Scientific insights Educational models Independently checked
Every calculator on this site, with its equations, limits and test results
The site contains 75 interactive models. Each one is listed here with what it calculates, the equations behind it, where it applies, what it does not establish, and how it was checked against an independent calculation.
How models are checked
Each calculator is compared with a separate implementation, in the real page
For every calculator a reference version was written independently in Python (NumPy, SciPy, statsmodels, CoolProp or the thermo package, depending on the model). A browser test sets the page's own controls to defaults, low and high values and published benchmarks, reads the displayed results, and compares them with the reference within a stated tolerance. A second test drives every slider on the site to its minimum, midpoint and maximum and fails on any NaN, infinity or script error.
75models with a methods panel
27reference case files (tests/numerics)
1705displayed values compared with the reference
0failing comparisons in the last full run
What this does and does not show. Passing these checks means the page computes the stated equations correctly within their stated range. It does not validate the equations for any particular botanical material or process; every model here is educational unless its panel says it reproduces published data. Unit tests: bioavail.test.js, clinical.test.js, coa.test.js, combo.test.js, contact.test.js, deliverydesign.test.js, designspace.test.js, lab.test.js, lbm.test.js, methoddev.test.js, operations.test.js, purify.test.js, qbd.test.js, safety.test.js, tox.test.js.
Commercialization & licensing
Commercialization & licensing
4 models · 54 reference checks, 0 failing · reference script tests/numerics/ref/applications-commercialization-licensing.py
| Model | Equations | Applies within | Does not establish |
|---|---|---|---|
| A license is priced from the specific deal: the royalty base, the rate, what else the licensee pays, and when the money …The net royalty rate after an anti-stacking credit and a floor; the net present value to the licensor of ten years of royalties plus an upfront fee and a year-3 milestone; and the licensee's total … | r net = max(min(r, f), r − a·s). S t = S₁(1 + g) t−1 ; R t = S t ·r net for t = 1…10. NPV = U + Σ (R t + M·[t = 3]) / (1 + d) t . Burden = r net + s. | The slider ranges. Outputs cannot be negative: the net rate is bounded below by min(r, f) ≥ 0. | Any typical or fair royalty rate. No verifiable data set of botanical ingredient license rates was found, so none is shown. Real contracts define the base, credits, caps, audits and minimums in detail. |
| Late attrition, not early attrition, is what makes each launch expensiveHow many of 100 screened ideas reach launch after four gates, the average spend per launch, and the share of total spend that goes to candidates later killed. | N₁ = 100·p₁ (feasibility), N₂ = N₁·p₂ (development), N₃ = N₂·p₃ (validation), N launch = N₃·p₄. Spend S = 1·N₁ + 5·N₂ + 15·N₃. Spend per launch = S / N launch . Share on failures = 1 − N launch ·(1 + 5 + 15) / S. | Pass rates from 5% to 100%, so at least 0.0625 launches result and no division by zero is possible. Results are expected values, so fractional candidates are meaningful. | Real pass rates or costs for botanical programs or for EUTECNIQ. The relative costs are invented to show the arithmetic. |
| An expected-value tree shows whether the next stage is worth funding, and how sensitive that answer isThe expected value of starting a three-stage program when the sponsor may stop before any stage, the probability of reaching launch, and the launch value at which starting breaks even. | EV₃ = max(0, −C₃ + p₃·V); EV₂ = max(0, −C₂ + p₂·EV₃); expected value of starting = −C₁ + p₁·EV₂ (shown before the final max, so a negative value means do not start). P(launch) = p₁·p₂·p₃. V* = (C₁ + p₁·C₂ + p₁·p₂·C₃) / (p₁·p₂·p₃). | The slider ranges; probabilities are at least 5%, so V* is always finite. At V = V* every continuation value is non-negative, so the closed form agrees with the rollback. V* above 600 is reported but not drawn. | The probability, cost or value of any real program. Real decisions also discount time, price risk and consider portfolio effects. |
| A long development program spends patent term before the product earns anythingYears of U.S. patent term left when a product launches L years after a provisional filing, with or without a regulatory-review extension. | Expiry without extension = 1 + 20 = 21 years after the provisional. Term left = max(0, 21 − L). Extension E = min(5, max(0, 14 − (21 − L))) if premarket approval was needed, else 0. Term left with extension = (21 − L) + E; expiry = 21 + E. | L up to 16 years, so term left is always positive. | The term of any real patent. An actual extension is computed from the testing and approval phases of the regulatory review (35 U.S.C. 156(c)) and may be shorter. Dietary supplements and cosmetics are not eligible. |
Ab initio reactivity
Ab initio reactivity
1 model · 62 reference checks, 0 failing · reference script tests/numerics/ref/insights-ab-initio-reactivity.py
| Model | Equations | Applies within | Does not establish |
|---|---|---|---|
| Comparing two computed BDEs predicts which way a hydrogen would move between them, with most of the method error …The gas-phase reaction enthalpy for moving a hydrogen atom from a phenolic O–H site to a radical. | ΔH = BDE(donor O–H) − BDE(acceptor Y–H). With a phenoxyl acceptor from this set, both BDEs are the computed B3LYP values. With an entered Y–H value, the donor BDE is anchored: 86.7 kcal/mol + ΔBDE(donor). | Hydrogen atom transfer between neutral species. Ionizing solvents can open the proton-loss pathway, which this calculation does not describe. | A rate constant, an antioxidant capacity or any biological effect. Rates also depend on steric hindrance, solvent hydrogen bonding and polar effects (Litwinienko and Ingold 2007). |
Botanical Intelligence
Botanical Intelligence
1 model · 25 reference checks, 0 failing · reference script tests/numerics/ref/insights-botanical-intelligence.py
| Model | Equations | Applies within | Does not establish |
|---|---|---|---|
| Expected improvement picks the next experiment by weighing a high predicted yield against high uncertaintyThe Gaussian-process posterior mean μ(x) and standard deviation σ(x) of a synthetic yield curve on a 0.2 °C grid from 30 to 80 °C, the expected improvement EI(x) over the best measured yield, and the … | Kernel k(x, x′) = s² exp(−(x − x′)²/(2ℓ²)), s² = 64 (yield points)², ℓ = 6 °C; noise variance σ n ² = 0.36. Prior mean = mean of the measured yields. μ(x) = m + k * ᵀ(K + σ n ²I)⁻¹(y − m); σ²(x) = s² − k * ᵀ(K + σ n ²I)⁻¹k * (Rasmussen & Williams 2006, chapter 2). EI as shown above with f* = best … | 30–80 °C. The demo is deterministic: the same button presses give the same runs (fixed random seed). | That Bayesian optimization outperforms classical design for any real botanical, or any yield value. Real campaigns have several inputs, refitted hyperparameters and batch effects. |
Combination science & synergy
Combination science & synergy
3 models · 54 reference checks, 0 failing · reference script tests/numerics/ref/insights-combination-science.py
| Model | Equations | Applies within | Does not establish |
|---|---|---|---|
| Whether a combination looks synergistic depends on the reference model, because Loewe additivity and Bliss independence …The effect expected if two agents do not interact, under Loewe additivity and under Bliss independence, and their difference, over a grid of doses from 0.03 to 30 times each agent's EC50. | Single agent: E ( u ) = u n /( u n + 1), inverse U ( E ) = ( E /(1 − E )) 1/ n . Loewe: the E that solves u / U ( E ) + v / V ( E ) = 1, found by bisection on logit( E ). Bliss: E A + E B − E A E B . Highest single agent: max( E A , E B ). | Equal maximum effects; monotonic Hill curves; no observed combination data. When maxima differ, Loewe additivity is not defined above the lower maximum. | That any real pair is synergistic, additive or antagonistic; a dose; or an effect in people. It shows only why the reference model must be stated. |
