Chapter 3-Geometric Properties (Size and Shape) - PDF

IMK114: Introduction to Food Physics IMK 114: Introduction to Food Physics Chapter 3: Geometric Properties- Size and Shape 1 Chapter Outline 1. The importance of size and shape...

IMK114: Introduction to Food Physics IMK 114: Introduction to Food Physics Chapter 3: Geometric Properties- Size and Shape 1 Chapter Outline 1. The importance of size and shape 2. Shape: Standard chart, Sphericity, Form Factor, Roundness 3. Techniques to measure size ▪ Instrument Geometric equivalent diameters ▪ Image analysis Physical equivalent diameters ▪ Equivalent diameters Individual particles Bulk materials ▪ Specific surface area 4. Particle size distribution 5. Shrinkage 2 2 IMK114: Introduction to Food Physics The importance of size and shape in food technology Heat and mass transfer calculations Screening solids to separate foreign materials Grading of fruits and vegetables Evaluating the quality of food materials. 3 3 Measurement tools: Vernier calliper 4 4 IMK114: Introduction to Food Physics Measurement tools: Micrometer 5 5 Shape – Standard Chart Apple Peach Potato Figure 3.2. Example of charted standards for describing shape of fruits and vegetable Source of Figure 3.2. Figura, L.O. and Teixera, A.A. (2007). Food Physics: Physical Properties - Measurement and Applications. Springer-Verlag Berlin 6 Heidelberg, Leipzig, Germany. 6 IMK114: Introduction to Food Physics Form Factor - Sphericity The shape of a food material is usually expressed in terms of its sphericity and aspect ratio →convenience for prediction and calculation. Sphericity is an important parameter used in fluid flow and heat and mass transfer calculations. Sphericity or shape factor can be defined in different ways. 7 7 Sphericity: Volume ratio Form Factor - Sphericity The most used definition → sphericity is the ratio of volume of solid to the volume of a sphere that has a diameter equal to the major diameter of the object so that it can circumscribe the solid sample. 8 8 IMK114: Introduction to Food Physics Sphericity: Volume ratio 9 9 Sphericity: Volume ratio 10 10 IMK114: Introduction to Food Physics Sphericity: Diameter ratio By using diameter → dimensionless ratio of any specified equivalent diameter for the particle (arithmetic or geometric mean, or other equivalent diameter from Table 3.3) divided by the particle’s major diameter (see Table 3.2). d Sphericity= e dc Note that the major diameter of a particle is the same as the diameter of the smallest circumscribing sphere (smallest sphere that can surround the particle). It is necessary to specify which equivalent diameter has been used for calculation 11 11 Characteristic diameters of particles Characteristic Description (based on fig. 3.7) Symbol diameter Feret diameter Height of projected area (top to xFe bottom) Martin Diameter which cuts projected area xMA diameter into equal area parts (but maybe different shape) Major diameter The longest diameter completely a inside the projected area stretching across opposite ends of the projected area Table 3.2. Some characteristic diameters of particles Figure 3.7. Characteristic length of particles (examples). Source of Table 3.2. and Figure 3.7. Figura, L.O. and Teixera, A.A. (2007). Food Physics: Physical Properties - Measurement and Applications. Springer-Verlag 12 Berlin Heidelberg, Leipzig, Germany. 12 IMK114: Introduction to Food Physics Image analysis 𝑎𝑟𝑖𝑡ℎ𝑚𝑒𝑡𝑖𝑐 𝑚𝑒𝑎𝑛 𝑑𝑖𝑎𝑚𝑒𝑡𝑒𝑟 𝑎+𝑏+𝑐 = 3 𝑔𝑒𝑜𝑚𝑒𝑡𝑟𝑖𝑐 𝑚𝑒𝑎𝑛 𝑑𝑖𝑎𝑚𝑒𝑡𝑒𝑟 = (𝑎 × 𝑏 × 𝑐)1/3 Figure 3.6. Different projections of particles Source of Figure 3.6. Figura, L.O. and Teixera, A.A. (2007). Food Physics: Physical Properties - Measurement and Applications. Springer-Verlag Berlin Heidelberg, 13 Leipzig, Germany. 13 Table 3.3. Geometric equivalent diameters Source of Table 3.3. Figura, L.O. and Teixera, A.A. (2007). Food Physics: Physical Properties - Measurement and Applications. Springer-Verlag Berlin Heidelberg, 14 Leipzig, Germany. 