^{1}

^{*}

^{1}

In manufacturing process, it is necessary to measure change in CSD (Crystal Size Distribution) with time accurately because CSD is one of the most important indices that evaluate quality of products. FBRM (Focused Beam Reflectance Measurement) can measure CLD (Chord Length Distribution) in line, but CLD is different from CSD because of principle of FBRM. However, if CSD is determined beforehand, CLD can be calculated from the CSD with statistical method. First, when crystal shape is defined from the characteristic crystal size, the matrix of each crystal shape which transforms CSD into CLD in a uniform manner is calculated with Monte Carlo analysis. Characteristic crystal size is added to the variables defining chord length in order to avoid complex integrals and apply the change in crystal shape with characteristic crystal size to the transforming matrix. Secondly, CSD and CLD are actually measured in suspension of acetaminophen in ethanol and suspension of L-arginine in water to demonstrate the validity of 2 matrices. Lastly, these matrices are multiplied by some simple CSD models to test the properties of these matrices and demonstrate the utility of this transformation.

Inmanufacturing process of crystal, powder, or granule products, it is necessary to measure change in CSD (crystal size distribution) or PSD (particle size distribution) with time accurately because CSD is one of the most important indices that evaluate quality of products [

FBRM (focused beam reflectance measurement) can measure CLD (chord length distribution) in line, but it is well known that CLD is different from CSD because of principle of FBRM [^{ }

In this paper, first, when crystal shape was defined from the characteristic crystal size, the matrix of each crystal shape which transforms CSD into CLD in a uniform manner was calculated with Monte Carlo analysis. Secondly, CSD and CLD were actually measured in suspension of acetaminophen (AAP) in ethanol and suspension of L-arginine (Arg) in water to demonstrate the validity of 2 matrices. Lastly, these matrices were multiplied by some simple CSD models to test the properties of these matrices and demonstrate the utility of this transformation.

Because this transformation is simply represented by a matrix, it is easy to apply the matrix to inverse transformation and this method is assumed to contribute significantly to in-line measurement of CSD. In some of previous studies [

Particle Track G400, which can measure CLD in line based on FBRM, was used in this paper. Focused beam from probe enters suspension in vessel, and chord length is measured within a range of 1 to 1000 μm based on detection time of backscattered light when beam runs cylindrically at the speed of 2 m/sec. The concept of FBRM is shown in

In

As seen from

First, it is considered that crystal shape P is defined only from vertex coordinates and that the vertex coordinates are mapping of characteristic crystal size L_{CS}. At this time, Equation (1) is established.

where _{CS} is represented by Equation (2) in order to adjust L_{CS} to the measuring range of FBRM.

Secondly, projection area

where the subscript rot means vertex coordinate after rotation, and projection area

Thirdly, because the beam scans crystals along the

where the subscript transl means vertex coordinate after translation, and

where it is desirable that

Lastly, chord length L_{CL} is defined as the length of line intersection of projection area _{CL} in a uniform manner, intersections

The set of _{m}C_{2} intersection coordinates

At this time, the chord length L_{CL} is represented by Equation (10).

However, when L_{CL} is smaller than the measuring lower limit or all of the intersection coordinates _{CL} is defined as 0, and the range of L_{CL} is represented by Equation (11).

A domain of crystal size in the jth fraction when the domain of crystal size in Equation (2) is divided into n equal parts by a logarithmic scale is represented by Equation (12).

Similarly, a range of chord length in the ith fraction when the range of chord length in Equation (11) except 0 is divided into n equal parts by a logarithmic scale is represented by Equation (13).

Originally, it is not necessarily required that the number of fractions on crystal size is the same to that on chord length. At this time, the probability that one of an infinitely large number of crystals in Equation (12) is measured as the chord length in Equation (13) is to be calculated. First of all, L_{CS} by a logarithmic scale and

Then, 5 variables are arranged to be denoted by a vector

Moreover, domains of 5 variables in Equations (12), (5), and (7) are arranged to be denoted by

where the integrated value in all of the domains K is independent of fraction number j because the integrated values in all of the fractions by a logarithmic scale are the same to one another when the domain of crystal size is divided into equal parts by a logarithmic scale. In the domains

Because chord length is clearly defined from 5 independent variables (see Section 2.2), the integrated value

where _{ }and

For the following discussion,

The integration range of Equation (17) is too complex for the exact solution to be obtained. Therefore, Monte Carlo analysis is performed with uniformly distributed pseudorandom number

At this time, to make 5 independent variables have the domains in Equations (12), (5), and (7), and to make Equation (14) about probability density established, 5 independent variables are defined as Equations (21), (22), and (23) with random number.

