Marie Luong

Didier Maquin

Chang Trung Huynh

José Ragot

José Ragot Observability

Trung Chanh

Huynh

Observability, redundancy, reliability and integrated design of measurement systems

Keywords: Degree of redundancy, observability, reliability, canonical matrix, cycle analysis

niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

objectives: 1) guaranteeing the observability of variable required for the control 2) guaranteeing degrees of redundancy of some variables 3) ensuring minimal cost of the instrumentation.

It is important for process control improvement to integrate the design of the measurement system while conceiving the process itself. The main problem to define a measurement system concerns the selection of variables to be measured; especially the state estimation of the process is influenced by this choice. Indeed, estimation techniques are only applicable for observable systems; one of the fundamental steps then consists in isolating observable parts of the process. Publications on observability concepts are numerous; (Vaclavek 1969) is one of the firsts to undertake the linear system study. (Mah et al., 1976) and (Gomolka et al., 1992) develop a technique based on the graph theory. One can also quote the works performed for bilinear systems in the domain of data reconciliation (Kretsovalis et al., 1988;Crowe 1989;Maquin et al., 1989 andRagot et al., 1991).

The proposed algorithm is based on the analysis of cycles of the graph associated with a process. It presents an interest as much for its facility of implementation as for its rapidity with regard to matrix methods (Darouach et al., 1986 andMaquin et al., 1987). Having concern for the simplicity, we limit this communication to the linear system study. It is organized as follows: the section 2 is relative to the process description; the section 3 presents the analysis of the process according to the concepts of observability, redundancy and reliability; using the analysis results of section 3, the fourth section proposes an algorithm to solve the problem of sensor placement and presents some numerical results.

Some synthesis works deal with this topic; indeed, in order to carry out a diagnosis, a minimum number of information provided from the sensors of the process is required. If the amount of information is not sufficient, a number of supplementary sensors is proposed so as to complete "poor" parts in information of the process (Vaclavek 1969;Ragot et al., 1986;Darouach et al., 1986 andMaquin et al., 1987). The conception of measurement system for bilinear models has also been published (Ragot et al., 1992). Thus, the observability concept has provided one of the criteria for the improvement of the measurement systems. More recently, optimization study for the global reliability of the measurement system taking into account simultaneously observability and sensor reliability has been studied (Turbatte et al., 1993).

DESCRIPTION OF A PROCESS

A process can be described by equations linking different variables; in a linear system, these equations may be represented by a graph made up of m arcs and n nodes. The complete set of equations of the process can be written as:

M X = 0 (1)
where M represents the (n.m) node incidence matrix of rank n, and X the (m.1) vector of variables of the system. Considering L, the set of variables of the process, it is divided into two following subsets: L 1 is the subset of variables required for the control and L 2 the subset of variables non required for the control. In the case of a process equipped with sensors, measured and unmeasured variables can be distinguished into the two lists X m and X m -.

Our purpose is focused on the conception of sensor placement by considering the reliability of sensors. Based on the concepts of observability, redundancy and reliability, this communication presents an automatic sensor placement technique allowing to minimize the global cost of the instrumentation. More precisely, given the list of variables required for the control, the maintenance and the safety of the process (in the following, we only use the word control instead), we seek the optimal sensor placement that meets the following As an example, let us consider the process network of figure 1; it can be described by the node incidence matrix M (table 1), where the first row contains the numbers of variables, and the first column the numbers of equations (nodes).

The subsets of variables required and non required for the control respectively are: chords. The number of these cycles is C c k . To obtain all the cycles, we use the recursive procedure: L 1 = {1, 4, 6, 9, 10} L 2 = {2, 3, 5, 7, 8} -by using the fundamental cycle matrix, we generate cycles containing 2 chords, -then, we generate cycles containing 3 chords from cycles containing 2 chords and fundamental cycles,

I II V VI IV 1 2 3 4 5 6 8
7 9

III

-more generally, we generate cycles containing k chords from cycles containing (k-1) chords and fundamental cycles.

The total number of cycles is given by Nbc:

Nbc = C c 1 + C c 2 +...+ C c c = ∑ i=1 c C c i
(5) Fig. 1. A process network This algorithm will be useful both for the analysis of a process equipped with sensors and for the conception of a measurement system. In this section, we introduce the concepts of observability, degree of redundancy of a variable and reliability. Table 1. Node incidence matrix M From this process network, the construction of a cyclic graph is possible by connecting arcs corresponding to its feeds (entries) and its products (exits) to a so-called environment node. This graph is interesting because it allows all the necessary cycles for the implementation of algorithms that we will present to be determined (Berge 1989).

Matrix of all the cycles of a graph

For an instrumented process, we define four lists whether a variable is measured or not and according to its membership to L 1 and L 2 : X 1m: measured variables required for the control, X 1m -: unmeasured variables required for the control, X 2m: measured variables non required for the control, The cycle matrix X 2m -: unmeasured variables non required for the control. It is important to note that a process graph is independent of its measurement system. Especially, cycles are the same whether the process is supplied with sensors or not. The research of cycles depends on the determination of a spanning tree of the graph. The literature proposes several methods among which the method of Gauss-Jordan is retained. This one uses linear equation combinations (rows) and variable exchanges (columns) in order to obtain a canonical form of the node incidence M (n.m) called the canonical node incidence matrix or the fundamental cutset matrix (Maquin et al., 1987;Gomolka et al., 1992):

These four lists allow the columns of the matrix M to be rearranged as follows: 2. Node incidence matrix M Then, this matrix is restructured. Using Gauss-Jordan elimination, we first construct the greatest identity matrix I C2 with unmeasured variables. Then, if the rank of this matrix is not equal to n, we reiterate the operation with measured variables, which generates the matrix I C1 . The matrix M 5 concerns the remaining unmeasured variables. The submatrices M 6 , M 7 and M 8 are constituted of remaining measured variables. Thus, we obtain the canonical matrix M C with the following form:

<---------measured--------><------unmeasured-------> X 1m X 2m X 1m - X 2m - M 1 M 2 M 3 M 4 Table
M C = [I C | M CS ]
(2)

where I C is the identity matrix (n.n) whose columns correspond to the branches of the spanning tree, and M CS the singular matrix (n.(m-n)) whose columns correspond to the chords. The fundamental cycle matrix C F ((m-n). m) is defined by

C F = [C FS | I F ] (3) me m - m -e - m e - m e (C FS ) T = -M CS (4) I C1 0 0 0 M 8 0 I C2 M 5 M 6
M 7 If we are solely interested in the occurrence of a variable in a cycle, the absolute value (of terms) of the matrix C F is used and noted C FA . The matrix containing all the cycles generated from the fundamental cycles will be noted C T . Table 3. Canonical matrix M C For each list L i (i = 1, 2), we define: X ime : variables corresponding to the identity submatrix Theorem 1: a cycle containing k different chords results from the ring sum of k fundamental cycles containing these k chords. I C1 and the submatrices M 8 and M 7 X im -: variables corresponding to the identity submatrix I C2 X ime -: variables corresponding to the submatrix M 5 Proof: If we consider the expressions (2) and (3), the variables of the identity submatrix I F columns correspond to the chords. Then, k distinct fundamental cycles are composed of k distinct chords, therefore k different arcs. The ring sum of these k fundamental cycles is the same operation as the deletion of common arcs of these cycles. As these k chords are distinct, there are no possible deletion of these k chords. = {3, 4, 5, 8, 9 10} andX m -= {1, 2, 6, 7} .

1 1 . . 1 . . 1 . 1 1 1 . 1 1 . . . 1 . 1 1 1 1 1 1 1 . . . 1 . 1 1 1 1 . 1 .
After having analysed this matrix, we can classify the variables in the following lists according to their observability state:

1 1 . 1 . 1 1 . . 1 . . 1 . . . . 1 1 . 1 . 1 . 1 . . 1 . 1 1 . . . 1 . . . 1 1 X m -e -= Ø X m -e = {1, 2, 6, 7} . . . 1 1 . 1 1 1 . X me -= {8} X me = {3, 4, 5, 9, 10} 1 . . 1 . . 1 1 . 1 1 . 1 1 . . 1 . 1 1 1 1 . . 1 1 . 1 1 1 3.3. Degree of redundancy 1 1 1 1 . 1 1 1 1 1
Definition 1: a redundant variable of degree k is an observable variable whose value remains deducible in case of simultaneous failures of any k sensors.

Table 5. Cycle matrix C T

Observability concepts

The observability analysis consists in classifying variables according to four categories. A deducible and measured variable (redundant) is a measured variable whose value can still be deduced from measurements of the other variables of an equation when the sensor measuring this variable fails. A measured and non deducible variable is a measured variable whose value cannot be deduced from other variables of an equation when the sensor measuring this variable fails. An unmeasured and deducible variable is an unmeasured variable whose value can be obtained from other variables of an equation. The equation in which a deducible variable occurs is a deduction equation. An unmeasured non deducible variable is an unmeasured variable appearing only in equations comprising two unmeasured variables at least.

Especially, a minimal observable variable is a variable of zero redundancy degree; a variable belonging to a redundancy equation is redundant of degree 1 at least.

Definition 2: an unmeasured variable is pseudo-redundant (the difference with a redundant variable is that a pseudoredundant variable is not measured) of degree 1 if it at least belongs to an equation where all the other variables are redundant or pseudo-redundant of degree 1.

By applying the next rules (V and VI), one can more generally deduce the redundancy degree of a measured variable, and similarly the pseudo-redundancy degree of an unmeasured variable: Rule V: a measured variable is redundant of degree k if, and only if, it only belongs to cycles where at least (k+1) variables are measured. Measured variables that are deducible or not and unmeasured deducible variables are observable variables. The canonical matrix structure of the table 3 leads to the variable classification by researching redundancy and deduction equations. However, we are going to show that the use of cycles facilitates this classification.

Rule VI: an unmeasured variable is pseudo-redundant of degree k if, and only if, it only belongs to cycles where at least (k+1) variables are measured.

The redundancy degree of a variable is found by computing the minimum number of measured variables contained in cycles (among these of C T ) to which this variable belongs. If this number is equal to k, then the redundancy degree is (k-1).

Rules of observability

The observability analysis is based on the study of cycles, using the four next rules: Rule I: a measured variable is deducible if, and only if, it only belongs to cycles where at least two variables are measured.

Again, consider the process of the figure 1. From the cycle matrix (table 5), we obtain the following results: Rule II: a measured variable is not deducible if, and only if, it at least belongs to a cycle where it is the only measured variable.

-the measured variable 8 that belongs to the only cycle (the first) containing one measurement is a minimal observable variable. Rule III: an unmeasured variable is deducible if, and only if, it only belongs to cycles comprising one measured variable at least.

-the unmeasured variables 1, 2 and 6 belong to the only cycle where one variable is measured. They are pseudoredundant variables of degree 0 or minimal observable variables. Rule IV: an unmeasured variable is not deducible if, and only if, it at least belongs to a cycle where no variable is measured.

-the measured variables 3, 9 and 10 are redundant of degree 1 because they belong to cycles whose minimum number of measured variables is two.

