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TIP 0502-17 ISSUED – 1999 CORRECTION – 2002 WITHDRAWN – 2005 REVISED AND REINSTATED – 2008 ©2008 TAPPI The information and data contained in this document were prepared by a technical committee of the Association. The committee and the Association assume no liability or responsibility in connection with the use of such information or data, including but not limited to any liability under patent, copyright, or trade secret laws. The user is responsible for determining that this document is the most recent edition published.

Papermaker’s formulas Scope This is a set of equations that can be used by paper mill superintendents and engineers during their day-to-day operation of the paper machine. Also included are general guidelines for the acceptable ranges of some of the variables being calculated. This is expected to be a dynamic list and the Papermakers Committee would welcome any additions or corrections that will make the list more useful. Safety precautions Anyone working around paper machines needs to be well trained in the hazard associated with operating machinery. The use of these equations will not cause hazard conditions but collection of data to make some of these calculations will present situations where expertise in the safety requirements of operating paper machines is absolutely necessary. Index to formulas 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.

Tank sizing and capacity Hydraulic pump power Pipeline and channel velocity Weir water flows Theoretical head (approximate headbox pressure required to achieve target jet speed) Approximate spouting velocity Headbox flow rate per unit width (slice method) Approximate headbox slice flow rate per unit width (consistency method) Tissue headbox flow rate per unit width Headbox free jet length Flow/tons/consistency relationship Retention Approximate stock thickness on forming fabric Fourdrinier forming length guidelines Formation – blade pulse frequency Fourdrinier shake number Dandy roll rotational speed Gas laws (commonly used in vacuum system applications) Tension power Drag load – conventional

TIP Category: Data and Calculations TAPPI

TIP 0502-17 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43.

Papermaker’s formulas / 2

Component drag load (wet end) Approximation for vacuum component line load when taking nip impressions Approximate method for determining proper change in total crown of two rolls from nip impression width Deflection of a roll over face (normally used for crown calculations) Press impulse Paper web draw Drying rate for uncoated paper Drying rate for coated paper Rimming speed (5-ft and 6-ft dryers) Dryer felt tension (hanging weight tensioners) Tissue crepe Instantaneous production rate (off reel) Lineal paper on roll Paper caliper Basis weight conversions Roll rotational speed Natural frequency of single degree of freedom system Critical speed of calender roll Approximate critical speed of a roll Inertia (WR2) of a roll Torque Power Common conversion factors

Formulas 1.

Tank sizing and capacity English Units

SI Units

lb × Volume 3 Tons = ft 2000 lb/ton

=

t = Volume × %B.D. / 100 Volume = t × 100 / %B.D.

%B.D. × Volume 1.6 × 2000

Volume = 3200 × tons / %B.D. US Gallons = Volume / 7.4805 where: = Weight of dry stock per lb/ft3 volume of slurry Volume = volume of tank (ft3) %B.D. = percent consistency of stock 1 US gallon = 231 in3

where: t = metric tons Volume = volume of tank (m3) %B.D. = percent consistency of stock

3 / Papermaker’s formulas 2.

TIP 0502-17

Hydraulic pump power English Units Hydraulic Pump Power = H × Q / 1714

Power (hp) H = Differential pressure from pump (psi) Q = flow (gal/min)

SI Units Hydraulic Pump Power = H × Q / 60000

Power (kW) H = Differential pressure from pump (kPa) Q = flow (l/min)

In centrifugal pumps or blowers A. Capacity varies directly with speed B. Head varies as the square of speed C. Power varies as the cube of speed

3.

Pipeline and channel velocity V = Q × k 1 / r2 V = Q × k2 /A English Units Where, V = velocity (ft/s) Q = flow (gal/min) k1 = 0.0007092 k2 = 0.321 r = pipe inside radius (ft) A = pipe or channel cross sectional area (in.2)

SI Units Where, V = velocity (m/s) Q = flow (L/s) k1 = 3142 k2 = 0.001 r = pipe inside radius (m) A = pipe or channel cross sectional area (m2)

Screen to headbox acceptable range is 7 to Screen to headbox acceptable range is 2.1 to 14 ft/s. 4.3 m/s. Note: These formulas are for savealls and general pipe flow, since there is no orifice coefficient included.

4.

