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PPT Contains basics of thermodynamics, laws of thermodynamics, SFEE

- 1. PREPARED BY: PRADEEP KUMAR GUPTA ASSISTANT PROFESSOR DEPARTMENT OF MECHANICAL ENGINEERING
- 2. Table of Content • Introduction • Microscopic and Macroscopic Approach • Microscopic Approach • Macroscopic Approach • Difference between Microscopic and Macroscopic Approach • Definitions of thermodynamics terminology • Surrounding • Boundary • Universe • Thermodynamic systems • State of system • Classification of thermodynamic systems • Concept of Continuum • Equilibrium • Thermodynamic Properties of a system • Classification of properties of a thermodynamic system • Path • Process • cycle • Quasi-Static Process • Energy and its forms • Work • Power • Heat • Types of Heat • Laws of thermodynamics • Zeroth law of thermodynamics • Thermometry • Definition of temperature • Temperature Scales • Conversion from Celsius to Fahrenheit or Fahrenheit to Celsius • Conversion from Celsius to Kelvin • Relation between Celsius, Fahrenheit and Kelvin • First law of thermodynamics • First law of thermodynamics for a cyclic process • Internal energy and Enthalpy • Thermodynamic Processes and calculation of work • Joule’s experiment • First law for non flow process
- 3. Table of Content continued…………… • Constant volume process (Isochoric process) • Constant pressure process (isobaric process) • Constant Temperature process (isothermal process) • Adiabatic process (isentropic process) • Polytropic process • Steady Flow Energy Equation (S.F.E.E.) • Applications of Steady flow energy equation • Throttling Process
- 4. Definition of Thermodynamics Thermodynamics is a branch of science which deals with energy, phenomena of energy and related properties of matter, especially of laws of transformation of heat into other forms of energy and vice versa. Microscopic and Macroscopic Approach Microscopic Approach This approach considers that the system is made up of a very large numbers of the discrete particles known as molecules. These molecules have different velocities and energies. The behaviour of system is found by using statistical method as the number of molecules is very large. The properties like velocity, momentum, impulse, kinetic energy etc, which describes the molecule cannot be easily measured by instruments. Large number of variables is needed to describe such a system. So approach is complicated.
- 5. Macroscopic Approach: In this approach, we do not follow the behavior of individual molecules but study the properties of particular mass of the substances. The analysis of macroscopic system requires simple mathematical formulae. The values of the properties of system are their average values. Only few properties are needed to describe such a system.
- 6. Difference between Microscopic and Macroscopic Approach Sr. No. Microscopic Approach Macroscopic Approach 1 This approach considers that the system is made up of a very large numbers of the discrete particles known as molecules. These molecules have different velocities and energies. In this approach, the behaviour of individual molecules is not considered but studies the properties of particular mass of the substances. 2 The behaviour of system is found by using statistical method as the number of molecules is very large. The analysis of macroscopic system requires simple mathematical formulae. 3 The properties like velocity, momentum, impulse, kinetic energy etc, which describes the molecule cannot be easily measured by instruments. The properties like temperature and pressure are required to describe the system can be easily measured. 4 Large number of variables is needed to describe such a system. So approach is complicated. Only few properties are needed to describe such a system.
- 7. Definitions of thermodynamics terminology Thermodynamic systems: A thermodynamic system may be defined as the quantity of matter or definite region in space upon which some thermodynamic process is taking place. Thermodynamic systems are defined by using a real or imaginary boundary. Anything beyond real or imaginary boundary is known as surroundings. Surrounding: The space outside the thermodynamic system is known as surrounding. Boundary: The line separating the system and surrounding is known as boundary. Universe: The combination of system , surrounding and boundary is known as universe. State of system: A state is a macroscopic condition of a thermodynamic system as described by its particular thermodynamic parameters. Some thermodynamic parameters are pressure, temperature, density, composition etc.
- 8. Classification of thermodynamic systems Thermodynamic systems may be broadly classified in three categories: 1. Open system 2. Closed system 3. Isolated system 1.Open system: Open system is one in which matter (mass of working substance) as well as energy (heat and work) crosses the boundary of the system. As shown by figure (a) energy as well as water vapour is coming out from the system. 2.Closed system: Closed system is one in which only energy (heat and work) crosses the boundary of the system without adding or losing of matter (mass of working substance). As shown by fig (b). 3.Isolated system: In an isolated system neither matter (mass of working substance) nor energy (heat and work) crosses the boundary of the system. As shown by fig (c).
