Air-blast freezing time of a partially frozen ice-cream brick: 22-Agric-B8 Food Process Engineering (Part 1), May 2017, Q4

Question 4: Air-blast freezing time of a partially frozen ice-cream brick (15 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.

Given. A rectangular ice-cream brick, already at its freezing point (-5°C, "partially frozen"), is blast-frozen in -25°C air to a target centre temperature of -18°C.

Given data
Quantity	Symbol	Value
Package dimensions	—	8 × 10 × 20 cm
Surface coefficient	h	50 W/(m²·K)
Product temperature in package	Tᵢ	−5°C
Air (freezing medium) temperature	Tₔ	−25°C
Density	ρ	700 kg/m³
Frozen thermal conductivity	k	1.2 W/(m·K)
Frozen specific heat	Cᶭᵩ	1.9 kJ/(kg·K)
Latent heat to remove	ΔHᶱᵏᵏ	100 kJ/kg
Final target temperature	Tƒᵣℼₕ	−18°C

Find. The blast-freezing time.

Rectangular ice-cream brick; heat is removed through all six faces, but the shortest dimension (8 cm) sets the fastest conduction path and dominates the freezing time.

Approach. Modified Plank's equation: total modified enthalpy change per kg (stated latent heat plus the frozen-phase sensible heat from −5°C down to −18°C) divided by the temperature driving force, times the classical Plank slab geometric factors (P=½, R=⅛) applied to the shortest (governing) dimension. The exam explicitly allows Levy's chart, Cleland's, or Pham's method with "any equation, assume any unknown" — the slab form is used here rather than reading the P–R chart for a brick, since it needs no additional graphical interpolation and is explicitly sanctioned by the question.

Check: Two engineering assumptions are made explicit here. (1) Since the source states the product is already "partially frozen" and specifies its starting temperature (−5°C) rather than a separate initial freezing point, this problem is solved as a freezing stage only (no above-freezing sensible-heat term is added — Cᶭᶨ is not needed for this sub-method). (2) The brick's heat loss is approximated through its shortest (8 cm) dimension only, using Plank's classical infinite-slab shape constants P=1/2, R=1/8, rather than the exam's page-9 P–R chart for a finite brick (β₁=10/8=1.25, β₂=20/8=2.5) — the slab approximation is a standard, explicitly-permitted fallback for a brick whose two larger faces (10×20 cm) are still much bigger than its 8 cm thickness.

Modified latent heat. $$\Delta H_1 = \Delta H_{lat} + C_{pf}(T_i-T_{final}) = 100 + 1.9(-5-(-18)) = 100+1.9(13) = \boxed{124.7\ \text{kJ/kg}}$$
Driving temperature difference and characteristic dimension. $$\Delta T = T_i - T_a = -5-(-25) = 20\ ^\circ\text{C}, \qquad a = 0.08\ \text{m (shortest dimension)}$$
Plank's slab equation. $$t=\frac{\rho\,\Delta H_1}{\Delta T}\left[\frac{Pa}{h}+\frac{Ra^2}{k}\right]=\frac{(700)(124\,700)}{20}\left[\frac{(0.5)(0.08)}{50}+\frac{(0.125)(0.08)^2}{1.2}\right]$$ $$=4\,364\,500\times[0.000800+0.000667]=4\,364\,500\times0.001467=\boxed{6401\ \text{s}\approx 1.78\ \text{h}}$$

Final results
Quantity	Value
Modified enthalpy, ΔH₁	124.7 kJ/kg
Estimated freezing time	6.4×10³ s ≈ 1.78 h

Topic: Modified Plank's equation for freezing-time estimation.

Key relations: t=(ρΔH₁/ΔT)[Pa/h+Ra²/k]; slab constants P=½, R=⅛; modified enthalpy ΔH₁ folds sensible heat either side of freezing into the latent-heat term so a single-pass calculation applies.

Why this works: Plank's original derivation assumes freezing proceeds as a sharp phase front; the "modified" enthalpy trick approximates the more realistic gradual freezing (mushy zone) of a real food by lumping the extra sensible heat removed below the initial freezing point into an equivalent single "latent heat," which experiment shows tracks measured freezing times much better than the unmodified equation.

Common pitfall: using the diameter/longest dimension instead of the shortest dimension as "a" — heat escapes fastest along the shortest path, so that is always the dimension that controls the freezing time for a brick or slab.

Source: Cleland, Food Refrigeration Processes, modified Plank equation and shape factors; Singh & Heldman, Introduction to Food Engineering, Ch. on freezing-time estimation.

Question 4 of 10: Air-blast freezing time of a partially frozen ice-cream brick

Question 4: Air-blast freezing time of a partially frozen ice-cream brick (15 marks)