| Whether a combination looks synergistic depends on the reference model, because Loewe additivity and Bliss independence …At effect level x , the dose pairs that give x under Bliss independence, plotted in units of each agent's single-agent dose for x ( A x , B x ), with the Loewe line and the highest-single-agent … | A x = EC 50,A ( x /(1 − x )) 1/ n A . For each s , w solves E Bliss ( s A x , w B x ) = x by bisection. On the ray s = w = t , the Loewe combination index of the Bliss-additive mixture is CI = 2 t . | — | A combination index from data; real CIs need replicated dose-response data and confidence intervals. For the Chou–Talalay calculator and a Bliss dose-matrix calculator, see Clinical research . |
| A 2 × 2 factorial trial is the design that can test an interaction, and testing an interaction as large as the main …Total participants for a 2 × 2 factorial trial with a continuous outcome to detect (a) the main effect of one ingredient and (b) the interaction between the two, and the power that the main-effect … | Equal allocation, N /4 per arm. Main effect of A (mean of the two A arms minus mean of the two non-A arms): variance 4 σ ²/ N . Interaction contrast (μ AB − μ B ) − (μ A − μ 0 ): variance 16 σ ²/ N . Required N from the normal approximation, rounded up to a multiple of 4. Interaction power = Φ( z I … | Reliable for totals above about 40; smaller trials need t-distribution methods. | The effect size, SD or dropout for any product; these must come from prior data. It does not replace a statistician's design. |
Design space & optimization
Design space & optimization
3 models · 57 reference checks, 0 failing · reference script tests/numerics/ref/insights-design-space-optimization.py
| Model | Equations | Applies within | Does not establish |
|---|---|---|---|
| Both responses peak near 60% ethanol and 45 °C, and the desirability optimum reproduces the authors' reported conditionsPredicted TPC and DPPH from the published coded models at any ethanol, temperature and time; Derringer–Suich individual and overall desirability; the slice of conditions meeting both specifications; … | Models as in the table above. d i as defined by Derringer and Suich (maximize, minimize or target, with exponents s and t ; the page uses t = s ); D = ( d 1 d 2 ) 1/2 . Prediction interval ± t 0.975, 6 s e √(1 + z ᵀ( X ᵀ X )⁻¹ z ), with t 0.975, 6 = 2.447 and residual standard deviations s e = … | 30–90% ethanol, 25–75 °C, 15–45 min. Inside the sphere of radius √2 the models interpolate between runs; at the cube's corners they extrapolate. Limits are reordered if a lower limit exceeds a target, and a warning appears. | Performance at other scales, other seed lots or other equipment; any health effect; a validated design space. Significant lack of fit means the surfaces are approximations. |
| Measuring a frontA three-objective Pareto front for the published models (maximize TPC, maximize DPPH, minimize sonication time), its hypervolume, and checks that its points are mutually non-dominated. | — | Objectives are mean model predictions; uncertainty is treated in the next section. Time stands in for cost because power was constant. | That front points meet specifications in practice. A point chosen at a specification limit has, at best, an even chance of meeting it. |
| Accounting for model uncertainty shrinks the region that meets both specifications, and the cheapest conditions on the …For each condition in the slice, the probability that a new run would give TPC and DPPH at or above the specifications, given the 16 published runs; the share of the slice meeting the specifications … | Both responses share the quadratic design matrix X (16 × 10). Under the noninformative prior p ( B , Σ) ∝ |Σ| −3/2 , the predictive distribution of a new pair of responses is bivariate t with ν = 16 − 10 − 2 + 1 = 5 degrees of freedom, location B̂ ᵀ z ( x ) and scale (1 + h ) S / ν , where S is the … | The quadratic model is correct; errors are normal, independent and of constant variance; no prior information. Significant lack of fit means the model is not fully correct, so these probabilities are optimistic in that respect. | A design space in the regulatory sense. ICH Q8(R2) requires demonstration, which means confirmation runs at the chosen conditions and, for scale-up, evidence that the region holds at production scale. Dietary supplements have no … |
Global trade data
Global trade data
1 model · 32 reference checks, 0 failing · reference script tests/numerics/ref/insights-global-trade.py
| Model | Equations | Applies within | Does not establish |
|---|---|---|---|
| U.S. vanilla imports depend on Madagascar for 80% of their value, while HS 09.10 spices and essential oils spread …Each origin's share of U.S. import value for one HS heading in 2024, the top-3 share, the Herfindahl–Hirschman index and its inverse, the number of equal-sized origins that would give the same index. | s i = value i / Σ value; HHI = Σ (100 s i ) 2 ; equivalent origins = 10,000 / HHI. | — | Measures country concentration only, not supplier concentration or the concentration of a single botanical inside a heading. No threshold on this page labels a heading safe or risky; merger-review thresholds for firms do not apply to … |
Large Botanical Model
Large Botanical Model
4 models · 178 reference checks, 0 failing · reference script tests/numerics/ref/insights-large-botanical-model.py
| Model | Equations | Applies within | Does not establish |
|---|---|---|---|
| A molecule becomes a sequence of chemically meaningful tokens before a language model can read itThe list of tokens produced by the published regular expression, and counts of characters, tokens, distinct tokens, atom tokens, bracketed atoms and ring-closure digits. | Tokens are the successive matches of the expression shown above, applied left to right with global matching. A string is reproduced exactly when the tokens joined together equal the input, the check the authors' code asserts. | Any text up to 2,000 characters. Characters the expression does not cover are listed in a warning, and the string is then not reproduced exactly. | That a token sequence is the best input for a model; graph representations and other tokenizers exist. Token counts depend on how the SMILES was written, as the two quercetin forms show. |
| Published scaling laws show loss falling with both parameters and data, so a small corpus caps what a larger model can …The compute-optimal model size and token count for a compute budget, the fitted loss there, the best loss reachable when unique tokens are capped, and the loss floor set by the data term alone. | L(N, D) = E + A/N α + B/D β ; N opt = G(C/6) a , D opt = G −1 (C/6) b with G, a and b as above (Hoffmann et al. 2022, Eq. 4). With a cap D avail < D opt , the one-epoch model uses D = D avail and N = C/(6D avail ), because loss falls monotonically with N. Floor = E + B/D avail β . | The original runs spanned 70 million to over 16 billion parameters and 5 to 500 billion tokens; values far outside are extrapolations. Budgets 10 17 –10 26 FLOPs; caps 10 6 –10 14 tokens. | Any loss, accuracy or benefit for a botanical model. Repeated data change the picture: Muennighoff et al. (2023) found up to 4 epochs of repetition gave negligible loss changes compared with unique data, while further repetition gave … |
| Adapting an open model takes a few GPU-hours to a few hundred, while pretraining one from scratch needs thousands of …Training FLOPs, GPU-hours and wall-clock days for a model of N parameters trained on D tokens, at a chosen GPU peak and utilization; and how D compares with COCONUT's 48.2 million SMILES tokens. | C = 6ND (2N per token forward, about twice that backward; Kaplan et al. 2020). GPU-hours = C/(P peak u · 3600 s). Days = GPU-hours/(n GPU · 24). | Dense transformers. Mixture-of-experts models use the active parameters per token instead of N. | A cost, schedule or budget. No prices are used, and EUTECNIQ has no compute contract. |
| At rank 16, low-rank adapters train about 0.5% of an open 7-billion-parameter model and cut model-state memory about …The number of trainable parameters LoRA adds to OLMo 2 7B for a rank and a set of adapted matrices, its share of the full model, and the memory for weights, gradients and optimizer states. | Each adapted matrix W ∈ ℝ d×k gains B ∈ ℝ d×r and A ∈ ℝ r×k , so r(d + k) parameters. Per layer: query, key, value and output projections are 4,096 × 4,096; gate and up projections 4,096 × 11,008; down projection 11,008 × 4,096. Full fine-tuning with mixed-precision Adam needs 2 + 2 + 12 = 16 bytes … | OLMo 2 7B only; other models need their own shapes. | That adapters reach full fine-tuning quality on any botanical task, or the memory for activations, which depends on sequence length and batch size. |
