117 Physics Statistical Mechanics & Thermodynamics
Ann Esin 8 Nov 2006 Due: 15 Nov 2006
Problem Set 10
Please staple problems 1 & 2 and 3 &4. 1. Stirling Engine Schroeder, 4.21 2. Cooling of Nonmetallic solid to T = 0
We know that the heat capacity of nonmetallic solids at sufficiently low temperatures is proportional to T 3 , as C = aT 3 . Assume that it were possible to cool a piece of such a solid to T = 0 by means of a reversible refrigerator that uses the solid specimen as its (varying!) low-temperature reservoir, and for which the high-temperature reservoir has a fixed temperature Th equal to the initial temperature Ti of the solid. Find an expression for the electrical energy required. 3. Heat Pumps A heat pump is an electrical device that heats a building by pumping heat from the cold outside. In other words, it is the same as a refrigerator, but its purpose is to warm the hot reservoir, rather than to cool the cold reservoir (even though it does both). Let us define the following standard symbols, all taken to be positive by convention: Th Tc Qh Qc W = = = = = temperature building inside temperature outside heat pumped into building heat taken from outdoors electrical energy used by heat pump
(a) Explain why the "coefficient of performance", , for the heat pump should be defined as Qh /W . (b) Derive an equation for in terms of h and c for a reversible heat pump. Explain what happens if the pump is not reversible. (c) Assume that the electricity consumed by a reversible heat pump must be generated by a Carnot engine operating between the temperatures hh and c . What is the ratio Qhh /Qh , of the heat consumed at hh , to the heat delivered at h ? Give numerical values for Thh = 600 K; Th = 300 K; Tc = 270 K. Explain why a heat pump is better than an electrical furnace, which simply converts electrical work directly into heat. (d) Draw an energy-entropy flow diagram for the combination heat engine - heat pump, involving no external work, only energy and entropy flows at three temperatures. 1
4. Second Law of Thermodynamics and Carnot Efficiency Schroeder, 4.16 (Hint: Start by drawing an energy flow diagram for the combined system.)
2