- 9. Equilibrium Equilibrium indicates the state of balance. In an equilibrium state there are no unbalanced potentials within the system. Equilibrium may be classified as: (i) Chemical Equilibrium (ii) Mechanical Equilibrium (iii) Thermal Equilibrium (i)Chemical Equilibrium: If there is no chemical reaction or diffusion of matter from one part of the system to another, the system is said to be in chemical equilibrium. (ii)Mechanical Equilibrium: If there are no unbalanced forces in the system, the system is said to be in mechanical equilibrium. (iii)Thermal Equilibrium: When a system is prevailing in chemical and mechanical equilibrium is separated from its surroundings by a diathermic wall and if no spontaneous change in any property of the system, the system is said to be in state of thermal equilibrium. Thermodynamic Properties of a system Properties are those characteristics of the system which can be used for defining the system. Such as volume, pressure, temperature, viscosity etc. Classification of properties of a thermodynamic system The thermodynamic properties may be classified into two categories: 1. Intensive property 2. Extensive property
- 10. 1.Intensive property: Intensive properties are those properties which have same value for any part of the system or these are those properties that are independent of the mass of the system. Such as temperature, pressure and density. 2.Extensive property: Extensive properties are those properties which depend upon the mass of the system and do not maintain the same value for any path of the system. Such as mass, enthalpy, volume and energy etc. Note: The ratio of extensive property of the system to the mass of the system gives the intensive property. Such as the ratio of total volume (V) of the system to its total mass (m) is known as specific volume. = V/m …………it is an intensive property. Path: If all the changes of states of the system are plotted, then line joining the change of states of the system is known as path. Process: A process is a complete description of change of state of a thermodynamic system through a specified path. cycle: A thermodynamic cycle is defined as the series of state of changes such that the intial state is identical with the final state. sv
- 11. Quasi-Static Process Consider a system which contained gas in a cylinder in fig. Initially it is in an equilibrium state, represents the properties P1, v1, T1. The weight on the piston just balance the force exerted by the gas. When weight is removed from the piston the system become unbalanced. The unbalanced force is between the system and the surrounding, and gas pressure will moves the piston in upward direction till it hits the stop.
- 12. The system again comes to an equilibrium states, being described by the properties P2, v2, T2. But the immediate states passed through by the system are non-equilibrium states which cannot be described by thermodynamic coordinates. Figure shows the points 1 and 2 as the initial and final equilibrium states joined by dotted line. Now if the single weight on the piston is made up of many very small pieces of weights and these weights are removed one by one very slowly, at any instant of the upward travel of the piston, if the gas is isolated, the departure of the state of the system from thermodynamic equilibrium state will be infinitesimally small. So every state passed through by the system will be an equilibrium state.
- 13. Energy and its forms Energy is defined as the capacity to do work or energy can also be defined as the capacity to exert a force in a given direction through a distance. The unit of energy in SI (System international) system is Nm or Joule (J). Forms of Energy 1. Work 2. Heat Work Work is one of the basic modes of energy transfer. In mechanics the action of a force on a moving body is identified as work. The work is done by a force as it acts upon a body moving in the direction of force. In thermodynamics, work transfer is considered as occurring between the system and the surroundings. Work is said to be done by a system if the sole effect on the things external to the system can be reduced to the raising of a weight. The work is done by a system, it is taken to be positive, and when work is done on a system, it is taken to be negative.
- 14. (a)Work is Positive (+ve) (b) Work is negative (-ve) Power: The rate of energy transfer is known as power or the rate of work transfer is known as power. The unit of power is J/s or Watt. Heat Heat is defined as the form of energy that is transferred across a boundary by virtue of a temperature difference. The temperature difference is the potential or force and heat transfer is the flux. Heat flow into a system is taken to be positive, and heat flow out of a system is taken as negative.
