TEST BANK accompanying the book THERMODYNAMICS - FUNDAMENTALS AND ENGINEERING APPLICATIONS P. Colonna (Complete Questions & Answers)

TEST BANK accompanying the book thermodynamics - fundamentals and engineering applications P. Colonna Delft University of Technology, The Netherlands     PREFACE 4 SAMPLE EXAM 1 5 Exercise 1 (true/false questions) 5 Exercise 2 5 Exercise 3 6 Exercise 4 6 Exercise 5 6 Exercise 6 7 SAMPLE EXAM 2 8 Exercise 1 (true/false questions, with justification) 8 Exercise 2 8 Exercise 3 9 Exercise 4 9 Exercise 5 10 Exercise 6 11 SAMPLE EXAM 3 12 Exercise 1 (true/false questions) 12 Exercise 2 13 Exercise 3 13 Exercise 4 14 Exercise 5 14 Exercise 6 16 SAMPLE EXAM 4 16 Exercise 1 (true/false questions) 16 Exercise 2 17 Exercise 3 17 Exercise 4 18 Exercise 5 18 Exercise 6 18 SAMPLE EXAM 5 19 Exercise 1 (true/false questions) 19 Exercise 2 20 Exercise 3 20 Exercise 4 21 Exercise 5 21 Exercise 6 21 Exercise 7 21 SAMPLE EXAM 6 23 Exercise 1 (true/false questions) 23 Exercise 2 23 Exercise 3 24 Exercise 4 24 Exercise 5 24 Exercise 6 25 Exercise 7 25   PREFACE This is a collection of exemplary exams aimed at testing the first five chapters of the book, therefore material typically covered in an undergraduate course. Each exercise is followed by its worked-out solution, in red. These exams vary in difficulty and should take three to four hours to solve, depending on the level of the student. I would much appreciate being notified of any error that may be detected by instructors and their students, as well as of any suggestion for improvements. P. Colonna  SAMPLE EXAM 1 Exercise 1 (true/false questions) 1) For a polytropic ideal gas, the specific internal energy difference between two states over an isobaric process (P=const.) is given by Δu= c_P (T_2-T_1 ). 2) A fluid can be condensed by isobaric cooling starting from a state defined by a pressure that is greater than the critical pressure. 3) The specific enthalpy at the outlet of a compressor is higher than that at its inlet. 4) The internal energy of an expanding gas contained in a non-adiabatic cylinder piston always decreases. 5) The rotor of an adiabatic turbine transfers entropy to the working fluid. 6) The c_P of a fluid depends only on temperature. 7) The accumulation term in an accounting equation is positive if the final value of the accounted variable is greater that the initial value. 8) A gas turbine engine can be more efficient than a Carnot engine if these engines do not operate between the same minimum and maximum temperatures. 9) A fluid kept at constant specific volume within a perfectly tight piston-cylinder system cannot transfer energy as work. 10) The internal energy of a macroscopic system is the sum of all types of microscopic energy modes and of macroscopic potential and kinetic energy. 11) Boiling a fluid in a closed rigid container increases its entropy. 12) The isentropic efficiency of a real compressor can reach 100% if the compressor is designed very well. Exercise 2 In the US, the efficiency of a power plant is often expressed in terms of “heat rate”. The heat rate is the amount of energy as heat input to a power plant to generate one kilowatt hour (kWh) of electricity, that is heat rate≡ (Thermal Energy in BTU)/(Electrical Energy out in kW h) Determine the thermal efficiency of a power plant defined as η≡W ̇_output/Q ̇_input , if its heat rate is 11000 BTU/kWh. Use the unit equivalent 1 BTU = 1055 J and show all the steps to obtain it. Exercise 3 A certain type of actuator consists of a closed piston-cylinder system in which the force on the piston due to the pressurized gas inside the cylinder is controlled by heating or cooling. The piston starts from state 1 and moves to another state. The gas obeys the ideal gas law. During the actuation, the heater is controlled such that the pressure varies with the position of the piston according to the prescribed function P(x)= P_1 V_1 (πrx)^(-1) Derive the expression for the energy transfer as heat Q(x) for the process in terms of piston position x , initial pressure P_1, initial position x_1, and internal radius of the cylinder r. Exercise 4 The volume of a piston-cylinder vessel containing 100 kg of CO2 is kept fixed at 1 m3, while the container is cooled from T = 354 K until it reaches saturated vapor conditions. After that, the fluid is cooled at constant pressure until it reaches the saturated liquid condition. Using the provided P-h thermodynamic chart for carbon dioxide, estimate the energy transfer as heat and indicate the process by drawing a line on the P-h chart. Exercise 5 A refrigeration system works according to the inverse Rankine cycle (or so-called heat pump cycle) using CO2 as the working fluid. The fluid undergoes the following processes: pressurization in a