1. (Problem 4.1 DS ) Recall problem 1.34, which concerned an ideal diatomic gas taken around a rectangular cycle on a PV diagram. Suppose now that this system is used as a heat engine, to convert the heat added into mechanical work. (total 10 points) a. Evaluate the efficiency of this engine for the case V,= 3V., P,=2P,. (7 points) b. Calculate the efficiency of an 'ideal engine' operating between the same temperature extremes. (3 points) 2. (Problem 4.3 DS ) A power plant produces 1 GW (G=10: W), at an efficiency of 40% (total 10 points) a. At what rate does the plant expel waste heat into its environment? (2 points) b . Assume first that the cold reservoir for this plant is a river whose flow rate is 100 m:/s. By how much will the temperature of the river increase? Assume this heating happens at constant volume and (specific heat), c, for water is 4190 J/kg K. Density of water is 1000 kg/m: (4 points) To avoid this thermal pollution of the river, the plant could instead be cooled by evaporation of river water. At what rate must the water evaporate? What fraction of the river must be evaporated ? Assume that in the cooling device or the evaporative cold reservoir', a thin layer of water evaporates and thus dissipates the heat rejected to the cold reservoir. This process is somewhat similar to the sense of chillness we experience as we step out after a hot shower. The latent heat of evaporation for water at 1 atm pressure and 100-C, or enthalpy of vaporization is AH, = 22.6*10: J/kg. Latent heat of vaporization is to convert saturated liquid water at 100=C, 1atm pressure to saturated vapor at 100=C, 1atm pressure. (4 points) 3. (Problem 4.5 DS ) (total 10 points ) a. Prove directly (by calculating the heat taken in and the heat expelled) that a Carnot engine using an ideal gas as a working substance has an efficiency of 1-T./T. (6 points) b. Draw the thermodynamic cycle on a PV diagram for an ideal Carnot engine. Clearly show the direction in which the cycle occurs and label each step of the cycle .(4 points) 4. (Problem 4.13 DS) (total 10 points ) a. Under many conditions, the rate at which heat enters an air conditioned building on a hot summer day is proportional to the difference in temperature between inside and outside (T.-T.). Show that, under these conditions the cost of air conditioning should be roughly proportional to the square of the temperature difference. (6 points) b. If the temperature maintained in a room is lowered from 20 deg C to 18 deg C with the temperature of the hot reservoir (typically outside ambient air) is 40 deg C, what would be the impact on the cost of air conditioning? (4 points)
