Tandem rolling mill

To obtain the true roll force between the Work Rolls one must consider the position of the Load Cells that measure the force; are they with the Filler Plates under the bottom BU Roll bearings, or on top of the top BU Roll bearings.

Another thing that must be considered (if it is present) is Work Roll Balance.

The Work Roll balance cylinders act to separate the Work Rolls (no force between them) when the Screw-downs are raised; that is, the force of the balance cylinders F_bal is just greater than the weight of the top roll set, (Wt_bu + Wt_wr).

With the Screws raised, they exert no force on the top BU Roll bearings. So the Load Cells output zero force, but the top rolls press down on the bottom Work Rolls; therefore the weight of the top roll set (Wt_bu + Wt_wr) must be added to the Load Cell reading.

With the Screws raised, the Load Cells have the top roll set pressed against them with a force of F_bal - (Wt_bu + Wt_wr) and this value must be subtracted from the Load Cell reading to obtain the actual force of zero.

Again with the Screws raised, they exert no force on the top BU Roll bearings. However, the Load Cells have the weight of all four Rolls pressing down on them, and so their output is 2•Wt_bu + 2•Wt_wr But the true force on the bottom Work Roll is just the weight of the top roll set, so the weight of the bottom Work Roll and the bottom BU Roll (Wt_bu + Wt_wr) must be subtracted from the Load Cell reading.

The Load Cells have the weight of the bottom roll set pressed onto them by the Work Roll balance cylinders, so they read F_bal + (Wt_bu + Wt_wr). However, this must be subtracted to obtain the true roll force of zero.

This was the condition of the Mill when Graph 2 was obtained. Note that, when the Work Roll balance was turned OFF, then F_bal became zero and so the correcting term was still correct.

The above Roll Weights Wt_bu and Wt_wr are only nominal values; the actual values will vary a little depending on how many times the Rolls have been ground down between campaigns.

Since the Roll Weights are only nominal values, any residual error should be slowly zeroed out whenever the roll balance is ON and the Screws are raised sufficiently.

Graph 4. Rolling Mill Force Change for a Screw-down movement down

If during rolling, it is necessary to move the Screw-downs to correct either the Rolling Force or the exit Strip Thickness, then consider the triangle (shown circled in the Graph on the right) created when the Screw-downs are moved down from the purple to the green.

Note that the Strip becomes thinner and the Rolling Force increases.

Sketch 2: Amplification of Area shown circled in Graph 4

ΔS = Δh + a  with ΔH = 0,  but the slope Q = ΔF / Δh,  and the slope M = ΔF / a

Therefore,  ΔS = ΔF / Q + ΔF / M

Which gives,  ΔS / ΔF = 1 / Q + 1 / M  equation 5

This term is used to ensure the control of the rolling force using the Screws is independent of the metal being rolled.

Using   ΔF = Q . Δh

Gives   ΔS / Δh = 1 + Q / M equation 6

This factor is used to guarantee that the control of the exit thickness by the Screws is independent of the metal being rolled.

Sketch 4. Flat verses Crowned Rolls

Sketch 3. The Forces and Tensions that act on a Strip during Rolling

One could say the steel is compressed by the force of the Work Rolls, equivalent to forging; however, if there are tensions present, then it could be said that the steel is stretched by the tension pulling it through the rotating Work Rolls, as in extruding through a die.

The tension effect should be represented in Graphs 1 and 3 above, by drawing the Steel Compression Curve with the Elastic region reduced by an amount equal to the Tension induced strain.

The relationship of the Rolling Force to the entry and exit Strip Tensions is important in determining the finished Strip Flatness.[3] Too much force produces strip with edge wave (often called "pressure wave"). Too much tension, that is too little force, can cause centre buckle (depending on the Crown of the Rolls).

The tension stress is 30% to 50% of the yield stress for Cold Mills and often higher in Hot Mills (which can result in heavy necking and even strip breaks).

Sketch 5. The Force Distribution through the Roll Bite during Rolling

Note that the force is offset from the Work Roll centres because the strip is thicker at the entry than at the exit; this is one component of the torque that the Main Drives must supply. The other component is the difference in the tension forces. If the exit tension force is much greater than the entry tension force, then the tension torque may be larger than the torque due to the rolling-force and the Main Drives will generate power.

