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Thermodynamics (ME22002) Spring 2006 Indian Institute of Technology Kharagpur Department of Mechanical Engineering Course Outline 1. Introduction: Fundamental Concepts: definitions of system and surrounding, concept of control volume, thermodynamic state, concepts of simple compressible substances, pure substance and phase, thermodynamic processes and thermodynamic equilibrium; Temperature and Zeroth law; Thermodynamic properties and use of tables of thermodynamic properties; Idea of a generalized chart and the law of corresponding states; Concept of ideal gases and their equations of state; Thermodynamic concept of energy; Modes of work and heat transfer. 2. First Law of Thermodynamics: The first law referred to cyclic and non-cyclic processes, concept of internal energy of a system, conservation of energy for simple compressible closed systems; Definitions of enthalpy and specific heats; Conservation of energy for an open system or control volume, steady & transient processes. 3. Second Law of Thermodynamics: The directional constraints on natural processes; Formal statements; Concept of reversibility; Carnot principle; Absolute thermodynamic temperature scale; Clausius Inequality, entropy, change in entropy in various thermodynamic processes, Tds relations, entropy balance for closed and open systems, Principle of increase- in- Entropy, entropy generation. 4. Exergy: Concept of reversible work & irreversibility; Second law efficiency; Exergy change of a system: closed & open systems, exergy transfer by heat, work and mass, exergy destruction, exergy balance in closed & open systems. 5. Thermodynamic Property Relations: Maxwell relations; Clausius-Clapeyron equation; Difference in heat capacities; Ratio of heat capacities; Joule-Thompson coefficient; a revisit to property diagrams of simple compressible substances. 6. Introduction to Properties of Mixtures and Phases: •

Ideal gas mixtures: Amagat and Dalton model for mixture of ideal gases, Equation of state and properties of ideal gas mixtures, Change in entropy on mixing.

•

Non ideal mixtures: Partial molal properties, fugacity of a component in a mixture, changes in property on mixing, free energy of mixing, concept of an ideal solution, conditions of phase equilibrium and chemical equilibrium of multi-component systems, Gibbs phase rule.

7. Thermodynamics of Reactive Systems: First law analysis of reactive system; Internal energy and enthalpy of reaction; Enthalpy of formation; Second law applied to a reactive system. 8. Air Standard Cycles: Carnot, Stirling, Ericssion, Otto, Diesel, and Dual cycles. Brayton cycle: intercooling, reheating and regeneration. 9. Vapour Cycles: Carnot cycle; Simple Rankine cycle, Reheat and Regenerative cycles with open & closed feedwater heater; Ideal vapour compression refrigeration cycle. Required Text Van Wylen, Sonntag and Borgnakke: Fundamentals of Thermodynamics, 6th Edition, 2004 (John Wiley). Assessment of Students 1. 20% credit from two class tests & teacher’s assessment 2. 30% credit from the mid-semester examination (tentative syllabus: up to and including Thermodynamic property relations) 3. 50% credit from end-semester examination

Thermodynamics: ME22002 Course Instructor: Suman Chakraborty

ASSIGNMENT PROBLEMS SET 1 Chapters 3-5 (Fundamentals of Thermodynamics, 6th Ed., by Sonntag, Borgnakke and Van Wylen)

Problem numbers Chapter 3: 41, 44, 49, 57, 60, 65, 73, 78, 97, 99, 101, 103, 105, 107, 109, 113 Chapter 4: 33, 41, 42, 53, 54, 61, 63, 64, 67, 73, 77, 110 Chapter 5: 35, 48, 50, 57, 58, 62, 70, 73, 95, 99, 101, 130, 132, 134

Thermodynamics: ME22002 Course Instructor: Suman Chakraborty SOLUTIONS TO ASSIGNMENT PROBLEMS FROM Chapter 3 (Fundamentals of Thermodynamics, 6th Ed., by Sonntag, Borgnakke and Van Wylen)

Thermodynamics: ME22002 Course Instructor: Suman Chakraborty SOLUTIONS TO ASSIGNMENT PROBLEMS FROM Chapter 4 (Fundamentals of Thermodynamics, 6th Ed., by Sonntag, Borgnakke and Van Wylen)

Thermodynamics: ME22002 Course Instructor: Suman Chakraborty SOLUTIONS TO ASSIGNMENT PROBLEMS FROM Chapter 5 (Fundamentals of Thermodynamics, 6th Ed., by Sonntag, Borgnakke and Van Wylen)

Thermodynamics: ME22002 Course Instructor: Suman Chakraborty

ASSIGNMENT PROBLEMS SET 2 Chapters 6-9 (Fundamentals of Thermodynamics, 6th Ed., by Sonntag, Borgnakke and Van Wylen)

