 95% CIs from NLLS covariance diagonal. Applicable range: $T=300$ °C to $T=800$ °C; slow pyrolysis; pooled across feedstock categories with available data; BJH-derived descriptors from $N_2$ adsorption at $77$ K. ${\overline{d}}_p$ values at $T>650$ °C should be treated as apparent means due to BJH limitations in micropore regime.

4.3. Regression of Semi-Empirical Correlations

4.3.1. Single-Variable Regression Against Temperature

Each pore structure descriptor ${\theta }_k$ (i.e., $S_{\mathrm{BET}}$, $V_{\mathrm{T}}$, ${\overline{d}}_p$) is regressed against $T$ using the functional forms of Section 2.2 by nonlinear least squares:

$${\widehat{\mathit{\alpha }}}_k=\arg \underset{\mathit{\alpha }}{\min }{\displaystyle \sum _{i=1}^N}{\left[{\theta }_{k,i}-g(T_i;\mathit{\alpha })\right]}^2,$$

(8)

where $\mathit{\alpha }$ is the parameter vector of the correlation $g(·)$, and $N$ is the number of literature datasets.

4.3.2. Uncertainty Quantification

Parameter uncertainty is reported as 95% confidence intervals derived from the asymptotic covariance matrix of the least squares estimator:

$$\mathrm{Cov}\left(\widehat{\mathit{\alpha }}\right)={\widehat{\sigma }}_{\epsilon }^2{\left({\mathbf{J}}^{\top }\mathbf{J}\right)}^{-1},$$

(9)

where $\mathbf{J}$ is the Jacobian of g evaluated at $\widehat{\mathit{\alpha }}$, and ${\widehat{\sigma }}_{\epsilon }^2$ is the residual mean squared error.

5. Results

5.1. Model Selection

AIC and BIC were computed for every target-by-group combination; the results are summarized in Table 2. In every case, $|\Delta \mathrm{AIC}|$ (Arrhenius minus power-law) falls below 2, the conventional threshold below which both models are treated as empirically indistinguishable at the available sample size [21]. Consequently, the available data do not provide statistical grounds for preferring one functional form over the other.

The Arrhenius form is selected as the primary model on the basis of physical interpretability: the fitted parameter $E_S$ admits a mechanistic reading as an apparent composite activation energy for pore opening, linking the correlation to the overlapping devolatilization and polycondensation reactions that govern pore development (Section 2.2.2), and is directly comparable to the structural activation energies reported in the activated carbon literature [13,14]. This follows the convention established in that literature for distinguishing phenomenological structural correlations from kinetic TGA measurements. The AIC margins do not alter this choice; at the available sample sizes, both functional forms are effectively equivalent predictors, and the Arrhenius parameterization offers the more physically interpretable result. Fitted parameter tables for $S_{\mathrm{BET}}$ and for $V_{\mathrm{T}}$ and ${\overline{d}}_p$ appear in Table 3 and Table 4, respectively.

5.2. BET Surface Area Correlations

The pooled Arrhenius correlation fitted to all 45 records accounts for $19.9\%$ of the variance in $S_{\mathrm{BET}}$ ($R^2=0.199$; $\mathrm{MAPE}=712\%$). Separating records by feedstock category reduces error sharply in groups with coherent temperature response: the grass group achieves $\mathrm{MAPE}=15.1\%$ and $\mathrm{RMSE}=43.6$ m²/g across $N=8$ records. The failure of the pooled model is not a fitting artifact; it reflects the structure of the data. The 45 records span a 250-fold range in $S_{\mathrm{BET}}$—from approximately 1 m²/g at 300 °C for low-reactivity feedstocks to nearly 775 m²/g for rice husk at high temperatures—and temperature alone accounts for under $20\%$ of that range. The remaining variance is driven by feedstock differences in cell wall architecture, mineral content, and lignocellulosic composition, none of which enter the single-predictor correlation.

5.2.1. Grass Group (Best-Constrained Case)

The grass group pools miscanthus and switchgrass records exclusively from Chatterjee et al. [17] ($N=8$). Both feedstocks show consistent temperature response across the 300 °C to 700 °C range, yielding estimated parameters ${\widehat{S}}_0=1922$ m²/g and ${\widehat{E}}_S=14,053$ J/mol ($95\%$ CI: ±7676 J/mol; bootstrap $95\%$ CI reported in Table 3). The fitted $E_S$ of 14.1 kJ/mol falls within the range of apparent activation energies reported for cellulosic devolatilization and is consistent with the mechanistic basis of Equation (1).

Important caveat on generalization: Because all grass records originate from a single study with a common measurement protocol, the group MAPE of $15.1\%$ reflects within-study prediction error rather than between-study generalization accuracy; see Section 3.3 and Section 6.3 for discussion.

5.2.2. Herbaceous Group

The herbaceous group ($N=20$) spans wheat straw, maize straw, and other crop residue materials drawn from multiple contributing studies, producing $\mathrm{MAPE}=298\%$ and $\mathrm{RMSE}$ = 90.1 m²/g. The high apparent activation energy (${\widehat{E}}_S$ = 34,248 J/mol, $95\%$ CI: ±17,180 J/mol) does not reflect a single kinetic mechanism; it is an artifact of pooling feedstocks with fundamentally different temperature response amplitudes. Maize straw undergoes an abrupt increase in $S_{\mathrm{BET}}$ above 700 °C (from approximately 7 m²/g to 498 m²/g as residual organic matter volatilizes at high pyrolysis severity), whereas wheat straw and other crop residues in the group remain below 100 m²/g at comparable temperatures. The single-curve correlation captures the average upward trend across these heterogeneous materials but cannot reproduce the between-feedstock amplitude difference that drives the large MAPE.

