UDK 621.771.28:621.89:519.8
Original scientific article/Izvirni znanstveni članek
ISSN 1580-2949 MTAEC9, 47(5)551(2013)
MODELLING OF THE LUBRICANT-LAYER THICKNESS ON A MANDREL DURING ROLLING SEAMLESS
TUBINGS
MODELIRANJE DEBELINE PLASTI MAZIVA NA TRNU PRI VALJANJU BREZSIVNIH CEVI
Du{an ]ur~ija, Ilija Mamuzi}
Croatian Metallurgical Society, Berislaviceva 6, Zagreb, Croatia ilija.mamuzic@public.carnet.hr
Prejem rokopisa - received: 2012-07-03; sprejem za objavo - accepted for publication: 2013-02-26
AutoCAD-a 3D modelling and the programs of Mathematica, MATLAB and MathCAD Professional were used to calculate the approximate solutions of the Reynolds differential equations for lubrication. Excel Software and Monte-Carlo solutions are compared. The results indicate a fast decrease in the lubricant-layer thickness in the stands for the continuous rolling of tubes. At the start of the rolling, the conditions for stable hydrodynamic lubrication are fulfilled for the tube-diameter-reduction processes. Theoretical calculations indicate that, on the finishing stands, "lubricant pickling" may occur on the mandrel. An approximate calculation is performed for the rolling point M and verified with the numerical Monte-Carlo method.
Keywords: Reynolds equation, AutoCad camera viewing, lubricant, continuous rolling line, seamless tubes
Za izračun približnih rešitev Reynoldosovih diferencialnih enačb za mazanje so bili uporabljeni modeliranje AutoCAD-a 3D in programi Mathematica, MATLAB in MathCAD Professional. Primerjane so rešitve, dobljene s programom Excel in z numerično metodo Monte-Carlo. Rezultati kažejo hitro zmanjšanje debeline plasti maziva na ogrodjih za kontinuirno valjanje cevi. Na začetku valjanja, pri postopku zmanjševanja premera cevi, so izpolnjeni pogoji za stabilno hidrodinamsko mazanje. Teoretični izračuni kažejo, da lahko pri končnih ogrodjih na trnu pride do odsotnosti mazanja. Izvršen je približen izračun pri točki valjanja M, ki je bil preverjen z numerično metodo Monte-Carlo.
Ključne besede: Reynoldsova enačba, opazovanje z AutoCad-kamero, mazivo, kontinuirna valjarniška linija, brezšivne cevi
1 INTRODUCTION
Tube rolling with round calibres1 is a variant of longitudinal rolling with a formation of a deformation zone and in Figure 1 the rolls and a mandrel are shown. The tubes rolling on a long floating mandrel use rolling lines with 7 to 9 working stands. Before inserting the rolling stock between the rolls, a cylindrical mandrel is
Figure 1: Scheme of rolling seamless tubes on a continuous rolling line
Slika 1: Shema valjanja brezšivnih cevi na kontinuirni valjarniški progi
put in, which then moves in the deformation zone jointly with the rolled tube. At the exit from the rolls, the mandrel speed is lower than that in the front tube part. The characteristic of the rolling is that the speeds of the tube and rolls are equal in the deformation zone only in two points of the roll groove. In recent years continuous rolling processes with a deformation of tubes with a stable, long, cylindrical cone mandrel or step-shaped mandrel were developed. The characteristic of this rolling is that there are two subzones of the reduction of the tube diameter and of the tube wall thickness, as shown in Figure 1. On the exterior contact of the tube and the rolls2,3 without a lubricant addition, friction occurs according to Kulon-Amonton laws, while in the tube interior, because of the lubricant presence, Newton's laws of fluid friction govern. The extent of the tangential stresses in the lubricant layer4,5 is calculated from equation (1):
ju(vC -vT) 1 dp
e( x)
2 £(X) dx
(1)
The change in the pressure in the lubricant layer4-7 is calculated from equation (2) and the volume of the lubricant used on the tube perimeter6,7 from equation (3):
dp dx
6ju(vC + vT) 12 juQ
e2 (x)
e3 (x)
(2)
x
£ ( X )
r 1 dp ,
Q(X) = J udy = — -fV(x) + 0 12 ^ dx
(vc + vT )
£(x) (3)
The geometry of the lubricant contact8-10 is calculated from equation (4) and the lubricant wedge lenght from equation (5):
£( x) = £ 0 + R0
cos aA -, 1-1 sinaA -
R
a = Rr
£ a + £
cos a 0- RT+R
1-
0
£(x) = £ o - a o x+
2R
a 0 x ■ 2 r2
- sin a
8 R l
(4)
(5)
(6)
Relation (6) is an evolution11-13 of equation (4) in a Maclaurin series. The lubricant characteristic for the theoretical investigation and the process geometry used are quoted from the references811 and are shown below.
