248 M.J. Lewis Fig 9.1.(a)Frequency distribution(F), (b)cumulative distribution(C): see also data in Table 9.5 Table 9. 5. Frequency distribution Size range Mean diameter Number Frequency Cumulative Cumulative (um) in range distributiondistribution volume 0to10 18 1.8 10to20 20to30 129 20.3 2.0 30to4035 18.4 38.7 7.0 40to5045 20.3 0.0 6055 18.5 41.0 640 70to8075 74 96.7 850 80to9085 98.0 90to10095 0.4 100.0 0 mean diameter =45.96 um; d2/ 1=53. 29 um; d3/2=58.86 Am Cumulative volume represents the percentage of the total volume less than the mean diameter of the range Other values which may be calculated from the distribution include the mean diameter nd the median diameter and the standard deviation, which gives an indication of the The simplest is the mean diameter, defined ∑n4/∑
248 M. J. Lewis 100 80 60 10 40 20 0 LL *OB 0 0 20 Size 40 (prn) 60 80 100 0 Fig. 9.1. (a) Frequency distribution (F), (b) cumulative distribution (0: see also data in Table 9.5. Table 9.5. Frequency distribution Size range Mean diameter Number Frequency Cumulative Cumulative OLm) of range (pm) in range distribution distribution volume distribution 0 to 10 5 5 1.8 1.8 0 10 to 20 15 15 5.6 7.4 0 20to30 25 35 12.9 20.3 2.0 30to40 35 50 18.4 38.7 7.0 40to50 45 55 20.3 59.0 20.0 50to60 55 50 18.5 77.5 41 .O 60to70 65 32 11.8 89.3 64.0 70to 80 75 20 7.4 96.7 85.0 80to90 85 8 2.9 99.6 98.0 90to 100 95 1 0.4 100.0 100.0 >loo 0 27 1 - mean diameter = 45.96 pm; d2/1 = 53.29 pm; d3/2 = 58.86 pm Cumulative number frequency indicates the percentage of the total number less than the mean diameter of the range. Cumulative volume represents the percentage of the total volume less than the mean diameter of the range. Other values which may be calculated from the distribution include the mean diameter and the median diameter and the standard deviation, which gives an indication of the spread. The simplest is the mean diameter, defined as C nidi /C ni 7
Solids separation processes 249 where ni is the number of particles in class i and d; is the mean diameter of class i. The median diameter is the diameter which cuts the cumulative distribution in half. The d2/1 ratios and d3/2 ratios are also calculated However, one widely used characteristic is the Sauter mean particle diameter(d3/2) This is calculated from This gives the diameter of the particle having the same surface area to volume ratio as the entire dispersion The surface area/volume ratio =6/d3/2 Rates of heat transfer and mass transfer are proportional to the surface area to volume ratio. Therefore the surface area exposed has a big influence on physical properties, e ettability, dispersion, dissolution and chemical reactions, such as oxidation, as we the forces acting at the surface of powders. Equation(9. 2) demonstrates that decreasing 3/2 will increase the surface area to volume ratio Such data can be converted to frequency or cumulative distribution based on surface area or volume, by calculating the surface area and volume of each range. These cumula tive distributions based on numbers and volume are compared in Fig. 9. 2. This distinction is made because the shape of a numbers distribution and a mass or volume distribution is quite different because the area and volume distributions are most influenced by the larger diameter particles, since the volume ==r. For example, it can be seen that only 10.7% of the particles are greater than 65 um, whereas on a volume basis, 36% by olume are greater than 65 um(Fig. 9.2). The weight fraction distribution would be similar to the volume fraction distribution, provided that the solid density is independent of particle size. The volume distribution is a common form of presentation in emulsion science, since it is often the larger particles which are likely to cause separation problems Therefore it can be very informative to know what fractions by volume are bigger than a particular size. For example, in cream separation in milk, problems may arise from a relatively small number of large fat globules g. 9.2. Comparison of volume distribution(V and cumulative number distribution(M). St also data in table 9.5
Solids separation processes 249 where ni is the number of particles in class i and di is the mean diameter of class i. The median diameter is the diameter which cuts the cumulative distribution in half. The d2/1 ratios and d3/2 ratios are also calculated. However, one widely used characteristic is the Sauter mean particle diameter (d& This is calculated from d3p = xnid:/xnid,? (9.1) This gives the diameter of the particle having the same surface area to volume ratio as the entire dispersion. The surface area/volume ratio = 6/d3/2. (9.2) Rates of heat transfer and mass transfer are proportional to the surface area to volume ratio. Therefore the surface area exposed has a big influence on physical properties, e.g. wettability, dispersion, dissolution and chemical reactions, such as oxidation, as well as the forces acting at the surface of powders. Equation (9.2) demonstrates that decreasing d312 will increase the surface area to volume ratio. Such data can be converted to frequency or cumulative distribution based on surface area or volume, by calculating the surface area and volume of each range. These cumulative distributions based on numbers and volume are compared in Fig. 9.2. This distinction is made because the shape of a numbers distribution and a mass or volume distribution is quite different because the area and volume distributions are most influenced by the larger diameter particles, since the volume = 4 m3. For example, it can be seen that only 10.7% of the particles are greater than 65 pm, whereas on a volume basis, 36% by volume are greater than 65 pm (Fig. 9.2). The weight fraction distribution would be similar to the volume fraction distribution, provided that the solid density is independent of particle size. The volume distribution is a common form of presentation in emulsion science, since it is often the larger particles which are likely to cause separation problems. Therefore it can be very informative to know what fractions by volume are bigger than a particular size. For example, in cream separation in milk, problems may arise from a relatively small number of large fat globules. 100 40 0 liL 20 0 20 40 60 80 100 Size (pm) Fig. 9.2. Comparison of volume distribution (V) and cumulative number distribution (N). See also data in Table 9.5
