NivaarExam PrepOfficial exam papers ↗

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

Question 5 of 10: Ball Process Time, Come-Up Correction and General-Method \(F_0\)

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 5: Ball Process Time, Come-Up Correction and General-Method \(F_0\) (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.

Ball process data
QuantitySymbolValue
Heating-rate index\(f_h\)5 min
Lag factor\(j\)0.8
Initial temperature\(T_i\)80°F
Retort temperature\(T_{ret}\)250°F
Target lethality\(F_0\)4 min
z-value\(z\)18°F
Come-up time (part b)CUT4 min

Find. (a) process time \(B\); (b) operator's steam-on to steam-off time; (c) delivered \(F_0\) for the missed-process temperature record.

Approach. Parts (a)-(b) apply the standard Stumbo \(f_h/U\)-vs-\(g\) table lookup and Ball's 42% come-up rule; part (c) abandons the formula method (there is no single \(f_h\) for an aberrant record) and instead integrates the lethal-rate curve \(L(t)=10^{(T(t)-250)/z}\) directly over the recorded time-temperature history (the "general method"), using Simpson's rule on the ramp/hold segments described by the chart.

  1. Part (a) — process time. Since the process reference temperature (250°F) equals the retort temperature, \(U=F_0=4\ \text{min}\), so \(f_h/U = 5/4 = 1.25\). Interpolating the Stumbo \(g\)-table at \(z=18^\circ\text{F}\) between \(f_h/U=1.0\) and \(2.0\), then correcting for \(j=0.8\) via \(g_j = g_{j=1}+(j-1)\Delta g/\Delta j\), gives \(g=\boxed{0.812^\circ\text{F}}\). The process time is \(B=f_h\log_{10}\!\left[\dfrac{j(T_{ret}-T_i)}{g}\right]=5\log_{10}\!\left[\dfrac{0.8(250-80)}{0.812}\right]=\boxed{11.1\ \text{min}}\).
  2. Part (b) — steam-on to steam-off time. \(B\) is measured from the moment the retort is AT temperature; Ball's come-up correction credits 42% of the come-up time (CUT) as equivalent full-temperature exposure, so the timer is effectively started \(0.42\,CUT\) before the retort is fully up to temperature, and the total steam-on-to-steam-off time is \(t_{total}=B + (1-0.42)\,CUT = 11.1+0.58(4)=\boxed{13.4\ \text{min}}\) after the steam is first turned on.
  3. Part (c) — segmenting the missed-process record. The chart gives a linear ramp \(70\to210^\circ\text{F}\) over 0–3 min, a flat hold at \(210^\circ\text{F}\) from 3–10 min, an instantaneous jump to \(250^\circ\text{F}\) at \(t=10\), and a flat hold at \(250^\circ\text{F}\) from 10–16 min (steam off).
  4. Part (c) — lethal-rate integral. \(F_0=\int_0^{16} 10^{(T(t)-250)/18}\,dt\). The 0–3 min ramp and the \(210^\circ\text{F}\) hold (3–10 min, where \(L=10^{(210-250)/18}=0.00599\)) contribute only \(F_{pre}=\boxed{0.043\ \text{min}}\) (Simpson's-rule integration; \(L\) is negligible on the ramp and small on the 210°F hold). The 10–16 min hold at the full \(250^\circ\text{F}\) reference gives \(L=1\) exactly, contributing \(F_{post}=6\times1=\boxed{6.000\ \text{min}}\) outright.
  5. Part (c) — total delivered lethality. \(F_0=F_{pre}+F_{post}=0.043+6.000=\boxed{6.04\ \text{min}}\).
Final results
QuantityValue
(a) Process time \(B\)11.1 min
(b) Steam-on to steam-off time13.4 min
(c) Delivered \(F_0\) (general method)6.04 min
Check: part (c) assumes \(z=18^\circ\text{F}\) (the value established for this product in part (a), since the record itself gives no independent z-value) and integrates only through \(t=16\) min (steam off) because the record supplies no cooling-curve data beyond that point — any further lethality picked up during the cool-down before the can drops below the lethal-rate threshold is not counted, which is the standard, conservative practice when a cooling curve is not given.