Problem numbers Chapter 6: 39, 57, 59, 75, 82, 88,105,107, 108,111, 115, 116, 117, 118,119, 120, 121, 122, 131,133, 134 Chapter 7: 38, 44, 48, 58, 62, 64, 65, 71, 76, 82, 84, 86, 88, 89, 90, 91 Chapter 8: 14, 15, 16, 33, 45, 52, 54, 65, 70, 71, 76, 82, 85, 86, 90, 98, 100, 101, 103, 109, 120, 126, 128, 129, 130, 131, 133, 134, 137 Chapter 9: 26, 28, 37, 38, 39, 41, 51, 68, 74, 77, 96, 101, 116, 119, 126, 127, 130, 132

Thermodynamics: ME22002 Course Instructor: Suman Chakraborty SOLUTIONS TO ASSIGNMENT PROBLEMS FROM Chapter 6 (Fundamentals of Thermodynamics, 6th Ed., by Sonntag, Borgnakke and Van Wylen)

Thermodynamics: ME22002 Course Instructor: Suman Chakraborty SOLUTIONS TO ASSIGNMENT PROBLEMS FROM Chapter 7 (Fundamentals of Thermodynamics, 6th Ed., by Sonntag, Borgnakke and Van Wylen)

Thermodynamics: ME22002 Course Instructor: Suman Chakraborty SOLUTIONS TO ASSIGNMENT PROBLEMS FROM Chapter 8(Fundamentals of Thermodynamics, 6th Ed., by Sonntag, Borgnakke and Van Wylen)

Thermodynamics: ME22002 Course Instructor: Suman Chakraborty SOLUTIONS TO ASSIGNMENT PROBLEMS FROM Chapter 9 (Fundamentals of Thermodynamics, 6th Ed., by Sonntag, Borgnakke and Van Wylen)

Thermodynamics: ME22002 Course Instructor: Suman Chakraborty SOLUTIONS TO ASSIGNMENT PROBLEMS FROM Chapter 10 (Fundamentals of Thermodynamics, 6th Ed., by Sonntag, Borgnakke and Van Wylen)

Thermodynamics: ME22002 Course Instructor: Suman Chakraborty SOLUTIONS TO ASSIGNMENT PROBLEMS FROM Chapter 12 (Fundamentals of Thermodynamics, 6th Ed., by Sonntag, Borgnakke and Van Wylen)

Thermodynamics: ME22002 Course Instructor: Suman Chakraborty SOLUTIONS TO ASSIGNMENT PROBLEMS FROM Chapter 13 (Fundamentals of Thermodynamics, 6th Ed., by Sonntag, Borgnakke and Van Wylen)

Thermodynamics: ME22002 Course Instructor: Suman Chakraborty

ASSIGNMENT PROBLEMS CHAPTER 15 (Fundamentals of Thermodynamics, 6th Ed., by Sonntag, Borgnakke and Van Wylen)

Problem numbers 32, 38, 49, 52, 54, 60, 62, 72, 80

Thermodynamics: ME22002 Course Instructor: Suman Chakraborty SOLUTIONS TO ASSIGNMENT PROBLEMS FROM Chapter 15 (Fundamentals of Thermodynamics, 6th Ed., by Sonntag, Borgnakke and Van Wylen)

Thermodynamics (ME22002), IIT Kharagpur, Spring 2006 Problem set on Properties of Mixtures and Phases Q1. Equation of state of a non-ideal 2-component gas mixture is given as: RT v= + y12 B11 + 2 y1 y2 B12 + y22 B22 , where yi is the mole fraction of the ith component in p the mixture. For a binary mixture of 10 mole % chloroform (component 1) in acetone (component 2) at 333K and 10 bar, the coefficients Bij are given as follows: B11= -910 cm3/mol, B22= -1330 cm3/mol, and B12 = -2005 cm3/mol. Determine: (i) Molar specific volume of pure chloroform (ii) Partial molal volume of chloroform in the mixture, and (iii) Change in volume on mixing. (Ans: (i) 1860 cm3/mol, (ii) 991 cm3/mol, (iii) -372 cm3/mol) Q2. (a) For a binary system (constituents A + B) at constant temperature and pressure, the molar volume (cm3/mol) is obtained in terms of the respective mole fractions as: v = 100yA + 80yB +2.5yAyB, where yA is the mole fraction of the A and yB is the mole fraction of the B in the mixture. (i) What is the partial molal volume of the component A in the mixture, as the mole fraction of A tends to zero, in a limiting sense? (ii) Illustrate with a sketch, how would you obtain the above result from a graphical plot of molar volume versus mole fraction of the component B? No explicit numerical calculations are necessary. (iii) Obtain an expression for change in volume due to mixing, as a function of mole fraction of individual constituents. What is the mole fraction for which this change is a maximum? Does the volume increase or decrease, on account of mixing? (Ans: (i) 102.5 cm3/mol, (iii) ∆Vmix = 2.5 y A yB > 0 ) Q3. At 21°C, the enthalpy of mixing of sulphuric acid and water can be fit to the following equation: ∆H mix = −74.4 y H 2 SO4 yH 2O (1 − 0.561 yH 2 SO4 ) kJ/mol. Molar enthalpies of pure sulphuric acid and water are 1.596 kJ/mol and 1.591 kJ/mol, respectively. For an equimolar mixture of the two components, obtain the partial molal enthalpies of sulphuric acid and water. (Ans: -6.6 kJ/mol, -17 kJ/mol)) Q4. For the van der Waals gas, ⎡ (v − b) p ⎤ ⎡f⎤ b 2a − ln ⎢ ⎥ = − ln ⎢ ⎥+ ⎣ p⎦ ⎣ RT ⎦ v − b RTv

