To tackle this question, we need to understand how heat transfer works in both series and parallel configurations of the cylinders. The rate of heat transfer through a material is influenced by its thermal conductivity, the temperature difference across it, and its geometry. Let's break this down step by step.
Heat Transfer in Series
When the two identical cylinders are connected in series, the total thermal resistance is the sum of the individual resistances. The heat transfer rate, denoted as H₀, can be expressed using Fourier's law of heat conduction:
H₀ = (T_hot - T_cold) / R_total
Here, R_total is the total thermal resistance, which for two identical cylinders in series is:
R_total = R₁ + R₂ = 2R
Where R is the thermal resistance of one cylinder. Therefore, we can rewrite the heat transfer rate as:
H₀ = (T_hot - T_cold) / (2R)
Heat Transfer in Parallel
Now, when the cylinders are connected in parallel, the situation changes significantly. In this configuration, the heat transfer rate through each cylinder adds up. The thermal resistance for each cylinder remains the same, but the total thermal resistance for the parallel arrangement is given by:
1/R_total_parallel = 1/R₁ + 1/R₂ = 1/R + 1/R = 2/R
Thus, the total thermal resistance for the parallel configuration is:
R_total_parallel = R/2
Now, we can express the new heat transfer rate, H_parallel, as:
H_parallel = (T_hot - T_cold) / R_total_parallel = (T_hot - T_cold) / (R/2)
This simplifies to:
H_parallel = 2(T_hot - T_cold) / R
Comparing Heat Transfer Rates
We can now relate H_parallel back to H₀. From our earlier expression for H₀, we have:
H₀ = (T_hot - T_cold) / (2R)
Thus, we can see that:
H_parallel = 2 * (T_hot - T_cold) / R = 4 * (T_hot - T_cold) / (2R) = 4H₀
Final Answer
Therefore, when the cylinders are connected in parallel, the rate of heat transfer increases to 4H₀. The correct answer is (B) 4H₀. This illustrates how the configuration of heat transfer elements can significantly impact the overall efficiency of heat conduction.