Observability algorithm

-the measured variables 4 and 5 are redundant of degree 2 because they belong to cycles whose minimum number of measured variables is three.

Using these four rules, we propose the following algorithm for the observability study: 1) computing the cycle matrix C T According to the preceding results, values of the variables 3, 9 and 10 are available even if any one sensor fails.

Values of the variables 4 and 5 remain always available in case of simultaneous failures of any two sensors.

2) computing cycles comprising from 0 to 2 measured variables among cycles of C T , (the knowledge of cycles with more than two measurements is not necessary according to the rule I) In practice, the obtaining of the redundancy degrees of variables is interesting since that allows the security of the process functioning to be guaranteed.

3) detecting cycles which do not contain measured variables: unmeasured variables belonging to these cycles are unmeasured and non deducible variables (rule IV). The other unmeasured variables of the system are unmeasured but deducible variables.

Reliability of a measurement system

The measurement system function is to provide necessary information for the control of the process. Then, this function is not fulfilled if the measurement of a required variable for the control is no longer available. The reliability of an instrumentation system is defined as the probability that information required for the control are available through measurements or deduction during the time interval [0, t]. Then, it is advisable to compute the 4) detecting cycles containing only one measured variable: measured variables belonging to these cycles are measured and non deducible variables (rule II). The other measured variables of the system are deducible measured variables.

The cycle matrix comprising a maximum number of two measured variables is obtained from the cycle matrix C T (table 5) and is presented in the table 6: number of sensor failures conserving the observability of the variables required for the control (list L 1 ).

Classification of variables required for the control

The canonical matrix structure (table 3) allows redundancy and deduction equations of variables required for the control to be determined (subset L 1 ). Indeed, variables are classified in the following classes (i=1,2):

Assumptions H0: sensors are considered irreparable. H1: there is no breakdown of common cause. X ime: deducible measured variables of L i H2: the failure rate λ of a sensor is independent of the time, which allows the reliability of a sensor to be defined analytically by using for example, the law of Poisson: X im -: unmeasured variables of L i X ime : unmeasured deducible variables of L i X ime -: unmeasured non deducible variables of L i X ime -: measured non deducible variables of L i r(t) = exp(-λ t) (6)

The unmeasured deducible variables of X im -belong to the rows of the submatrix I C2 . These rows correspond to null rows of M 5 . From the canonical matrix (table 3), variables of the list X ime are obtained by rearranging M 5 to make its null rows appear. As the system is initially observable (according to H3), non deducible variables of X 1me -do not exist. Since we are interested in the observability of variables of the list L 1 , the equations containing variables of the list L 2 as well as the columns corresponding to variables of the lists X 2me , X 2me -are suppressed. We obtain the following reduced matrix:

H3: initially, the system is observable.

By definition, the reliability of a sensor i is the probability r i (t) that no failure occurs during the time interval [0, t]. More generally, the reliability of a system constituted of p sensors with respective reliabilities r i (t) may be expressed by: R(t) = f(r 1 (t), r 2 (t), ..., r p (t)) ( 7)

We also define the MTTF function (Mean Time To Failure) by:

(a) (b) (c) (d) (e) (f) (g) X 1me X 2me X 1m -e X 1me -X 2me -X 1me X 2me 1 I C11 M 81 MTTF = ∞ ∫ 0 R(t)dt (8) 2 I C12
The calculation of the MTTF may or may not take into account the redundancy equations. If we do not consider the model, the failure of a sensor measuring a value required for the control cannot be admitted. The reliability R 0 (t) is the probability that all sensors well function. Finally, columns of bloc (g) are reordered to obtain zero submatrix in the zone (g1): If the model is taken into account, some sensor failures can be tolerated. Indeed, one might estimate the value of a variable whose sensor has failed through redundant equations owing to the model. Practically, it is necessary to determine:

(a) (b) (c) (d) (e) (f) (ga) (gb) X 1me X 2me X 1m -e X 1me - X 2me - X 1me X 2me X 2me I C11 M 811 I C12
M 812 -the maximum number v of admissible failed sensors conserving the observability of the variables required for the control.

I C211

M 61 M 711 M 712 Table 9. Reduced canonical matrix M C -the probability R i (t) so that the variables required for the control are available when i sensors (i =1,..,v) fail.

If we consider the theorem 2 and the "horizontal bands" (1) and (3) of the table 8, the knowledge of the values of variables corresponding to the columns of (d), (e), (f), (ga) and (gb) allows the deduction of necessary variables (columns of (a) and (c)) in case of simultaneous failure of sensors measuring variables corresponding to the columns of (a) and (b). The number v is equal to (m a +m b ) where m a et m b are respective numbers of columns of (a) and (b). If the submatrix M 712 of dimension (n 3 .m gb ) is a null matrix (n 3 is the total number of lines of the matrix I C211 ), the equations (2) of the table 8 will no longer be useful for the deduction of necessary variables. Then, the number v is equal to (m a +m b +m gb ).

Finally, the reliability of the measurement system is expressed by:

R(t) = ∑ i=0 v R i (t) (9)
It is obvious from (9) that the reliability increases if analytic relationships of the process are taken into account. It is shown that there is a relationship linking the observability, the degree of redundancy and the reliability.

The computation of the reliability necessitates to enumerate the cases where the failure of sensors can be tolerated. The value of a variable whose sensor fails is no longer known if, among all cycles to which this variable belongs, there is at least one cycle which do not contain measured variables (Rule IV). The availability of the value of a variable therefore depends on the number of measured variables in the cycle possessing the less ones; the more this cycle has measured variables, the more the measurement system can tolerate sensor breakdowns.

Reliability under constraint

The problem consists in determining the number of failures conserving the observability of variables required for the control (list L 1 ). If equations of the process are not considered, the probability that all sensors function is equal to: R 0 (t) = r(t) p (10)

On the other hand, if we take into account these equations, the probability that a sensor fails while (p-1) others (p > 1) well function is: (1-r(t))r(t) p-1 . If α 1 is the number of possibilities to consider a failed sensor while ensuring the observability of necessary variable, the probability that the system is operational while any one sensor fails is

The computation of the maximal number v of admissible failed sensor is based on the following theorem.

Theorem 2: in a connected graph of (n+1) nodes, the maximal number of deducible variables through others is n; these variables are carried by the branches of the considered spanning tree. R 1 (t) = α 1 (1-r(t))r(t) p-1 (11) Proof: considering the expression (2) and the canonical matrix (table 3) whose regular part variables are carried by the branches of the spanning tree. It is obvious that these variables are deducible through the knowledge of variable values of the singular part of the matrix.

More generally, we have:

R i (t) = α i (1 -r(t)) i r(t) p-i (12)
where the coefficient α i is the total number of configurations that admit a number i of failed sensors while ensuring the observability of variables required for the control of the process.

with regard to the criterion of the minimum sensor cost so as to ensure:

-the minimal observability of variables required for the control (L 1 ), The problem consists in computing R i (t). We therefore compute the coefficients α i for i = 0,...,v. According to the definition of the coefficient α i , cycles comprising only variables non required for the control (list L 2 ) have not to be counted. Furthermore, we eliminate cycles whose number of measured variables is higher than v. These cycles are not useful for the proposed analysis in the case where the number of simultaneous failures is less or equal to v. We finally obtain a so-called C Tv cycle matrix whose analysis comprises three steps:

-redundancy degrees of variables of the sets Ld k .

Having concern for the simplicity and without loss of generality, the value of k is limited to 1.

Algorithm

It comprises three steps: the first consists to create the matrix of all cycles, the second to select an optimal set of variables to be measured with respect to the criterion of cost and minimal observability of necessary variables (list L 1 ), the third to choose supplementary variables to be measured in order to ensure redundancy degree 1 of variables of the set Ld 1 while minimizing the corresponding sensor cost and taking into account measured variables given by the second step.

step 1: extracting among cycles of the matrix C Tv those whose number of measured variables varies from 1 to i (cycles with k measured variables such that k is higher than i will have one measured variable at least when i sensors fail simultaneously, which is not interesting for the research of non admitted combinations), step 2: forming C p i combinations of i failed sensors among p sensors. For each combination, variables measured by failed sensors are considered as unmeasured ones. We then calculate the number of measured variables for each cycle. According to the observability rule III, if all cycles include a measured variable at least, the considered combination is admissible, step 1: useful cycle matrix From the lists L 1 and L 2 , the procedure of researching a spanning tree is the same as that of section 3. The measured and unmeasured variables might be replaced by those of the lists L 1 and L 2 respectively. Then, we deduce the fundamental cycle matrix C FA and the cycle matrix C T . step 2: minimal observability of variables required for the control. step 3: deducing α i which is equal to the number of admissible combinations, then R i (t).

According to the rule V, we need to measure a variable at least per cycle containing a variable of L 1 . We simplify this research by eliminating, among cycles of C T , those containing only variables of the subset L 2 . The resulting matrix C 1 of dimension (n c1 .m c1 ) is obtained. During this step, we first compute all possible sets of variables to be measured L int , then select one, L s1 which satisfies the cost criterion.

Consider the example of the figure 1 and the following lists:

L 1 = {1, 4, 6, 9, 10} L 2 = {2, 3, 5, 7, 8} X m = {3, 4, 5, 8, 9, 10} X m -= {1, 2, 6, 7}
The greatest admissible number of failed sensors is equal to 2. Cycles of the graph are presented in the table 6. The matrix C T2 whose cycles contain 2 measured variables at the most, is given in the table 10:

step 3: redundancy of variables of the set Ld 1 .

According to the rule V, we have to measure two variables at least per cycle containing a variable of Ld 1 . First, all possible sets of variables to be measured are computed. Before selecting a list satisfying the cost criterion, we take into account the proposed measured variables of the precedent list L s1 . Then, we select the list L s2 fulfilling the cost criterion.

4 9 1 6 2 7 8 10 3 5 N .

.

Results

Let us note λ the common failure rate of all sensors. The coefficient α 0 is equal to 1. Among six possibilities to consider a sensor failing, only the possibility to consider the failure of the sensor measuring the variable 8 is not admitted (according to the cycle 1). Thus, α 1 is equal to 5. Combinations including the variable 8 are not retained. Besides, next combinations are not accepted either: (9,10), (9,3), (10, 3) according to the last three cycles. We have 7 admissible combinations for α 2 . Therefore, the function of reliability is:

Consider the process network of the figure 1 with L 1 = {1, 4, 6, 9, 10} and L 2 = {2, 3, 5, 7, 8}, we wish to define the measurement system with a redundancy degree 1 for variables of the list Ld 1 = {1, 9}. To each variable is associated a weight representing the cost of the sensor (table 11).

Variable

Cost

Variable Cost 1 3 6 4 2 3 7 8 3 4 8 7 4
1 9 2 R(t) = e -6λt + 5(1-e -λt )e -5λt + 7(1-e -λt ) 2 e -4λt 5 9 10 5 Table 11. Costs associated with variables. Then, the value of the MTTF is 0.45/λ step 1: the cycle matrix is the same as that presented in the table 6. It only includes cycles containing at least a variable of the list L 1 ; let us note it by C 1 .