show

that

the

fugacity is

given by

Q5. For a binary gas mixture composed of species a and b, an equivalent form of van der RT ( na + nb ) na2 aa + 2na nb aa ab + nb2 ab Waals equation of state is given as p = − . V − ( na ba + nb bb ) V2 Show that the fugacity of the component a in the gas mixture is given by

(

)

2 ya na + yb aa ab ⎡ ( v − bmix ) p ⎤ ⎡ f ⎤ ba , where v is the mixture + − ln ⎢ a ⎥ = − ln ⎢ ⎥ RT RTv ⎣ ya p ⎦ ⎣ ⎦ v − bmix specific volume and bmix = ya ba + yb bb .

Q6. Derive the Clapeyron equation from the considerations of equilibrium of a two-phase single component system. Q7. What pressure is required to make diamond from graphite at 25°C? For your calculations, following data are given for a temperature of 25°C and pressure of 0.1 MPa: Graphite Diamond g 0 2867.8 kJ/kmol 3 0.000284 m /kg 0.000284 m3/kg v 0.016×10-6 1/MPa 0.304×10-6 1/MPa βT (Ans: 1493 MPa) Q8. Air (~21% O2, 79% N2) is cooled to 80 K, 0.1 MPa. Calculate the composition of liquid and vapour phases in this condition. Given: saturation pressure of O2 = 0.137 MPa and saturation pressure of N2 = 0.03006 MPa. (Ans: mole fraction of O2 in liquid phase =0.654 and in vapour phase = 0.896) Q9. Determine the mole fraction of air at the surface of a lake whose temperature is 17°C. Take the atmospheric pressure at the lake level to be 92 kPa. Given, Henry’s constant for air dissolved in water at 290 K= 62000 bar. (Ans: 1.45×10-5) Q10. Fresh water is to be obtained from sea water at 15°C, with a salinity of 3.48% on mass basis. Determine the minimum work input required to separate 1 kg of sea water completely into pure water and pure salt. State any assumptions you make. Given: molecular weight of water= 18, molecular weight of salt= 58.44. (Ans: 7.87 kJ/kg of seawater, assumption: the mixture behaves as an ideal solution)

Thermodynamics (ME22002), IIT Kharagpur, Spring 2006

Problem Set on Air Standard Cycles 1.

A stoichiometric mixture of gasoline and air has an energy release upon combustion of approximately 2800 kJ/kg of the mixture. To approximate an actual spark- ignition engine using such a mixture, consider an air-standard Otto cycle that has a heat addition of 2800 kJ/kg of air, a compression ratio of 7, and a pressure and temperature at the beginning of the compression process of 90 kPa, 10°C. Assuming constant specific heat, with the value from Table A.l0, determine a) The maximum pressure and temperature of the cycle. b) The thermal efficiency of the cycle. c) The mean effective pressure.

2. In the air-standard Otto cycle, all the heat transfer qh occurs at constant volume. It would be more realistic to assume that part of qh occurs after the piston has started its downward motion in the expansion stroke. Therefore, consider a cycle identical to the Otto cycle, except that the first two-thirds of the total qh occurs at constant volume and the last one-third occurs at constant pressure. Assume that the total qh is 2400 kJ/kg, that the pressure and temperature at the beginning of the compression process are 90 kPa, 20°C, and that the comparison ratio is 7 Calculate the maximum pressure and temperature and the thermal efficiency of this cycle. Compare the results with those of a conventional Otto cycle having the same given variables. 3. Consider an ideal air-standard diesel cycle in which the state before the compression process is 95 kPa, 290 K, and the compression ratio is 20. What maximum temperature must the cycle have to have a thermal efficiency of 60%? 4. An air-standard Ericsson cycle has an ideal regenerator. Heat is supplied at 1000°C and heat is rejected at 20°C. Pressure at the beginning of the isothermal compression process is 70 kPa. The heat added is 600 kJ/kg. Find the compressor work, the turbine work, and the cycle efficiency. 5. Consider an ideal Stirling-cycle engine in which the pressure and temperature at the beginning of the isothermal compression process are 100kPa, 25°C, the compression ratio is 6, and the maximum temperature in the cycle is 1100°C. Calculate a) The maximum pressure in the cycle. b) The thermal efficiency of the cycle with and without regenerators. 6. Consider a large stationary gas-turbine power plant that operates on the ideal Brayton cycle and delivers a power output of 100 MW to an electric generator. The minimum temperature in the cycle is 300 K, and the maximum temperature is