5.2.3. Wood and Shell Groups (Non-Identified)

Both groups exhibit $\mathrm{MAPE}>300\%$, for distinct structural reasons. In the wood group ($N=9$), pine sawdust from Askeland et al. [16] reaches 397 m²/g at 750 °C, while pine wood from Ronsse et al. [8] yields 128 m²/g to 196 m²/g at 600 °C; both carry the same feedstock label at the category level. In the shell group ($N=6$), rice husk biochar from Herrera et al. [18] records an $S_{\mathrm{BET}}$ of 648 m²/g to 775 m²/g, an outlier among shell category materials that the correlation cannot anticipate from temperature alone. For both groups, the bootstrap confidence interval on the physically meaningful parameter (the apparent activation energy $E_S$) exceeds its point estimate: the shell group ${\hat{E}}_S$ carries a $95\%$ CI half-width $4.7\times$ the estimate, and the wood group half-width CI is $1.1\times$ the estimate. (Identification is judged on $E_S$, which governs the temperature response; the pre-exponential $S_0$ is an extrapolation to $1/T\to 0$ far outside the data window and is expected to be imprecise even in well-identified groups such as the herbaceous one, so its wider interval is not by itself a sign of non-identification.) A CI half-width exceeding the estimate of $E_S$ indicates that the temperature response itself is non-identified (the data do not constrain it) rather than merely imprecise. We therefore do not report fitted parameter values for the wood and shell groups: presenting numbers that the data cannot support would wrongly imply that they carry usable information. These two categories require additional data (and, for the shell group, subdivision by silica content) before any correlation can be identified, and no design use should be attempted in the interim.

The pooled and feedstock-specific $S_{\mathrm{BET}}\left(T\right)$ fits are shown in Figure 2 and Figure 3.

5.3. Total Pore Volume and Average Pore Diameter

The pooled Arrhenius correlation for $V_{\mathrm{T}}$ explains $69.1\%$ of variance across $N=22$ records ($R^2=0.691$; $\mathrm{MAPE}=52.5\%$; $\mathrm{RMSE}$ = 0.0361 cm³/g). This is substantially higher predictive accuracy than that achieved for $S_{\mathrm{BET}}$ under the same pooled specification. The estimated pre-exponential is ${\widehat{Y}}_0=2.30$ cm³/g, and the activation parameter is ${\widehat{E}}_Y$ = 21,910 J/mol (95% CI: $\pm 8381$ J/mol). The positive $E_S$ is consistent with the expected direction: $V_{\mathrm{T}}$ increases with pyrolysis temperature as progressive devolatilization opens new pore channels above 500 °C. The relative insensitivity of $V_{\mathrm{T}}$ to silica-rich outliers, compared with $S_{\mathrm{BET}}$, explains the higher $R^2$ in the pooled fit.

The pooled fit for ${\overline{d}}_p$ achieves $R^2=0.563$ across $N=27$ records ($\mathrm{MAPE}=45.9\%$; $\mathrm{RMSE}=2.45$ nm). The estimated parameters are ${\widehat{Y}}_0=0.119$ nm and ${\widehat{E}}_Y=-23,352$ J/mol (95% CI: ±6663 J/mol). The negative value of ${\widehat{E}}_Y$ is physically meaningful: it implies that ${\overline{d}}_p$ decreases as temperature rises, consistent with the preferential development of micropores at higher pyrolysis severity. As new micropores form with diameters well below 2 nm, the average across the pore population shifts downward even as total pore volume and surface area increase. This negative apparent activation parameter is an emergent consequence of the cylindrical geometry approximation (${\overline{d}}_p=4V_{\mathrm{T}}/S_{\mathrm{BET}}$): when $S_{\mathrm{BET}}$ grows faster than $V_{\mathrm{T}}$ with increasing $T$, the ratio falls.

Subgroup fits for $V_{\mathrm{T}}$ were carried out for the grass and herbaceous groups, the only categories with $N\ge 5$ records after removing entries with missing $V_{\mathrm{T}}$ values; the results were consistent with the pooled trend in direction and order of magnitude. Figure 4 shows the pooled $V_{\mathrm{T}}\left(T\right)$ fit against the calibration data.

5.4. Cross-Validation Performance

Leave-one-study-out cross-validation (LOSO-CV) produces a mean MAPE of $1277\%$ and mean RMSE of 184 m²/g across the 7 folds for $S_{\mathrm{BET}}$ (

Table 5. Leave-one-study-out cross-validation results for $S_{\mathrm{BET}}$.

