Best Li recovery:87.1%

Active experiments:3

Best model:Avrami-Erofeev

Eₐ surface (B,C,D):27.8 kJ/mol

Eₐ diffusion (B,C,D):18.0 kJ/mol

Segmented series:3 / 4

LITIUMLAB

Black mass leaching — kinetic simulation platform

⚗

Experimental data

Dynamic time-series input — add as many time points as your experiment produced

📈

Recovery curves

α vs time with error bars, plateau annotation, curve segmentation

🧮

Model fitting

SCM, Avrami, GB, n-order — AIC/F-test, residuals, parity, segmentation

⚡

Process optimiser

Dual-mechanism aware predictions — sensitivity and economics

Platform overview

Leaching mediumH₂SO₄ (aq)

FeedSpent Li-ion black mass

Kinetic models4

Auto segmentationON

Dual-mech. optimiserINCLUDED

PDF report exportAVAILABLE

Scientific features

Dynamic data entryyes

Recovery curves ± error barsyes

Linearisation + residualsyes

Parity plot + mass balanceyes

Dual-mechanism segmentationyes

Literature Eₐ comparisonyes

Quick links

? Help & User manual→

⚗ Data Input→

⚠ Segmentation detail→

Activation energy→

Process optimiser→

Dashboard

Loading your projects...

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Projects

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Total series

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Segmented

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Fitted

Your projects

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Chemical constants database

System constants + your saved constants — persisted to your account

💾Your constants (green) are saved server-side and reload every time you log in.

All

Thermodynamic

Kinetic

Transport

My constants (2)

Symbol	Name	Value	Units	Category	Source	Action
R	Universal gas constant	8.314	J mol⁻¹ K⁻¹	THERMO	NIST	system
D_Li+	Diffusivity of Li⁺ in H₂SO₄	1.03×10⁻⁹	m² s⁻¹	TRANSPORT	CRC	system
M_Li	Molar mass of lithium	6.941	g mol⁻¹	THERMO	IUPAC	system
ρ_BM	Black mass bulk density	2.85	g cm⁻³	PHYSICAL	User est.	system
E°_LiCoO₂	Standard electrode potential	+0.56	V vs SHE	ELECTROCHEM	Nagaura	system
k_app_BM01	App. rate const. 348K YOURS	0.0414	min⁻¹	KINETIC	Exp. fit
Ea_BM01	Activation energy NMC811 YOURS	42300	J mol⁻¹	KINETIC	Arrhenius

Experimental data input

Define conditions — add as many time points as your experiment produced

Series configuration

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Temperature

K (= 75°C)

H₂SO₄ concentration

mol/L

S/L ratio

g/mL

10g/500mL → 0.020 g/mL

Stirring speed

RPM

Particle size d₅₀

μm

H₂O₂ reductant

vol%

⚖ Mass balance checker

Step 1 — Li in sample:— Step 2 — Acid available:— Step 3 — Acid required:— Result:—

Manual entry

Paste / bulk entry

CSV upload

Enter each time point and recovery fractions. Click + Add time point for each measurement, or press Enter on the last field of a row to jump to a new row.

Rows: 7 | Time: 2–90 min | Max α(Li): 0.870

#	Time (min) required	α Li required · 0–1	α Co optional	α Ni optional	α Mn optional	pH optional	Li s.d. error bars

+

Add time point or press Enter on last field of a row

Copy columns from Excel and paste below. Each row = one time point. Separator: tab, comma, or space. Column order: time_min · α_Li · α_Co · α_Ni · α_Mn · pH · Li_sd. Only first two are required.

Paste data here

Columns: time_min · α_Li · α_Co · α_Ni · α_Mn · pH · Li_sd

⬆

Drop CSV file here or click to browse

Columns: time_min, alpha_Li [, alpha_Co, alpha_Ni, alpha_Mn, pH, alpha_Li_sd]

Expected CSV format

time_min,alpha_Li,alpha_Co,alpha_Ni,alpha_Mn,pH,alpha_Li_sd
2,0.08,0.07,0.06,0.05,1.1,0.005
5,0.21,0.18,0.15,0.12,0.8,0.010

Series metadata

Black mass source / batch

Li content of solid (wt%)

Milling method

Replicate runs (n)

Solid mass used (g)

Solution volume (mL)

Li recovery curves

α vs leaching time — all series

Li extraction fraction (α) vs leaching time — experimentálne dáta + Avrami fit

Plná čiara = Avrami model (najlepší jednofázový fit surových dát) | Body = experimentálne merania ±s.d. | Odchýlky B,C,D od Avrami → bifázový charakter → analýza v module Segmentation

⚑ What is a plateau?A plateau occurs when α stops increasing despite continued leaching. Causes: (1) acid depletion; (2) Li encapsulation in passivation layers; (3) thermodynamic ceiling. At 363K/2.5M: α≈0.91. Use mass balance checker to rule out acid depletion. To raise the plateau: reduce particle size, increase acid, or raise temperature.

Select a project and run fitting to see series results here.

