# Mathematical Analysis of the Effect of Iron and Silica on the Reduction Performance of Manganese Ores

Authors: S. Ghali; E. A. Mousa
Journal: Journal of Metallurgy
 (2015)

---

## Abstract

In the current study, a factorial design is used to investigate the effect of total iron and silica on the metallurgical performance of different grades of manganese ores. The derived mathematical formulations are applied to estimate the reduction disintegration index (RDI+6.3, RDI+3.15, and RDI−0.5), reduction index (total reduction index (RIT), manganese reduction index (RIM), and iron reduction index (RIF)), and softening-melting property (start of softening (T
S
1), end of softening (T
S
2), start of melting (T
m
1), and end of melting (T
m
2)) of manganese ores. The RDI+6.3 and RDI+3.15 are increased with the individual effect of SiO2 and the interaction effect of iron with silica, while they are decreased as the total iron increased. The high-Fe high-SiO2 manganese ore showed the highest RIT and RIF. The RIM was almost identical in all manganese ores. The presence of high content of SiO2 resulted in a narrow softening range (62–83°C), while the high-Fe high-SiO2 manganese ore exhibited a wider softening range (135–140°C). The melting range was very small in high-Fe low-SiO2 (3–16°C) and high-Fe high-SiO2 (6–8°C) manganese ores, while the low-Fe low-SiO2 manganese ore showed wider melting range (72–74°C). The derived mathematical models are in a good agreement with the experimental results. The calculations are carried out using Matlab program.


---

## Body

## 1. Introduction

Manganese is considered to be one of the most important alloying elements in different grades of steel and cast iron. Manganese improves the tensile strength, machinability, toughness, hardness, and abrasion resistance of steel. In addition, manganese has favourable influence on forging, welding, and grain refining properties in steel casting. Manganese can be used for production of less expensive austenitic steel grades by replacing the expensive alloying elements such as nickel [1–4]. Moreover, manganese is important to control the sulfur content control in steel and it is commonly used as oxidizers of molten steels [5]. Manganese is usually added in the form of ferromanganese (FeMn) to steel during its treatment in the ladle furnace. About 90–95% of manganese production is used in steelmaking in the form of alloys such as ferromanganese and silicomanganese while ~30% of manganese production is used for desulfurization and deoxidation of steel while the other 70% of manganese is used purely as alloying element in steel [6]. Ferromanganese alloy is relatively expensive because it is produced by carbothermic reduction of manganese ores in the submerged electric arc furnace (SAF) which is highly energy consuming [7, 8]. Energy input to SAF for production of high carbon ferromanganese (HC FeMn) is 18400–24000 MJ/ton, including 300–360 kg of fixed carbon and 2400–3400 kWh electrical power [9]. In addition, fluxing materials (e.g., limestone and dolomite) are usually added in amount of 200–450 kg/ton HC FeMn to adjust the basicity of slag [10]. Smelting reduction processes require relatively high temperature (~1500°C) for production of HC FeMn and higher temperature (above 1600°C) is required for silicomanganese (SiMn) production. Manganese ores are commonly contaminated with many impurities (e.g., Fe2O3, SiO2, CaO, Al2O3, P, etc.) which are normally present in the form of complex minerals (e.g., Pyrolusite, bixbyite, braunite, manganite, hausmannite, tephroite, and rhodonite) [9]. Manganese ores can be classified according to their contents of manganese into different categories. The ores containing at least 35% manganese are defined as manganese ores while the ores having 10–35% Mn are known as ferruginous manganese ores [9]. The ores containing 5–10% manganese are defined as manganiferous ores while the ores containing less than 5% manganese with the balance of iron are classified as iron ores. The manganese ores are often found contaminated with iron and silica which affect the reduction process, the energy consumption, and the operation stability. Intensive work has been carried out to identify the reduction kinetics of manganese ores and the influence of different impurities on the high temperature metallurgical properties. The influence of iron and silica on the metallurgical properties of four different grades of manganese ores is discussed [11]. The ores are classified as high-iron high-silica, low-iron low-silica, high-iron low-silica, and low-iron high-silica ores. It was reported that the high-iron low-silica one exhibited a good reducibility and narrow melting range which is required for FeMn alloy and manganese-rich slag production. The low iron-low silica manganese ores showed that the highest melting temperature makes it suitable for SiMn alloy production. The low iron-high silica manganese ores exhibited low melting temperatures; therefore, it could be used for manganese-rich slag production. The high iron-high silica manganese ore showed good reducibility with low melting temperature which makes it suitable for the production of manganese-rich slag. The investigation carried out on the reduction of siliceous manganese ore by graphite indicated that the manganese oxide is firstly dissolved into the molten MnO-SiO2-Al2O3-CaO-MgO slag, and then it is reduced from the slag [12, 13]. The reduction of manganese oxide in slag is strongly retarded by silica. The carbothermal reduction of manganese oxides was studied in presence of hydrogen, helium, and argon at different temperatures [14, 15]. It was found that the carbothermal reduction of MnO at constant temperature was the fastest in hydrogen followed by helium and the slowest in argon. This magnitude effect of the surrounding atmospheres on the reduction rate was decreased as the temperature increased from 1275°C to 1400°C. The manganese oxides were reduced to α-Mn and Mn23C6 and Mn7C3 depending on the carbon to ore ratio. A reduction retardation was accompanied with the reduction of manganese ores which are contaminated with iron oxides [16, 17]. The reduction of Fe2O3-MnO2-SiO2 mixed oxides with CO gas is investigated [18]. It was found that the formation of hard reducible fayalite-manganoan [(Fe,Mn)2SiO4] phase resulted in a retardation of the reduction process [15]. The utilization of composite briquettes consisting of manganese ore, coke fines, and organic binder improved the thermal stability, softening property, and reducibility of manganese ore in smelting arc furnace [19].The previous survey indicates that, in order to maintain a stable operation of ferromanganese production with lowest energy consumption, it is important to keep the metallurgical properties of the applied manganese ores at the optimum conditions. Although many experimental studies were carried out to estimate the effect of different parameters on the smelting reduction of manganese ores, few studies tried to estimate the magnitude’s effect of the individual and interaction parameters on the overall reduction process. The factorial design provides a novel approach to precisely estimate the effect of different parameters either individually or collectively on the process [20–23]. The current study aims to investigate the effect of total iron and silica on the metallurgical properties of manganese ores using factorial design approach. Regression models are derived based on the experimental results of different manganese ores grades which are contaminated with unequal proportions of iron and silica [11]. The magnitude effects of the individual and combinations parameters on the low-temperature reduction disintegration, reduction index, and softening-melting range are evaluated.


## 2. Materials and Methods

### 2.1. Source and Analysis of Manganese Ores

A 22 factorial design is used to determine the main effect of total iron and silica and their interactions on the low-temperature reduction disintegration index (RDI), reduction index (RI), and softening-melting property (SMP) of different grades of manganese ores. The testing methods of the reduced manganese ores including RDI, RI, and SMP are reported by Zhang et al. elsewhere [11]. The RDI is classified into RDI+6.3 (the ratio of reduced manganese ores with size larger than 6.3 mm after tumbling test, %), RDI+3.15 (the ratio of reduced manganese ores with size larger than 3.15 mm after tumbling test, %), and RDI−0.5 (the ratio of reduced manganese ores with size smaller than 0.5 mm after tumbling test, %). The reduction index (RI) is classified into RIT (reduction index of total Fe-Mn oxides), RIM (reduction index of only manganese oxide), and RIF (reduction index of only iron oxide). The softening-melting property (SMP) is classified into T
S
1 (start of softening), T
S
2 (end of softening), T
m
1 (start of melting), and T
m
2 (end of melting). The chemical composition of the manganese ores is given in Table 1 [11].Table 1
Chemical composition of different grades of manganese ores.
Ore description
T Fe
T Mn
SiO2
Al2O3
CaO
MgO
P
LOI
High Fe-low Si
23.48
34.73
6.75
0.56
0.38
0.16
0.051
8.29
Low Fe-low Si
3.9
47.8
7.93
4.35
0.26
0.18
0.063
9.67
Low Fe-high Si
2.72
36.66
27.26
2.49
1.97
0.26
0.062
8.88
High Fe-high Si
33.24
13.67
16.32
1.32
0.2
0.06
0.12
10.03


### 2.2. Statistical Design

By convention, the total iron in manganese ores is denoted by “A.” The average of iron content in the low iron-manganese ores is equal to 3.31% and the average of iron content in the high iron-manganese ores is equal to 28.36%. Similarly, the silica will be denoted by “B.” The average of silica content in the low silicon-manganese ores is equal to 7.34% and the average of silica content in the high silica-manganese ores is equal to 21.79%.Based on this concept, the effect of a factor is donated by a capital Latin letter. Thus “A” refers to the effect of total iron, “B” refers to the effect of SiO2, and “A
B” refers to the interaction combination effect of total Fe and SiO2. The low and high level of “A” and “B” are denoted by “−” and “+,” respectively. The four treatment combinations in the design are usually represented by lowercase letters, where the high level of any factor in the treatment combination is denoted by the corresponding lowercase letter and the low level of a factor in the treatment combination is denoted by the absence of the corresponding letter. Thus, “a” represents the treatment combination of Fe “A” at high level and SiO2 “B” at low level, “b” represents “A” at low level and “B” at high level, and “a
b” represents both factors Fe and SiO2 (A and B) at the high levels, while “(
1
)” is used to denote both factors at low levels.