DESIGN OF MEASUREMENT SYSTEM

step 2: the whole possible measurement lists for the selection (lists of intersection L int ) is not presented here due to their sizes. Taking into account the constraint of minimum cost, the following variables are to be measured to guarantee the minimal observability of the list L 1 : L s1 = {1, 2, 4, 9}

Position of the problem and objective

We are provided with the subset of variables required for the control of the process (list L 1 ). A weight, that is proportional to the cost of sensor, is assigned to each variable. Among the variables of the subset L 1 , some are important for the functioning security; these are "high availability" variables. This demand may be ensured by analytic redundancy. According to the previous rules and especially the rule V, we can obtain a redundancy degree k for a high availability variable solely if the minimum number of measurable variables of all cycles to which this variable belongs is greater or equal to k+1. Necessary variables (subset L 1 ) are assumed measurable. Therefore, the purpose is the selection of variables to be measured step 3: the variables 1, 2, 4 and 9 are considered as measured ones, the matrix C 11 contains cycle number 2, 8, 11 and 12 of the matrix C 1 (because they contain less than two measured variables). Besides, we have suppressed columns of C 1 corresponding to variables 1 and 9 (list Ld 1 ). The resultant matrix C 11 is the following. We have presented an analysis of the observability, the redundancy degree and the reliability; this technique is based on the analysis of cycles of the graph associated with the process. This analysis has been applied to the conception of the measurement system of a process. The proposed solution gives the optimum number of sensors and their location in order to: obtain (1) the minimal observability of variables required for the control, (2) redundancy degrees of variable of high degree of availability while minimizing the cost of measurements. In the case where several solutions provide a same minimum cost, we can use a complementary criterion taking into account the reliability. The use of cycles is all the more interesting for the conception of measurement system as it allows one to obtain a list of measurements easily, according to one or several given criteria. The solution of a multicriterion problem including precision of estimation, sensitivity to a modification of precision, observability, redundancy degree and reliability is a future direction of research.

We need to measure one variable of each list of the lists (corresponding to a cycle of C 11 ): {7,10}, {10,3}, {6,2,8,10,3}, {4,6,8,10,5}. Finally, the supplementary variable to be measured is the variable 10.

In conclusion, the variables to be measured that meet the objective while minimizing the sensor cost are {1, 2, 4, 9, 10}. The corresponding total cost is 14. The reader may verify that cycles (of the matrix C 1 ) containing variables 1 and 9 comprise at least two measured variables. We can calculate the reliability of this measurement system with the help of the algorithm of the section 3. The evolution of the reliability is represented by R(t). The calculation is carried out with an identical failure rate for each sensor being equal to 1.25 E -4 . The calculation of MTTF is in fact represented by the area of the surface limited between the curve R(t) and the time axis. In the figure 3, the curve in solid line represents the reliability R(t) of the system with the measured variable list {1, 2, 4, 9, 10}. As a comparison, the curve in dashed line represents the reliability of the system obtained with the measured variable list {1, 2, 4, 9}, that only ensures the minimal observability of variables required for the control. For the measurement system ensuring only the minimal observability of variables required for the control, we obtain a value of MTTF being equal to 0.25/1.25 E 4 . The measurement system satisfying the objective gives a better value of MTTF with 0.45/1.25.E 4 . We observe that the system ensuring the redundancy degree 1 of variables 1 and 9 is more reliable than that not providing redundancy equations. If the cost of sensors is not taken into account, several configurations may satisfy the objectives of minimal observability and degrees of redundancy but present a less important reliability with a same number of sensors. Let us consider a measurement system (number 1) involving the measurements of the variables {1, 2, 4, 9, 10} and the second one (number 2) involving the measurements of variables {1, 4, 6, 9, 10}. It is easy to verify that the second measurement system meets the previous requirements. All the variables required for the control are observable and variables 1 and 9 are redundant of degree 1. The figure 4 shows the reliability of the two measurement systems. Fig. 4. Comparison of the evolution of the reliability From this figure, it is obvious that the reliability of the measurement system 2 is inferior than that of measurement system 1 and so is the MTTF (0.4/1.25.E 4 , for the

  us denote c = m-n, the number of fundamental cycles of a connected graph and C c k the binomial number of k elements among c. According to the theorem 1, each of the c fundamental cycles contains a chord. Cycles formed by the addition of k distinct fundamental cycles contain k Table 4. Fundamental cycle matrix For the process of the figure 1 where X m

  6

Fig. 3 .

 3 Fig. 3. Evolution of the reliability.

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  Maquin D., Darouach M., Ragot J. (1989), "Observability and data validation of bilinear systems". Congrès IFAC AIPAC' 89, Nancy, 3-5 juillet. Ragot J., Darouach M., Maquin D., Bloch G. (1990), "Validation de données et diagnostic". Traité des nouvelles technologies, série diagnosis et maintenance, Hermès. Ragot J., Maquin D., Darouach M. (1991), "Analysis of generalized bilinear systems -application to diagnostic". IMACS Symposium on Modelling an Control of Technological Systems. Ragot J., Maquin D., Bloch G. (1992), "Sensor positioning for processes described by bilinear equations". Revue Diagnostic et Sûreté de Fonctionnement, 2 (2), pp. 115-132. Schrick D. V. (1992), "Investigations of reliability for instrument fault detection state-estimator schemes". Revue Diagnostic et Sûreté de FonctionnementM., Mah R. S. H. (1981), "Observability and redundancy classification in process networks." Chemical Engineering Science, 36, pp. 1941-1954. Turbatte H. C., Maquin D., Huynh C. T., Ragot J. (1993), "Conception de l'architecture d'un système d'instrumentation -Application à un procédé pétrochimique". Proceedings of the International Conference on Fault Diagnostic, 1, pp. 406-415, Toulouse, France, avril 5-7. Vaclavek V. (1969), "Studies on system engineering III. Optimal choice of the balance measurements in complicated chemical systems". Chemical Engineering Science , 24, pp. 947-955. Villemeur A. (1988), "Sûreté de fonctionnement des systèmes industriels".Eyrolles, 795 p.

Table 6 .

 6 , the cycle matrix, where measured variables are in bold type, is presented in the table 5 (the fundamental cycle matrix corresponds to the four first rows of this matrix). Matrix of cycles comprising two measured variables at the most per cycle.

	4	9	1	6	2	7	8	10 3	5
	.	.	1	1	1	.	1	.	.	.

Table 8 .

 8 Reduced canonical matrix M C

Table 10 .

 10 Matrix of cycles C T2 .

	. . .	1 1 .	1 1 . . 1 . 1 .	1 . . 1 . 1 . .	1 . . 1 . . . . 1 . 1 1	. . . .	1 2 2 2	Finally, the set of variables to be measured fulfilling the objective is the union of the lists L s1 and L s2 .

Table 12 .

 12 Cycle matrix C 11

			2	7	8	10	3	5	measurement system 2) although the number of
	. . . 1	. . 1 1	. . 1 .	1 . . .	. . 1 1	1 1 1 1	. 1 1 .	. . . 1	measurements is the same. 5. CONCLUSION


Jon Sertucha
email: jsertucha@azterlan.es

Garikoitz Artola
email: gartola@azterlan.es

Urko De

La Torre
email: udelatorre@azterlan.es

Jacques Lacaze
email: jacques.lacaze@ensiacet.fr

Chunky Graphite in Low and High Silicon Spheroidal Graphite Cast Irons-Occurrence, Control and Effect on Mechanical Properties

Keywords: spheroidal graphite, chunky graphite, spiky graphite, cast iron, tin, antimony, mechanical properties

Chunky graphite appears easily in heavy-section spheroidal graphite cast irons and is known to affect their mechanical properties. A dedicated experiment has been developed to study the effect of the most important chemical variables reported to change the amount of chunky graphite, namely the content in silicon and in rare earths. Quite unexpectedly, controlled rare earths contents appear beneficial for decreasing chunky graphite when using standard charge materials. Tin is shown to decrease chunky graphite appearance and it is evidenced that this effect is not related to rare earths. Finally, the effect of tin and antimony are compared and it is noticed that both suppress chunky graphite but also lead to some spiky graphite when no rare earth is added. Chunky graphite negatively affects the room temperature mechanical properties, though much more in the case of low silicon spheroidal graphite cast irons than in high silicon ones. Spiky graphite has been found to be much more detrimental and should thus be avoided.

Introduction

Spheroidal graphite cast iron has rapidly become one of the most important engineering materials after its discovery some 70 years ago [START_REF] Millis | Cast Ferrous Alloy[END_REF]. Since then, the casting of spheroidal graphite cast irons has been successfully carried out by adding to the base melt an appropriate quantity of magnesium whose first action is to decrease the amount of sulfur and oxygen. Calcium and rare earths (RE) may be added together with magnesium for supporting this purification treatment with the further advantage of lowering the necessary amount of magnesium and thus the pyrotechnic effect of the spheroidization treatment. Spheroidal graphite is effectively obtained when the final amount of dissolved magnesium is about 0.020-0.025 wt.% [START_REF] Tybulczuk | Etude sur les formes dégénérées du graphite en vue du contrôle magnétique des pièces moulées en fonte à graphite sphéroïdal[END_REF][START_REF] Suárez | Active Mg estimation using thermal analysis: A rapid method to control nodularity in ductile cast iron production[END_REF] while compacted graphite corresponds to a level of 0.008-0.017 wt.% [START_REF] Dawson | Compacted graphite iron: A viable alternative[END_REF] in casting sections lower than about 60-70 mm.

Even if the spheroidization treatment has been carried out correctly, the presence of trace elements in the charge materials for the preparation of the melt may hinder the production of spheroidal graphite and then might affect the service properties of the cast parts. Such impurities lead to spheroidal graphite degeneracy known as spiky, crab-like and eventually mesh graphite as previously reviewed [START_REF] Lacaze | Effects of impurities on graphite shape during solidification of spheroidal graphite cast ions[END_REF]. These different kinds of degenerate graphite first appear as elongated protrusions growing out from the spheroids, suggesting that they develop in the prismatic a direction of the graphite as does lamellar graphite. In his pioneering work on the degeneration of spheroidal graphite, Thielemann [START_REF] Thielemann | Zur Wirkung von Spurennelementen im Gusseisen mit Kugelgraphit[END_REF] quantified the deleterious effect of Al, As, Bi, Pb, Sb, Sn and Ti, but many other trace elements have been identified, see the review by Reynaud [START_REF] Reynaud | Oligo-éléments et Fontes[END_REF]. It has been realized since a long time that adding rare earths (RE) offsets the problem in most cases [START_REF] Morrogh | Influence of some residual elements and their neutralization in magnesium-treated nodular cast iron[END_REF][START_REF] Sawyer | Effects and neutralization of trace elements in gray and ductile irons[END_REF][START_REF] Buhr | Vermiculite graphite formation in heavy section nodular iron castings[END_REF] which is thought to be firstly due to RE catching up impurities as stable compounds [START_REF] Zhao Bofa | The mechanism of interaction of Mg, Sb and Ce in ductile iron[END_REF][START_REF] Udomon | Comments concerning the interaction of rare earths with subversive elements in cast irons[END_REF][START_REF] Javaid | Quality control of heavy-section ductile cast irons[END_REF]. Amongst others, Zhao Bofan and Langer [START_REF] Zhao Bofa | The mechanism of interaction of Mg, Sb and Ce in ductile iron[END_REF] did effectively evidence Ce-Sb compounds using melts that contained high levels of these elements (>500 ppm), see the review by Javaid and Loper [START_REF] Javaid | Quality control of heavy-section ductile cast irons[END_REF].

