Beedy Academy · Calculators

Free calculators for pulp and paper

Consistency, oven-dry tonnes, paper machine, cooking, bleaching and liquors: the everyday calculations, each with its formula and practice problems.

Stock and consistency

Consistency of a sample

FormulaC = OD ÷ W × 100 · water = (100 − C) ÷ C

Wet sample mass (W)g

Oven-dry mass after 105 °C (OD)g

Consistency (C)—%

Water per oven-dry tonne—t/ODt

Low consistency: below 6 %; medium: 8 to 15 %; high: above 20 %. Filter the whole sample and dry the pad to constant mass.

Oven-dry tonnes from stock flow

FormulaP = Q × C ÷ 100 × 24

Stock flow (Q)m³/h

Consistency (C)%

Oven-dry production—ODt/d

Oven-dry production—ODt/h

Assumes a density of 1 t/m³, fine at low consistency. Air-dry tonne (90 % dry) = oven-dry tonne ÷ 0.9.

Dilution to a target consistency

FormulaQ₂ = Q₁ × C₁ ÷ C₂ · W = Q₂ − Q₁

Incoming stock flow (Q₁)m³/h

Consistency (C₁)%

Target consistency (C₂)%

Diluted flow (Q₂)—m³/h

Dilution water to add (W)—m³/h

Assumes clear dilution water (0 %). With loaded white water you need more: Q₂ = Q₁ × (C₁ − Cw) ÷ (C₂ − Cw).

Tower retention time

FormulaQ_s = P ÷ 24 ÷ (C ÷ 100) · t = V × N ÷ 100 ÷ Q_s × 60

Tower volume (V)m³

Level (N)%

Consistency (C)%

Pulp production (P)ODt/d

Stock flow (Qs)—m³/h

Retention time (t)—min

Density taken as 1 t/m³. Theoretical plug-flow time: channelling shortens it. In bleaching, reaction time matters as much as temperature.

Specific refining energy

FormulaP_net = P_tot − P₀ · E = P_net ÷ (F ÷ 24)

Total power (Ptot)kW

No-load power (P0)kW

Oven-dry fibre flow (F)ODt/d

Net power (Pnet)—kW

Net specific energy (E)—kWh/ODt

Order of magnitude at low consistency: 50 to 150 kWh/t for softwood kraft. No-load power is measured with the plates backed off, in water.

Paper machine

Basis weight (grammage)

FormulaBW = m ÷ A × 10 000 · lb/3000 ft² = BW × 0.6145

Sample mass (m)g

Sample area (A)cm²

Basis weight (BW)—g/m²

Imperial basis weight—lb/3000 ft²

Weigh at standard conditioning (23 °C, 50 % RH). The 3000 ft² ream is used for newsprint and tissue; board often uses 1000 ft².

Paper machine production

FormulaP_h = v × L × G × 60 ÷ 10⁶ · P_j = P_h × 24 × E ÷ 100

Reel speed (v)m/min

Trim width (L)m

Basis weight (G)g/m²

Overall efficiency (E)%

Gross hourly production (Ph)—t/h

Net daily production (Pd)—t/d

Tonnes at reel basis weight and moisture. Efficiency covers breaks, lost time and winder rejects; 85 to 95 % is common.

First-pass retention

FormulaFPR = (C_c − C_b) ÷ C_c × 100 · FPAR = (A_c − A_b) ÷ A_c × 100

Headbox consistency (Cc)%

White water consistency (Cb)%

Headbox ash (Ac)%

White water ash (Ab)%

Total retention (FPR)—%

Ash retention (FPAR)—%

Ash is expressed as % of the slurry (consistency × ash content). Typical: 50 to 90 % depending on grade; a sudden drop warns of deposits and holes.

Jet-to-wire ratio

Formulav_j = √(2 × P ÷ ρ) × 60 · r = v_j ÷ v_t · Δv = v_j − v_t

Headbox total pressure (P)kPa

Wire speed (vt)m/min

Jet speed (vj)—m/min

Jet-to-wire ratio (r)—

Rush (+) or drag (−) (Δv)—m/min

Theoretical speed (ρ = 1000 kg/m³, no losses): the real jet is a little slower, so it is calibrated with a jet-speed measurement. Usual ratio: 0.97 to 1.03 depending on the formation wanted.