1

1600 K. The minimum pressure in the cycle is 100 kPa, and the compressor pressure ratio is 14 to 1. a) Calculate the power output of the turbine. What fraction of the turbine output is required to drive the compressor? b) What is the thermal efficiency of the cycle? 7. Repeat Problem 6, but assuming that the compressor has an isentropic efficiency of 85% and the turbine an isentropic efficiency of 88%. 8. The gas turbine cycle shown in the figure below is to be used as an automotive engine. In the first turbine, the gas expands to a pressure P5, just low enough for this turbine to drive the compressor. The gas is then expanded through the second turbine connected to the drive wheels. The data for this engine are shown in the figure. Consider the working fluid to be air throughout the entire cycle, and assume that all processes are ideal. Determine. a) The intermediate pressure P5. b) The net specific work output of the engine, and the mass flow rate through the engine. c) The air temperature entering the burner T3, and the thermal efficiency of the engine. 7

=150 kW

9. Consider an ideal gas turbine cycle with two stages of compression and two stages of expansion. The pressure ratio across each compressor stage and each turbine stage is 8 to 1. The pressure at the entrance to the first compressor is 100 kPa, the temperature entering each compressor is 20°C, and the temperature entering each turbine is 1100°C. An ideal regenerator is also incorporated into the cycle. Determine the compressor work, the turbine work, and the thermal efficiency of the cycle. 10.

Repeat Problem 9, but assume that each compressor stage and each turbine stage has an isentropic efficiency of 85%. Also assume that the regenerator has an efficiency of 70%.

2

Thermodynamics (ME22002), IIT Kharagpur, Spring 2006

Problem Set on Vapour Cycles 1. A steam power plant has a boiler exit at 4MPa, 5000C and a condenser exit temperature of 450C. Assume all components are ideal and find the cycle efficiency and the specific work and heat transfer in the components. 2. Consider a simple ideal Rankine cycle that uses steam as the working fluid. The high–pressure side of the cycle is at a supercritical pressure. Such a cycle has a potential advantage of minimizing local temperature differences between the fluids in the steam generator, such as the instance in which the high–temperature energy source is the hot exhaust gas from a gas–turbine engine. Calculate the thermal efficiency of the cycle if the state entering the turbine is 25 MPa, 5000C, and the condenser pressure is 5 kPa. What is the steam quality at the turbine exit? 3. Consider an ideal steam regenerative cycle in which steam enters the turbine at 3.5 MPa, 400°C, and exhausts to the condenser at 10 kPa. Steam is extracted from the turbine at 0.8 MPa and also at 0.2 MPa for heating the boiler feed water in two open feed water heaters. The feed water leaves each heater at the temperature of the condensing steam. The appropriate pumps are used for the water leaving the con- denser and the two feed water heaters. Calculate the thermal efficiency of the cycle and the net work per kilogram of steam. 4. Consider an ideal steam combined reheat and regenerative cycle in which steam enters the high-pressure turbine at 3.5 MPa, 400°C, and is extracted for feed water heating at 0.8 MPa. The remainder of the steam is reheated to 400°C at this pressure, 0.8 MPa, and is fed to the low-pressure turbine. Steam is extracted from the low-pressure turbine at 0.2 MPa for feed water heating. The condenser pressure is 10 kPa. Both feed water heaters are open heaters. Calculate the thermal efficiency of the cycle and the net work per kilogram of steam. 5. An ideal steam power plant is designed to operate on the combined reheat and regenerative cycle and to produce a net power output of 10 MW. Steam enters the high-pressure turbine at 8 MPa, 550°C, and is expanded to 0.6 MPa, at which pressure some of the steam is fed to an open feed water heater, and the remainder is reheated to 550°C. The reheated steam is then expanded in the low-pressure turbine to 10 kPa. a. Determine the steam flow rate to the high-pressure turbine. b. Determine the size of motor required to drive each of the pumps. c. If the increase in the condenser cooling water temperature is restricted to a maximum of 10°C, what is the flow rate of the cooling water? d. If the steam velocity in the turbine-condenser connecting pipe is restricted to a maximum of 100 m/s, what is the diameter of the connecting pipe?

6.

Steam leaves a power plant steam generator at 3.5 MPa, 400°C, and enters the turbine at 3.4 MPa, 375°C. The isentropic turbine efficiency is 88%, and the turbine exhaust pressure is 10 kPa. Condensate leaves the condenser and enters the pump at 35°C, 10 kPa. The isentropic pump efficiency is 80%, and the discharge pressure is 3.7 MPa. The feed water enters the steam generator at 3.6 MPa, 30°C. Calculate the following. a. The thermal efficiency of the cycle. b. The irreversibility of the process in the line between the steam generator exit and the turbine inlet, assuming an ambient temperature of 25°C.

7.

For the steam power plant described in Problem 1, assume the isentropic efficiencies of the turbine and pump are 85% and 80%, respectively. Find the component specific work and heat transfers and the cycle efficiency.

8.

Find the availability .of the water at all the states in the steam power plant described in the previous problem. Assume. The heat source in the boiler is at 600"C and the low-temperature reservoir is at 25°C. Give the second law efficiency of all the components.