compressor; cooling by ambient air in a heat exchanger until it reaches saturated liquid conditions, expansion to a liquid-vapor state at low pressure and temperature through a throttling valve until it reaches a given pressure, and, from this liquid-vapor state, fluid receives energy as heat from the environment to be cooled until it vaporizes completely thus reaching the compressor inlet state. The diagram of the system is The system operates at steady state and the pressure drop in the heat exchangers can be neglected. The working fluid enters the compressor in saturated vapor conditions at a pressure of 15 bar and leaves the compressor at a pressure of 50 bar and temperature of 340 K. Use the tables and the P-h chart provided in the appendix to evaluate the needed thermodynamic properties. Hint: there is no need to perform interpolation for the property value of state 1, as the values of the two closest states in the table are almost identical. a) For each component (Compressor, Condenser, Valve, Evaporator) make a separate sketch to support the derivation of its energy balance; b) write the energy balance for each component; c) calculate the coefficient of performance of the system defined as COP≡Q ̇_evaporator/W ̇_compressor d) Determine the mass flow rate of CO2 that allows to obtain a cooling power of 20 kW. Exercise 6 Gas enters a device operating at steady state at a mass flow rate of M ̇ and at temperature T_1and pressure P_1, and exits the device at T_2 and P_2. Both the pressure and the temperature at the inlet are higher than at the outlet. The device converts the energy of the gas into shaft power W ̇ and is cooled by heat transfer at the rate (Q_0 ) ̇ to the environment at T_0. The properties of the gas at the conditions inside the device can be calculated assuming that it is an ideal gas with constant specific heat. a) Make a sketch of the system to support the derivation of the energy balance; b) make a sketch of the system to support the derivation of the entropy balance; c) derive an expression for the maximum shaft power W ̇ that can be obtained from the system in terms of M ̇, T_0, T_1, pressure ratio β_P=P_1/P_2 , temperature ratio β_T=T_1/T_2 and gas constants. SAMPLE EXAM 2 Exercise 1 (true/false questions, with justification) Provide a theoretical explanation of why the answer to the question is true or false, with mathematical relations if needed. 1) Temperature is an intensive property 2) A gas is compressed in a cylinder. Energy is transferred as heat from the gas to the surroundings of the cylinder. The internal energy accumulation of the gas is always lower than zero. 3) The state of a simple compressible substance is fixed if at least three independent thermodynamic variables are specified. 4) The kinetic energy of the gases expelled by the nozzle of a rocket is the same in a reference frame moving with the rocket and in the one fixed to the earth, considered as inertial. 5) The temperature in the inside of a fridge decreases, thus the entropy of the substances within it also decreases. The global production of entropy associated with this process is negative. Exercise 2 When considering the motion of a fluid on a molecular level, the state of the fluid can no longer be described by the Navier-Stokes equations. By integrating the phase space of possible states of the system, both over velocity and location, one can find mean properties. One of these properties is the shear stress τ. The solution of the Couette-Poiseuille flow problem with a heated wall provides the expression for shear stress as τ=ρV/(√π β_w ) where ρ is the number density (number of moles per volume), V is the bulk velocity of the fluid and β_w some thermodynamic parameter. a) Find the unit of the parameter β_w. Express it only in SI units of primary quantities. For example, Newton should be expressed kg∙m/s^2 . b) Find the value of parameter β_w using the data V=1.86 miles/h, ρ=1.157 mol/ft^3 , τ=20 Pa, 1 km=0.621 miles, 1 m=3.28 ft. Exercise 3 An ideal gas is heated in a piston-cylinder system. The friction between the piston and the cylinder can be neglected. The piston is initially locked. The energy is transferred as heat in such a way that the temperature of the gas remains constant. The system has a circular cross-section with radius r and the surrounding environment is at sea-level atmospheric conditions. The pressure in the cylinder is much higher than that of the environment. At a certain time instant the lock is removed and the piston accelerates. Obtain an expression for the velocity of the piston head as a function of the piston head mass M_p, the gas constant R, the temperature of the gas, the radius of the piston r, the mass of the gas in the cylinder, the atmospheric pressure, the change in piston head position