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Surface area production during grinding
Minerals Engineering 22 (2009) 587–592
Contents lists available at ScienceDirect
Minerals Engineering journal homepage: www.elsevier.com/locate/mineng
Surface area production during grinding Elias Stamboliadis *, Olga Pantelaki, Evangelos Petrakis Technical University of Crete, Mineral Resources Engineering, University Campus, 73100 Chania, Crete, Greece
a r t i c l e
i n f o
Article history: Received 2 August 2008 Accepted 11 December 2008 Available online 12 January 2009 Keywords: Comminution Grinding Size analysis Specific surface area
a b s t r a c t During grinding, in cascading mills, the mass distribution of the material moves continuously to finer sizes. Grinding models are usually designed to predict the size distribution of the mill products either as a function of effective breakage events and the time or the energy consumed by the mill. Mill products are always tested for their size analysis and their fineness is usually expressed as the size d through which a certain amount of material passes, i.e. d80. However, particulate materials have some supplementary properties that a grinding model should predict as well. The specific surface area is such an important supplementary property. Initially, the present work examines the relationship between mass distribution and the surface area of ground materials and determines the conditions under which, mass distribution can be used to determine the surface area. Based on these findings the work further examines the operating conditions of a cascading mill under which the surface area of the material increases at the highest rate. The operating parameter examined is the mill load ratio expressed as the mass ratio of mill grinding media and the material present in the mill. Ó 2008 Elsevier Ltd. All rights reserved.
1. Introduction Grinding models are usually designed to predict the size distribution of the mill products either as a function of effective breakage events, the time or the energy consumed by the mill. They also take into consideration the size distribution of the mill feed. The models are tested against experimental data and are approved if the agreement is acceptable. The test is always based on the size analysis of the mill products. This analysis gives the mass distribution of the material at predetermined size classes and the criterion is the correlation between experimental and theoretical data. However, particulate materials have some supplementary properties that a reliable model should predict as well. An important supplementary property of particulate materials is their surface area, usually expressed as the specific surface area. Almost all grinding models fail to predict the specific surface area of the mill products, especially at intensive grinding conditions, when the products are much finer than the feed. An attempt to relate the surface area of a material to the energy consumed is described by Stamboliadis (2004), but this is still at a theoretical level and requires experimental justification. The specific surface area of particulate materials is usually calculated either by measurements of nitrogen adsorption, using the BET method, or by the size analysis of the material, using screens or the laser scattering method. For non porous materials both
* Corresponding author. Tel.: +30 28210 37601; fax: +30 28210 37884. E-mail address: [email protected] (E. Stamboliadis). 0892-6875/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.mineng.2008.12.001
methods should give comparative results but for materials with open pores the BET method gives a much larger surface area than the one corresponding to the size analysis. If one is interested to relate the size analysis to the specific surface area of a particulate material then the latter must be calculated from the size analysis. In this case the results represent the external surface area, regardless of any existing pores. Almost all the grinding models assume that in any mill product there exist particles down to zero size and as grinding proceeds the mass of the material in the n + 1 size class increases. Actually the number n, of reference sizes, does not affect the result of calculating the mass distribution and is selected to satisfy the practical needs and to simplify the experimental technique. This is so because the total mass of the material is conserved regardless of the number of size classes. However, as will be seen further down, this is not so with the calculation of the surface area of the material and the result depends on the number of size classes used. As a consequence most of the grinding models although they are useful in predicting the mass distribution they cannot be used for analytical work and determine other properties that depend on the size analysis. The present work explains the reason of such a discrepancy and determines the conditions under which the size analysis of a material can be used to determine its surface area. After this, it treats grinding as a surface area creating process and examines the effect of mill load on the surface creation efficiency of the mill. The results obtained conform to the theory of energy distribution in comminution, Stamboliadis (2007). Other operating parameters are also under investigation and the results will be published when available.