9.

In a particular reheat-cycle power plant, steam enters the high-pressure turbine at 5 MPa, 450°C and expands to 0.5 MPa, after which it is reheated to 450°C. The steam is then expanded through the low-pressure turbine to 7.5 kPa. Liquid water leaves the condenser at 30°C, is pumped to 5 MPa, and then returned to the steam generator. Each turbine is adiabatic with an isentropic efficiency of 87% and the pump efficiency is 82%. If the total power output of the turbines is 10 MW, determine a. The mass flow rate of steam b. The pump power input c. The thermal efficiency of the power plant

10.

Consider an ideal refrigeration cycle that has a condenser temperature of 45°C and an evaporator temperature of -15°C. Determine the coefficient of performance of this refrigerator for the working fluids R-12 and R-22.

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Ideal gas mixtures: Amagat and Dalton model for mixture of ideal gases, Equation of state and properties of ideal gas mixtures, Change in entropy on mixing.

•

Non ideal mixtures: Partial molal properties, fugacity of a component in a mixture, changes in property on mixing, free energy of mixing, concept of an ideal solution, conditions of phase equilibrium and chemical equilibrium of multi-component systems, Gibbs phase rule.

7. Thermodynamics of Reactive Systems: First law analysis of reactive system; Internal energy and enthalpy of reaction; Enthalpy of formation; Second law applied to a reactive system. 8. Air Standard Cycles: Carnot, Stirling, Ericssion, Otto, Diesel, and Dual cycles. Brayton cycle: intercooling, reheating and regeneration. 9. Vapour Cycles: Carnot cycle; Simple Rankine cycle, Reheat and Regenerative cycles with open & closed feedwater heater; Ideal vapour compression refrigeration cycle. Required Text Van Wylen, Sonntag and Borgnakke: Fundamentals of Thermodynamics, 6th Edition, 2004 (John Wiley). Assessment of Students 1. 20% credit from two class tests & teacher’s assessment 2. 30% credit from the mid-semester examination (tentative syllabus: up to and including Thermodynamic property relations) 3. 50% credit from end-semester examination

Thermodynamics: ME22002 Course Instructor: Suman Chakraborty

ASSIGNMENT PROBLEMS SET 1 Chapters 3-5 (Fundamentals of Thermodynamics, 6th Ed., by Sonntag, Borgnakke and Van Wylen)

Problem numbers Chapter 3: 41, 44, 49, 57, 60, 65, 73, 78, 97, 99, 101, 103, 105, 107, 109, 113 Chapter 4: 33, 41, 42, 53, 54, 61, 63, 64, 67, 73, 77, 110 Chapter 5: 35, 48, 50, 57, 58, 62, 70, 73, 95, 99, 101, 130, 132, 134

Thermodynamics: ME22002 Course Instructor: Suman Chakraborty SOLUTIONS TO ASSIGNMENT PROBLEMS FROM Chapter 3 (Fundamentals of Thermodynamics, 6th Ed., by Sonntag, Borgnakke and Van Wylen)

Thermodynamics: ME22002 Course Instructor: Suman Chakraborty SOLUTIONS TO ASSIGNMENT PROBLEMS FROM Chapter 4 (Fundamentals of Thermodynamics, 6th Ed., by Sonntag, Borgnakke and Van Wylen)

Thermodynamics: ME22002 Course Instructor: Suman Chakraborty SOLUTIONS TO ASSIGNMENT PROBLEMS FROM Chapter 5 (Fundamentals of Thermodynamics, 6th Ed., by Sonntag, Borgnakke and Van Wylen)

Thermodynamics: ME22002 Course Instructor: Suman Chakraborty

ASSIGNMENT PROBLEMS SET 2 Chapters 6-9 (Fundamentals of Thermodynamics, 6th Ed., by Sonntag, Borgnakke and Van Wylen)

Problem numbers Chapter 6: 39, 57, 59, 75, 82, 88,105,107, 108,111, 115, 116, 117, 118,119, 120, 121, 122, 131,133, 134 Chapter 7: 38, 44, 48, 58, 62, 64, 65, 71, 76, 82, 84, 86, 88, 89, 90, 91 Chapter 8: 14, 15, 16, 33, 45, 52, 54, 65, 70, 71, 76, 82, 85, 86, 90, 98, 100, 101, 103, 109, 120, 126, 128, 129, 130, 131, 133, 134, 137 Chapter 9: 26, 28, 37, 38, 39, 41, 51, 68, 74, 77, 96, 101, 116, 119, 126, 127, 130, 132

Thermodynamics: ME22002 Course Instructor: Suman Chakraborty SOLUTIONS TO ASSIGNMENT PROBLEMS FROM Chapter 6 (Fundamentals of Thermodynamics, 6th Ed., by Sonntag, Borgnakke and Van Wylen)

Thermodynamics: ME22002 Course Instructor: Suman Chakraborty SOLUTIONS TO ASSIGNMENT PROBLEMS FROM Chapter 7 (Fundamentals of Thermodynamics, 6th Ed., by Sonntag, Borgnakke and Van Wylen)