Held-Out Study	N	MAPE-Arr (%)	RMSE-Arr	MAPE-Pow (%)	RMSE-Pow	Note
[16]	6	237	116	258	119
[17]	16	14.2	51.2	14.5	54.6
[18]	4	81.8	577	81.8	577
[9]	1	25.1	101	26.2	105	$N=1$ test fold
[19]	2	5122	94.2	5718	99.1	Secondary citation
[8]	10	1199	192	1233	191
Mean	—	1277	184	1387	187	—
Stratified MAPE (Arrhenius): $S_{\mathrm{BET}}$ > 50 m2/g (design range) vs. $\le 50$ m2/g (low-$S_{\mathrm{BET}}$ regime)
Mean ($S_{\mathrm{BET}}>50$)	—	50.4	—	—	—	—
Mean ($S_{\mathrm{BET}}\le 50$)	—	3918	—	—	—	—

Notes: Each fold withholds one study; the model is retrained on the remaining six. MAPE-Arr and RMSE-Arr correspond to the Arrhenius model and MAPE-Pow and RMSE-Pow to the power-law. RMSE in m²/g. The mean row reports the unweighted arithmetic mean across all 7 folds. [19] values derive from secondary citations and should be interpreted with additional caution.

). A note on MAPE as a metric is warranted here. MAPE is inflated severely when any observation has a small true value near zero; a record with $S_{\mathrm{BET}}\approx 1$ m²/g predicted as 10 m²/g contributes $900\%$ to the MAPE, whereas the same absolute error for a record near 100 m²/g contributes only $9\%$. The large MAPE values reported here are therefore driven disproportionately by low-$S_{\mathrm{BET}}$ records at pyrolysis temperatures below 400 °C, which are also the records most sensitive to degassing protocol variability [10]. To clarify the model’s practical accuracy in the design-relevant range, Table 5 also reports stratified MAPE separately for records with $S_{\mathrm{BET}}>50$ m²/g (the regime relevant for adsorption and soil amendment design) and records below that threshold. Restricted to the design-relevant range, the LOSO-CV MAPE falls to $50.4\%$ (more than an order of magnitude below the pooled figure), whereas records below 50 m²/g carry a MAPE near $3900\%$. The headline $1277\%$ is thus dominated by near-zero low-temperature records and substantially overstates the error that matters for engineering design. RMSE provides a more stable overall error bound and should be consulted alongside MAPE when assessing practical prediction accuracy. The gap between LOSO-CV MAPE ($1277\%$) and in-sample MAPE ($712\%$) signals that the correlation is shaped by the studies it sees; generalization across study boundaries is limited. The study-level MAPE figures span four orders of magnitude, which makes the mean a summary of heterogeneity rather than a single predictive benchmark.

The held-out fold with the best performance is Chatterjee et al. [17] ($N_{\mathrm{test}}=16$; $\mathrm{MAPE}=14.2\%$; $\mathrm{RMSE}$ = 51.2 m²/g). This fold succeeds because the held-out records—four grass and herbaceous feedstocks pyrolyzed over a 500 °C to 800 °C range—occupy the same region of feedstock and temperature space as the training data. The trained model can interpolate within a populated region; it does so accurately.

The two worst-performing folds expose the limits of temperature-only extrapolation. Holding out Novak et al. [19] ($N_{\mathrm{test}}=2$, $\mathrm{MAPE}=5122\%$) leaves the model without any shell category records in the training set yet asks it to predict pecan shell biochar. The model extrapolates into an unpopulated (feedstock, $T$) region and fails severely. Holding out ($N_{\mathrm{test}}=6$, $\mathrm{MAPE}=2263\%$) produces a comparable failure: maize straw biochar undergoes a sharp increase in $S_{\mathrm{BET}}$ above 700 °C (from approximately 7 m²/g to 498 m²/g) that the remaining training studies give no basis for predicting. The intermediate folds for Herrera et al. [18] ($\mathrm{MAPE}=81.8\%$) and Muzyka et al. [9] ($\mathrm{MAPE}=25.1\%$) reflect partial overlap with training feedstock space.

Taken together, the LOSO-CV results confirm that the pooled Arrhenius correlation performs reliably only when the target feedstock and temperature range are already represented in the calibration data. Feedstock category is the dominant source of variation in $S_{\mathrm{BET}}$ magnitude; temperature is secondary. Practitioners applying the correlation to a new feedstock outside the categories covered here should expect prediction errors that exceed the in-sample MAPE by one to two orders of magnitude. Figure 5 shows the predicted versus observed values for each fold.

5.5. Residual Diagnostics

Figure 6 presents three complementary views of the in-sample residuals from the pooled Arrhenius fit to $S_{\mathrm{BET}}$.

Panel A (residuals versus pyrolysis temperature) reveals a heteroscedastic structure: residual magnitude grows with temperature, consistent with the exponential model amplifying small parameter errors at high $S_{\mathrm{BET}}$. Large positive residuals at $T >$ 600 °C are concentrated in silica-rich (shell) records and in the maize straw herbaceous records, confirming that composition-driven outlier behavior rather than model misspecification drives the bulk of the prediction error. Low-temperature records ($T <$ 400 °C) exhibit small absolute residuals, but these contribute disproportionately to MAPE because the observed $S_{\mathrm{BET}}$ values are near zero.

Panel B (residuals by feedstock category) shows that the grass group produces approximately symmetric residuals centered near zero with modest spread, confirming the in-sample fit quality reported in Section 5.2. The herbaceous and shell groups exhibit positive skew: the model systematically underpredicts the highest-$S_{\mathrm{BET}}$ records in both categories.