588
E. Stamboliadis et al. / Minerals Engineering 22 (2009) 587–592
For k < 1, if xmin = 0 then xk1 min ¼ 1 and Dss becomes infinite. For k > 1, if xmin = 0 then xk1 min ¼ 0 and Dss becomes
2. Calculation of the surface area from the size analysis The size analysis actually gives the cumulative mass fraction P(x) of particles finer than size x according to the integral of Eq. (1)
PðxÞ ¼
Z
Dss ¼
x
dPðxÞ
ð1Þ
RðxÞ ¼ 1 PðxÞ
ð2Þ
and the mass fraction dP(x) of particles at size x within the size interval dx is
dPðxÞ ¼ P0 ðxÞ dx
ð3Þ
Nhe external surface area S(x) of the single particle of size x is
SðxÞ ¼ f x2
ð4Þ
where f is the surface coefficient (for spheres f = p). The mass m(x) of the particle is
mðxÞ ¼ k q x3
ð5Þ
where k is the volume coefficient (for spheres k = p/6) and q is the density of the material. The specific surface area s(x) per unit mass of the particle is inversely proportional to its size
SðxÞ f 1 ¼ mðxÞ k q x
sðxÞ ¼
ð6Þ
dSðxÞ ¼ sðxÞ dPðxÞ or dSðxÞ ¼ sðxÞ P 0 ðxÞ dx
ð7Þ
The total surface area SS of the particulate material is
Z
xmax
dSðxÞ or SS ¼
0
Z
xmax
sðxÞ P0 ðxÞ dx
ð8Þ
0
The overall specific surface area ss of the material is
ss ¼
SS 1 ¼ M M
Z
xmax
ð9Þ
0
Dss ¼ ssðxmin Þ ssðxmax Þ
ð10Þ
In many cases for xmin = 0 then ss(0) = 0 as well and Dss acquires a definite maximum value ss(xmax). In many other cases ss(0) is indefinable and Dss cannot be calculated. As an example let us consider the case of a Gates–Gaudin–Schuhman (GGS) distribution where
PðxÞ ¼ M
x
k
ð11Þ
xmax
M = 1 but is not erased in order to denote that the unit of P(x) is mass. The corresponding equation for the over all specific surface area ss is found to be
ss ¼
f k k q xkmax
Z
xmax
xk2 dx
ð12Þ
0
For k – 1
Dss ¼
f ðln ðxmax Þ ln ðxmin ÞÞ k q xkmax
ð15Þ
if xmin = 0 then ln (xmin) = 1 and Dss becomes infinite. For most comminution products k < 1 and Dss cannot be calculated unless there is a minimum size below which no particles can exist. In practice the methods used to measure the size analysis give the mass fraction DPi of the material in the size range between xi and xi1. The size xi is one of the n reference sizes (screens) used. The n reference sizes determine n + 1 size classes. It is convenient to have a constant pffiffiffi ratio between reference sizes, xi /xi + 1 = constant, usually 2 or 2. The (n + 1)th size class corresponds to the mass fraction DPn + 1 of the material with size between zero and xn. The total mass M of the material is
M¼
nþ1 X
DP i ¼ 1
ð16Þ
1
After this the mass fraction of the material finer than size xi is
Pi ¼
nþ1 X
DP i
ð17Þ
and the mass fraction Ri of the material coarser than xi is
Ri ¼ 1 P i
ð18Þ
The average size di of the particles in the ith size class is pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi di ¼ xi xi1 and for the last class dn+1 = xn/2. The mass mi of the average particle, its external surface area Si and its specific surface area si are calculated from Eq. (4)–(6) replacing x by di. The surface area DSi of the material in the ith size class is
DSi ¼ DPi si sðxÞ P0 ðxÞ dx
Depending on the function P(x) the integral of Eqs. (8) and (9) is indefinable and one has to integrate from a minimum size xmin to the maximum xmax and calculate the specific surface area Dss of the material between the minimum and the maximum size
Dss ¼
iþ1
The surface area dS(x) of the particulate material of size x in the range dx is
SS ¼
ð14Þ
For k = 1
0
The total mass of the material is P(xmax) = M = 1, when x is equal to the size of the coarser particle. Nhe cumulative mass fraction R(x) of particles coarser than size x is
f k xk1 k q xkmax k 1 max
f k k1 ðxk1 max xmin Þ k q xkmax k 1
ð13Þ
ð19Þ
the total area SS of the material is
SS ¼
nþ1 X 1
DSi ¼
nþ1 X
DP i si
ð20Þ
1
the specific surface area ss is
ss ¼
SS M
ð21Þ
Taking into consideration Eqs. (6), (16), (20), and (21) the overall specific surface area of a particulate material is given by Eq. (22), which is equivalent to the mathematical form of Eq. (9)
ss ¼
Pnþ1 DPi f 1 d Pnþ1 i kq D Pi 1
ð22Þ
Eq. (22) indicates that the calculated value of the specific surface area of a particulate material containing particles down to almost zero size, depends on the size xn of the finer screen used. This means that the specific surface area of a particulate material cannot be calculated from the size analysis unless the minimum particle size is known together with the mass distribution of the material down to this size. The fact that the specific surface area of all fine products measured by the BET method give a definite value indicates that in all cases there is a corresponding minimum size in the particle population.
E. Stamboliadis et al. / Minerals Engineering 22 (2009) 587–592
589
3. Experimental procedure The material used is hand selected quartz from Assyros, Macedonia, Northern Greece, crushed to 0.85 mm, using a laboratory jaw crusher followed by cone crusher. The grinding equipment is a laboratory rod mill with internal dimensions D L 203 280 mm, rotating at 72 rpm, loaded with 9 kg of rods between 12 mm and 20 mm in diameter. There were three grinding tests each one using 0.5 kg, 1.0 kg and 1.5 kg of crushed quartz respectively. In these tests the feed ratio is defined as the mass ratio of feed material to grinding media, and it is 0.056, 0.111 and 0.167, respectively. During each test the mill was stopped at j predetermined intervals of cumulative time tj and a representative sample of about 5 g was collected for size analysis using a laser beam analyzer. The surface area of the sample is calculated from the size analysis and for some samples it is also measured by the BET nitrogen adsorption method. The energy Ej consumed by the mill at the time interval tj is given by
Ej ¼ P t j
ð23Þ
where Ej = energy, in Joules, P = power of the mill, in Watt and tj = time at the interval j, in seconds. The mill power is supposed to be the one used for the breaking action and does not include the transmission and friction losses. It is a function of the mechanical design, the rotation speed and the load of balls or rods in the mill. For laboratory cascading mills the net power can be calculated by the following formula, Stamboltzis (1990–1991).
