Question 2 of 9: Bench-Scale Determination of Thermal Properties and of the Maximum Tolerable Freezing Rate

Question 2: Bench-Scale Determination of Thermal Properties and of the Maximum Tolerable Freezing Rate (20 marks)

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) Thermal conductivity and diffusivity. Thermal conductivity \(k\) is most conveniently measured with a line-heat-source probe: a thin needle containing both a resistance heater and a thermocouple is inserted into a large, uniform sample held at a stable initial temperature. A constant heater power per unit length \(q'\) is applied, and the probe temperature rise is recorded against \(\ln t\). For the ideal infinite-line-source solution, temperature rises linearly with \(\ln t\) once early transients decay, with slope \(q'/(4\pi k)\), so $$k = \frac{q'}{4\pi}\cdot\frac{\ln(t_2/t_1)}{T_2-T_1}$$ read directly off the straight-line portion of the plot. The method is fast (minutes per sample), needs only a small sample volume, and works on pastes, gels and particulate foods that would be awkward to cast into a slab or cylinder for a steady-state method. A steady-state alternative (the guarded or modified Fitch method) sandwiches the sample between a heated and a cooled plate of known area and measures the steady heat flux and the temperature drop across a known sample thickness, giving \(k\) directly from Fourier's law; it is slower and needs a larger, more uniform sample, but avoids any transient-fit uncertainty.

Thermal diffusivity \(\alpha\) is best obtained from an unsteady-state immersion test: a sample of simple, known geometry (sphere, infinite cylinder, or slab) at a uniform initial temperature is suddenly immersed in a well-stirred bath held at a constant surface temperature (a very high surface Biot number, so the surface is effectively "instantly" at the bath temperature). The centre temperature is logged against time and compared with the one-term analytical (or Gurney–Lurie/Heisler chart) solution for unsteady conduction in that geometry; matching the measured centre-temperature ratio at a known time to the chart's dimensionless temperature ratio fixes the Fourier number \(\text{Fo}=\alpha t/L^2\), from which \(\alpha\) follows directly since \(t\) and the characteristic dimension \(L\) are both known. As a cross-check, \(\alpha\) can also be built up from separately measured bulk properties, \(\alpha = k/(\rho c_p)\), using the probe value of \(k\) from part (a), a pycnometer or displacement measurement of density \(\rho\), and a calorimetric \(c_p\) (Question 1's method of mixtures is exactly this kind of measurement).

(b) Maximum tolerable freezing rate. The maximum tolerable freezing rate is found empirically by freezing replicate batches of the product across a deliberately wide range of rates — from slow still-air freezing through blast freezing to rapid cryogenic (liquid-nitrogen) immersion — and correlating the measured freezing rate (or, equivalently, the time spent crossing the −1 to −5°C "zone of maximum ice crystal formation") against a set of measurable post-thaw quality attributes: percentage drip (thaw) loss from gravity or centrifuge drainage of weighed samples, instrumental texture (firmness/shear force on a texture analyzer), colour, and, where practical, microscopy of the mean ice-crystal size and location (intra- versus extracellular). Slow freezing produces large, predominantly extracellular ice crystals that rupture cell membranes and osmotically dehydrate cells, producing high drip loss and soft, mushy texture on thaw; increasing the freezing rate produces progressively smaller, more numerous, more intracellular crystals and better-retained texture. Plotting each quality attribute against freezing rate typically shows steep improvement at low rates that flattens into a plateau — the "maximum tolerable" (i.e. most economical) freezing rate is the rate at the knee of that curve, beyond which further increases in freezing rate cost more (refrigeration duty, cryogen consumption) without a measurable further gain in quality, and, for some structured products (whole fruit, some fish fillets), can even reintroduce a new defect (surface cracking from extreme thermal shock) that sets a practical upper bound on rate as well.