Δx and the initial position of the piston head x_1. Exercise 4 To boil pasta for dinner, we introduce 1.5 litres of saturated liquid water in a special pot with a massless lid that can move freely (no friction) and is perfectly tight. This way saturated conditions can be maintained. The environment is at sea level. In order to transfer energy as heat to it, we place the pot on top of a stove which has the following power settings A: 1300 W B: 1400 W C: 1500 W At the same time, energy as heat is transferred from the pot to the kitchen (surroundings) at a rate of 100 W. The cooking time is 8 minutes. To ensure that all the pasta is submerged in the water, it is necessary that the volume of water is always at least 1.2 litres. The effect of the pasta in the water can be neglected in this simple analysis. a) What is the power setting providing the highest rate of energy transfer heat that can be used? b) Starting from the saturated liquid state, how much time would it take for the water to completely evaporate if the stove is kept at the power setting calculated as the answer to question a)? Exercise 5 A combined cycle power plant is a system formed by two thermal engines, which is commonly used for electricity generation. The operation of the two thermal engines is based on two different thermodynamic cycles. The working principle of the combined cycle is that the high temperature cycle discharges thermal energy as heat to the low temperature cycle. Consider a concentrated solar power combined cycle system whose simplified system diagram is reported in the figure below. The high-temperature cycle is called Brayton cycle, while the low temperature cycle is called Rankine cycle. The working fluid of the high-temperature cycle is air and that of the low temperature cycle is water. Specifications: Environment: P_1=1 bar,T_1 = 25 ℃; Air turbine inlet: P_3 = 17 bar,T_3 = 1233 ℃; Air turbine isentropic efficiency: η_34s = 0.85; Air compressor efficiency: η_12s = 0.80; Energy transfer as heat to the air Brayton cycle: Q ̇_in=Q ̇_23=96.44 MW Air temperature at the stack: T_5 = 120 ℃; Steam turbine inlet (superheated): P_6 = 20 bar,T_6 = 477 ℃; Steam turbine isentropic efficiency: η_67s = 0.82; Pressure in the steam condenser: P_7 =P_8= 0.2 bar; The power consumption of the feed-water pump can be neglected: state 8 = state 9. The thermodynamic state of the pump inlet/condenser outlet is saturated liquid water. a) In addition to the usual sketches of the system which are part of the 8-step method for energy analysis, draw a qualitative sketch of the processes forming the two cycles in the two Ts charts below, one of air, and one of water. b) Calculate the thermal efficiency η≡W ̇_net⁄Q ̇_in of the combined cycle. Calculate also the thermal efficiency of the two cycles individually. Assume the following: Air can be considered a polytropic ideal gas with constant isobaric specific heat, c_P=1.0 kJ/(kg∙K). The heat exchangers are adiabatic (only energy transfer as heat between fluids and not with the surroundings). No state change of the working fluids within the ducts connecting the components. Pressure losses in all heat exchangers can be neglected. The power consumption of the feed-water pump can be neglected: state 8 = state 9. The pump inlet/condenser outlet states are saturated liquid water. c) Compute the second law efficiency based on the maximum and minimum cycle temperature of the combined cycle and those of the two cycles and discuss these results. Exercise 6 A tank is separated into two compartments by an adiabatic and frictionless piston. One compartment contains 0.2 m^3 of air and the other compartment contains 0.1 kg of oxygen, both initially at 27 ℃ and 90 kPa. The compartment of the tank containing oxygen is insulated. The compartment of the tank containing air is slowly supplied with energy as heat through a wall whose temperature is controlled and kept constant at 300 ℃. During the process the two compartments are in mechanical equilibrium. The energy transfer as heat stops when the pressure of the air rises to 150 kPa. The molar mass of oxygen is 32 g/mol and its specific heat at constant pressure is 940 J/(kg∙K). Assume that the ideal gas assumption holds for both gases. a) Determine the temperature of the air and of the oxygen at the end of the heating process. b) Calculate the energy transferred as heat into the system and the entropy production during the process. c) Suppose you could alter the temperature of the wall where the heat is transferred. What would be the optimal temperature to minimize entropy production? SAMPLE EXAM 3 Exercise 1 (true/false questions) 1) While heating a saturated liquid at constant volume, its pressure and temperature remain constant. 