TMCnet News
A Guide to Characterizing Particle Size and Shape [Chemical Engineering Progress](Chemical Engineering Progress Via Acquire Media NewsEdge) Knowledge of particle properties Is essential for understanding how the particles will Impact your process. Learn how to identify the relevant properties for a particular application, select the best measurement technique, and analyze the data to extract meaningful Information. (ProQuest: ... denotes formulae omitted.) Particle science and technology is relevant to a broad range of chemical processes, including those for manufacturing paints and coatings, pharmaceuticals, catalysts, fertilizers, food products, and cosmetics. For processes that have particles as a raw material, intermediate, or final product, the ability to measure, monitor, and characterize fundamental particle properties, such as size, size distribution, shape, density, and surface texture, is essential. A particle can be defined as a solid with well-defined boundaries; its linear dimensions can range from nanometers (colloids) to millions of meters (extraterrestrial objects). Whether for an industrial process or a natural process, measuring particle properties (e.g., size and shape) is a means to an end - the ultimate objective is to understand the effect of morphological parameters on particle behavior. Particle properties can affect absorption, agglomeration, aggregation, bioavailability, compressibility, combustibility, entrainment, fluid-particle interactions, packing, particle trajectories, permeability, reactivity, segregation, separation, settling, and toxicity. This article reviews different ways to define particle size and explains how to decide which size variable is appropriate for a particular process and objective. It describes techniques for measuring particle size and provides key advantages and disadvantages of each. Finally, it discusses particle shape, as it is difficult to separate the effect of particle size and particle shape on the macroscopic response of a complex particulate process. The many ways to define particle size The size of a spherical particle is unambiguously defined by its diameter. This diameter, which can be considered the particle's characteristic dimension, can be directly measured. However, particles encountered in most industrial processes are rarely spheres, and very rarely of the same size. For particles that deviate from the spherical shape, relevant characteristic dimensions (e.g., length, width, diameter) or derived dimensions (e.g., aspect ratio) that relate to a size-dependent property must be identified. These so-called equivalent spherical diameters are often used to translate a certain property (e.g., surface area, volume, perimeter) of an irregular particle to a spherical dimension (Figure 1, Table 1). Each derived equivalent diameter represents a mechanism or characteristic relevant to the process of interest. Therefore, it is critically important to understand the relevance of these definitions, and select wisely. Particle size data If every particle in the population were identical in size and shape, only one particle would need to be analyzed. However, that is extremely rare, so to get a full, accurate, and statistically representative profile of the size or shape distribution of a sample, a large number of particles, often as many as hundreds of thousands, must be analyzed. The data are then typically sorted into a series of successive size intervals characterized by the number of particles, surface area, or mass of each interval. The entire size range, which can span up to several orders of magnitude, can be covered with a relatively small number of intervals. For samples with a narrow size distribution, it may be appropriate to group the data into linear intervals, such as 0-1 pm, 1-2 pm, 2-3 pm, etc., and express the contents of each interval as a percentage of the whole population. One disadvantage of grouping into linear intervals is that the resolution (i.e., the ratio of the interval width to the mean size of that interval) is not constant across the entire size distribution. For example, if the particles are classified into ten intervals from 0 pm to 10 pm with each interval 1 pm wide, the resolution of the first interval (0-1 pm) is 2, while for the last interval (9-10 pm) the resolution is 0.1. The data can also be organized on a geometric basis, with the interval widths increasing by the same factor (e.g., 1-2 pm, 2-4 pm, 4-8 pm, 8-16 pm). Grouping the particles in this way maintains a constant resolution over the entire distribution. Table 2 summarizes a population of particles whose size intervals are a geometric progression increasing by a factor of 1.5. The fourth column shows the number of particles in each interval. The remaining columns show the proportion of the total population in each interval, which can be expressed in terms of the number of particles, surface area, volume, or any other basis upon which data are acquired. Once the data have been collected and grouped into the