Weir water flows

TIP 0502-17

Papermaker’s formulas / 4

Rectangular weir with end contractions English Units

SI Units 2 × 2g × L × H 3 / 2 3 H⎞ ⎛ Where C d = 0.622 × ⎜1 − 0.2 × ⎟ L⎠ ⎝ Q = Cd ×

Q = 3.33 × (L – 0.2 × H) × H1.5 Q = Flow (ft3/s) L = length of weir opening (ft) (should be 4-8 times H) H = head on weir (ft) (~6 ft back of weir opening) a = at least 3H (side of chamber to edge of weir opening)

Q = 1.837 × (L – 0.2 × H) × H1.5 Q = Flow (m3/s) L = length of weir opening (m) (should be 4-8 times H) H = head on weir (m) (~2 m back of weir opening) a = at least 3H (side of chamber to edge of weir opening)

Triangular Notch Weir with End Contractions English Units

SI Units

Q = C × (4 / 15) × L × H × 2 × g × H 3

Q = Flow (ft /s) L = width of notch at H distance above apex (ft) H = head of water above apex of notch (ft) C = 0.57 a = should be not less than ¾L (side of chamber to edge of weir opening) g = 32.174 ft/s2

Q = Flow (m3/s) L = width of notch at H distance above apex (m) H = head of water above apex of notch (m) C = 0.57 a = should be not less than ¾L (side of chamber to edge of weir opening) g = 9.81 m/s2

For 90° notch, the formula becomes: Q = 2.4381 × H

5/2

Q = 1.4076 × H

5/2

Q = 1.3466 × H5/2 For 60° notch, the formula becomes: Q = 0.7776 × H5/2

5 / Papermaker’s formulas

5.

TIP 0502-17

Theoretical head (approximate headbox pressure required to achieve target jet speed)

English Units Theoretical Head = (V / 100) 2 / K

Theoretical Head = (V) 2 / 70610

V = spouting velocity (ft/min) K = constant (see table)

Head (m of H2O) V = spouting velocity (m/min)

Units for Head in. of H2O ft. of H2O in. of Hg PSIG

6.

SI Units

K 1.9304 23.165 26.196 53.336

Approximate spouting velocity

English Units V=K h V = spouting velocity (ft/min) h = head (units consistent with table for K) K = constant (see table below)

Head K

in. of H2O 139.2

ft. of H2O 481.5

SI Units V = 265.7 h V = spouting velocity (m/min) h = head (m H2O)

in. of Hg 513.3

PSIG 732.3

TIP 0502-17

7.

Papermaker’s formulas / 6

Headbox flow rate per unit width (slice method)

English Units

SI Units

gal / min/ in. = S.O. × V × 0.052 × C c

V = spouting velocity (ft/min) S.O. = slice opening (in.) Cc = orifice (contraction) coefficient (See table for approximate values) Type Nozzle A B C

8.

L / min/ m = S.O. × V × C c

V = spouting velocity (m/min) S.O. = slice opening (mm) Cc = orifice (contraction) coefficient (See table for approximate values)

Cc 0.95 0.75 0.70 0.60

Approximate headbox slice flow rate per unit width (consistency method)

English Units gal / min/ in. =

(B.D. Ton / 24 hr / in.)(16.76)(1.5 − Tray Consistency) 1.5 × Net Consistency

SI Units L / min/ m =

(B.D.MT / d / m)(70)(1.5 − Tray Consisitency) 1.5 × Net Consistency

Net Consistency = Headbox Consistency - Tray Consistency

9.

Tissue headbox flow rate per unit width

English Units gal / min/ in. = T.O. × V / 19.25 = T.O. × V × 0.052

T.O. = throat opening (in.) V = spouting velocity (ft/min) Note: assumes contraction (orifice) coefficient = 1.0 10. Headbox free jet length

SI Units L / min/ m = T.O. × V

T.O. = throat opening (mm) V = spouting velocity (m/min)

7 / Papermaker’s formulas

TIP 0502-17

x=

υ cosA ⎛ 2 2 ⎜ υ sin A + 2gh - υ sinA ⎞⎟ ⎠ g ⎝

Notes: a) Applies for case of level jet landing surface (fabric). b) Use positive value for angle A with jet downward from horizontal. c) See TIPs 0410-02, 0410-03, and 0410-04 for estimating jet angle, A. English Units

SI Units

υ cosA ⎛ 2 2 ⎜ υ sin A + 19304h - υ sinA ⎞⎟ ⎠ 9652.5 ⎝ υ = initial jet velocity (ft/min) A = jet angle (degrees) g = 32.174 ft/s2 h = height of apron tip to wire (in.) x = jet length, apron to landing (in.) x=