## 2.1. Source and Analysis of Manganese Ores

A 22 factorial design is used to determine the main effect of total iron and silica and their interactions on the low-temperature reduction disintegration index (RDI), reduction index (RI), and softening-melting property (SMP) of different grades of manganese ores. The testing methods of the reduced manganese ores including RDI, RI, and SMP are reported by Zhang et al. elsewhere [11]. The RDI is classified into RDI+6.3 (the ratio of reduced manganese ores with size larger than 6.3 mm after tumbling test, %), RDI+3.15 (the ratio of reduced manganese ores with size larger than 3.15 mm after tumbling test, %), and RDI−0.5 (the ratio of reduced manganese ores with size smaller than 0.5 mm after tumbling test, %). The reduction index (RI) is classified into RIT (reduction index of total Fe-Mn oxides), RIM (reduction index of only manganese oxide), and RIF (reduction index of only iron oxide). The softening-melting property (SMP) is classified into T
S
1 (start of softening), T
S
2 (end of softening), T
m
1 (start of melting), and T
m
2 (end of melting). The chemical composition of the manganese ores is given in Table 1 [11].Table 1
Chemical composition of different grades of manganese ores.
Ore description
T Fe
T Mn
SiO2
Al2O3
CaO
MgO
P
LOI
High Fe-low Si
23.48
34.73
6.75
0.56
0.38
0.16
0.051
8.29
Low Fe-low Si
3.9
47.8
7.93
4.35
0.26
0.18
0.063
9.67
Low Fe-high Si
2.72
36.66
27.26
2.49
1.97
0.26
0.062
8.88
High Fe-high Si
33.24
13.67
16.32
1.32
0.2
0.06
0.12
10.03


## 2.2. Statistical Design

By convention, the total iron in manganese ores is denoted by “A.” The average of iron content in the low iron-manganese ores is equal to 3.31% and the average of iron content in the high iron-manganese ores is equal to 28.36%. Similarly, the silica will be denoted by “B.” The average of silica content in the low silicon-manganese ores is equal to 7.34% and the average of silica content in the high silica-manganese ores is equal to 21.79%.Based on this concept, the effect of a factor is donated by a capital Latin letter. Thus “A” refers to the effect of total iron, “B” refers to the effect of SiO2, and “A
B” refers to the interaction combination effect of total Fe and SiO2. The low and high level of “A” and “B” are denoted by “−” and “+,” respectively. The four treatment combinations in the design are usually represented by lowercase letters, where the high level of any factor in the treatment combination is denoted by the corresponding lowercase letter and the low level of a factor in the treatment combination is denoted by the absence of the corresponding letter. Thus, “a” represents the treatment combination of Fe “A” at high level and SiO2 “B” at low level, “b” represents “A” at low level and “B” at high level, and “a
b” represents both factors Fe and SiO2 (A and B) at the high levels, while “(
1
)” is used to denote both factors at low levels.


## 3. Results and Discussions

### 3.1. Mathematical Formulations

Mathematical formulations are driven to estimate the effect of Fe, SiO2, and their interaction (Fe-SiO2) on the metallurgical properties of manganese ores. The effect of “A” at low level of B is [
a
-
(
1
)
] and the effect of “A” at high level of B is [
a
b
-
b
]. The main effect of “A” is the average of its effect at low and high level of B as given in
(1)
A
=
1
2
a
b
-
b
+
a
-
1
=
1
2
a
b
+
a
-
b
-
(
1
)
.
The average effect of “B” can be calculated from the effect of “B” at low level of “A” ([
b
-
(
1
)
]) and at the high level of “A” (i.e., [
a
b
-
a
]) as given in
(2)
B
=
1
2
a
b
-
a
+
b
-
(
1
)
=
1
2
a
b
+
b
-
a
-
(
1
)
.
The interaction effect “A
B” is defined as the average difference between the effect of “A” at the high level of “B” and the effect of “A” at the low level of “B” as given in
(3)
A
B
=
1
2
a
b
-
b
-
a
-
(
1
)
=
1
2
a
b
+
(
1
)
-
a
-
b
.
The sum of squares (SS) of A, B, and A
B can be calculated as given in (4)–(6), respectively. Consider
(4)
S
S
A
=
a
b
+
a
-
b
-
(
1
)
2
4
,
(5)
S
S
B
=
a
b
+
b
-
a
-
(
1
)
4
2
,
(6)
S
S
A
B
=
a
b
+
(
1
)
-
a
-
b
4
2
.
The total sum of squares (S
S
T) and sum of squares (S
S
E) can be calculated using (7) and (8) respectively. Consider
(7)
S
S
T
=
∑
i
=
1
2
∑
j
=
1
2
∑
k
=
1
n
y
i
j
k
2
-
y
2
4
,
(8)
S
S
E
=
S
S
T
-
S
S
A
-
S
S
B
-
S
S
A
B
.


### 3.2. Application of Factorial Design

#### 3.2.1. Reduction Disintegration Index (RDI)

The complete analyses of the effect of Fe (A) and/or SiO2 (B) on reduction disintegration index (RDI) of manganese ores can be calculated using (1)–(8). The effects of Fe and/or SiO2 on RDI+6.3, RDI+3.15, and RDI−0.5 are given in Tables 2–4, respectively.Table 2
Effect of total iron (A), silica (B), and their interaction (A
B) on  on RDI+6.3.
Source of variance
The average effect
Sum of square (SS)
Mean square (MS)(MS = SS/degree of freedom)
Magnitude effect,
F
o (MSVariance/M
S
E)
A
−1.1518
2.653286
2.653286
9.096124
B
5.4581
59.58171
59.58171
204.2609
A
B
−9.64705
186.1311
186.1311
638.1037
Error
1.166777
0.291694
Total
249.5329Table 3
Effect of total iron (A), silica (B), and their interaction (A
B) on RDI+3.15.
Source of variance
The average effect
Sum of square (SS)
Mean square (MS)(MS = SS/degree of freedom)
Magnitude effect,
F
o (MSVariance/M
S
E)
A
1.765325
6.232745
6.232745
17.63527
B
4.867275
47.38073
47.38073
134.0617
A
B
−6.47918
83.95942
83.95942
237.5594
Error
1.4137
0.353425
Total
138.9866Table 4
Effect of total iron (A), silica (B), and their interaction (A
B) on RD
I
-
0.5.
Source of variance
The average effect
Sum of square (SS)
Mean square (MS)(MS = SS/degree of freedom)
Magnitude effect,
F
o (MSVariance/M
S
E)
A
0.185425
0.068765
0.068765
37.04831
B
0.893125
1.595345
1.595345
859.5207
A
B
2.370475
11.2383
11.2383
6054.839
Error
0.007424
0.001856
Total
12.90984In Table2, it can be seen that the highest negative effect on RDI+6.3 is exhibited by the interaction of iron and silica followed by the individual effect of iron. On the other hand, silica exhibited a positive effect on RDI+6.3 of the reduced manganese ores. The results in Table 3 indicate that both of iron and silica have an individual positive effect on RDI+3.15 while the interaction coefficient of iron and silica affects negatively the RDI+3.15 of the reduced manganese ores. Table 4 indicates that the individual and collective parameters have positive effects on RDI−0.5 with relatively higher magnitude for the interaction coefficient parameter of iron with silica.The experimental results can be generally expressed in terms of regression model as given in (9) for RDI+6.3, RDI+3.15, and RDI−0.5, respectively. Consider
(9)
Y
=
β
0
+
β
1
x
1
+
β
2
x
2
+
β
12
x
1
x
2
+
ε
,
where Y refers to RDI+6.3, RDI3.15, or RDI−0.5, x
1 is a coded variable representing the total iron, x
2 is a coded variable representing the SiO2, β
s
′ is regression coefficient, and ε is the residual (the difference between observed and fitted point of the design). β
0 is the intercept which is the grand average of all observations; the regression coefficients β
1
,
β
2 and β
12 are one-half the corresponding factor while ε is the residual. The values of β
0, β
1, β
2, and ε are given in Table 5. In all cases, the residuals are very small and can be neglected.Table 5
Regression coefficient values for RDI+6.3, RDI+3.15, and RDI
-
0.5.
Y
β
0
β
1
β
2
β
12
ε
RDI+6.3
76.18
−0.5759
2.729
−4.82353
±0.419
RDI+3.15
83.96099
0.882662
2.433638
−3.23959
±0.441
RDI
-
0.5
5.95688
0.092713
0.446563
1.185238
±0.0386The relation between the natural variables and the coded variable is given as follow: the coded variable is equal to [(natural variable − 1/2(variable at high level + variable at low level))/1/2(variable at high level − variable at low level)]. Consequently, the RDI+6.3, RDI+3.15, and RDI−0.5 can be predicted as a function of total iron and SiO2 as given in (10)–(12), respectively. Consider
(10)
RDI
+
6.3
=
59.11397
-
0.0533
[
TotalFe
]
+
0.730374
Si
O
2
+
1.221772
TotalFe
Si
O
2
,
(11)
RD
I
+
3.15
=
69.68242
-
0.0358
TotalFe
+
0.591889
Si
O
2
+
0.903718
TotalFe
Si
O
2
,
(12)
RD
I
-
0.5
=
7.960216
-
0.013098
TotalFe
-
0.18336
Si
O
2
-
0.14559
TotalFe
Si
O
2
.The RDI+6.3, RDI+3.15, and RDI−0.5 for the different grades of manganese ores (low-Fe low-Si, high-Fe low-Si, low-Fe high-Si, and high-Fe high-Si manganese ores) are calculated based on (10)–(12). The calculated values of RDI are compared to the experimental results as can be seen in Figures 1, 2, and 3. The given notations summarized the grade of ores as follows: refers to low-Fe low-Si ore; a refers to high-Fe low-Si ore; b refers to low-Fe high-Si ore; a
b refers to high-Fe high-Si ore. It can be seen that, in all cases, the coded and actual variables are in good agreement with those of the experimental results. The RDI+6.3 and RDI+3.15 are the highest in low-Fe high-Si ores and the lowest in low-Fe low-Si ores. This indicates that silica has the ability to improve the strength of manganese ore during the reduction at 500°C [11]. In the iron-manganese oxides, manganese ferrite (Fe2MnO4) is formed in vicinity of large pores as a result of solid solution reaction between MnO and Fe2O3. On the other hand, the presence of silica in iron-manganese ore is able to diminish the formation of manganese ferrite (MnFe2O4), consequently decreasing the porosity and improving the strength of the ore [18, 24]. The iron content has a significant effect on the reduction disintegration index. The ore disintegration increases with iron due to the crystal distortion which is accompanied by the transformation of hematite to magnetite. Such disintegration is caused on one hand by lattice transformations and on the other hand by an anisotropic reaction rate [25]. Hematite crystallizes in hexagonal rhombohedral lattice while magnetite has an inverse spinel lattice structure. During the transformation from hematite to magnetite, a layer of close magnetite grows on the surface of the porous hematite and results in cracks and disintegrations. The unit cell volume of magnetite is equal to 592.07 Å3 which it is equal to 302.72 Å3 for hematite. This results in a disintegration of the ore during reduction. In addition, the transformation of MnO2 into Mn2O3 is accompanied by sharp increase in the cell volume from 55.64 to 834.56 [11]. Therefore, the index of reduction disintegration is decreased as the iron and/or manganese content in the ore increased.Figure 1
The experimental and predicted RDI+6.3 by using coded and actual variables.Figure 2
The experimental and predicted RDI+3.15 by using coded and actual variables.Figure 3
The experimental and predicted RDI−0.5 by using coded and actual variables.