appropriate size intervals, they are usually tabulated and then transformed into a graphical representation. The size data can be plotted as a differential frequency plot (the percentage of the total relevant quantity, e.g., number of particles, surface area, or mass, in each interval) or a cumulative frequency plot, as illustrated in Figure 2. Differential frequency plots allow for the direct comparison of different distributions, provided the intervals of the distributions are identical. Cumulative distributions are useful for comparing several distributions that have intervals of different widths. What does "mean diameter" mean? Size distributions can be reduced to a single average diameter, such as the mean, median, or mode. Distribution averages are often a source of confusion when data from different instruments that may have calculated the average diameter differently are being compared. An average particle diameter is meaningless without specifying how it was calculated. The mean diameter is a calculated statistic that represents the size of the entire particle population. Several definitions of the mean are commonly used for various types of comparisons. The most appropriate mean diameter for a particular application is the one that corresponds most closely to the relevant property of the particle system. The simplest mean is the arithmetic mean, which is the sum of all of the diameters in the population divided by the total number of particles: ... where ni is the number of particles in group i and di is the midpoint diameter of the particles in group This diameter is commonly referred to as D[ l ,0] because the diameter term in the numerator of the equation is of the power 1 and the diameter term in the denominator is of the power 0. The mean surface-area diameter and volume (or mass) diameter reflect the fact that surface area is proportional to the square of the diameter and the volume and mass are proportional to the cube of the diameter. The mean surface-area diameter is used when the particle surface behavior is important. It is referred to as D[2,0] and is the arithmetic mean of the particle surface areas: ...(2) Similarly, the mean volume diameter (or mean mass diameter) is used when the volume (or mass) of the particles is important. Referred to as D[3,0], it is the mean diameter of the particle volumes or masses within the population: ...(3) It is not unusual for a population with a wide size distribution to have a mean mass diameter that is one or two orders of magnitude larger than the arithmetic mean diameter based on the number of particles. In any polydisperse system, the diameters always follow this relationship: ...(4) Equations 1-3 are based on the number of particles being measured, and therefore are limited to particle-sizing techniques that provide information on individual particles. The most common techniques measure particle parameters related to bulk quantities, such as surface area, volume, or mass, and do not provide information on the number of particles. Thus, other ways to calculate averages must be used. The two most important means calculated from bulk quantities are the surface-to-volume mean (i.e., the Sauter mean diameter, D[3,2]), and the volume mean diameter (i.e., the De Broukere mean diameter, D[4,3]). The Sauter mean diameter is: ...W where is the total surface area of the particles in group /' (.s\ = rind1-) and S is the total surface area of the entire population. The Sauter mean diameter can be defined as the diameter of a sphere having the same surface-area-to-volume ratio as the entire particle population. In the calculation of the mean, most particle-size analyzers assume that the particles are smooth spheres, which may lead to significant errors for populations of highly irregular particles. In such cases, the Brunauer-EmmettTellet (BET) gas adsorption technique, which estimates the total surface area from the number of gas molecules required to cover the surface of the particles, can be used to measure the surface area; this surface area can then be used to calculate the Sauter mean diameter: ...(6) where p^ is the density of the particles, M is the total mass, and S is the surface area measured by the BET technique. The De Brouckere diameter (also referred to as the mass mean diameter) is: ...