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Definition and conversion of the mean particle diameter referring to mixing homogeneity
Technology. 31 (1982) 231 - 232 0 Elsevier Sequoia, S-A.. Lausanne - Printed in The Netherlands
231
Powder
Short
Communication
Definition and Conversion of the Mean Particle Diameter Referring to ~Mixing Homogeneity
number, length, surface or volume of the particles_ For d,,, the indices p and f assume the values 3 and 0 respectively; for d,, p and f are both 3 (see Table l)_
H. EGERMANN Pharmaceutical Technology Division. Organic and Pharmaceutical Chemistry. Innsbruck, A-6020 Innsbruck, Innmin (Received
September
Institute of Unirrersity of 52~ (Austria)
133,
1
Power index, frequency index and Hatch-Choatc coefficient of some mean diameters
18.1981)
The homogeneity of a random mlsture is dependent on the particle size of the miving components. For a particle size distribution, a representative mean diameter may be calculated from the detailed results of the particle size analysis [1, 2]_ According to Edmundson
TABLE
this
representative
mean
may be called the ‘volume-weighted/volumenumber mean diameter’, and for convenience in notation it was symbolized & in earlier work [ 2]_ Obviously, some uncertainties exist as to the definition and the physical meaning of this type of average. Most of the authors dealing with mixing problems avoid using a mean size, preferring to present the full particle size distribution of the powders they used [4 - 7]_ In some recent papers [7 - 91, tKe volume-number mean d,, is supposed to be representative of the quality of pharmaceutical powder mixtures. For an unambiguous definition of d, as well as of d,, the general equation of Edmundson [.3] for mean diameters may be applied:
where n is the number of particles in a size range the mid-point of which is d. The index p is the diameter power function witl~ values of 1, 2 or 3, corresponding to the length, surface or volume of the particles. f is the frequency index, which may be 0, 1, 2 or 3, depending on whether the size frequency distribution is expressed in terms of r;he total
Hatch-Choate
d -;lll
Power
index. p
Frequency index, f
coefficient,
4m d,w d_“”
0 0 3
0 3 0
0 3 1.5
3
3
4.5
d,
Q
is inversely related to the specific of particles per unit weight [ 33 ) and the fact that the index f is 0 implies that d,, is derived from the number frequency distribution_ According to this, d,, is representative of the mixing homogeneity only if the uniformity of the number of particles per sample is used as a measure of the mixing quality_ For pharmaceutical preparations, however, it is not the number of drug particles per se which is of interest, but the weight uniformity of the drug content per dose unit. If the particles are not uniform in size, they are not equivalent with respect to the weight variation of drug content [l, IO] _ Therefore the weight differences of the particles must be taken into account, which is effected by deriving the representative mean from the weight frequency distribution. This is true for & with an index f of 3. The correct definition of the representative mean is of practical importance, if this diameter is calculated from other kinds of average, as attempted previously [7 - 91. For powders which approach a log-normal distribution the Hatch-Choate equation may be applied, but if one assumes d,, to be the representative average [7 - 91, the results will be erroneous. d yn
number
232
The Hatch-Choate equation in this general form [3] :
In Gem
=Ind,+@
may be written
In2a,
(2)