#### 3.2.2. Reduction Index (RI)

The effect of total Fe and/or silica on the reduction index is calculated based on the total Fe-Mn oxides (RIT), Mn oxide (RIM), and Fe oxide (RIF) as given in Tables6–8. Table 6 indicates that total Fe has the highest positive effect on the total reduction RIT. Both of the silica and the interaction of total Fe with silica have almost equal positive effect which is lower than the individual effect of Fe. Table 7 shows that all parameters have negative effect on the reduction of manganese (RIM) with relatively higher negative effect of total Fe. Table 8 indicates that the total iron has a relatively high positive effect on the reduction of iron oxide (RIF). On the other hand, the interaction coefficient between Fe and silica exhibited a negative effect on the reduction of iron oxides.Table 6
Effect of total iron (A), silica (B), and their interaction (A
B) on the RIT.
Source of variance
The average effect
Sum of square (SS)
Mean square (MS)(MS = SS/degree of freedom)
Magnitude effect,
F
o (MSVariance/M
S
E)
A
19.687
775.1559
775.1559
3994.748
B
6.3685
81.11558
81.11558
418.0273
A
B
6.268
78.57565
78.57565
404.9378
Error
0.776175
0.194044
Total
935.6233Table 7
Effect of total iron (A), silica (B), and their interaction (A
B) on the RIM.
Source of variance
The average effect
Sum of square (SS)
Mean square (MS)(MS = SS/degree of freedom)
Magnitude effect,
F
o (MSVariance/M
S
E)
A
−2.3
10.58
10.58
2116
B
−0.3
0.18
0.18
36
A
B
−1.4
3.92
3.92
784
Error
0.02
0.005
Total
14.7Table 8
Effect of total iron (A), silica (B), and their interaction (A
B) on the RIF.
Source of variance
The average effect
Sum of square (SS)
Mean square (MS)(MS = SS/degree of freedom)
Magnitude effect,
F
o (MSVariance/MS
E)
A
21.4555
920.677
920.677
2537.032
B
0.613
0.751538
0.751538
2.07095
A
B
−2.603
13.55122
13.55122
37.34195
Error
1.451581
0.362895
Total
936.4313The results of experiments can be expressed in terms of regression models of RIT, RIM, and RIF as given in (13). The values of β
0, β
1, β
2, and ε are given in Table 9. In all cases, the residuals are very small and can be neglected. Consider
(13)
Z
=
β
0
+
β
1
x
1
+
β
2
x
2
+
β
12
x
1
x
2
+
ε
,
where Z refers to RIT, RIM, or RIF.Table 9
Regression coefficient values for RIT, RIM, and RIF.
Z
β
0
β
1
β
2
β
12
ε
RIT
61.24925
9.8435
3.18425
3.134
±0.389
RIM
98.15
−1.15
−0.15
−0.7
±0.05
RIF
84.39425
10.7277
0.3065
−1.3015
±0.483The RIT, RIM, RIF can be predicted as a function of total iron and SiO2 as given in (14)–(16), respectively. The calculated values of RIT, RIM, and RIF are compared to the experimetal results as can be seen in Figures 4–6. It can be seen that, in all cases, the coded and actual variables are in a good agreement with the experimental data. Consider
(14)
RI
=
50.37273
+
0.034632
TotalFe
+
0.281486
Si
O
2
-
0.10768
TotalFe
Si
O
2
,
(15)
RIM
=
98.12223
-
0.00774
TotalFe
+
0.02085
Si
O
2
+
0.101729
TotalFe
Si
O
2
,
(16)
RIF
=
66.89649
-
0.01438
TotalFe
+
1.065985
Si
O
2
+
0.270166
TotalFe
Si
O
2
.
It can be seen in Figure 4 that high-Fe high-Si and high-Fe low-Si manganese ores exhibited the highest reduction index. On the other hand, the lowest reduction index was exhibited in low-Fe low-Si and low-Fe high-Si ones. This indicates that the reduction index of manganese ore increases as iron oxide content increases. The reduction index based on manganese oxide (RIM) is almost identical for the different grades of manganese ores as shown in Figure 5. This can be attributed to the simple reduction of MnO2 to MnO. The reduction index based on iron oxide (RIF) is high in the ores rich with iron and low in the ore poor in iron oxides as shown in Figure 6. This is attributed to the higher reducibility of iron oxides compared to that of manganese oxides. Although the reduction of wüstite (FexO) to metallic iron required relatively high potential of reducing gas (η
CO
≈
30%, η
H
2
≈
30%), the reduction of MnO to Mn metal is more complicated and can be only proceeded by solid carbon at very high temperature (≈1423°C). Therefore, the reduction index decreased as manganese content increased.Figure 4
The experimental and predicted RIT by using coded and actual variables.Figure 5
The experimental and predicted RIM by using coded and actual variables.Figure 6
The experimental and predicted RIF by using coded and actual variables.