14 IMK114: Introduction to Food Physics Exercise 3.1. Equivalent diameter of a sphere with same volume Calculate the volume equivalent diameter of a hypothetical sphere (a) for a cube with (edge) length of a = 0.70 mm (b) for a tetrahedral regular shape with an (edge) length of a = 0.70 mm (c) for an octahedral regular shape with an (edge) length of a = 0.70 mm Solution: for a hypothetical sphere the volume is: Octahedron So, Answers: (a) 0.87 mm, (b) 0.43 mm, (c) 0.68 mm 15 15 Exercise 3.2. Equivalent diameter of sphere with same area Calculate the surface equivalent diameter of a hypothetical sphere (a) for a cube with (edge) length of a = 0.70 mm (b) for a tetrahedral regular shape with an (edge) length of a = 0.70 mm (c) for an octahedral regular shape with an (edge) length of a = 0.70 mm Acube = 6a2 Solution: 𝐴 = 𝜋𝑑𝐴2 for a hypothetical sphere, the area is Atetrahedon = 3𝑎2 so, 𝐴 𝑑𝐴 = Aoctagon = 2(1+ 2 )𝑎2 𝜋 Replace A in above equation with A of respective shape. 16 Answers: (a) 0.97 mm, (b) 0.52 mm, (c) 0.74 mm 16 IMK114: Introduction to Food Physics Form Factor Depending on the type of samples and relevant factors involved in the quality control or processing line, besides sphericity other form factor can be defined as listed in Table 3.4. Table 3.4. Definitions of form factors (examples) Source of Table 3.4. Figura, L.O. and Teixera, A.A. (2007). Food Physics: Physical Properties - Measurement and Applications. Springer-Verlag Berlin Heidelberg, Leipzig, Germany. 17 17 Form Factor When the definition of a form factor contains the “diameter x,” this diameter can be any one of the various equivalent diameters defined earlier in Table 3.2 or Table 3.3. Therefore, it is necessary to specify which equivalent diameter has been used for calculation Focusing on the Wadell sphericity we see its value ranges from 0 to 1. Φ = 1 →spherical particles , Φ=0 → needle shape. Like the particle size, the particle shape is not the same for all particles but is distributed. 18 18 IMK114: Introduction to Food Physics Table 3.6. Examples of sphericity Source of Table 3.6. Figura, L.O. and Teixera, A.A. (2007). Food Physics: Physical Properties - Measurement and Applications. Springer-Verlag Berlin Heidelberg, Leipzig, Germany. 19 19 Roundness When a particle is observed by optical means, we see the projection of the particle (projected area). To estimate the shape of the particle from that two-dimensional view there are different roundness factors that can be used. The most useful definitions for the roundness factor is 𝐴𝑝 𝑅= 𝐴𝑐 with R roundness factor Ap largest projected area of particle Ac area of smallest circumscribed circle Fig. 3.1. Roundness definition 20 20 IMK114: Introduction to Food Physics Exercise 3.3 a) Calculate the sphericity of a potato having major, intermediate, and minor diameters of 60.2 mm, 57.2 mm, and 50.8 mm, respectively, by using i. volume ratio ii. diameter ratio iii.Wadell’s sphericity factor b) Based on Wadell’s sphericity factor, state the shape of the potato. 21 21 Exercise 3.3 (i) 1/3 𝑉𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑠𝑜𝑙𝑖𝑑 𝑠𝑎𝑚𝑝𝑙𝑒 𝑠𝑝ℎ𝑒𝑟𝑖𝑐𝑖𝑡𝑦, ∅ = 𝑉𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑐𝑖𝑟𝑐𝑢𝑚𝑠𝑐𝑟𝑖𝑏𝑒𝑑 𝑠𝑝ℎ𝑒𝑟𝑒 1/3 30.1 × 28.6 × 25.4 ∅= 30.13 ∅ = 0.929 𝑜𝑟 92.9% 22 22 IMK114: Introduction to Food Physics Exercise 3.3 (ii) 𝑑𝑒 𝑠𝑝ℎ𝑒𝑟𝑖𝑐𝑖𝑡𝑦, ∅ = 𝑑𝑐 60.2 + 57.2 + 50.8 ∅𝑎𝑟𝑖𝑡ℎ𝑚𝑒𝑡𝑖𝑐 = 3 60.2 𝑠𝑝ℎ𝑒𝑟𝑖𝑐𝑖𝑡𝑦, ∅ = 0.931 𝑜𝑟 93.1% 23 23 Exercise 3.3 (iii) 𝜋𝑑 2 ∅𝑊𝐴 = 𝜋𝑑2𝑣 𝐴 3 6∙𝑉 𝑑𝑣 = 𝜋 4 𝑉= × 𝜋 × 30.1 × 28.6 × 25.4 = 91591.43317 𝑚𝑚3 3 3 6×91591.43317 𝑑𝑣 = =55.9266 mm 𝜋 55.926662 ∅𝑊𝐴 = = 𝟎. 𝟖𝟔𝟑 𝒐𝒓 𝟖𝟔. 𝟑% 60.22 24 24 IMK114: Introduction to Food Physics Exercise 3.3 (b) cylinder shape (if obtained 86.3% in 2 a iii→ taking major diameter as denominator) 25 25 Shrinkage Shrinkage is the decrease in volume of the food during processing such as drying. When moisture is removed from food during drying, there is a pressure imbalance between inside and outside of the food. This generates contracting stresses leading to material shrinkage or collapse Shrinkage affects the diffusion coefficient of the material and therefore influences the drying rate. 26 26 IMK114: Introduction to Food Physics Shrinkage Apparent shrinkage is defined as the ratio of the apparent volume at a given moisture content to the initial apparent volume of materials before processing: 27 27 Shrinkage Shrinkage is also defined as the percent change from the initial apparent volume. Two types of shrinkage are usually observed in food materials. If there is a uniform shrinkage in all dimensions of the material→called isotropic shrinkage. If there is a nonuniform shrinkage in all dimensions of the material →anisotropic shrinkage 28 28 IMK114: Introduction to Food Physics Particle size distribution (PSD) The PSD of flour is known The range of particle size Example: Flour. The to play an important role in foods depends on the hardness of grain is a in its functional cell structure and the significant factor in the properties and the degree of processing. PSD of flour. quality of end products. 