Thermodynamics: ME22002 Course Instructor: Suman Chakraborty SOLUTIONS TO ASSIGNMENT PROBLEMS FROM Chapter 8(Fundamentals of Thermodynamics, 6th Ed., by Sonntag, Borgnakke and Van Wylen)

Thermodynamics: ME22002 Course Instructor: Suman Chakraborty SOLUTIONS TO ASSIGNMENT PROBLEMS FROM Chapter 9 (Fundamentals of Thermodynamics, 6th Ed., by Sonntag, Borgnakke and Van Wylen)

Thermodynamics: ME22002 Course Instructor: Suman Chakraborty SOLUTIONS TO ASSIGNMENT PROBLEMS FROM Chapter 10 (Fundamentals of Thermodynamics, 6th Ed., by Sonntag, Borgnakke and Van Wylen)

Thermodynamics: ME22002 Course Instructor: Suman Chakraborty SOLUTIONS TO ASSIGNMENT PROBLEMS FROM Chapter 12 (Fundamentals of Thermodynamics, 6th Ed., by Sonntag, Borgnakke and Van Wylen)

Thermodynamics: ME22002 Course Instructor: Suman Chakraborty SOLUTIONS TO ASSIGNMENT PROBLEMS FROM Chapter 13 (Fundamentals of Thermodynamics, 6th Ed., by Sonntag, Borgnakke and Van Wylen)

Thermodynamics: ME22002 Course Instructor: Suman Chakraborty

ASSIGNMENT PROBLEMS CHAPTER 15 (Fundamentals of Thermodynamics, 6th Ed., by Sonntag, Borgnakke and Van Wylen)

Problem numbers 32, 38, 49, 52, 54, 60, 62, 72, 80

Thermodynamics: ME22002 Course Instructor: Suman Chakraborty SOLUTIONS TO ASSIGNMENT PROBLEMS FROM Chapter 15 (Fundamentals of Thermodynamics, 6th Ed., by Sonntag, Borgnakke and Van Wylen)

Thermodynamics (ME22002), IIT Kharagpur, Spring 2006 Problem set on Properties of Mixtures and Phases Q1. Equation of state of a non-ideal 2-component gas mixture is given as: RT v= + y12 B11 + 2 y1 y2 B12 + y22 B22 , where yi is the mole fraction of the ith component in p the mixture. For a binary mixture of 10 mole % chloroform (component 1) in acetone (component 2) at 333K and 10 bar, the coefficients Bij are given as follows: B11= -910 cm3/mol, B22= -1330 cm3/mol, and B12 = -2005 cm3/mol. Determine: (i) Molar specific volume of pure chloroform (ii) Partial molal volume of chloroform in the mixture, and (iii) Change in volume on mixing. (Ans: (i) 1860 cm3/mol, (ii) 991 cm3/mol, (iii) -372 cm3/mol) Q2. (a) For a binary system (constituents A + B) at constant temperature and pressure, the molar volume (cm3/mol) is obtained in terms of the respective mole fractions as: v = 100yA + 80yB +2.5yAyB, where yA is the mole fraction of the A and yB is the mole fraction of the B in the mixture. (i) What is the partial molal volume of the component A in the mixture, as the mole fraction of A tends to zero, in a limiting sense? (ii) Illustrate with a sketch, how would you obtain the above result from a graphical plot of molar volume versus mole fraction of the component B? No explicit numerical calculations are necessary. (iii) Obtain an expression for change in volume due to mixing, as a function of mole fraction of individual constituents. What is the mole fraction for which this change is a maximum? Does the volume increase or decrease, on account of mixing? (Ans: (i) 102.5 cm3/mol, (iii) ∆Vmix = 2.5 y A yB > 0 ) Q3. At 21°C, the enthalpy of mixing of sulphuric acid and water can be fit to the following equation: ∆H mix = −74.4 y H 2 SO4 yH 2O (1 − 0.561 yH 2 SO4 ) kJ/mol. Molar enthalpies of pure sulphuric acid and water are 1.596 kJ/mol and 1.591 kJ/mol, respectively. For an equimolar mixture of the two components, obtain the partial molal enthalpies of sulphuric acid and water. (Ans: -6.6 kJ/mol, -17 kJ/mol)) Q4. For the van der Waals gas, ⎡ (v − b) p ⎤ ⎡f⎤ b 2a − ln ⎢ ⎥ = − ln ⎢ ⎥+ ⎣ p⎦ ⎣ RT ⎦ v − b RTv

show

that

the

fugacity is

given by

Q5. For a binary gas mixture composed of species a and b, an equivalent form of van der RT ( na + nb ) na2 aa + 2na nb aa ab + nb2 ab Waals equation of state is given as p = − . V − ( na ba + nb bb ) V2 Show that the fugacity of the component a in the gas mixture is given by

(

)

2 ya na + yb aa ab ⎡ ( v − bmix ) p ⎤ ⎡ f ⎤ ba , where v is the mixture + − ln ⎢ a ⎥ = − ln ⎢ ⎥ RT RTv ⎣ ya p ⎦ ⎣ ⎦ v − bmix specific volume and bmix = ya ba + yb bb .