P ¼ 9:9 W L N D
ð24Þ
where WL = mass of the load, media plus feed, in Kg, N = rotation frequency, in Hertz and D = diameter, in m. The mass specific energy ej in Joule/kg is the ratio of the energy to the mass M of the feed material in the mill
ej ¼ Ej =M
ð25Þ
4. The reliability of the surface area measurements Some results for surface areas, calculated both by the size analysis and at the same time by the BET nitrogen adsorption method, are compared in Fig. 1. The correlation between the two methods is good above 600 m2/kg and theoretically the relationship should be a straight line of the form y = ax with the slope a = 1. In practice, for the particular devices used, the relationship is of the form ax + b with a = 0.5 and b = 590. The explanation for this deviation could be the following. The calculation of the specific surface area from the size analysis, obtained by the laser method, assumes that particles are spherical and the specific surface area of each size class is given by Eq. (8) where f/k = 6. In practice this is not so because the particles are not spherical and for the same volume they have a larger area than the equivalent spheres. The slop of the line in Fig. 1 gives the deviation from sphericity and the actual geometrical ratio should be f/k = 6/0.5 = 12. The parameter b = 590 in the line of Fig. 1 suggests that the calibration of the particular BET instrument used gives specific surface values that are lower than the actual ones and the value 590 must be added to the BET results. Practically one can say that the BET method with nitrogen is not reliable for specific surface area measurements below 600 m2/kg or 0.600 m2/g. The lower particle size given by the laser beam analyzer is about 0.06 lm and the results show that there is no material below
Fig. 1. Correlation of the two surface area measuring methods.
it. The existing correlation between the two surface area measuring methods actually, supports the argument that there is a minimum size and practically the lower limit of the laser size analyzer used is adequate for the scope of the present work. 5. Experimental results 5.1. Mass and surface area distributions The results of the size analysis obtained for the case of 1.5 kg mill feed (load ratio 6) are presented in Fig. 2 for different residence times. As expected the median of the distribution curve moves to finer sizes as grinding proceeds but the area below each curve remains always constant and equal to the initial mass of the material in the mill that is 1.5 kg. The corresponding distribution of the surface area of the material in the mill versus the size is presented in Fig. 3. The area below each curve of Fig. 3 gives the total surface area of the 1.5 kg material in the mill. It is clear that the surface area of the material increases as grinding proceeds. However the median of the distribution curves remains practically constant at about 0.3 lm. A first explanation could be that as grinding proceeds fragments of broken larger particles are accumulated around the median size but they are not further broken. 5.2. The effect of feed ratio The variation of the specific surface area, versus energy input, is presented in Fig. 4 for the three tests of different feed ratios. The graph gives the total surface of the product in the mill, versus
Fig. 2. Mass distribution density for 1.5 kg material.
590
E. Stamboliadis et al. / Minerals Engineering 22 (2009) 587–592
Fig. 5. Specific surface versus specific energy. Fig. 3. Surface area distribution density for 1.5 kg material.
the energy consumed. For high feed ratios the surface area produced is larger and the productivity of the mill in producing surface area is higher. The specific surface area (m2/kg) produced per specific energy input (kJ/kg) is presented in Fig. 5. All three curves of Fig. 5 have the same intercept, corresponding to the specific surface are of the feed material. Each one crosses the horizontal axis at slightly different point corresponding to e0. The value e0 is the theoretical specific energy required to prepare the mill feed in a similar manner and if added to the actual specific energy consumed gives the potential specific energy eP of the material
eP ¼ e þ e0
ð26Þ
Fig. 6 gives the relationship of the specific surface area and the specific potential energy. The intercept is now at zero. It appears that as grinding proceeds, at the lower feed ratio, the specific surface area of the material is higher for the same specific energy input. In other words, although the total surface in the mill is lower at low feed ratios, however, it exists in a state of higher specific area. Generally speaking the value of a powder is higher when its specific surface area is high and one could use the value of the specific surface area as an index of its quality. Accordingly one can attribute a cost index (money per kg) to the potential specific energy. In doing so, one can plot the cost of a material (potential specific energy, kJ/kg) versus its quality (specific surface area, m2/kg) and obtain Fig. 7. This is produced by simple rotation of the previous Fig. 6.
Fig. 4. Total surface area versus energy.
Fig. 6. Specific surface versus specific potential energy.
For low quality materials (low specific surface areas) it is better to have a high feed ratio in order to increase the production rate. However if high specific area products are required there is a substantial benefit in running the mill at low feed ratio (less material in the mill for the same grinding media). 5.3. Energy size relationship It is well known that the higher the specific surface area of a product, the finer its particle size. The particle size of a particulate material is usually expressed by a size modulus below which a pre-
Fig. 7. Product cost as a function of its quality.