The Neutral Point, or no-slip point[4] is the point within the roll bite where the Work Rolls and the Strip are doing the same speed.

The position of the Neutral Point is influenced by the entry and exit tensions.

Sketch 6: Distribution of oil in Backup Roll White-metal Bearings

Shudder occurs when the Neutral Point is at an edge of the Roll Bite; that is the Work Rolls are alternately grabbing the Strip and letting it slip.

Forward Slip (1+f) is the ratio of the Exit Strip Speed to the Work Rolls peripheral speed. Backward slip (1-b) is the ratio of the Entry Strip Speed to the Work Rolls peripheral speed

The Back-up Roll bearings are usually white-metal bearings which rely on a film of oil between the shaft and the white-metal to reduce the friction; see in Sketch 6.

As the speed increases more oil is dragged into the active region of the bearing and this increases the thickness of the oil film in this region. This pushes the top Work Roll down and the bottom Work Roll up, which reduces the roll gap in the same manner as running the Screws down. To compensate for this, most Screw-down control loops include a feedforward parameter derived from either; an equation of rolling speed, or a value extracted from a lookup table using interpolation.

To ensure an oil film exists even at zero speed; pumps are often used to force oil through very tiny holes into the bearing’s active region; this is referred to as hydrostatics.

Chart Recording 1: Start of a Coil being rolled with a light elongation

Chart Recording 2: Rolling Mill variables at start of SR coil

In Chart 1, the scale for the Screw-downs position (mauve trace) was 0.2 mm per division; this was too coarse. Consequently Chart 2 was created from a similar coil, but with a Screw-down position scale of 0.06 mm per division; that is, from 5.8 mm to 6.4 mm.

In the chart recordings, notice that the force (light green trace) has been held constant by the automatic control, which has raised the Screw-downs (mauve trace) as the speed (red trace) has increased. This increase in Screw position is a measure of the White Metal Bearing speed effect.

Excel plot of the Measured Bearing Speed Effect data and curve matching those points

For a more accurate measurement, the force of each Mill Stand should be measured as it is run through its speed range without strip present.

The values measured from Chart 2 were plotted in an Excel spread-sheet. The equation that was used to match the measured points was 680*POWER((speed/1200),0.225)-285.

Note that the use of oil hydrostatics can hold the oil film nearly constant up to about 20% of full speed; hence no Screw-down movement would be required in that low speed range (this is shown as the red line in the graph of the measured points).

Now recall the Gaugemeter equation in its simplest form: -

h  =  S – S_d  –   (F – F_d) / M

This equation must be modified to include the Backup Roll bearing speed effect S_v especially when rolling Product which has a thickness similar to the speed effect (~400 μm at some Temper Mills).

thus h = S – S_d – S_v  –  (F – F_d) / M  equation 7

Virtually all of the strip thickness variation is the result of the eccentricity and out-of-roundness of the Back-up Rolls from about Stand 3 of the Hot Strip Mill through to the Finished Product.

Hydraulic piston correcting out-of-round BU Roll

The Back-up Roll eccentricity can be up to 100 μm in magnitude per stack. The eccentricity can be measured off-line by plotting the force variation against time with the Mill on creep, no strip present, and the Mill Stand below face.

A modified Fourier analysis was employed by the 5 Stand Cold Mill at Bluescope Steel, Port Kembla from 1986 until that Cold Mill ceased production in 2009. Within each coil, the exit thickness deviation times 10 for every meter of strip was stored in a file. This file was analyzed separately for each frequency/wavelength from 5 m to 60 m in steps of 0.1 m. To improve the accuracy, care was taken to use a full multiple of each wavelength (100*). The calculate amplitudes were plotted against the wavelength, so that the spikes could be compared to the expected wavelengths created by the Backup Rolls of each Stand.