Water evaporated in the dryer section

Formulaw = 100 ÷ s₁ − 100 ÷ s₂ · w_p = w × s₂ ÷ 100 · E = w_p × P

Dryness into the dryers (s₁)%

Dryness at the reel (s₂)%

Production at the reel (P)t/h

Water evaporated per oven-dry tonne (w)—t/ODt

Water evaporated per tonne of paper (wp)—t/t

Evaporation rate (E)—t/h

Every point of dryness gained at the presses saves a lot of steam. Order of magnitude: 1.2 to 1.5 t of steam per tonne of water evaporated.

Cooking, bleaching and liquors

Pulping yield

FormulaY = P ÷ W × 100 · wood = W ÷ P

Oven-dry wood fed (W)ODt/d

Oven-dry pulp produced (P)ODt/d

Yield (Y)—%

Wood per tonne of pulp—t/t

Unbleached kraft: 45 to 55 %; bleached kraft: 40 to 48 %; thermomechanical pulp: 90 to 97 %. Use wood and pulp on the same oven-dry basis.

Chemical charge on oven-dry pulp

Formulakg/t = c × 10 · kg/h = P × c ÷ 100 × 1000 ÷ 24 · L/h = kg/h ÷ (x ÷ 100) ÷ ρ

Pulp production (P)ODt/d

Charge on oven-dry pulp (c)%

Solution strength (x)% m/m

Solution density (ρ)kg/L

Specific charge—kg/ODt

Active chemical—kg/h

Solution flow—L/h

Charge and strength must be on the same basis (active product, or expressed as NaOH, Na₂O, ClO₂…). 1 % on pulp = 10 kg/t.

Kappa number: lignin and ClO₂ charge

Formulalignin ≈ 0.15 × κ · aCl = KF × κ · ClO₂ = aCl ÷ 2.63

Kappa number (κ)

Kappa factor (KF)

Residual lignin (approx.)—%

Active chlorine charge on pulp—%

ClO₂ charge on pulp—%

Lignin ≈ 0.15 × kappa: approximation for chemical pulps (0.13 to 0.165 in the literature). Usual kappa factor in the first D₀ stage: 0.18 to 0.25. 1 kg ClO₂ = 2.63 kg active chlorine.

White liquor: AA, EA, sulfidity

FormulaAA = NaOH + Na₂S · EA = NaOH + ½ Na₂S · TTA = AA + Na₂CO₃ · S = Na₂S ÷ AA × 100

NaOH (as Na₂O)g/L Na₂O

Na₂S (as Na₂O)g/L Na₂O

Na₂CO₃ (as Na₂O)g/L Na₂O

Active alkali (AA)—g/L Na₂O

Effective alkali (EA)—g/L Na₂O

Total titratable alkali (TTA)—g/L Na₂O

Sulfidity (S)—%

Everything is expressed as Na₂O (as NaOH, multiply by 1.29). Typical sulfidity: 25 to 35 %; some compute sulfidity on TTA instead of AA, check your mill's convention.

Causticizing and reduction efficiency

FormulaCE = NaOH ÷ (NaOH + Na₂CO₃) × 100 · RE = Na₂S ÷ (Na₂S + Na₂SO₄) × 100

NaOH (as Na₂O) — white liquorg/L Na₂O

Na₂CO₃ (as Na₂O) — white liquorg/L Na₂O

Na₂S (as Na₂O) — green liquorg/L Na₂O

Na₂SO₄ (as Na₂O) — green liquorg/L Na₂O

Causticizing efficiency—%

Reduction efficiency—%

Typical causticizing: 78 to 85 % (equilibrium limits it near 85 to 90 %); overliming makes mud that is hard to wash. Recovery boiler reduction: 90 to 97 %.

Learning and checking tools. The ranges shown are general guidance: always follow your plant's procedures, permits and set points. Nothing is sent anywhere: the calculations run in your browser.