E. Stamboliadis et al. / Minerals Engineering 22 (2009) 587–592
591
determined fraction of the material passes. It is customary to use the d90 or d80 sizes but in any case such a size modulus is not adequate to express its fineness. There are an infinite number of materials that might have the same size modulus but still have different surface areas depending on the distribution of fine particles. A safer size modulus to express the fineness of a material is the surface equivalent particle size dS that has the same specific surface area as the total material. The size dS is easily calculated by the following equation derived from Eq. (6) presented above.
dS ¼
f 1 k q ss
ð27Þ
According to the distribution theory of comminution, Stamboliadis (2007) the relationship between the specific surface area ss and the specific potential energy eP of the material is given by l
ss ¼ a eP
ð28Þ
The corresponding values for l and a, as well as the correlation R2, between theoretical and experimental values, are calculated from the data of Fig. 6 and are presented in Table 1 for each test. The parameter a is related to the surface specific energy, or surface tension c of the material as well as to the initial efficiency Fo of the mil, according to
c ¼ F 0 =a
ð29Þ
The efficiency is defined as the ratio of energy used to create new surfaces to the total energy input. The parameters of Table 1 depend on the material and the mill type. Assuming Fo = 1, at the beginning of each test, the values of c can be calculated and are given in Table 1. The combination of Eq. (28) and (29) gives
eP ¼
l1
f
kqa
1
ð30Þ
1
dlS
This equation produces a straight line, in a log–log diagram of eP versus dS. The measured data of the three tests are presented in Fig. 8 together with the values predicted by the distribution theory of comminution, using the measured parameters a and l. This type of relationship between specific energy and material size is found in the case of previous theories like those of Rittinger (1867), Bond (1952), Charles (1957) and Stamboliadis (2002, 2003). In the later cases the size modulus is usually the d80 for example and the exponent 1/l can vary between 0.5 (Bond) to 1 (Rittinger). In the present case the size modulus is the dS and the exponent 1/l is of the order of 2. 6. Discussion and conclusions As it appear, from the above analysis, the surface area of a particulate material can be calculated from its size analysis only under the condition that there is no material below a certain lower size. The fact that the measurements of the specific surface areas by the BET process give defined values supports this condition. It appears that this lower size is close to the detection limits of the laser beam size analyzer. It also appears that BET is not reliable for relatively coarse materials that have specific surface areas below 0.6 m2/g. The laser
Table 1 Values of theoretical parameters. Feed ratio
l
a (m2/kJ)
R2
1/l
c = Fo/a (J/m2)
0.056 0.111 0.167 Average
0.497 0.469 0.442
141 139 173 151
0.995 0.996 0.974
2.013 2.134 2.260
7.11 7.18 5.78 6.62
Fig. 8. Potential specific energy eP versus equivalent size dS.
beam scattering method used for size analysis appears to be a reliable method for the particular case under investigation because the grinding technique used does not produce particles below the detection limits of the laser beam size analyzer. The correlation of the results obtained by the two surface area measuring techniques makes possible to calculate the ratio f/k that is related to the sphericity of the particles. It is reminded that in order to calculate the surface area of a particle from its equivalent diameter it is usually assumed that the particle is a sphere and that f/k = 6. However, this is not true for broken particles with irregular shapes that have a much larger specific surface area than the corresponding spheres. The justification of the use of size analysis for surface area measurements provides a simple tool for the study of grinding procedures. In the present case the parameter studied is the feed ratio defined as the mass ratio of the material present in the mill to that of the grinding media. It is understood that a low feed ratio represents a more intensive energy operating condition per unit mass of the material. The total surface area production rate in m2/kJ is faster at high feed ratios, when there is plenty of material in the mill. In this case the surface area produced exists in coarser particles with low specific surface area. However if one is not interested for the total surface area produced but for its quality then it appears that it is better to run the mill at low feed ratios, when there is less material in the mill. It appears that there is a lower size limit of particles that can be produced from the specific grinding technique used and this is also supported by the work of Petrakis (2004). As an absolute value this limit may probably depend on the type of mill used but there exists a lower limit in any case. Even more the production cost kJ/kg increases rapidly as the required specific area m2/kg of the product (quality) increases. This means that one can optimize the operating conditions of an existing industrial procedure by proper design of its equipment but can not surpass the limits of the process itself. Obviously for economic production of high specific surface area products one should investigate other techniques with more intensive energy input conditions. The results obtained are in agreement with the energy distribution theory of comminution and suggest that the surface equivalent size modulus dS provides a better index of the material size distribution than the size modulus d80, or a similar one, used to describe the size of a particulate material. References Bond, F.C., 1952. The third theory of comminution. Trans. AIME, Min. Eng. 193, 484– 494. Charles, R.J., 1957. Energy-size reduction relationships in comminution. Trans. AIME, Min. Eng. 208, 80–88.
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Petrakis, E., 2004. Energy-size relationship during grinding of brittle materials. Dissertation, Technical University of Crete, Mineral Resources Engineering, Chania, Greece. Rittinger, P.R., 1867. Lehrbuch der Aufbereitungskunde, Berlin. Stamboliadis, E., 2002. A contribution to the relationship of energy and particle size in the comminution of brittle particulate materials. Miner. Eng. 15 (10), 707– 713. Stamboliadis, E., 2003. Impact crushing approach to the relationship of energy and particle size in comminution. Eur. J. Min. Proc. Env. Prot. 3 (2), 160–166.
Stamboliadis, E., 2004. Surface energy potential of single particles and size distribution of broken particulate materials. Miner. Metall. Process., SME 21 (1), 52–56. Stamboliadis, E.Th., 2007. The energy distribution theory of comminution specific surface energy, mill efficiency and distribution mode. Miner. Eng. 20 (2), 140– 145. Stamboltzis, G., 1990–1991. Calculation of the net power of laboratory ball mills. Min. Metall. Annals, Greece (76), 47–55.