υ cosA ⎛ 2 2 ⎜ υ sin A + 70610h - υ sinA ⎞⎟ ⎠ 35305 ⎝ υ = initial jet velocity (m/min) A = jet angle (degrees) g = 9.807 m/s2 h = height of apron tip to wire (m) x = jet length, apron to landing (m) x=

11. Flow/tons/consistency relationship English Units

SI Units

Ton/d = C × Q / K

t/d = C × Q × 4.1727/ K

Where, C = consistency (%) Q = flow (gal/min) K = a temperature related factor (see below)

T (°F) 100 120 140

Where, C = consistency (%) Q = flow (l/min) K = a temperature related factor (see below) T (°C) 37.8 48.9 60

K 16.76 16.83 16.93

TIP 0502-17

Papermaker’s formulas / 8

12. Retention Retention (%) = (Net Consistency / Headbox Consistency) × 100 Retention (%) = [(Headbox Consistency – Tray Consistency) / Headbox Consistency] × 100

13. Approximate stock thickness on forming fabric English Units

SI Units

BW × 0.1925 C × R × Re am × (J / W ) T = thickness of stock on table (in.) BW = basis weight (lb) Ream = ream size (ft2) C = consistency (%/100) R = retention from that point down the rest of the machine (%/100) J/W = jet/wire ratio = 1.0 except at slice

BW / 10000 C × R × (J / W ) T = thickness of stock on table (cm) BW = basis weight (g/m2) C = consistency (%/100) R = retention from that point down the rest of the machine (%/100) J/W = jet/wire ratio = 1.0 except at slice

T=

T=

Note: Result T for headbox slice is after vena contracta. Example: Determine the overall retention of a machine with slice opening of 0.5 in. (1.27 cm) making 50 g/m2 at 0.6% slurry and jet/wire ratio of 0.95. Assume the headbox jet contraction coefficient is 0.75 yielding final jet thickness after vena contracta of 0.375 in. (0.952 cm). R=

50 / 10000 = 0.921 , or 92.1% 0.0060 × 0.952 × 0.95

14. Fourdrinier forming length guidelines Dwell Time (sec) (headbox slice to Machine Speed that can be Wire Speed or Grade first flatbox or Supported dandy roll) 1.5 Forming Length × 40

View more...
Papermaker’s formulas Scope This is a set of equations that can be used by paper mill superintendents and engineers during their day-to-day operation of the paper machine. Also included are general guidelines for the acceptable ranges of some of the variables being calculated. This is expected to be a dynamic list and the Papermakers Committee would welcome any additions or corrections that will make the list more useful. Safety precautions Anyone working around paper machines needs to be well trained in the hazard associated with operating machinery. The use of these equations will not cause hazard conditions but collection of data to make some of these calculations will present situations where expertise in the safety requirements of operating paper machines is absolutely necessary. Index to formulas 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.

Tank sizing and capacity Hydraulic pump power Pipeline and channel velocity Weir water flows Theoretical head (approximate headbox pressure required to achieve target jet speed) Approximate spouting velocity Headbox flow rate per unit width (slice method) Approximate headbox slice flow rate per unit width (consistency method) Tissue headbox flow rate per unit width Headbox free jet length Flow/tons/consistency relationship Retention Approximate stock thickness on forming fabric Fourdrinier forming length guidelines Formation – blade pulse frequency Fourdrinier shake number Dandy roll rotational speed Gas laws (commonly used in vacuum system applications) Tension power Drag load – conventional

TIP Category: Data and Calculations TAPPI

TIP 0502-17 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43.

Papermaker’s formulas / 2

Component drag load (wet end) Approximation for vacuum component line load when taking nip impressions Approximate method for determining proper change in total crown of two rolls from nip impression width Deflection of a roll over face (normally used for crown calculations) Press impulse Paper web draw Drying rate for uncoated paper Drying rate for coated paper Rimming speed (5-ft and 6-ft dryers) Dryer felt tension (hanging weight tensioners) Tissue crepe Instantaneous production rate (off reel) Lineal paper on roll Paper caliper Basis weight conversions Roll rotational speed Natural frequency of single degree of freedom system Critical speed of calender roll Approximate critical speed of a roll Inertia (WR2) of a roll Torque Power Common conversion factors

Formulas 1.