(7) where mj is the total mass of the particles in group i and M is the total mass of the entire population. Instrument selection To obtain the information required to calculate these mean diameters, the proper instrument must be selected. When choosing an instrument, consider the type of data needed, including the required accuracy and precision. Consider a fouling problem in a heat exchanger handling the stream characterized by Table 2. Since smallparticle behavior tends to be dominated by surface forces, fouling is typically associated with small particles sticking to the surface of the heat exchanger. In this example, the experimental data show that only the particles with a diameter of <2 pm are causing the fouling. However, the surface-area-based particle-size distribution, as well as the volume-based curve (Figure 2), indicate that 100% of the particles have diameters larger than 2 pm - which cannot be correct since there is a fouling problem. Instruments that measure particle size on a number basis are needed to detect those smaller particles. As a general rule of thumb, when a small percentage of small particles in a distribution is of interest, single-particle counting methods should be used. When a number-based distribution is needed, laser diffraction and dynamic light scattering should not be used. They should only be used for surface-areaor volume (mass)-based measurements. Consider another example in which one 820-pm particle is added to the population in Table 2, which is now represented by Figure 3. This one particle accounts for roughly 30% of the volume distribution of the population, but does not effect the number-based distribution. Surface-area-based measurements should be considered when the surface area of the particulate system plays an important role, such as in catalysis, fluidization, dust explosion hazard assessment, or droplet formation. In cases where the particle shapes are very irregular, the BET technique may be required to accurately determine the surface area of the particles. The most common size-measurement basis in the chemical process industries is mass (or volume). For example, a wastewater treatment plant is subject to regulations that limit the total mass of suspended solids in the effluent water, so a distribution or average based on mass or volume is the relevant quantity of interest. Particle-sizing instruments With the exception of imaging technologies, every particle characterization technology provides the measurement of an equivalent spherical diameter. This equivalent spheri- cal diameter is deduced indirectly from the behavior of the particles as they pass through restricted volumes or channels under the influence of gravity or centrifugal force fields, or from interactions with some form of radiation or ultrasonic waves (l, 2). Therefore, data for irregular-shaped particles obtained from different instruments are typically not in good agreement. Table 3 summarizes some of the advantages and disadvantages of each of the technologies described in the following sections. Particle characterization technologies in use today fall into one of three categories: ensemble techniques, fractionation techniques, and single-particle counting techniques. Ensemble techniques Ensemble techniques, which include laser diffraction, dynamic light scattering, and ultrasonic spectroscopy, can measure large numbers of particles simultaneously. Laser diffraction. These instruments became commercially available in the 1970s, and soon were considered the workhorse particle analyzer for industrial applications. As illustrated in Figure 4, light from a laser source is collimated (i.e., its rays are made parrallel) and transmitted through the particle dispersion. As the light passes through the dispersion, it is diffracted by the particles. The diffracted light passes through a Fourier lens that causes rays scattered at the same angle to converge at specific locations on a series of photodetectors. The detectors are strategically positioned to reveal the angular diffraction pattem, which corresponds to the size of the particles (larger particles diffract light at low angles, smaller particles at higher angles). While the position of the diffraction pattem provides information on the size of the particles, the intensity of the diffracted light relates to the total volume of particles of a particular size. A mathematical model then converts the diffraction pattern into a particle-size distribution. The mathematical models are based on the Fraunhofer and the Mie theories. The Fraunhofer model is a simple approximation that accounts for the diffraction phenomenon, but ignores reflection, transmission, and refraction, which may become significant for small particles (<20 pm). The more-sophisticated Mie scattering model takes into account all light-particle interactions. To use the Mie model, the refractive index of the particles and of the suspension medium must be known. The Fraunhofer approximation is typically used when these optical parameters are not known and lower accuracy is acceptable. Over the years, instrument manufacturers have sought to provide a wider analytical range, the ability to switch from liquid dispersions to dry-powder dispersions, and the ability to measure high-concentration dispersions. The combination of the Fraunhofer model with Mie theory has expanded the analytical range to roughly 0.04 pm (some vendors claim as low as 0.02 pm) to several millimeters, although the reliability at both ends of the range is somewhat questionable. Even with the combination of these two models, the validity of any multimodal distribution with particle sizes below 2 pm should be carefully verified. Several vendors offer dry-powder dispersion instruments capable of dispersing