#### 3.2.3. Softening-Melting Property (SMP)

A mathematical regression model is derived to estimate the effect of total iron and/or silica on the softening-melting property of manganese ores during reduction. The softening ranges can be estimated based on the determination of temperature at which the reduced ores start to soften (T
S
1: starting of softening) and the temperature at which the softening finsihed (T
S
2: end of softening). The melting range can be determined based on the identification of the start of melting (T
m
1) and the end of melting (T
m
2). The effects of total iron and/or silica on the softening property of reduced manganese ores are given in Tables 10 and 11, respectively. It can be seen that the iron and silica affect negatively the start and the end softening temperature.Table 10
Effect of total iron (A), silica (B), and their interaction (A
B) on T
S
1.
Source of variance
The average effect
Sum of square (SS)
Mean square (MS)(MS = SS/degree of freedom)
Magnitude effect,
F
o (MSVariance/M
S
E)
A
−29.5
1740.5
1740.5
386.7778
B
−47.5
4512.5
4512.5
1002.778
A
B
−42.5
3612.5
3612.5
802.7778
Error
18
4.5
Total
9883.5Table 11
Effect of total iron (A), silica (B), and their interaction (A
B) on T
S
2.
Source of variance
The average effect
Sum of square (SS)
Mean square (MS)(MS = SS/degree of freedom)
Magnitude effect,
F
o (MSVariance/M
S
E)
A
−8
128
128
10.24
B
−64
8192
8192
655.36
A
B
−10
200
200
16
Error
50
12.5
Total
8570The effect of iron and/or silica content on the melting property of manganese ores includingT
m
1 and T
m
2 is given in Tables 12 and 13, respectively. As can be seen in Table 12, the iron and the iron with silica affect positively the start temperature of melting while silica exhibited a negative effect on the start temperature of melting. Both of iron and silica decreased the end temperature of melting of manganese ore, while the interaction of iron and silica has positive effect on the end of melting as given in Table 13.Table 12
Effect of total iron (A), silica (B), and their interaction (A
B) on T
m
1.
Source of variance
The average effect
Sum of square (SS)
Mean square (MS)(MS = SS/degree of freedom)
Magnitude effect,
F
o (MSVariance/M
S
E)
A
44
3872
3872
1936
B
−79
12482
12482
6241
A
B
22
968
968
484
Error
8
2
Total
17330Table 13
Effect of total iron (A), silica (B), and their interaction (A
B) on T
m
2.
Source of variance
The average effect
Sum of square (SS)
Mean square (MS)(MS = SS/degree of freedom)
Magnitude effect,
F
o (MSVariance/M
S
E)
A
−19
722
722
361
B
−82
13448
13448
6724
A
B
30
1800
1800
900
Error
8
2
Total
15978The relation between the natural variables and the coded variable forT
S
1, T
S
2, T
m
1, and T
m
2 can be summarized in (17). The values of β
0, β
1, β
2, and ε are given in Table 14. Consider
(17)
H
=
β
0
+
β
1
x
1
+
β
2
x
2
+
β
12
x
1
x
2
+
ε
,
where H refers to T
S
1, T
S
2, T
m
1, or T
m
2.Table 14
Values of regression coefficient forT
S
1, T
S
2, T
m
1, and T
m
2.
β
0
β
1
β
2
β
12
ε
T
S
1
1123.25
−14.75
−23.75
−21.25
±1.5
T
S
2
1240.5
−4.0
−32.0
−5.0
±2.5
Δ
T
S
119.6888
11.65125
−8.33125
16.29125
±0.7
T
m
1
1229.5
22.0
−39.5
11.0
±1.0
T
m
2
1266.5
−9.5
−41.0
15.0
±1.0
Δ
T
m
36.66
−32.1825
−2.0175
3.51
±0.37TheT
S
1, T
S
2, T
m
1, and T
m
2 can be predicted as a function of total iron and SiO2 as given in (18)–(21), respectively. The calculated values of T
S
1, T
S
2, T
m
1, and T
m
2 are compared to the experimental results as can be seen in Figures 7–10. It can be seen that, in all cases, the coded and actual variables are in a good agreement with the experimental results. Consider
(18)
T
S
1
=
1135.617
-
0.23482
TotalFe
+
2.242574
Si
O
2
+
0.431248
TotalFe
Si
O
2
,
(19)
T
S
2
=
1297.323
-
0.05525
TotalFe
+
0.485396
Si
O
2
-
3.55414
TotalFe
Si
O
2
,
(20)
T
m
1
=
1309.35
+
0.121556
TotalFe
-
0.01398
Si
O
2
-
7.39197
TotalFe
Si
O
2
,
(21)
T
m
2
=
1399.393
+
0.165758
TotalFe
-
3.17275
Si
O
2
-
8.29953
TotalFe
Si
O
2
.Figure 7
The experimental and predictedT
S
1 by using coded and actual variables.Figure 8
The experimental and predictedT
S
2 by using coded and actual variables.Figure 9
The softening range (T
Δ
S) for experimental, coded, and actual variables.Figure 10
The experimental and predictedT
m
1 by using coded and actual variables.Figures7 and 10 showed that the lowest start temperature of softening (T
S
1) and melting (T
m
1) is exhibited in high-Fe high-Si manganese ore which is equal to 1062–1089°C and 1199.5–1225°C, respectively. Figures 8 and 11 clarify that the highest end of softening (T
S
2) and melting (T
m
2) is exhibited in high-Fe low-Si manganese ore which is equal to 1152–1154°C and 1273–1276°C, respectively. This indicates that the presence of relatively high percentage of silica resulted in a decrease of the start softening and melting temperatures. This can be attributed to the formation of relatively low melting rhodonite phase (MnSiO3; m.p. 1242°C) as a result of the reaction between SiO2 and MnO [26]. The formation of low melting rhodonite phase resulted in a narrow softening range (T
Δ
S) in low-Fe high-SiO2 manganese ore as can be seen in Figure 9. The high-Fe low-si and high-Fe high-si manganese ores showed a narrow melting range as can be seen in Figure 12. This can be attributed to the formation Fe-Mn olivine in the presence of relatively high concentration of iron [11].Figure 11
The experimental and predictedT
m
2 by using coded and actual variables.Figure 12
The melting range (T
Δ
m) for experimental, coded, and actual variables.Based on the previous findings, it can be concluded that the factorial design is very useful approach to predict and precisely estimate the effect of different impurities such as Fe and/or Si which commonly contaminate the manganese ores and affect negatively the smelting reduction process. The derived mathematical regression models are able to predict the reduction disintegration index, reduction index, and the softening-melting property of manganese ores as a function of the content of total iron and silica.


## 3.1. Mathematical Formulations

Mathematical formulations are driven to estimate the effect of Fe, SiO2, and their interaction (Fe-SiO2) on the metallurgical properties of manganese ores. The effect of “A” at low level of B is [
a
-
(
1
)
] and the effect of “A” at high level of B is [
a
b
-
b
]. The main effect of “A” is the average of its effect at low and high level of B as given in
(1)
A
=
1
2
a
b
-
b
+
a
-
1
=
1
2
a
b
+
a
-
b
-
(
1
)
.
The average effect of “B” can be calculated from the effect of “B” at low level of “A” ([
b
-
(
1
)
]) and at the high level of “A” (i.e., [
a
b
-
a
]) as given in
(2)
B
=
1
2
a
b
-
a
+
b
-
(
1
)
=
1
2
a
b
+
b
-
a
-
(
1
)
.
The interaction effect “A
B” is defined as the average difference between the effect of “A” at the high level of “B” and the effect of “A” at the low level of “B” as given in
(3)
A
B
=
1
2
a
b
-
b
-
a
-
(
1
)
=
1
2
a
b
+
(
1
)
-
a
-
b
.
The sum of squares (SS) of A, B, and A
B can be calculated as given in (4)–(6), respectively. Consider
(4)
S
S
A
=
a
b
+
a
-
b
-
(
1
)
2
4
,
(5)
S
S
B
=
a
b
+
b
-
a
-
(
1
)
4
2
,
(6)
S
S
A
B
=
a
b
+
(
1
)
-
a
-
b
4
2
.
The total sum of squares (S
S
T) and sum of squares (S
S
E) can be calculated using (7) and (8) respectively. Consider
(7)
S
S
T
=
∑
i
=
1
2
∑
j
=
1
2
∑
k
=
1
n
y
i
j
k
2
-
y
2
4
,
(8)
S
S
E
=
S
S
T
-
S
S
A
-
S
S
B
-
S
S
A
B
.


