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Alloying elements 1 2 3 …… n
Concentration inside the
tolerance limits, X (%)
Concentration outside the
tolerance limits, X (%)
sumed scrap streams or, in other words, the
necessary level of scrap sorting. Such prediction
algorithms for wrought aluminium alloys
have been reported by several authors-for a
review see Ref. [5], but are not focused on the
recycling of wrought aluminium alloys from
post-consumed scrap.
Thus, the purpose of this paper is to present
possibilities for numerical modelling 1 of both
technological options for increasing the amount
of post-consumed aluminium scrap in wrought
aluminium alloys. Based on the model developed,
the optimal solution was suggested as
the starting point for further development and
implementation of the appropriate technology
of wrought aluminium alloy recycling.
2. Modelling of wrought aluminium
alloy properties as a function of
their chemical composition
X 1 ±∆X 1 X 2 ±∆X 2 X 3 ±∆X 3 …… X n ±∆X n
X 1 ±∆X 1 X 2 ±∆X 2 X 3 ±∆X 3 …… X n ±∆X n
Table 1: Wrought aluminium alloy compositions considered in the model
Generally, the selected properties of a wrought
aluminium alloy (e. g. yield strength – YS, ultimate
strength – US, elongation – L and hardness
– H) can be all expressed as different
functions of the alloy composition:
YS = F (X 1 , X 2 , X 3 , … , X n ) (1)
US = G(X 1 , X 2 , X 3 , … , X n ) (2)
L = L(X 1 , X 2 , X 3 , … , X n ) (3)
H = H(X 1 , X 2 , X 3 , … , X n ) (4)
Here, X 1 , X 2 , X 3 , … , X n represent the concentrations
of particular alloying elements.
On the other hand, the concentrations of
alloying elements in wrought aluminium alloys,
especially in recycled ones, are most often
designed for achieving maximal strength.
To achieve the proper combination of properties
(not only mechanical but also electrical,
thermal, corrosion resistant, etc.), the concentrations
of alloying elements should be inside
the standard tolerance limits.
However, in wrought compositions containing
an increased amount of scrap, usually
it is not easy and certainly not cost-effective
to assure such narrow compositions. Therefore,
producers of recycled wrought alloys
try to develop so-called ‘recycling friendly’
compositions with broader tolerance limits,
which at the same time do
not significantly influence
the selected (usually some
of the mechanical) properties
of the alloys.
The achievement of
standard wrought alloy
composition by mixing various fractions of
scrap with different chemical composition is
practically impossible. Statistically, in the real
mixture obtained by combining such different
fractions of scrap from the scrap yard, the
concentration of some of the alloying elements
will be higher than those prescribed by
the standard, the concentration of others will
be lower and there will also be some alloying
elements whose concentrations will fit the
standard requirements.
The situation is illustrated in Fig. 1 where
the concentration X 1 of the alloying element
1 in the scrap mixture prepared for melting is
inside the standard interval of concentrations,
the concentration X 2 of the alloying element
2 is higher and the concentration X 3 of the alloying
element 3 is lower. However, all three
concentrations are inside the alternative interval
of concentrations formulated for a ‘recycling
friendly’ composition.
The mathematical condition for ‘recycling
friendly’ alloy compositions under which the
selected alloy properties will all remain the
same is expressed by Eqs.(5) - (8):
dYS = 0 (5)
dUS = 0 (6)
dL = 0 (7)
dH = 0 (8)
In this way, the model developed gives the
combination of non-standard and standard
tolerance limits (∆X i ) under which the selected
alloy’s properties YS, US, A and H remain the
same. The determination of such a combination
of non-standard and standard tolerance
limits (i.e. intervals of concentrations for each
of alloying elements appearing in the alloy)
proceeds in two steps. In the first step, the
intervals ∆X i for alloying elements are determined
by considering each of the properties
individually. After that, in the second step,
the limits obtained for alloying element were
reduced to the intersection of particular intervals,
under which all the selected properties
(YS, US, A and H) are to remain constant simultaneously.
Mathematically speaking, in the first step
we solve the individual equations (5)-(8). Note
that the solution of each of these equations is
the enlistment of the intervals of concentration
(∆X i ). In the second step, the solution of the
system of Eqs. (5)-(8) under which the selected
properties remain constant is obtained as
the intersection of these various intervals obtained
for each particular alloying element.
3. Practical application of the model
Let us consider a wrought alloy with standard
composition and concentrations of alloying
elements inside the tolerance limits, and the
alternative (‘recycling friendly’) alloy with
concentrations of alloying elements slightly
outside the standard tolerance limits (Table 1).
Experimentally available data are collected
in Table 2 where the yield strength (YS) was
measured and correlated with the actual alloy
composition determined by emission spectroscopy.
Let us further assume that the selected alloy
properties (e. g. yield strength – YS, ultimate
strength – US, elongation – L and hardness
– H) are polynomial functions of the alloy
composition.
Note that in each of the equations in the
system of Eqs. (5) - (8), the tolerance limits, ∆X i
(i = 1, 2, 3, … , n) of the individual alloying
elements appear as n independent variables.
Hence, the exact solution of these equations is
not possible. The particular solution of whichever
equations of the system of Eqs. (5) - (8) is,
theoretically speaking, the randomly selected
combination of tolerance limits ∆X i (i = 1, 2,
3, … , n) for which the right hand side of the
equation is equal to zero. However, in the
practical case the values of the tolerance limits
of recycling-friendly wrought aluminium alloy
cannot be selected randomly but should be as
close as possible to the standard ones. Note
that these minimal deviations of each alloying
element from the standard concentrations
Modelling of the degree of post-consumed scrap sorting for recycling-friendly wrought
compositions
Sample
Yield strength
(MPa)
Ultimate
strength(MPa)
Elongation(%)
Harness
Alloying elements (1,2,…n)
and their concentrations (%)
1 2 ….. n
1
Following the editorial request to avoid extended formulas
and demanding mathematical explanations, the
model developed is presented in this article only in a
descriptive way. However, the detailed mathematical approach
can be sent by the author on request.
1 YS 1 US 1 L 1 H 1 X 1,1 X 2,1 X n,1
2 YS 2 US 2 L 2 H 2 X 1,2 X 2,2 X n,2
m YS m US m L m H m X 1,m X 2,m X n,m
Table 2: Experimentally measured data for yield strength (YS), ultimate strength (US), elongation (L) and
hardness (H) as a function of the concentrations of alloying elements
ALUMINIUM · 7-8/2013 59