(a) Sketch a Carnot cycle of a gas on a T-S diagram. Show the direction of the state change along the cycle such that the engine can actually do some useful work. (b) Calculate the efficiency of the cycle shown in Fig. la and compare to the efficiency of an ideal Carnot cycle working between T and T. Show the direction of the cycle (clockwise or anti-clockwise) such that the engine can be used to lift a weight. Explain your choice. (c) Consider the cycle in Fig. 1b. What cycle direction would yield useful work? Calculate the efficiency of the engine in Fig. 1b. Which heat engine is more efficient, a) or b)? a) b) S2 S2 4. S1 T1 T2 T1 T2 Figure 1: T-S cycles of two heat engines

(a) Sketch a Carnot cycle of a gas on a T-S diagram. Show the direction of the state change along the cycle such that the engine can actually do some useful work. (b) Calculate the efficiency of the cycle shown in Fig. la and compare to the efficiency of an ideal Carnot cycle working between T and T. Show the direction of the cycle (clockwise or anti-clockwise) such that the engine can be used to lift a weight. Explain your choice. (c) Consider the cycle in Fig. 1b. What cycle direction would yield useful work? Calculate the efficiency of the engine in Fig. 1b. Which heat engine is more efficient, a) or b)? a) b) S2 S2 4. S1 T1 T2 T1 T2 Figure 1: T-S cycles of two heat engines

University Physics Volume 2

18th Edition

ISBN:9781938168161

Author:OpenStax

Publisher:OpenStax

Chapter4: The Second Law Of Thermodynamics

Section: Chapter Questions

Problem 48P: For the Carnot cycle of Figure 4.12, what is the entropy change of the hot reservoir, the cold...

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The gasoline internal combustion engine operates in a cycle consisting of six parts. Four of these parts involve, among other things, friction, heat exchange through finite temperature differences, and accelerations of the piston; it is irreversible. Nevertheless, it is represented by the ideal reversible Otto cycle, which is illustrated below. The working substance of the cycle is assumed to be air. The six steps of the Otto cycle ale as follows: i. Isobaric intake stroke (OA). A mixture of gasoline and air is drawn into the combustion chamber at atmospheric pressure P0 as the piston expands, increasing the volume of the cylinder from zero to VA . ii. Adiabatic compression stroke (AB). The temperature of the mixture rises as the piston compresses it adiabatically from a volume VA to VB . iii. Ignition at constant volume (BC). The mixture is ignited by a spark. The combustion happens so fast that there is essentially no motion of the piston. During this process, the added heat Q1 causes the pressure to increase from pB to pc at the constant volume VB(=Vc) . iv. Adiabatic expansion (CD). The heated mixture of gasoline and air expands against the piston, increasing the volume from VC to VD . This is called the power stroke, as it is the part of the cycle that delivers most of the power to the crankshaft. v. Constant-volume exhaust (DA). When the exhaust valve opens, some of the combustion products escape. There is almost no movement of the piston during this part of the cycle, so the volume remains constant at VA(=VD) . Most of the available energy is lost here, as represented by the heat exhaust Q2 . vi. Isobaric compression (AO). The exhaust valve remains open, and the compression from VA to zero drives out the remaining combustion products. (a). Using (i)e=W/Q1; (ii)w=Q1Q2; and (iii)Q1=nCv(TCTB),Q2=nCv(TDTA), Show that e=1TDTATCTB. (b). Use the fact that steps (ii) and (iv) are adiabatic to show that e=11r1 where r=VA/VB . The quantity r is called the compression ratio of the engine. (c) In practice, r is kept less than around 7. For larger values, the gasoline-air mixture is compressed to temperatures so high that it explodes before the finely timed spark is delivered. This preignition causes engine knock and loss of power. Show that for r=6 and =1.4 (the value for air), e=0.51 , or an efficiency of 51%. Because of the many irreversible processes, an actual internal combustion engine has an efficiency much less than this ideal value. A typical efficiency for a tuned engine is about 25% to 30%.
A Carnot engine operates between 550 and 20 baths and produces 300 kJ of energy in each cycle. Find the change in entropy of the (a) hot bath and (b) cold bath, in each Carnot cycle?
For the Carnot cycle of Figure 4.12, what is the entropy change of the hot reservoir, the cold reservoir, and the universe? Figure 4.11 The four processes of the Carnot cycle. The working substance is assumed to be an ideal gas whose thermodynamic path MNOP is represented in Figure 4.12. Figure 4.12 The total work done by the gas in the Carnot cycle is shown and given by the area enclosed by the loop MNOPM.
The Carnot cycle is represented by the temperature-entropy diagram shown below. (a) How much heat is absorbed per cycle at the high-temperature reservoir? (b) How much heat is exhausted per cycle at the low-temperature reservoir? (c) How much work is done per cycle by the engine? (d) What is the efficiency of the engine?
A diatomic ideal gas is brought from an initial equilibrium state at p1=0.50 atm and T1=300K to a final stage with p2=0.20 atm and T1=500K. Use the results of the previous problem to determine the entropy change per mole of the gas.
Check Your Understanding A quantity of heat Q is absorbed from a reservoir at a temperature Th by a cooler reservoir at a temperature Tc . What is the entropy change of the hot reservoir, the cold reservoir, and the universe?
Two hundred joules of heat are removed from a heat at a temperature of 200 K. What is the entropy change of the reservoir?
A Carnot engine is used to measure the temperature of a heat reservoir. The engine operates between the heat reservoir and a reservoir consisting of water at its triple point. (a) If 400 J per cycle are removed from the heat reservoir while 200 J per cycle are deposited in the triple-point reservoir, what is the temperature of the heat reservoir? (b) If 400 J per cycle are removed from the triple-point reservoir while 200 J per cycle are deposited in the heat reservoir, what is the temperature of the heat reservoir?
One mole of an ideal monatomic gas is confined to a rigid container. When heat is added reversibly to die gas, its temperature changes from T1 to T2 . (a) How much heat is added? (b) What is the change in entropy of the gas?
Does the entropy increase for a Carnot engine for each cycle?
Two hundred grams of water at 0 is brought into contact with a heat reservoir at 80 . After thermal equilibrium is reached, what is the temperature of the water? Of the reservoir? How much heat has been transferred in the process? What is the entropy change of the water? Of the reservoir? What is the entropy change of the universe?
A Carnot engine operating between heat reservoirs at 500 and 300 K absorbs 1500 J per cycle at the high-temperature reservoir. (a) Represent the engine's cycle on a temperature-entropy diagram. (b) How much per cycle is done by the engine?