- 15. A process in which no transfer of heat through boundary is known as adiabatic process. (a)Heat transfer is Positive (+ve) (b) Heat transfer is negative (-ve) The symbol used for heat transfer is Q. The unit of heat transfer in SI (System international) system is Nm or Joule (J). The rate of heat transfer is given by W or kW. Types of Heat 1. Specific Heat: Specific heat is defined as the amount of heat required to raise the temperature of a unit mass (1kg) of the substance by unit degree (1oC or 1K) change in temperature. The quantity of heat absorbed or rejected by a system during heating or cooling is measured by the formula as given below: Q=m×c×(T2-T1) Where, Q= heat gain or loose by the system in kJ, m= mass of the substance in kilograms (kg), c= specific heat in kJ/kgK (T2-T1)= Temperature rise or drop in degree Celsius or Kelvin.
- 16. Types of specific heat: Basically there are two types of specific heats as given below: (i)Specific heat at constant pressure (cp) (ii)Specific heat at constant volume (cv) (i)Specific heat at constant pressure (cp): It is defined as the amount of heat required to raise the temperature of a unit mass (1kg) of the substance by unit degree (1oC or 1K) change in temperature when the pressure is constant. It is represented by cp. Its unit is kJ/kgK. (ii)Specific heat at constant volume (cv): It is defined as the amount of heat required to raise the temperature of a unit mass (1kg) of the substance by unit degree (1oC or 1K) change in temperature when the volume is constant. It is represented by cv. Its unit is kJ/kgK. Specific heat of water: c=4.186 kJ/kgK Specific heats of air: cp=1.005 kJ/kgK cv=1.005 kJ/kgK 2.Latent heat of vaporization: It is defined as the amount of heat required to evaporated one kilogram of water at its saturation temperature (boiling point) without change of temperature. It is represented by hfg. Its unit is kJ/kg. The latent heat of vaporization of water or latent heat of steam is 2257 kJ/kg.
- 17. Laws of thermodynamics There are three laws of thermodynamics given as under: 1.Zeroth law of thermodynamics 2.First law of thermodynamics 3.Second law of thermodynamics Zeroth law of thermodynamics: Zeroth law states that if two systems are at same time in thermal equilibrium with a third system, they are in equilibrium with each other. If the system A and B are in thermal equilibrium with a third system C separately then the two systems A and B will also be in thermal equilibrium with each other.
- 18. Thermometry: Thermometry is defined as that branch of science, in which the temperature is measured with accuracy and precision. Definition of temperature: Temperature is defined as the measure of hotness and coldness of a substance with reference to a standard value. Temperature Scales There are three types of temperature scales for the measurement of temperature. (a) Celsius, (b) Fahrenheit and (c) Kelvin. Celsius: Swedish astronomer Anders Celsius in 1742. It is also called as centigrade temperature scale, in this scale freezing point of water is represented by 0 degree and boiling point is represented by 100 degree. It has 100-degree intervals between the defined points so that sometimes it is called the centigrade scale. Fahrenheit: German physicist Daniel Gabriel Fahrenheit in 18th century. In this scale freezing point of water is 32 and boiling point of water is 212. The interval between the two (32-212) being divided into 180 parts. Kelvin: British physicist William Thomson, Baron Kelvin. It is defined as 1/ 273.16 of the triple point (equilibrium among the solid, liquid, and gaseous phases) of pure water. The Kelvin is written by symbol K with using degree (o). This scale has as its zero point absolute zero, the theoretical temperature at which the molecules of a substance have the lowest energy. The difference between the freezing and boiling points of water is 100 degrees in each, so that the Kelvin has the same magnitude as the degree Celsius.
- 20. Relation between Celsius, Fahrenheit and Kelvin 𝐶 100 = (𝐹 − 32) 180 = (𝐾 − 273) 100 𝐶 5 = (𝐹 − 32) 9 = (𝐾 − 273) 5
- 21. First law of thermodynamics First law of thermodynamics also states that, “the energy can neither be created nor be destroyed it can only be transformed from one form to another.” According to this law, when a system undergoes a thermodynamic process, both heat and work transfer takes place. The net energy is stored within the system and is termed as stored energy or total energy of the system. Mathematically it is written as: δQ-δW=dE First law of thermodynamics for a cyclic process A process is cyclic if the initial and final states of the system are identical. A system represented by state 1 undergoes a process 1-r-2 and returns to the initial state following the path 2-s-1. All the properties of the system are restored, when the initial and final state is reached. During the completion of these processes: (a) Area 2-3-4-1-s-2 denotes the work done W1 by the system during expansion process 2-s-1. (b) Area 4-3-1-s-4 denotes the work done W2 supplied to the system during the compression process 4-s-1. (c) Area 1-r-2-s-1 denotes the net work done (W1-W2) delivered by the system.
