Contents

1 Draft / Draught ....................................................................................................3

1.1 Absolute Draught .........................................................................................3

1.2 Relative Draught ..........................................................................................3

1.3 Actual Draught .............................................................................................3

1.4 Maximum Draft ............................................................................................3

1.5 Maximum Draft (Bakhtinov’s Formula) ......................................................4

2 Elongation ...........................................................................................................4

2.1 Elongation Factor .........................................................................................4

2.2 Elongation Percentage ..................................................................................4

3 Spread ..................................................................................................................4

3.1 Absolute Spread ...........................................................................................4

3.2 Percentage Spread ........................................................................................4

3.3 Actual Spread (Bakhtinov’s Formula) .........................................................4

3.4 Actual Spread (Grishkov’s Formula) ...........................................................5

3.5 Actual Spread (SIMAC Formula) ................................................................5

3.6 Maximum Spread (oval – round) .................................................................6

4 Contact Area........................................................................................................9

5 Bite Angle or Contact Angle .............................................................................10

6 Coff of Friction .................................................................................................11

6.1 Neutral Point ..............................................................................................12

6.2 Forward Slip ...............................................................................................13

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7 Working Diameter .............................................................................................13

8 Reduction ..........................................................................................................14

8.1 Relative Reduction .....................................................................................14

8.2 Percentage Reduction .................................................................................14

9 Roll Peripheral Speed........................................................................................14

10 Rolling Load ..................................................................................................15

11 Power and Rolling Toque ..............................................................................18

11.1 Frictional power ......................................................................................20

12 Productivity....................................................................................................21

13 Number of Bars in One Stock........................................................................22

14 Billet Time in Furnace ...................................................................................22

15 Cooling Time of Rebars on The Cooling Bed ...............................................23

16 Aprons Roller Distance .................................................................................23

17 Maximum Number of Bars Can Cold Shear Cut ..........................................23

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 Rolling Mill Equations

1 Draft / Draught

Figure 1 Computation of Draught

1.1 Absolute Draught

∆𝐻 = 𝐻1 − 𝐻2

1.2 Relative Draught

𝐻1 − 𝐻2
∆𝐻% =
𝐻1

1.3 Actual Draught

∆𝐻 = 0.85𝐻1 − 0.79𝐻2

1.4 Maximum Draft

∆𝐻𝑀𝑎𝑥 = 𝜇2 𝑅

𝝁: Coefficient of Friction.

𝑹: Roll Radius.

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1.5 Maximum Draft (Bakhtinov’s Formula)
1
∆𝐻𝑀𝑎𝑥 = 𝐷 (1 − )
√1 + 𝜇2

𝝁: Coefficient of Friction.

𝑫: Roll Diameter.

2 Elongation
2.1 Elongation Factor
𝐿2
𝐸= (𝐸 𝑖𝑠 𝑎𝑙𝑤𝑎𝑦𝑠 𝑔𝑟𝑒𝑎𝑡𝑒𝑟 𝑡ℎ𝑎𝑛 1)
𝐿1

2.2 Elongation Percentage

𝐿2 − 𝐿1
%𝐸 =
𝐿2

3 Spread
3.1 Absolute Spread
∆𝐵 = 𝐵2 − 𝐵1

3.2 Percentage Spread

𝐵2 − 𝐵1
∆𝐵% = × 100%
𝐵2

3.3 Actual Spread (Bakhtinov’s Formula)

∆𝐻 ∆𝐻
∆𝐵 = 1.15 (𝐿 − )
2𝐻1 2𝜇

𝜟𝑩: Total Spread.

𝜟𝑯: Draft.

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𝜟𝑩: Total Spread.

𝜟𝑯: Draft.

𝑹: Radius of Roll.

𝑯𝟏 : Initial height of metal stock.

𝑳: Length of Contact Area.

𝝁: Coefficient of Friction.

3.4 Actual Spread (Grishkov’s Formula)

∆𝐻 𝐻1
𝛥 𝐵 = Қ × 𝐴 × 𝐶 (𝐿 − ) × 2.3 log
2𝜇 𝐻2

𝝁: Coefficient of Friction.