## 3.2. Application of Factorial Design

### 3.2.1. Reduction Disintegration Index (RDI)

The complete analyses of the effect of Fe (A) and/or SiO2 (B) on reduction disintegration index (RDI) of manganese ores can be calculated using (1)–(8). The effects of Fe and/or SiO2 on RDI+6.3, RDI+3.15, and RDI−0.5 are given in Tables 2–4, respectively.Table 2
Effect of total iron (A), silica (B), and their interaction (A
B) on  on RDI+6.3.
Source of variance
The average effect
Sum of square (SS)
Mean square (MS)(MS = SS/degree of freedom)
Magnitude effect,
F
o (MSVariance/M
S
E)
A
−1.1518
2.653286
2.653286
9.096124
B
5.4581
59.58171
59.58171
204.2609
A
B
−9.64705
186.1311
186.1311
638.1037
Error
1.166777
0.291694
Total
249.5329Table 3
Effect of total iron (A), silica (B), and their interaction (A
B) on RDI+3.15.
Source of variance
The average effect
Sum of square (SS)
Mean square (MS)(MS = SS/degree of freedom)
Magnitude effect,
F
o (MSVariance/M
S
E)
A
1.765325
6.232745
6.232745
17.63527
B
4.867275
47.38073
47.38073
134.0617
A
B
−6.47918
83.95942
83.95942
237.5594
Error
1.4137
0.353425
Total
138.9866Table 4
Effect of total iron (A), silica (B), and their interaction (A
B) on RD
I
-
0.5.
Source of variance
The average effect
Sum of square (SS)
Mean square (MS)(MS = SS/degree of freedom)
Magnitude effect,
F
o (MSVariance/M
S
E)
A
0.185425
0.068765
0.068765
37.04831
B
0.893125
1.595345
1.595345
859.5207
A
B
2.370475
11.2383
11.2383
6054.839
Error
0.007424
0.001856
Total
12.90984In Table2, it can be seen that the highest negative effect on RDI+6.3 is exhibited by the interaction of iron and silica followed by the individual effect of iron. On the other hand, silica exhibited a positive effect on RDI+6.3 of the reduced manganese ores. The results in Table 3 indicate that both of iron and silica have an individual positive effect on RDI+3.15 while the interaction coefficient of iron and silica affects negatively the RDI+3.15 of the reduced manganese ores. Table 4 indicates that the individual and collective parameters have positive effects on RDI−0.5 with relatively higher magnitude for the interaction coefficient parameter of iron with silica.The experimental results can be generally expressed in terms of regression model as given in (9) for RDI+6.3, RDI+3.15, and RDI−0.5, respectively. Consider
(9)
Y
=
β
0
+
β
1
x
1
+
β
2
x
2
+
β
12
x
1
x
2
+
ε
,
where Y refers to RDI+6.3, RDI3.15, or RDI−0.5, x
1 is a coded variable representing the total iron, x
2 is a coded variable representing the SiO2, β
s
′ is regression coefficient, and ε is the residual (the difference between observed and fitted point of the design). β
0 is the intercept which is the grand average of all observations; the regression coefficients β
1
,
β
2 and β
12 are one-half the corresponding factor while ε is the residual. The values of β
0, β
1, β
2, and ε are given in Table 5. In all cases, the residuals are very small and can be neglected.Table 5
Regression coefficient values for RDI+6.3, RDI+3.15, and RDI
-
0.5.
Y
β
0
β
1
β
2
β
12
ε
RDI+6.3
76.18
−0.5759
2.729
−4.82353
±0.419
RDI+3.15
83.96099
0.882662
2.433638
−3.23959
±0.441
RDI
-
0.5
5.95688
0.092713
0.446563
1.185238
±0.0386The relation between the natural variables and the coded variable is given as follow: the coded variable is equal to [(natural variable − 1/2(variable at high level + variable at low level))/1/2(variable at high level − variable at low level)]. Consequently, the RDI+6.3, RDI+3.15, and RDI−0.5 can be predicted as a function of total iron and SiO2 as given in (10)–(12), respectively. Consider
(10)
RDI
+
6.3
=
59.11397
-
0.0533
[
TotalFe
]
+
0.730374
Si
O
2
+
1.221772
TotalFe
Si
O
2
,
(11)
RD
I
+
3.15
=
69.68242
-
0.0358
TotalFe
+
0.591889
Si
O
2
+
0.903718
TotalFe
Si
O
2
,
(12)
RD
I
-
0.5
=
7.960216
-
0.013098
TotalFe
-
0.18336
Si
O
2
-
0.14559
TotalFe
Si
O
2
.The RDI+6.3, RDI+3.15, and RDI−0.5 for the different grades of manganese ores (low-Fe low-Si, high-Fe low-Si, low-Fe high-Si, and high-Fe high-Si manganese ores) are calculated based on (10)–(12). The calculated values of RDI are compared to the experimental results as can be seen in Figures 1, 2, and 3. The given notations summarized the grade of ores as follows: refers to low-Fe low-Si ore; a refers to high-Fe low-Si ore; b refers to low-Fe high-Si ore; a
b refers to high-Fe high-Si ore. It can be seen that, in all cases, the coded and actual variables are in good agreement with those of the experimental results. The RDI+6.3 and RDI+3.15 are the highest in low-Fe high-Si ores and the lowest in low-Fe low-Si ores. This indicates that silica has the ability to improve the strength of manganese ore during the reduction at 500°C [11]. In the iron-manganese oxides, manganese ferrite (Fe2MnO4) is formed in vicinity of large pores as a result of solid solution reaction between MnO and Fe2O3. On the other hand, the presence of silica in iron-manganese ore is able to diminish the formation of manganese ferrite (MnFe2O4), consequently decreasing the porosity and improving the strength of the ore [18, 24]. The iron content has a significant effect on the reduction disintegration index. The ore disintegration increases with iron due to the crystal distortion which is accompanied by the transformation of hematite to magnetite. Such disintegration is caused on one hand by lattice transformations and on the other hand by an anisotropic reaction rate [25]. Hematite crystallizes in hexagonal rhombohedral lattice while magnetite has an inverse spinel lattice structure. During the transformation from hematite to magnetite, a layer of close magnetite grows on the surface of the porous hematite and results in cracks and disintegrations. The unit cell volume of magnetite is equal to 592.07 Å3 which it is equal to 302.72 Å3 for hematite. This results in a disintegration of the ore during reduction. In addition, the transformation of MnO2 into Mn2O3 is accompanied by sharp increase in the cell volume from 55.64 to 834.56 [11]. Therefore, the index of reduction disintegration is decreased as the iron and/or manganese content in the ore increased.Figure 1
The experimental and predicted RDI+6.3 by using coded and actual variables.Figure 2
The experimental and predicted RDI+3.15 by using coded and actual variables.Figure 3
The experimental and predicted RDI−0.5 by using coded and actual variables.


### 3.2.2. Reduction Index (RI)

The effect of total Fe and/or silica on the reduction index is calculated based on the total Fe-Mn oxides (RIT), Mn oxide (RIM), and Fe oxide (RIF) as given in Tables6–8. Table 6 indicates that total Fe has the highest positive effect on the total reduction RIT. Both of the silica and the interaction of total Fe with silica have almost equal positive effect which is lower than the individual effect of Fe. Table 7 shows that all parameters have negative effect on the reduction of manganese (RIM) with relatively higher negative effect of total Fe. Table 8 indicates that the total iron has a relatively high positive effect on the reduction of iron oxide (RIF). On the other hand, the interaction coefficient between Fe and silica exhibited a negative effect on the reduction of iron oxides.Table 6
Effect of total iron (A), silica (B), and their interaction (A
B) on the RIT.
Source of variance
The average effect
Sum of square (SS)
Mean square (MS)(MS = SS/degree of freedom)
Magnitude effect,
F
o (MSVariance/M
S
E)
A
19.687
775.1559
775.1559
3994.748
B
6.3685
81.11558
81.11558
418.0273
A
B
6.268
78.57565
78.57565
404.9378
Error
0.776175
0.194044
Total
935.6233Table 7
Effect of total iron (A), silica (B), and their interaction (A
B) on the RIM.
Source of variance
The average effect
Sum of square (SS)
Mean square (MS)(MS = SS/degree of freedom)
Magnitude effect,
F
o (MSVariance/M
S
E)
A
−2.3
10.58
10.58
2116
B
−0.3
0.18
0.18
36
A
B
−1.4
3.92
3.92
784
Error
0.02
0.005
Total
14.7Table 8
Effect of total iron (A), silica (B), and their interaction (A
B) on the RIF.
Source of variance
The average effect
Sum of square (SS)
Mean square (MS)(MS = SS/degree of freedom)
Magnitude effect,
F
o (MSVariance/MS
E)
A
21.4555
920.677
920.677
2537.032
B
0.613
0.751538
0.751538
2.07095
A
B
−2.603
13.55122
13.55122
37.34195
Error
1.451581
0.362895
Total
936.4313The results of experiments can be expressed in terms of regression models of RIT, RIM, and RIF as given in (13). The values of β
0, β
1, β
2, and ε are given in Table 9. In all cases, the residuals are very small and can be neglected. Consider
(13)
Z
=
β
0
+
β
1
x
1
+
β
2
x
2
+
β
12
x
1
x
2
+
ε
,
where Z refers to RIT, RIM, or RIF.Table 9
Regression coefficient values for RIT, RIM, and RIF.
Z
β
0
β
1
β
2
β
12
ε
RIT
61.24925
9.8435
3.18425
3.134
±0.389
RIM
98.15
−1.15
−0.15
−0.7
±0.05
RIF
84.39425
10.7277
0.3065
−1.3015
±0.483The RIT, RIM, RIF can be predicted as a function of total iron and SiO2 as given in (14)–(16), respectively. The calculated values of RIT, RIM, and RIF are compared to the experimetal results as can be seen in Figures 4–6. It can be seen that, in all cases, the coded and actual variables are in a good agreement with the experimental data. Consider
(14)
RI
=
50.37273
+
0.034632
TotalFe
+
0.281486
Si
O
2
-
0.10768
TotalFe
Si
O
2
,
(15)
RIM
=
98.12223
-
0.00774
TotalFe
+
0.02085
Si
O
2
+
0.101729
TotalFe
Si
O
2
,
(16)
RIF
=
66.89649
-
0.01438
TotalFe
+
1.065985
Si
O
2
+
0.270166
TotalFe
Si
O
2
.
It can be seen in Figure 4 that high-Fe high-Si and high-Fe low-Si manganese ores exhibited the highest reduction index. On the other hand, the lowest reduction index was exhibited in low-Fe low-Si and low-Fe high-Si ones. This indicates that the reduction index of manganese ore increases as iron oxide content increases. The reduction index based on manganese oxide (RIM) is almost identical for the different grades of manganese ores as shown in Figure 5. This can be attributed to the simple reduction of MnO2 to MnO. The reduction index based on iron oxide (RIF) is high in the ores rich with iron and low in the ore poor in iron oxides as shown in Figure 6. This is attributed to the higher reducibility of iron oxides compared to that of manganese oxides. Although the reduction of wüstite (FexO) to metallic iron required relatively high potential of reducing gas (η
CO
≈
30%, η
H
2
≈
30%), the reduction of MnO to Mn metal is more complicated and can be only proceeded by solid carbon at very high temperature (≈1423°C). Therefore, the reduction index decreased as manganese content increased.Figure 4
The experimental and predicted RIT by using coded and actual variables.Figure 5
The experimental and predicted RIM by using coded and actual variables.Figure 6
The experimental and predicted RIF by using coded and actual variables.