250 M.J. Lewis Most of the discussion has focused upon spherical particles or those closely approxi mating to these. However, the particle shape is also very likely to be important and a ide variety of shapes are also found Irregular-shaped objects are more complicated define and a number of characteristic dimensions have been used to represent them. Some are given in Table 9.6 Table 9.6. Characteristic diameters for irregular shaped particles Surface diameter The diameter of a sphere having the same surface area Volume diameter he diameter of a sphere having the same volume as dd Drag The diameter of the particle having the same resistance to motion as the particle in a fluid of the same density and viscosity Sieve diameter The width of the minimum square aperture through which the particle will pass Other dimensions include the free-falling diameter and Stokes diameter, the projected area diameter and the specific surface diameter. In many cases the shape is more complex and a large number of dimensions would be required to describe the size and shape Image analysis methods, whereby an image of the object is transferred to a computer screen and software is available to do any number of manipulations and calculations on the shape, are useful for this The particle size and distribution has a pronounced effect on interparticle adhesion, which will affect some of the bulk properties, such as bulk density, porosity, flowability and wettability(see Section 9.2.5) 9.2.3 Particle density The density of an individual particle is important as it will determine whether the compo- nent will float or sink in water or any other solvent; the particle may or may not contain air. It can be measured using a specific gravity bottle, using a fluid in which it will not issolve, Alternatively, it may be measured by flotation principles. However, surface forces may start to predominate for fine powders In the absence of air, the particle density can be estimated from the following equation, based on the mass fractions and densities of the food ce M1/p1+M2/p2+…+M/Pn where Mi is the mass fraction of component 1, P1 is the density of component 1 and n is the number of components. Data on mass fractions can be found from the Composition of
250 M. J. Lewis Most of the discussion has focused upon spherical particles or those closely approximating to these. However, the particle shape is also very likely to be important and a wide variety of shapes are also found. Irregular-shaped objects are more complicated to define and a number of characteristic dimensions have been used to represent them. Some are given in Table 9.6. Table 9.6. Characteristic diameters for irregular shaped particles 4 Surface diameter dv Volume diameter dd Drag diameter The diameter of a sphere having the same surface area as the particle The diameter of a sphere having the same volume as the particle The diameter of the particle having the same resistance to motion as the particle in a fluid of the same density and viscosity The width of the minimum square aperture through which the particle will pass. 4 Sieve diameter Other dimensions include the free-falling diameter and Stokes diameter, the projected area diameter and the specific surface diameter. In many cases the shape is more complex and a large number of dimensions would be required to describe the size and shape. Image analysis methods, whereby an image of the object is transferred to a computer screen and software is available to do any number of manipulations and calculations on the shape, are useful for this. The particle size and distribution has a pronounced effect on interparticle adhesion, which will affect some of the bulk properties, such as bulk density, porosity, flowability and wettability (see Section 9.2.5). 9.2.3 Particle density The density of an individual particle is important as it will determine whether the component will float or sink in water or any other solvent; the particle may or may not contain air. It can be measured using a specific gravity bottle, using a fluid in which it will not dissolve. Alternatively, it may be measured by flotation principles. However, surface forces may start to predominate for fine powders. In the absence of air, the particle density can be estimated from the following equation, based on the mass fractions and densities of the food components. P = 1/ [ (M1 lP1+ M2 / P2 + * * * + Mtl lPn ,I (9.3) where Ml is the mass fraction of component 1, p1 is the density of component 1 and n is the number of components. Data on mass fractions can be found from the Composition of
