JEE Advanced 2013
LEVELJEE Main
Visualized Solution
The Sigma Insight: Heat Transfer
The journey of heat through different materials is a fascinating phenomenon, much like water flowing through pipes or electricity traveling through wires. In this problem, we are tasked with comparing the time it takes for a specific amount of heat to travel through two different arrangements of identical blocks. Let's dive into the physics of thermal conduction and see how the configuration of materials drastically alters the rate of heat transfer.
Analyzing the Setup
We are given two rectangular blocks with identical dimensions—let's say they have a length and a cross-sectional area . One block has a thermal conductivity of , and the other has .
In Configuration I, the blocks are placed end-to-end along the X-axis. This means the heat must flow through the first block and then through the second block. This is a classic series combination.
In Configuration II, the blocks are placed side-by-side (stacked vertically in the diagram) such that heat flows through both of them simultaneously along the X-axis. This represents a parallel combination.
We are told that the temperature difference across the ends is the same in both configurations, and we need to transport the same amount of heat .
The Master Equation
The rate of heat flow through a material is given by Fourier's law of thermal conduction:
where is the equivalent thermal resistance of the setup. The thermal resistance of a single block is defined as:
From our master equation, we can express the time taken to transfer heat as:
Since and are constant for both configurations, we can clearly see that the time taken is directly proportional to the equivalent thermal resistance:
This is a crucial insight! It means that to compare the times, we simply need to compare the thermal resistances of the two configurations.
Resistance of Configuration I (Series)
In a series combination, the equivalent thermal resistance is simply the sum of the individual resistances, just like resistors in an electrical circuit.
Substituting the values for our two blocks:
To add these, we find a common denominator:
This is the total resistance opposing the heat flow in the first setup.
Resistance of Configuration II (Parallel)
In a parallel combination, the reciprocal of the equivalent resistance is the sum of the reciprocals of the individual resistances.
Substituting our values:
Adding these together:
Taking the reciprocal gives us the equivalent resistance for the second setup:
Final Calculation
Now that we have the equivalent resistances for both configurations, we can find the ratio of the times taken. Since , we have:
Substituting the expressions we derived:
The terms beautifully cancel out, leaving us with a simple numerical ratio:
We are given that the time taken in Configuration I is . Plugging this into our ratio:
The heat is transported much faster in the parallel configuration! This makes perfect physical sense. By placing the blocks in parallel, we have effectively increased the total cross-sectional area available for heat flow, providing multiple paths and drastically reducing the overall resistance.
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