𝜟𝑯: Absolute Draft.

𝑳: Length of Contact Area.

8√3.5
Қ=
0.5 + 𝑉

𝑽: 𝑅𝑜𝑙𝑙 𝑃𝑒𝑟𝑖𝑝ℎ𝑒𝑟𝑎𝑙 𝑆𝑝𝑒𝑒𝑑 𝑖𝑛 𝑚/𝑠𝑒𝑐.

∆𝐻
𝐴= + 0.5
3𝐻1

4(1 − ∆𝐻) 𝐵 𝐵 ∆𝐻
𝐶=( × − 0.15) × 𝑒 1.5 (0.15 − ) +
𝐻 𝐿 𝐿 𝐻1

3.5 Actual Spread (SIMAC Formula)

∆𝐻
∆𝐵 = 0.4 × × √𝑅∆𝐻
𝐻1

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3.6 Maximum Spread (oval – round)

̅𝑖 − 𝐻
√𝑅𝑒𝑓𝑓 (𝐻 ̅𝑜 )
𝐴ℎ
𝑊𝑚𝑎𝑥 = 𝑊𝑖 1 + 𝛾 ×
𝑊𝑖 + 0.5𝐻𝑖 𝐴𝑜
[ ]
𝐴𝑜 − 𝐴𝑠 − 𝐴ℎ
̅𝑜 =
𝐻
𝐵𝑐

𝐴𝑜 − 𝐴𝑠
̅𝑖 =
𝐻
𝐵𝑐

Where:

𝑾𝒎𝒂𝒙 : Maximum spread.

̅ 𝒐 : Effective height of outgoing work piece.

𝑯

̅ 𝒊 : Effective height of incoming work piece.

𝑯

𝑯𝒊 : Effective height of incoming work piece.

𝑹𝒆𝒇𝒇 : Effective roll radius.

𝑾𝒊 : Width of incoming work piece.

𝑩𝒄 : Effective Width of work piece.

𝑨𝒐 : Area of Incoming workpiece.

𝑨𝒉 : is the part of the cross-section area of the metal before rolling that is
outside the groove.

𝑨𝒔 : Area extra from mean width.

𝜸: Constant depends on the shape of the pass.

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 Figure 3 Spread from round to oval. Figure 2 Spread from oval to round.

Pass 𝜸
Square – Oval 0.92
Round - Oval 0.97
Square - Diamond 0.83
Oval - Square 1.06
Oval - Round 0.83
Diamond - Square 0.83
Diamond - Diamond 0.95
Oval - Oval 0.95
Table 1 The value of 𝜸 for different passes

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Calculation of Arc Area (As):

Figure 4 Arc Area Calculation

𝐻 𝑊2
𝑅𝑎𝑑𝑖𝑢𝑠 = +
2 8𝐻

Where:

W: is the length of the chord defining the base of the arc.

H: is the height measured at the midpoint of the arc's base.

Figure 5 Circle Equations

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4 Contact Area

Figure 6 Computation of Contact Area

∆𝐻2
𝐿𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝐶𝑜𝑛𝑡𝑎𝑐𝑡 𝐴𝑟𝑒𝑎 (𝐿)(𝑚) = √𝑅∆𝐻 − ≈ √𝑅∆𝐻
4

𝐿𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝐶𝑜𝑛𝑡𝑎𝑐𝑡 𝐴𝑟𝑒𝑎 (𝐿)(𝑚) = 𝑅 × sin 𝛼

𝐵1 + 𝐵2
𝑀𝑒𝑎𝑛 𝑊𝑖𝑑𝑡ℎ (𝑏)(𝑚) =
2
2
𝑀𝑒𝑎𝑛 𝑊𝑖𝑑𝑡ℎ (𝑏)(𝑚) = 𝐵1 + (𝐵2 − 𝐵1 )
3

𝐷𝑒𝑓𝑜𝑟𝑚𝑎𝑡𝑖𝑜𝑛 𝑍𝑜𝑛𝑒 𝑜𝑟 𝐶𝑜𝑛𝑡𝑎𝑐𝑡 𝐴𝑟𝑒𝑎 (𝐹𝑑 )(𝑚2 ) = 𝐿 × 𝑏

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5 Bite Angle or Contact Angle

