22-Agric-A6 Physical Properties of Biological Materials and Food Products · May 2016
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Paper format. 04-Agric-A6 Physical Properties of Biological Materials and Food Products, National Exams May 2016 — a three-hour closed-book exam (approved calculator permitted; one aid sheet, both sides). Nine questions are set and candidates answer any five, each worth 20 marks, for a 100-mark paper. All nine are worked here so the set is a complete study resource.
Reference texts. M.A. Rao, S.S.H. Rizvi, A.K. Datta and J. Ahmed, Engineering Properties of Foods, 4th ed. (rheology of fluid and semisolid foods, particle size, surface/interfacial properties); N.N. Mohsenin, Physical Properties of Plant and Animal Materials, 2nd ed. (thermal properties, calorimetry, texture and rheological testing); R.P. Singh and D.R. Heldman, Introduction to Food Engineering, 5th ed. (thermal-property measurement, freezing-point depression, particle size); J.F. Steffe, Rheological Methods in Food Process Engineering, 2nd ed. (viscometry, viscoelasticity, the Kelvin-Voigt model, time-dependent flow behaviour); R.L. Earle, Unit Operations in Food Processing, 2nd ed. (particle-size averages, specific surface from sieve data).
Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.
(a) Given. The particle-count size distribution \((D_{pi}, N_i)\) tabulated above; eight size classes from 2 to 30 µm.
Find. The length mean diameter \(\bar D_L\) and the volume mean diameter \(\bar D_v\).
Approach. The length mean diameter is the number-weighted first moment of the distribution; the volume mean diameter is the diameter of the (spherical) particle whose volume equals the number-average particle volume, i.e. the cube root of the number-weighted third moment.
| \(D_{pi}\), µm | \(N_i\) | \(N_iD_{pi}\) | \(N_iD_{pi}^3\) |
|---|---|---|---|
| 2 | 30 | 60 | 240 |
| 6 | 40 | 240 | 8,640 |
| 10 | 90 | 900 | 90,000 |
| 14 | 100 | 1,400 | 274,400 |
| 18 | 120 | 2,160 | 699,840 |
| 22 | 80 | 1,760 | 851,840 |
| 26 | 65 | 1,690 | 1,142,440 |
| 30 | 15 | 450 | 405,000 |
| Σ | 540 | 8,660 | 3,472,400 |
| Quantity | Value |
|---|---|
| Length mean diameter, \(\bar D_L\) | 16.04 µm |
| Volume mean diameter, \(\bar D_v\) | 18.60 µm |
(b) Drag coefficient, frictional drag, and terminal velocity. The drag coefficient \(C_D\) is a dimensionless ratio of the actual drag force on a particle to the dynamic-pressure force that the free-stream flow exerts on the particle's projected area, $$C_D = \frac{F_D}{\tfrac{1}{2}\rho_f v^2 A_p},$$ and it collapses the drag behaviour of particles of different size/shape/Reynolds number onto a common, correlatable curve (\(C_D\) versus particle Reynolds number). Frictional (viscous) drag is the component of the total drag force arising from fluid viscosity acting tangentially on the particle surface (skin friction); at low particle Reynolds number (creeping/Stokes flow) frictional drag dominates the total drag, whereas at high Reynolds number form (pressure) drag from the wake dominates instead, so "frictional drag" and "total drag force" are only interchangeable in the Stokes regime. Terminal velocity \(v_t\) is the constant velocity a particle falling (or rising) freely through a fluid eventually reaches once the net driving force (gravity minus buoyancy) is exactly balanced by drag, so the particle's acceleration is zero; for a small spherical particle in the Stokes regime it is given in closed form by $$v_t = \frac{gD_p^2\left(\rho_p-\rho_f\right)}{18\mu},$$ and at higher Reynolds number \(v_t\) must instead be solved iteratively from the force balance \(F_{\text{gravity}}-F_{\text{buoyancy}} = C_D\cdot\tfrac12\rho_f v_t^2 A_p\) using the appropriate \(C_D\)-Reynolds-number correlation.
