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Midterm_F09_Solution

Midterm_F09_Solution - MAE118A Introduction to Energy...

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MAE118A: Introduction to Energy Systems Midterm Exam Fall 2009 Kowsik Bodi November 6, 2009 1. The rate of consumption of useful work at t = 0 is given as P 0 . This work is provided by converting a primary energy source with an efficiency η 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, Q 0 , available in this primary energy resource which can be accessed, find how long the primary energy resource will last. 25 pts. Total energy reserve is at Q 0 . Conversion efficiency of this resource for use is η 0 . So total available energy is η 0 Q 0 . Rate of consumption of energy is the Power demand. At t = 0, power demand is P 0 . If the fractional increase in power demand is r , 1 P d P d t = r = P ( t ) = P 0 e rt (1) The total amount of consumed resources, at a given time τ , is the sum of consumption from t = 0 till the year t = τ , Q ( τ ) = t = τ integraldisplay t =0 d t P ( t ) = t = τ integraldisplay t =0 d t P 0 e rt = P 0 r [ e - 1] (2) The resource is exhausted at a time τ = T if all available energy has been consumed: Q ( T ) = η 0 Q 0 , η 0 Q 0 = P 0 r bracketleftbig e rT - 1 bracketrightbig T = 1 r bracketleftbigg ln parenleftbigg 0 Q 0 P 0 + 1 parenrightbiggbracketrightbigg (3) 1
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2. Consider a heat engine that functions by first compressing a gas from an initial state 1 at pressure p 1 to a state 2 with pressure p 2 > p 1 via a reversible adiabatic compression. This gas then has a quantity of heat, q h , 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 p 4 = p 1 . The gas at state 4 is then cooled down back to the initial temperature T 1 by rejecting a quantity of heat q l to a cold reservoir. A quantity of work, w c , obtained from the expansion stage is used to operate the compressor. The remainder of the work, w , is then available for useful purposes. 25 pts. 1 2: Adiabatic, reversible (isentropic) compression. s 2 = s 1 , p 2 > p 1 , c p r ( T 2 - T 1 ) = h 2 - h 1 = w c , where c p r is the specific heat at const. pressure of the reactants, and w c is the work done by the compressor. 2 3: Heat addition due to ideal (const. pressure) combustion. p 3 = p 2 , c p pr ( T 3 - T 2 ) = h 3 - h 2 = q h , where c p pr is the specific heat at const.
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