29 29 30 30 IMK114: Introduction to Food Physics Images of steamed bun made by different particle size wheat flour. (A: D50 = 108.89 μm; B: D50 = 88.13 μm; C: D50 = 78.47 μm; D: D50 = 65.73; E: D50 = 52.36 μm) 31 31 Screening A set of standard screens is stacked one upon the other with the smallest opening at the bottom and the largest at the top placed on an automatic shaker for screen analysis (sieve analysis). In screen analysis, the sample is placed on the top screen and the stack is shaken mechanically for a definite time. The particles retained on each screen are removed and weighed. Then, the mass fractions of particles separated are calculated. Any particles that pass through the finest screen are collected in a pan at the bottom of the stack 32 32 IMK114: Introduction to Food Physics Sieve number also referred as mesh number 33 33 American Society For Testing and Materials (ASTM) 34 34 IMK114: Introduction to Food Physics ASTM Standard Test Sieve More info: https://www.globalgilson.com/blog/sieve-sizes 35 35 Particle Size Analyzer 36 36 IMK114: Introduction to Food Physics Particle Size Analyzer https://www.azom.com/article.aspx?ArticleID=14762 37 37 38 38 IMK114: Introduction to Food Physics Particle size distribution (PSD) Since the particles on any one screen are passed by the screen immediately ahead of it, two numbers are required to specify the size range of an increment: one for the screen through which the particles passes. one for the screen which the particles are retained. For example, 6/8 refers to the particles passing through the 6-mesh and remaining on an 8-mesh screen. 39 39 Particle size distribution (PSD) Differential Analysis Mass or number fraction in each size increment is plotted as a function of average particle size or particle size range. The results are often presented as a histogram as shown with a continuous curve to approximate the distribution 𝑋𝑖𝑤 Average particle size versus 𝐷𝑃𝑖+1 −𝐷𝑃𝑖 Or mass fraction 𝑋𝑖𝑤 vs 𝐷𝑃𝑖+1 − 𝐷𝑃𝑖 (the particle 40 size range in the increment i) 40 IMK114: Introduction to Food Physics Particle size distribution (PSD) Cumulative analysis Cumulative analysis is obtained by adding, consecutively, the individual increments, starting with that containing the smallest particles. Cumulative sums of the particle size vs the maximum particle diameter in the increment. In a cumulative analysis, the data may appropriately be represented by a continuous curve 41 41 Table 3.7: Typical screen analysis-differential analysis 42 42 IMK114: Introduction to Food Physics Table 3.7: Typical screen analysis-Cumulative analysis 43 43 Sauter Mean Diameter , D(3,2) Differential analysis Cumulative analysis This diameter represents the size of a particle that would have the same ratio of volume to surface area as the entire sample of particles, effectively providing an "average" diameter based on geometry rather than mass or number. 44 44 IMK114: Introduction to Food Physics Mass Mean Diameter Differential analysis Cumulative analysis If mass fraction is known 45 45 Arithmetic Mean Diameter Differential analysis Cumulative analysis If number of particles is known 46 46 IMK114: Introduction to Food Physics Volume Mean Diameter Differential analysis Cumulative analysis 47 47 Exercise 3.4 Wheat flour is made by grinding the dry wheat grains. Particle size is an important characteristic in many of the wheat products. For example, in making wafers, if the flour is too fine, light and tender products are formed. On the other hand, incomplete sheets of unsatisfactory wafers are formed if the flour is too coarse. Therefore, it is important to test the grinding performance of flour by sieve analysis in wafer producing factories. Determine the volume surface mean diameter (Sauter mean diameter), mass mean diameter, and volume mean diameter of wheat flour by differential analysis using the data given in Table 3.7 48 48 IMK114: Introduction to Food Physics Table 3.7: Sieve analysis of wheat flour 49 49 50 50 IMK114: Introduction to Food Physics Calculate the average particle size, 𝐷𝑝𝑖 Refer to the Excel file on e-learn for solution 51 51