### 3.2.3. Softening-Melting Property (SMP)

A mathematical regression model is derived to estimate the effect of total iron and/or silica on the softening-melting property of manganese ores during reduction. The softening ranges can be estimated based on the determination of temperature at which the reduced ores start to soften (T
S
1: starting of softening) and the temperature at which the softening finsihed (T
S
2: end of softening). The melting range can be determined based on the identification of the start of melting (T
m
1) and the end of melting (T
m
2). The effects of total iron and/or silica on the softening property of reduced manganese ores are given in Tables 10 and 11, respectively. It can be seen that the iron and silica affect negatively the start and the end softening temperature.Table 10
Effect of total iron (A), silica (B), and their interaction (A
B) on T
S
1.
Source of variance
The average effect
Sum of square (SS)
Mean square (MS)(MS = SS/degree of freedom)
Magnitude effect,
F
o (MSVariance/M
S
E)
A
−29.5
1740.5
1740.5
386.7778
B
−47.5
4512.5
4512.5
1002.778
A
B
−42.5
3612.5
3612.5
802.7778
Error
18
4.5
Total
9883.5Table 11
Effect of total iron (A), silica (B), and their interaction (A
B) on T
S
2.
Source of variance
The average effect
Sum of square (SS)
Mean square (MS)(MS = SS/degree of freedom)
Magnitude effect,
F
o (MSVariance/M
S
E)
A
−8
128
128
10.24
B
−64
8192
8192
655.36
A
B
−10
200
200
16
Error
50
12.5
Total
8570The effect of iron and/or silica content on the melting property of manganese ores includingT
m
1 and T
m
2 is given in Tables 12 and 13, respectively. As can be seen in Table 12, the iron and the iron with silica affect positively the start temperature of melting while silica exhibited a negative effect on the start temperature of melting. Both of iron and silica decreased the end temperature of melting of manganese ore, while the interaction of iron and silica has positive effect on the end of melting as given in Table 13.Table 12
Effect of total iron (A), silica (B), and their interaction (A
B) on T
m
1.
Source of variance
The average effect
Sum of square (SS)
Mean square (MS)(MS = SS/degree of freedom)
Magnitude effect,
F
o (MSVariance/M
S
E)
A
44
3872
3872
1936
B
−79
12482
12482
6241
A
B
22
968
968
484
Error
8
2
Total
17330Table 13
Effect of total iron (A), silica (B), and their interaction (A
B) on T
m
2.
Source of variance
The average effect
Sum of square (SS)
Mean square (MS)(MS = SS/degree of freedom)
Magnitude effect,
F
o (MSVariance/M
S
E)
A
−19
722
722
361
B
−82
13448
13448
6724
A
B
30
1800
1800
900
Error
8
2
Total
15978The relation between the natural variables and the coded variable forT
S
1, T
S
2, T
m
1, and T
m
2 can be summarized in (17). The values of β
0, β
1, β
2, and ε are given in Table 14. Consider
(17)
H
=
β
0
+
β
1
x
1
+
β
2
x
2
+
β
12
x
1
x
2
+
ε
,
where H refers to T
S
1, T
S
2, T
m
1, or T
m
2.Table 14
Values of regression coefficient forT
S
1, T
S
2, T
m
1, and T
m
2.
β
0
β
1
β
2
β
12
ε
T
S
1
1123.25
−14.75
−23.75
−21.25
±1.5
T
S
2
1240.5
−4.0
−32.0
−5.0
±2.5
Δ
T
S
119.6888
11.65125
−8.33125
16.29125
±0.7
T
m
1
1229.5
22.0
−39.5
11.0
±1.0
T
m
2
1266.5
−9.5
−41.0
15.0
±1.0
Δ
T
m
36.66
−32.1825
−2.0175
3.51
±0.37TheT
S
1, T
S
2, T
m
1, and T
m
2 can be predicted as a function of total iron and SiO2 as given in (18)–(21), respectively. The calculated values of T
S
1, T
S
2, T
m
1, and T
m
2 are compared to the experimental results as can be seen in Figures 7–10. It can be seen that, in all cases, the coded and actual variables are in a good agreement with the experimental results. Consider
(18)
T
S
1
=
1135.617
-
0.23482
TotalFe
+
2.242574
Si
O
2
+
0.431248
TotalFe
Si
O
2
,
(19)
T
S
2
=
1297.323
-
0.05525
TotalFe
+
0.485396
Si
O
2
-
3.55414
TotalFe
Si
O
2
,
(20)
T
m
1
=
1309.35
+
0.121556
TotalFe
-
0.01398
Si
O
2
-
7.39197
TotalFe
Si
O
2
,
(21)
T
m
2
=
1399.393
+
0.165758
TotalFe
-
3.17275
Si
O
2
-
8.29953
TotalFe
Si
O
2
.Figure 7
The experimental and predictedT
S
1 by using coded and actual variables.Figure 8
The experimental and predictedT
S
2 by using coded and actual variables.Figure 9
The softening range (T
Δ
S) for experimental, coded, and actual variables.Figure 10
The experimental and predictedT
m
1 by using coded and actual variables.Figures7 and 10 showed that the lowest start temperature of softening (T
S
1) and melting (T
m
1) is exhibited in high-Fe high-Si manganese ore which is equal to 1062–1089°C and 1199.5–1225°C, respectively. Figures 8 and 11 clarify that the highest end of softening (T
S
2) and melting (T
m
2) is exhibited in high-Fe low-Si manganese ore which is equal to 1152–1154°C and 1273–1276°C, respectively. This indicates that the presence of relatively high percentage of silica resulted in a decrease of the start softening and melting temperatures. This can be attributed to the formation of relatively low melting rhodonite phase (MnSiO3; m.p. 1242°C) as a result of the reaction between SiO2 and MnO [26]. The formation of low melting rhodonite phase resulted in a narrow softening range (T
Δ
S) in low-Fe high-SiO2 manganese ore as can be seen in Figure 9. The high-Fe low-si and high-Fe high-si manganese ores showed a narrow melting range as can be seen in Figure 12. This can be attributed to the formation Fe-Mn olivine in the presence of relatively high concentration of iron [11].Figure 11
The experimental and predictedT
m
2 by using coded and actual variables.Figure 12
The melting range (T
Δ
m) for experimental, coded, and actual variables.Based on the previous findings, it can be concluded that the factorial design is very useful approach to predict and precisely estimate the effect of different impurities such as Fe and/or Si which commonly contaminate the manganese ores and affect negatively the smelting reduction process. The derived mathematical regression models are able to predict the reduction disintegration index, reduction index, and the softening-melting property of manganese ores as a function of the content of total iron and silica.