2) A thermal energy reservoir (TER) can supply or absorb a finite amount of energy as heat without undergoing any change in temperature. 3) An isolated system consists of two volumes made of different materials with different initial temperatures, and in contact with one another. After some time, the system has reached an equilibrium state. Both materials have the same final temperature and the same final specific internal energy. 4) A thermodynamic engine can function without discharging energy as heat. 5) The coefficient of performance of a system can be obtained by dividing the useful energy output by the total energy input. 6) Any intensive property of matter depends on the considered mass. 7) The specifications sheet of a car engine shows that its net efficiency is 45% at an ambient temperature of 23 ℃. Your friend Jeremy tells you that the peak temperature of the gases in the cylinders is 230 ℃. Is this true or false? 8) The internal energy of a gas is related to the energy associated with the microscopically disorganized motion of its molecules. 9) Energy is transferred as heat to a saturated fluid mixture that is stored in a rigid container. The quality of the initial state is 0.2, and it is 0.8 in the final state. The pressure and temperature of the fluid do not change during this process. 10) The entropy change of any fluid can be evaluated analytically if only the specific heats at the initial and final states are given. 11) An isolated control volume containing an ideal gas undergoes an isothermal process. No internal energy gets accumulated within the control volume. 12) An adiabatic mixer merges two streams of a certain fluid entering into the mixer at the different temperatures. The entropy of the outlet stream is equal to the sum of the entropy at the inlets. Exercise 2 Non-dimensional quantities play an important role in energy systems modeling and design. An example of a dimensionless quantity used for pumps is the pump specific speed N_s, and its definition is N_s=(N√Q)/(gH)^(3/4) Where N is the rotational speed of the pump, in rad/s, Q is the volume flow rate, in m^3/ s, g is the gravitational acceleration, in m/s^2, H is the head of the pump, in m. a) Prove that the specific speed is a dimensionless quantity. b) Calculate the specific speed of the liquid oxygen (LOX) pump used in the SpaceX Raptor Engine, given the following information: g = 9.81 m/s^2, Q = 346 000 gal/hr, N = 14 000 rpm, H = 7 250 ft. Use the following unit conversions: 1 ft = 0.3048 m 1 gal = 0.00379 m^3 Exercise 3 An ideal gas is heated in a piston-cylinder system where the piston is able to move freely. The system has a circular cross-section with radius r and the surrounding environment is at sea-level atmospheric conditions. The mass of the piston can be considered negligible. Obtain an expression for the change in piston head position as a function of the amount of energy transferred as heat, the pressure, the isochoric specific heat c_v, the gas constant R, and the radius r. Exercise 4 A constant-volume vessel of 20 liters holds 2 kg of oxygen in thermodynamic equilibrium at an initial temperature of 102 K. The vessel is heated until there is only saturated vapor oxygen. a) Determine the temperature and pressure once the oxygen has reached its final equilibrium state. b) Determine the amount of energy transferred as heat for this process c) Sketch the process on the provided P-h diagram for oxygen Exercise 5 Consider the following simple Rankine cycle system operating with water as the working fluid, Pressure losses in pipes, ducts and heat exchangers are negligible. The turbine and pump isentropic efficiencies are reported in the table. Pressure losses are negligible everywhere in the system. Turbine isentropic efficiency 0.6 Pump isentropic efficiency 0.8 The following tables list the information needed to perform the computation of the cycle thermodynamic states at the inter-component interfaces for two different cases. Case A ID state T / ℃ P / bar x s / kJ/(kg∙K) h / kJ/kg 1 Pump inlet 49.3 0 2 Pump outlet 4 0.6966 207.412 3 Turbine inlet 150 4 4 Turbine outlet Case B ID state T / ℃ P / bar x s / kJ/(kg∙K) h / kJ/kg 1 Pump inlet 151.85 0 2 Pump outlet 140 3 Turbine inlet 376.85 140 4 Turbine outlet 6.348 2547.35 NOTE: NOT ALL THE CELLS IN THE TABLES NEED TO BE FILLED IN TO SOLVE THE PROBLEM. The two thermodynamic cycles in the temperature-entropy diagram of water are Hint: turbine thermodynamic states The turbine isentropic efficiency is η_(t,s)=∆h/∆h_s Where ∆h is the turbine specific work, and ∆h_s is the turbine isentropic specific work. Hint: second law efficiency The second law efficiency is η_II≡η_th/(1-T ̅_C/T ̅_H ) Where η_th is the cycle thermal efficiency. T ̅_C is the so-called equivalent thermodynamic temperature for thermal energy rejection (condenser). T ̅_H is the equivalent thermodynamic temperature for thermal energy transfer to the steam (boiler). They are computed as T ̅_C=(h_4-h_1)/(s_4-s_1 ) and T ̅_H=(h_3-h_2)/(s_3-s_2 ). In order to grasp the need for introducing these temperatures, look at the shape of the cycles in the temperature-entropy diagrams. Do they look similar to a Carnot cycle? a) Calculate the cycle thermal efficiency for both case A and case B. Discuss the difference between these values. b) Calculate the cycle second law efficiency for both case A and case B. Exercise 6 A metal wall is exposed to temperature T_1 at one side and to temperature T_2 at the other side. Demonstrate which of the two temperatures must be higher for the energy transfer as heat to occur in the direction indicated in the figure (i.e., from left to right). Assume that the process is steady-state, that the wall is infinitely long, and that no other energy transfer occurs. SAMPLE EXAM 4 Exercise 1 (true/false questions) 1) A reversible process is isentropic. 2) A person in a raising elevator lifting a box over a distance of half meter will do more work than a person on the ground lifting the same box over the same distance. 3) The terms of the equation for the steady-state energy balance of an open system are the same as those for a closed system. 4) The efficiency of a wind turbine cannot be larger than the Carnot cycle efficiency. 5) The critical point can be evaluated using the ideal gas model. 6) An explosion can be regarded as a sudden entropy production. 7) When mixing adiabatically hot and cold gases to a new equilibrium state, the entropy increases. 8) In a polytropic process the pressure can change independently of the volume. 9) A system formed by a simple compressible substance enclosed in a constant volume can transfer energy as work and heat 10) Removing energy as heat from a closed system reduces its entropy. 11) At ambient room conditions, methane is liquid. 12) Energy transfer as heat under a finite temperature difference is reversible. 13) Cooling a gas in closed volume at constant pressure increases its volume. 14) An ideal valve transfers more energy as work than a non-ideal valve. Exercise 2 (Same as Exercise 1.1 of the book) An 80 kg student wishes to walk on snow without sinking and decides to make a pair of snow shoes. The student determined experimentally that the maximum pressure the snow can withstand is 0.6 kPa. What is the minimum surface required by one snow shoe? Would the student be able to sell snow shoes to his fellow classmates? Explain your answer. Exercise 3 The temperature of 10 kg of air inside a piston-cylinder is increased from 25 ℃ to 80 ℃ using an electrical heater, while the pressure inside the cylinder is maintained constant. During this process 50 kJ of energy is transferred as heat from the system to the environment. Find the required amount of energy that must be added to the system by the heater, for cylinder pressures of 1, 2 and 5 bar. Assume the air inside the cylinder as an ideal gas with c_P=1.006 kJ⁄((kg∙K) ) Exercise 4 A constant volume vessel of 100 liters holds 1 kg of water at an initial pressure of 30 bar and initial temperature T_1. The vessel is then cooled to 350 K. a) Determine the initial temperature of the system T_1 and the pressure after cooling. b) Determine the energy transferred as heat during the process. Make a sketch of the process on the provided P-h chart. Exercise 5 A small solar engine is used for power conversion. Water enters the pump as saturated liquid at 50 ℃ and is pumped up to 2 bar by a small centrifugal pump (process 1-2). Solar energy is gathered using solar collectors, connected to the boiler. The boiler evaporates the water at 2 bar (process 2-3), and saturated vapor at this pressure enters a small turbine (process 3-4). The steam leaves the turbine at 50 ℃ and with a liquid fraction of 6 %, and is subsequently condensed (process 4-1). The mass flow rate is 140 kg/hr and the pump is driven by a 0.4 kW motor. a) Calculate the net mechanical power output of this solar engine. b) Compute the energy conversion efficiency taken as the net shaft work output divided by the energy transfer as heat to the fluid in the boiler. c) Estimate the area of the solar collectors, assuming the collectors can absorb 800 W/m^2 . Exercise 6 Consider the two closed adiabatic boxes illustrated in the following sketch. The first box accommodates a frictionless shaft transferring mechanical work to a spinning propeller. The second box accommodates a frictionless shaft transferring mechanical work to lift a weight. Assume that the same amount of energy as work is provided to the two shafts for a fixed amount of time starting at t_s and ending at t_e. a) Describe in conceptual terms (in words) the changes occurring in the boxes during the transfer of energy as work to the shafts (t_s t t_e). b) Describe in conceptual terms (words) the changes occurring in the boxes once the transfer of energy as work to the shafts stops (t te). SAMPLE EXAM 5 Exercise 1 (true/false questions) 1) Volume is an intensive property. 