The effect of these trace elements becomes more and more evident as the casting size is increased which seems to have been first related to the fading of the spheroidizing and inoculation treatments because of the lengthening of the solidification time [START_REF] Javaid | Quality control of heavy-section ductile cast irons[END_REF]. An obvious action would thus be to increase the amount of magnesium and RE but this leads to the so-called over-treatment which has eventually been associated with the occurrence of a specific form of degenerate spheroidal graphite, namely chunky graphite [START_REF] Javaid | Quality control of heavy-section ductile cast irons[END_REF][START_REF] Hoover | A literature survey on degenerate graphite in heavy section ductile iron[END_REF]. From a metallurgical point of view, the main feature of chunky graphite is that its overall growth direction is along the basal c direction of graphite as it is for spheroidal graphite. Thus, the growth mechanism of chunky graphite certainly differs from the one mentioned above for protrusions. From a practical point of view, chunky graphite appears either as interconnected thin strings of graphite in large cells or as coarser strings in the last to freeze areas [START_REF] Xi | Formation and prevention of chunky graphite in slowly cooled nodular irons[END_REF]. On 2D metallographic sections, chunky graphite may easily be confused with other graphite degeneracies and, in particular, compacted graphite [START_REF] Buhr | Vermiculite graphite formation in heavy section nodular iron castings[END_REF] as shown with the wording "grouped vermicular" graphite used by Hoover [START_REF] Hoover | A literature survey on degenerate graphite in heavy section ductile iron[END_REF]. What is anyway clear is that it is a cause of rejection of castings because it is expected to lead to a significant decrease of the mechanical properties, see the recent review by Baer [START_REF] Baer | Chunky graphite in ferritic spheroidal graphite cast iron: Formation, prevention, characterization, impact on properties: An overview[END_REF].

Amongst the various factors that increase the risk of chunky graphite are casting size (or low cooling rate), high silicon content and over-treatment with spheroidizers, in particular, RE as reviewed a long time ago by Hoover [START_REF] Hoover | A literature survey on degenerate graphite in heavy section ductile iron[END_REF] and again demonstrated since then by Asenjo et al. [START_REF] Asenjo | Microstructure investigation of small-section nodular iron castings with chunky graphite[END_REF] although rejected by Gagné and Argo [START_REF] Gagné | Heavy section ductile iron castings-Part I: Structure and properties[END_REF]. An increase in the casting size does enhance all types of spheroidal graphite degeneracies, and this may be due in part to the fact that nodules are fewer and larger. This led to the suggestion of enhancing graphite nucleation by higher cooling rates or increased inoculation. While the effect of increased cooling rate, in particular by the use of chills, seems clear and accepted [START_REF] Baer | Chunky graphite in ferritic spheroidal graphite cast iron: Formation, prevention, characterization, impact on properties: An overview[END_REF], the beneficial effect of increased inoculation is controversial [START_REF] Karsay | Control of graphite structure in heavy ductile iron castings[END_REF][START_REF] Ignaszak | The risk of ductile iron post-inoculation for heavy section castings[END_REF][START_REF] Asenjo | Effect of mould inoculation on the formation of chunky graphite in heavy-section spheroidal graphite cast iron parts[END_REF] though it is still considered [START_REF] Ferro | Effect of inoculant containing rare earth metals and bismuth on microstructure and mechanical properties of heavy-section near-eutectic ductile iron castings[END_REF][START_REF] Mihalic Pokopec | Effect of inoculation on the formation of chunky graphite in ductile-iron castings[END_REF]. That fewer and larger nodules are more prone to degenerate is well known, this has been recently evidenced by X-ray computerized tomography (4D-XCT) [START_REF] Azem | Synchrotron quantification of graphite nodule evolution during the solidification of cast iron[END_REF] and theorized for the case of exploded graphite [START_REF] Ghergu | A possible mechanism for the formation of exploded graphite in nodular cast irons[END_REF]. Finally, using the fact that chunky graphite precipitates during coupled growth with austenite, it has been suggested that the effect of increased silicon could be related to a change in the austenite/liquid surface energy [START_REF] De La Torre | Chunky graphite formation in ductile cast irons: Effect of silicon, carbon and rare earths[END_REF].

The practical way to counteract the detrimental effect of RE is to add antimony, though other elements have been mentioned such as As, Pb and Sn [START_REF] Karsay | Control of graphite structure in heavy ductile iron castings[END_REF]. It is of prime interest to stress that these trace elements are otherwise considered as promoting degeneracy of spheroidal graphite when present above a critical level and should be balanced by RE addition in that case. This ambivalent role of trace elements and RE is quite striking and would lead to the conclusion that chunky graphite should not appear if pure charge materials and no RE were used for melt preparation. That this is so has been recently reported by Jia-xin Bai [START_REF] Bai | Selection of raw materials and control of trace elements for production of high-quality SG iron[END_REF] for low-silicon cast irons prepared with high-purity charge materials, but such a way certainly leads to a dramatic cost increase. It remains unclear if an optimum addition of elements such as Sb and Sn could be found that would be below the critical limit for graphite degeneracy when no RE are present, but high enough for eliminating the risk of chunky graphite.

Although the effect of silicon on the formation of chunky graphite has long been documented [START_REF] Xi | Formation and prevention of chunky graphite in slowly cooled nodular irons[END_REF][START_REF] Karsay | Control of graphite structure in heavy ductile iron castings[END_REF][START_REF] Prinz | Untersuchung von Ursachen von Graphitentartungen bei Gusseisen mit Kugelgraphit in Form von Chunky-Graphit[END_REF][START_REF] Hamberg | Chunky graphite in ductile iron, a literature survey[END_REF][START_REF] González-Martínez | Effects of high silicon contents on graphite morphology and room temperature mechanical properties of as-cast ferritic ductile cast irons. Part I-Microstructure[END_REF], much less is known about its appearance in the high-silicon spheroidal graphite cast irons that are currently under development [START_REF] De La Torre | Chunky graphite formation in ductile cast irons: Effect of silicon, carbon and rare earths[END_REF][START_REF] Bauer | Effect of bismuth on preventing chunky graphite in high-silicon ductile iron castings[END_REF]. Such an interest is also triggered by the fact that it has been demonstrated that chunky graphite does not affect room temperature mechanical properties of high-silicon spheroidal graphite cast irons contrary to the case of low-silicon alloys [START_REF] González-Martínez | Effects of high silicon contents on graphite morphology and room temperature mechanical properties of as-cast ferritic ductile cast irons. Part II  Mechanical properties[END_REF]. Accordingly, the present investigation was dedicated to: (i) the development of an instrumented small scale casting giving low cooling rate to generate conditions prone for chunky graphite formation; (ii) further investigating the effect of high Si as compared to standard Si content, and the effect of a low level of RE and Sb on chunky graphite occurrence; (iii) documenting the effect of Sn in eliminating chunky graphite and confronting it to that of Sb; (iv) characterizing the impact of chunky graphite on mechanical properties of low and high silicon spheroidal graphite iron.

Experimental

For studying the formation of chunky graphite in a foundry laboratory, the basic idea was to define a casting with a size as small as possible but with a long enough solidification time to promote heavy section castings behavior. A first trial (P1) was performed using an exothermic mold (based on an exothermic feeder) but an important part of the melt together with slag came out of the mold just after pouring. Due to this fact, exothermic molds were rejected and were replaced by isolating ones with satisfactory results. Nine further experimental trials have then been carried out which are chronologically denoted by the trial number following the capital letter "P". Table 1 shows a summary of the trials. The melts were prepared as described previously [START_REF] De La Torre | Chunky graphite formation in ductile cast irons: Effect of silicon, carbon and rare earths[END_REF] in a medium frequency induction furnace. When needed a FeSi75 alloy was used for adjusting the silicon content. The base melt held in the furnace was transferred at about 1500 °C to a ladle with a chamber in which FeSiMg spheroidizer and inoculant were located and covered with steel cuttings. The amount of spheroidizer and inoculant was 1.2-1.3% and 0.30% of the melt weight, respectively. Two FeSiMg alloys and two inoculants have been used that contain or not some RE. The inoculant without RE was the same FeSi75 alloy as the one mentioned above, after crushing and proper size selection (0.2-0.7 mm). A commercial product, here after denoted as Anoc, was used as RE-bearing inoculant. The compositions of all these treatment alloys are listed in Table 2. Intermediate RE levels were obtained by appropriate mixing of products. Once the Mg treatment had finished, a chemical composition sample was obtained and analyzed by spark emission spectroscopy for controlling and validating that the target composition was obtained. The alloy was then poured in two thermal analysis (TA) cups (for reproducibility) and in the mold illustrated in Figure 1. This mold was instrumented with four Ø3 mm stainless steel sheathed K-type thermocouples whose locations are indicated in the figure. These thermocouples were covered with a further external ceramic sheath which was in contact with the alloy cast into the molds. Exceptionally, K-type thermocouples with a metallic sheath 1.5 mm in diameter were employed in P8 trial while keeping the same experimental method. Cooling curves from the TA cups and from the thermocouples were recorded in all cases.

After cooling down, the TA cups and the cylindrical castings were cut and prepared for metallographic analysis to check their microstructure and to determine the area fraction of the different graphite particles present in the central zone of the TA cups (three different fields per cup) and in locations close to the tip of the four thermocouples in the cylindrical castings (five different fields in each location). The magnification of all images used in these analyses was x100 corresponding to an area of 0.757 mm 2 each. The micrographs were then binarized and all particles with an area less than 25 µm 2 were disregarded. The total count, Ncount, and area fraction, fA, of graphite particles were then evaluated. Then, the different types of graphite particles were sorted into three categories following criteria described in Table 3. The circularity and aspect ratio are respectively defined as 4π• P and ma min , where A is the area of a given graphite particle, P is its perimeter, and Lmin and Lmax are its minimum and maximum lengths, respectively. Table 3. Classification criteria which are used to determine the different area fractions. Types V and VI relate to regular and highly regular nodules, respectively, while type III corresponds to chunky graphite in most cases (trials P2 to P5 and P7), except in the case of addition of Sn or Sb when it was also used to quantify spiky graphite (trials P6, P8 and P9). In trial P10, both chunky graphite and spiky graphite were present, so their area fractions were separately determined. This differentiation between chunky and spiky graphite was performed by means of visual inspection after the type III particles had been extracted by automated image analysis. The area fractions of graphite type III, V and VI were evaluated and then expressed as the relative value of the total graphite area fraction, fIII_A, fV_A and fVI_A, with fIII_A + fV_A + fVI_A = 1. After graphite characterization, the samples were etched with Nital for evaluating the constitution of the matrix, i.e., the fraction of pearlite and ferrite.