The scheme of the tube-wall deformation geometry is shown in Figure 2. The cone shape of the tube is representative of the continuous rolling-line outset.
2 DISCUSSION, SOLUTIONS AND GRAPHIC INTERPRETATION
The equations for the approximate analytical solutions are listed in Table 1. The solution for the ^ range is achieved using the gripping angle a from references1114, being the known solution of Grudev, Mazur and Kolmogorov14.
The solutions for the Q range, M point and OS are derived in this work.
Figure 3 represents a comprehensive understanding of the solutions from Table 1. The M point is shown in a set of relations (£aA(1/3); A£o/£a}. For series 1, A£o is the difference from the pilot £%, i.e., the lubricant-layer thickness for every gripping a angle, minus the lubricant layer thickness for the pilot at 0.1 rad. Similarly, series 2
Figure 2: Stand with three rolls used in the analysis of tribomecha-nical systems: 1- mandrel, 2- lubricant, 3- tube, 4- roll Slika 2: Ogrodje s tremi valji, uporabljeno pri analizi tribomehanskih sistemov: 1- trn, 2- mazivo, 3- cev, 4- valj
Figure 3: Distribution of the areas along the rolling line according to the solutions from Table 1
Slika 3: Porazdelitev področij po dolžini valjarniške proge skladno z rešitvami v tabeli 1
is calculated for £0 -£*0 / £ a and pilot £0, with £*0 as the variable thickness of the lubricant layer.
Figure 4 shows a model of the lubricant layer on the mandrel calculated using the numerical Monte-Carlo method. At the entry (I) the lubricant is in surplus, while, at the exit of the continuous rolling line (II), the lubricant layer on the mandrel is picked because of the absence of fresh lubricant.
Table 1: Approximate solutions of equation (2) Tabela 1: Približne rešitve enačbe (2)

0
9
4
Zone of Fig. 3	Approximate analytical solution
	£0 = 0.7726£ £0 = 0.5Ro(a* )2 a = i £a >> £o y 15R0A
Q	a 0 = 2 I £aV2 R0£a = R)( a *)2 y 10 R02 + 15R02A£aA/ 2 R0£a 2
M point	a; =f ]2 — = ^^ £aMAx = 0.28674- V 5 y A£aMAX 2 »A
O 2	£° = «aÎ1 - °'57348£a 1 £aMAX = 0.28674-^|5 £a ^ £0 V £aMAX y » A
Figure 4: AutoCAD-modelled lubricant layer on the mandrel along the rolling line
Slika 4: AutoCAD modelirana plast maziva na trnu po dolžini valjar-niške proge
In Figure 5 approximate analytical solutions and numerical calculations are compared. Series 1 relates to the O area; series 2 relates to the 2 area, the marker of square M is the solution for the M point from Table 1; series 3 is the Monte-Carlo solution and numbers 1, 2, 3 and 4 indicate the positions of the AutoCAD viewing
Figure 5: Comparison of the approximate analytical solutions from Table 1 and numerical Monte-Carlo solutions for the area between the approximate center of the rolling line and the exit stands Slika 5: Primerjava približnih analitičnih rešitev iz tabele 1 in nume-ričnih rešitev Monte-Carlo za področje od približno sredine valjar-niške proge do končnih ogrodij
Table 2: 3D cameras for Figure 6, viewing left to W on Figure 5 Tabela 2: 3D-kamere za sliko 6, pogled levo od W na sliki 5
Figure Marking
Remark
Common markings
Fig. 6 Fig. 7
Mark 1
Mark 2
Mark 3
Mark 4
Ln(£A), Ln(£o)
Q- area in Figure 3 and Table 1
2- area in Figure 3 and Table 1
Numerical Monte-Carlo method
Point M in Figure 3, mark M in Figure 5, point M in Table 1
Abscissa and ordinate in 2D, for the AutoCAD command Revsurf
Figure
Fig. 6
Fig. 7
Specific markings
Marking
Markers 1, 4
Remark
"Hat" obtained with the rotation of the arc P-Pi in Figure 5
Drawing of pyramid for point P in Figure 3 (cross-section of Q and 2)
Figure 6: Viewing position 1 from Figure 5. The camera is covering the area between the center and the exit of the rolling line. Slika 6: Pogled s pozicije 1 na sliki 5. Kamera obsega področje od sredine do konca valjarniške proge.