However, Equation (22) is established only if the directions of crystals are uniformly distributed regardless of crystal shape and the direction of the suspension flow. 5 random numbers change for each trial and the dependent variable L_{CL} is calculated on each trial. By using the total number of trials K_{MC} instead of sample space K in Equation (17) and the number of times

where the subscript MC means the value about Monte Carlo analysis. If r is true random number, the approximate probability _{MC} increases. In this paper, pseudorandom number was created with MATLAB 7.5.0 (R2007b).

In this paper, the verification experiment was performed with acetaminophen (CH_{3}CONHC_{6}H_{4}OH, abbreviated to AAP) and L-arginine (C_{6}H_{14}N_{4}O_{2}, abbreviated to Arg). AAP, the molecular weight of which is 151.16, is a white crystalline compound, hardly soluble in water and readily soluble in ethanol. AAP has 3 kinds of polymorphs. AAP is often used as an analgesic antipyretic or a cold medicine. In the verification experiment, ethanol was purchased from Wako Pure Chemical Industries, Ltd. (Osaka, Japan) and AAP from Tokyo Chemical Industry Co., Ltd. (Tokyo, Japan). Then, Arg, the molecular weight of which is 174.02, is a white crystalline basic amino acid, readily soluble in water and hardly soluble in ethanol. Arg has 2 kinds of pseudo polymorphs: anhydrate and dehydrate. Arg also activates immune function and accelerates cell proliferation. In the verification experiment, Arg was purchased from Wako Pure Chemical Industries, Ltd. (Osaka, Japan).

Solution temperature and CLD were measured with the apparatus shown in

Solution temperature was measured with platinum electrode (Pt100, JISC1604-1997/IEC 751). CLD was measured with FBRM (made by Mettler-To- ledo, model G400).

Measurement conditions of FBRM are described below.

・ The measuring range is 1 - 1000 μm.

・ The measurement mode is Macro.

・ The measuring range is divided into 30 equal parts by a logarithmic scale.

・ The wavelength of the laser beam is 780 μm.

AAP (45 g) was added to ethanol (300 mL) to prepare a saturated solution at 20˚C. Then, with the solution held at 20˚C, AAP seed crystals were added to the solution under 5 conditions. The suspension of AAP in ethanol was stirred and the crystals were washed for about 30 min with CLD from FBRM measured. After it was confirmed that CLD was steady, the suspension was sampled at the same time that CLD was recorded and CSD was measured with an optical microscope. The experimental condition is showed in

In

The experiment with suspension of Arg in water was performed likewise in section 3.3.1. The experimental condition is showed in

Shape transformation matrix

Cond. No. | Substance | Mass of solute [kg] | Mass of solvent [kg] | Agitation rate [rpm] | Saturation temperature [˚C] | Mass of fine seed [g] | Mass of coarse seed [g] |
---|---|---|---|---|---|---|---|

1 | 2 | 2 | |||||

2 | 4 | 2 | |||||

3 | AAP | 0.045 | 0.237 | 300 | 20 | 6 | 2 |

4 | 2 | 4 | |||||

5 | 2 | 6 | |||||

6 | 2 | 2 | |||||

7 | 4 | 2 | |||||

8 | Arg | 0.048 | 0.3 | 300 | 20 | 6 | 2 |

9 | 2 | 6 | |||||

10 | 2 | 8 |

reference, and crystal shape

In addition, the characteristic crystal size of each substance was defined as the black line of each model shape in

The absolute value of CLD hardly has quantitative information, because ^{3}-weighted distribution is usually used, and so the shape transformation matrix ^{3}-weighted distribution is practically small. L^{3}-Weighted distribution

where L is a diagonal matrix the ith diagonal element of which is the average of the ith fraction ^{3}-weighted distributions were calculated by using Equation (25). Then, every L^{3}-weighted distribution was normalized and the total amount of every L^{3}-weighted distribution was adjusted to 1 to exclude quantitative discussion. Normalized L^{3}-weighted distribution

At this time, CSD measured with an optical microscope was assumed to be a calculated vector, CLD from FBRM a observed vector, and relative error E was defined as a 2-norm of difference between a calculated vector and observed one. E is represented by Equation (27).

E from CSD after the transformation was compared with that before the transformation, and the validity of

The properties of the shape transformation matrices of AAP and Arg, which were created in section 3.3.3 and the validity of which was demonstrated in section 3.4, were tested by being multiplied by the following 2 CSD models to demonstrate the utility of the transformation.