Q6. Derive the Clapeyron equation from the considerations of equilibrium of a two-phase single component system. Q7. What pressure is required to make diamond from graphite at 25°C? For your calculations, following data are given for a temperature of 25°C and pressure of 0.1 MPa: Graphite Diamond g 0 2867.8 kJ/kmol 3 0.000284 m /kg 0.000284 m3/kg v 0.016×10-6 1/MPa 0.304×10-6 1/MPa βT (Ans: 1493 MPa) Q8. Air (~21% O2, 79% N2) is cooled to 80 K, 0.1 MPa. Calculate the composition of liquid and vapour phases in this condition. Given: saturation pressure of O2 = 0.137 MPa and saturation pressure of N2 = 0.03006 MPa. (Ans: mole fraction of O2 in liquid phase =0.654 and in vapour phase = 0.896) Q9. Determine the mole fraction of air at the surface of a lake whose temperature is 17°C. Take the atmospheric pressure at the lake level to be 92 kPa. Given, Henry’s constant for air dissolved in water at 290 K= 62000 bar. (Ans: 1.45×10-5) Q10. Fresh water is to be obtained from sea water at 15°C, with a salinity of 3.48% on mass basis. Determine the minimum work input required to separate 1 kg of sea water completely into pure water and pure salt. State any assumptions you make. Given: molecular weight of water= 18, molecular weight of salt= 58.44. (Ans: 7.87 kJ/kg of seawater, assumption: the mixture behaves as an ideal solution)

Thermodynamics (ME22002), IIT Kharagpur, Spring 2006

Problem Set on Air Standard Cycles 1.

A stoichiometric mixture of gasoline and air has an energy release upon combustion of approximately 2800 kJ/kg of the mixture. To approximate an actual spark- ignition engine using such a mixture, consider an air-standard Otto cycle that has a heat addition of 2800 kJ/kg of air, a compression ratio of 7, and a pressure and temperature at the beginning of the compression process of 90 kPa, 10°C. Assuming constant specific heat, with the value from Table A.l0, determine a) The maximum pressure and temperature of the cycle. b) The thermal efficiency of the cycle. c) The mean effective pressure.

2. In the air-standard Otto cycle, all the heat transfer qh occurs at constant volume. It would be more realistic to assume that part of qh occurs after the piston has started its downward motion in the expansion stroke. Therefore, consider a cycle identical to the Otto cycle, except that the first two-thirds of the total qh occurs at constant volume and the last one-third occurs at constant pressure. Assume that the total qh is 2400 kJ/kg, that the pressure and temperature at the beginning of the compression process are 90 kPa, 20°C, and that the comparison ratio is 7 Calculate the maximum pressure and temperature and the thermal efficiency of this cycle. Compare the results with those of a conventional Otto cycle having the same given variables. 3. Consider an ideal air-standard diesel cycle in which the state before the compression process is 95 kPa, 290 K, and the compression ratio is 20. What maximum temperature must the cycle have to have a thermal efficiency of 60%? 4. An air-standard Ericsson cycle has an ideal regenerator. Heat is supplied at 1000°C and heat is rejected at 20°C. Pressure at the beginning of the isothermal compression process is 70 kPa. The heat added is 600 kJ/kg. Find the compressor work, the turbine work, and the cycle efficiency. 5. Consider an ideal Stirling-cycle engine in which the pressure and temperature at the beginning of the isothermal compression process are 100kPa, 25°C, the compression ratio is 6, and the maximum temperature in the cycle is 1100°C. Calculate a) The maximum pressure in the cycle. b) The thermal efficiency of the cycle with and without regenerators. 6. Consider a large stationary gas-turbine power plant that operates on the ideal Brayton cycle and delivers a power output of 100 MW to an electric generator. The minimum temperature in the cycle is 300 K, and the maximum temperature is

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1600 K. The minimum pressure in the cycle is 100 kPa, and the compressor pressure ratio is 14 to 1. a) Calculate the power output of the turbine. What fraction of the turbine output is required to drive the compressor? b) What is the thermal efficiency of the cycle? 7. Repeat Problem 6, but assuming that the compressor has an isentropic efficiency of 85% and the turbine an isentropic efficiency of 88%. 8. The gas turbine cycle shown in the figure below is to be used as an automotive engine. In the first turbine, the gas expands to a pressure P5, just low enough for this turbine to drive the compressor. The gas is then expanded through the second turbine connected to the drive wheels. The data for this engine are shown in the figure. Consider the working fluid to be air throughout the entire cycle, and assume that all processes are ideal. Determine. a) The intermediate pressure P5. b) The net specific work output of the engine, and the mass flow rate through the engine. c) The air temperature entering the burner T3, and the thermal efficiency of the engine. 7

=150 kW

9. Consider an ideal gas turbine cycle with two stages of compression and two stages of expansion. The pressure ratio across each compressor stage and each turbine stage is 8 to 1. The pressure at the entrance to the first compressor is 100 kPa, the temperature entering each compressor is 20°C, and the temperature entering each turbine is 1100°C. An ideal regenerator is also incorporated into the cycle. Determine the compressor work, the turbine work, and the thermal efficiency of the cycle. 10.

