&L=extension of the tube segment of length L
φn=nominal diameter of the tube (expressed in mm)
φm= average diameter of the tube after soldering (expressed in mm)
s= thickness of the tube
θ=diameter of the center of the first calibration roller
During the calibration step, the pipe, after having been soldered and placed in a cooling pool, undergoes a diameter restriction by the use of motorized horizontal rollers alternated to vertical rollers. This generates a progressive variation of the diameter of the core of the rollers, assuming that the angular speed is constant for all the calibration heads.
The starting hypothesis is the constancy of the tube section during the diameter restriction phase from φm value to the nominal value of φn. This hypothesis is empirically proved for thin pipes (φn/s>40/60), but it is not completely valid for thicker pipes that have a ration of φn/s<40
In the calibration of thick tubes, the section transformation occurs together with the tube thickness variation, and therefore the extension and thickening processes are connected to φn/s and to the mechanical characteristics of the material.
After these first considerations, we want to discuss the mathematical analysis of transformation with s as constant - the calibration of thin tubes.
In this situation, it is easily figured out:
&L = L*( φm- φn)/( φn-s) and since n &θ/θ =&L/L by which the incremental ratio is &θ/ θ = 1/n*( φm- φn)/( φn-s) where n is the number
of calibration heads in which the diameter restriction from φm to φn occurs.
Then since φm = φn + k with k = 08/1,5 mm The following incremental ratio is obtained n&θ/θ = k/n(φn-s)
The graphs of the tubes with thickness between 0,4 and 2,5 mm are
shown below, for the following range of pipes from 30 to 127 mm:

1st graph
φn = 30/60 mm
k = 1 mm
n = 4

2nd graph
φn =70/90 mm
k = 1,2 mm
n = 4

3rd graph
φn = 90/127 mm
k =1,5 mm
n = 4

Explicitating &θ = θn+1 - θ n it is obtained:
θn+1 = θn +&θ and from this:
θn+1= θn(1 +&θ/ θn)
Therefore it is possible to calculate the progression of the cores diameters after the various runs of the calibrating rollers for a given tube φn, assuming that the diameter of the core after the last run is known.
Final Considerations
Analyzing the diagrams described above, the following conclusions can be drawn:
1) The incremental ratio does not vary very much when the thickness of a given tube diameter φn varies, and therefore it is possible to use the same rollers without penalizing the surface of the tube for a vast range of thicknesses.
2) The incremental ration increases at the decrease of the nominal diameter of the tube.
3) The curve derivative decreases at the increase of the tube diameter, and therefore it is proven that it is better to use the lowest k as possible for every given tube.
(This article was written using some excerpts from the projecting archive of Celozzi & C.'s Europrogetto s.a.s. )

By Ing. Celozzi