For example, normalized L^{3}-weighted distributions under cond. 3 and cond. 9 are shown in

In

At this time, the rate of change in relative error obtained before and after shape transformation was calculated. In addition, the average of the rate of change by each substance was calculated to demonstrate the validity of the shape transformation matrix on each substance.

Then, CLDs which

Cond. No. | Substance | Mass of fine seed [g] | Mass of coarse seed [g] | E_{before}_{ } [-] | E_{after} [-] | Changing rate [%] | Ave. [%] |
---|---|---|---|---|---|---|---|

1 | 2 | 2 | 0.731 | 0.494 | −32.41 | ||

2 | 4 | 2 | 0.615 | 0.464 | −24.45 | ||

3 | AAP | 6 | 2 | 0.440 | 0.362 | −17.77 | −24.46 |

4 | 2 | 4 | 0.681 | 0.491 | −27.83 | ||

5 | 2 | 6 | 0.577 | 0.463 | −19.86 | ||

6 | 2 | 2 | 0.126 | 0.124 | −1.52 | ||

7 | 4 | 2 | 0.208 | 0.212 | 2.01 | ||

8 | Arg | 6 | 2 | 0.168 | 0.205 | 22.28 | −4.22 |

9 | 2 | 6 | 0.375 | 0.292 | −22.71 | ||

10 | 2 | 8 | 0.232 | 0.182 | −21.71 |

were sometimes measured as a longer chord length than the characteristic crystal size only for Arg. This was because the chord length around the body diagonal line was longer than the characteristic crystal size. The effect that crystals were measured around the edge or at a slant affected CSD complexly, depending on crystal shape. For example,

Then, _{d}. However, within the ranges of large fraction numbers, the trend was reversed. This was assumed to occur because of the effect that crystals were measured around the edge or at a slant, which is mentioned above.

Consequently, the state of CSD cannot be discussed from CLD without using shape transformation matrix, and the utility of the shape transformation matrix calculated in this paper was assumed to be demonstrated.

By using Monte Carlo analysis, shape transformation matrices which transformed CSD into CLD for the crystal shape defined beforehand were created. The validity of these shape transformation matrices were tested with the suspension of AAP in ethanol and the suspension of Arg in water. The verification experiments show that the relative error between CLD and CSD after transformation was significantly smaller than that between CLD and CSD before transformation only in the case that the actual crystal shapes corresponded with the definition. Therefore, the validity of this transformation method of CSD with the shape transformation matrix was demonstrated. Then, the virtual experiments in which the CLDs were obtained by the shape transformation matrices multiplied by some CSD models show that the trend and the statistics of CSD greatly differed from those of CLD and that the degree of the difference depended on the crystal shape. In other words, the state of CSD cannot be discussed from CLD without using shape transformation matrix, and the utility of the shape transformation matrix calculated in this paper was demonstrated.

In this paper, the crystal shape was assumed to be similar regardless of crystal size for simplicity, but actually, shape transformation matrix can be created even if crystal shape is defined as a mapping of the crystal size. This mapping is accurately researched beforehand and inserted in the shape transformation matrix to enable the matrix to shape-transform for more general cases. In addition, by using the shape transformation matrix with the method in this paper for inverse transformation, the algorithm transforming CLD into CSD is created to realize real-time monitoring of CSD with FBRM. Many of the operations containing matrix can be performed in a very short time with numerical analysis software. In other words, the fact that shape-transforming operator was obtained as matrix in this paper seems to contribute to transforming CLD into CSD with the quality of in-line in FBRM remaining.

We express thanks to Mettler-Toledo K.K. BU AutoChem for technical support.

Unno, J. and Hirasawa, I. (2017) Transformation of CSD When Crystal Shape Changes with Crystal Size into CLD from FBRM by Using Monte Carlo Analysis. Advances in Chemical En- gineering and Science, 7, 91-107. https://doi.org/10.4236/aces.2017.72008

E domain of

E relative error [-]

K sample space [

_{R} in diagonal matrix [m]

L_{CL} chord length [m]

L_{CS} crystal size [m]

L_{R} geometric average of both ends of fraction [m]

M probability event [

N number of crystals [#]

N number of fractions [-]

r pseudorandom number [-]

X x-coordinate [m]

_{ }intersection of

Y y-coordinate [m]

y_{d} translation toward y-axis [m]

y_{d, }_{max} required minimax value of y_{d} [m]

z z-coordinate [m]

^{3}-weighted distribution [# m^{3}]

after after transformation

before before transformation

calc calculated value

CLD chord length distribution

CSD crystal size distribution

MC Monte Carlo analysis

obs observed value

prj projection

rot after rotation

transl after translation

AAP acetaminophen

Arg L-arginine

FBRM focused beam reflectance measurement