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Unformatted text preview: 1. The rate of consumption of useful work at t=0 is given as P0. This work is provided by converting a primary energy source with an efficiency 0. The power demand, P, is growing at a fixed rate r per year and the conversion efficiency is fixed. If at t=0 there is a fixed quantity of energy, Q0, available in this primary energy resource which can be accessed, find how long the primary energy resource will last. 2. Consider a heat engine that functions by first compressing a gas from an initial state 1 at pressure p1 to a state 2 with pressure p2>p1 via a reversible adiabatic compression. This gas then has a quantity of heat, qh, added to it via an ideal combustion process. Let this state be denoted as state 3. The gas is then goes through a reversible adiabatic expansion leaving it at state 4. This expansion leaves p4=p1. The gas at state 4 is then cooled down back to the initial temperature T1 by rejecting a quantity of heat ql to a cold reservoir. A quantity of work, wc, obtained from the expansion stage is used to operate the compressor. The remainder of the work, w, is then available for useful purposes. a. Draw the T‐s diagram for this heat engine, labeling states 1 through 4, and noting where the heat exchange events qh and ql occur and where the work w is extracted. b. Derive (do not simply write down) the ideal thermal efficiency of this cycle in terms of the pressure ratio p2/p1. c. What real thermodynamic cycle does this configuration correspond to? 3. Derive (do not simply write down) the Carnot efficiency using the following procedure. a. Draw a schematic of an ideal heat engine showing the hot and cold reservoirs, the heat, qh, extracted from the hot reservoir, the heat rejected to the cold reservoir, qc, and the work, w, done by the engine. b. Apply the first law of thermodynamics to find w in terms of qh and qc. c. Let us assume that the heat exchange processes are ideal reversible processes. Draw the T‐s diagram for the working fluid of this engine, making sure to label where the heat qh and qc is exchanged and where the work is extracted. d. Derive the thermal efficiency of this engine using the definition that relates the change in entropy to the heat exchange and temperature in a reversible process. e. Explain why the Carnot efficiency is higher than that found in problem 2 in terms of the working fluid temperature during the heat exchange processes (Hint: compare the closed curves in the two T‐s diagrams). UCSD Department of Mechanical and Aerospace Engineering MAE 118A MIDTERM Professor G.R. Tynan CLOSED BOOK CLOSED NOTES. NO CALCULATORS OR OTHER ELECTRONIC DEVICES PERMITTED. ALL QUESTIONS ARE EQUALLY WEIGHTED. 4. You are given the job of evaluating the carbon emissions that will result from producing a given amount of power, P, using a fossil fuel powered heat engine. You are asked to evaluate two different power plant choices: a steam‐based Rankine cycle with an efficiency R, and a natural gas‐fired combined cycle power plant that has a gas turbine with efficiency T, and a steam‐based cycle with efficiency R that is identical to that of the pure Rankine cycle choice. The fuel heating value of coal qc and the fuel heating value of natural gas be given as qg. For simplicity assume that qc=qg/2. By what factor is CO2 emission reduced if you choose the combined cycle system to produce the power demand P ? ...
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- Fall '09