Operational excellence
Operational excellence
5 models · 82 reference checks, 0 failing · reference script tests/numerics/ref/insights-operational-excellence.py
| Model | Equations | Applies within | Does not establish |
|---|---|---|---|
| A value stream map shows that a botanical lot spends most of its lead time waiting, not being processedLead time, process-cycle efficiency and rolled throughput yield for one lot. | Lead time (d) = Σ waits (d) + Σ cycle times (h) / 24. Process-cycle efficiency = value-adding time / lead time. Rolled throughput yield = Π FPY i . | One lot moves through every step; waits are averages; changeover time is shown but not added to the lot's lead time. | Capacity, cost or the cause of yield loss. A map describes one value stream at one time. |
| Capability indices compare the spread of a process with its specification, but only a stable, normal, adequately …Short-term capability (C p , C pk ) from the within-subgroup spread, overall performance (P p , P pk ) from the sample standard deviation, an approximate 95% interval for C pk , the expected … | σ within = R̄/d 2 with d 2 = 2.059 for subgroups of four. C pk 95% interval = Ĉ pk ± 1.96 √(1/(9N) + Ĉ pk 2 /(2(N − 1))), N = 100. Expected rate = Φ(−(USL − x̄)/s) + Φ(−(x̄ − LSL)/s). X̄-chart limits = x̄ ± A 2 R̄ with A 2 = 0.729. | Mean 96–104% LC, within SD 0.30–2.20, drift 0–1.60; inputs outside are clamped. The interval approximation holds for N ≥ 25 and 0.75 ≤ C pk ≤ 4 (NIST); outside that range the page reports it as not valid. | Normality. The indices and the ppm estimate assume normal data; skewed assays (for example, results bounded by a detection limit) need a transformation or percentile-based indices. Stability. A process out of statistical control has no … |
| An individuals and moving-range chart flags a process change within a few lots, before results reach the specification …Center line and 3σ limits for individual results and their moving ranges from Phase I, then the Western Electric and Nelson run rules for all 40 lots. | σ̂ = MR̄/d 2 , d 2 = 1.128; limits x̄ ± 3σ̂; UCL MR = D 4 MR̄ with D 4 = 3.267; zone lines at ±1σ̂ and ±2σ̂. | Shift 0–2.5σ, trend 0–0.2σ per lot, variation ×1.0–2.5; inputs outside are clamped. | That results meet specification. A process can be in control and still incapable, or capable and out of control. |
| A failure mode and effects analysis ranks where an extract process can fail, so controls go first where severity and …Risk priority number for each failure mode and the ranking by RPN. | RPN = S × O × D, each scored 1–10 (range 1–1,000). Severity: effect on the customer or product; occurrence: likelihood of the cause; detection: chance that current controls miss it (10 = almost certainly missed). | Scores come from a cross-functional team using written scales; the values here are invented. | Product safety, regulatory compliance or the effectiveness of a control. Equal RPNs do not mean equal risk. |
| Overall equipment effectiveness splits lost capacity into availability, performance and quality losses, so the biggest …Availability, performance, quality and their product, OEE, for one week of one machine, and the hours lost at each step. | Run time = planned − downtime − changeovers. A = run/planned. Net run time = min(output/ideal rate, run time). P = net/run. Q = (output − rejects)/output. Fully productive time = net × Q = good output/ideal rate. | Downtime 0–40 h, changeovers 0–30 h, output 1,000–6,000 kg, rejects 0–600 kg. Rejects above output are capped at the output. Output above what the ideal rate allows in the run time is physically impossible; performance is then capped at … | Product quality beyond the reject count, the cost of each loss or a benchmark for botanical dryers. |
Quantum chemistry
Quantum chemistry
2 models · 87 reference checks, 0 failing · reference script tests/numerics/ref/insights-quantum-chemistry.py
| Model | Equations | Applies within | Does not establish |
|---|---|---|---|
| A single surface descriptor captures part of a solvent ranking; the full model is needed for the restHow closely one descriptor, computed from the σ-profiles of the solute and solvent, reproduces the order in which COSMO-SAC ranks eight solvents for one botanical molecule at 25 °C. | Descriptors as defined above the explorer. Rank agreement ρ = Spearman's rank correlation between the descriptor ranking (most favorable first: smallest mismatch, or largest complementarity) and the COSMO-SAC ranking (lowest ln γ∞ first), with average ranks for ties. | The five botanical molecules and eight solvents computed here, at 298.15 K. ρ is not defined when a descriptor has the same value for every solvent (limonene has no hydrogen-bonding surface, so its complementarity is zero everywhere). | It does not test either ranking against measurements, and it does not show that any descriptor predicts solubility. Solubility also depends on the solid's melting properties and on concentration. |
| Every number on this page can be regenerated from the archived scripts | C-PCM with the dielectric constant set to infinity (the COSMO conductor limit); smooth (SWIG) surface discretization with a 194-point Lebedev grid per atom; radii H 1.30, C 2.00, N 1.83, O 1.72, Cl 2.05 Å (Klamt et al. 1998). | — | — |
Interactive science lab
Interactive science lab
5 models · 56 reference checks, 0 failing · reference script tests/numerics/ref/insights-science-lab.py
| Model | Equations | Applies within | Does not establish |
|---|---|---|---|
| Extraction: halving the particle diameter cuts the time to 90% extraction fourfoldThe fraction of the extractable solute that has left a spherical particle after time t , the time to reach 90%, and the radial concentration profile shown in the cross-section. | Released fraction M t / M ∞ = 1 − (6/π²) Σ (1/ n ²) exp(− n ²π²τ) with τ = D eff t / R ². For τ < 0.01 the equivalent short-time form 6(τ/π) 1/2 − 3τ is used (the neglected terms are below 10 −40 ). Concentration C / C 0 = (2/πρ) Σ ((−1) n +1 / n ) sin( n πρ) exp(− n ²π²τ), ρ = r / R . τ 90 = … | Diffusion-controlled extraction where external mass transfer is fast. Very short computed times signal that mixing or external film transfer, which the model ignores, would control the rate; a warning appears below 10 s. Plant material … | Yield, purity, selectivity, solvent choice, thermal degradation or the performance of any EUTECNIQ process. No EUTECNIQ data are used. |
| Emulsions: over one day, droplets below about 400 nm move more by Brownian motion than by creaming; over two years the …The creaming (or settling) velocity of a droplet, the distance it would move under gravity over the chosen time window, its one-dimensional root-mean-square Brownian displacement over the same … | v = d ²|Δρ| g /(18η) (Stokes); D = k B T /(3πη d ) (Stokes–Einstein); x rms = (2 Dt ) 1/2 . Setting vt = x rms gives the crossover diameter d * = [216 k B T η / (π Δρ² g ² t )] 1/5 , so d * falls as t −1/5 . | Reynolds numbers stay far below 0.1 across the slider range, so Stokes drag applies. Creaming distances longer than the container height mean the droplets would already have reached the top; a warning appears above 10 cm. Concentrated … | Physical shelf life, which also depends on coalescence and Ostwald ripening; chemical stability; or absorption in the body. A nanoemulsion is thermodynamically unstable and only kinetically stabilized, unlike a microemulsion (McClements … |
| Freeze-drying: ice sublimes only while the chamber pressure is below its vapor pressure, and the product must stay …The equilibrium vapor pressure of ice at the chosen product temperature, the difference between that pressure and the chamber pressure, and the margin between the product temperature and an example … | ln p ice = 9.550426 − 5723.265/ T + 3.53068 ln T − 0.00728332 T , with T in K and p in Pa. 1 Pa = 7.50062 mTorr. Margin to collapse = T c − T p . | The Murphy & Koop equation is valid above 110 K; over this slider range it agrees with the IAPWS 2011 sublimation equation to within 0.05%. Outside 110–273.16 K the calculator returns no value. | A drying cycle, drying time, shelf temperature, residual moisture or product quality. Cycle design keeps the product below its collapse temperature with a margin chosen for the formulation and equipment (Tang & Pikal 2004). |
