Rolling mill is a very simple device, that is mechanically speaking. However, design of a rolling mill to insure ease of use and durability, is a bit involved. To chose rolling mill intelligently and to be able to troubleshoot problems, a goldsmith should have some understanding of the forces which are involved in rolling of metal. In this article we shall explore some of these forces to gain better understanding of the process of rolling.
We will use some equations in this text, but set your mind at ease. I am fully aware of dislike for anything which looks like even remotely mathematical. So no effort was spared to make it as painless as possible. Just read the explanations! The knowledge gained should serve you well when you will be shopping for your next rolling mill.
Here is schematic of rolling process.
We have two rolls which are set up with a gap between them. Rolls are rotating in directions indicated by black arrows. When metal is fed into the gap, the thickness of the metal is reduced. Blue arrows indicate force exerted by rolls on metal. This force, let’s call it useful because we want as much of it as possible ( without putting too much strain on bearings, of course ). Force indicated by red arrows, let’s call it parasitic, because we do not want at all. Alas, it is always present, but we should try to make it as small as possible. Excessive parasitic force can be caused by wrong mill operation, or wrong mill design, or both. The worst case is when incompetent goldsmith is operating badly designed mill. Let’s hope that after reading this article you would know how to avoid such situation.
Metal can be deformed homogeneously and non-homogeneously. The former is the desired outcome. There is a relation predicting under what conditions rolled metal deformed homogeneously, or otherwise. If bite ( thickness reduced in one pass ) divided by 2 is larger than roller contact length, the deformation is non-homogeneous. This relation shows importance of roll contact length, which in turn depends on roll diameter. The larger roll diameter, the greater roll contact length, and the better chance of homogeneous deformation. It is not necessary to compute actual length of roll contact. But if someone wants to know, than here it is:
ac – radius of the roll; bc – radius of the roll – 1/2 bite. Since abc is a right triangle, then angle c = cos^-1( bc / ac ), and length of roll contact = Cos^-1(bc/ac) * 2 * Pi * radius / 360. We have to use cos^-1 to convert radians to degrees, Pi is equal to 3.14, 360 is number of degrees in a circle.
Roll diameter is 65 mm. Bite ( reduction of thickness in one pass ) is 0.1mm. Then ac is 32.50mm, bc is 32.45 mm, angle C = cos^-1(32.45/32.5) = 3.17 degree, and length of roll contact = 3.17 * 2 * 3.14 * 32.5 / 360 = 1.797mm
As you see contact length is 18 times larger than the bite and deformation will be homogeneous. It is a fair question to ask “what if contact length is equal or only slightly larger than bite ?” The answer is that while deformation still will be homogeneous, it would take more power to drive such mill. The closer we get to the point where bite approaching contact length, the more power it takes to roll the metal.
Another point is that it seems that it would take very small roll diameter or very large bite to create conditions of non-homogeneous deformation. Well, it is true that for manually operated mills it is largely theoretical, aside of power requirements. Motorized mills are another matter. If ingot is not forged prior to rolling, than the first pass should be 1/3 of ingot thickness. That is where non-homogeneous deformation can manifest itself. The large first bite is required to change crystalline structure throughout the thickness of an ingot. But if it is done non-homogeneously, the ingot will be ruined. The only way to fix it would be to anneal it and forge all over with a hammer. This is kinds of ironic because first large bite is used to avoid forging.
Ease of rolling, which comes with large diameter rolls, has a price. Remember force indicated by red arrows, we called it parasitic force. People who design rolling mills for a living call it Roll Separating Force. The symbol for it is Pr and equation describing this phenomenon is
Pr = 1.15 * Qp * Gfm * L
It looks a bit intimidating and we can ignore most of it because variables Qp and Gfm are related to width and quality of ingot. The part that we are interested in is L, which stands for familiar length of roll contact. From this equation we can see that the larger the length, the greater the roll separating force and mill design must have provisions to deal with it. Such provisions manifest as massive frame and substantial bearings to withstand increased load. Adequacy of frame must be considered as it relates to roll width. As roll width increases, so is the Qp, which is pressure intensification factor per unit of width. One method of comparing between rolling mills of different design is to divide roll width by weight of the mill. The one with greater weight per unit of width is probably a better design. I say probably because mill design is alway a compromise of competing priorities. If mill will not be pushed to extremes, it is not that important to maximize each and every parameter of mill operation.