Tank sizing and capacity English Units

SI Units

lb × Volume 3 Tons = ft 2000 lb/ton

=

t = Volume × %B.D. / 100 Volume = t × 100 / %B.D.

%B.D. × Volume 1.6 × 2000

Volume = 3200 × tons / %B.D. US Gallons = Volume / 7.4805 where: = Weight of dry stock per lb/ft3 volume of slurry Volume = volume of tank (ft3) %B.D. = percent consistency of stock 1 US gallon = 231 in3

where: t = metric tons Volume = volume of tank (m3) %B.D. = percent consistency of stock

3 / Papermaker’s formulas 2.

TIP 0502-17

Hydraulic pump power English Units Hydraulic Pump Power = H × Q / 1714

Power (hp) H = Differential pressure from pump (psi) Q = flow (gal/min)

SI Units Hydraulic Pump Power = H × Q / 60000

Power (kW) H = Differential pressure from pump (kPa) Q = flow (l/min)

In centrifugal pumps or blowers A. Capacity varies directly with speed B. Head varies as the square of speed C. Power varies as the cube of speed

3.

Pipeline and channel velocity V = Q × k 1 / r2 V = Q × k2 /A English Units Where, V = velocity (ft/s) Q = flow (gal/min) k1 = 0.0007092 k2 = 0.321 r = pipe inside radius (ft) A = pipe or channel cross sectional area (in.2)

SI Units Where, V = velocity (m/s) Q = flow (L/s) k1 = 3142 k2 = 0.001 r = pipe inside radius (m) A = pipe or channel cross sectional area (m2)

Screen to headbox acceptable range is 7 to Screen to headbox acceptable range is 2.1 to 14 ft/s. 4.3 m/s. Note: These formulas are for savealls and general pipe flow, since there is no orifice coefficient included.

4.

Weir water flows

TIP 0502-17

Papermaker’s formulas / 4

Rectangular weir with end contractions English Units

SI Units 2 × 2g × L × H 3 / 2 3 H⎞ ⎛ Where C d = 0.622 × ⎜1 − 0.2 × ⎟ L⎠ ⎝ Q = Cd ×

Q = 3.33 × (L – 0.2 × H) × H1.5 Q = Flow (ft3/s) L = length of weir opening (ft) (should be 4-8 times H) H = head on weir (ft) (~6 ft back of weir opening) a = at least 3H (side of chamber to edge of weir opening)

Q = 1.837 × (L – 0.2 × H) × H1.5 Q = Flow (m3/s) L = length of weir opening (m) (should be 4-8 times H) H = head on weir (m) (~2 m back of weir opening) a = at least 3H (side of chamber to edge of weir opening)

Triangular Notch Weir with End Contractions English Units

SI Units

Q = C × (4 / 15) × L × H × 2 × g × H 3

Q = Flow (ft /s) L = width of notch at H distance above apex (ft) H = head of water above apex of notch (ft) C = 0.57 a = should be not less than ¾L (side of chamber to edge of weir opening) g = 32.174 ft/s2

Q = Flow (m3/s) L = width of notch at H distance above apex (m) H = head of water above apex of notch (m) C = 0.57 a = should be not less than ¾L (side of chamber to edge of weir opening) g = 9.81 m/s2

For 90° notch, the formula becomes: Q = 2.4381 × H

5/2

Q = 1.4076 × H

5/2

Q = 1.3466 × H5/2 For 60° notch, the formula becomes: Q = 0.7776 × H5/2

5 / Papermaker’s formulas

5.

TIP 0502-17

Theoretical head (approximate headbox pressure required to achieve target jet speed)

English Units Theoretical Head = (V / 100) 2 / K

Theoretical Head = (V) 2 / 70610

V = spouting velocity (ft/min) K = constant (see table)

Head (m of H2O) V = spouting velocity (m/min)

Units for Head in. of H2O ft. of H2O in. of Hg PSIG

6.

SI Units

K 1.9304 23.165 26.196 53.336

Approximate spouting velocity

English Units V=K h V = spouting velocity (ft/min) h = head (units consistent with table for K) K = constant (see table below)

Head K

in. of H2O 139.2

ft. of H2O 481.5

SI Units V = 265.7 h V = spouting velocity (m/min) h = head (m H2O)

in. of Hg 513.3

PSIG 732.3

TIP 0502-17

7.