Class Circularity Aspect Ratio

A representative piece of each produced cylindrical casting was also used to determine the final composition of the alloys. Carbon and sulfur contents were determined by combustion (LECO CS300) while the inductively coupled plasma mass spectrometer, ICP-MS, (Agilent 7500ce) was used for other elements after dissolving a metallic sample in a mixture of acids. Finally, silicon contents were determined by gravimetric techniques.

For mechanical testing, a series of eight additional castings were prepared with the mold shown in Figure 1 but without thermocouples. These castings will be referred to as PM1-PM8 in Section 3.4. All tensile tests were performed on specimens with a gage 5 mm in diameter and 30 mm in length with threaded ends (Figure 2). On these specimens, the ultimate tensile strength (UTS), the elongation (A) and the yield strength (YS) values were measured using a Zwick Z100 tensile testing equipment at a constant strain rate of 2.5•× 10 -4 s -1 .

Results and Discussion

Table 4 contains the results of the chemical analyses carried out on samples obtained from the produced cylindrical castings. It was verified that the medal sample and the casting itself gave very similar chemical analysis results, thus ensuring that the whole casting procedure was correct. Values of the carbon equivalent (CE) that were determined by using the expressions proposed by Castro et al. [START_REF] Castro | Simulation of thermal analysis applied to the description of the solidification of hypereutectic SG cast irons[END_REF], CE99, and in ASM handbook [START_REF] Stefanescu | Thermodynamics principles as applied to cast iron[END_REF], CEASM, have been also included in Table 4. The calcium and bismuth contents were checked also and found below 0.0010 wt.% and 0.0005 wt.%, respectively, in all cases.

Effect of Si and RE Contents

The first four castings (P2 to P5) were prepared for studying the effect of silicon (2.3 and 4.0 wt.%) and that of RE when added at the level of 100-150 ppm. Figure 3 illustrates the temperature recorded with the thermocouples T1-T4 in casting P5. These records were similar for all four trials P2-P5 and show that solidification proceeded in two steps, namely a pro-eutectic reaction and then a eutectic plateau. The pro-eutectic reaction is hardly visible for T4, while it is clearly sensed with all three other thermocouples. While for T1 it leads only to a slope change, the pro-eutectic thermal arrest shows recalescence for T2 and T3 suggesting equiaxed solidification of austenite in these two locations. It may thus be postulated that T4 is in an area with a significant temperature gradient where austenite grows as columnar dendrites, T1 at a location where the temperature gradient is lower but still present, while T2 and T3 are within or close to the thermal center of the casting. In relation with this, it is seen that the eutectic recalescence is the highest for T2 and T3, lower in T1 and is very limited in T4. The above description agrees with usual observation and analysis of casting solidification, see e.g., Dioszegi et al. [START_REF] Dioszegi | Prediction of volume fraction of primary austenite at solidification of lamellar graphite cast iron using thermal analyses[END_REF]. The results obtained from the metallographic characterization are reported in Table 5. The total area fraction, fA (%), and total particle count, Ncount (mm -), of graphite are listed as well as the relative area fraction of each type of graphite. For these four castings, all graphite of class III was associated with chunky graphite. Chunky graphite was observed in the four locations T1-T4 of all cylindrical castings P2-P5, though in an amount significantly higher for P2 and P3 than for P4 and P5 in relation with the higher silicon content in these former castings. When analyzing the microstructure, it was noted that the amount of chunky graphite could in several cases vary a lot within the 3 (TA cups) or 5 (cylinders) measurement fields made on a given location. This led to the standard deviation being sometimes equal or even higher than the mean value. This suggested indicating between brackets the minimum and maximum values of fIII_A. Such variability affects also the number of class V and class VI spheroids and also the total count of particles, Ncount. The standard deviation for this latter is given for illustration.

It must be remarked that the quantification of chunky graphite can be troublesome due to the fact that chunky cells have millimeter size, i.e., a size similar to the size of the micrographs used in the present work. Using lower magnification micrographs would have led to other difficulties such as not separating properly the various shapes of graphite. In practice area fraction of class III graphite will be measured with a low standard deviation if it is either higher than 0.80 or lower than 0.20, but the scattering could be very high for intermediate values. Once this was realized during the first characterizations for this study, it was decided to select the location of the three (resp. five) micrographs: one should show the highest amount, one the lowest one and the third (resp. last three) should be representative of the area under investigation. In this way, it was thought that the error on the average value would be minimized.

Pro-eutectic reaction

Eutectic reaction

Owing to the much higher cooling rate, the thermal cups showed no chunky graphite for P4 and P5 and a low amount for P2 and P3. Thus, as expected, they are of little use for assessing the tendency of a melt to give chunky graphite. Nonetheless, it was of interest to notice a change in distribution. The micrographs in Figure 4 illustrate the graphite distribution in casting P2 for the location T3 and for the central part of the TA cup. It is seen that the decrease in the chunky graphite fraction corresponds to a significant change in its distribution, with large cells of interconnected strings replaced by isolated coarser strings. Cells are expected to have appeared early during the eutectic reaction while isolated strings relate to areas where solidification ended [START_REF] Xi | Formation and prevention of chunky graphite in slowly cooled nodular irons[END_REF]. Finally, in low silicon alloys, some rare special features were observed that are illustrated in Figure 4c. The shape suggests a two-phase growth around a dendrite arm. The area fraction of chunky graphite particles classified as type III has been plotted in Figure 5 for the four thermocouple locations and the TA cups of all P2-P5 castings. The most relevant chemical information is also indicated for each casting to ease reading. It is seen that:

1. Increasing the silicon content from 2.3 wt.% to about 4.0 wt.% leads effectively to a significant increase of fIII_A.

2. Though TA cups show little or no chunky graphite, they appear sorted as the castings are. The melts giving the highest fIII_A values in cups are also those giving the highest fIII_A in the cast cylinders.

3. For both high and low silicon alloys, the addition of RE to alloys free of these elements shows a decrease in chunky graphite. Furthermore, this effect seems stronger for the low silicon alloys than for the high silicon alloys. This beneficial effect of RE goes against the trend generally considered but must be associated with the low level of the addition and has been observed previously [START_REF] De La Torre | Chunky graphite formation in ductile cast irons: Effect of silicon, carbon and rare earths[END_REF]. A possible reason is that RE counteract impurities that are at very low levels such as Al, P, S and Ti, see Table 4.

Figure 5. Evolution of the chunky graphite fraction fIII_A with the chemical composition for the four thermocouple locations in the cylindrical castings and for the thermal analysis (TA) cups.

In Figure 3, it appeared that the cooling curves recorded with thermocouples T2 and T3 are capable of sensing effects that were not recorded with thermocouples T1 and T4. Owing to the larger changes in chunky graphite fraction seen in location T3 (Figure 5), this thermocouple was selected for comparing the cooling records for all four castings and this is shown in Figure 6. The corresponding chunky graphite fractions have been indicated in the figure caption.

In cast iron, the equilibrium eutectic temperature depends firstly on the alloy's silicon content, increasing by about 4.246 °C per wt.% of silicon [START_REF] Castro | Simulation of thermal analysis applied to the description of the solidification of hypereutectic SG cast irons[END_REF][START_REF] Regordosa | Quantitative analysis of the effect of inoculation and magnesium content on compacted graphite irons-Experimental approach[END_REF]. Hence, high-Si castings P2 and P3 should have a very similar equilibrium eutectic temperature whose value is higher by about 7 °C than that for low-Si castings P4 and P5. It is seen that the sorting of the curves in Figure 6 agrees with this predicted change in eutectic temperature. Further, it is noticed that the maximum plateau temperature, TER, increases with the chunky graphite fraction. This increase is in agreement with the fact that the growth rate of chunky graphite cells is much higher than that of spheroidal graphite eutectic cells, being in between undercooled and lamellar graphite growth rates according to Källbom et al. [START_REF] Källbom | On the solidification sequence of ductile iron castings containing chunky graphite[END_REF].

Effect of Intermediate Level of RE and of Antimony

It has been seen in the previous section that adding some RE decreased the amount of chunky graphite though its level remained quite high. This prompted a trial with an intermediate level of RE in an attempt to find an optimum value for high silicon alloys. In effect, Skaland [START_REF] Skaland | Nucleation mechanism in ductile iron[END_REF] has reviewed a number of works suggesting there is an optimum RE content for either graphite shape and nodularity or nodule count, and it was wondered if such a relation could be similarly found for chunky graphite. The amount of RE that was targeted was half that added in P2 and the corresponding trial has been numbered P7. The total amount of RE in P7 is about 50 ppm close to the value of 60-80 ppm achieved in a previous study [START_REF] De La Torre | Chunky graphite formation in ductile cast irons: Effect of silicon, carbon and rare earths[END_REF] and slightly lower than half the 110-165 ppm in P2 and P4.

The results of the metallographic characterization of casting P7 are given in Table 6. As before, minimum and maximum values for fIII_A and standard deviation for Ncount are listed which show marked variability in some locations. By comparison with trials P2 and P3, it was observed that the intermediate value of RE did effectively decrease the amount of chunky graphite in the locations T1-T3 but surprisingly not in T4, see Figure 7. It is also noticed that the amount of chunky graphite remains quite high even in the cups where 10% of the graphite is classified as chunky. In this respect, it may be noted that the total amount of deleterious elements, Al, P, S and Ti, at 330 ppm is far above the amount of RE measured in casting P7. Accordingly, it seems unlikely that there exists an optimum amount of RE that would allow suppressing chunky graphite.

The above conclusion as well as the literature review suggested another trial consisting in adding antimony at the usual level used to avoid chunky graphite but without adding RE. This goes in line with the observation that a very low level of Sb improves nodularity in RE-free standard spheroidal graphite cast irons [START_REF] Tybulczuk | Etude sur les formes dégénérées du graphite en vue du contrôle magnétique des pièces moulées en fonte à graphite sphéroïdal[END_REF][START_REF] Campomanes | Effect of minute additions of antimony on structure and properties of ductile iron[END_REF][START_REF] Wessén | Influence of antimony on microstructure and mechanical properties in thick-walled ductile iron castings[END_REF][START_REF] Dekker | Effect of antimony on graphite growth in ductile iron[END_REF] though there is a risk of appearance of spiky graphite if the Sb addition is higher than 100 ppm [START_REF] Javaid | Quality control of heavy-section ductile cast irons[END_REF] or 200 ppm [START_REF] Gilbert | A new look at subversive elements in nodular irons[END_REF]. In the present work, an addition of 39 ppm of antimony without RE was done in casting P9, and this casting was found to be totally free of chunky graphite. However, some spiky graphite appeared instead which is illustrated with the micrographs in Figure 8. The metallographic values are listed in Table 6 and plotted in Figure 7 where it is seen that up to 20% of graphite is spiky in location T3. Figure 7. Evolution of fIII_A for the four thermocouple locations in the cylindrical castings P2, P3, P7 and P9 and for the TA cups. In P9, this is spiky graphite that was observed instead of chunky graphite.