2) A simple Brayton cycle engine has a higher efficiency than the Carnot engine. 3) When compressing a gas adiabatically its internal energy increases. 4) Color is a thermodynamic property. 5) A closed system allows mass exchange with the surroundings. 6) The production of entropy in the universe is zero. 7) The production of energy in the universe is zero. 8) Temperature has the same dimensional units as energy. 9) The temperature is proportional to volume in an isobaric process of an ideal gas. 10) The value of the internal energy of an ideal gas is always larger than the value of its enthalpy at a given temperature. 11) Extracting energy as heat from a closed system reduces its entropy. 12) Energy transfer as heat is a reversible process. 13) A refrigerator is a heat pump. 14) In a polytropic process the pressure can change independently of the volume. Exercise 2 (Similar to exercise 1.2 of the book) A wind farm of 30 identical windmills is converting the kinetic energy of the wind into 50 MW of electrical power. The velocity of the wind blowing through the rotor disc of each windmill is V = 7 m/s. The air density is ρ = 1.2 kg/m^3. If the efficiency of each windmill is 0.63, what is the windmills rotor radius R? Exercise 3 (Similar to exercise 4.13 of the book) The throttling calorimeter is a device for determining the thermodynamic state of a liquid vapour mixture by means of temperature and pressure measurements. The procedure is to deviate a small amount of the mixture from the main flow, pass it through a valve, and take the measurements as shown in the figure below. Assume that the fluid flowing through the calorimeter is water. a) Make a sketch of the process between state 1 and 2 on a T-s diagram, indicating the state points and the line corresponding to the throttling process. b) Compute the quality of the mixture at state 1, given the pressure and temperature measurements displayed in the figure below. Exercise 4 13 kg of ammonia at its critical state is placed in a constant volume container, which is then cooled until the temperature is 310 K. a) How much energy was transferred as heat from the ammonia in this process? b) How much of the tank volume is occupied by liquid at the end of the process? c) Sketch the process on the provided P-h diagram for ammonia. Exercise 5 Consider the intake stroke of an internal combustion engine under the following idealizations. Negligible kinetic and potential energy everywhere 
 Uniform state inside the cylinder at any instant
 One-dimensional flow
 Ideal gas Non-adiabatic cylinder walls a) Make a sketch of the system, including i) its boundary, ii) all the quantities involved in the mass and energy balance, and arrows indicating the positive direction for the energy transfer terms. b) Write the mass balance on the control volume. c) Write the energy balance on the control volume in terms of temperature T, mass M, volume V, energy transfer rate as heat Q ̇ and gas constants c_P and c_v. Exercise 6 Ten grams of computer chips with a specific heat c_P=c_v=0.3 kJ/(kg∙K) are initially at 20 ℃ . These chips are placed in an adiabatic container with 5 grams of saturated liquid R134a at -40 ℃ in order to cool them down. The fluid remains saturated over the process. Assume that during the cooling process the pressure remains constant. a) Calculate the entropy change of the chips; b) Calculate the entropy change of the R134a; c) Compute the entropy production due to this process; d) state whether this process is possible or not. Exercise 7 The majority of current aircraft engines employ gas turbines based on the open Brayton thermodynamic cycle. Hence, being able to analyze the performance of this type of thermal engines is important for the aspiring aerospace engineer, and it is easier to start from the analysis of the ideal Brayton thermodynamic cycle. The sketch of the process flow diagram and of the temperature-entropy diagram for the ideal Brayton cycle are The corresponding processes are as follows: 1-2: Isentropic compression 2-3: Isobaric transfer of energy as heat 3-4: Isentropic expansion 4-1: Isobaric transfer of energy as heat Assume that air is the working fluid and that it obeys the ideal gas law with constant specific heats (polytropic ideal gas). Derive the efficiency of an engine operating according to the ideal Brayton cycle, as a function of the cycle pressure ratio Π≡P_2/P1 and the ratio of the specific heats γ≡c_p/c_v . Hint: for a polytropic ideal gas T_1/T_2 =(P_2/P_1 )^((γ-1)/γ)   SAMPLE EXAM 6 Exercise 1 (true/false questions) 1) A modern petrol engine cannot be more efficient than a Carnot engine. 