Figure 7 Bite Angle

∆𝐻
𝐵𝑖𝑡𝑒 𝐴𝑛𝑔𝑙𝑒 (𝛼)(𝐷𝑒𝑔𝑟𝑒𝑒) = cos −1 (1 − ) (1)
𝐷

∆𝐻
𝐵𝑖𝑡𝑒 𝐴𝑛𝑔𝑙𝑒 (𝛼)(𝑅𝑎𝑑) = √ (2)
𝑅

180
𝐷𝑒𝑔𝑟𝑒𝑒 = 𝑅𝑎𝑑 ×
𝜋

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6 Coff of Friction
tan 𝛽 = 𝜇 = 𝐾1 𝐾2 𝐾3 (1.05 − 0.0005 × 𝑡) (1)

𝒕: Workpiece Temperature.

𝑲𝟏 : Co-efficient for condition of surface of rolls.

= 1.0 for steel rolls.

= 0.8 for cast iron rolls.

𝑲𝟐 : Dynamic Friction.

Figure 8 Computation of Value of 𝑲𝟐

𝑲𝟑 : Co-efficient depending upon the composition of rolled metal.

= 1.0 for carbon steel.

= 1.24 – 1.63 for alloy steel.

For steel rolls:

𝜇 = 1.05 − 0.0005𝑡 − 0.056𝑣 (2)

For cast iron rolls:

𝜇 = 0.92 − 0.0005𝑡 − 0.0056𝑣 (3)

𝒗: Roll Peripheral Speed in m/sec. 𝒕: Workpiece Temperature.

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6.1 Neutral Point
√∆𝐻 1 ∆𝐻
𝛿= − ×
2𝐷 𝜇 2𝐷

𝜹: no slip angle in radian.

∆𝑯: Draft.

𝝁: co-efficient of friction.

Figure 9 Forward Slip and Neutral Point Position

ℎ𝑛 = 𝐷𝑤 (1 − cos 𝛿) + ℎ1

𝑉𝑛 = 𝑉𝑟 cos 𝛿

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6.2 Forward Slip
𝑉1 − 𝑉0 ℎ𝑛 cos 𝛿
𝐹𝑜𝑟𝑤𝑎𝑟𝑑 𝑆𝑙𝑖𝑝 (𝑠𝑓 ) = = −1
𝑉1 ℎ1

𝑏 𝐷𝑤 𝑏 𝐷𝑤
𝐹𝑜𝑟𝑤𝑎𝑟𝑑 𝑆𝑙𝑖𝑝 (𝑠𝑓 ) = × (1 + ) cos 𝛿 − × cos 2 𝛿 − 1
𝐵2 ℎ2 𝐵2 ℎ2

Where:

𝜹: no slip angle (radian).

𝑫𝒘 : Working Diameter (mm).

𝒉𝟐 : The height of the outlet stock (mm).

𝒃: Mean Width (mm).

𝑩𝟐 : Exit width of the stock (mm).

7 Working Diameter

Figure 10 Roll Pass

𝑊𝑜𝑟𝑘𝑖𝑛𝑔 𝐷𝑖𝑎𝑚𝑒𝑡𝑒𝑟 (𝐷𝑤) = 𝑅𝑜𝑙𝑙 𝐷𝑖𝑎𝑚𝑒𝑡𝑒𝑟 (𝐷𝑒) − 𝐺𝑟𝑜𝑜𝑣𝑒 𝐹𝑎𝑐𝑡𝑜𝑟

𝐴𝑟𝑒𝑎
𝐺𝑟𝑜𝑜𝑣𝑒 𝐹𝑎𝑐𝑡𝑜𝑟 = − 𝐺𝑎𝑝
𝑊𝑖𝑑𝑡ℎ

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8 Reduction
8.1 Relative Reduction
𝐴𝑖𝑛
𝑅=
𝐴𝑜𝑢𝑡

𝑨𝒊𝒏 : Cross-sectional area of inlet stock.