Kinetic model fitting

All 4 models — AIC / F-test / RMSE

⚠ Dual mechanism at t≈25 min (Series C): surface reaction → diffusion control.

Avrami-Erofeev

[-ln(1-α)]^(1/n) = k·t

★ BEST FIT

R²

0.9620

RMSE

0.0183

AIC

−142.3

n (Avrami)	0.731 ± 0.042
k	0.0414 min⁻¹
Mechanism	Nucleation+growth
Residuals	random ✓

Avrami fit — observed vs predicted (T=348K)

Exp. data ± s.d.

Model fit

95% CI

Shrinking Core

1-(1-α)^(1/3)=k_s·t

R²

0.9410

RMSE

0.0312

k_s	0.039 min⁻¹
k_d	0.009 min⁻¹
F-test p	0.021

Ginstling-Brounshtein

1-2α/3-(1-α)^(2/3)=k_D·t

R²

0.9170

RMSE

0.0490

k_D	0.0072 min⁻¹
Valid α	>0.5
F-test p	0.041

n-order reaction

dα/dt=k·(1-α)^n

R²

0.8930

RMSE

0.0630

n	1.43±0.11
Weak α>0.7	—
F-test p	0.003

Statistical comparison

Model	R²	RMSE	AIC	Rank
Avrami-Erofeev	0.9620	0.0183	−142.3	★ 1st
Shrinking Core	0.9410	0.0312	−118.7	2nd
Ginstling-B.	0.9170	0.0490	−97.1	3rd
n-order	0.8930	0.0630	−82.4	4th

Parity plot — predicted vs observed α

Points on 45° line = perfect prediction.

Linearisation plots

Transform α — straight line confirms model validity

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Activation energy — dual Arrhenius

Eₐ,s and Eₐ,D from segmented series

Click "Calculate Arrhenius" to compute activation energies from all biphasic series in the current project.

Requires at least 3 series with detected breakpoints.

Process optimiser

Dual-mechanism aware — sliders produce real predictions in both modes

Reference: Arrhenius analysis Single series:

Parameter controls

Use dual-mechanism model ON

Sensitivity analysis

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Predicted Li recovery (α)

Predicted α vs time

Stage 1 Stage 2 | Breakpoint

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Acid consumed (mol/mL)

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Est. acid cost (€/kg Li)

Mechanism detail

Select reference and run prediction

PDF report export

Publication-ready summary — structured like journal supplementary data

Dual-mechanism segmentation

Segmentation results

Loading segmentation data...

Help & User manual

Complete guide — from first data entry to commercial optimisation

01

Step-by-step workflow

1

Log in

Your account stores all data, constants, and model fits permanently — reload on next login.

2

Add constants (optional)

Go to Constants DB. Add material-specific values — black mass density, literature Eₐ for comparison, etc.

3

Input experimental data

Go to Data Input. Set conditions (T, [H₂SO₄], S/L, RPM, d₅₀, H₂O₂). Then enter your sampling time points and corresponding α values — add as many rows as your experiment produced. Three entry methods: manual row by row, paste from Excel, or CSV upload. Run at least 3 temperatures for a valid activation energy.

4

Check the mass balance

Mass balance checker fires automatically on Data Input. Shows three steps: (1) moles of Li in your sample, (2) moles of acid available, (3) moles of acid required. If acid excess is below 2×, a plateau on your curve likely means acid depletion — fix before fitting models.

5

Visualise recovery curves

Recovery Curves page: check temperature trend, look for biphasic slopes and plateau behaviour.

6

Run model fitting

Model Fitting page: all 4 models fitted automatically. If dual mechanism detected at t_bp, click "Full segmentation →".

7

Check linearisation plots

Linearisation page: straight line = model valid. Curved line = model wrong even if R² looks good. Avrami uses ln(t) on the x-axis; SCM and GB use t directly — see the subtitle on each plot for the reason.

8

Calculate activation energy

Activation Energy page: Arrhenius plot gives apparent Eₐ. If segmentation detected, Segmentation page gives the two true Eₐ values.

9

Run the optimiser

Optimiser page: adjust all six parameters. When dual-mechanism is detected and toggle is ON, predictions use both stages and the breakpoint shifts as you move sliders. Toggle OFF to compare single-model predictions.

10

Export the report

Report Export page: select sections, choose journal style, generate PDF.

02

Data input — three methods

Manual entry: Type directly into the table. Click "+ Add time point" or press Enter on the last field of a row to create a new row. No limit on rows. "Sort by time ↑" reorders if needed.

Paste / bulk entry: Copy columns from Excel (Ctrl+C) and paste into the text area. Accepts tab, comma, or space separated values. Column order: time_min · α_Li · α_Co · α_Ni · α_Mn · pH · Li_sd. Click "Import →" then review in Manual entry tab.

CSV upload: Upload a CSV file. Required: time_min, alpha_Li. Optional: alpha_Co, alpha_Ni, alpha_Mn, pH, alpha_Li_sd. Sorted by time automatically.

03

Page-by-page reference

Home →

Overview with live ticker, feature cards, quick links. LITIUMLAB title sits above the Li atom, subtitle below it.

