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 $[S]^1$:

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

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.