𝑨𝒐𝒖𝒕 : Cross-sectional area of outlet stock.

8.2 Percentage Reduction

𝐴𝑖𝑛 − 𝐴𝑜𝑢𝑡
𝑅 (%) = ( ) × 100%
𝐴𝑖𝑛

9 Roll Peripheral Speed

𝜋𝐷𝑤 𝑛
𝑉𝑟 (𝑚/𝑠) =
60

𝑫𝒘 : Working Diameter.

𝒏: Rolling Speed (RPM) = 𝑁/𝑖

𝒊: Gear Box Ratio.

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10 Rolling Load

Figure 11 Calculation of rolling load.

𝑃 = 𝐹𝑑 × 𝑃𝑚

𝑷: Total rolling load (Kg. f).

𝑭𝒅 : Projected contact area (mm2).

𝑷𝒎 : Mean unit pressure of metal on rolls (Kg/mm2).

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 𝑃𝑚 = (κ + 𝜂𝜆)(1 + 𝑚)

𝛋: Unit resistance to static compression (yield point in static compression at rolling

temperature) in kg/mm².

𝜼: Viscosity of metal being rolled kg. s/mm².

𝝀: Mean rate of deformation 1/sec.

𝒎: Co-efficient accounting for increase in resistance to deformation of metal due

to friction between stock and the rolls.

κ = (14 − 0.01 × t)(1.4 + %C + %Mn + 0.3 %Cr)

𝐭: Rolling Temperature

𝜂 = 0.01(14 − 0.01 × 𝑡) × 𝐶𝑣

𝑪𝒗 is a co-efficient, whose value depends upon peripheral speed of rolls.

Figure 12 Cv and peripheral velocity of the roll.

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 2 × 𝑉 × √∆𝐻⁄𝑅
𝜆=
ℎ0 + ℎ1

∆𝑯: Draft.

𝒉𝟎 : The height of inlet stock.

𝒉𝟏 : The height of the outlet stock.

𝜋𝐷𝑤 𝑛
V: Peripheral Roll Speed. = 𝑉 = .
60

n: Rolling Speed (RPM) = 𝑁/𝑖

𝒊: Gear Box Ratio.

1.6 × 𝜇 × 𝐿 − 1.2 × ∆𝐻
𝑚=
ℎ0 + ℎ1

𝝁: Coefficient of friction.

𝑳: Length of Contact Area.

Figure 13 Equivalent Rectangles

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 𝐴0
Equivalent entry height (ℎ0𝑒𝑞 ) =
𝑊0𝑚𝑎𝑥

𝐴1
Equivalent exit height (ℎ1𝑒𝑞 ) =
𝑊1𝑚𝑎𝑥

Equivalent draught (∆𝐻𝑒𝑞 ) = ℎ0𝑒𝑞 − ℎ1𝑒𝑞

Equivalent spread (∆𝑊𝑒𝑞 ) = 𝑤1𝑚𝑎𝑥 − 𝑤0𝑚𝑎𝑥

11 Power and Rolling Toque

𝑁𝑟 + 𝑁𝑓
𝑁𝑚 =
𝜂𝑑

𝑵𝒎 : Required power of the drive motor on a non-reversible motor.

𝑵𝒓 : The power required for rolling process (effective power required to deform the
metal with rolls).

𝑵𝒇 : Frictional power is power loss due to frictional loss in roll bearings.

𝜼𝒅 : Efficiency of roll drive.

𝜂𝑑 = 𝑛1 × 𝑛2 × 𝑛3

𝒏𝟏 : Co-efficient for frictional losses in spindles = 0.95 – 0.97.

𝒏𝟐 : Co-efficient for frictional losses in reducing gears and coupling = 0.93 – 0.96.

𝒏𝟑 : Co-efficient for frictional losses in pinion stand 0.93–0.95.

𝑁𝑟 = 𝑀𝑟 × 𝜔

𝑴𝒓 : Rolling Torque.

