A cubic box is being used as an insulator to store soup and keep it hot on a cold day. The wall thickness of the insulator is halved. The exposed area is decreased to a quarter. What would happen to the rate of heat loss?

Question

BlackTom AI · Accepted Answer

Fourier’s law for steady-state conduction gives the heat-loss rate Q̇ = k A ΔT / L, where k is thermal conductivity, A is exposed area, ΔT is temperature difference, and L is wall thickness.

1) Relevant knowledge points and concepts
- Heat transfer by conduction through a flat wall: Q̇ ∝ A/L (for fixed k and ΔT).
- If A or L change, Q̇ changes in proportion to A and inversely to L.
- The problem states the same material and temperature difference, so k and ΔT are constant.

2) Complete reasoning process step by step
- Let initial area be A1 and initial thickness be L1. Initial heat-loss rate: Q̇1 = k A1 ΔT / L1.
- New area A2 = A1/4 (exposed area decreased to a quarter).
- New thickness L2 = L1/2 (thickness halved).
- New heat-loss rate: Q̇2 = k A2 ΔT / L2.

3) Calculations with intermediate steps
- Substitute A2 and L2: Q̇2 = k (A1/4) ΔT / (L1/2).
- Simplify denominator: divide by (L1/2) equals multiply by 2/L1, so Q̇2 = k A1 ΔT * (1/4) * (2/L1).
- Multiply factors: (1/4)*(2) = 2/4 = 1/2.
- Therefore Q̇2 = (1/2) * [k A1 ΔT / L1] = (1/2) Q̇1.

4) Why the correct answer is correct
- Algebra shows Q̇ decreases by a factor of 1/2 because the area reduction (×1/4) and thickness reduction (÷1/2) combine to give an overall factor (1/4)/(1/2) = 1/2 on the original heat-loss rate.

5) Why the other options are incorrect
- (a) Doubled: would require net factor ×2, but calculation gives ×1/2, so incorrect.
- (b) No change: would require the area and thickness changes to cancel exactly, they do not; net effect is 1/2.
- (c) Increase by factor of six: wrong sign and magnitude; the rate actually decreases, not increases.

6) Conclusion confirming final answer
- The rate of heat loss is halved, so the correct choice is d. The rate of heat loss will be halved.

A cubic box is being used as an insulator to store soup and keep it hot on a cold day. The wall thickness of the insulator is halved. The exposed area is decreased to a quarter. What would happen to the rate of heat loss?单项选择题

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