particles as small as 10 pm (and in some cases, smaller). It is good practice to validate measurements based on dry dispersion by comparing them to data obtained by the more-traditional wet-dispersion method. Some laser-diffraction instruments designed for highconcentration measurements have been used for online applications. Online applications should be thoroughly validated. It is also important to note that the measurement basis for laser diffraction is mass (or volume) and conversions to number-based distributions should never be used even when offered by instrument vendor software. Dynamic light scattering. The early particle-size instruments from the 1960s and 70s had limited capabilities and required expert operators. It took almost two decades of technical improvements, including major advances in digital autocorrelators, microprocessors, and lasers, to bring dynamic light-scattering technology to a viable existence in the marketplace. Today, dynamic light-scattering instruments require a low skill level for operation. In dynamic light scattering, the size distribution of sub-micrometer particles dispersed in a liquid medium is deduced from the random movements of particles undergoing Brownian diffusion. Unlike laser diffraction, where the static intensity of light is measured, a dynamic lightscattering instrument records the fluctuations in scattered light and processes these data using an autocorrelator, as illustrated in Figure 5. The smaller particles have a higher diffusion constant and generate a higher-frequency signal than the lower-frequency signal of the larger particles. The autocorrelator analyzes the periodicity of these signals to extract the particle-size distribution. The measurement basis is intensity weighted, which makes the measurement more sensitive to the larger particles in the population. This type of instrument is ideal for quality control applications involving unimodal colloidal dispersions. The analytical range is about 3 nm-1 pm. The lower size limit is reached as the intensity of scattered light gets below the detection limit, and the upper size limit is reached when the particles start to settle. Ultrasonic spectroscopy. This technique uses ultrasound to probe particles that are homogeneously suspended in a liquid, thereby exploiting the inherent advantage that sound waves (unlike light waves) can propagate through opaque dispersions. It can therefore handle suspensions with much higher particle concentrations than optical techniques. In ultrasonic spectroscopy, sound waves are passed through a suspension. As they travel through the sample and interact with the particles, the sound waves lose energy via scattering and absorption. Mathematical models based on fundamental theory convert this energy loss (or attenuation) as a function of sound frequency into a size distribution. A key advantage of ultrasonic spectroscopy over optical particle analysis is its ability to analyze high-concentration samples. However, it is generally more difficult to implement because, for both phases, numerous physical constants - speed of sound, density, thermal coefficient of expansion, heat capacity, thermal conductivity, viscosity of the fluid phase, shear rigidity of the solid phase - must be known for accurate size determination. Thus, this method is best suited for online applications involving the analysis of high-concentration samples where other technologies are not practical. The analytical range of ultrasonic spectroscopy is 0.01-1,000 pm, with concentrations ranging from a few volume percent to 80 vol% for emulsions. The concentration limits are highly dependent on the material. Fractionation techniques Fractionation methods - which include sieving, sedimentation, field-flow fractionation, hydrodynamic chromatography, and capillary hydrodynamic chromatography - separate particles by size prior to detection and measurement. Some fractionation techniques are based on physical phenomena related to first principles and do not require calibration (others are not and do need calibration). Both the analytical range and the resolution of these methods are governed by the efficiency of the separation process they utilize. Sieving. This technique, which has become the workhorse method for analyzing coarse particles, is straightforward and the equipment is simple. There are two types of sieving - dry sieving and wet sieving - both of which filter particles through a series of sieves of different mesh sizes. Dry sieving is typically carried out by shaking a stack of sieves using a mechanical vibrator, air pulses of sonic frequency, or rotating air jets. Dry sieving works well for particles with diameters larger than about 45 pm. Wet sieving, with water or another liquid facilitating particle passage through the sieves, is used for finer powders, down to roughly 5 pm. The equivalent diameter obtained from this method is the sieve diameter, which is defined as the size of the sieve aperture through which the particle being measured just passes (i.e., the particle