mean and where d,, isthe geometric-number ug is the geometric standard deviation as a measure of the distribution about the mean. Q symbolizes the Hatch-Choate coefficient, which varies according to the kind of mean that is required. It is related to the indices p and f of eqn. (1) and has the value (l/2 p + f) [3] _ Table 1 gives the coefficients Q and the corresponding indices p and f of some types
of d,,,,For d,,, 0 amounts
to 1.5, and for dgn @ is equal to O_ Thus eqn_ (2) takes the special form T
In 6,
= In d,
+ 1.5 ln2 ue
(3)
More frequently, the geometric-weight mean d,, is used for the calculation of d, [7 - 9]_ Then Q must be reduced by 3, since 0 = 3 when d gw is calculated from d,, (Table 1): ln d,,
=ln
d,, -11.51n2a,
(4)
For &, however, Q amounts to 4.5, and fare both 3 (Table 1). Accordingly,
as p the following equations are the correct ones for calculating the representative mean Tom dgn:
In & = In d,, + 4.5 ln2 u=
(5)
and from dgu_: In & = In d,,
+ l-5
ln2 ug
(6)
The coefficients @ corresponding to other common types of d,,, were given by
Edmundson 133 _ Substituting these coefficients into eqn. (2), and a value of 4.5 for equatio_n provides a a,, the Hatch-Choate simple method of converting 4, to any type of average, or uice uer5a, provided that the distribution is log-normal_ The practical applicatiocs and limitations of these conversions with respect to mixing problems will be considered later_
REFERENCES
1 K. R. Poole, R. F. Taylor and G. P. WaU, Trans. InsL Chem. Eng.. 42 (1964) 305 - 315_ 2 H_ Egermann, Sci_ Pharm.. 42 (1974) 1 - 19. analysis, Advances 3 J. C_ Edmundson, Particl-ize in PharmaceulicaI Sciences. Vol_ 2. Academic Press, London and New York, 1967, pp_ 100 117_ 4 J. C_ Samyn and K_ S. Murthy, J. Pharm. Sci.. 63 (1974) 370 - 375. 5 I?_ Cook and J_ A_ Heaey, Powder Technol.. 9 (1974) 257 - 261. 6 M_ C_ R. Johnson, Pharm. Acfn Helu.. 50 (1975) 60 - 63. 7 M_ J. Crooks and R. Ho. Powder Technol.. 14 (lSi6) 161- 167. 8 C. C. Yeung and J. A_ Heney, Powder Technol. 22 (1979) 127 - 131. 9 P_ Thanomkiat. P_ J_ Stewart and P. S. Grover. Powder Technol.. 24 (1979) 97 - 98. 10 H Egermann. Acla Pharm. Technol.. 22 (1976) 207 - 215.
231
Powder
Short
Communication
Definition and Conversion of the Mean Particle Diameter Referring to ~Mixing Homogeneity
number, length, surface or volume of the particles_ For d,,, the indices p and f assume the values 3 and 0 respectively; for d,, p and f are both 3 (see Table l)_
H. EGERMANN Pharmaceutical Technology Division. Organic and Pharmaceutical Chemistry. Innsbruck, A-6020 Innsbruck, Innmin (Received
September
Institute of Unirrersity of 52~ (Austria)
133,
1
Power index, frequency index and Hatch-Choatc coefficient of some mean diameters
18.1981)
The homogeneity of a random mlsture is dependent on the particle size of the miving components. For a particle size distribution, a representative mean diameter may be calculated from the detailed results of the particle size analysis [1, 2]_ According to Edmundson
TABLE
this
representative
mean
may be called the ‘volume-weighted/volumenumber mean diameter’, and for convenience in notation it was symbolized & in earlier work [ 2]_ Obviously, some uncertainties exist as to the definition and the physical meaning of this type of average. Most of the authors dealing with mixing problems avoid using a mean size, preferring to present the full particle size distribution of the powders they used [4 - 7]_ In some recent papers [7 - 91, tKe volume-number mean d,, is supposed to be representative of the quality of pharmaceutical powder mixtures. For an unambiguous definition of d, as well as of d,, the general equation of Edmundson [.3] for mean diameters may be applied:
where n is the number of particles in a size range the mid-point of which is d. The index p is the diameter power function witl~ values of 1, 2 or 3, corresponding to the length, surface or volume of the particles. f is the frequency index, which may be 0, 1, 2 or 3, depending on whether the size frequency distribution is expressed in terms of r;he total