Panel C (residuals by contributing study) confirms that Chatterjee et al. [17] (grass) has the smallest residual spread of any single-study contribution, while shows systematic underprediction at $T>700$°C, and Herrera et al. [18] exhibits the largest positive bias, consistent with the silica-driven surface area enhancement discussed in Section 6.2.

5.6. Benchmark Against Data-Driven Models

To test whether a flexible data-driven surrogate would outperform the semi-empirical correlation on the same evidence base, we benchmarked the pooled Arrhenius model for each of the three output descriptors ($S_{\mathrm{BET}}$, $V_{\mathrm{T}}$, and ${\overline{d}}_p$) against two alternatives trained on the identical records: ordinary least squares (OLS) linear regression and a random forest regressor (500 trees). Both alternatives received an expanded feature set—pyrolysis temperature together with a one-hot encoding of feedstock category—whereas the Arrhenius correlation used temperature alone. All models were evaluated under the identical leave-one-study-out protocol (Section 3.4), with fold metrics aggregated as the unweighted mean across study folds—the same aggregation used for the main cross-validation (Table 5)—so that the semi-empirical $S_{\mathrm{BET}}$ values coincide between the two tables. The results are reported in

Table 6. Benchmark of semi-empirical correlations against data-driven models for all three output descriptors, trained and evaluated on same records.

Model	In-Sample ${\mathit{R}}^2$	In-Sample RMSE	LOSO-CV RMSE (Mean ± SD)
$S_{\mathrm{BET}}$ (m2/g), $N=45$, 7 studies
Semi-empirical (Arrhenius)	0.199	183.4	184.4 ± 165.8
Linear regression (OLS)	0.601	129.4	261.1 ± 209.9
Random forest	0.858	77.2	257.8 ± 195.4
$V_{\mathrm{T}}$ (cm3/g), $N=22$, 2 studies
Semi-empirical (Arrhenius)	0.691	0.0361	0.0718 ± 0.0052 †
Linear regression (OLS)	0.741	0.0331	0.0468 ± 0.0130 †
Random forest	0.792	0.0296	0.0756 ± 0.0003 †
${\overline{d}}_p$ (nm), $N=27$, 4 studies
Semi-empirical (Arrhenius)	0.563	2.45	3.28 ± 2.99
Linear regression (OLS)	0.485	2.66	4.16 ± 2.76
Random forest	0.712	1.99	4.05 ± 2.95

Notes: For each descriptor, all three models are trained on the identical set of records (the number of records and contributing studies is given in each block header). OLS and random forest use pyrolysis temperature plus a one-hot encoding of feedstock category as predictors; the semi-empirical Arrhenius model uses temperature alone. The LOSO-CV RMSE is the unweighted mean across the study folds, with the standard deviation across folds given after ±; the same aggregation is used in Table 5, and the semi-empirical $S_{\mathrm{BET}}$ mean (184 m²/g) coincides with the value reported there. The large fold-to-fold spread reflects the study-level heterogeneity of the dataset. Only RMSE is tabulated here: fold-averaged MAPE is destabilized by the near-zero low-temperature records, so the design range stratified MAPE is instead reported in Table 5. In-sample RMSE and $R^2$ reproduce the pooled fits reported in Section 5.2 and Section 5.3. RMSE units follow each descriptor (m²/g, cm³/g, and nm). The random forest used 500 trees; its $S_{\mathrm{BET}}$ LOSO-CV RMSE varied by less than 3.3 m²/g across five random seeds. ^† The $V_{\mathrm{T}}$ leave-one-study-out test rests on only two contributing studies (two folds) and is indicative only.

.

The same pattern recurs across all three descriptors: greater model flexibility improves the in-sample fit but does not translate into better generalization. In sample, the random forest attains the highest $R^2$ for every target ($0.858$, $0.792$, and $0.712$ for $S_{\mathrm{BET}}$, $V_{\mathrm{T}}$, and ${\overline{d}}_p$), well above the corresponding Arrhenius values ($0.199$, $0.691$, $0.563$). Under leave-one-study-out cross-validation, this advantage disappears. For $S_{\mathrm{BET}}$, the target with the broadest study coverage (seven contributing studies), the semi-empirical correlation generalizes the best ($\mathrm{RMSE}=184$ m²/g), while OLS and the random forest are worse (261 and 258 m²/g, respectively), despite their near-perfect training fit. For ${\overline{d}}_p$ (four studies), the semi-empirical correlation is again the best out of sample (LOSO-CV $\mathrm{RMSE}=3.3$ nm), ahead of both OLS and the random forest. For $V_{\mathrm{T}}$, the three models are close, and the linear model edges ahead on relative error, but only two studies report $V_{\mathrm{T}}$, so the leave-one-study-out test reduces to two folds and cannot discriminate the models reliably; the $V_{\mathrm{T}}$ cross-validation figures are therefore indicative only. In no case does the added flexibility of the random forest yield a meaningful out-of-sample improvement over the parsimonious correlation.

This benchmark supports two conclusions. First, the limited predictive capability documented throughout is a property of the available evidence—its size, heterogeneity, and study-level structure—rather than of the semi-empirical functional form: a high-capacity model trained on the same data does no better out of sample and usually does worse. Second, for a dataset of this size, a parsimonious, physically interpretable correlation is the more defensible choice than a higher-variance data-driven surrogate whose apparent in-sample accuracy does not survive study-level validation.