| Stability: accelerated data predict shelf life only if one degradation mechanism holds across temperaturesThe first-order rate constant implied by a measured time to 90% assay at 40 °C, its value at 25 °C and 30 °C under the Arrhenius equation, and the resulting predicted times to 90%. | k 40 = ln(1/0.9)/ t 90,40 ; k ( T )/ k 40 = exp[−( E a / R )(1/ T − 1/313.15 K)]; t 90 ( T ) = ln(1/0.9)/ k ( T ); remaining assay = 100% × exp(− kt ). | Interpolation and short extrapolation of chemical degradation in a stable matrix. Under ICH Q1A(R2), a 5% change in assay from the initial value is a "significant change" for a drug product; when it occurs at the accelerated condition, ICH … | A validated shelf life or an expiration date. ICH Q1E limits extrapolation beyond long-term data (for room-temperature products with little change, up to twice the period covered, and not more than 12 months beyond it) and requires … |
| Chromatography: selectivity moves resolution far more than plate count, which is why method development starts thereThe resolution of two adjacent peaks from plate count, selectivity and retention, the plate count needed to reach R s = 1.5, and a simulated chromatogram. | R s = (√ N /4)((α − 1)/α)( k 2 /(1 + k 2 )); N req = [4 R s α(1 + k 2 )/((α − 1) k 2 )]². Peaks: k 1 = k 2 /α, t i = t 0 (1 + k i ), σ i = t i /√ N . | Retention factors between about 1 and 10 are usually targeted in method development; outside that range the calculator warns. At α = 1 no plate count separates the peaks. | A validated analytical procedure (specificity, accuracy, precision, range and robustness under ICH Q2(R2)), the identity of a peak, or the absence of co-eluting compounds. |
Thermodynamics
Thermodynamics
2 models · 111 reference checks, 0 failing · reference script tests/numerics/ref/insights-thermodynamics.py
| Model | Equations | Applies within | Does not establish |
|---|---|---|---|
| Recovering extraction ethanol: each step on the diagram is one equilibrium stageThe number of ideal equilibrium stages, the feed stage, the minimum and actual reflux ratios, the ethanol recovery and the vapor boil-up for a continuous column that concentrates an ethanol–water … | Equilibrium y*(x) from NRTL bubble points (dense precomputed grid, linear interpolation). Rectifying line y = R/(R + 1)·x + x D /(R + 1). Feed line x = z F (q = 1). Stripping line through (x B , x B ) and the intersection of the other two. Minimum reflux from the steepest line through (x D , x D ) … | Feed 5–60 wt%, distillate 60–94 wt% (below the NRTL azeotrope at 95.0 wt%; the measured azeotrope is at 95.4 wt%), bottoms 0.1–2 wt%, reflux 1.05–3 times minimum. Near 94 wt% the stage count becomes very sensitive to the equilibrium model; … | Real tray counts or packing heights (these need efficiencies measured at pilot scale), column diameter, energy cost, or the effect of other extract components on the equilibrium. |
| Two ways to crystallize salicylic acid: cool a hot water solution, or add water to an ethanol solutionThe fraction of dissolved salicylic acid that crystallizes at equilibrium when a saturated water solution is cooled, or when water is added to a saturated ethanol or ethanol–water solution at 25 °C. | Cooling: c*(T) = x/(1 − x) · M SA /M W with ln x = A + B/T, A = 5.7554, B = −4,161.44 K (least-squares fit to 20 measured points); yield = 1 − c*(T end )/c*(T start ), because the water mass is unchanged. Antisolvent: solubility S(w) in kg acid per kg solute-free solvent, from the measured 25 °C … | Cooling 10–75 °C, the range of the measurements (the fit is dotted outside it); antisolvent only at 25 °C and between 0 and 100 wt% ethanol in the solvent. | Crystal size, purity, filtration behavior, batch time or a validated process yield. The metastable band is illustrative and must be measured for any real process. |
QbD & control strategy
QbD & control strategy
1 model · 54 reference checks, 0 failing · reference script tests/numerics/ref/platform-qbd-control-strategy.py
| Model | Equations | Applies within | Does not establish |
|---|---|---|---|
| An accelerated result predicts shelf life only through an assumed activation energy, and the same 6-month result can …The first-order rate constant implied by a stated loss after 6 months at 40 °C, the rate at other temperatures for an assumed activation energy, and the time to 90% of the initial content ( t 90 ). | k 40 = −ln(1 − f )/6 per month; k ( T ) = k 40 exp[−( E a / R )(1/ T − 1/313.15)]; t 90 = ln(10/9)/ k ; lowest E a for a target t * at 25 °C: E a = R ln( t * k 40 /ln(10/9)) / (1/298.15 − 1/313.15). | Loss 1–25% at 40 °C; E a 50–150 kJ/mol; 15–40 °C. The range of activation energies is illustrative, chosen to show sensitivity. | A shelf life, a retest period or an expiration date. For drugs, ICH Q1E caps extrapolation and requires confirmation with long-term data. |
Clinical research
Clinical research
4 models · 60 reference checks, 0 failing · reference script tests/numerics/ref/regulatory-clinical-research.py
| Model | Equations | Applies within | Does not establish |
|---|---|---|---|
| Sample size follows from the smallest difference worth detecting: halving that difference roughly quadruples the trialThe number of evaluable participants per group that gives the chosen power for a two-sided, two-sample t test at a family-wise significance level α split across m primary comparisons, optionally … | Power( n ) = 1 − F ( t c ; ν, δ) + F (− t c ; ν, δ), with F the noncentral t distribution, ν = 2 n − 2, δ = d √( n /2), t c the 1 − α′/2 quantile of the central t with ν degrees of freedom, and α′ = α/ m . The search starts from the closed form 2( z 1−α′/2 + z 1−β )²/ d ² + z ² 1−α′/2 /4 (Guenther … | Standardized effects from about 0.03 to 15 over the slider range. Effects above about 1 imply very small trials, where the normality assumption and chance baseline imbalance matter more; a warning appears. | An approved or finalized protocol. A real sample-size justification also covers the estimand (ICH E9(R1)), the source of the assumed difference and variance, the handling of missing data, interim analyses, and review by the study … |
| For a responder endpoint, the sample size depends on both response rates, not only on their differenceThe number of evaluable participants per group needed to detect the difference between two response rates with the chosen power, with and without a continuity correction, and the number to randomize … | n = [ z 1−α′/2 √(2 p̄ (1 − p̄ )) + z 1−β √( p 1 (1 − p 1 ) + p 2 (1 − p 2 ))]²/( p 1 − p 2 )², p̄ = ( p 1 + p 2 )/2, α′ = α/ m ; corrected n cc = ( n /4)[1 + √(1 + 4/( n | p 1 − p 2 |))]². Both are rounded up; randomized per arm = ⌈ n cc /(1 − dropout)⌉. | Normal approximation; when n p or n (1 − p ) is below about 5 in either group, exact or simulation-based methods are needed, and a warning appears. Equal rates have no finite sample size. | An approved protocol, the clinical relevance of the responder threshold, or the expected rates themselves, which must come from prior data. |
| Dose–response: potency (EC50) and steepness (Hill coefficient) are separate properties, and a steep curve narrows the …The concentration–effect curve of the sigmoid Emax (Hill) model and the concentrations giving 10% and 90% of the maximal effect. | E = E max C n /( EC 50 n + C n ); EC x = EC 50 ( x /(100 − x )) 1/ n , so EC 10 = EC 50 ·9 −1/ n , EC 90 = EC 50 ·9 1/ n and their ratio is 81 1/ n . EC values are fractions of E max , so they do not change with E max . | Single-site, equilibrium response with no tolerance, receptor reserve or time dependence. Fitted EC50 values depend on the assay system and exposure time. | A human dose, efficacy or safety. |
