Industrial Bioreactor Output Bottlenecks (Chemical Engineering) Solution

Part 1: Degree Comparison & Growth Dominance

The degree of $[S]$ in the numerator is $1$ (leading term $450[S]$ ), and the degree of $[S]$ in the denominator is $1$ (leading term $[S]$ ). Because the powers are equal ( $n = m = 1$ ), this rational function is in the balanced degrees regime, meaning the limit at infinity is dictated by the ratio of the leading coefficients.

Part 2: Analytical Limit Calculation

Divide numerator and denominator by the highest power of $[S]$ in the denominator, which is $[S]^1$ :

\lim_{[S] \to \infty} \frac{450[S]}{18 + [S]} = \lim_{[S] \to \infty} \frac{\frac{450[S]}{[S]}}{\frac{18}{[S]} + \frac{[S]}{[S]}} = \lim_{[S] \to \infty} \frac{450}{\frac{18}{[S]} + 1}

As $[S] \to \infty$ , the term $\frac{18}{[S]} \to 0$ :

\frac{450}{0 + 1} = 450\text{ kg/h}

Part 3: Horizontal Asymptote & Biochemical Interpretation

Horizontal Asymptote: $y = 450$
Physical Interpretation: The bioreactor has a strict upper throughput capacity of $V_{\max} = 450\text{ kg/h}$ . Regardless of how much additional substrate is pumped into the vessel, the enzymatic binding sites become fully saturated, making further feedstock expenditure economically ineffective beyond the asymptotic knee.