That antimony suppresses chunky graphite even if no RE is present demonstrates that its action is not only to fix excess RE when present as most often assumed. Further, that this finding applies to high silicon cast irons which are so prone to chunky graphite formation suggests that even a low level of antimony, as the 39 ppm added here, is sufficient to strongly affect the growth mechanism of spheroidal graphite. As above, one may wonder if an optimum addition of antimony could be found which would lead to no chunky graphite and no spiky graphite. However, any attempt to find an optimal addition of antimony at some tens of ppm would not be of practical use and this was thus not carried out.

Effect of Sn on Chunky Graphite in High-Silicon Cast Irons

In contrast with antimony, tin addition for suppressing chunky graphite in low silicon cast irons has been reported to be much higher at some 250-500 ppm in 5-8 inches side cubes [START_REF] Karsay | Control of graphite structure in heavy ductile iron castings[END_REF]. Considering the detrimental effect of silicon, three RE-free high silicon castings were prepared to study the effect of Sn at the level of 250 (P10), 500 (P8) and 1000 (P6) ppm. Table 7 gives the full record of data of graphite characterization in these castings which shows that chunky graphite totally disappeared at 0.051 wt.% (casting P8) and 0.094 wt.% (casting P6) of Sn but also that spiky graphite appeared in all castings with added Sn. In case of P10 at 230 ppm Sn, both chunky graphite and spiky graphite were observed. Note again that the scattering is quite high for some of the measurements. These results are compared in Figure 9 to values for casting P3 without RE and no Sn.

Table 7. Castings P6, P8 and P10. Average of measured values for total area fraction of graphite (fA), relative area fractions for each type of graphite particles (fIII_A, fV_A and fVI_A) and total particle counts, Ncount. fIII_A is split in either or both fCHG_A and fspiky_A fractions. Between brackets are listed the minimum and maximum values of fIII_A and the standard deviation for Ncount. Looking at Figure 9 suggests comparing the microstructures in locations T1-T3 where both chunky graphite and spiky graphite formed in casting P10 to location T4 where only spiky graphite was observed in the three castings with added Sn. This comparison is illustrated in Figure 10 with micrographs in location T1 in the column to the left and in location T4 in the column to the right. Though this may need further investigation, it seems that spiky graphite appeared late during solidification in all three castings and both locations. In location T1 of casting P10, chunky graphite seems to have precipitated at an intermediate solidification stage. Such characteristics are guessed to relate to microsegregation building up during solidification.

Trial

Figure 9.

Evolution of fIII_A for the four thermocouple locations in the cylindrical castings P3, P6, P8 and P10, and for the TA cups. Chunky graphite relates to solid lines and spiky graphite to dotted lines; note that both were present in casting P10.

Whatever the amount of added Sn, it is of interest to notice that the highest amount of spiky graphite that has been measured is much lower than when as little as 39 ppm of antimony was added in casting P9. Furthermore, based on Figure 9 and the micrographs in Figure 10, the intermediate level of Sn at 500 ppm would appear satisfactory for avoiding chunky graphite and minimizing spiky graphite.

Pursuing the above cooling curve analysis, Figure 11 compares the cooling curves at T3 for castings P3 without the addition of Sn and P6 at 0.094 wt.% Sn. The decrease in the temperature of the first arrest associated with precipitation of austenite is in line with the change in the carbon equivalent of these alloys, but the more striking feature is that the eutectic plateau differs significantly. For P3 which has been shown to present 87% of chunky graphite, there is a marked recalescence as already pointed out. With the addition of 0.094 wt.% Sn in P6, this recalescence has nearly disappeared while the temperature of the start of the eutectic plateau is the same as in P3. Due to the use of different thermocouples for P8, the cooling curve at T3 cannot be compared because the eutectic was recorded at a lower temperature, but again it did present only very little recalescence in relation to the low level of chunky graphite found in this casting. The whole set of available data of chunky graphite fraction versus recalescence for the cylindrical castings is plotted in Figure 12. As expected, and described above, there is a clear trend for an increased recalescence when chunky graphite becomes more and more prevalent. However, there are outliers on or close to the axes which would hinder using such a relationship for prediction purposes, apart maybe if considering the thermocouple located in T1. It may be worth noting that the maximum recalescence recorded at the center of square blocks with a 5 cm thermal modulus and solidifying in 2.5 h was twice as much at 8 °C [START_REF] Sertucha | Experimental investigation by thermal analysis of the formation of chunky graphite during solidification of heavy-section spheroidal graphite cast iron parts[END_REF] than the value recorded in the cylindrical castings of this study. This is a clear indication that even with the molds used in the present study local thermal effects are very much dependent on the heat transfer conditions of the whole casting. values are associated with casting P7 and moreover the highest of them relate to location T4 which is expected to be in a well-established temperature gradient and thus should be associated with a decreased amount of chunky graphite according to the literature review. As a matter of fact, Figure 7 shows that another casting, namely P3, presented also a very high level of chunky graphite in position T4. The first idea was that the heat transfer conditions were different for these two castings than for all others. To verify this, the cooling curves were compared and are plotted in Figure 13. Unexpectedly, these cooling curves differ significantly from each other, apparently because the pouring temperature differed significantly. Thus, the cooling curves do not give any hint that could explain why these two castings P3 and P7 showed so much chunky graphite in location T4. On the other hand, this comparison points out an important fact: even in a positive temperature gradient-the cooling curve for P7 does certainly correspond to such conditions-chunky graphite can develop during the whole eutectic solidification process. In other words, the use of chills may not always be a remedy against the formation of chunky graphite. To explain the maximum in chunky graphite fraction at location T4 in castings P3 and P7, the only correlation that we found is that these are the two alloys with the lowest CE values of the series, 4.11 and 4.05 wt.% respectively. Accordingly, the eutectic reaction takes place while an austenite dendrite network is well established, and this may be particularly true within a temperature gradient. This certainly points out the conditions for the early growth of graphite which finds some support in the schematic proposed by Zhou et al. [START_REF] Zhou | Untersuchung der Gefgebildung von Gusseisen mit Kugelgraphit bei langsamer Erstarrung[END_REF] who suggested that chunky graphite nucleates at the interface between austenite and liquid where the carbon content is highest.

One important output of the present study is that Sn acts as Sb in suppressing chunky graphite formation even when RE are not involved. There is however a major difference between these elements in the needed level to be effective. While 20-50 ppm of Sb appear sufficient, ten times more are needed if Sn is employed. From a practical point of view, it may be expected an easier control with tin than with antimony. From a theoretical point of view, this strongly suggests that the mechanism by which these elements affect graphite growth differs from each other. The risk when using either Sn or Sb is to favor pearlite formation, see Appendix A.

At the highest amount of Sn they added, namely 1000 ppm, Karsay and Campomanes [START_REF] Karsay | Control of graphite structure in heavy ductile iron castings[END_REF] noticed the presence of intercellular inclusions and assumed this could affect mechanical properties. Similarly, it has been shown that spiky graphite may be even worse than chunky graphite [START_REF] Foglio | Fatigue characterization and optimization of the production process of heavy section ductile iron castings[END_REF]. This triggered the mechanical tests that are presented in the next section.

Mechanical Properties

The above microstructure results suggested studying the effect of chunky graphite and of spiky graphite on the mechanical properties. To ascertain the findings, it was thought useful to compare results for both low-Si and high-Si spheroidal graphite irons. Accordingly, four alloys were planned for each type of cast iron, with the intention to have: (1) fully spheroidal graphite; (2) a medium quantity of chunky graphite; (3) a high quantity of chunky graphite; (4) some spiky graphite. Eight new castings were thus produced with the same mold as before (Figure 1) but with no thermocouples, denoted as PM1--PM8. Five samples for the tensile test were machined out from each casting at locations 1-5 as shown in Figure 14. Table 8 shows the achieved values for composition and microstructure characteristics, while the whole compositions are listed in Appendix B. The metallographic characteristics have been determined in a cylindrical cross-section positioned as close as possible to the fracture surface of the tensile specimen. In Table 8, the results for the two specimens from casting PM1 with no CHG or spiky graphite but with elongation below 6% were greyed due to the presence of defects in the fracture surface. Hence, these samples were removed from further analysis. Looking at all other results in Table 8, it is seen that for each casting the UTS and YS values vary little. However, it is also evident that there is an important scattering of the A values.

For analyzing this scattering, it appeared of interest to look first at the best of the tensile curves for each of the castings, this is done in Figure 15 for low-silicon (a) and high-silicon (b) alloys. Along each of these curves, the amount of chunky graphite or spiky graphite is indicated. For both types of alloys, it is seen that spiky graphite fractions above 0.11-0.12 are highly effective in decreasing the elongation at rupture. Chunky graphite is far less effective on high-silicon alloys, while the results for low-silicon alloys need further analysis, see later. It must be noted that PM5-1 showed a 100% pearlitic microstructure in the spiky graphite areas and 40-50% of pearlite in the rest of the inspected cross sections, thus explaining its higher tensile strength. The presence of pearlite also contributes to the decrease in the elongation at rupture together with the spiky graphite. Focusing on high-silicon alloys, Figure 16 shows the change in UTS (a) and A (b) as a function of fIII_A for the alloys investigated in the present work. In this figure, our results are compared with those of two studies that investigated the effect of chunky graphite in an extended domain of degeneracy. Nakayama et al. [START_REF] Nakayama | Effect of amount of chunky graphite on mechanical properties of spheroidal-graphite cast iron[END_REF] have studied this effect with fully ferritic cast iron at 3.5 wt.% Si that was cast as a large cylinder. The samples were machined out from this cylinder in locations having chunky graphite fraction varying between 0% and 82%. Källbom et al. [START_REF] Källbom | Chunky graphite-Formation, and influence on mechanical properties in ductile cast iron[END_REF] prepared five melts denoted A to E with 3.2-3.4 wt.% Si that were cast in a mold with plates of varying thickness. The chunky fraction changed from 0% in the thin plates to 100% in some of the thicker plates. The UTS and A values from these two studies were picked up on the provided graphs, excluding those results where shrinkage was evidenced by the authors. These values are reported in Figure 16 as a function of the chunky graphite fraction which was measured in both cases on the fracture surfaces and thus probably over-estimated.

In the graphs of Figure 16, the values of a cast iron with 4.2 wt.% Si and no chunky graphite from previous work [START_REF] González-Martínez | Effects of high silicon contents on graphite morphology and room temperature mechanical properties of as-cast ferritic ductile cast irons. Part II  Mechanical properties[END_REF] are also reported, namely 580 MPa for UTS and 15.4% for A. It is thus seen that 30% of chunky graphite decreases the UTS value to about 540 MPa, but that a further increase in chunky graphite has no effect. This agrees with Nakayama et al. [START_REF] Nakayama | Effect of amount of chunky graphite on mechanical properties of spheroidal-graphite cast iron[END_REF] who suggested an abrupt decrease in UTS between 20% and 40% chunky graphite, while the results by Källbom et al. [START_REF] Källbom | Chunky graphite-Formation, and influence on mechanical properties in ductile cast iron[END_REF] suggest a slight but continuous decrease. In contrast, it is most often reported that chunky graphite does not affect the yield stress, though the results by Nakayama et al. [START_REF] Nakayama | Effect of amount of chunky graphite on mechanical properties of spheroidal-graphite cast iron[END_REF] and Källbom et al. [START_REF] Källbom | Chunky graphite-Formation, and influence on mechanical properties in ductile cast iron[END_REF] show a very slight decrease. The present YS values for the high-silicon alloys vary between 425 MPa and 480 MPa with a mean at 460 MPa which is only slightly lower than the previously reported value of 480 MPa [START_REF] González-Martínez | Effects of high silicon contents on graphite morphology and room temperature mechanical properties of as-cast ferritic ductile cast irons. Part II  Mechanical properties[END_REF].