gets caught by the next sieve, which has a smaller aperture). Since irregular particles tend to orient themselves along their smallest dimension to pass through a two-dimesional aperture, the sieve diameter is typically related to the second-smallest dimension of the particle. The data from sieving are mass-based measurements - i.e., the mass of the material retained on each sieve is measured to obtain a mass fraction, which is then presented in tabular or graphical form. Sedimentation. This mass-based method measures the settling rate of particles dispersed in a liquid that is subjected to gravitational or centrifugal forces. At the start of the measurement, the particles may be uniformly distributed throughout the liquid (homogeneous start) or concentrated within a narrow band at the liquid's surface (line start). Particle movement can be monitored using optical or X-ray detectors. Regardless of the system configuration (line/homogeneous start, gravitational/centrifugal force, optical/X-ray detector), the Stokes equation applies to all sedimentation equipment: ... (8) where v is the terminal velocity of a particle, g is the gravitational constant, d is the particle diameter, pp is the particle density, p, is the liquid density, and r\ is the liquid viscosity. For centrifugal sedimentation, the constant g is replaced with the actual acceleration of the system. Sedimentation instruments have high resolution because the separation technique is based on terminal velocity, which is proportional to the square of the particle diameter. However, this high resolution comes with a disadvantage: obtaining results for a sample with a wide size distribution generally takes a long time, thereby reducing the throughput of the technique. Field-flow fractionation (FFF). Although field-flow fractionation methods were first described in 1967, the technology transfer from an academic laboratory technique to a viable commercial instrument took more than 20 years, after which rapid proliferation took place. The term field-flow fractionation refers to a family of mass-based techniques in which particles dispersed in a liqNomenclature uid flow through a narrow channel that is subjected to forces perpendicular to the flow (Figure 6). The applied forces (e.g., cross-flow stream, temperature gradient, centrifugal force) cause the particles to move toward the bottom of the channel, where diffusion opposes the applied forces. The opposing forces confine particles of different sizes to specific regions of the channel - smaller particles have higher diffusion coefficients and thus flow near the center of the channel, while larger particles, with lower diffusion coefficients, travel near the edges. Since a parabolic flow profile exists in the channel, particles of different sizes are transported at different rates, allowing them to be separated according to size. These techniques provide a high-resolution separation over the size range of 3 nm-3 pm. Hydrodynamic chromatography (HDC) and capillary hydrodynamic chromatography (CHDC). In both HDC and CHDC (Figure 7), the particle-containing liquid travels through microchannels; in HDC, the microchannels are in the interstices between beads packed in a column, while CHDC utilizes straight microchannels. Under laminar conditions, a parabolic flow profile arises, in which the smaller particles travel closer to the walls in the lower-velocity zones and the larger particles travel farther away from the walls at higher velocities. HDC and CHDC provide a high-resolution separation over the analytical size range of 0.02-1 pm. The first CHDC instruments became commercially available around 1990, about 30 years after the discovery of this fractionation technique. Meanwhile, HDC was developed in an industrial laboratory to fulfill a sub-micron analytical need; although it provides high-quality sizing data, HDC has seen limited success in the market, mainly because a high skill level is required to operate the instrument and the packed column has a tendency to retain particles and plug over time. Single-particle counting methods Particle-counting instruments (also referred to as stream-counting methods), including dynamic image analysis, optical counters, and electrozone counters, detect and count particles one at a time. These methods are not based on first principles and therefore require calibration. Particle counters offer the ultimate in resolution, but can suffer from poor statistical accuracy when the data are converted from a number-based distribution to a massor volume-based one, especially when the size range is larger than two orders of magnitude. Because of the inherently narrow analytical range of these techniques, most counting devices come equipped with several (typically two or three) distinct sensors or sets of magnifying optics to increase their analytical ranges. Dynamic image analysis. The pioneering studies of particle characterization by imaging technologies were carried out in the late 1960s and early 1970s. Increasing dataprocessing power coupled