Hatch-Choate
d -;lll
Power
index. p
Frequency index, f
coefficient,
4m d,w d_“”
0 0 3
0 3 0
0 3 1.5
3
3
4.5
d,
Q
is inversely related to the specific of particles per unit weight [ 33 ) and the fact that the index f is 0 implies that d,, is derived from the number frequency distribution_ According to this, d,, is representative of the mixing homogeneity only if the uniformity of the number of particles per sample is used as a measure of the mixing quality_ For pharmaceutical preparations, however, it is not the number of drug particles per se which is of interest, but the weight uniformity of the drug content per dose unit. If the particles are not uniform in size, they are not equivalent with respect to the weight variation of drug content [l, IO] _ Therefore the weight differences of the particles must be taken into account, which is effected by deriving the representative mean from the weight frequency distribution. This is true for & with an index f of 3. The correct definition of the representative mean is of practical importance, if this diameter is calculated from other kinds of average, as attempted previously [7 - 91. For powders which approach a log-normal distribution the Hatch-Choate equation may be applied, but if one assumes d,, to be the representative average [7 - 91, the results will be erroneous. d yn
number
232
The Hatch-Choate equation in this general form [3] :
In Gem
=Ind,+@
may be written
In2a,
(2)
mean and where d,, isthe geometric-number ug is the geometric standard deviation as a measure of the distribution about the mean. Q symbolizes the Hatch-Choate coefficient, which varies according to the kind of mean that is required. It is related to the indices p and f of eqn. (1) and has the value (l/2 p + f) [3] _ Table 1 gives the coefficients Q and the corresponding indices p and f of some types
of d,,,,For d,,, 0 amounts
to 1.5, and for dgn @ is equal to O_ Thus eqn_ (2) takes the special form T
In 6,
= In d,
+ 1.5 ln2 ue
(3)
More frequently, the geometric-weight mean d,, is used for the calculation of d, [7 - 9]_ Then Q must be reduced by 3, since 0 = 3 when d gw is calculated from d,, (Table 1): ln d,,
=ln
d,, -11.51n2a,
(4)
For &, however, Q amounts to 4.5, and fare both 3 (Table 1). Accordingly,
as p the following equations are the correct ones for calculating the representative mean Tom dgn:
In & = In d,, + 4.5 ln2 u=
(5)
and from dgu_: In & = In d,,
+ l-5
ln2 ug
(6)
The coefficients @ corresponding to other common types of d,,, were given by
Edmundson 133 _ Substituting these coefficients into eqn. (2), and a value of 4.5 for equatio_n provides a a,, the Hatch-Choate simple method of converting 4, to any type of average, or uice uer5a, provided that the distribution is log-normal_ The practical applicatiocs and limitations of these conversions with respect to mixing problems will be considered later_
REFERENCES
1 K. R. Poole, R. F. Taylor and G. P. WaU, Trans. InsL Chem. Eng.. 42 (1964) 305 - 315_ 2 H_ Egermann, Sci_ Pharm.. 42 (1974) 1 - 19. analysis, Advances 3 J. C_ Edmundson, Particl-ize in PharmaceulicaI Sciences. Vol_ 2. Academic Press, London and New York, 1967, pp_ 100 117_ 4 J. C_ Samyn and K_ S. Murthy, J. Pharm. Sci.. 63 (1974) 370 - 375. 5 I?_ Cook and J_ A_ Heaey, Powder Technol.. 9 (1974) 257 - 261. 6 M_ C_ R. Johnson, Pharm. Acfn Helu.. 50 (1975) 60 - 63. 7 M_ J. Crooks and R. Ho. Powder Technol.. 14 (lSi6) 161- 167. 8 C. C. Yeung and J. A_ Heney, Powder Technol. 22 (1979) 127 - 131. 9 P_ Thanomkiat. P_ J_ Stewart and P. S. Grover. Powder Technol.. 24 (1979) 97 - 98. 10 H Egermann. Acla Pharm. Technol.. 22 (1976) 207 - 215.