If a Mill Stand is fitted with Hydraulic Pistons in series with, or instead of the electrically driven Mechanical Screws, then it is possible to eliminate the effect of that Stands Back-up Roll eccentricity.[5][6] While rolling, the eccentricity of each Back-up Roll is determined by sampling the roll force and assigning it to the corresponding portion of each Back-up Roll’s rotational position. These recordings are then used to operate the Hydraulic Piston so as to neutralize the eccentricities.

5 Stand Temper Mill (strip goes right to left)

A Rolling Mill does not create nor destroy steel during normal steady state rolling.

That is; the same mass of steel leaves the Mill as entered it.

And so; expressing the entry volume as H . W_n. ℓ, and the exit volume h . W_x . L

But the entry length ℓ = v . t and the exit length L = V t where t is the total rolling time.

Therefore ρ . H . W_n . v . t  =  ρ . h . W_x . V . t

The density ρ is unaffected by the rolling process and can be cancelled out. The width may change, but it does so by an insignificant amount (only a fraction of the strip thickness), and so the change may be ignored when rolling thin (< 1 mm) strip. The Roll Force tends to widen the strip, while the entry and exit tensions (when present) tend to make the strip narrower.

Chart Recording 3 made at the Head-end of a Double Reduced Coil with 25% reduction

So, cancelling the density ρ, the width W, and the time t, gives

h . V  =  H . v  equation 8

This can be used in a Rolling Mill to calculate the exit thickness h that the X-ray gauge will measure when the corresponding portion of the Strip finally reaches the Gauge.

By assuming all of the Cold Mill head-end off-gauge has been completely removed by the previous Continuous Annealing Line; the scheduled entry thickness can be substituted in place of the actual entry thickness, H.[7] Then the Entry Bridle and Exit Bridle speeds can be used as the measurements of entry speed, v and exit speed, V respectively.

The resulting Calculated Thickness Deviation can be seen as the light blue trace in Chart Recording 3. Notice that the thickness control was working at thread speed (red trace). In the Block Diagram, the calculated gauge (thickness) is q62 and the thickness error is q66. Note the use of the Sensitivity Factor dS/dh as q2. There are two other interesting factors with this control: -

Block Diagram for the Thickness Control of a 2 Stand Rolling Mill

Initially the control appears to be a PD control with q18 containing a P term equal to q16 times the gain q4, plus a D term being the constant q10 times the change in q16. However, since q20 is effectively added to itself, this summation converts the P term into an Integral, and the D term becomes a Proportional term. This arrangement has the advantage that the gains q4 and q10 can be changed while the control loop is active without causing a step in the output q20; that is, it’s a Bumpless control. The rate-of-change limit is designed to prevent the equivalent of integral windup.

Normally moving the Screw-downs to correct the Strip Thickness would perturb the Inter-stand Tension; its control would then need to trim the speed of the appropriate Stand to restore the tension. So, what is required, is a compensating trim applied to the Tension Control at the same time as the Thickness trim goes to the Screw-downs. This is referred to as a Non-Interactive Control;[8] that is, the Thickness correction no-longer disturbs the Tension. In the Block Diagram, the Screw Trim q20 is converted into a compensating IS Tension trim using the sensitivity factor dT/dS (the value of this was measured by applying a small step change to the thickness reference and looking for any change in the IS Tension).

For the coil in Chart Recording 1 above, the Cold Mill Head-end off-gauge was not fully removed at the CA line; this can be seen as the difference between the X-ray Deviation (green trace) and the calculated thickness deviation (light blue trace).

Bridle Rolls are used to increase or decrease the strip tension in a Processing Line or Rolling Mill.

Sketch 7 shows 2, 3 and 4 Roll Bridles indicating the variables

The Bridle Rolls normally come in a set of 2, 3, or 4 rolls of equal diameter, with each Roll individually powered by an electric motor/generator.[9] The drives of the Exit Bridle generate power as they decrease the Strip tension. This power is nearly equal to that used by the Entry Bridle to increase the Strip tension; the difference is just the mechanical and electrical losses.