## 3.2.1. Reduction Disintegration Index (RDI)

The complete analyses of the effect of Fe (A) and/or SiO2 (B) on reduction disintegration index (RDI) of manganese ores can be calculated using (1)–(8). The effects of Fe and/or SiO2 on RDI+6.3, RDI+3.15, and RDI−0.5 are given in Tables 2–4, respectively.Table 2
Effect of total iron (A), silica (B), and their interaction (A
B) on  on RDI+6.3.
Source of variance
The average effect
Sum of square (SS)
Mean square (MS)(MS = SS/degree of freedom)
Magnitude effect,
F
o (MSVariance/M
S
E)
A
−1.1518
2.653286
2.653286
9.096124
B
5.4581
59.58171
59.58171
204.2609
A
B
−9.64705
186.1311
186.1311
638.1037
Error
1.166777
0.291694
Total
249.5329Table 3
Effect of total iron (A), silica (B), and their interaction (A
B) on RDI+3.15.
Source of variance
The average effect
Sum of square (SS)
Mean square (MS)(MS = SS/degree of freedom)
Magnitude effect,
F
o (MSVariance/M
S
E)
A
1.765325
6.232745
6.232745
17.63527
B
4.867275
47.38073
47.38073
134.0617
A
B
−6.47918
83.95942
83.95942
237.5594
Error
1.4137
0.353425
Total
138.9866Table 4
Effect of total iron (A), silica (B), and their interaction (A
B) on RD
I
-
0.5.
Source of variance
The average effect
Sum of square (SS)
Mean square (MS)(MS = SS/degree of freedom)
Magnitude effect,
F
o (MSVariance/M
S
E)
A
0.185425
0.068765
0.068765
37.04831
B
0.893125
1.595345
1.595345
859.5207
A
B
2.370475
11.2383
11.2383
6054.839
Error
0.007424
0.001856
Total
12.90984In Table2, it can be seen that the highest negative effect on RDI+6.3 is exhibited by the interaction of iron and silica followed by the individual effect of iron. On the other hand, silica exhibited a positive effect on RDI+6.3 of the reduced manganese ores. The results in Table 3 indicate that both of iron and silica have an individual positive effect on RDI+3.15 while the interaction coefficient of iron and silica affects negatively the RDI+3.15 of the reduced manganese ores. Table 4 indicates that the individual and collective parameters have positive effects on RDI−0.5 with relatively higher magnitude for the interaction coefficient parameter of iron with silica.The experimental results can be generally expressed in terms of regression model as given in (9) for RDI+6.3, RDI+3.15, and RDI−0.5, respectively. Consider
(9)
Y
=
β
0
+
β
1
x
1
+
β
2
x
2
+
β
12
x
1
x
2
+
ε
,
where Y refers to RDI+6.3, RDI3.15, or RDI−0.5, x
1 is a coded variable representing the total iron, x
2 is a coded variable representing the SiO2, β
s
′ is regression coefficient, and ε is the residual (the difference between observed and fitted point of the design). β
0 is the intercept which is the grand average of all observations; the regression coefficients β
1
,
β
2 and β
12 are one-half the corresponding factor while ε is the residual. The values of β
0, β
1, β
2, and ε are given in Table 5. In all cases, the residuals are very small and can be neglected.Table 5
Regression coefficient values for RDI+6.3, RDI+3.15, and RDI
-
0.5.
Y
β
0
β
1
β
2
β
12
ε
RDI+6.3
76.18
−0.5759
2.729
−4.82353
±0.419
RDI+3.15
83.96099
0.882662
2.433638
−3.23959
±0.441
RDI
-
0.5
5.95688
0.092713
0.446563
1.185238
±0.0386The relation between the natural variables and the coded variable is given as follow: the coded variable is equal to [(natural variable − 1/2(variable at high level + variable at low level))/1/2(variable at high level − variable at low level)]. Consequently, the RDI+6.3, RDI+3.15, and RDI−0.5 can be predicted as a function of total iron and SiO2 as given in (10)–(12), respectively. Consider
(10)
RDI
+
6.3
=
59.11397
-
0.0533
[
TotalFe
]
+
0.730374
Si
O
2
+
1.221772
TotalFe
Si
O
2
,
(11)
RD
I
+
3.15
=
69.68242
-
0.0358
TotalFe
+
0.591889
Si
O
2
+
0.903718
TotalFe
Si
O
2
,
(12)
RD
I
-
0.5
=
7.960216
-
0.013098
TotalFe
-
0.18336
Si
O
2
-
0.14559
TotalFe
Si
O
2
.The RDI+6.3, RDI+3.15, and RDI−0.5 for the different grades of manganese ores (low-Fe low-Si, high-Fe low-Si, low-Fe high-Si, and high-Fe high-Si manganese ores) are calculated based on (10)–(12). The calculated values of RDI are compared to the experimental results as can be seen in Figures 1, 2, and 3. The given notations summarized the grade of ores as follows: refers to low-Fe low-Si ore; a refers to high-Fe low-Si ore; b refers to low-Fe high-Si ore; a
b refers to high-Fe high-Si ore. It can be seen that, in all cases, the coded and actual variables are in good agreement with those of the experimental results. The RDI+6.3 and RDI+3.15 are the highest in low-Fe high-Si ores and the lowest in low-Fe low-Si ores. This indicates that silica has the ability to improve the strength of manganese ore during the reduction at 500°C [11]. In the iron-manganese oxides, manganese ferrite (Fe2MnO4) is formed in vicinity of large pores as a result of solid solution reaction between MnO and Fe2O3. On the other hand, the presence of silica in iron-manganese ore is able to diminish the formation of manganese ferrite (MnFe2O4), consequently decreasing the porosity and improving the strength of the ore [18, 24]. The iron content has a significant effect on the reduction disintegration index. The ore disintegration increases with iron due to the crystal distortion which is accompanied by the transformation of hematite to magnetite. Such disintegration is caused on one hand by lattice transformations and on the other hand by an anisotropic reaction rate [25]. Hematite crystallizes in hexagonal rhombohedral lattice while magnetite has an inverse spinel lattice structure. During the transformation from hematite to magnetite, a layer of close magnetite grows on the surface of the porous hematite and results in cracks and disintegrations. The unit cell volume of magnetite is equal to 592.07 Å3 which it is equal to 302.72 Å3 for hematite. This results in a disintegration of the ore during reduction. In addition, the transformation of MnO2 into Mn2O3 is accompanied by sharp increase in the cell volume from 55.64 to 834.56 [11]. Therefore, the index of reduction disintegration is decreased as the iron and/or manganese content in the ore increased.Figure 1
The experimental and predicted RDI+6.3 by using coded and actual variables.Figure 2
The experimental and predicted RDI+3.15 by using coded and actual variables.Figure 3
The experimental and predicted RDI−0.5 by using coded and actual variables.


## 3.2.2. Reduction Index (RI)

The effect of total Fe and/or silica on the reduction index is calculated based on the total Fe-Mn oxides (RIT), Mn oxide (RIM), and Fe oxide (RIF) as given in Tables6–8. Table 6 indicates that total Fe has the highest positive effect on the total reduction RIT. Both of the silica and the interaction of total Fe with silica have almost equal positive effect which is lower than the individual effect of Fe. Table 7 shows that all parameters have negative effect on the reduction of manganese (RIM) with relatively higher negative effect of total Fe. Table 8 indicates that the total iron has a relatively high positive effect on the reduction of iron oxide (RIF). On the other hand, the interaction coefficient between Fe and silica exhibited a negative effect on the reduction of iron oxides.Table 6
Effect of total iron (A), silica (B), and their interaction (A
B) on the RIT.
Source of variance
The average effect
Sum of square (SS)
Mean square (MS)(MS = SS/degree of freedom)
Magnitude effect,
F
o (MSVariance/M
S
E)
A
19.687
775.1559
775.1559
3994.748
B
6.3685
81.11558
81.11558
418.0273
A
B
6.268
78.57565
78.57565
404.9378
Error
0.776175
0.194044
Total
935.6233Table 7
Effect of total iron (A), silica (B), and their interaction (A
B) on the RIM.
Source of variance
The average effect
Sum of square (SS)
Mean square (MS)(MS = SS/degree of freedom)
Magnitude effect,
F
o (MSVariance/M
S
E)
A
−2.3
10.58
10.58
2116
B
−0.3
0.18
0.18
36
A
B
−1.4
3.92
3.92
784
Error
0.02
0.005
Total
14.7Table 8
Effect of total iron (A), silica (B), and their interaction (A
B) on the RIF.
Source of variance
The average effect
Sum of square (SS)
Mean square (MS)(MS = SS/degree of freedom)
Magnitude effect,
F
o (MSVariance/MS
E)
A
21.4555
920.677
920.677
2537.032
B
0.613
0.751538
0.751538
2.07095
A
B
−2.603
13.55122
13.55122
37.34195
Error
1.451581
0.362895
Total
936.4313The results of experiments can be expressed in terms of regression models of RIT, RIM, and RIF as given in (13). The values of β
0, β
1, β
2, and ε are given in Table 9. In all cases, the residuals are very small and can be neglected. Consider
(13)
Z
=
β
0
+
β
1
x
1
+
β
2
x
2
+
β
12
x
1
x
2
+
ε
,
where Z refers to RIT, RIM, or RIF.Table 9
Regression coefficient values for RIT, RIM, and RIF.
Z
β
0
β
1
β
2
β
12
ε
RIT
61.24925
9.8435
3.18425
3.134
±0.389
RIM
98.15
−1.15
−0.15
−0.7
±0.05
RIF
84.39425
10.7277
0.3065
−1.3015
±0.483The RIT, RIM, RIF can be predicted as a function of total iron and SiO2 as given in (14)–(16), respectively. The calculated values of RIT, RIM, and RIF are compared to the experimetal results as can be seen in Figures 4–6. It can be seen that, in all cases, the coded and actual variables are in a good agreement with the experimental data. Consider
(14)
RI
=
50.37273
+
0.034632
TotalFe
+
0.281486
Si
O
2
-
0.10768
TotalFe
Si
O
2
,
(15)
RIM
=
98.12223
-
0.00774
TotalFe
+
0.02085
Si
O
2
+
0.101729
TotalFe
Si
O
2
,
(16)
RIF
=
66.89649
-
0.01438
TotalFe
+
1.065985
Si
O
2
+
0.270166
TotalFe
Si
O
2
.
It can be seen in Figure 4 that high-Fe high-Si and high-Fe low-Si manganese ores exhibited the highest reduction index. On the other hand, the lowest reduction index was exhibited in low-Fe low-Si and low-Fe high-Si ones. This indicates that the reduction index of manganese ore increases as iron oxide content increases. The reduction index based on manganese oxide (RIM) is almost identical for the different grades of manganese ores as shown in Figure 5. This can be attributed to the simple reduction of MnO2 to MnO. The reduction index based on iron oxide (RIF) is high in the ores rich with iron and low in the ore poor in iron oxides as shown in Figure 6. This is attributed to the higher reducibility of iron oxides compared to that of manganese oxides. Although the reduction of wüstite (FexO) to metallic iron required relatively high potential of reducing gas (η
CO
≈
30%, η
H
2
≈
30%), the reduction of MnO to Mn metal is more complicated and can be only proceeded by solid carbon at very high temperature (≈1423°C). Therefore, the reduction index decreased as manganese content increased.Figure 4
The experimental and predicted RIT by using coded and actual variables.Figure 5
The experimental and predicted RIM by using coded and actual variables.Figure 6
The experimental and predicted RIF by using coded and actual variables.