Contents lists available at ScienceDirect
Minerals Engineering journal homepage: www.elsevier.com/locate/mineng
Surface area production during grinding Elias Stamboliadis *, Olga Pantelaki, Evangelos Petrakis Technical University of Crete, Mineral Resources Engineering, University Campus, 73100 Chania, Crete, Greece
a r t i c l e
i n f o
Article history: Received 2 August 2008 Accepted 11 December 2008 Available online 12 January 2009 Keywords: Comminution Grinding Size analysis Specific surface area
a b s t r a c t During grinding, in cascading mills, the mass distribution of the material moves continuously to finer sizes. Grinding models are usually designed to predict the size distribution of the mill products either as a function of effective breakage events and the time or the energy consumed by the mill. Mill products are always tested for their size analysis and their fineness is usually expressed as the size d through which a certain amount of material passes, i.e. d80. However, particulate materials have some supplementary properties that a grinding model should predict as well. The specific surface area is such an important supplementary property. Initially, the present work examines the relationship between mass distribution and the surface area of ground materials and determines the conditions under which, mass distribution can be used to determine the surface area. Based on these findings the work further examines the operating conditions of a cascading mill under which the surface area of the material increases at the highest rate. The operating parameter examined is the mill load ratio expressed as the mass ratio of mill grinding media and the material present in the mill. Ó 2008 Elsevier Ltd. All rights reserved.
1. Introduction Grinding models are usually designed to predict the size distribution of the mill products either as a function of effective breakage events, the time or the energy consumed by the mill. They also take into consideration the size distribution of the mill feed. The models are tested against experimental data and are approved if the agreement is acceptable. The test is always based on the size analysis of the mill products. This analysis gives the mass distribution of the material at predetermined size classes and the criterion is the correlation between experimental and theoretical data. However, particulate materials have some supplementary properties that a reliable model should predict as well. An important supplementary property of particulate materials is their surface area, usually expressed as the specific surface area. Almost all grinding models fail to predict the specific surface area of the mill products, especially at intensive grinding conditions, when the products are much finer than the feed. An attempt to relate the surface area of a material to the energy consumed is described by Stamboliadis (2004), but this is still at a theoretical level and requires experimental justification. The specific surface area of particulate materials is usually calculated either by measurements of nitrogen adsorption, using the BET method, or by the size analysis of the material, using screens or the laser scattering method. For non porous materials both
* Corresponding author. Tel.: +30 28210 37601; fax: +30 28210 37884. E-mail address: [email protected] (E. Stamboliadis). 0892-6875/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.mineng.2008.12.001
methods should give comparative results but for materials with open pores the BET method gives a much larger surface area than the one corresponding to the size analysis. If one is interested to relate the size analysis to the specific surface area of a particulate material then the latter must be calculated from the size analysis. In this case the results represent the external surface area, regardless of any existing pores. Almost all the grinding models assume that in any mill product there exist particles down to zero size and as grinding proceeds the mass of the material in the n + 1 size class increases. Actually the number n, of reference sizes, does not affect the result of calculating the mass distribution and is selected to satisfy the practical needs and to simplify the experimental technique. This is so because the total mass of the material is conserved regardless of the number of size classes. However, as will be seen further down, this is not so with the calculation of the surface area of the material and the result depends on the number of size classes used. As a consequence most of the grinding models although they are useful in predicting the mass distribution they cannot be used for analytical work and determine other properties that depend on the size analysis. The present work explains the reason of such a discrepancy and determines the conditions under which the size analysis of a material can be used to determine its surface area. After this, it treats grinding as a surface area creating process and examines the effect of mill load on the surface creation efficiency of the mill. The results obtained conform to the theory of energy distribution in comminution, Stamboliadis (2007). Other operating parameters are also under investigation and the results will be published when available.