Parasitic force can be the cause of roll bending during milling. It may be difficult to believe that massive steel cylinders can bend, but it is true. Here is equation describing this phenomenon:
Delta = ( P*L^3 ) / E*I + 0.2( P*L / A*G )
- Delta is amount of deflection across central axis, which is an indication of roll bending.
- L is width of rolls here, not length of roll contact.
- P is roll pressure.
- E is elastic modulus of steel that rolls are made from.
- I is the moment of inertia.
- G is shear modulus of steel that rolls are made from.
- A is cross-section of the rolls.
We do not need to compute actual Delta to make use of this equation in selecting rolling mill. Parameter like elastic modulus ( Young modulus) and shear modulus are characteristics of steel and it’s heat treatment. Moment of inertia describes speed of rolling. That can be ignored for purposed of comparison, as well as roll pressure. Mill operator can adjust these parameter. What is pertinent is A, which is cross-section of roll and L, which width of rolls. Since A is part of denominator, the larger it is, the smaller the delta, which is highly desirable. Width of rolls is part of numerator and therefore increases value of delta, which is not desirable. So we can see that wider rolls require larger cross-section to keep deflection at a minimum. Cross-section is just another way of indicating roll diameter. To compare different mill designs we can compute ratio of roll diameter to roll width. Larger ratio means less deflection and therefore better mill design.
Another phenomenon in rolling is Hitchcook radius. Hitchcook radius is radius of the roll deformed by rolling pressures. Such deformation is temporary because it does not exceed limit of elasticity of the roll shape is restored once pressure abates. Nevertheless, when it is present even in minute quantity, the quality of rolled metal suffers. Hitchcook radius is denominated by R‘ and equation
R’ = R [ 1 + ( 16 ( 1 - v^2 )) / ( Pi * E * bite )) * Pr ]
Most of parameters should be familiar by now except R which is roll radius when roll is not under pressure, and v which stands for Poisson Ratio. The only parameters that goldsmith has control over is Pr and E. E stands for Young modulus and the only way to affected it is to insure that rolls are made from quality steel and properly tempered. Pr was discussed before and we know that large roll diameter increases it’s value. Since Pr used as magnifying factor in this equation, large roll diameter increases Hitchcook radius as well. So, it appears that large roll diameter is positive under some circumstances, but negative under others. That said, let’s recall that Pr is also dependent on width of ingot been rolled, and on the quality of ingot. These parameters are under goldsmith control. So, I would still opt for the largest roll diameter possible and control Hitchcook radius by correctly manipulating ingot. Excessive Hitchcook radius is the cause of ingot curving either up or down during rolling, and sometime even creates the wave effect.
The last item to consider is torque, which is denominated by M and expressed by equation
M = Pr * L / 2,
where L is length of roll contact, and Pr is roll separating force. All these parameter quite familiar by now. So there is no need to rehash it. However, subject of torque brings attention to reduction gears and length of handle. Torque of any mill can be increased by lengthening it’s handle. However it is not recommended, because by doing so you can exceed load specification for bearings. Length of the handle can be used as an indicator of magnitude of stress that bearings can take. There is another aspect to reducing gears and length of handle. Very frequently mill manufacture has several models in different roll width. While extend of forces in mills of different width are remarkably different, all mills are equipped with the same gear ratios and the same handles. I understand that it is easier that way, but from the point of design it is absurd. I would seriously question commitment to quality by such manufactures. But if you have no choice but to deal with one, select the mill in mid-range. Chances are that such mill would have the best characteristics.
We have come to the end of this article. If you feel a bit disappointed, I can understand that. After all, the title is “Demystifying Rolling Mill”, but after re-reading it myself, I feel that mystery only got deeper. Look at it this way. There are no perfect mills because there are no perfect materials. No matter how well chosen steel is and how well tempered, it is still a subject to deformation with all it’s consequences. I do hope that you would approach selection of rolling mill and it’s operation with more attention to details and better understanding of the process.
© 2013, Leonid Surpin, all right reserved.