| Combination index: a mixture tested at a fixed ratio can be synergistic at one effect level and antagonistic at anotherThe Chou–Talalay combination index of an invented fixed-ratio mixture at three effect levels, the corresponding isobologram points, and the Fa–CI curve. | Median-effect dose D x = D m ( f a /(1 − f a )) 1/ m . The dose-additive mixture at f a = 0.5 is 1/( p / D A,50 + (1 − p )/ D B,50 ); the mixture's own D m is that dose divided by the potency factor k . CI = p D mix / D A,x + (1 − p ) D mix / D B,x , which equals 1/ k at f a = 0.5. | Each agent and the mixture follow the median-effect equation; the mixture ratio is constant; effects are measured in one system. CI values without replicate-based confidence intervals are not interpretable as evidence. | Synergy in people, a dose, or that the ratio can be achieved in tissue. |
NDIN strategy & dossiers
NDIN strategy & dossiers
1 model · 35 reference checks, 0 failing · reference script tests/numerics/ref/regulatory-ndin.py
| Model | Equations | Applies within | Does not establish |
|---|---|---|---|
| Safety calculations: margin of safety, ADI and the EDI/ADI ratio Educational calculationThe estimated daily intake (EDI), margin of safety, acceptable daily intake (ADI), the EDI/ADI ratio, and the intake at which EDI/ADI equals 1, as described in Section IX of FDA's October 2026 final … | EDI = I ÷ BW. Margin = RP ÷ EDI. ADI = RP ÷ SF. Ratio = EDI ÷ ADI. Intake at ratio 1 = ADI × BW. FDA uses the NOAEL as the reference point (RP). | — | It is not a safety determination. It does not choose or validate a NOAEL, weigh study quality, severity or human data, add intake from other sources, or quantify risk. EFSA's margin-of-exposure benchmark of 10,000 applies to substances … |
USP, ICH & COA
USP, ICH & COA
3 models · 74 reference checks, 0 failing · reference script tests/numerics/ref/regulatory-quality-standards.py
| Model | Equations | Applies within | Does not establish |
|---|---|---|---|
| No single document qualifies a supplier, so EUTECNIQ would score seven independent lines of evidence and let three …A weighted mean of seven scores, rescaled to 0–100%, and a decision band. Three gates are evaluated separately; if any gate fails the decision is "Not approved" whatever the score. | S = Σ(w i s i ) ÷ (5 Σ w i ) × 100, with weights w i and scores s i integers from 0 to 5. If all weights are 0, S is reported as 0. | Scores are judged on a written rubric by the quality unit; the lines of evidence are treated as independent and compensatory above the gates. The examples are invented to show how gates and weights interact. | It is not a supplier qualification under § 111.75(a)(2)(ii), which requires confirming the supplier's test results. It does not replace an audit, testing, or legal review of origin and sanctions. |
| Sampling √N + 1 containers finds a 10% adulterated consignment of 40 drums only about half the timeThe probability that a simple random sample of n containers from a consignment of N contains at least one of K adulterated containers and that the test flags it; the smallest n that reaches the … | Hypergeometric: P(k) = C(K,k) C(N−K, n−k) / C(N,n). P detect = 1 − Σ k P(k)(1 − s) k ; with s = 1 this reduces to 1 − C(N−K, n)/C(N, n). WHO TRS 929 Annex 4: n = 1 + √N "obtained by simple rounding", every container if N ≤ 4; p = 0.4√N and r = 1.5√N, "rounding up to the next highest integer". | Containers are either adulterated or not; samples are drawn at random without replacement; each sampled container is tested separately; test results are independent. Compositing samples, as WHO's n plan does, dilutes an adulterated portion … | It does not measure within-container heterogeneity, which WHO treats as intrinsic for "partially processed natural products … herbal (dried plants and their parts)". It does not replace the § 111.75(a)(1)(i) identity test of each dietary … |
| USP ⟨2232⟩ caps lead at 5 µg per day from a supplement; California's Proposition 65 reproductive-toxicity safe harbor …Daily exposure to one element from a supplement, its share of the USP ⟨2232⟩ permitted daily exposure, the highest concentration that keeps exposure at the PDE for this intake, and for lead its share … | E = c × m (µg/day). Share of PDE = 100 E / PDE. c max = PDE / m. Share of MADL = 100 E / 0.5. | For inorganic arsenic, ⟨2232⟩ allows a total-arsenic result as a conservative estimate; if it exceeds the limit, speciation is required. Methylmercury testing is unnecessary when total mercury is below the methylmercury limit. | A Proposition 65 warning decision depends on exposure assessment rules, other chemicals and counsel's advice; the MADL comparison is a screening figure. It does not address intake from other foods. |
Toxicology & exposure
Toxicology & exposure
1 model · 18 reference checks, 0 failing · reference script tests/numerics/ref/regulatory-toxicology.py
| Model | Equations | Applies within | Does not establish |
|---|---|---|---|
| The rules, from TG 423 Annex 2The sequence of doses and the GHS acute oral category that OECD TG 423's flow charts (Annex 2a–2d) assign for the outcomes you enter, and the number of animals used. | — | "Deaths" means animals found dead or humanely killed as moribund, as in the annex. The 5000 mg/kg limit test of Annex 3 is not included. | A hazard category is not a reference point for intake. It gives no NOAEL, no target organ information from repeated exposure and no basis for a daily dose; those come from repeat-dose studies and the margin calculations above. |
Analytical method development
Analytical method development
2 models · 56 reference checks, 0 failing · reference script tests/numerics/ref/technologies-analytical-method-development.py
| Model | Equations | Applies within | Does not establish |
|---|---|---|---|
| Gradient time and column temperature together move the critical pair, and the MODR is the region where every pair is …Retention times and peak widths of five constituents in a linear gradient, the minimum resolution between adjacent peaks (the critical pair), the last retention time, and the region of gradient time … | log k = log k w − S·C ; b = S Δ C t 0 / t G ; t R = ( t 0 / b ) log 10 [ln10· k i b (1 − t D /( t 0 k i )) + 1] + t 0 + t D (2.303 in the source is ln 10); k e = 1/(ln10· b + 1/ k i ); w = 4 t 0 (1 + k e )/√ N ; log 10 k w = A + B / T . A constituent with k i t 0 ≤ t D elutes before the gradient … | The slider and map ranges. Retention can deviate from LSS and van 't Hoff linearity; Tanase et al. (2019) attribute nonlinear van 't Hoff behavior largely to multiple retention mechanisms. | A method for any real extract; robustness to pH, column lot or instrument; peak purity. ICH Q14 requires the routine operating point to be covered by validation data, and model predictions must be confirmed experimentally. |
| Detection and quantitation limits estimated from a calibration line are only as good as the standards near the limitAn ordinary least-squares line through the standards, r², the residual standard deviation, the standard error of the intercept, DL and QL from each, and the largest relative error when each standard … | S = Σ( x − x̄ )( y − ȳ )/Σ( x − x̄ )²; intercept a = ȳ − S x̄ ; s y/x = √(Σ residual²/( n − 2)); s a = s y/x √(1/ n + x̄ ²/Σ( x − x̄ )²); DL = 3.3σ/ S ; QL = 10σ/ S . | At least three points with different concentrations and a positive slope; non-numeric entries are ignored. ICH recommends at least five concentrations for linearity. | That the limits are achieved. ICH Q2(R2) section 3.2.3.5: an estimated QL "should be subsequently validated by the analysis of a suitable number of samples known to be near or at the QL". |
Bioavailability by design
Bioavailability by design
1 model · 40 reference checks, 0 failing · reference script tests/numerics/ref/technologies-bioavailability-design.py
| Model | Equations | Applies within | Does not establish |
|---|---|---|---|
| When solubility limits absorption, the absorbed amount stops rising with dose, and the maximum absorbable dose …The maximum absorbable dose, the resulting ceiling on the fraction of a dose absorbed when solubility is limiting, the absorption rate constant, and the dose, absorption and dissolution numbers. | MAD = S · K a ·SIWV·SITT; K a = 2 P eff / R ; ceiling = min(1, MAD/dose); Do = dose/(250 mL × S ); An = P eff · T si / R ; Dn = 3 D S T si /( r ² ρ ), with T si = SITT. | A screening estimate only. Fasted small-intestinal water measured by MRI averaged 105 ± 72 mL in 12 volunteers and 54 ± 41 mL after a meal (Schiller et al. 2005), well below the 250 mL convention. Ding et al. (2012) found the equation gave … | Absorption of any real ingredient, the effect of food or formulation, first-pass metabolism ( F T ), or efficacy. ICH M9 criteria apply to drug biowaivers, not to supplements. |