Figure 7: Viewing point 4 from Figure 5. The camera is directed to the approximate analytical solutions from Table 1. Slika 7: Pogled s točke 4 na sliki 5. Kamera je usmerjena k približnim analitičnim rešitvam v tabeli 1.
Figure 8: Comparison of the analytical and Monte-Carlo solutions for the area between the entering stands and the approximate center of the rolling line
Slika 8: Primerjava analitičnih rešitev in rešitev Monte-Carlo za področje od vhodnih ogrodij do približno sredine valjarniške proge
P
Table 4: Comparison of the approximate analytical solutions from Table 1 with the numerical Monte-Carlo solution from around the middle to the exit of the rolling line
Tabela 4: Primerjava približnih analitičnih rešitev v tabeli 1 z Monte-Carlo numerično metodo, od približno sredine valjarniške proge do izhodnih ogrodij
Table 3: Explanations of the markers from Figure 9 Tabela 3: Pojasnila k markerjem na sliki 9
Marker	Link	Remark
Series 1	Monte-Carlo	Numerical calculations
Series 2	Q- area in Table 1	
Series 3	Solution according to M point in Table 1	Control of Monte-Carlo
Series 4	2- area in Table 1, left of point P	Graph agrees with Monte-Carlo
Point P	Equal significance as in Figure 7	Cross-section of solutions Q and 2
Q-I	The transition area around the M point where Q no longer agrees with Monte-Carlo and 2 formula is not yet in accord with the Monte-Carlo solution. Acceptable for interpolations of the polygonal method.14 In the interval, the section point P is a polynomial of degree 12.	
£a /m	Monte-Carlo e0/m	Zone Q e0/m	M point e0/m	Zone $ and 2 eI0/m
9.420E-04	1.225E-05	1.225E-05	-	-
8.735E-05	1.136E-05	1.129E-05	-	-
1.069E-05	5.801E-06	-	5.801E-06	-
7.425E-06	4.618E-06	-	-	4.466E-06
cameras. According to solution 2, ln(gA) ~ -15.414043 for £o^0.
The explanation of Figures 6 and 7 is given in Table
2.
The numerical Monte-Carlo method and approximate analytical solutions for about half of the rolling line, according to references 7-9, are shown in Figure 8, where the calculated lubricant-layer thickness is shown for the area from the middle part to the exit of the rolling line. The linear correction of the Q area was obtained with a solution in the M point according to Table 1 and is also shown in Figure 8b.
An integral comparison for the whole continuous line is shown in Figure 9 and, accordingly, for the abscissa as well in Figure 5. The markers are explained in Table 3.
An important characteristic of the M point is the control of the Monte-Carlo numerical method in this
-,-,-1-1-1-1-1-1--4G-
5 -23 -21 -19 -17 -15 -13 -11	-9
-24
Ln(£A)
Figure 9: Comparison of the analytical solutions from Table 1 with the numerical Monte-Carlo solutions for the whole rolling line Slika 9: Primerjava analitičnih rešitev iz tabele 1 in numeričnih rešitev Monte-Carlo vzdolž cele valjarniške proge
point. The conditions from Figure 9 are shown in Table 4.15
The solution for the M point has been confirmed. The numerical methods should be controlled in a narrow interval, at least in one initial point, with the approximate analytical solutions. In the case of the approximate analytical solutions with simplified mathematical solutions, the numerical methods are not reliable.
