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The Wusatowski formula
10 input data (4+6):
H = entry thickness, mm
h = exit thickness, mm
B = entry width, mm
D = roll diameter, mm
plus
t = rolling temperature, șC
v = exit speed, m/s
C = Carbon content %
Mn = Manganese content %
Cr = Chrome content %
KSC = roll material factor
The value for KSC can be found in the following table:
0 (Non-graphitic steel); 0.5 (Graphitic steel);
1 (Nodular pearlitic cast iron); 1.5 (Nodular acicular cast iron);
2 (Nodular martensitic cast iron); 2.5 (Chilled cast iron);
3 (Indefinite Chill cast iron); 3.5 (Hi-Chrome cast iron);
4 (Tungsten carbide).
Calculated data:
GAM = h/H
DEL = B/H
EPS = H/D
W = 10^(-1.269*DEL*EPS^0.56)
a = temperature correction
c = speed correction
d = stock material correction
f = roll material correction
BET = coefficient of spread (uncorrected)
BET = GAM^(-W)
BETA = coefficient of spread (corrected)
b = exit width
b = B*BETA
BETA=a*c*d*f*BET <<<<
FORMULA! (1)
where:
a=1 to 1.005 (IP to t)
c=0.98 to 1.02 (IP to v)
d=0.99 to 1.02 (may be DP to % of alloying elements)
f=0.98 to 1.02 (IP to roll surface smoothness)
(DP and IP stand for Directly Proportional and Inversely Proportional.)
Now let's see two examples:
EXA: flat 200x40 reduced to 30 mm
EXB: square 40x40 reduced to 30 mm
In both examples, D=280, t=1000, v=14, C=1, Mn=1, Cr=1.
In both examples, roll material is pearlitic cast iron (KSC=1).
The results of the calculations are:
EXA EXB
GAM
0.7500 0.7500
DEL
5 1
EPS
0.1429 0.1429
W
0.0073
0.3743
a
1.00
1.00
c
0.99 0.99
d
1.01 1.01
f
1.01 1.01
BET
1.0021 1.1137
a*c*d*f 1.0099 1.0099
BETA 1.0120
1.1247 Table I
b
202.40 44.988
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From these results we can infer that there is something odd
in the Wusatowski spread correction method. In fact, it may
lead to applying a 1% correction (1.0099) to a .2% coefficient
(1.0021). It is rather like trying to fix your glasses with
a carpenter's screwdriver.
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The Suppo/Izzo formula
This inconvenience was noticed by Ugo Suppo in the late '60s.
In the early '70s we developed and published the Suppo/Izzo
spread formula (see "La Metallurgia Italiana", Feb. 1972):
BETA=BET+(BET-1)*cw
<<<< FORMULA! (2)
where cw is the global spread correction coefficient:
cw = a'+ b'+ c'+ d'
Coefficients a', b', c' and d' account for the same physical
parameters considered by Wusatowski's a, c, d, f: only, they are
addends in an addition, rather than factors in a multiplication.
And while the "no correction" value for Wusatowski is 1, for
Suppo/Izzo the "no correction" value is 0.
The human designer can assign the value of cw in the range -0.25
to +0.25. In the LINEBOW programs the electronic designer calculates
cw with the following expression:
cw=.05*(C+Mn+.3*Cr)-.005*v-.03*KSC-.08*(t-800)/400+.13
In our example, the human designer would say cw=0-0.1+0.1+0.1=+0.1
while the electronic designer calculates cw=0.105.
If we apply the Suppo/Izzo spread formula, we obtain
BETA=1.0021+(1.0021-1)*0.105=1.0023 (vs. 1.0120)
BETA=1.1137+(1.1137-1)*0.105=1.1256 (vs. 1.1247)
The Suppo/Izzo formula version (2) is only recommended for
EPS>=0.1. In fact, Suppo also noticed that for EPS<0.1 the
Wusatowski formula tends to underestimate the value of BET.
The complete Suppo/Izzo formula
La Metallurgia Italiana (quoted issue) actually published a
complete version of the Suppo/Izzo spread formula:
BETA=K*BET+(K*BET-1)*cw
<<<< FORMULA! (3)
where
K=1
for EPS>=0.1
K=1+RHO*(0.1-EPS)^2 for EPS<0.1
with
RHO=14
for BET>=1.18
RHO=16.935*EXP[-54.1*(BET-1.18)^2]-2.935
for BET<1.18
Let us consider the example with square 40 (EXB), assuming that there
are no corrections (cw=0 or a*c*d*f=1) and that we work with doubled
roll diameter (560 vs. 280 mm) - thus halving EPS.
Rebuilding Table I we get:
Formula (1) Formula (3)
GAM
0.7500 0.7500
DEL
1
1
EPS
0.0714 0.0714
W
0.5135 0.5135
BET
1.1592 1.1592
RHO
13.608
K
1.0111 Table II
BETA
1.1592
1.1721 --------
From Table II we can see how the "geometrical" spread coefficient
is raised by use of formula (3) to better match the experimental
results for very low values of EPS.
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