To assist with threading there are normally Guides and even a Pinch Roll or Rolls, as shown for the 2 Roll Bridle in Sketch 7

To determine the size of the electrical drives, it is necessary to calculate the values of the intermediate tension or tensions.[10]

The maximum tension difference ΔT across a single Bridle Roll is determined by the wrap angle α (in radians) of the Strip around that Roll, and the roll-to-strip sliding friction μ, i.e.  T₂  =  T₁ • e^μ•α.[9]  equation B-1

The power required to drive such a Bridle is (T₂ – T₁) • (R + h/2) • ω   i.e.   (T₂ – T₁) • v

where   v   is the strip speed in m/sec

  h   is the strip thickness in meters

  R   is the diameter of the Bridle Rolls in meters

and  ω  is the Bridle’s angular speed in radians per second.

The electrical power required by the Drive Motor = Volts • Amps. The voltage can be regulated according to the strip speed, leaving the current to be proportional to the required tension change.

To prevent slippage, the Bridle Rolls within a set are operated at only a fraction p of the maximum tension difference, so the actual tension difference across each Bridle Roll will be e^p•μ•α. That is, a lower value of friction is used in the calculations.

Consider the simplest case: -

A 2 Roll Bridle set with both Rolls having the same wrap angle α. Then T₂ = T₁ • e ^p•μ•α,  and T₃ = T₂ • e ^p•μ•α.

Therefore  T₂ / T₁ = T₃ / T₂  which gives , T₂² = T₁ • T₃

And so  ${\displaystyle T_{2}={\sqrt {T_{1}.T_{3}}}}$  equation B-2

Now consider an example: -

Let  T₃ = 2.0 • T₁,  then,  T₂ = 1.4142 • T₁

so the tension across the first Bridle will be  ( 1.4142 – 1.0) • T₁ = 0.4142 • T₁

and the Tension across the second Bridle will be ( 2.0 – 1.4142) • T₁ = 0.5858 • T₁

And so, the second Bridle requires just over 40 % more motor power compared to the first.

If one wishes to reduce the number of spares; then it is desirable to have motors of the same power.

To do that, the wrap angle on the first Bridle must be increased so that the tension difference across both Bridles is the same;

T₃ - T₂ = T₂ - T₁   That is  T₂ = 1.5 • T₁  equation B-3

Let the wrap angle of Bridle Roll 1 be (α+Δ), where α is the wrap angle of Bridle Roll 2.

That is  T₂ / T₁ = e ^{p•μ•(α+Δ)} = 1.5

Taking logarithms of both sides gives  p•μ•(α+Δ) = ln_e(1.5)  equation B-4

For Bridle Roll 2:  T₃ / T₂ = e ^p•μ•α = 2.0 / 1.5

Again, taking logarithms gives   p•μ•α = ln_e(1.3334)  equation B-5

From equations B-4 and B-5:  Δ = 0.4094 • α  equation B-6

Now consider a 4 Roll Bridle with Tensions T₁ through to T₅. Normally in such a Bridle set all of the Rolls have the same Strip wrap angle, as shown in Sketch 7

The wrap angle from T₁ to T₃ is the same as that from T₃ to T₅.

Therefore, using equation B-2   ${\displaystyle T_{3}={\sqrt {T_{1}.T_{5}}}}$ equation B-7

Similarly,   ${\displaystyle T_{4}={\sqrt {T_{3}.T_{5}}}}$ equation B-8

So if we let   T₅ = 4.0 • T₁  then   T₃ = 2.0 • T₁  which gives   T₂ = 1.4142 • T₁

And finally   ${\displaystyle T_{4}={\sqrt {8}}T_{1}}$ = 2.8284 • ${\displaystyle T_{1}}$ equation B-9

Graph 6: Discontinuity at Yield Point

The discontinuity in the stress/strain of annealed steel[11] makes it impossible to create round tinned steel-cans. Wherever the steel bends first is where most of the bending will occur, rather than uniformly.

The discontinuity is shown within the red circle in Graph 6. It is the reason the Strip is given a light reduction (~1.3%) normally referred to as an elongation or extension.

Since it is referred to as an elongation and not a reduction, this Strip is said to have been reduced only once (at the Cold Mill prior to annealing); hence the term Single Reduced, SR.

After the elongation the discontinuity is no longer present.

