GATE BT 2026 • Q54 NAT • 2 Marks

Section 2: General Biotechnology

Biomolecules, enzyme kinetics

Subtopic: Fundamentals of Biomolecules, enzyme kinetics

Question Statement:

A heat exchanger during operation in a bioprocess has a steady temperature of 90 oC. After completion of its operation, it was shut down and it was observed that the rate of decrease of temperature at any time was directly proportional to the difference T(t) - 30 oC, where T(t) denotes temperature at time t. It was observed that it took 30 min for the temperature to drop to 70 oC. The temperature after 51.5 min will be_________ oC. (rounded off to the nearest integer)

Official Answer Key:

59 to 61

Step-by-Step Solution & Explanation

### **Step-by-Step Solution & Mathematical Derivation (GATE BT 2026 Q.54)** **1. Governing Concept & Definition**: - **Topic**: Heat Transfer & Newton's Law of Cooling - Newton's law of cooling states: $\frac{dT}{dt} = -k(T - T_\infty) \implies \ln\left(\frac{T(t) - T_\infty}{T(0) - T_\infty}\right) = -k t$ **2. Detailed Step-by-Step Calculation**: - **Given**: - Initial temperature $T(0) = 90^\circ\text{C}$ - Surrounding temperature $T_\infty = 30^\circ\text{C}$ - Temperature difference at $t=0$: $\Delta T(0) = 90 - 30 = 60^\circ\text{C}$ - At $t = 30\text{ min}$: $T(30) = 70^\circ\text{C} \implies \Delta T(30) = 70 - 30 = 40^\circ\text{C}$ - **Step 1: Determine cooling rate constant $k$**: $\frac{\Delta T(30)}{\Delta T(0)} = \frac{40}{60} = \frac{2}{3} \implies e^{-30k} = \frac{2}{3} \implies k = \frac{\ln(1.5)}{30} \approx \frac{0.405465}{30} \approx 0.0135155\text{ min}^{-1}$ - **Step 2: Find temperature at $t = 51.5\text{ min}$**: $\ln\left(\frac{\Delta T(51.5)}{60}\right) = -0.0135155 \times 51.5 \approx -0.69605$ $\frac{\Delta T(51.5)}{60} = e^{-0.69605} \approx 0.49855$ $\Delta T(51.5) \approx 60 \times 0.49855 \approx 29.91^\circ\text{C}$ $T(51.5) = 30 + 29.91 \approx 59.91^\circ\text{C} \approx 60^\circ\text{C}$ **3. Final Answer**: - **Calculated Value**: **60°C** - **Official GATE Accepted Range**: **59 to 61**

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