Computational science & AI
Computational science & AI
1 model · 5 reference checks, 0 failing · reference script tests/numerics/ref/technologies-computational-science-ai.py
| Model | Equations | Applies within | Does not establish |
|---|---|---|---|
| A model is judged on botanicals it has never seen, against independent benchmarks, and it must say when it does not knowFor 14 synthetic query points, whether each lies inside the applicability domain, defined as a distance to its fifth-nearest training sample below the threshold. | Euclidean distance in the two-dimensional projection; threshold = 0.25 + 1.35 × slider/100 (synthetic units). | 108 synthetic training points from three Gaussian clusters (fixed seed); a real model would measure distance in its own learned representation. | It shows how abstention works; it says nothing about any real model's domain or accuracy. |
Advanced delivery
Advanced delivery
1 model · 28 reference checks, 0 failing · reference script tests/numerics/ref/technologies-delivery.py
| Model | Equations | Applies within | Does not establish |
|---|---|---|---|
| Results appear here when the page's script runs.Four published quantities for the model compound, and a sorting of ten delivery families into "worth evaluating", "possible, with conditions" and "not indicated by these inputs" using the rules in … | Dose number Do = D /( V 0 · S ) with V 0 = 250 mL, the volume in the ICH M9 definition of a highly soluble drug. General solubility equation: log S 0 = 0.5 − 0.01(MP − 25) − log P (mol/L). Potts–Guy: log k p = −2.7 + 0.71 log P − 0.0061 MW (cm/h). β-Cyclodextrin for a 1:1 complex: D × 1,135/MW … | The GSE was validated on organic nonelectrolytes; the Potts–Guy data set spanned MW 18 to about 750 and log P −3 to +6, and the tool warns outside it. | It does not recommend a formulation, predict bioavailability or show that any carrier is clinically superior. "Worth evaluating" means a family addresses the barrier the inputs suggest; laboratory performance and human pharmacokinetics … |
Discovery & characterization
Discovery & characterization
1 model · No numerical calculator on this page (descriptive model)
| Model | Equations | Applies within | Does not establish |
|---|---|---|---|
| Green tea Camellia sinensis (L.) Kuntze, unfermented leafFor each botanical: identity methods, constituent classes, characterization methods, and published safety and quality questions, each traced to an agency document or peer-reviewed source listed under … | — | — | No compound concentrations are given, because they depend on the lot, the extract and the method. Nothing here is a safety assessment, a dose recommendation or an efficacy claim; traditional use and in vitro activity are not clinical proof. |
Drying & particle engineering
Drying & particle engineering
5 models · 74 reference checks, 0 failing · reference script tests/numerics/ref/technologies-drying-particle-engineering.py
| Model | Equations | Applies within | Does not establish |
|---|---|---|---|
| The Péclet number says whether a droplet dries into a dense bead or a hollow shellThe first-stage drying time of one droplet, the diffusion coefficient of a dissolved molecule, the Péclet number and the surface enrichment of that solute. | τ D = d 0 ²/ κ ; D = k B T /(6π ηr ); Pe = κ /(8 D ); E = 1 + Pe/5 + Pe²/100 − Pe³/4000 (Vehring 2008, eq. 8). | The enrichment formula is accurate to about ±1% for Pe below 20 (Vehring 2008); above that the readout says so instead of extrapolating. | τ D covers only the first drying stage, before the shell locks up; the second stage through the crust is slower. The model does not predict particle morphology for a real extract, which is checked by electron microscopy and density … |
| Primary drying is a trade-off: a warmer product dries faster, until it crosses its collapse temperatureThe vapor pressure of ice at the product temperature, the sublimation driving force against the chamber pressure, and the margin between product and collapse temperature. | ln p ice = 9.550426 − 5723.265/ T + 3.53068 ln T − 0.00728332 T (Murphy & Koop 2005, eq. 7; T in K, p in Pa); Δ p = p ice − p chamber . | −45 to −10 °C and 2 to 30 Pa, inside the range of the Murphy–Koop equation (above 110 K). | It does not compute sublimation rate, which also needs the dried-layer resistance and shelf heat transfer. Collapse temperatures are measured for each product, for example by freeze-drying microscopy. The cycle is illustrative, not a … |
| A steeper sorption isotherm means more water taken up per step in humidityEquilibrium water content from the GAB isotherm at the chosen humidity, the glass-transition temperature of the wetted solid from the Gordon–Taylor equation, and the critical humidity at which T g … | As shown above, with w s = 1/(1 + W /100) and w w = 1 − w s ; the critical point is found by bisection. | 2–70% relative humidity, inside the measured range of the source data. | Caking also depends on time above T g , particle size and pressure in the pack. The values describe mango-based powders, not any EUTECNIQ material. |
| Packaging sets how fast a powder reaches its critical water activity, so the film is chosen from the powder's isothermHow the water activity inside a sealed package rises over time, and when it reaches the powder's critical water activity at 25 °C. | Film permeance Π = WVTR/(0.90 p sat (38 °C)); flux J = Π A p sat (25 °C)(RH − a w ); time to critical t = ( m dry /100)/(Π A p sat ) ∫ (d W /d a w )/(RH − a w ) d a w , integrated with Simpson's rule from the packed to the critical water activity. | Outside humidity 40–85%; packed water activity 0.02–0.30; inside the GAB parameter range of the source data. | It ignores oxygen ingress, light, chemical degradation, temperature cycling and seal failure. It predicts a physical limit, not a shelf life; that comes from stability testing in the final package. |
| Stability data support a shelf life only as far as the lower confidence bound stays inside the specification, and only …A least-squares line through illustrative assay results, its lower one-sided 95% confidence bound for the mean, the time at which that bound meets a 90% acceptance criterion, and the ICH Q1E cap on … | As shown above. S tt = Σ( t i − t̄ )²; s is the residual standard deviation with n − 2 degrees of freedom; cap = min(2 t max , t max + 12 months). The supported period is the smaller of the crossing time and the cap, rounded down to whole months. | Loss rates 0.05–0.8% per month; scatter 0.2–2%; 6 to 24 months of data. | A real program would test at least three batches (ICH Q1A(R2)) in the final package, check whether batches can be pooled (ICH Q1E), and track several attributes. For a multi-constituent botanical, the marker chosen may not represent the … |
Emulsions & colloids
Emulsions & colloids
2 models · 28 reference checks, 0 failing · reference script tests/numerics/ref/technologies-emulsions-colloids.py
| Model | Equations | Applies within | Does not establish |
|---|---|---|---|
| Creaming speed rises with the square of droplet size, so a tenfold smaller droplet creams a hundred times more slowlyThe rise (or settling) speed of one droplet, the time to cross the liquid height at that speed, and the gravitational length at 25 °C. | v = 2 r ²Δ ρ g /(9 η ); t = H / v ; l g = k B T /((4/3)π r ³Δ ρ g ). The concentration of droplets at equilibrium falls with height as exp(− z / l g ). | Particle Reynolds number far below 1, true for every setting here. Droplet diameters 20 nm to 20 µm; viscosity 0.89 to 1,000 mPa·s. | Concentrated emulsions cream more slowly than Stokes predicts; flocculated droplets cream faster as clusters; thickened or gelled water phases need measured rheology. It says nothing about coalescence, ripening or chemical stability. |
| Small droplets feed large ones, and the oil's water solubility sets the paceThe LSW ripening rate for a limonene-like oil of chosen water solubility, the mean diameter over time and the time for the mean diameter to double. | ω = (8/9)( c (∞)/ ρ )( γ V m / RT ) D ; r ³( t ) = r 0 ³ + ω t ; doubling the diameter needs t = 7 r 0 ³/ ω . | Solubility 10⁻⁴ to 100 mg/L; starting diameter 60–300 nm. The curve is clipped at 1 µm. | Absolute times depend on the example γ and D ; real emulsions are measured. Mixtures of oils, surfactant micelles and polydisperse droplets can ripen faster or slower than LSW predicts. |