Figure 16b shows the change of A with the amount of chunky graphite. The value previously evaluated [START_REF] González-Martínez | Effects of high silicon contents on graphite morphology and room temperature mechanical properties of as-cast ferritic ductile cast irons. Part II  Mechanical properties[END_REF] for a cast iron with 4.2 wt.% Si and no chunky graphite was 15.4% and is indicated with the large red dot along the Y-axis in Figure 16b. All studies agree that 30-40% of chunky graphite lead to an abrupt decrease of A with respect to the value for a sound casting. Such a conclusion could not be drawn in our previous work but is now evidenced. [46] and crosses from Källbom et al. [START_REF] Källbom | Chunky graphite-Formation, and influence on mechanical properties in ductile cast iron[END_REF]. Outliers in the present results are due to spiky graphite (arrow). The large red dots along the Y-axes locate the value of UTS and A for a 4.2 wt.% Si cast iron without chunky graphite as previously reported [START_REF] González-Martínez | Effects of high silicon contents on graphite morphology and room temperature mechanical properties of as-cast ferritic ductile cast irons. Part II  Mechanical properties[END_REF].

Finally, both graphs in Figure 16 show a strong effect of spiky graphite. Though the number of samples investigated during the present study is not very large, an attempt was made to sort the results in such a way that statistical trends could be deduced concerning the effect of chunky and spiky graphite, comparing low and high silicon alloys. The analysis was performed by defining sets of specimens depending on their content of chunky graphite and the presence of spiky graphite, which were compared by means of hypothesis testing on the variance and the average of a normal population. The population of the results matching each case of interest submitted to the hypothesis test was split into two sets with a variable number of samples each. The size of each sample depended on a threshold value for which a significant difference was found. The significance level was set to 5% in all cases.

To begin with the high silicon specimens with less than 8% of spiky graphite, all the sets that were tested gave a homogeneous outcome in terms of YS and UTS: the presence of chunky graphite at an amount higher than about 30% does not lead to a statistically significant variation of these two mechanical parameters. On the other hand, the presence of chunky graphite in high silicon specimens from Table 8 did affect elongation as observed by Källbom and Nakayama. Being more specific on this effect, it was found that a statistically significant difference of average elongation values is found when comparing the population with fCHG_A values below 49% and the population with fCHG_A values over 69%. The corresponding average values for the interval of 95% confidence are 6.7% ± 1.0% and 5.0% ± 1.1%, respectively. Thus, the observations conclude that high silicon ductile irons do not reduce its YS nor its UTS by the only effect of chunky graphite and that a threshold value can be established below which the effect of chunky graphite in the elongation is not significant.

Analyzing the rest of the dataset with no spiky graphite, YS strength showed to be independent of the chunky graphite content. Nevertheless, opposed to what was observed with high silicon alloys, UTS of low silicon alloys was affected by chunky graphite. This is evidenced when the set of specimens with fCHG_A < 17% is compared to the set with fCHG_A > 22%. In this case, the average UTS of each set is significantly different, giving values for the interval of 95% confidence of 395 ± 11 MPa and 342 ± 76 MPa, respectively. Using the same two sets for hypothesis testing with the elongation gave the same results in terms of significant variations of average elongation: 12.0% ± 2.2% and 4.0% ± 4.0% respectively. The confidence interval of the set with fCHG_A > 22% has been narrowed since, given the high variance of the data, the population was too small and the Student's t multiplier led to values without physical meaning. With the information above as a reference, it can be stated that, for the material employed in this study, the high silicon ductile irons show more tolerance to the presence of chunky graphite than the low silicon ones. On one hand, UTS do not show any reduction on high silicon alloys due to chunky graphite when above about 30%, while low silicon alloys do. On the other hand, the sensitivity threshold at which the elongation dropped was close to 50% chunky graphite for the high silicon alloys while it was around 20% for the low silicon ones. Thus, the mechanical properties of these last group of alloys are affected by a smaller amount of chunky graphite in the microstructure. Furthermore, the average elongation drop for the high silicon alloys was 1.7% while it was 8% for the low silicon ones when the chunky graphite sensitivity threshold was exceeded.

To complement the results on the effect of chunky graphite, the tensile data on specimens containing spiky graphite were also analyzed, leading to the fact that even small spiky graphite contents cause a strong change in mechanical properties. Again, a threshold was detected for the high silicon alloys, as the results for the set of specimens with fspiky_A values below 10% are not significantly different from the set with fCHG_A < 49% and free of spiky graphite. On the low silicon alloys though, the mechanical properties varied significantly with the slightest content of spiky graphite. In both cases, high and low silicon alloys, the elongation drop that is caused by the presence of spiky graphite is caused by twice as much content in chunky graphite. Thus, as a rule of thumb, the results in Table 8 indicate that spiky graphite is twice as bad as chunky graphite when the sensitivity threshold is exceeded. This confirms the conclusion of Foglio et al. [START_REF] Foglio | Fatigue characterization and optimization of the production process of heavy section ductile iron castings[END_REF].

The concomitant decrease of A and UTS is illustrated in Figure 17 by plotting one versus the other for the high silicon alloys presently investigated (4.2 wt.% Si) and the ones studied by Nakayama et al. [START_REF] Nakayama | Effect of amount of chunky graphite on mechanical properties of spheroidal-graphite cast iron[END_REF] and Källbom et al. [START_REF] Källbom | Chunky graphite-Formation, and influence on mechanical properties in ductile cast iron[END_REF] (3.2-3.5 wt.% Si). The two sets of data are clearly differentiated owing to the increase of the maximum UTS value with silicon. The shape of the evolution suggests comparing it with stress-strain curve and this has been done by superimposing the tensile test curve for sample PM8-5. This fact suggests that the presence of spiky graphite or chunky graphite in the high silicon ductile iron samples studied here lead to two distinct failure mechanisms. On the one hand, the five datapoints below 500 MPa correspond to samples containing spiky graphite but no chunky graphite, for which the specimens have failed in the elastic region of the tensile curve. On the other hand, the datapoints over 500 MPa correspond to specimens containing chunky graphite, for which the specimens have failed after elastic yield and with no apparent necking. Provided the matrix is similarly 100% ferritic for both cases, it is reasonable to also consider the toughness to be similar. This points to an elastic fracture mechanics failure driven by spiky graphite acting as the stress intensity raiser feature in the microstructure. The effect of chunky graphite as a stress intensity raiser is thus much less severe, allowing elastoplastic fracture mechanics to intervene, as specimen collapse is found well beyond yield strength.

Conclusions

A dedicated experiment mimicking heavy-section solidification has been developed to study the effect of the most important chemical variables reported to change the amount of chunky graphite, namely the content in silicon and in rare earths. Quite unexpectedly, controlled rare earth contents appear beneficial for decreasing chunky graphite when using standard charge materials. As does antimony, tin is shown to decrease chunky graphite appearance when added at a level of 500 ppm. It is also evidenced that this effect of tin and antimony is not related to rare earths which calls for further study on the role of these elements on graphite growth. However, if no rare earths are present, the addition of tin or antimony might lead to spiky graphite. Work is going on to determine if a critical level of tin at about 250-500 ppm would give sound castings, i.e., without chunky and spiky graphite. Chunky graphite affects the room temperature mechanical properties, though much more in the case of low silicon spheroidal graphite cast irons than in the alloys with high silicon contents. Spiky graphite is found to be much more detrimental than chunky graphite and its avoidance is considered critical. However, it has been observed that the mechanical properties of high silicon cast irons are not much sensitive to the exact amount of chunky graphite when above 30%.

Figure 1 .

 1 Figure 1. Schematic of the isolating mold with the indication of the location of the thermocouples (a) and a photograph of the mold just before removing the casting (b).

Figure 2 .

 2 Figure 2. Sketch of the specimens employed in mechanical testing. Dimensions are given in millimeters.

Figure 3 .

 3 Figure 3. Example of the temperature evolution in the four locations T1-T4 of casting P5. The insert shows an enlargement of the beginning of solidification.

Figure 4 .

 4 Figure 4. Micrographs of high-silicon casting P2 in location T3 of the cylindrical casting (a) and in the center of the thermal cup (b). Complex graphite string in location T3 of low silicon casting P4 (c).

Figure 6 .

 6 Figure 6. Cooling curves were recorded by thermocouple T3 in the four castings P2-P5. The corresponding fractions of chunky graphite, fIII_A, are indicated in the caption.

Figure 8 .

 8 Figure 8. Micrographs illustrating the appearance of spiky graphite due to the addition of Sb in casting P9, locations T1 (a) and T3 (b).

Figure 10 .Figure 11 .

 1011 Figure 10. Micrographs of castings P6 (a,b), P8 (c,d) and P10 (e,f), locations T1 (column to the left, (a,c,e)) and T4 (column to the right, (b,d,f)).

Figure 12 .

 12 Figure 12. The relation between chunky graphite area fraction and recalescence in the four locations T1-T4 of the cylindrical castings.

Figure 13 .

 13 Figure 13. Cooling curves of castings P3 and P7 in location T4.

Figure 14 .

 14 Figure 14. The area used for tensile testing (a) and location of the specimens in this area (b).

Figure 15 .

 15 Figure 15. Best stress/strain curves for the low-silicon (a) and high-silicon (b) castings. The amount of chunky (CHG) or spiky graphite is indicated for each sample along the corresponding curve.

Figure 16 .

 16 Figure 16. Change of UTS (a) and A (b) as a function of the fraction of chunky graphite (%) for the high-silicon alloys. Solid dots are for results from the present work, open dots from Nakayama et al.[46] and crosses from Källbom et al.[START_REF] Källbom | Chunky graphite-Formation, and influence on mechanical properties in ductile cast iron[END_REF]. Outliers in the present results are due to spiky graphite (arrow). The large red dots along the Y-axes locate the value of UTS and A for a 4.2 wt.% Si cast iron without chunky graphite as previously reported[START_REF] González-Martínez | Effects of high silicon contents on graphite morphology and room temperature mechanical properties of as-cast ferritic ductile cast irons. Part II  Mechanical properties[END_REF].

Figure 17 .

 17 Figure 17. The plot of UTS values versus A values for high silicon alloys from the present work and from literature. The solid curve is the tensile curve for sample PM8-5.

Table 1 .

 1 Description of the trials.

	Trial	Alloy	FeSiMg	Inoculant	Observation
	P2 P3	High Si ductile iron	with RE no RE	Anoc FeSi75 (no RE)	--
	P4 P5	Low Si ductile iron	with RE no RE	Anoc FeSi75 (no RE)	--
	P6		with RE	Anoc	+0.100 wt.% Sn
	P7				Intermediate level of RE
	P8 P9	High Si ductile iron	no RE	FeSi75 (no RE)	+0.050 wt.% Sn +0.004 wt.% Sb
	P10				+0.025 wt.% Sn

Table 2 .