588
E. Stamboliadis et al. / Minerals Engineering 22 (2009) 587–592
For k < 1, if xmin = 0 then xk1 min ¼ 1 and Dss becomes infinite. For k > 1, if xmin = 0 then xk1 min ¼ 0 and Dss becomes
2. Calculation of the surface area from the size analysis The size analysis actually gives the cumulative mass fraction P(x) of particles finer than size x according to the integral of Eq. (1)
PðxÞ ¼
Z
Dss ¼
x
dPðxÞ
ð1Þ
RðxÞ ¼ 1 PðxÞ
ð2Þ
and the mass fraction dP(x) of particles at size x within the size interval dx is
dPðxÞ ¼ P0 ðxÞ dx
ð3Þ
Nhe external surface area S(x) of the single particle of size x is
SðxÞ ¼ f x2
ð4Þ
where f is the surface coefficient (for spheres f = p). The mass m(x) of the particle is
mðxÞ ¼ k q x3
ð5Þ
where k is the volume coefficient (for spheres k = p/6) and q is the density of the material. The specific surface area s(x) per unit mass of the particle is inversely proportional to its size
SðxÞ f 1 ¼ mðxÞ k q x
sðxÞ ¼
ð6Þ
dSðxÞ ¼ sðxÞ dPðxÞ or dSðxÞ ¼ sðxÞ P 0 ðxÞ dx
ð7Þ
The total surface area SS of the particulate material is
Z
xmax
dSðxÞ or SS ¼
0
Z
xmax
sðxÞ P0 ðxÞ dx
ð8Þ
0
The overall specific surface area ss of the material is
ss ¼
SS 1 ¼ M M
Z
xmax
ð9Þ
0
Dss ¼ ssðxmin Þ ssðxmax Þ
ð10Þ
In many cases for xmin = 0 then ss(0) = 0 as well and Dss acquires a definite maximum value ss(xmax). In many other cases ss(0) is indefinable and Dss cannot be calculated. As an example let us consider the case of a Gates–Gaudin–Schuhman (GGS) distribution where
PðxÞ ¼ M
x
k
ð11Þ
xmax
M = 1 but is not erased in order to denote that the unit of P(x) is mass. The corresponding equation for the over all specific surface area ss is found to be
ss ¼
f k k q xkmax
Z
xmax
xk2 dx
ð12Þ
0
For k – 1
Dss ¼
f ðln ðxmax Þ ln ðxmin ÞÞ k q xkmax
ð15Þ
if xmin = 0 then ln (xmin) = 1 and Dss becomes infinite. For most comminution products k < 1 and Dss cannot be calculated unless there is a minimum size below which no particles can exist. In practice the methods used to measure the size analysis give the mass fraction DPi of the material in the size range between xi and xi1. The size xi is one of the n reference sizes (screens) used. The n reference sizes determine n + 1 size classes. It is convenient to have a constant pffiffiffi ratio between reference sizes, xi /xi + 1 = constant, usually 2 or 2. The (n + 1)th size class corresponds to the mass fraction DPn + 1 of the material with size between zero and xn. The total mass M of the material is
M¼
nþ1 X
DP i ¼ 1
ð16Þ
1
After this the mass fraction of the material finer than size xi is
Pi ¼
nþ1 X
DP i
ð17Þ
and the mass fraction Ri of the material coarser than xi is
Ri ¼ 1 P i
ð18Þ
The average size di of the particles in the ith size class is pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi di ¼ xi xi1 and for the last class dn+1 = xn/2. The mass mi of the average particle, its external surface area Si and its specific surface area si are calculated from Eq. (4)–(6) replacing x by di. The surface area DSi of the material in the ith size class is
DSi ¼ DPi si sðxÞ P0 ðxÞ dx
Depending on the function P(x) the integral of Eqs. (8) and (9) is indefinable and one has to integrate from a minimum size xmin to the maximum xmax and calculate the specific surface area Dss of the material between the minimum and the maximum size
Dss ¼
iþ1
The surface area dS(x) of the particulate material of size x in the range dx is
SS ¼
ð14Þ
For k = 1
0
The total mass of the material is P(xmax) = M = 1, when x is equal to the size of the coarser particle. Nhe cumulative mass fraction R(x) of particles coarser than size x is
f k xk1 k q xkmax k 1 max
f k k1 ðxk1 max xmin Þ k q xkmax k 1
ð13Þ
ð19Þ
the total area SS of the material is
SS ¼
nþ1 X 1
DSi ¼
nþ1 X
DP i si
ð20Þ
1
the specific surface area ss is
ss ¼
SS M
ð21Þ
Taking into consideration Eqs. (6), (16), (20), and (21) the overall specific surface area of a particulate material is given by Eq. (22), which is equivalent to the mathematical form of Eq. (9)
ss ¼
Pnþ1 DPi f 1 d Pnþ1 i kq D Pi 1
ð22Þ
Eq. (22) indicates that the calculated value of the specific surface area of a particulate material containing particles down to almost zero size, depends on the size xn of the finer screen used. This means that the specific surface area of a particulate material cannot be calculated from the size analysis unless the minimum particle size is known together with the mass distribution of the material down to this size. The fact that the specific surface area of all fine products measured by the BET method give a definite value indicates that in all cases there is a corresponding minimum size in the particle population.
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3. Experimental procedure The material used is hand selected quartz from Assyros, Macedonia, Northern Greece, crushed to 0.85 mm, using a laboratory jaw crusher followed by cone crusher. The grinding equipment is a laboratory rod mill with internal dimensions D L 203 280 mm, rotating at 72 rpm, loaded with 9 kg of rods between 12 mm and 20 mm in diameter. There were three grinding tests each one using 0.5 kg, 1.0 kg and 1.5 kg of crushed quartz respectively. In these tests the feed ratio is defined as the mass ratio of feed material to grinding media, and it is 0.056, 0.111 and 0.167, respectively. During each test the mill was stopped at j predetermined intervals of cumulative time tj and a representative sample of about 5 g was collected for size analysis using a laser beam analyzer. The surface area of the sample is calculated from the size analysis and for some samples it is also measured by the BET nitrogen adsorption method. The energy Ej consumed by the mill at the time interval tj is given by
Ej ¼ P t j
ð23Þ
where Ej = energy, in Joules, P = power of the mill, in Watt and tj = time at the interval j, in seconds. The mill power is supposed to be the one used for the breaking action and does not include the transmission and friction losses. It is a function of the mechanical design, the rotation speed and the load of balls or rods in the mill. For laboratory cascading mills the net power can be calculated by the following formula, Stamboltzis (1990–1991).
