A steel sphere of radius 0.1 m at 400°K is immersed in an oil at 300°K. If the centre of the sphere reaches 350°K in 20 minutes, how long will it take for a 0.05 m radius steel sphere to reach the same temperature (at the centre) under identical conditions ? Assume that the conductive heat transfer co-efficient is infinitely large.

ANS:A - 5 minutes

To solve this problem, we can use Newton's Law of Cooling, which states that the rate of change of temperature of an object is proportional to the difference between its temperature and the surrounding temperature. The equation for Newton's Law of Cooling is: dtdT​=−k⋅(T−Tambient​) where:

• dtdT​ is the rate of change of temperature with respect to time,
• k is the constant of proportionality (heat transfer coefficient),
• T is the temperature of the object,
• ambientTambient​ is the temperature of the surroundings.

Given that the conductive heat transfer coefficient is infinitely large, k is effectively infinite. This means that the rate of change of temperature with respect to time is solely dependent on the temperature difference between the object and its surroundings. Let's denote:

• T1​ as the initial temperature of the sphere (400 K),
• T2​ as the final temperature of the sphere (350 K),
• ambientTambient​ as the temperature of the surrounding oil (300 K).

For the given scenario, we can set up the following equation: tdT​=−k⋅(T−Tambient​) tdT​=−k⋅(T−300) −300=−T−300dT​=−k⋅dt Now, we can integrate both sides of the equation: ∫T1​T2​​T−3001​dT=−∫0t​kdt ∣T2​−300∣−ln∣T1​−300∣=−kt ln(T1​−300T2​−300​)=−kt Since we want to find the time it takes for the temperature to decrease from T1​ to T2​, we can rearrange the equation to solve for t: t=−k1​⋅ln(T1​−300T2​−300​) Now, we'll use this equation to find the time it takes for a steel sphere with a radius of 0.05 m to reach the same temperature at the center under identical conditions: ln⁡(350−300400−300)t=−k1​⋅ln(400−300350−300​) t=−k1​⋅ln(10050​) t=−k1​⋅ln(0.5) Since k is infinitely large, k1​ tends to zero. Therefore, the time t becomes effectively zero. Hence, the correct answer is not 5 minutes. It would take a negligible amount of time for the smaller sphere to reach the same temperature under the given conditions.