𝝎: Angular Rolling Speed (Rad/s).

18 | A d e l E z z
 𝑉
𝜔=
𝑅

𝑽: Roll Peripheral Speed.

𝑹: Roll radius.

Figure 14 Rolling Torque Calculation

𝑀𝑟 = 𝑀1 + 𝑀2

𝑴𝒓 : Rolling torque required to drive rolls (without bearing loss).

𝑴𝟏 and 𝑴𝟐 are rolling torque are required for the top and bottom rolls respectively.
If both roll diameters are equal, then, 𝑴𝟏 = 𝑴𝟐

𝑀𝑟 = 2 × 𝑎 × 𝑃

𝑷: Rolling load.

𝒂: The lever arm of the resultant total load applied to the arc of contact.

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 𝑎 =𝛼×𝑅×𝜓

𝜶: Angle of bite (Radian).

𝑹: Roll radius.

𝝍: Torque arm co-efficient. which is equal to the ratio between the length of the
torque arm and that of contact area. On the basis of the experience based on the
experimental data, the value of ψ is taken as: For hot rolling.

ψ = (0.45 to 0.50)

11.1 Frictional power

𝑁𝑓 = 𝑀𝑓 × 𝜔

𝑵𝒇 : Frictional power is power loss due to frictional loss in roll bearings.

𝑀𝑓 = 𝑓 × 𝑑𝑛𝑒𝑐𝑘 × 𝑃

𝒇: Frictional co-efficient for Roll neck bearing.

= 0.003 to 0.005 for anti-friction roller bearing.

= 0.005 to 0.01 for textolite bearing.

𝒅𝒏𝒆𝒄𝒌 : Diameter of roll neck.

𝑷: Rolling load.

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12 Productivity
Example on determination of tonnage rolled per hour:

Given:

Billet Size = 130 × 130 × 4000 mm3 , Billet Weight = 520 Kg

Finished Product Size = 16 mm2, Speed at the last stand = 12 m/s

Inter Billet Time (Time between charging out billets from furnace) = 5 second

Solution:

𝑚 520 𝑘𝑔
𝐷𝑒𝑛𝑠𝑖𝑡𝑦 (𝜌) = = 3
= 7692.3 𝐾𝑔/𝑚3
𝑉 0.13 × 0.13 × 4 𝑚
𝐾𝑔 𝜋
𝑀𝑎𝑠𝑠 𝐹𝑙𝑜𝑤 𝑅𝑎𝑡𝑒 (𝑚̇) = 𝜌 × 𝐴 × 𝑉 = 7692.3 3
× ( (0.0162 )𝑚2 ) × 12 𝑚/𝑠
𝑚 4
= 18.56 𝐾𝑔/𝑠 = 66.816 𝑡𝑜𝑛/ℎ

𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝐵𝑖𝑙𝑙𝑒𝑡𝑠 𝑅𝑜𝑙𝑙𝑒𝑑 𝑝𝑒𝑟 𝐻𝑜𝑢𝑟 (𝑤𝑖𝑡ℎ𝑜𝑢𝑡 𝑖𝑛𝑡𝑒𝑟 𝑏𝑖𝑙𝑙𝑒𝑡 𝑡𝑖𝑚𝑒)

66.816 𝑡𝑜𝑛⁄ℎ
= = 128.5 𝐵𝑖𝑙𝑙𝑒𝑡/ℎ
0.52 𝑡𝑜𝑛
3600
𝑇𝑖𝑚𝑒 𝑓𝑜𝑟 𝑅𝑜𝑙𝑙𝑖𝑛𝑔 𝑂𝑛𝑒 𝐵𝑖𝑙𝑙𝑒𝑡 = = 28.02 𝑆𝑒𝑐𝑜𝑛𝑑𝑠
128.5