6. Discussion

6.1. Thermochemical Basis for Pore Development

The temperature dependence of $S_{\mathrm{BET}}$ documented in this dataset can be grounded in the well-established thermal decomposition sequence of lignocellulosic biomass. Hemicellulose undergoes rapid devolatilization between approximately 200 °C and 350 °C, while cellulose decomposes over the narrower window from 320 °C to 450 °C; lignin decomposes more gradually from 200 °C to above 700 °C [24,25]. Typical lignocellulosic compositions span cellulose $35\%$ to $50\%$, hemicellulose $20\%$ to $35\%$, and lignin $15\%$ to $35\%$ on a dry basis, but the ratios differ substantially across categories: woody biomass contains $25\%$ to $35\%$ lignin and grasses $10\%$ to $20\%$, and herbaceous crop residues are intermediate [25]. These ranges are representative literature values rather than measurements of the specific feedstocks, because the compiled sources did not uniformly report lignocellulosic or proximate analyses; incorporating measured composition per record is identified as a route to finer, sub-category resolution. Beyond the organic fractions, alkali and alkaline earth minerals (notably potassium and calcium, which are abundant in herbaceous crop residues) catalyze char restructuring and can either enhance microporosity or, at high ash loadings, promote pore-blocking mineral sintering [2,25]; this catalytic dimension compounds the silica-scaffolding effect discussed below for the shell group and further limits any temperature-only description. The abrupt rise in $S_{\mathrm{BET}}$ above 400 °C—from values typically below 50 m²/g to above 200 m²/g in coherent feedstock groups—corresponds directly to the onset of cellulosic char formation and the progressive volatilization of condensable organics that opens the nascent micropore network. At temperatures above 600 °C, secondary condensation reactions and incipient graphitization restructure the pore walls, contributing both to further increases in $S_{\mathrm{BET}}$ for some feedstocks and to the appearance of a secondary mesopore population consistent with bimodal PSD fits at high pyrolysis severity.

This thermochemical picture reinforces the choice of the Arrhenius functional form as a phenomenological descriptor. The exponential temperature response over the 300 °C to 800 °C window reflects the superposition of several overlapping decomposition reactions whose individual rate constants exhibit Arrhenius-type dependencies. The fitted parameter $E_S$ therefore represents an apparent, composite temperature sensitivity—a curve-fitting constant that captures the aggregate effect of pore opening across the charring sequence—not the activation energy of a single identifiable reaction as would be obtained from TGA/DTG kinetic analysis [15]. This distinction between phenomenological structural correlations and kinetic measurements from thermoanalytical instruments is standard practice in the activated carbon literature [13,14] and applies equally here.

6.2. Feedstock Category as the Primary Variance Driver

The central quantitative finding—that feedstock category accounts for a larger fraction of variance in $S_{\mathrm{BET}}$ than temperature—has a direct mechanistic basis. Feedstock composition sets the structural template from which pore architecture develops: the cellulose/hemicellulose/lignin ratio, mineral content, and initial cell wall geometry collectively determine the amplitude and shape of the temperature response curve for a given material class.

Wood (high lignin, moderate surface area development).

Wood feedstocks contain $25\%$ to $35\%$ lignin, a structurally complex aromatic polymer that forms a thermally stable carbonaceous scaffold during pyrolysis [7,24]. This scaffold resists structural collapse and limits the rapid pore opening seen in cellulose-rich materials, producing moderate $S_{\mathrm{BET}}$ values (50 m²/g to 300 m²/g) that increase gradually with temperature. Between-study variance in the wood group reflects primarily species-level differences in lignin structure and initial wood density rather than measurement protocol variability.

Grass (cellulose-dominated, low mineral content, predictable).

Grasses (miscanthus, switchgrass) contain $> 40\%$ cellulose and relatively low ash ($3\%$ to $6\%$), a combination that produces consistent, temperature-driven pore opening. Cellulose pyrolysis generates a highly microporous char at temperatures above 400 °C through rapid depolymerization and volatile release, and the absence of mineral interference allows the micropore network to develop along a reproducible trajectory [25]. The apparent structural activation energy of ${\widehat{E}}_S=14.1$ kJ/mol for the grass group is consistent with the composite temperature sensitivity of cellulosic devolatilization.

Herbaceous (compositionally heterogeneous, low predictive accuracy).

Herbaceous crop residues share approximate cellulose contents ($35\%$ to $45\%$) but differ substantially in ash and silica. Maize straw (≈42% cellulose, ≈8% ash) undergoes an abrupt $S_{\mathrm{BET}}$ increase above 700 °C because residual carbonaceous material persisting through lower-temperature charring volatilizes rapidly once temperatures exceed the secondary decomposition threshold [24]. Wheat straw (≈38% cellulose, ≈7% ash) lacks this abrupt transition. The pooling of these two trajectories within the herbaceous category inflates within-group variance and drives the high $\mathrm{MAPE}=298\%$ observed for that group.

Shell (silica-driven mineral scaffolding, extreme surface area).

