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Unformatted text preview: MECH 310 Thermodynamics I American University of Beirut, Fall 2006 Final Exam January 22, 2007 Handout # Final Exam • This is a 180 minutes exam. • You are allowed to bring in 3 cheat sheets in addition to the thermodynamic tables. • You are advised to read the whole exam before you start. • Make sure you state all the assumptions you make and that you clearly identify any control mass or control volume you utilize in your analysis. • Good luck! Name: Section: Solve 5 of the following 6 problems. Make sure your choice is clear. 1 Problem 1 [20 points] One ton of water is in a ﬁxed-volume container. Initially, the temperature is T1 = 190 ◦ C and the pressure is p1 = 10 bar. It is desired to raise the pressure to p2 = 30 bar. This is achieved by interactions with another system that transfer energy and 1000 kJ/K of entropy into the container. (a) How much energy is transferred? (b) How entropy is generated by irreversibility? (c) What are the possible types of interactions that could result in the given transfers of energy and entropy? correction: [control mass : 2 pts] [state 1: 2 pts], [state 2: 3 pts], [energy: 5 pts], [irrversibility : 5 pts], [part (c): 3 pts]. Problem 1 Solution (a) Take the container as the control mass. Initial state is p1 = 10 bar T1 = 190 ◦ C s1 = 6.641 kJ/kg.K v1 = 0.2002 m3 /kg u1 = 2603 kJ/kg Since the container is closed, the speciﬁc volume remains unchanged, so that the ﬁnal state is given p2 = 30 bar v2 = 0.2002 m3 /kg T2 = 1030 ◦ C s2 = 8.461 kJ/kg.K u2 = 4108.1 kJ/kg Applying the ﬁrst law for a control mass, the energy transferred into the container is W ← + Q← = U2 − U1 = m(u2 − u1 ) = 1000(4108.1 − 2603) = 1505100 kJ (b) Applying the second law for a control mass, ∆S = S ← + Sirr ⇒ Sirr = m(s2 − s1 ) − S ← = 1000(8.461 − 6.641) − 1000 = 820 kJ/K (c) The energy transferred into the container has to be heat since it is the only way to transport entropy across the boundary of a closed system. 2 Problem 2 [20 points] Consider a gas turbine consisting of a compressor, combustion chamber and a turbine as shown in the ﬁgure below. The compressor and turbine are coupled by a common shaft as shown in the ﬁgure. If the net work of the cycle is zero meaning that the work produced by the turbine is equal to the work consumed by the compressor, ﬁnd the pressure of the air leaving given the following information. The air enters the compressor at 0.1 MPa, 15 ◦ C . The pressure leaving the compressor is 1.0 MPa and the temperature of air entering the turbine is 1100 ◦ C . The pressure drop in the combustion chamber is negligible. Assume the eﬃciency of the compressor 80% and the turbine eﬃciency 85%. 3 Problem 3 [20 points] Geothermal saturated steam at 100 ◦ C is being considered for heating a building. You are only to consider the latent heat of the geothermal steam in the heating schemes you are asked to come up with. The temperature of the building is to be maintained at 20 ◦ C and the average outdoor temperature is 5 ◦ C . Also available to you is the electricity grid so that you can connect it to any machinery you decide to employ. Electricity from the grid is valued at 10x Lebanese pounds per kWh. Energy from the steam is valued at x Lebanese pounds per kWh. (a) What is the smallest energy cost per unit of energy delivered to the building? How is the smallest energy cost achieved? Make a sketch of the interacting systems involved. (b) Under the conditions of part (a), how much electricity is transferred per unit energy delivered to the building? Is this electricity transferred to the building or to the grid? Problem 3 Solution What we can do is to employ a reversible cyclic power engine between the saturated steam and the building (which is considered as thermal reservoir at 20 ◦ C ), and in the process we produce work, i.e. we provide electricity to the grid. See the sketch below. Using this design, we provide power to the building from steam at the cost of x Lebanese pounds per kWh. At the same time we produce work that is equal to ˙ ˙ steam ˙ ˙← W → = Q← − Q→ ˙ building = ηcarnot Qsteam = ηcarnot msteam hf g @100◦ C Noting that ηcarnot = 1 − Then ˙ W→ = msteam ˙ = 1− Tbuilding hf g@100◦ C Tsteam 273 + 20 1− 2257 kJ/kg = 484 kJ/kg 273 + 100 Tbuilding Tsteam 4 Problem 4 [20 points] Consider an air compressor that receives air at 100 kPa, 25 ◦ C . It compresses the air to a pressure of 1 MPa, where it exits at a temperature of 540 K. The compressor heat losses are 50 kJ for each kg of air ﬂowing through the compressor. Find the following: a) Work of the compressor assuming variable speciﬁc heats. b) The eﬃciency of the compressor assuming variable speciﬁc heat. c) The amount of entropy generated. d) Draw the T − s diagram. 