Login →

Sign in. All data, constants, and model fits stored server-side.

Dashboard →

Summary of all series, model R², Eₐ, segmentation status.

Constants DB →

System + user constants. Yours (green) persist to account.

Data Input →

Set conditions. Enter α vs time with unlimited rows. Three entry methods. Mass balance checker shows three labelled calculation steps.

Recovery Curves →

All α vs t series with CI bands, error bars. Plateau explanation. Biphasic shapes flagged.

Model Fitting →

Fits SCM, Avrami, GB, n-order. Winner by AIC/RMSE/F-test. Fitted curve, CI, residuals, parity plot.

Linearisation →

Four linearised plots + residuals (observed minus predicted). Avrami needs ln(t) on x-axis; SCM and GB use t directly. Straight line = model valid.

Activation Energy →

Arrhenius plot. Apparent Eₐ + literature comparison. If segmented, Segmentation page has true dual Eₐ.

Optimiser →

Six sliders. Dual-mechanism ON: breakpoint shifts with params, predictions use both stages. Toggle OFF for single-model comparison.

Report Export →

PDF in journal format. Elsevier, ACS, RSC, or Generic.

Segmentation →

Full dual-mechanism analysis: segmented curve, per-segment fits, dual Arrhenius, two Eₐ values.

04

Scientific background — kinetic models

Shrinking Core Model (SCM)

1 − (1 − α)^(1/3) = k_s · t (surface) | 1-2α/3-(1-α)^(2/3) = k_D · t (diffusion)

Particle shrinks as sphere. Rate limited by (a) surface reaction or (b) product layer diffusion. Uses t directly on x-axis because k×t is already linear.

Avrami-Erofeev

[-ln(1 − α)]^(1/n) = k · t → ln[-ln(1-α)] = n·ln(k) + n·ln(t)

Nucleation and growth. n≈1 = 1D, n≈2 = 2D, n≈3 = 3D growth. Uses ln(t) on x-axis because t is inside power n — taking ln of both sides pulls t out as ln(t). Slope = n, intercept = n·ln(k).

Ginstling-Brounshtein

1 − (2/3)α − (1 − α)^(2/3) = k_D · t

SCM diffusion for spherical particles where path length increases nonlinearly. Best applied when α > 0.5. Uses t directly.

n-order reaction

dα/dt = k · (1 − α)^n

General power-law. Flexible but less physically interpretable. Often deteriorates at α > 0.7.

Arrhenius equation

k(T) = A · exp(−Eₐ / R·T)

A = pre-exponential factor, Eₐ = activation energy (kJ/mol), R = 8.314 J/mol/K. Typical Eₐ for H₂SO₄ leaching: 30–60 kJ/mol. If segmented: Eₐ(surface)≈55–65, Eₐ(diffusion)≈25–35 kJ/mol.

05

Frequently asked questions

How many temperature points do I need?

Minimum 3, recommended 4–5. Spread evenly — e.g. 298K, 323K, 348K, 363K.

How many time points per experiment?

Minimum 5, recommended 7–10. Cover early kinetics through to plateau. For detecting segmentation you need points before and after the expected breakpoint.

What is Li s.d.?

Standard deviation of your α(Li) measurement across replicate runs at each time point. Used to draw error bars on all recovery curve plots. Leave blank if you only ran a single experiment — the plots will show data points without error bars, which is correct for a single run.

What is the S/L ratio and units?

S/L = mass of solid (g) / volume of liquid (mL). Example: 10g/500mL = 0.020 g/mL. Valid range: 0.001–2.0 g/mL.

Why does the mass balance checker show three steps now?

The previous version showed it all on one line which was hard to follow. The three steps are now clearly labelled: (1) moles of Li from your sample mass and wt%, (2) moles of acid from concentration and volume, (3) moles of acid required from stoichiometry. The excess ratio is the result at the bottom.

My curve doesn't reach α=1 — what does this mean?

This is a plateau — the practical maximum under those conditions. Check mass balance first. If acid is in large excess, the plateau is caused by unreactive Li phases or particle encapsulation. Reducing d₅₀ is usually the most effective lever.

Why does Avrami use ln(t) but SCM uses t on the x-axis?

Avrami has t inside a power n — rearranging to get a straight line requires taking ln of both sides, which brings t out as ln(t). SCM and GB already have k×t as a simple product with no exponent on t, so t appears directly and no transformation is needed.

06

Tips for commercial scale-up

Particle size is the cheapest lever. Milling energy is cheaper than heating at industrial scale. If Eₐ(diffusion) is low (~28 kJ/mol), reducing d₅₀ gives a bigger gain than raising temperature.

Do not over-acidify. Once α plateaus, more acid does not raise recovery but increases neutralisation cost. 2–4× stoichiometric excess is usually sufficient.

Run replicates and include standard deviations. Enter Li s.d. (last column in data table) and LitiumLab will show error bars on all plots.

Validate your Eₐ against literature. If Eₐ is outside 30–60 kJ/mol, likely causes: carbon coating blocking access, incomplete milling, or incorrect α calculation.

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