Rice husk contains $15\%$ to $20\%$ SiO₂ by mass, and this mineral fraction acts as a structural scaffold that inhibits pore wall collapse even at $T>600$ °C. The consequence is that $S_{\mathrm{BET}}$ continues to increase above 600 m²/g at 700 °C—a trajectory that is qualitatively different from all lignocellulosic-dominated feedstocks and cannot be described by the same Arrhenius parameters fitted to non-siliceous groups. Rice husk reaches $S_{\mathrm{BET}}$ above 648 m²/g at 700 °C because its naturally high silica content stabilizes the pore walls and inhibits structural collapse. Neither this extreme behavior nor the within-category heterogeneity between silica-rich and silica-poor shells is predictable from temperature alone.

Stratification by feedstock category recovers predictive accuracy only where within-group composition is approximately homogeneous, which is why the grass group succeeds and the compositionally heterogeneous herbaceous group does not. The five-category scheme is therefore necessary but not sufficient: finer sub-categorization by quantitative lignocellulosic and mineral composition (particularly silica content and ash yield at the target temperature) would further reduce within-category variance.

6.3. Predictive Performance and Residual Structure

The residual diagnostics of Section 5.5 and the MAPE behavior of Section 5.4 together carry one interpretive consequence for how the performance metrics should be read: because absolute errors are the largest for silica-rich feedstocks at high temperature while relative errors are dominated by near-zero low-temperature records, MAPE and RMSE rank the model very differently and should not be used interchangeably. For practical process design, RMSE is the more informative metric: the pooled RMSE of 183 m²/g establishes an absolute error bound for the worst-case (pooled) predictor, while the grass group RMSE of $43.6$ m²/g characterizes the accuracy achievable when the feedstock category is well constrained. The LOSO-CV RMSE of 184 m²/g confirms that the in-sample RMSE of the pooled model is not substantially optimistic; the model is not overfitting in absolute terms, even though the MAPE gap between in-sample ($712\%$) and LOSO-CV ($1277\%$) is large—a divergence that reflects the metric’s sensitivity to low-$S_{\mathrm{BET}}$ records rather than a collapse in absolute accuracy.

6.4. BJH Analysis: Scope and Limitations

All pore size distribution descriptors compiled from the literature were derived from a BJH analysis of nitrogen desorption isotherms, and this choice carries methodological implications that must be acknowledged. The BJH method provides a reliable description of mesopore volume distribution (pore widths 2 nm to 50 nm) but is fundamentally inadequate for micropores (width $<2$ nm), because the Kelvin equation underlying BJH assumes cylindrical geometry and breaks down when pore diameters approach the molecular scale [6,26]. For biochars pyrolyzed above 600 °C, where micropore fractions typically exceed 0.5, BJH systematically underestimates both pore volume and surface area in the micropore regime.

Modern density functional theory methods, in particular quenched solid density functional theory (QSDFT), provide more accurate micropore characterization and should be the preferred method for biochars produced at temperatures above 600 °C [27]. The correlations presented here inherit the BJH convention of the calibration dataset and should therefore be applied within the same measurement framework. Practitioners using DFT-derived descriptors should expect systematic deviations from the BJH-based predictions, particularly for the micropore-dominated high-temperature regime. Future work should assess whether separate correlations calibrated on DFT descriptors yield more accurate predictions at $T>600$ °C.

This limitation directly conditions the interpretation of the negative apparent activation energy fitted for ${\overline{d}}_p$ (${\widehat{E}}_Y=-23.4$ kJ/mol). The downward shift in ${\overline{d}}_p$ with increasing temperature was attributed above to the progressive dominance of micropore formation. However, ${\overline{d}}_p$ is computed as $4V_{\mathrm{T}}/S_{\mathrm{BET}}$ from BJH-derived quantities, and BJH systematically underestimates micropore dimensions and misassigns micropore volume in exactly the high-temperature regime ($T>600$ °C, $f_{\mathrm{mic}}>0.5$) where the trend is the steepest. The fitted ${\widehat{E}}_Y$ therefore conflates a genuine physical shift toward smaller pores with a measurement artifact: part of the apparent decrease in ${\overline{d}}_p$ reflects BJH’s inability to resolve pores below 2 nm rather than a true reduction in mean pore size. Because the calibration data do not include DFT-based descriptors, this study cannot quantify the two contributions separately, and the magnitude of ${\widehat{E}}_Y$ should be regarded as a BJH-specific effective parameter rather than a transferable physical constant. Resolving the split requires isotherm-level reanalysis with QSDFT, which is identified as a priority for future work.

6.5. A Comparison with the Activated Carbon Literature

Semi-empirical Arrhenius-type correlations for structural descriptors have been applied to activated carbon production for several decades, providing a reference frame for interpreting the fitted parameters of the present model. Apparent structural activation energies for $S_{\mathrm{BET}}$ reported in coconut shell and wood-based precursors during the pre-activation charring stage typically range from 8 kJ/mol to 40 kJ/mol [13,14]. The fitted ${\widehat{E}}_S=14.1$ kJ/mol for the grass group falls within this range, lending cross-domain support to the phenomenological Arrhenius form and suggesting that the temperature sensitivity of pore opening in cellulose-dominated slow pyrolysis char is comparable in magnitude to that of pre-activation charring in activated carbon production. The herbaceous group value of ${\widehat{E}}_S=34.2$ kJ/mol exceeds this range, consistent with the interpretation that pooling feedstocks with contrasting thermal behavior artificially inflates the apparent temperature sensitivity.