Papermaker’s formulas / 6

Headbox flow rate per unit width (slice method)

English Units

SI Units

gal / min/ in. = S.O. × V × 0.052 × C c

V = spouting velocity (ft/min) S.O. = slice opening (in.) Cc = orifice (contraction) coefficient (See table for approximate values) Type Nozzle A B C

8.

L / min/ m = S.O. × V × C c

V = spouting velocity (m/min) S.O. = slice opening (mm) Cc = orifice (contraction) coefficient (See table for approximate values)

Cc 0.95 0.75 0.70 0.60

Approximate headbox slice flow rate per unit width (consistency method)

English Units gal / min/ in. =

(B.D. Ton / 24 hr / in.)(16.76)(1.5 − Tray Consistency) 1.5 × Net Consistency

SI Units L / min/ m =

(B.D.MT / d / m)(70)(1.5 − Tray Consisitency) 1.5 × Net Consistency

Net Consistency = Headbox Consistency - Tray Consistency

9.

Tissue headbox flow rate per unit width

English Units gal / min/ in. = T.O. × V / 19.25 = T.O. × V × 0.052

T.O. = throat opening (in.) V = spouting velocity (ft/min) Note: assumes contraction (orifice) coefficient = 1.0 10. Headbox free jet length

SI Units L / min/ m = T.O. × V

T.O. = throat opening (mm) V = spouting velocity (m/min)

7 / Papermaker’s formulas

TIP 0502-17

x=

υ cosA ⎛ 2 2 ⎜ υ sin A + 2gh - υ sinA ⎞⎟ ⎠ g ⎝

Notes: a) Applies for case of level jet landing surface (fabric). b) Use positive value for angle A with jet downward from horizontal. c) See TIPs 0410-02, 0410-03, and 0410-04 for estimating jet angle, A. English Units

SI Units

υ cosA ⎛ 2 2 ⎜ υ sin A + 19304h - υ sinA ⎞⎟ ⎠ 9652.5 ⎝ υ = initial jet velocity (ft/min) A = jet angle (degrees) g = 32.174 ft/s2 h = height of apron tip to wire (in.) x = jet length, apron to landing (in.) x=

υ cosA ⎛ 2 2 ⎜ υ sin A + 70610h - υ sinA ⎞⎟ ⎠ 35305 ⎝ υ = initial jet velocity (m/min) A = jet angle (degrees) g = 9.807 m/s2 h = height of apron tip to wire (m) x = jet length, apron to landing (m) x=

11. Flow/tons/consistency relationship English Units

SI Units

Ton/d = C × Q / K

t/d = C × Q × 4.1727/ K

Where, C = consistency (%) Q = flow (gal/min) K = a temperature related factor (see below)

T (°F) 100 120 140

Where, C = consistency (%) Q = flow (l/min) K = a temperature related factor (see below) T (°C) 37.8 48.9 60

K 16.76 16.83 16.93

TIP 0502-17

Papermaker’s formulas / 8

12. Retention Retention (%) = (Net Consistency / Headbox Consistency) × 100 Retention (%) = [(Headbox Consistency – Tray Consistency) / Headbox Consistency] × 100

13. Approximate stock thickness on forming fabric English Units

SI Units

BW × 0.1925 C × R × Re am × (J / W ) T = thickness of stock on table (in.) BW = basis weight (lb) Ream = ream size (ft2) C = consistency (%/100) R = retention from that point down the rest of the machine (%/100) J/W = jet/wire ratio = 1.0 except at slice

BW / 10000 C × R × (J / W ) T = thickness of stock on table (cm) BW = basis weight (g/m2) C = consistency (%/100) R = retention from that point down the rest of the machine (%/100) J/W = jet/wire ratio = 1.0 except at slice

T=

T=

Note: Result T for headbox slice is after vena contracta. Example: Determine the overall retention of a machine with slice opening of 0.5 in. (1.27 cm) making 50 g/m2 at 0.6% slurry and jet/wire ratio of 0.95. Assume the headbox jet contraction coefficient is 0.75 yielding final jet thickness after vena contracta of 0.375 in. (0.952 cm). R=

50 / 10000 = 0.921 , or 92.1% 0.0060 × 0.952 × 0.95

14. Fourdrinier forming length guidelines Dwell Time (sec) (headbox slice to Machine Speed that can be Wire Speed or Grade first flatbox or Supported dandy roll) 1.5 Forming Length × 40