3 CONCLUSION
It is shown in several figures that the approximate analytical solutions of equation (2) from Table 1 are in accordance with the numerical Monte-Carlo solutions. The representations in AutoCAD provide a refined understanding and a comparison, giving also realistic images of the lubricant on the mandrel for rolling the seamless tubes on a continuous line with several stands.
The solution developed for the M point15,16 is of special importance since it enables a linear correction of the lubricant layer in the area of the Q formula and the correction can also be extended to the area Q-I.
4 SYMBOLS
Symbol	Unit	Description
£0	m	Lubricant thickness in the entry section of the deformation zone (Figure 1)
£10	m	Lubricant thickness for the gripping angle ao tending to zero
	m	Lubricant-layer thickness for the gripping angle 0.1 rad, Figure 3 (series 1)
		Characteristic lubricant thickness for the
£*0	m	square trinominal in relation (6), with zero as a discriminant of the square equation
£(x)	m	Lubricant-layer thickness in the range [-a : 0], Figure 1, equation (4)
£sr	m	Average mandrel-lubricant thickness after a pass
		Lubricant-layer thickness on the
£a	m	mandrel after the entry section of the deformation zone
£aMAX	m	Characteristic lubricant-layer thickness on the mandrel in point M
a	m	Length of the lubricant wedge (Figure 1), equation (5)
a - a0	rad	Reduction-tube-diameter angle (Figure 1)
a0	rad	Tube-wall deformation angle (Figure 1)
a*0	rad	Characteristic angle added to £*0
VR	m/s	Circumferential roll speed
VC	m/s	Tube speed
vT	m/s	Mandrel speed
dT/2	m	Mandrel radius
R	m	Stand-roll radius
R0	m	Ro = R + Scx
Sc	m	Tube-wall thickness on the mandrel after the deformation zone
Sc1	m	Rolled-tube wall thickness
Lr	m	Abscissa projection of the tube-diameter-reduction zone
Ls	m	Abscissa projection of the tube-wall-projection zone
Ld	m	Ld = Lr + Ls
Tx	Pa	Tangential stress in the lubricant layer
	Pa s	Dynamic lubricant viscosity under rolling pressure
0	Pa s	Dynamic lubricant viscosity under atmospheric pressure
u	m/s	Lubricant speed along the abscissa
r	Pa-1	Lubricant-viscosity piesocoefficient
p	Pa	Rolling pressure
Q	m2/s	Lubricant use on the mandrel perimeter: one-dimensional model
dp/dx	Pa/m	Pressure gradient in the lubricant layer, equation (2)
dp/dx	Pa/m	Partial-pressure differential along the abscissa, equation (1)
sina... ln (£a)	rad...	Symbol for the trigonometric functions and natural logarithm
		Symbols for the viewing locations of the AutoCAD cameras. After the
O	i	approximate calculations according to Table 2 the obtained data are compared to the numerical Monte-Carlo solutions obtained with the mathematical programs and then modelled with AutoCAD, Figures 6 and 7.
		Rolling-line zone described with the
V	m	solutions from Table 1 and Figure 3, according to4,5, for the lubricant surplus on the mandrel.
Q	m	Rolling line after the V zone where £a influences £o and the solutions are
		presented, in this work, as the zone of tube-diameter reduction.
2	m	Part of the rolling line with the start and finish of intensive lubricant pickling on the mandrel and in the place where a tube-wall deformation takes place.
$	m	Area around point M from Figure 3 where interpolation polynomials can be used to connect the lubricant-layer calculations with the Q and 2 equations from Table 1.
Q -1	m	Larger area around point M, shown in Figure 9 and suitable for using the polygonal method14
M-Point	m	Point of the approximate analytical solution from Table 1 controlling the numerical Monte-Carlo method, allowing a linear correction of the lubricant layer in Figure 8b. The effect is noticeable in Figures 3 and 5.
W	-	Reference point in Figure 3; right of W - reduction of the tube diameter; left of W - deformation of the tube wall. It explains the topics and has no analytical expression.
>>, ® ,0 A, E	-	Mathematical symbols for much greater, approximate, step mark, tending, mark for an exponent, base of number 10
A	m-1	Technological parameter: A= [1- exp(- p)] / [6 o ( vc + vt) ]
exp 16	-	Base of natural logarithm, a reference
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