 2 Composition (wt.%) of the FeSi75 alloy for adjustment of Si, of the FeSiMg spheroidizers, and of the Anoc inoculant.

	Product	FeSi75 FeSiMg (No RE) FeSiMg (with RE) Anoc
	Size (mm) 0.2-0.7	5-25	5-25	0.2-0.5
	Si	74.40	43.20	46.60	69.90
	Mg	-	6.26	6.00	-
	Al	0.95	0.25	0.71	0.93
	C	0.25	-	-	-
	Ca	1.53	0.83	0.96	1.38
	Ce	-	<0.05	-	-
	La	-	0.16	-	-
	RE	-	-	0.92	0.37
	Bi	-	-	-	0.49

Table 4 .

 4 Chemical analysis of the castings (wt.%).

	Trial	P2	P3	P4	P5	P6	P7	P8	P9	P10
	C	3.11	3.01	3.77	3.72	3.04	2.92	3.00	3.03	3.08
	Si	3.97	3.91	2.27	2.31	4.20	4.00	4.20	4.10	4.07
	Mn	0.17	0.18	0.19	0.18	0.13	0.18	0.21	0.20	0.19
	P	0.021	<0.010	<0.010	<0.010	<0.010	0.010	<0.010	<0.010	<0.010
	S	0.005	0.006	0.006	0.006	0.006	0.008	0.007	0.008	<0.005
	Cr	0.035	0.027	0.036	0.039	0.030	0.036	0.035	0.041	0.0029
	Ni	0.051	0.038	0.044	0.041	0.083	0.045	0.046	0.079	0.0038
	Cu	0.025	<0.020	0.030	0.052	<0.020	0.026	0.021	0.038	0.0022
	Ti	0.018	0.008	0.007	0.007	0.016	0.007	0.008	0.006	0.011
	Mg	0.037	0.040	0.046	0.041	0.044	0.039	0.036	0.040	0.045
	Ce	0.0067 <0.0005 0.0100 <0.0005 <0.0005 0.0034 <0.0005 <0.0005 <0.0005
	La	0.0040 <0.0005 0.0055 <0.0005 <0.0005 0.0017 <0.0005 <0.0005 <0.0005
	Sb	<0.0005 <0.0005 <0.0005 <0.0005 <0.0005 <0.0005 <0.0005 0.0039 <0.0005
	Sn	<0.005	<0.005	<0.005	<0.005	0.094	<0.005	0.051	<0.005	0.023
	Al	0.007	0.008	0.007	0.008	0.009	0.008	0.009	0.009	0.008
	CE99	4.23	4.11	4.41	4.37	4.22	4.05	4.18	4.18	4.23
	CEASM	4.35	4.22	4.47	4.44	4.34	4.16	4.30	4.30	4.34

Table 5 .

 5 Castings P2-P5. Average of measured values for total area fraction of graphite (fA), relative area fraction for each type of graphite particles (fIII_A, fV_A and fVI_A) and total particle count, Ncount. Type III particles correspond to chunky graphite. Between brackets are listed the minimum and maximum values of fIII_A and the standard deviation for Ncount.

					1175
					1170
		1300
				T1	1165
				T2
				T3
					1160
	(°C)	1250		T4	1155
	Temperature	1200			1150	0	500	1000	1500	2000
		1150
		1100
		0	500 1000 1500 2000 2500 3000 3500 4000
								Time (s)
		Trial Zone	fA	fIII_A	fV_A fVI_A	Ncount
			Cup 0.11 0.03 (0.01-0.06) 0.20 0.77 369 (22)
			T1	0.09 0.73 (0.08-1.00) 0.10 0.17 617 (287)
		P2	T2	0.08 0.63 (0.07-1.00) 0.10 0.27 509 (152)
			T3	0.08 0.77 (0.36-1.00) 0.08 0.15 670 (229)
			T4	0.09 0.57 (0.01-1.00) 0.29 0.14 451 (229)
			Cup 0.09 0.10 (0.06-0.15) 0.21 0.69 360 (13)
			T1	0.10 0.73 (0.63-0.82) 0.05 0.22 530 (102)
		P3	T2	0.10 0.76 (0.51-0.96) 0.07 0.17 520 (43)
			T3	0.09 0.87 (0.69-1.00) 0.06 0.07 687 (150)
			T4	0.09 0.89 (0.80-0.95) 0.09 0.02 526 (69)
			Cup 0.11	0.00	0.21 0.79 354 (29)
			T1	0.11 0.07 (0.00-0.20) 0.55 0.38 133 (31)
		P4	T2	0.10 0.35 (0.07-0.66) 0.40 0.25 170 (35)
			T3	0.11 0.12 (0.02-0.22) 0.56 0.32 128 (24)
			T4	0.11 0.10 (0.00-0.33) 0.52 0.38 152 (55)
			Cup 0.10	0.00	0.22 0.78 276 (67)
			T1	0.11 0.43 (0.01-0.98) 0.16 0.41 333 (142)
		P5	T2	0.12 0.49 (0.00-0.87) 0.08 0.43 330 (104)
			T3	0.11 0.51 (0.00-0.95) 0.19 0.30 315 (189)
			T4	0.09 0.42 (0.00-0.88) 0.17 0.41 254 (110)

Table 6 .

 6 Castings P7 and P9. Average of measured values for total area fraction of graphite (fA), relative area fractions for each type of graphite particles (fIII_A, fV_A and fVI_A) and total particle counts, Ncount. fIII_A is reported as either fCHG_A (P7) or fspiky_A (P9). Between brackets are listed the minimum and maximum values of fIII_A and the standard deviation for Ncount.

	Trial Zone	fA	fCHG_A	fspiky_A	fV_A fVI_A	Ncount
		Cup 0.12 0.12 (0.10-0.15)	0.00	0.22 0.66 371 (44)
		T1	0.14 0.53 (0.02-0.87)	0.00	0.16 0.32 384 (155)
	P7	T2	0.11 0.53 (0.32-0.97)	0.00	0.20 0.27 405 (152)
		T3	0.12 0.63 (0.52-0.81)	0.00	0.14 0.23 514 (125)
		T4	0.11 0.98 (0.88-1.00)	0.00	0.01 0.01 742 (136)
		Cup 0.12	0.00	0.00	0.23 0.77 380 (39)
		T1	0.13	0.00	0.15 (0.07-0.23) 0.26 0.59 136 (13)
	P9	T2	0.11	0.00	0.08 (0.05-0.13) 0.40 0.52 107 (13)
		T3	0.11	0.00	0.20 (0.08-0.38) 0.39 0.41 100 (12)
		T4	0.11	0.00	0.08 (0.00-0.15) 0.61 0.31	52 (8)

Table 8 .

 8 Composition, tensile testing results and microstructure characteristics of the alloys. fCHG_A stands for area fraction of chunky graphite, fSpiky_A stands for area fraction of spiky graphite, fA stands for total graphite area fraction and Ncount stands for graphite count. Ferrite and pearlite contents are reported in Appendix A, TableA2, for reference.

	Alloy	C	Si	Composition (wt.%) Ce La	Sb	Tensile Tests (MPa, A in %) Sample UTS YS A	Microstructure (Average of Three Fields) fCHG_A fSpiky_A Ncou fA nt
								1	315	285	1.0	0.78	0.00	0.16	614
	PM1	3.78 2.46 <0.0005 <0.0005 <0.0005	2 3 4	372 335 420	286 274 283 11.0 5.5 3.0	0.00 0.22 0.17	0.00 0.00 0.00	0.15 0.12 0.13	47 185 166
								5	402	284	6.0	0.00	0.00	0.14	44
								1	367	282	8.0	0.06	0.00	0.12	136
	PM2	3.88 2.47	0.0041	0.0021	<0.0005	2 3	375 381	296 271 10.0 8.0	0.30 0.10	0.00 0.00	0.13 0.14	165 131
								4	400	299 12.5	0.00	0.00	0.14	84
								5	399	280 10.0	0.00	0.00	0.16	91
								1	530	470	5.0	0.98	0.00	0.12	705
	PM3	3.15 4.17 <0.0005 <0.0005 <0.0005	2 3 4	533 529 545	474 470 466	3.0 2.5 5.0	0.59 0.69 0.69	0.00 0.00 0.00	0.12 0.12 0.12	660 577 574
								5	529	468	4.5	0.97	0.00	0.13	564
								1	547	468	5.0	0.83	0.00	0.15	455
	PM4	3.20 4.20	0.0038	0.0020	<0.0005	2 3 4	534 537 541	466 467 470	5.0 6.0 6.0	0.84 0.37 0.88	0.00 0.00 0.00	0.14 0.13 0.13	493 264 476
								5	542	462	7.0	0.84	0.00	0.14	379
								1	449	295	6.5	0.00	0.11	0.14	126
	PM5	3.79 2.44 <0.0005 <0.0005	0.0079	2 3 4	445 449 425	296 298 291	6.5 5.5 6.5	0.00 0.00 0.00	0.06 0.05 0.18	0.12 0.14 0.14	104 95 105
								5	422	291	5.5	0.00	0.06	0.12	97
								1	393	268 11.5	0.01	0.00	0.13	121
	PM6	3.80 2.43	0.0075	0.0040	0.0072	2 3 4	402 396 396	268 19.0 292 12.5 270 13.0	0.00 0.02 0.07	0.00 0.00 0.00	0.18 0.15 0.16	137 119 101
								5	395	270 13.0	0.16	0.00	0.14	98
								1	444	441	1.0	0.00	0.24	0.12	128
	PM7	3.27 4.20 <0.0005 <0.0005	0.0043	2 3 4	451 459 487	444 425 430	0.5 1.5 3.5	0.00 0.00 0.00	0.28 0.27 0.12	0.13 0.15 0.12	88 85 144
								5	486	430	3.0	0.00	0.18	0.11	98
								1	523	455	4.5	0.35	0.22	0.12	269
	PM8	3.22 4.22	0.0073	0.0039	0.0057	2 3 4	537 533 535	453 480 474	6.5 6.0 7.0	0.33 0.49 0.29	0.04 0.08 0.00	0.13 0.15 0.15	144 230 166
								5	538	447	8.0	0.33	0.00	0.12

Appendix A

Due to the low cooling rates achieved in the present work, the matrices were essentially ferritic in all castings, though pearlite is also present in some areas. Tables A1 andA2 show the results of the structure characterization in the two series of alloys prepared, namely P2-P10 and PM1-PM8, respectively. Notice that pearlite is always located in areas free of chunky graphite (last solidification areas or those with spiky graphite).

Table A1. Pearlite contents and its location in the P2-P10 cylindrical castings (%, balance ferrite). Pearlite randomly distributed out of chunky graphite areas (a), in grain boundaries of areas free of chunky graphite (b), in areas that contain spiky graphite (c).

Alloy

As an example, Figure A1 shows the microstructures after etching of castings P10 and PM1 which contained pearlite in the areas with spiky graphite and with nodules, respectively.