Extraction & purification
Extraction & purification
5 models · 56 reference checks, 0 failing · reference script tests/numerics/ref/technologies-extraction.py
| Model | Equations | Applies within | Does not establish |
|---|---|---|---|
| Pressure turns CO₂ from a thin gas into a dense, liquid-like solventThe density of pure CO₂ at the chosen temperature and pressure from the Peng–Robinson cubic equation, and its error against the Span–Wagner reference equation of state. | a = 0.45724 R ² T c ²/ P c · α( T ); b = 0.07780 RT c / P c ; α = [1 + κ(1 − √( T / T c ))]², κ = 0.37464 + 1.54226ω − 0.26992ω². The cubic in the compressibility factor Z is solved analytically; where three roots exist, the root with the lowest fugacity coefficient is kept. Density ρ = PM /( ZRT ). | 32–80 °C and 1–40 MPa; inputs are clamped to that range. Every state shown is above the critical temperature, so no phase boundary is crossed. | It does not predict solubility, extraction yield or the behavior of CO₂ with a co-solvent or dissolved extract. It shows only why process design uses a reference equation of state near the critical point. |
| Pressure turns CO₂ from a thin gas into a dense, liquid-like solventHow many times more of a solute CO₂ can dissolve at 30 MPa than at 10 MPa, both at 40 °C, for an assumed exponent k. | At fixed temperature, S ₂/ S ₁ = (ρ₂/ρ₁) k , because the a / T + b terms cancel. ρ from the Span–Wagner reference grid. | The Chrastil form holds for the solute over this density range; k is a fitted constant for each solute and is illustrative here. | It does not give the solubility of any named botanical compound. Fitted Chrastil parameters from measured solubility data are needed for that. |
| A supercritical extraction curve has a fast straight start and a slow tailThe share of the extractable oil recovered after a given mass of CO₂ per kilogram of oil-free plant solids has passed through a packed bed. | First period: e = y s q until the broken-cell oil r x u is exhausted at q 1 = r x u / y s . Second period: first-order release from intact cells, e = x u [1 − (1 − r )exp(− C 2 ( q − q 1 ))]. CO₂ for a fraction f > r : q = q 1 + ln[(1 − r )/(1 − f )]/ C 2 . | r 10–90%, y s 1–15 g/kg, C 2 0.005–0.08 (clamped). Plug flow, constant solubility, no co-solvent. | It does not predict the yield of any real material. Pilot runs at fixed pressure, temperature and flow would be needed to fit y s , r and C 2 . |
| Recovery, RFrom the yield, the plant's target content and the recovery, the target content of the extract; and from the solvent-to-feed ratio and yield, the mass of solvent and the latent heat needed to … | P = R x f / Y ; enrichment = P / x f ; native DER = 1/ Y ; solvent per kg extract m = ( S / F )·ρ/ Y ; heat Q = m ·Δ h vap ( T b ); 1 kWh = 3.6 MJ. | Inputs are clamped to the slider ranges. If R x f > Y , the purity would exceed 100%, which is impossible; the tool says so instead of showing a number as valid. | It is not an energy or cost estimate for any plant. Multiple-effect evaporation and vapor recompression supply far less steam per kilogram evaporated than a single-effect evaporator, solvent retained in the spent plant must also be … |
| Solvent choice is a safety and quality decision, not only a yield decisionThe Hansen parameters of an ethanol–water blend and its Hansen distance to soybean oil, as a guide to how alike the two liquids are. | Blend: δ i = φ E δ i,E + (1 − φ E )δ i,W for i = D, P, H, with φ E the ethanol volume fraction. Distance: R a = [4(ΔδD)² + (ΔδP)² + (ΔδH)²] ½ . | Ideal volume-fraction mixing (no volume change on mixing); 25 °C parameter values. | A smaller distance indicates likely miscibility; it does not give the solubility of any botanical compound or the selectivity of an extraction. Strongly hydrogen-bonding mixtures such as ethanol–water deviate from ideal blending. |
Process engineering & scale-up
Process engineering & scale-up
11 models · 269 reference checks, 0 failing · reference script tests/numerics/ref/technologies-process-engineering.py
| Model | Equations | Applies within | Does not establish |
|---|---|---|---|
| The same recipe gives a lower yield and more degradation at plant scale, unless the process is redesigned around heat …For one recipe at two scales: the batch temperature history, the fraction of extractable solids in solution, the loss of a heat-labile marker, the mixing time, and energy per kilogram of extract for … | Vessel: cylinder with liquid height equal to diameter, jacket on wall and flat bottom, A = 1.25π T ². Heating with steam at 120 °C from 20 °C, then a hold, then cooling with 20 °C water to 35 °C, each segment exponential with time constant Mc p / UA . Extraction: F = ( α /(1+ α ))·[1 − Σ 6 α (1+ α … | Plant 0.1–10 m³; particles 0.5–5 mm; 4–20 kg solvent per kg herb; 40–80 °C; 15–240 min hold; diffusivity 10⁻¹² to 10⁻¹⁰ m²/s at 60 °C. Very short times use the short-time limit of the series. | It does not predict a yield for any real botanical. Plants recover yield with countercurrent stages (see the Kremser calculator) and recover energy with multiple-effect or vapor-recompression evaporation; neither is modeled here. Real … |
| Spray drying spends roughly half its heat on evaporation; most of the rest leaves in the exhaust airWater evaporated, powder made, dry-air rate, inlet air volume, heater duty and its split, heat per kilogram of water, thermal efficiency and the exhaust dew point. | Solids balance P = F x s /(1 − x p ); water E = F (1 − x s ) − P x p ; the chamber balance above solved for G ; heater duty Q = G c H Δ T in ; humid heat c H = 1.006 + 1.86 W 0 ; λ 0 = 2,501 − (4.184 − 1.86) T 0 ; exhaust humidity W 1 = W 0 + E / G ; dew point where the Murphy–Koop liquid vapor … | A solution exists only when (1 − f )Δ T in exceeds Δ T out ; otherwise the readouts say "no solution" and a warning explains why. | Stickiness, glass transition, volatile losses, particle size and residence time, which set the real operating window, are outside the balance. |
| Counterflow recovers more heat than parallel flow from the same exchanger, and the gap widens as the exchanger growsHeat duty, outlet temperatures, effectiveness and the log-mean temperature difference of a two-stream exchanger in counterflow or parallel flow, and the same duty for the other arrangement. | Counterflow ε = (1 − e −NTU(1− C r ) )/(1 − C r e −NTU(1− C r ) ), and NTU/(1 + NTU) when C r = 1; parallel ε = (1 − e −NTU(1+ C r ) )/(1 + C r ); Q = ε C min ( T h,in − T c,in ); the profiles integrate the local balances with a fourth-order Runge–Kutta step. | Single-phase liquids within the slider ranges. The UA·ΔT lm check equals the duty to rounding. | Exchanger area, pressure drop, fouling margin or mechanical design. |
| A 2,000 L batch takes about thirteen times as long to heat as 1 L, because jacket area per liter falls as volume risesThe time for a well-mixed batch to heat from its starting temperature to a target with condensing steam in the jacket, at 1 L, 100 L and 2,000 L. | T ( t ) = T s − ( T s − T 0 )e − t /τ , τ = Mc p /( UA ); time to target t = τ ln[( T s − T 0 )/( T s − T )]; vessel diameter T = (4 V /π) 1/3 , jacket area 1.25π T ². | Target temperature below the steam temperature (the target is held at least 0.5 K below it). | Real U depends on agitation, fouling and the jacket side; internal coils change the area-to-volume ratio. |
| Three ideal countercurrent stages at a solvent ratio of 1.5 recover 88% of the solute, and each added stage buys lessThe fraction of solute recovered after N ideal countercurrent washing stages at solvent ratio S , the stages needed for 99% and the gain from one more stage. | Fraction left in the marc φ = ( S − 1)/( S N +1 − 1), and 1/( N + 1) when S = 1; recovery = 1 − φ. | S 0.5–4, N 1–10; with S < 1 recovery approaches S and 99% may be unreachable, which the readout states. | Diffusion out of particles makes real stages less than ideal; stage efficiency is measured in pilot runs. |