P ¼ 9:9 W L N D
ð24Þ
where WL = mass of the load, media plus feed, in Kg, N = rotation frequency, in Hertz and D = diameter, in m. The mass specific energy ej in Joule/kg is the ratio of the energy to the mass M of the feed material in the mill
ej ¼ Ej =M
ð25Þ
4. The reliability of the surface area measurements Some results for surface areas, calculated both by the size analysis and at the same time by the BET nitrogen adsorption method, are compared in Fig. 1. The correlation between the two methods is good above 600 m2/kg and theoretically the relationship should be a straight line of the form y = ax with the slope a = 1. In practice, for the particular devices used, the relationship is of the form ax + b with a = 0.5 and b = 590. The explanation for this deviation could be the following. The calculation of the specific surface area from the size analysis, obtained by the laser method, assumes that particles are spherical and the specific surface area of each size class is given by Eq. (8) where f/k = 6. In practice this is not so because the particles are not spherical and for the same volume they have a larger area than the equivalent spheres. The slop of the line in Fig. 1 gives the deviation from sphericity and the actual geometrical ratio should be f/k = 6/0.5 = 12. The parameter b = 590 in the line of Fig. 1 suggests that the calibration of the particular BET instrument used gives specific surface values that are lower than the actual ones and the value 590 must be added to the BET results. Practically one can say that the BET method with nitrogen is not reliable for specific surface area measurements below 600 m2/kg or 0.600 m2/g. The lower particle size given by the laser beam analyzer is about 0.06 lm and the results show that there is no material below
Fig. 1. Correlation of the two surface area measuring methods.
it. The existing correlation between the two surface area measuring methods actually, supports the argument that there is a minimum size and practically the lower limit of the laser size analyzer used is adequate for the scope of the present work. 5. Experimental results 5.1. Mass and surface area distributions The results of the size analysis obtained for the case of 1.5 kg mill feed (load ratio 6) are presented in Fig. 2 for different residence times. As expected the median of the distribution curve moves to finer sizes as grinding proceeds but the area below each curve remains always constant and equal to the initial mass of the material in the mill that is 1.5 kg. The corresponding distribution of the surface area of the material in the mill versus the size is presented in Fig. 3. The area below each curve of Fig. 3 gives the total surface area of the 1.5 kg material in the mill. It is clear that the surface area of the material increases as grinding proceeds. However the median of the distribution curves remains practically constant at about 0.3 lm. A first explanation could be that as grinding proceeds fragments of broken larger particles are accumulated around the median size but they are not further broken. 5.2. The effect of feed ratio The variation of the specific surface area, versus energy input, is presented in Fig. 4 for the three tests of different feed ratios. The graph gives the total surface of the product in the mill, versus
Fig. 2. Mass distribution density for 1.5 kg material.
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Fig. 5. Specific surface versus specific energy. Fig. 3. Surface area distribution density for 1.5 kg material.
the energy consumed. For high feed ratios the surface area produced is larger and the productivity of the mill in producing surface area is higher. The specific surface area (m2/kg) produced per specific energy input (kJ/kg) is presented in Fig. 5. All three curves of Fig. 5 have the same intercept, corresponding to the specific surface are of the feed material. Each one crosses the horizontal axis at slightly different point corresponding to e0. The value e0 is the theoretical specific energy required to prepare the mill feed in a similar manner and if added to the actual specific energy consumed gives the potential specific energy eP of the material
eP ¼ e þ e0
ð26Þ
Fig. 6 gives the relationship of the specific surface area and the specific potential energy. The intercept is now at zero. It appears that as grinding proceeds, at the lower feed ratio, the specific surface area of the material is higher for the same specific energy input. In other words, although the total surface in the mill is lower at low feed ratios, however, it exists in a state of higher specific area. Generally speaking the value of a powder is higher when its specific surface area is high and one could use the value of the specific surface area as an index of its quality. Accordingly one can attribute a cost index (money per kg) to the potential specific energy. In doing so, one can plot the cost of a material (potential specific energy, kJ/kg) versus its quality (specific surface area, m2/kg) and obtain Fig. 7. This is produced by simple rotation of the previous Fig. 6.
Fig. 4. Total surface area versus energy.
Fig. 6. Specific surface versus specific potential energy.
For low quality materials (low specific surface areas) it is better to have a high feed ratio in order to increase the production rate. However if high specific area products are required there is a substantial benefit in running the mill at low feed ratio (less material in the mill for the same grinding media). 5.3. Energy size relationship It is well known that the higher the specific surface area of a product, the finer its particle size. The particle size of a particulate material is usually expressed by a size modulus below which a pre-
Fig. 7. Product cost as a function of its quality.
