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22-Agric-B8 Food Process Engineering (Part 1) · December 2013

Question 7 of 10: Freezing Time in Liquid Nitrogen — Surface Heat Transfer Coefficient

Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)

Notes on this paper

Paper format. 04-Agric-B8 Food Process Engineering (Part 1), National Exams December 2013 — a three-hour open-book exam (any non-communicating calculator permitted). Ten questions are set in four sections (I–IV), each with a "choose N of M" instruction; candidates who follow the choice rule answer six questions for a 100-mark paper. All ten are worked here so the set is a complete study resource.

Reference texts. R.T. Toledo, Fundamentals of Food Process Engineering, 3rd ed. (thermal-process lethality, D and z values, Ball/Stumbo process calculation, aseptic holding-tube residence time — this is the exam's own appendix source); C.J. Geankoplis, Transport Processes and Separation Process Principles, 4th ed. (evaporator heat and mass balances, multiple-effect steam economy, vapour recompression); R.P. Singh and D.R. Heldman, Introduction to Food Engineering, 5th ed. (freezing-time estimation, modified Plank equation, unsteady-state heat transfer in canned foods); A.C. Cleland, Food Refrigeration Processes: Analysis, Design and Simulation (Plank/Cleland-Earle freezing-time correlations); F.P. Incropera and D.P. DeWitt, Fundamentals of Heat and Mass Transfer (transient conduction, Heisler charts, composite-wall resistance).

Question 7: Freezing Time in Liquid Nitrogen — Surface Heat Transfer Coefficient (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.

LN2 immersion freezing data
QuantitySymbolValue
Cake diameter / thickness—0.23 m / 0.04 m
Freezing time\(t\)1.7 min = 102 s
Product mass\(m\)0.372 kg
LN2 usage rate—0.665 kg N₂/kg product
Temperatures\(T_i,\,T_f,\,T_{final}\)22°C \(\to\) -2°C (freezing pt.) \(\to\) -18°C
LN2 medium temperature\(T_m\)-196°C
LN2 latent / specific heat\(L_{N_2},\,c_{p,N_2}\)197.98 kJ/kg, 1.044 kJ/(kg·K)
Frozen-cake conductivity\(k\)1.731 W/(m·K)

Find. The surface convective heat transfer coefficient, \(h\).

Liquid N2 bath, T_m = -196 degCPecan coffee-cake(infinite slab)a = 0.04 m thickq (h, surface)q (h, surface)22 degC -> -18 degC through T_f = -2 degC
Immersion freezing modelled as an infinite slab, cooled from both faces.

Approach. Get the total heat removed per kg of product from an energy balance on the nitrogen itself (it boils off at -196°C, then the cold gas warms toward ambient, absorbing sensible heat too); back out the cake's bulk density from its mass and geometry; solve the modified Plank equation (infinite slab, \(P=\tfrac12\), \(R=\tfrac18\), with \(\Delta H\) replacing the pure latent heat to also cover the pre- and sub-cooling sensible loads) for \(h\).

  1. Cake density. \(V=\dfrac{\pi}{4}(0.23)^2(0.04)=1.662\times10^{-3}\ \text{m}^3\); \(\rho = 0.372/1.662\times10^{-3}=\boxed{223.8\ \text{kg/m}^3}\) (a light, aerated baked good).
  2. Total heat removed per kg product, from the nitrogen side. Each kg of N2 absorbs its latent heat at -196°C, then the resulting cold gas is credited with warming back to the 22°C ambient (a standard LN2 mass-balance shortcut, since the vented gas ultimately equilibrates with the room): \(\Delta h_{N_2} = L_{N_2}+c_{p,N_2}(T_{amb}-T_{N_2}) =197.98+1.044(22-(-196))=\boxed{425.6\ \text{kJ/kg N}_2}\). Per kg product: \(\Delta H = 0.665\times425.6=\boxed{283.0\ \text{kJ/kg product}}\).
  3. Modified Plank equation, solved for \(h\). With \(a=0.04\) m (full slab thickness), \(P=\tfrac12\), \(R=\tfrac18\), \(T_f-T_m=-2-(-196)=194\ \text{K}\): \(t=\dfrac{\rho\Delta H\,a}{T_f-T_m}\left(\dfrac{P}{h}+\dfrac{Ra}{k}\right)\). The conduction term alone accounts for \(37.7\ \text{s}\) of the 102 s total, leaving the surface term to supply the rest; solving, \(\boxed{h \approx 101.6\ \text{W/(m}^2\text{K)}}\).
Final results
QuantityValue
Cake bulk density223.8 kg/m³
Total heat removed per kg product283.0 kJ/kg
Surface heat transfer coefficient, \(h\)≈ 101.6 W/(m²·K)
Check: crediting the vented nitrogen gas with warming fully to ambient (22°C) before leaving the freezing tunnel is the standard textbook shortcut for this class of problem; a real tunnel recovers less of that sensible enthalpy, which would lower \(\Delta H\) per kg product and hence lower the back-calculated \(h\) somewhat. The resulting \(h\approx 100\ \text{W/(m}^2\text{K)}\) sits in the expected range for LN2 immersion/spray freezing.