5 Problem 5 [20 points] Consider two large blocks of copper, A and B , of masses mA and mB (kg). Initially, block A is at temperature of TA,1 K and block B at TB,1 . The speciﬁc heat of copper is c (kJ/kg K). For this problem, we neglect expansion of copper as a function of temperature. (a) If perfect machinery is available, what is the largest work that can be extracted using the two copper blocks? (b) If perfect machinery and a reservoir at To are available, what is the largest work that can be extracted from the two blocks? (c) Which scenario produces more work (a) or (b)? Explain. Problem 5 Solution (a) As seen from quiz 2 solution, the maximum work that can be obtained from employing a carnot engine between the two blocks and noting that work will continue to be produced until both blocks reach the same temperature. Assuming that TA,1 > TB,1 , we get W → = Q← − Q→ = mA c [TA,1 − Tf ] − mB c [Tf − TB,1 ] H L where the ﬁnal temperature is
mA m Tf = TB,B c TA,1 c 1
1 c(mA +mB ) (b) In this case, we employ a carnot engine between A and atmosphere and B and atmosphere, assuming TA > T0 and TB > T0 , then
→ → W → = WA,atm + WB,atm = (mA c) (TA,1 − To ) − To ln TB,1 TA,1 + (mB c) (TB,1 − To ) − To ln To To (c) If TA > TB > T0 , then scenario (b) produces more work than scenario (a) because on way to achieve scenario (b) is to employ ﬁrst scenario (a) at the end of which TA = TB = Tf and then put a cyclic engine between Tf and environment at T0 < Tf , in this case,
→ W(→ = W(→) + WTf ,T0 b) a m mA Actually, as long as Tf = TB,B TA,1 1 more work than scenario (a).
1 mA +mB is less larger than T0 , scenario (b) produces 6 Problem 6 [20 points] Two tanks contain steam, and they are both connected to a piston/cylinder, as shown in the ﬁgure below. Initially the piston is at the bottom and the mass of the piston is such 1that a Two tanks of 1.4 MPa isand they are both connected to a piston/cAlinder,kg shown in 700 pressure contain steam, required to lift it. Steam in tank y is 4 as at 7 MPa, the figure below. Initially the piston is at the bottom and the mass of the piston is such ◦ C andthat a pressure of 1.4 MPa iMPa, 350 ◦lC .it. Steam invalves are kg at 7 MPa, 700 ºC water tank B has 2 kg at 3 s required to ift The two tank A is 4 opened, and the comes anda uniform 2state. 3Find the ºﬁnal he two valves are opened,total entropy generation to tank B has kg at MPa, 350 C. T temperature and the and the water comes assumingano heat state. Find the final temperature and the total entropy generation assuming to uniform transfer.
no heat transfer. 2- A spring-loaded piston cylinder, shown in the figure below contains water at 100 kPa with v = 0.07237 m3/kg. The water is now heated to a pressure of 3 MPa by a reversible heat pump extracting Q from a reservoir at 300 K. It is known that the water will pass through saturated vapor at 1.5 MPa and that pressure varies linearly with volume. Find the final temperature, the specific heat transfer to the water and the work input to the heat pump. 3- Consider a gas turbine consisting of a compressor, combustion chamber and a turbine as shown in the figure below. The compressor and turbine are coupled by a common shaft as shown in the figure. If the net work of the cycle is zero meaning that the work produced by the turbine is equal to the work consumed by the compressor, find the pressure of the air leaving given the following information. The air enters the compressor at 0.1 MPa, 15 ºC. The pressure leaving the compressor is 1.0 MPa and the temperature of air entering the turbine is 1100 ºC. The pressure drop in the combustion chamber is negligible. Assume the efficiency of the compressor 80% and the turbine efficiency 85%. 7 ...
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