with the high performance and falling costs of video cameras and high-speed frame grabbers have enabled the development of sophisticated and powerful image processing and analysis systems. Modem instruments are capable of acquiring tens or even hundreds of frames per second with simultaneous image processing to extract size and shape information. This new breed of instruments has gained considerable popularity within the past decade. As illustrated in Figure 8, particles flow through the sensing zone, where they are illuminated by a light source (e.g., strobe, laser, or incandescent), magnified, and then detected by a charge-coupled device (CCD) camera. The particles are dynamically presented to the instrument - as dry particles carried in a gas stream or falling from a vibratory feeder, or as a liquid dispersion moving through thin rectangular glass cells. Dynamic image analysis systems are normally used for particles larger than roughly 5 pm in dia. Several manufacturers have exaggerated claims regarding the lower limit of the analytical range for these instruments. The smallest particle measured should be represented by at least 7-10 pixels - not a single pixel, which would imply a square or rectangular particle. The measurement basis is the number of particles, but the data can be converted to a volume-based distribution. Optical particle counting. The first optical particle counters were developed in the late 1950s and early 1960s to measure the size distribution of aerosols. Instruments for liquid-based systems were subsequently developed and used extensively to measure the cleanliness of hydraulic fluids. Optical particle counters (Figure 9) use the principle of light blockage to count particles and measure their size. As a particle passes through the illuminated sensing zone, it casts a shadow onto the photodetector, which converts the shadow into an electrical signal, or pulse. The photodetector sends this pulse to an analyzer, where the height of the pulse, which is proportional to the cross-section of the particle, is converted into an equivalent diameter, referred to as the projected-area diameter (da). The analytical range of optical particle counting is roughly 0.2-2,500 pm, depending on the manufacturer; two or three sensors, each with a limited analytical range, are required to cover the full analytical range. The numberbased measurement can be converted to a volume-based distribution. Electrozone counters. The first electrozone counter was introduced by Wallace Coulter in 1954 to count blood cells. Since then, this technology has found a wide range of applications, and after more than half a centuiy, the Coulter counter is still the preferred technique for obtaining blood counts. Particles suspended in an electrolyte are drawn through an orifice situated between two electrodes (Figure 10), altering the electrical resistance of the electrolyte. The change in the resistance experienced by the electrolyte is proportional to the volume of the particle (i.e., the volume of electrolyte displaced). These instruments measure the volume equivalent diameter (dv) and provide a numberbased measurement that can be converted to a volumebased distribution. They have an analytical range of 0.4-1,200 pm using multiple sensors. Particle shape Irregularly shaped particles are much more common than spherical particles. Two irregular (nonspherical) particles with the same equivalent spherical diameter can have very different shapes. British Standard BS 2955 (3) includes a list of terms related to particle shape, some of which are shown in Table 4. While these terms may be useful descriptors, their lack of quantification does not lend them to use in mathematical equations. Shape factors and shape coefficients provide quantitative information about particle shape and can serve many purposes, such as: * relating various definitions of derived particle-size measurements * relating particle volume or surface area to a characteristic dimension (e.g., length, diameter, etc.) * generating a 2D outline of the original particle. The shape of a large particle can be physically measured and expressed as linear dimensions on orthogonal axes. For example, geologists routinely take direct measurements to define the shapes of rocks and pebbles. However, the study of microscopic particles must rely on silhouettes or outlines. The orientation of the particles is usually random, so particles of identical form often have different outlines. Thus, it is necessary to use statistics to obtain a representative characterization of particle shape. Common shape factors References 4-6 provide excellent reviews on shape measurement. However, clear guidance on the selection of suitable shape factors for various applications is not available. The underlying physics of the macroscopic behavior of interest dictates the appropriate morphological properties for consideration. Before selecting a shape factor, ask the following questions to narrow the choices: * Is the shape factor independent of