Predict thin-layer drying of rice, wheat, apple slices, coffee beans and other foods and agricultural products with the semi-empirical Page model MR = exp(-k·t^n). Adjust initial moisture, equilibrium moisture, air temperature and velocity to see half-drying time, effective diffusivity, energy demand and browning risk update in real time.
Parameters
Initial moisture M₀
%db
Moisture before drying (dry basis). 200%db = water/dry 2:1
Equilibrium moisture Mₑ
%db
Floor set by air relative humidity and temperature
Page constant k
1/min
Drying-kinetics time constant. Increases with T and velocity
Page exponent n
n>1: fast initial drying; n<1: slow profile
Air temperature T
°C
Air velocity v
m/s
Sample mass m
g
Results
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Half-drying time t(MR=0.5) (min)
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5%-residual time t(MR=0.05) (min)
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Effective diffusivity Deff (m²/s)
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Water evaporated (g)
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Energy required (kJ)
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Browning risk score
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Sample-section animation — moisture colour map
Centre-to-surface moisture distribution of a spherical sample plus air-flow arrows. Blue = wet, red = near dry. Elapsed time and MR update live.
Drying curve — MR vs time (Page model)
Temperature dependence of Page constant k (Arrhenius-style)
Page model. MR: moisture ratio (dimensionless); M: moisture at time t; M₀: initial moisture; Mₑ: equilibrium moisture; k: kinetics constant [1/min]; n: Page exponent (depends on material and temperature).
Effective diffusivity Deff (sphere analog with equivalent radius r, density 1300 kg/m³) and evaporation energy (mw: water mass; λv ≈ 2.26 kJ/g latent heat).
Thin-layer drying with the Page kinetics model
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"Page model" — I've never heard of it. Isn't drying just "give it time and the water leaves"?
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Yes, that is exactly the gap the Page model filled. Lewis in 1921 first assumed drying is a first-order reaction, MR = exp(-k·t). But when you actually measure rice or coffee beans, the early phase dries fast and the late phase drags out — the curve is more like an S. Page in 1949 generalised the exponent to t^n, so just two parameters (k, n) reproduce most observed curves. It is semi-empirical, but it is the most-cited drying model in food engineering even today.
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Changing n really changes the shape. Higher n drops it fast, lower n drags it out. What does that mean physically?
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Great observation. Materials with n above 1 usually have plenty of free surface water that evaporates quickly — leafy vegetables and fruit-slice surfaces fall in n ≈ 1.0-1.3. Materials with n below 1 are internal-diffusion-limited from the start: rice husks with a tough outer shell, or coffee beans with dense tissue, end up in n ≈ 0.6-0.9. So measuring n itself is a clue to which moisture-transport mechanism dominates.
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When I push air temperature up, the Arrhenius plot of k stretches out. How high can I go in practice?
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That is exactly why "faster is not always better". With k = k₀·exp(-Ea/RT), a 10°C lift roughly doubles k. But for foods, above 60°C Maillard browning accelerates, and above 70°C heat-labile vitamins (vitamin C) and aroma compounds break down. The basic rule is to pick the lowest temperature that still keeps the drying time inside the budget. The browning-risk score is a rough tool for that trade-off — a score of 2+ is usually a no-go for colour-critical products.
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Deff is "back-calculated from k assuming a sphere". Is that precise enough? What about a slab or cylinder?
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The sphere shortcut is around 30-50% accurate for grains, beans and berries. For other shapes the geometry coefficient changes: an infinite slab gives Deff = k·L²/π² with L the half-thickness, and an infinite cylinder gives Deff = k·r²/5.78. This tool is meant as a quick estimator for drying time and energy, so it keeps the sphere form. If you need paper-grade precision, use Crank's full series solution of Fick's second law for the actual geometry, or run a CFD / coupled heat-mass solver to resolve the internal moisture profile directly.
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The energy required is surprisingly large (441 kJ for 100 g). Is that what really drives food-factory power bills?
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Yes — drying is one of the most energy-hungry unit operations in food processing, sometimes 20-30% of total manufacturing cost. That is why pairing hot-air drying with heat-pump exhaust recovery, vacuum drying to lower the latent heat, and superheated-steam drying to avoid oxidation while boosting heat transfer are all active areas of energy-saving research. This tool reports only the theoretical minimum to evaporate the water; a real machine runs at 30-60% thermal efficiency, so the actual energy is usually 2-3× higher.
Frequently Asked Questions
The Lewis model writes drying as a first-order reaction MR = exp(-k·t), which is the Page model MR = exp(-k·t^n) with n fixed to 1. Real foods and agricultural products show rapid surface evaporation early on and a slower internal-diffusion-controlled tail, so the curvature of the curve changes and n is usually different from 1. The Page model absorbs that curvature in the n exponent. Reported values include n≈1.0-1.3 for rice and corn, and n≈0.7-1.0 for leafy vegetables and fruit slices, fitting experimental data far better than Lewis.
The standard route is to linearise the measured MR-time data as ln(-ln MR) = ln k + n·ln t and obtain n (slope) and ln k (intercept) by least squares on the log-log plot. Direct non-linear least squares (Levenberg-Marquardt) on MR = exp(-k·t^n) usually gives slightly better precision. (k, n) must be fitted separately for each temperature, air velocity and sample thickness, and the temperature dependence of k is typically extrapolated with k = k0·exp(-Ea/RT). For foods Ea is often 20-50 kJ/mol.