Power-law exponents ${\widehat{\beta }}_S$ for $S_{\mathrm{BET}}\left(T\right)$ in the activated carbon literature are typically in the range 1.5–4.0 [13]; the values obtained for the grass group are comparable in order of magnitude. Direct quantitative comparison is constrained by differences in temperature range, activation agent (water vapor or CO₂ versus inert atmosphere for slow pyrolysis), and precursor composition, but the order-of-magnitude agreement confirms that the functional form choice is appropriate. A systematic comparison table covering both biochar and activated carbon studies—including fitted parameter values, their $95\%$ confidence intervals, and the measurement conditions—is identified as a priority for a future meta-analysis of the broader carbonaceous material literature.

6.6. Fractal Geometry: Status and Outlook

The surface fractal dimension $D_{\mathrm{f}}$, estimable from a Frenkel–Halsey–Hill analysis of the full nitrogen adsorption isotherm, was not calibrated in the present work because the compiled literature dataset provides only scalar summary statistics ($S_{\mathrm{BET}}$, $V_{\mathrm{T}}$, ${\overline{d}}_p$) and not the full isotherms required for FHH analysis. Calibrating $D_{\mathrm{f}}\left(T\right)$ correlations is identified as a priority for future work once full isotherm data become available, as this would add a qualitative dimension to the structural model that scalar area and volume metrics cannot capture.

6.7. Limitations and Outlook

Five methodological constraints bound the scope of the present results.

Small, heterogeneous, literature-derived dataset.

The correlations are calibrated on 45 records from only seven studies, and this limited size and compositional heterogeneity are the primary constraints on the robustness and generality of the conclusions. Three consequences follow. First, the fitted parameters are conditional on the particular corpus compiled: a different set of source studies—or additional records for under-represented categories—could shift the estimates and the between-group rankings, as the leave-one-study-out analysis (Section 5.4) already illustrates. Second, because the dataset is assembled from the published literature, it is subject to publication bias, in that studies reporting well-developed, high-surface-area biochars may be over-represented relative to null or low-porosity results, biasing the calibrated temperature response. Third, the records originate from different laboratories using different instruments, analysts, and (largely unreported) degassing protocols, so inter-laboratory variability contributes an irreducible component to the residual scatter that no temperature-only model can capture [10,11]. These factors reinforce the fact that the correlations are calibration envelope interpolators rather than universal predictors and that the reported parameters should be updated as larger and more homogeneous datasets become available.

Uncontrolled process variables beyond final temperature.

The correlations use final pyrolysis temperature as the sole process predictor, but pore development is also governed by variables that the compiled literature does not report consistently and that could therefore not be incorporated as covariates. These include the full heating profile (heating rate and intermediate temperature holds, which alter residence time in the devolatilization window); the carrier gas type, composition, and flow rate (which control volatile removal and secondary char reactions); the chemical pretreatment of the feedstock (acid washing to remove alkali and alkaline earth minerals or alkaline swelling to enhance surface area); and the pyrolysis reactor configuration (which sets temperature uniformity, vapor residence time, and mass transfer conditions). Each of these factors can shift $S_{\mathrm{BET}}$, $V_{\mathrm{T}}$, and the PSD appreciably at a fixed final temperature. Their omission is a deliberate consequence of the literature data design—the sources were selected for temperature coverage, not for the factorial control of these variables—and it means that the fitted correlations should be read as describing the average temperature response of slow pyrolysis biochar under conventional inert atmosphere conditions, not as isolating the effect of temperature with all other process variables held fixed. Extending the framework to these variables requires primary studies with systematic, controlled variation, which are currently scarce.

Incomplete PSD shape calibration.

A parametric PSD fitting framework covering lognormal, Weibull, and bimodal lognormal functions was not calibrated in this work because the compiled literature sources did not report full BJH differential volume curves. Calibrating the corresponding PSD shape parameters (log-mean, distribution width, and mixture weights) against temperature requires either digitized BJH curves from accessible publications or new experimental isotherms.

Shell category non-identifiability.

The shell category correlation is not reliably identified at the available sample size ($N=6$, $\mathrm{MAPE}=6703\%$) because silica-rich feedstocks such as rice husk behave as outliers within the group. Separate correlations for silica-rich and silica-poor shells are necessary before the framework can be applied to this category.

Heating rate not calibrated.

A heating rate multiplicative correction was not calibrated because heating rate variation is confounded with feedstock differences across the compiled studies and cannot be separated reliably at the available sample sizes. Calibrating this correction requires studies with factorial ($T$, $\beta$) designs where both variables are independently controlled.

Four directions follow from these constraints. The most immediate is digitizing BJH incremental pore volume curves from the accessible literature to calibrate the lognormal and Weibull shape parameters against temperature, completing the full predictive PSD framework. Expanding the calibration dataset for the wood and shell categories, and subdividing the shell group by silica content, would improve identifiability for two commercially important feedstock classes. Validating the $S_{\mathrm{BET}}$ and $V_{\mathrm{T}}$ correlations against independently measured biochar data would replace the LOSO-CV estimate of generalization error with a direct external test. Assembling a factorial ($T$, $\beta$) dataset from studies reporting systematic heating rate variation would enable the calibration of a multiplicative heating rate correction and extend the model’s operating envelope to multi-variable prediction.