𝑇𝑜𝑡𝑎𝑙 𝑇𝑖𝑚𝑒 𝑓𝑜𝑟 𝐶ℎ𝑎𝑟𝑔𝑖𝑛𝑔 𝑂𝑛𝑒 𝐵𝑖𝑙𝑙𝑒𝑡 = 5 sec + 28.02 sec = 33.02 𝑠𝑒𝑐𝑜𝑛𝑑𝑠

3600
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝐵𝑖𝑙𝑙𝑒𝑡𝑠 𝐵𝑒𝑖𝑛𝑔 𝑅𝑜𝑙𝑙𝑒𝑑 𝑝𝑒𝑟 𝐻𝑜𝑢𝑟 = = 109 𝐵𝑖𝑙𝑙𝑒𝑡𝑠/ℎ
33.02

𝑃𝑟𝑜𝑑𝑢𝑐𝑡𝑖𝑣𝑖𝑡𝑦 = 109 × 0.52 = 𝟓𝟔. 𝟕 𝒕𝒐𝒏/𝒉

21 | A d e l E z z
13 Number of Bars in One Stock
The following formula is used to calculate the nominal weight of the bar:

𝐷𝑖𝑎𝑚𝑒𝑡𝑒𝑟 2
𝑁𝑜𝑚𝑖𝑛𝑎𝑙 𝑊𝑒𝑖𝑔ℎ𝑡 =
162.163

𝑆𝑡𝑜𝑐𝑘 𝑊𝑒𝑖𝑔ℎ𝑡
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝐵𝑎𝑟𝑠 𝑖𝑛 𝑆𝑡𝑜𝑐𝑘 (𝑛) =
𝐵𝑎𝑟 𝑊𝑒𝑖𝑔ℎ𝑡

Example:

162
𝑁𝑜𝑚𝑖𝑛𝑎𝑙 𝑊𝑒𝑖𝑔ℎ𝑡 𝑓𝑜𝑟 𝑃𝑟𝑜𝑑𝑢𝑐𝑡 𝑆𝑖𝑧𝑒 16 𝑚𝑚 = = 1.579 𝐾𝑔/𝑚
162.163

2 × 103
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝐵𝑎𝑟𝑠 𝑖𝑛 𝑆𝑡𝑜𝑐𝑘 (𝑛) = = 106 𝑏𝑎𝑟
1.579 × 12

14 Billet Time in Furnace

𝐵𝑖𝑙𝑙𝑒𝑡 𝑇𝑖𝑚𝑒 𝑖𝑛 𝐹𝑢𝑟𝑛𝑎𝑐𝑒 (𝑇) (ℎ𝑜𝑢𝑟𝑠) = 𝛼 𝑥 𝐾 𝑥 𝐷 𝑥 √𝐷

Where:
− T: Time Duration.
− 𝑫: Billet Dimensions.
− 𝑲: Constant depends on carbon percentage in billet. K = 12.5 for low and
intermediate carbon steel and K = 20:25 for high carbon steel.
− 𝜶: Constant depends on the shape of billets in the reheating furnace.

Figure 15 Calculation of α

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15 Cooling Time of Rebars on The Cooling Bed
𝑙𝐶𝐵 × 𝑛
𝑇=
𝑉𝑓 × 60

𝑻: Cooling Time (Min).

𝒍𝑪𝑩 : Cooling Bed Useful Length (m).

𝒏: Cooling Bed Teeth Numbers of The Rack.

𝑽𝒇 : Speed at The Last Stand (m/s).

16 Aprons Roller Distance

𝑉𝑓 2
𝐴𝑝𝑟𝑜𝑛𝑠 𝑅𝑜𝑙𝑙𝑒𝑟 𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒 (𝑁𝑜 𝑜𝑓 𝑐𝑙𝑜𝑠𝑒𝑑 𝑟𝑜𝑙𝑙𝑒𝑟 𝑏𝑜𝑥𝑒𝑠) (𝑚) =
8

𝑽𝒇 : Speed at The Last Stand (m/s).

17 Maximum Number of Bars Can Cold Shear Cut

𝐹
𝑁=
0.5181 × 𝑅 × 𝐷2

𝑵: Maximum Number of Bars Can Cold Shear Cut.

𝑭: Cutting force of the flying shear (tons).

𝑹: Shear Strength (Kg. f/mm2).

𝑫: Bar Diameter (mm).

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