Solids separation processes Foods Tables(Paul and Southgate, 1978). A simple two-component model can be used (n= 2; water and solids) or a multicomponent system. The density of the major ce nts are given as(kg m -)(Peleg, 1983) 1000 salt 2160 900950 citric acid 154 protein 1400 cellulose 1270-1610 sucrose 1500 glucose It is noteworthy that all solid components except fat are substantially more dense than water. However the differences between protein and the various types of carbohydrates are less marked, although minerals are much higher. In comparison air has a density of 1.27 kg m This equation is not applicable where there is a substantial volume fraction of air in the particle. Any deviation between the experimentally determined value and the value calculated from the above equation may mean that there is substantial air within the solid. An estimate of the volume fraction of air( va) can be made from p=VaPa +vsPs= vaPa+(l-va)Ps where Pa= density of air; Ps =density of solid(estimated using eq (9.2))and p=true lid density, measured experimentally. This volume fraction (Va) of air is sometimes known as the intenal porosity Many other foods contain substantial amounts of air, for example mechanically worked doughs. One solution to determine the unaerated density is to measure the dough density at different pressures and extrapolate back to zero pressure(absolute) to obtain the unaerated density. This methodology could then be used to determine the extent of aeration during the mixing process Note that from the compositional data, the calculated particle density of an apple is about 1064 kg m, Most apples float in water, indicating a density less than 1000 kg m-3 Mohsenin(1986)quotes a value of 846 kg m-3, suggesting an air content of about 20%. One important objective of blanching is to remove as much air as possible from fruit and vegetables prior to heat-treatment in sealed containers, to prevent exces- sive pressure development during their thermal processing. Data on the amount of air in fruits and vegetables are scarce in the food literature. There is evidence that this air is quickly displaced by water during soaking Data on particle densities are provided by Lewis(1990), Mohsenin(1986), and Hayes (916 kg m3 at 0 C). However, not all the water is likely to be frozen, even at-300 ted (1987). Note that if the food is frozen, the density of ice should be substi The particle density of dehydrated powders is considerably affected by the conditions of spray drying. Increasing the solids content of the feed to the drier will result in higher particle densities and bulk densities. High particle densities will enhance sinkability and ing and separation techniques, e.g. flotation, sedimentation and air classification ]l clean reconstitution properties. Differences in particle densities are exploited for seve
Solids separation processes 25 1 Foods Tables (Paul and Southgate, 1978). A simple two-component model can be used (n = 2; water and solids) or a multicomponent system. The density of the major components are given as (kg m-3) (Peleg, 1983): water 1000 salt 2160 fat 900-950 citric acid 1540 protein 1400 cellulose 1270-1 6 10 sucrose 1590 starch 1500 glucose 1560 It is noteworthy that all solid components except fat are substantially more dense than water. However the differences between protein and the various types of carbohydrates are less marked, although minerals are much higher. In comparison air has a density of 1.27 kg m-3. This equation is not applicable where there is a substantial volume fraction of air in the particle. Any deviation between the experimentally determined value and the value calculated from the above equation may mean that there is substantial air within the solid. An estimate of the volume fraction of air (V,) can be made from P= &pa + V,P~ =VaPa +(l-va)Ps (9.4) where pa = density of air; ps = density of solid (estimated using eq. (9.2)) and p = true solid density, measured experimentally. This volume fraction (V,) of air is sometimes known as the internal porosity. Many other foods contain substantial amounts of air, for example mechanically worked doughs. One solution to determine the unaerated density is to measure the dough density at different pressures and extrapolate back to zero pressure (absolute) to obtain the unaerated density. This methodology could then be used to determine the extent of aeration during the mixing process. Note that from the compositional data, the calculated particle density of an apple is about 1064 kg m-3. Most apples float in water, indicating a density less than 1000 kg m-3, Mohsenin (1986) quotes a value of 846 kg m-3, suggesting an air content of about 20%. One important objective of blanching is to remove as much air as possible from fruit and vegetables prior to heat-treatment in sealed containers, to prevent excessive pressure development during their thermal processing. Data on the amount of air in fruits and vegetables are scarce in the food literature. There is evidence that this air is quickly displaced by water during soaking. Data on particle densities are provided by Lewis (1990), Mohsenin (1986), and Hayes (1987). Note that if the food is frozen, the density of ice should be substituted (916 kg m-3 at 0°C). However, not all the water is likely to be frozen, even at -30°C. The particle density of dehydrated powders is considerably affected by the conditions of spray drying. Increasing the solids content of the feed to the drier will result in higher particle densities and bulk densities. High particle densities will enhance sinkability and reconstitution properties. Differences in particle densities are exploited for several cleaning and separation techniques, e.g. flotation, sedimentation and air classification