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determined fraction of the material passes. It is customary to use the d90 or d80 sizes but in any case such a size modulus is not adequate to express its fineness. There are an infinite number of materials that might have the same size modulus but still have different surface areas depending on the distribution of fine particles. A safer size modulus to express the fineness of a material is the surface equivalent particle size dS that has the same specific surface area as the total material. The size dS is easily calculated by the following equation derived from Eq. (6) presented above.
dS ¼
f 1 k q ss
ð27Þ
According to the distribution theory of comminution, Stamboliadis (2007) the relationship between the specific surface area ss and the specific potential energy eP of the material is given by l
ss ¼ a eP
ð28Þ
The corresponding values for l and a, as well as the correlation R2, between theoretical and experimental values, are calculated from the data of Fig. 6 and are presented in Table 1 for each test. The parameter a is related to the surface specific energy, or surface tension c of the material as well as to the initial efficiency Fo of the mil, according to
c ¼ F 0 =a
ð29Þ
The efficiency is defined as the ratio of energy used to create new surfaces to the total energy input. The parameters of Table 1 depend on the material and the mill type. Assuming Fo = 1, at the beginning of each test, the values of c can be calculated and are given in Table 1. The combination of Eq. (28) and (29) gives
eP ¼
l1
f
kqa
1
ð30Þ
1
dlS
This equation produces a straight line, in a log–log diagram of eP versus dS. The measured data of the three tests are presented in Fig. 8 together with the values predicted by the distribution theory of comminution, using the measured parameters a and l. This type of relationship between specific energy and material size is found in the case of previous theories like those of Rittinger (1867), Bond (1952), Charles (1957) and Stamboliadis (2002, 2003). In the later cases the size modulus is usually the d80 for example and the exponent 1/l can vary between 0.5 (Bond) to 1 (Rittinger). In the present case the size modulus is the dS and the exponent 1/l is of the order of 2. 6. Discussion and conclusions As it appear, from the above analysis, the surface area of a particulate material can be calculated from its size analysis only under the condition that there is no material below a certain lower size. The fact that the measurements of the specific surface areas by the BET process give defined values supports this condition. It appears that this lower size is close to the detection limits of the laser beam size analyzer. It also appears that BET is not reliable for relatively coarse materials that have specific surface areas below 0.6 m2/g. The laser
Table 1 Values of theoretical parameters. Feed ratio
l
a (m2/kJ)
R2
1/l
c = Fo/a (J/m2)
0.056 0.111 0.167 Average
0.497 0.469 0.442
141 139 173 151
0.995 0.996 0.974
2.013 2.134 2.260
7.11 7.18 5.78 6.62
Fig. 8. Potential specific energy eP versus equivalent size dS.
beam scattering method used for size analysis appears to be a reliable method for the particular case under investigation because the grinding technique used does not produce particles below the detection limits of the laser beam size analyzer. The correlation of the results obtained by the two surface area measuring techniques makes possible to calculate the ratio f/k that is related to the sphericity of the particles. It is reminded that in order to calculate the surface area of a particle from its equivalent diameter it is usually assumed that the particle is a sphere and that f/k = 6. However, this is not true for broken particles with irregular shapes that have a much larger specific surface area than the corresponding spheres. The justification of the use of size analysis for surface area measurements provides a simple tool for the study of grinding procedures. In the present case the parameter studied is the feed ratio defined as the mass ratio of the material present in the mill to that of the grinding media. It is understood that a low feed ratio represents a more intensive energy operating condition per unit mass of the material. The total surface area production rate in m2/kJ is faster at high feed ratios, when there is plenty of material in the mill. In this case the surface area produced exists in coarser particles with low specific surface area. However if one is not interested for the total surface area produced but for its quality then it appears that it is better to run the mill at low feed ratios, when there is less material in the mill. It appears that there is a lower size limit of particles that can be produced from the specific grinding technique used and this is also supported by the work of Petrakis (2004). As an absolute value this limit may probably depend on the type of mill used but there exists a lower limit in any case. Even more the production cost kJ/kg increases rapidly as the required specific area m2/kg of the product (quality) increases. This means that one can optimize the operating conditions of an existing industrial procedure by proper design of its equipment but can not surpass the limits of the process itself. Obviously for economic production of high specific surface area products one should investigate other techniques with more intensive energy input conditions. The results obtained are in agreement with the energy distribution theory of comminution and suggest that the surface equivalent size modulus dS provides a better index of the material size distribution than the size modulus d80, or a similar one, used to describe the size of a particulate material. References Bond, F.C., 1952. The third theory of comminution. Trans. AIME, Min. Eng. 193, 484– 494. Charles, R.J., 1957. Energy-size reduction relationships in comminution. Trans. AIME, Min. Eng. 208, 80–88.
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Petrakis, E., 2004. Energy-size relationship during grinding of brittle materials. Dissertation, Technical University of Crete, Mineral Resources Engineering, Chania, Greece. Rittinger, P.R., 1867. Lehrbuch der Aufbereitungskunde, Berlin. Stamboliadis, E., 2002. A contribution to the relationship of energy and particle size in the comminution of brittle particulate materials. Miner. Eng. 15 (10), 707– 713. Stamboliadis, E., 2003. Impact crushing approach to the relationship of energy and particle size in comminution. Eur. J. Min. Proc. Env. Prot. 3 (2), 160–166.
Stamboliadis, E., 2004. Surface energy potential of single particles and size distribution of broken particulate materials. Miner. Metall. Process., SME 21 (1), 52–56. Stamboliadis, E.Th., 2007. The energy distribution theory of comminution specific surface energy, mill efficiency and distribution mode. Miner. Eng. 20 (2), 140– 145. Stamboltzis, G., 1990–1991. Calculation of the net power of laboratory ball mills. Min. Metall. Annals, Greece (76), 47–55.