| In drying, heat supply to the surface limits the constant-rate period and moisture moving out of the solid limits the …Constant-rate and falling-rate drying times for a tray or bed to reach a target moisture. | t c = L s ( X 0 − X c )/( AN c ); t f = L s ( X c − X e )/( AN c ) · ln[( X c − X e )/( X − X e )]. | The critical moisture is held at least 0.05 below the initial moisture and the target at least 0.005 above equilibrium; a warning appears when a slider is clamped. | Real materials can show several falling-rate zones, shrinkage and case hardening; drying curves are measured. |
| A concentrated extract can flow laminar where water would be turbulent, which changes pressure drop, pump choice and …Velocity, Reynolds number, friction factor, pressure drop, head loss and pump shaft power for flow in a straight pipe. | Re = ρvD / μ ; laminar f = 64/Re below Re 2,300; otherwise Colebrook 1/√ f = −2 log 10 ( ε /(3.7 D ) + 2.51/(Re√ f )), solved by iteration from the Haaland estimate; Δ p = f ( L / D )( ρv ²/2); head = Δ p /( ρg ); shaft power = Q Δ p /η. | Transitional flow (Re 2,300–4,000) is flagged because the friction factor there is uncertain. | Concentrated extracts are often shear-thinning and need measured rheology; pump selection also needs NPSH and the pump curve. |
| A continuous step with a broad residence-time distribution under-processes some material and over-exposes the restThe exit-age distribution of N equal well-mixed tanks, the fractions leaving before and after chosen times, and the Péclet number of a closed vessel with the same variance. | As above, with θ = t /τ; fractions from the cumulative distribution F ( θ ) = 1 − e − Nθ Σ k =0 N −1 ( Nθ ) k / k !; Pe solved by bisection. | 1 to 50 tanks; the Péclet equivalence is exact in variance only. | Real equipment has dead zones and bypassing that this model cannot show; the distribution is measured with a tracer test. |
| No single scale-up rule keeps every mixing quantity constantThe plant impeller speed under four scale-up rules and the resulting power, power per volume, tip speed, Reynolds number and blend time relative to the laboratory. | Scale ratio k = ( V 2 / V 1 ) 1/3 ; power P = Po ρN ³ D ⁵; tip speed π ND ; Re = ρND ²/ μ ; rules: constant P/V gives N ∝ k −2/3 , constant tip speed N ∝ k −1 , constant Re N ∝ k −2 , constant blend time N unchanged. | Laboratory 1–50 L; plant 100 L to 20 m³ (logarithmic slider, shown to two significant figures). | Solids suspension, gas dispersion and heat transfer have their own correlations; local shear near the impeller differs from the averages shown. |
| A valve homogenizer concentrates energy that a stirred tank spreads over hours, and the product heats with every passEnergy density and adiabatic temperature rise of valve homogenization; for a stirred tank, the mean dissipation rate, the Kolmogorov length and the Hinze largest stable droplet; the stirring time … | As above, with ε = ( P / V )/ ρ and ν = μ / ρ ; stirring time = n Δ p /( P / V ). | 10–200 MPa; 1–10 passes; 10 W/m³ to 10 kW/m³. | It does not predict homogenizer droplet size, which depends on valve design, emulsifier adsorption speed and recoalescence, and is measured. Shear-sensitive ingredients need their own damage tests. |
| Response-surface optimization finds a region where high yield and low degradation overlap, not a single pointLeast-squares quadratic models for simulated yield and degradant data from a 20-run central composite design, the slice of the experimental region meeting both limits at the chosen time, and the best … | ŷ = b 0 + Σ b i x i + Σ b ij x i x j + Σ b ii x i ²; b = ( X ᵀ X )⁻¹ X ᵀ y ; R ² = 1 − SSE/SST; t = b /se( b ) with 10 residual degrees of freedom. | Inside the experimental region; outside it the model extrapolates and the plot is shaded. | Regions drawn from mean predictions overstate assurance; an ICH Q8(R2) design space has to be demonstrated, with prediction uncertainty and confirmation runs. |
Purification & standardization
Purification & standardization
4 models · 89 reference checks, 0 failing · reference script tests/numerics/ref/technologies-purification-standardization.py
| Model | Equations | Applies within | Does not establish |
|---|---|---|---|
| When two peaks overlap, a higher purity specification is paid for in lost recovery, and the exchange rate rises steeply …For a target peak (12.00 min) and one impurity peak, the fraction of each collected between cut start a and cut end b , the resulting purity and recovery, the resolution, and the cut window with the … | Fraction of a Gaussian peak (center t , standard deviation σ ) inside [ a , b ]: f = ½[erf(( b − t )/( σ √2)) − erf(( a − t )/( σ √2))]. Recovery = f T . Purity = f T /( f T + ρ f I ), where ρ is the impurity-to-target mass ratio in the load. R s = |Δ t |/(2( σ 1 + σ 2 )). | The slider ranges. At preparative loads the isotherm is nonlinear, peak shapes then depend on it, and this Gaussian model no longer applies (Guiochon 2002 reviews the nonlinear models). | The purity or recovery of any real separation; the effect of other impurities; detector response factors. |
| A resin column is stopped at breakthrough, long before its capacity is used up, so flow rate and bed size set how much …The breakthrough curve of a packed adsorbent bed, the throughput and time at which the outlet reaches the chosen fraction of the feed, the mass retained per liter of bed at that point, and the share … | C / C 0 = 1/(1 + e a − bV ) with a = kN 0 / F and b = kC 0 / F . Stop point V c = ( a − ln(1/ p − 1))/ b . Retained mass per mL of bed = C 0 [ V − ∫ 0 V ( C / C 0 )d V ], with the integral in closed form, (ln(1 + e bV − a ) − ln(1 + e − a ))/ b . | The slider ranges. When the outlet already exceeds the stop fraction at the start, the tool reports that the bed is overloaded. | Resin choice, capacity or selectivity for any extract; competitive adsorption; elution recovery. |
| Blending three assayed lots can bring two markers inside their ranges at the lowest cost, and the feasible blends form …The set of blends of three lots that meets two marker ranges and stock limits, the least-cost blend in that set, and the markers and price of a blend you choose. | Blend content Σ x i g i for each marker; price Σ x i c i ; fractions sum to 1 and lot i cannot exceed its stock s i divided by batch size M . With x C = 1 − x A − x B , every constraint is a half-plane in ( x A , x B ); the feasible set is the triangle clipped by all of them (Sutherland–Hodgman), … | Linear mixing by mass, homogeneous lots, assay values without uncertainty, and no losses. Real decisions would use the assay uncertainty and a guard band inside each range. | EMA allows blending batches to adjust quantified extracts and blending or excipients for standardised ones. Blending must not be used to dilute a contaminant: FDA's food CGMP rule states that mixing a food with defects at adulterating … |
| A fingerprint similarity of 0.99 can hide a missing minor constituent, because large peaks dominate the scoreThe congruence coefficient and Pearson correlation between a reference fingerprint (nine peak areas) and a test lot in which the largest peak, one minor marker, an extra peak and the overall dilution … | As shown; vectors include a tenth position that is zero in the reference, so an unknown peak enters the comparison. | Peaks already matched by retention time; equal weighting of peak areas; no normalization other than that inside each coefficient. | An acceptance threshold for any product; the identity of an unknown peak. |
Translational research
Translational research
1 model · 14 reference checks, 0 failing · reference script tests/numerics/ref/technologies-translational-research.py
| Model | Equations | Applies within | Does not establish |
|---|---|---|---|
| Oral bioavailability is a chain of fractions, and the weakest link sets the ceilingThe fraction of an oral dose that reaches the systemic circulation unchanged, and the highest value that fraction could reach if absorption were complete. | F = F a · F g · F h ; ceiling = F g · F h ; loss at each step = remaining before the step × (1 − its fraction). | Linear (dose-independent) kinetics; the fractions act in sequence and independently; parent compound only. Real values depend on dose, formulation, food and the individual, and saturation of first-pass elimination can make them … | It does not predict the bioavailability of any botanical constituent. It shows why a formulation that raises absorption cannot lift exposure above the ceiling set by first-pass metabolism, and why metabolites, not only the parent compound, … |