Repeat Problem 9, but assume that each compressor stage and each turbine stage has an isentropic efficiency of 85%. Also assume that the regenerator has an efficiency of 70%.

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Thermodynamics (ME22002), IIT Kharagpur, Spring 2006

Problem Set on Vapour Cycles 1. A steam power plant has a boiler exit at 4MPa, 5000C and a condenser exit temperature of 450C. Assume all components are ideal and find the cycle efficiency and the specific work and heat transfer in the components. 2. Consider a simple ideal Rankine cycle that uses steam as the working fluid. The high–pressure side of the cycle is at a supercritical pressure. Such a cycle has a potential advantage of minimizing local temperature differences between the fluids in the steam generator, such as the instance in which the high–temperature energy source is the hot exhaust gas from a gas–turbine engine. Calculate the thermal efficiency of the cycle if the state entering the turbine is 25 MPa, 5000C, and the condenser pressure is 5 kPa. What is the steam quality at the turbine exit? 3. Consider an ideal steam regenerative cycle in which steam enters the turbine at 3.5 MPa, 400°C, and exhausts to the condenser at 10 kPa. Steam is extracted from the turbine at 0.8 MPa and also at 0.2 MPa for heating the boiler feed water in two open feed water heaters. The feed water leaves each heater at the temperature of the condensing steam. The appropriate pumps are used for the water leaving the con- denser and the two feed water heaters. Calculate the thermal efficiency of the cycle and the net work per kilogram of steam. 4. Consider an ideal steam combined reheat and regenerative cycle in which steam enters the high-pressure turbine at 3.5 MPa, 400°C, and is extracted for feed water heating at 0.8 MPa. The remainder of the steam is reheated to 400°C at this pressure, 0.8 MPa, and is fed to the low-pressure turbine. Steam is extracted from the low-pressure turbine at 0.2 MPa for feed water heating. The condenser pressure is 10 kPa. Both feed water heaters are open heaters. Calculate the thermal efficiency of the cycle and the net work per kilogram of steam. 5. An ideal steam power plant is designed to operate on the combined reheat and regenerative cycle and to produce a net power output of 10 MW. Steam enters the high-pressure turbine at 8 MPa, 550°C, and is expanded to 0.6 MPa, at which pressure some of the steam is fed to an open feed water heater, and the remainder is reheated to 550°C. The reheated steam is then expanded in the low-pressure turbine to 10 kPa. a. Determine the steam flow rate to the high-pressure turbine. b. Determine the size of motor required to drive each of the pumps. c. If the increase in the condenser cooling water temperature is restricted to a maximum of 10°C, what is the flow rate of the cooling water? d. If the steam velocity in the turbine-condenser connecting pipe is restricted to a maximum of 100 m/s, what is the diameter of the connecting pipe?

6.

Steam leaves a power plant steam generator at 3.5 MPa, 400°C, and enters the turbine at 3.4 MPa, 375°C. The isentropic turbine efficiency is 88%, and the turbine exhaust pressure is 10 kPa. Condensate leaves the condenser and enters the pump at 35°C, 10 kPa. The isentropic pump efficiency is 80%, and the discharge pressure is 3.7 MPa. The feed water enters the steam generator at 3.6 MPa, 30°C. Calculate the following. a. The thermal efficiency of the cycle. b. The irreversibility of the process in the line between the steam generator exit and the turbine inlet, assuming an ambient temperature of 25°C.

7.

For the steam power plant described in Problem 1, assume the isentropic efficiencies of the turbine and pump are 85% and 80%, respectively. Find the component specific work and heat transfers and the cycle efficiency.

8.

Find the availability .of the water at all the states in the steam power plant described in the previous problem. Assume. The heat source in the boiler is at 600"C and the low-temperature reservoir is at 25°C. Give the second law efficiency of all the components.

9.

In a particular reheat-cycle power plant, steam enters the high-pressure turbine at 5 MPa, 450°C and expands to 0.5 MPa, after which it is reheated to 450°C. The steam is then expanded through the low-pressure turbine to 7.5 kPa. Liquid water leaves the condenser at 30°C, is pumped to 5 MPa, and then returned to the steam generator. Each turbine is adiabatic with an isentropic efficiency of 87% and the pump efficiency is 82%. If the total power output of the turbines is 10 MW, determine a. The mass flow rate of steam b. The pump power input c. The thermal efficiency of the power plant

10.

Consider an ideal refrigeration cycle that has a condenser temperature of 45°C and an evaporator temperature of -15°C. Determine the coefficient of performance of this refrigerator for the working fluids R-12 and R-22.

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