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22-Agric-B8 Food Process Engineering (Part 1) · May 2014

Question 4 of 10: Sweet Cherry Freezing Time — Cleland-Earle Method

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 May 2014 — a three-hour open-book exam (any non-communicating calculator permitted). Ten questions are set in four sections (I–IV), each with a "do one/any N of M" instruction; a candidate following the choice rules answers 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, evaporator design — 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); R.P. Singh and D.R. Heldman, Introduction to Food Engineering, 5th ed. (freezing-time estimation, modified Plank and Cleland-Earle equations, 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).

Check: this paper's four roman-numeral section headers ("I. Heat transfer", "II. Food freezing and freeze concentration", "III. Thermal processing", "IV. Several assumptions (retort come-up correction factor, reference temperature for spore D-values, evaporator steam temperature reused for Question 9) are flagged inline where the source leaves a value implicit.

Question 4: Sweet Cherry Freezing Time — Cleland-Earle Method (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.

Cherry freezing data (sphere)
QuantitySymbolValue
Diameter (sphere)\(D\)1.5 cm → \(a=0.75\) cm
Surface coefficient\(h\)50 W/(m²·K)
Initial temperature\(T_i\)5°C
Air temperature\(T_a\)-30°C
Initial freezing point\(T_{fi}\)-2.5°C
Target centre temperature\(T_f\)-15°C
Volumetric enthalpy change\(\Delta H_1\)278 kJ/kg
Frozen conductivity\(k_1\)1.108 W/(m·K)
Unfrozen specific heat\(C_{PU}\)0.22 kJ/(kg·K)
Frozen specific heat\(C_{PI}\)2.05 kJ/(kg·K)
Density\(\rho\)1050 kg/m³

Find. Freezing time to -15°C by the Cleland-Earle method.

Approach. Cleland & Earle (1982) keep Plank's algebraic form but replace the fixed geometry fractions with regression functions of two dimensionless groups — the Plank number \(Pk\) (precooling sensible heat relative to \(\Delta H_1\)) and the Stefan number \(Ste\) (subcooling sensible heat relative to \(\Delta H_1\)) — so the geometry factors \(P,R\) are evaluated first, then substituted into the same Plank-style time equation used in Question 3.

  1. Plank and Stefan numbers. \(Pk=\dfrac{C_{PU}(T_i-T_{fi})}{\Delta H_1}=\dfrac{0.22(5-(-2.5))}{278}=0.00594\); \(Ste=\dfrac{C_{PI}(T_{fi}-T_a)}{\Delta H_1}=\dfrac{2.05((-2.5)-(-30))}{278}=0.2028\).
  2. Cleland-Earle geometry factors for a sphere. With \(E=3\) for a sphere, \(P=\dfrac{1}{2E}\big[1.026+0.5808\,Pk+Ste(0.2296\,Pk+0.1050)\big] = \boxed{0.1752}\) (vs. Plank's constant \(1/6=0.1667\)); \(R=\dfrac{1}{8E}\big[1.202+Ste(3.410\,Pk+0.7336)\big] = \boxed{0.0565}\) (vs. Plank's constant \(1/24=0.0417\)) — both correctly larger than the pure-Plank values, since the extra sensible-heat load increases the effective resistance term.
  3. Freezing time. \(t=\dfrac{\rho\,\Delta H_1}{T_{fi}-T_a}\left[\dfrac{Pa}{h}+\dfrac{Ra^2}{k_1}\right]\) \(=\dfrac{1050\times278{,}000}{-2.5-(-30)}\left[\dfrac{0.1752\times0.0075}{50}+\dfrac{0.0565\times0.0075^2}{1.108}\right] = \boxed{309\ \text{s} = 5.16\ \text{min}}\).
Final results
QuantityValue
\(Pk\), \(Ste\)0.00594, 0.2028
\(P\), \(R\)0.1752, 0.0565
Freezing time \(t\)309 s (5.16 min)
Check: the exam supplies no chart/table for the Cleland-Earle regression coefficients (unlike Fig. 1 for Question 1), so the standard published coefficients (Cleland & Earle, 1982, as reproduced in Toledo's Fundamentals of Food Process Engineering) are used directly; the small ~5-minute answer is physically reasonable given the cherry's small (1.5 cm) size and moderate \(h\).