Hot Pocket - Thermodynamics and Convection · USAPhO 2021 #B2
Physics · thermodynamics
Problem
This question consists of two independent parts.
a. It's winter and you want to keep warm. The temperature is T₀ = 263 K outside and T₁ = 290 K in your room. You have started a fire, which acts as a hot reservoir at temperature T₂ = 1800 K. You want to add a small amount of heat dQ₁ to your room. The simplest method would be to extract heat −dQ₂,dump = dQ₁ from the fire and directly transfer it to your room. However, it is possible to heat your room more efficiently. Suppose that you can exchange heat between any pair of reservoirs. You cannot use any external source of work, such as the electrical grid, but the work extracted from running heat engines can be stored and used without dissipation.
i. What is the minimum heat extraction −dQ₂,min required by the laws of thermodynamics to heat up the room by dQ₁?
ii. Let the "efficiency gain" be the ratio G = dQ₂,dump/dQ₂,min. Assuming T₁ is fixed at 298 K, make a graph whose axes are T₀ and T₂, where T₀ varies from 230 K to 290 K, and T₂ varies from 300 K to 2000 K. On the graph, sketch curves corresponding to gain G = 2, 5, and 12.
b. When the air at the bottom of a container is heated, it becomes less dense than the surrounding air and rises. Simultaneously, cooler air falls downward. This process of net upward heat transfer is known as convection.
Consider a closed, rectangular box of height h filled with air initially of uniform temperature T₀. Next, suppose the bottom of the box is heated so that the air there instantly reaches temperature T₀ + ΔT. The hot parcel of air at the bottom rises upward until it hits the top of the box, where its temperature is instantly reduced to T₀.
You may neglect any heat transfer and friction between the parcel of air and the surrounding air, and assume that the temperature difference is not too large. In addition, you may assume the height h is small enough so that the pressure P₀ and density ρ₀ of the surrounding air are very nearly constant throughout the container. More precisely, assume that ρ₀gh/P₀ << ΔT/T₀ << 1. Express your answers in terms of P₀, g, h, ΔT, and T₀.
i. As a parcel of air moves upward, it accelerates. Find a rough estimate for the average speed v₀ during its upward motion.
ii. In the steady state, warm parcels of air are continuously moving upward from the bottom, and cold parcels of air are continuously moving downward from the top. Find a rough estimate for the net rate of upward energy transfer per area.