2) The value of the internal energy of an ideal gas is always lower than the value of the enthalpy at a given temperature. 5) Temperature is an intensive property. 4) The specific heat of air does not vary with the temperature. 5) In an adiabatic process energy as heat is transferred from high to low temperature. 6) A diesel engine with high compression ratio can be more efficient than a Carnot engine. 7) An explosion can be regarded as a sudden entropy production. 8) By definition c_p is always smaller than c_v. 9) The production of entropy in the universe is zero. 10) In a polytropic process the pressure can change independently of the volume. 11) Removing energy as heat from a closed system reduces its entropy. 12) A system formed by a simple compressible substance enclosed in a constant volume cannot transfer energy as work. 13) In a gas turbine engine the fluid enthalpy at the outlet of the turbine is larger than at the inlet. Exercise 2 What is the kinetic energy in MJ of an aircraft weighing 150 000 lbm (pound mass) and flying at 880 km/h? For the conversion of units, use the method of multiplying by suitable units, e.g., 1=(1 kg)/(2.205 lbm) Exercise 3 Consider the isochoric (constant-volume) compression of a gas from pressure P_1 to P_2 in a piston-cylinder system. The increase of pressure is achieved by controlled energy transfer as heat Q to the gas, while the piston is kept fixed. a) Make a sketch of the considered system with its control volume, the appropriate nomenclature for all quantities involved in the energy balance, and arrows indicating the positive direction of energy transfers. b) Treating the gas as polytropic and ideal, derive the expression for the energy input as heat Q as a function of the pressure ratio P_2/P_1. Exercise 4 5 kg of CO2 at its critical state is placed in a perfectly rigid container, which is then cooled until the temperature is 260 K. a) What is the volume occupied by liquid at the end of the process? b) How much energy is transferred as heat during the process? Exercise 5 Consider the simple steam Rankine cycle system operating at steady state, whose process flow diagram is depicted in the figure with the corresponding processes in the T-s diagram. Water is compressed in a pump (1-2, single point in the T-s diagram as the temperature and entropy increase are negligible), receives energy as heat, which first increases its temperature and then evaporates it and superheats it (2-3), steam is expanded in a turbine (3-4) and then condensed (4-1). Assume that the flow is one-dimensional at all intercomponent thermodynamic states, that the kinetic energy and potential energy of the fluid at the state points is negligible. Assume in addition that the pump and the turbine are adiabatic. The following assumptions are valid: The work of the pump is negligible (compressing liquid water does not require much work compared to the energy converted into mechanical work by the turbine) The enthalpy of the water at the outlet of the pump is 138 kJ/kg. The evaporation pressure is 80 bar, the condensation temperature is 30 oC The steam at the inlet of the turbine at 400 oC The isentropic efficiency of the turbine is 0.85 For each of the 4 components, 1) Make a sketch of the pertinent control volume, with the appropriate nomenclature for all quantities involved in the energy balance and arrows indicating the positive direction of energy transfers and derive the steady-state energy balance per unit mass flow of working fluid; 2) Calculate the efficiency of the system. Exercise 6 Consider a turbine operating at steady state. The working fluid enters the control volume in state 1 and leaves it in state 2. The boundary of the control volume that separates the turbine from a large environment is at temperature T_0. The turbine reversibly rejects energy as heat to the environment. Assume that the kinetic and potential energy of the entering and leaving flow can be neglected. a) Derive the expression for the mechanical power output as a function of thermodynamic properties of the inlet and outlet states, of the temperature of the environment and of the rate of entropy production within the process. b) Derive the expression for the maximum mechanical power of the turbine. Exercise 7 Consider a two-part isolated system whereby the total volume of the system and of the parts are constant. Demonstrate that if the two parts are approaching thermal equilibrium, the energy transfer as heat between the two parts must be from the part at higher temperature to the part at lower temperature. The relation between the thermodynamic temperature and the entropy is T=(∂u/∂s)_v