size, magnification, orientation, and rotation? * How does the measurement method affect the shape factor or shape coefficient? * Does the resolution of the instrument affect the measurement, especially at the ends of the particle-size distribution? * Is the measurement based on a 3D measurement or a 2D profile or outline? * Is the shape factor relevant to the macroscopic property under investigation? Broadly speaking, the quantification of particle shape falls into four categories (5): 1. dimensional ratios 2. sphericity, which depicts form or overall shape 3. roundness, or circularity, which indicates angularity or the sharpness of comers 4. roughness, which characterizes surface texture. Heywood (6) recognized that the definition of shape must include its form and the relative proportions of its dimensions. Form refers to the particle's resemblance to a defined geometry (e.g., sphere, cube, tetrahedron). He proposed the use of an elongation ratio and a flakiness ratio as shape factors. Dimensional ratios. The form of a particle can be defined in terms of ratios of length (L), breadth (B), and thickness (7). As shown in Figure 11, T is the minimum distance between two parallel planes that are tangential to the opposite surfaces of the particle when the particle is in its most stable orientation; B is the minimum distance between two parallel planes that are tangential to the particle surface and perpendicular to the planes defining particle thickness; and L is the distance between two parallel planes that are perpendicular to the planes defining T and B. These three dimensions can be expressed as ratios: * elongation ratio = L/B (ranges from 1 to infinity) * flakiness ratio = BIT (ranges from 1 to infinity) * chunkiness ratio = B/L (ranges from 0 to 1 ). For a 2D particle outline, the aspect ratio (AR) may be defined in terms of the minimum distance (F . ) and the v min' maximum distance (Fmax) between pairs of tangents to the particle profile: AR = FmJFmax (ranges from 0 to 1 ). Sphericity. This property indicates how closely the particle resembles a sphere. Wadell defined sphericity (4/) as the surface area of a sphere having the same volume as the particle divided by the actual surface area of the particle. Owing to the difficulty in measuring the surface area of an irregular particle, Wadell proposed an approximation, in which 4* is the square of the diameter of a sphere having the same volume as the particle (dv) divided by the diameter of a circumscribing sphere (ds): ... (9) Riley proposed a quicker method to estimate 4/ as the square of the diameter of the largest inscribed circle divided by the diameter of the smallest circle circumscribing the particle outline. Circularity. Circularity (<j>) is often used as the shape factor for 2D planar profiles. The circularity of a particle can be defined as the circumference of a circle whose area is equal to that of the particle's projected area divided by the perimeter of the actual particle. The circularity shape factor (<J>jjr) is defined as: ... (10) where A is the projection area of the particle outline and P is the perimeter of the particle outline. Note that the circularity shape factor and circularity are different. Roundness. Roundness is a measure of the angularity of a particle profile, and an indicator of the presence of sharp comers: ... (11) where ri is the radius of curvature of a comer, N is the total number of round comers in the particle profile, and R is the maximum inscribed radius of the particle. Convexity. The convex hull perimeter of a particle can be visualized as the length of an elastic rubber band that fits around the particle. Convexity is the convex hull perimeter divided by the perimeter of the particle excluding roughness. It is a measure of the compactness of a particle profile, and is affected by surface irregularities, spikes, and concave regions of the perimeter. Convexity is sometimes measured as the projected area of the particle divided by the area of the convex hull, which are easier to measure. Surface texture. The surface texture of the particle can be quantified by rugosity, which is the perimeter of the particle outline including roughness divided by the perimeter of a smooth curve circumscribing the particle profile. Shape coefficients Surface area and volume are related to the square and the cube of the characteristic linear dimension of the particle, respectively. The shape coefficient expresses this relationship. The value of the shape coefficient depends on the definition of particle size (e.g., projected-area diameter, volume-equivalent diameter, Stokes diameter) and the bulk property of interest {i.e., volume or surface area). The measurement method must be reported with a shape coefficient. For example, the volume-area shape coefficient (ava) relates the volume diameter to the projected-area diameter: ...