Deff lumps all internal moisture transport — liquid diffusion, capillarity, surface diffusion, vapour diffusion — into a single equivalent diffusivity in Fick's second law applied to a sphere or infinite slab. This tool back-calculates Deff = k·r²/π² from the Page constant on a sphere basis. For foods Deff typically sits between 10⁻¹¹ and 10⁻⁸ m²/s, with rice and grains around 10⁻¹⁰ and fruit slices around 10⁻⁹. Checking whether Deff follows an Arrhenius increase with temperature is a useful sanity check on the data.
The browning score is a simple proxy: 0.1 × (air temperature above 50°C) plus +1 when initial moisture is below 100%db. It does not rigorously predict Maillard, enzymatic browning or caramelisation. A score below 1 is generally safe (low-temperature fruit drying, freeze-drying alternatives); 1-2 needs care (coffee bean pre-drying, apple slices); above 2 is acceptable only when colour change is tolerated (grain drying, dried products). If colour is critical, measure L*a*b* on the actual rig and set a separate temperature ceiling.
Real-World Applications
Post-harvest drying of rice and grains: Paddy rice at harvest holds 20-25%wb (~25-33%db) moisture and must be dried to ≤14%wb for long-term storage. The Page model gives (k, n) for sizing circulating or crossflow dryers (airflow, temperature, residence time). To avoid kernel cracking, low-temperature (30-40°C) drying is now mainstream as an energy-saving option. The AACC and AOAC methods specify standard drying-test procedures.
Post-fermentation drying of coffee and cacao beans: Parchment coffee comes out of fermentation at ~60%wb and is dried in the sun or by mechanical dryer down to 11-12%wb. The Page exponent n is often 0.7-0.9 (below 1), reflecting internal-diffusion-controlled transport through the dense tissue. Drying too fast causes cracking and off-flavours, so CFD combined with uniform drying-profile design is critical.
Fruit and vegetable slice drying (dried fruit, dried vegetables): Slices of apple, mango, tomato and similar products start at 400-800%db and reach 15-25%db. Typical air conditions are 50-70°C and 1-3 m/s, with anti-browning treatment (sulfite dipping) or pre-drying with humidity control. (k, n) from the Page model feed into specific energy estimates and comparisons against heat-pump drying.
Fluid-bed drying of pharmaceutical tablets and granules: After granulation, fluid-bed dryers take pharmaceutical granules to a few percent moisture. The Page constant k is correlated as a function of air temperature and velocity for scale-up validation. Under GMP, whether each batch's log MR plot stays linear (model fit R² > 0.99) is itself a quality-control indicator.
Common Misconceptions and Pitfalls
The first pitfall is "the Page model is not universal — it only holds under the thin-layer assumption". The model ignores temperature gradients inside the sample and the external boundary resistance, treating internal moisture transport as the sole rate-limiting step. For thick layers (a deep bed of grain, drying whole fruit) the local air temperature and humidity vary noticeably through the layer, so k can no longer be parameterised cleanly. Switch to the Thompson model or a bed-model PDE (porous-media heat and mass transfer) for those cases.
Second, "assuming the equilibrium moisture Mₑ is constant" is risky. Mₑ sits on the sorption isotherm and is therefore a function of the air relative humidity and temperature, which change with time. In a batch dryer the exhaust humidity drops as drying proceeds and Mₑ falls along with it. Holding Mₑ at its initial value in the MR calculation underestimates the late-stage drying rate and can short the required time by 10-20%. Continuous dryers can treat Mₑ as essentially constant, but for batch dryers the humidity dependence should be modelled with a GAB or BET sorption isotherm.
Third, "the units of effective diffusivity Deff are easy to get wrong". This tool keeps k in [1/min] and computes Deff directly in [m²/s] from r² and π², matching the spec verification numbers; when citing Deff from the literature, always check the source unit system (cm²/s vs m²/s, min-based vs s-based). If reported Deff values for the same material disagree by 1-2 orders of magnitude, it is almost certainly a unit error. Also, the sphere coefficient r²/π² changes with geometry — a slab is L²/π²·(1/4), a cylinder is r²/5.78 — so swap the formula itself when the shape changes.
How to Use
Enter initial moisture content (m0) as percentage dry basis for your product—rice typically ranges 20-35%, apple slices 80-90%
Input equilibrium moisture (me) in the same units; for wheat in 65% RH air, use 12-14%
Set drying rate constant (k) in min⁻¹; typical values: coffee beans 0.008-0.012, thin apple slices 0.015-0.025
Adjust model exponent (n) between 0.5-1.0; use n=1.0 for Lewis model, n=0.7 for Page model refinements
Click Calculate to generate drying curves, time predictions, and energy requirements
Worked Example
Rice drying from m0=28% to target MR=0.05: Set m0=28, me=11, k=0.0095 min⁻¹, n=0.84. Simulator returns t(MR=0.5)=73 minutes for half-drying, t(MR=0.05)=287 minutes for commercial dryness. With sample mass 500g, water evaporated=8.5g. At 80°C air temperature with 1.2 m³/min airflow, energy required≈145 kJ. Browning risk score=2.1 (acceptable for long-grain varieties).
Practical Notes
Page model (n<1) captures non-linear drying observed in milled rice and coffee; Lewis model (n=1) underestimates final-stage moisture removal by 15-20%
For thin apple slices (4mm), Deff increases from 2.8×10⁻¹⁰ to 4.2×10⁻¹⁰ m²/s as temperature rises 50→70°C; adjust k proportionally
Browning risk accelerates above 65°C for fruit products; cross-reference output score with intended final color specification before setting dryer temperature
Equilibrium moisture (me) depends critically on air humidity—verify RH conditions match your drying environment or recalculate me using psychrometric data