252 M.J. Lewis 9. 2. 4 Forces of adhesion There will be interactions between particles, known as forces of adhesion and also be- tween particles and the walls of containing vessels. These forces of attraction will influ ence how the material packs and how it will flow. Some of the mechanisms for adhesive forces have been described as liquid bridging by surface moisture or melted fat: electrostatic charges molecular forces, such as van der Waals and electrostatic forces crystalline surface energy. Schubert(1987a) describes some of the models that have been used to quantify these forces, and the limitations of such models There is some indication that interparticle adhesion increases with time, as the material onsolidates. Flowability may be time-dependent and decrease with time 9. 2.5 Bulk properties Although the discussion so far has focused on individual particles, the behaviour of the collective mass of particles or bulk is very important in most operations. The bulk properties of fine powders are dependent upon geometry, size, surface characteristics, chemical composition, moisture content and processing history. Therefore it is difficult to put precise values on them and any cited values should be regarded as applying only to that specific circumstance, Peleg(1983) The term cohesive is used to describe the behaviour of powders, as they are influenced by forces of attraction(or repulsion) between particles. For powders that are cohesive, the ratio of the interparticle forces to the particles'own weight is large. This ratio is alse inversely proportional to the square of the particle size, which explains why small articles adhere to each other more strongly than large particles. Schubert(1987a)states that the majority of food particles are non-cohesive(and thus free flowing)only when the particle size exceeds 100 um. Increase in moisture content makes powders more cohesive and increases the size at which the transition from cohesive to non-cohesive takes place Some of the bulk properties will be considered in more detail 9.2.6 Bulk density and porosity The bulk density is an important property, especially for storage and transportation, rather than separation processes. It is defined as the mass divided by the total volume occupi by the material. This total volume includes air trapped between the particles. The volume fraction trapped between the particles is known as the porosity (e), where where ps and pb are measured solid and bulk densities. Methods for determining bulk density are described by the Society of Dairy Technology(1980)and Niro(1978). Terms used depend upon the method of determination and include loose bulk density and com- Some bulk densities of powders are given in Table 9.7. Further values are given by Peleg(1983), Hayes(1987)and Schubert(1987a). Peleg(1983)argues that the relatively
252 M. J. Lewis 9.2.4 Forces of adhesion There will be interactions between particles, known as forces of adhesion and also between particles and the walls of containing vessels. These forces of attraction will influence how the material packs and how it will flow. Some of the mechanisms for adhesive forces have been described as liquid bridging by surface moisture or melted fat; electrostatic charges; molecular forces, such as Van der Waals and electrostatic forces; crystalline surface energy. Schubert (1987a) describes some of the models that have been used to quantify these forces, and the limitations of such models. There is some indication that interparticle adhesion increases with time, as the material consolidates. Flowability may be time-dependent and decrease with time. 9.2.5 Bulk properties Although the discussion so far has focused on individual particles, the behaviour of the collective mass of particles or bulk is very important in most operations. The bulk properties of fine powders are dependent upon geometry, size, surface characteristics, chemical composition, moisture content and processing history. Therefore it is difficult to put precise values on them and any cited values should be regarded as applying only to that specific circumstance, Peleg (1983). The term cohesive is used to describe the behaviour of powders, as they are influenced by forces of attraction (or repulsion) between particles. For powders that are cohesive, the ratio of the interparticle forces to the particles’ own weight is large. This ratio is also inversely proportional to the square of the particle size, which explains why small particles adhere to each other more strongly than large particles. Schubert (1987a) states that the majority of food particles are non-cohesive (and thus free flowing) only when the particle size exceeds 100 pm. Increase in moisture content makes powders more cohesive and increases the size at which the transition from cohesive to non-cohesive takes place. Some of the bulk properties will be considered in more detail. 9.2.6 Bulk density and porosity The bulk density is an important property, especially for storage and transportation, rather than separation processes. It is defined as the mass divided by the total volume occupied by the material. This total volume includes air trapped between the particles. The volume fraction trapped between the particles is known as the porosity (E), where E = PS - pb/ps (9.5) where ps and pb are measured solid and bulk densities. Methods for determining bulk density are described by the Society of Dairy Technology (1980) and Niro (1978). Terms used depend upon the method of determination and include loose bulk density and compacted and compressed bulk densities. Some bulk densities of powders are given in Table 9.7. Further values are given by Peleg (19831, Hayes (1987) and Schubert (1987a). Peleg (1983) argues that the relatively