Alternatively; after annealing, the steel strip can be reduced a second time (by up to 30%) to make it both thinner and work hardened. [12] When this is done, the strip is said to have been reduced twice; that is, Doubled Reduced, DR.

Consider a Violin String: -

The Tension Force F established in the String of length ℓ is related to e, the amount the String is stretched: -

( F . ℓ ) / ( e . A )  =  E   equation 9

Where F / A is Tension Stress (Tension force / unit area)

e / ℓ is the Strain (amount of stretch, in PU)

E   is Young’s modulus of elasticity[11]

and the area A is equal to the strings cross-sectional area.

Now consider the strip between the Stands of a multi-stand Mill: -

In time Δt an extra length of ℓ = V.Δt leaves the previous Stand, where V is the strip’s exit speed.

To establish the tension in this piece of strip, requires it to be stretched by an amount equal to F.V.Δt / ( E.A ) in the time interval of Δt; that is, a speed of F.V / E.A where the strip’s cross-sectional area A is equal to its width W times its thickness h,

If the speed difference between the next stand’s entry and the previous stand’s exit is different to this, then the strip of length L between the two stands will stretch or relax according to the integral of speed difference, and this will change the actual tension as shown in the Block Diagram.

Chart Recording 4 Start of Rolling a Coil showing Tension Components

The overall transfer function from speed-difference to tension is: -

E . W . h / (sL + V)  equation 10

Therefore, the control loop must divide the tension error by the strip width W and the strip thickness h in order to have a consistent response.

In Chart Recording 4, the two components necessary to change the Tension (brown trace) can be seen in the Speed Trim (light blue trace). There is the extra speed difference necessary to stretch the Strip already in the Inter-stand Gap (large trim); and the slight increase in speed difference to maintain the new level of Tension.

This definition only applies to the rolling of slabs, plates and strip in a Rolling Mill.

Reduction, r is defined as the per-unit change in thickness with respect to the entry thickness H, and so   r = (H – h) / H   where h is the exit thickness.[13][14]

Note that, as the material is reduced, its length becomes proportionately longer; this can be seen in the attached GIF movie above.

There are many other definitions of the word Reduction; such as in Chemistry, Medicine, Surgery, Safety, Investment, and in a more general sense, such as in cooking and waste reduction, etc.

If the required reduction is very small (< 2%) it is normally measured as elongation, e; which is defined as the per-unit increase in length with respect to the original length.

Given an entry length ℓ, then  e = (L – ℓ) / ℓ where L is the final length.

Sketch 8 Shows symbols related to thickness during rolling

If the width is unaffected (as is the case when rolling thin strip <2mm), then the Mass Flow concept, gives H . ℓ = h . L

Thus elongation,  e = (H – h) / h

Note that the difference (H – h) is now with respect to the exit thickness h and not the entry thickness H; so, reduction and elongation are not exactly equal.

Return to the main body of the text.

When the reduction is small (< 2%), it is normally referred to as an elongation or an extension.

Sketch 9 illustrates the elongation nomenclature

Graph 7: Elongation in a Round Bar

Graph 8: Discontinuity at Yield Point

Elongation, e is defined as the per-unit increase in length due to a decrease in area with respect to the entry, regardless of shape (as shown for a round bar in Graph 7 to the right).

Given an entry length ℓ, then  e = (L – ℓ) / ℓ where L is the final length.

If the width is unaffected (as is the case when rolling thin strip <2mm), then the Mass Flow concept, gives H . ℓ = h . L

Thus elongation,  e = (H – h) / h

When the elongation is large it is normally measured as reduction, r which is defined as the per-unit change in thickness with respect to the entry thickness H; and so if h is the exit thickness,[13][14]

Reduction,   r = (H – h) / H

These definitions apply to the rolling of thin strip in a Rolling Mill.

An elongation of typically 1.3% is performed to eliminate the discontinuity (seen at yield point in Graph 8) in the stress verses strain reaction of thin steel strip[15] before it is tinned ready for making cans intended for containing preserved foods.

There are many other definitions of the word Elongation; such as in Astronomy, Plasma physics, Biology (genetics), and in a more general sense, such as referring to the lengthening of an elastic band.

Return to the main body of the text.