(12) where V is the volume of the particle, dv is the volumeequivalant diameter, and da is the projected-area diameter. In the same way, the surface-area-projected-area shape coefficient (as a) relates the surface-area diameter (ds ) to the projected-area diameter: ... (13) where S is the surface area of the particle. Closing thoughts Knowledge of particle size, size distribution, and shape is critical for designing and operating complex particulate processes. While the modem age of electronics and computers made the measurement techniques more robust and less onerous, the amount of data that can be generated has also proliferated. Such data can result in misleading conclusions unless you clearly understand the fundamentals. ES A = projected area of the particle's outline d = mean diameter dj = midpoint diameter of the particles in group / d m = De Brouckere diameter ds = surface-area equivalent diameter ds = diameter of a sphere circumscribing the particle dsv = surface-area-to-volume equivalent diameter dv = volume equivalent diameter g = gravitational constant m . = total mass of particles in group i M = total mass of the entire population of particles nj = number of particles in group i P = perimeter of a particle's outline r = radius of curvature of a comer of a particle's profile R = maximum inscribed radius of a particle s. = total surface area of particles in group i S = total surface area of the entire particle population v = terminal velocity of a particle V = volume of a particle Greek Letters asa = surface-area-projected-area shape coefficient ava = volume-area shape coefficient r\ = viscosity of the liquid pp = density of the particles p, = density of the liquid <j> . = circularity shape factor = sphericity Literature Cited 1. Allen, T., "Powder Sampling and Particle Size Determination," Elsevier, Philadelphia, PA (2003). 2. Dalla Valle, J. IM., "Micromeritics - The Technology of Fine Particles," Pitman Publishing Corp., New York, NY (1943). 3. British Standards Institution, "Glossary of Terms Relating to Particle Technology," BS 2955, BS1, London, U.K. (1993). 4. Hawkins, A. E., "The Shape of Powder-Particle Outlines," John Wiley and Sons, New York, NY (1993). 5. Rodriguez. J. M., "Particle Shape Quantities and Measurement Techniques - A Review," Electronic Journal of Geotechnical Engineering. 18 (A), pp. 169-198 (2013). 6. Singh, P., and P. Ramakrishnan. "Powder Characterization by Particle Shape Assessment," KONA Powder and Particle Journal. 14, pp. 16-30 (1996). Additional Resources Davies, R., "A Simple Feature-Space Representation of Particle Shape," Powder Technology12 (2), pp. 111-124 (Sept-Oct 1975). Gy, P., "Sampling Theory and Sampling Practice: Heterogeneity, Sampling Correctness, and Statistical Process Control," 2nd ed., CRC Press, Boca Raton, FL(1993). Heywood, H., "Symposium on Particle Size Analysis," Institution of Chemical Engineers. London, U.K. (Feb. 4. 1947). Jilavenkatesa, A., et at.. "NIST Recommended Practice Guide: Particle Size Characterization," U.S. Government Printing Office, Washington, DC (Jan. 2001). Kaye, B., and R. A. Trottier, "The Many Measures of Fine Particles," Chem. Eng.. 102 (4), pp. 78-86 (Apr. 1995). Leschonski, K., "Sieve Analysis, the Cinderella of Particle Size Analysis Methods?," Powder Technology. 24 (2), pp. 115-124 (Nov-Dec 1979). McGHnchey, D., ed., "Characterisation of Bulk Solids." WileyBlackwell. Hoboken, NJ (2005). Trottier, R. A., and S. Dhodapkar, "Sampling Particulate Materials the Right Way," Chem. Eng.. 119 (4), pp. 42-49 (Apr. 2012). RemiTrottier Shrikant Dhodapkar Dow Chemical Co. REMI TROTTIER is a research scientist in the solids processing discipline of engineering sciences at Dow Chemical (Email: [email protected]). He has over 20 years of industrial experience in particle characterization, aerosol science, air filtration, and solids processing technology. He has authored roughly 20 papers and has served as an instructor for several short courses on particle characterization. He received BS and MS degrees in applied physics from Laurentian Univ., Sudbury, ON, Canada, and his PhD in chemical engineering from Loughborough Univ. of Technology, U.K. SHRIKANT DHODAPKAR is a Fellow in the Process Fundamentals Group, Performance Plastics Process R&D at Dow Chemical (Email: [email protected]). He has been with Dow for 22 years and has worked closely with the elastomers business for the past 15 years to develop, design, and implement technologies related to material handling. He is an expert in the field of solids processing and bulk solids handling and has extensive industrial experience in powder characterization, fluidization, pneumatic conveying, silo design, gassolid separation, mixing, coating, computer modeling, and scaleup. He received his BTech from Indian Institute of Technology in chemical engineering, and an MS and PhD from the Univ. of Pittsburgh, both in chemical engineering. He is a senior member of AlChE and past chair of the Particle Technology Forum. He has authored or co-authored over 35 external publications and four chapters in handbooks. He is also an adjunct professor of chemical engineering at the Univ. of Pittsburgh. (c) 2014 American Institute of Chemical Engineers |