Question 7 of 8: Batch Mold Growth Kinetics — Specific Growth Rate, Yield, and Scale-Up
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Notes on this paper
National Exams — December 2016 — 04-BS-13, Biology. Three-hour, closed-book exam (one double-sided aid sheet permitted, approved calculator allowed). Format: Part I offers 5 questions (any 3 constitute a complete answer, 20 marks each) and Part II offers 3 questions (any 2 constitute a complete answer, 20 marks each) — a full paper is 5 questions. All 8 numbered questions are solved below for completeness. Q2–Q5 and Q7 are calculation/derivation questions; Q1, Q6, and Q8 are essay questions.
Reference texts: Shuler & Kargi, Bioprocess Engineering: Basic Concepts (2nd ed., Prentice Hall) — elemental/electron balances, yield coefficients, fermenter mass and energy balances, growth kinetics; Madigan et al., Brock Biology of Microorganisms (15th ed., Pearson) — bacterial/eukaryotic cell structure, fungi, protozoa/algae, Gram-stain cell envelope; Toledo, Fundamentals of Food Process Engineering (3rd ed., Springer) — plant tissue structure and cereal grain morphology.
Question 7: Batch Mold Growth Kinetics — Specific Growth Rate, Yield, and Scale-Up (20 marks)
Given. Batch growth data (table below); the last two pages of the source exam are blank regular-grid and semilog graph paper provided for the required plot (no additional data on them).
Time (h)
Cell conc. (g/L)
Glucose conc. (g/L)
0
1.25
100
9
2.45
97
16
5.1
90.4
23
10.5
76.9
30
22
48.1
34
33
20.6
36
37.5
9.38
40
41
0.63
Find. (a) $\mu_{max}$; (b) apparent yield $Y_{xs}$; (c) $X_{max}$ if the initial glucose were 150 g/L (same inoculum).
Approach. Plot $\ln X$ (or $X$ on a log axis) vs. $t$, identify the linear (exponential-phase) window and its slope ($=\mu_{max}$); use the overall cell/substrate change for the apparent yield; then use that yield to scale the substrate-limited maximum cell concentration to a new initial glucose loading.
Figure 4. Semilog plot of cell concentration vs. time. The exponential-phase window (t = 9–34 h, five interior points) regresses to a slope of 0.104 h−1 with $R^2>0.9999$; growth visibly decelerates once glucose is nearly exhausted (t > 34 h).
(a) Maximum net specific growth rate — regress $\ln X$ vs. $t$ over the exponential window. Computing the pairwise slope $\Delta\ln X/\Delta t$ between every consecutive pair of points shows a consistently steep, near-constant slope ($\approx0.101$–$0.106\ \mathrm h^{-1}$) from $t=9$ through $t=34$ h, then a visible fall-off as glucose nears exhaustion (consistent with the glucose concentration dropping from 90 g/L to under 21 g/L over that same window and then to near-zero by t=40 h). A least-squares regression of $\ln X$ on $t$ over that five-point window ($t=9,16,23,30,34$) gives:
$$\boxed{\mu_{max}\approx0.104\ \mathrm h^{-1}}\qquad(R^2=0.99998,\text{ an excellent exponential fit}).$$
(b) Apparent growth yield. Using the overall change in cell and glucose concentration across the whole batch ($t=0$ to $t=40$ h):
$$Y_{xs}=\frac{\Delta X}{\Delta S}=\frac{41-1.25}{100-0.63}=\frac{39.75}{99.37}=\boxed{0.400\ \mathrm{g\ cells/g\ glucose}}.$$
(c) Scale-up to 150 g/L initial glucose. Assuming the same inoculum size ($X_0=1.25$ g/L) and that the apparent yield $Y_{xs}=0.400$ g/g holds at the higher substrate loading (glucose again driven essentially to exhaustion, $S_f\approx0$):
$$X_{max}=X_0+Y_{xs}(S_0-S_f)=1.25+0.400(150-0)=\boxed{61.25\ \mathrm{g/L}}.$$
