Process Deviation — Stumbo g-Table Procedure: 22-Agric-B8 Food Process Engineering (Part 1), December 2013, Q2

Question 2: Process Deviation — Stumbo g-Table Procedure (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.

Design vs. actual process conditions
Quantity	Design (as calculated)	Actual (as used/true)
Retort temperature	250°F (121°C)	248°F (120°C)
Organism z-value	18°F assumed	14°F actual
\(f_h\)	30 min
\(j_c=j_h\)	1.07
Initial temperature \(T_i\)	150°F
Design \(F_0\)	6 min	—
\(D_{121^\circ C}\) (organism)	1.2 min
Initial spore load, \(N_0\)	56/can

Find. The probability of spoilage actually delivered, given the true z-value and the retort's true operating temperature.

Approach. Two-step Ball/Stumbo procedure: (1) back out the physical process time that the design step (z = 18°F, retort 250°F) actually specified, using the \(f_h/U\) vs. \(g\) table; (2) re-evaluate that same physical time under the true retort temperature (248°F) and true z-value (14°F) to get the delivered lethality, then convert to spoilage probability with the organism's own \(D_{121}\).

Step 1 — recover the process time from the design calculation. Because the design retort (250°F) equals the reference temperature, \(U = F_0\,10^{(250-250)/18}=F_0=6\ \text{min}\), so \(f_h/U = 30/6 = 5.00\). Reading the \(z=18^\circ\text{F}\) column of the g-table at \(f_h/U=5.0\) gives \(g_{j=1}=5.40^\circ\text{F}\), \(\Delta g/\Delta j = 1.59\); correcting for \(j_c=1.07\): \(g = 5.40+(0.07)(1.59)=\boxed{5.511^\circ\text{F}}\). The Ball heating-curve relation \((T_{ret}-T_i)\,j_c\,10^{-t/f_h}=g\) then gives the actual heating time used: \(t = f_h\log_{10}\!\left[\dfrac{j_c(T_{ret}-T_i)}{g}\right] = 30\log_{10}\!\left[\dfrac{1.07(250-150)}{5.511}\right] = \boxed{38.64\ \text{min}}\).
Step 2 — re-evaluate that time at the true retort temperature. With the same \(t\), \(f_h\), \(j_c\), \(T_i\), but \(T_{ret}=248^\circ\text{F}\): \(g_{actual} = j_c(T_{ret}-T_i)\,10^{-t/f_h} = 1.07(248-150)\times10^{-38.64/30} = \boxed{5.401^\circ\text{F}}\).
Step 3 — back into the g-table at the true z (14°F). Interpolating the \(z=14^\circ\text{F}\) column (with the \(j=1.07\) correction applied at each trial \(f_h/U\)) for \(g=5.401\) gives \(f_h/U \approx 7.21\), so \(U_{actual}=f_h/7.21 = \boxed{4.161\ \text{min}}\).
Step 4 — convert to \(F_0\)-equivalent lethality and spoilage probability. \(F_{actual} = U_{actual}\,10^{(T_{ret}-250)/z}=4.161\times10^{(248-250)/14}=\boxed{2.995\ \text{min}}\) (equivalent minutes at 250°F on the organism's own \(z=14^\circ\text{F}\) basis). Log-reduction \(= F_{actual}/D_{121} = 2.995/1.2 = 2.496\), so \(N = 56\times10^{-2.496} = \boxed{0.179\ \text{spores/can}}\).

Final results
Quantity	Value
Recovered physical process time	38.64 min
Delivered lethality, \(F_{actual}\)	2.995 min (\(z=14^\circ\text{F}\) basis)
Probability of spoilage	\(\approx 0.179\) (17.9%)

Check: the reduced retort temperature (248 vs. 250°F) barely changes \(g\), but the reduced z-value materially raises \(f_h/U\) and hence lowers the delivered lethality — the process, though only 1°F cooler, was designed for a spore population that is more heat-resistant at high temperature (smaller z) than assumed, so it under-processes relative to the original 6-min \(F_0\) target (2.995 < 6 min delivered on a like-for-like z basis).

Concept & Theory

Topic: Ball/Stumbo process-deviation evaluation using the \(f_h/U\) vs. \(g\) table.

Key relations: \(U=F_i\,10^{(250-T_{ret})/z}\); heating curve \((T_{ret}-T)/(T_{ret}-T_i)=j_c\,10^{-t/f_h}\); \(g_j=g_{j=1}+(j-1)\Delta g/\Delta j\).

Why this works: \(g\) (the retort-to-product-centre temperature difference at the end of heating) is a bridge between the physical heating curve (governed by \(f_h\), \(j_c\), \(T_i\)) and the process's delivered lethality (governed by \(U\), hence \(F_0\)) — holding the physical process time fixed while swapping in the corrected z and retort temperature isolates exactly what changes when the original design assumptions turn out to be wrong.

Common pitfall: assuming the "same process" means the same \(F_0\); it means the same physical time in the retort, whose delivered \(F_0\) must be re-derived because \(g\), \(U\) and \(F\) all depend on the z-value and retort temperature actually in play.

Source: Toledo, Fundamentals of Food Process Engineering, Ch. 9 (Ball/Stumbo process calculation, process deviations); Stumbo, Thermobacteriology in Food Processing, 2nd ed. (source of the g-table reproduced on the exam).

Question 2 of 10: Process Deviation — Stumbo g-Table Procedure

Question 2: Process Deviation — Stumbo g-Table Procedure (15 marks)