## 3.2.3. Softening-Melting Property (SMP)

A mathematical regression model is derived to estimate the effect of total iron and/or silica on the softening-melting property of manganese ores during reduction. The softening ranges can be estimated based on the determination of temperature at which the reduced ores start to soften (T
S
1: starting of softening) and the temperature at which the softening finsihed (T
S
2: end of softening). The melting range can be determined based on the identification of the start of melting (T
m
1) and the end of melting (T
m
2). The effects of total iron and/or silica on the softening property of reduced manganese ores are given in Tables 10 and 11, respectively. It can be seen that the iron and silica affect negatively the start and the end softening temperature.Table 10
Effect of total iron (A), silica (B), and their interaction (A
B) on T
S
1.
Source of variance
The average effect
Sum of square (SS)
Mean square (MS)(MS = SS/degree of freedom)
Magnitude effect,
F
o (MSVariance/M
S
E)
A
−29.5
1740.5
1740.5
386.7778
B
−47.5
4512.5
4512.5
1002.778
A
B
−42.5
3612.5
3612.5
802.7778
Error
18
4.5
Total
9883.5Table 11
Effect of total iron (A), silica (B), and their interaction (A
B) on T
S
2.
Source of variance
The average effect
Sum of square (SS)
Mean square (MS)(MS = SS/degree of freedom)
Magnitude effect,
F
o (MSVariance/M
S
E)
A
−8
128
128
10.24
B
−64
8192
8192
655.36
A
B
−10
200
200
16
Error
50
12.5
Total
8570The effect of iron and/or silica content on the melting property of manganese ores includingT
m
1 and T
m
2 is given in Tables 12 and 13, respectively. As can be seen in Table 12, the iron and the iron with silica affect positively the start temperature of melting while silica exhibited a negative effect on the start temperature of melting. Both of iron and silica decreased the end temperature of melting of manganese ore, while the interaction of iron and silica has positive effect on the end of melting as given in Table 13.Table 12
Effect of total iron (A), silica (B), and their interaction (A
B) on T
m
1.
Source of variance
The average effect
Sum of square (SS)
Mean square (MS)(MS = SS/degree of freedom)
Magnitude effect,
F
o (MSVariance/M
S
E)
A
44
3872
3872
1936
B
−79
12482
12482
6241
A
B
22
968
968
484
Error
8
2
Total
17330Table 13
Effect of total iron (A), silica (B), and their interaction (A
B) on T
m
2.
Source of variance
The average effect
Sum of square (SS)
Mean square (MS)(MS = SS/degree of freedom)
Magnitude effect,
F
o (MSVariance/M
S
E)
A
−19
722
722
361
B
−82
13448
13448
6724
A
B
30
1800
1800
900
Error
8
2
Total
15978The relation between the natural variables and the coded variable forT
S
1, T
S
2, T
m
1, and T
m
2 can be summarized in (17). The values of β
0, β
1, β
2, and ε are given in Table 14. Consider
(17)
H
=
β
0
+
β
1
x
1
+
β
2
x
2
+
β
12
x
1
x
2
+
ε
,
where H refers to T
S
1, T
S
2, T
m
1, or T
m
2.Table 14
Values of regression coefficient forT
S
1, T
S
2, T
m
1, and T
m
2.
β
0
β
1
β
2
β
12
ε
T
S
1
1123.25
−14.75
−23.75
−21.25
±1.5
T
S
2
1240.5
−4.0
−32.0
−5.0
±2.5
Δ
T
S
119.6888
11.65125
−8.33125
16.29125
±0.7
T
m
1
1229.5
22.0
−39.5
11.0
±1.0
T
m
2
1266.5
−9.5
−41.0
15.0
±1.0
Δ
T
m
36.66
−32.1825
−2.0175
3.51
±0.37TheT
S
1, T
S
2, T
m
1, and T
m
2 can be predicted as a function of total iron and SiO2 as given in (18)–(21), respectively. The calculated values of T
S
1, T
S
2, T
m
1, and T
m
2 are compared to the experimental results as can be seen in Figures 7–10. It can be seen that, in all cases, the coded and actual variables are in a good agreement with the experimental results. Consider
(18)
T
S
1
=
1135.617
-
0.23482
TotalFe
+
2.242574
Si
O
2
+
0.431248
TotalFe
Si
O
2
,
(19)
T
S
2
=
1297.323
-
0.05525
TotalFe
+
0.485396
Si
O
2
-
3.55414
TotalFe
Si
O
2
,
(20)
T
m
1
=
1309.35
+
0.121556
TotalFe
-
0.01398
Si
O
2
-
7.39197
TotalFe
Si
O
2
,
(21)
T
m
2
=
1399.393
+
0.165758
TotalFe
-
3.17275
Si
O
2
-
8.29953
TotalFe
Si
O
2
.Figure 7
The experimental and predictedT
S
1 by using coded and actual variables.Figure 8
The experimental and predictedT
S
2 by using coded and actual variables.Figure 9
The softening range (T
Δ
S) for experimental, coded, and actual variables.Figure 10
The experimental and predictedT
m
1 by using coded and actual variables.Figures7 and 10 showed that the lowest start temperature of softening (T
S
1) and melting (T
m
1) is exhibited in high-Fe high-Si manganese ore which is equal to 1062–1089°C and 1199.5–1225°C, respectively. Figures 8 and 11 clarify that the highest end of softening (T
S
2) and melting (T
m
2) is exhibited in high-Fe low-Si manganese ore which is equal to 1152–1154°C and 1273–1276°C, respectively. This indicates that the presence of relatively high percentage of silica resulted in a decrease of the start softening and melting temperatures. This can be attributed to the formation of relatively low melting rhodonite phase (MnSiO3; m.p. 1242°C) as a result of the reaction between SiO2 and MnO [26]. The formation of low melting rhodonite phase resulted in a narrow softening range (T
Δ
S) in low-Fe high-SiO2 manganese ore as can be seen in Figure 9. The high-Fe low-si and high-Fe high-si manganese ores showed a narrow melting range as can be seen in Figure 12. This can be attributed to the formation Fe-Mn olivine in the presence of relatively high concentration of iron [11].Figure 11
The experimental and predictedT
m
2 by using coded and actual variables.Figure 12
The melting range (T
Δ
m) for experimental, coded, and actual variables.Based on the previous findings, it can be concluded that the factorial design is very useful approach to predict and precisely estimate the effect of different impurities such as Fe and/or Si which commonly contaminate the manganese ores and affect negatively the smelting reduction process. The derived mathematical regression models are able to predict the reduction disintegration index, reduction index, and the softening-melting property of manganese ores as a function of the content of total iron and silica.


## 4. Conclusions

In the current study, a factorial design is built on the experimental data of four grades of manganese ores containing different percentages of iron and silica (low-Fe high-Si, high-Fe low-Si, low-Fe high-Si, and high-Fe high-Si manganese ores). The main findings can be summarized as follow.(1)
Regression formulations are derived to estimate the effect of total Fe and/or SiO2 on the reduction disintegration indexes (RDI+6.3, RDI+3.15, and RDI−0.5) of manganese ores. The RDI+6.3 and RDI+3.15 increased with the individual effect of SiO2 and the interaction effect of Fe-SiO2 while they decreased as the total Fe increased. The RDI−0.5 increased with Fe and decreased with individual effect of silica and the interaction effect of Fe-SiO2 in the manganese ores.
(2)
The effect of total Fe and/or SiO2 on the reduction indexes (total reduction of manganese and iron oxides (RIT), manganese oxides reduction (RIM), and iron oxides reduction (RIF)) is developed. The RIT and RIF increased as the iron oxide content in manganese ore increased. The RIM was almost identical due to the simple conversion of MnO2 to MnO.
(3)
The effect of total iron and SiO2 on the softening-melting property (start of softening (T
S
1) and end of softening (T
S
2), start of melting (T
m
1), and end of melting (T
m
2)) is mathematically derived. The developed formulations could be used to precisely predict the effect of total Fe and silica on the softening-melting property of manganese ores.
(4)
The validation of regressions formulations was found to be in a good agreement with the experimental data which indicates the efficiency of the factorial design to predict the metallurgical properties of manganese ores under the influence of different impurities.



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*Source: 679306-2015-01-01.xml*