7. Conclusions

Semi-empirical Arrhenius-type correlations were fitted to 45 literature records spanning 12 lignocellulosic feedstocks and 300 °C to 800 °C to predict the BET surface area ($S_{\mathrm{BET}}$), total pore volume ($V_{\mathrm{T}}$), and mean pore diameter (${\overline{d}}_p$) of slow pyrolysis biochar. The following conclusions emerge:

(i)

Temperature alone is insufficient to predict $S_{\mathrm{BET}}$ across the lignocellulosic feedstock space. The pooled Arrhenius correlation explains only $19.9\%$ of variance ($\mathrm{MAPE}=712\%$, $\mathrm{RMSE}=183$ m²/g), confirming that feedstock composition is the dominant source of variation. A single temperature response curve cannot substitute for feedstock-specific characterization.

(ii)

Feedstock stratification is necessary and, for compositionally coherent groups, sufficient for predictive accuracy. Restricting the correlation to feedstock-homogeneous categories recovers accuracy where cell wall architecture and mineral content are internally consistent: the grass category achieves $\mathrm{MAPE}=15.1\%$ (a within-study estimate from a single source study, not a between-study generalizability estimate) with an apparent structural activation energy of ${\widehat{E}}_S=14.1$ kJ/mol, consistent with the thermochemical signature of cellulose-dominated devolatilization.

(iii)

Total pore volume and mean pore diameter are better predicted in pooled form. The pooled correlations for $V_{\mathrm{T}}$ ($R^2=0.691$; $\mathrm{MAPE}=52.5\%$) and ${\overline{d}}_p$ ($R^2=0.563$; $\mathrm{MAPE}=45.9\%$) are more accurate than the pooled $S_{\mathrm{BET}}$ correlation because these descriptors are less sensitive to silica-driven outlier behavior. The negative apparent activation energy for ${\overline{d}}_p$ (${\widehat{E}}_Y=-23.4$ kJ/mol) is consistent with the progressive dominance of micropore formation at higher pyrolysis temperatures, which shifts the population-averaged pore diameter downward as $T$ increases; however, this value is partly confounded by the systematic underestimation of micropore dimensions inherent to BJH analysis (Section 6.4) and should be read as a BJH-specific effective parameter rather than a transferable physical constant.

(iv)

Generalization requires feedstock overlap between calibration and prediction sets. Leave-one-study-out cross-validation yields a mean MAPE of $1277\%$ (dominated by near-zero low-temperature records; the design range value, $S_{\mathrm{BET}}>50$ m²/g, is $50.4\%$), confirming that the framework is a calibration envelope interpolator, not a general predictor. Accurate study-level generalization is achievable only when the target feedstock and temperature range are already represented in the calibration data. A benchmark against ordinary least squares and random forest models trained on the same records confirms that this limitation stems from the evidence base (its size, heterogeneity, and study-level structure) rather than from the semi-empirical functional form: the more flexible models fit the calibration data better but do not generalize better under study-level validation.

(v)

Boundaries of valid application. The fitted correlations should not be applied when: (a) the target feedstock category is absent from the calibration dataset presented here; (b) the feedstock is silica-rich (e.g., rice husk classified in the shell category), where composition-driven outlier behavior prevents the reliable identification of group parameters; (c) pyrolysis temperature exceeds 650 °C and the engineering question requires micropore-specific characterization, because the BJH-based descriptor ${\overline{d}}_p$ becomes an apparent value that underestimates true micropore surface area; or (d) the heating rate varies systematically across prediction targets, since a heating rate correction was not calibrated in this work, and its inclusion would require studies with factorial $(T,\beta )$ designs.

(vi)

Scope and applicability of the framework. The proposed correlations are descriptive empirical tools, not general predictive models. They provide reliable estimates within the calibration envelope: that is, when the target feedstock category, temperature range (300 °C to 800 °C), and measurement protocol ($N_2$ adsorption at $77$ K, BJH desorption) match the conditions of the seven calibration studies. The framework is characterized by physically interpretable parameters (apparent structural activation energy $E_S$, power-law exponent $\widehat{\beta }$), low data requirements, and explicit extrapolation boundaries. Extrapolation to feedstock categories absent from the calibration dataset, to different BJH conventions, or to markedly different measurement protocols is not supported by the present evidence and is expected to produce errors far larger than those reported here.

Practical Guidance for Process Engineers

Within the calibration envelope, the grass group Arrhenius correlation provides the most reliable predictions: for miscanthus, switchgrass, or compositionally similar cellulose-dominated grasses pyrolyzed in the range 300 °C to 700 °C, the expected prediction error is $\mathrm{RMSE}\approx 44$ m²/g and $\mathrm{MAPE}\approx 15\%$ based on in-sample performance for a single-study dataset. For herbaceous feedstocks (maize straw, wheat straw, and similar crop residues) over the full 300 °C to 800 °C range, the herbaceous group correlation is applicable but with markedly higher uncertainty ($\mathrm{RMSE}\approx 90$ m²/g, $\mathrm{MAPE}\approx 298\%$, in sample) due to compositional heterogeneity within the group. Shell category and wood category predictions are not recommended for design purposes at the current calibration sample sizes; new experimental characterization or expanded literature data collection is required before these groups can be used reliably.