- 22. According to first law of thermodynamics, “when a closed system undergoes a thermodynamic cycle, the net heat transfer is equal to net work done.” Or “The cyclic integral of heat transfer is equal to cyclic integral of work done.” Mathematically it is written as: 𝛿𝑄 = 𝛿𝑊 On integrating the above equation for a thermodynamic state 1 to 2, we get, 1 2 𝛿𝑄 − 1 2 𝛿𝑊 = 1 2 𝑑𝐸 𝑄1−2 − 𝑊1−2 = 𝐸2 − 𝐸1 Where, Q1-2 = heat transferred to the system during the process 1 to 2. W1-2= Work transfer by the system during the process 1 to 2. E1 = Total energy of the system at state 1 E2 = Total energy of the system at state 2
- 23. Note: The total energy is the sum of potential energy, kinetic energy and internal energy of the system. It is mathematically written as: 𝐸 = 𝑃. 𝐸. +𝐾. 𝐸. +𝑈 𝐸 = 𝑚𝑔𝑧 + 𝑚𝑣2 2 + 𝑈 Where, P.E. = Potential energy, K.E. = Kinetic energy, U = Internal Energy. Internal Energy: Internal energy of steam is define as the energy stored in the steam, above 0oC (freezing point) of water. It may be obtained by subtracting the work done during evaporation to the enthalpy of steam. It is represented by U. Mathematically it is written as, Internal energy of steam=Enthalpy of steam-Workdone during evaporation Enthalpy: It is defined as the amount of heat absorbed by water from 0oC (freezing point) to saturation point (sensible heat) plus heat absorbed during evaporation (latent heat). It is represented by hg. So that, Enthalpy=sensible heat + latent heat
- 25. First law for non flow process In thermodynamics there are number of processes where in one or another state parameter remains constant. The basic thermodynamic processes used to analyze, (i) Relationship between various parameters such as temperature, pressure and volume, (ii) For obtaining the work and heat in the process, and (iii) For obtaining the alteration in internal energy.
- 26. Constant volume process (Isochoric process) • An Isochoric process is a process during which the specific volume v remains constant. Some facts about constant volume process Pressure, volume and temperature relationship For the initial state 1: 𝑃1 𝑣1 = 𝑚𝑅𝑇1 For the final state 2: 𝑃2 𝑣2 = 𝑚𝑅𝑇2
- 27. We know that from general gas equation, 𝑃1 𝑣1 𝑇1 = 𝑃2 𝑣2 𝑇2 Since during the process specific volume is constant (v1=v2), so that 𝑃1 𝑇1 = 𝑃2 𝑇2 𝑜𝑟 𝑃 𝑇 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 Since there is no expansion of gas (dV=0), no work is done on the system or by the system. From non flow energy equation 𝛿𝑄 = 𝛿𝑊 + 𝛿𝑈 Since 𝛿𝑊 = 0 all the heat is used to change the internal energy of the system. Therefore, 𝛿𝑄 = 𝛿𝑈 Heat added during a constant volume process is given by 𝛿𝑄 = 𝑚𝑐 𝑣(𝑇2 − 𝑇1) Or it may be written as 𝛿𝑄 = 𝛿𝑈 = 𝑚𝑐 𝑣(𝑇2 − 𝑇1) Where cv= specific heat at constant volume For unit mass (i.e. m=1) 𝑑𝑈 = 𝑐 𝑣 𝑑𝑇 Or 𝑐 𝑣 = 𝑑𝑈 𝑑𝑇
- 28. Constant pressure process (isobaric process) An Isobaric process is one during which the pressure P remains constant. Pressure, volume and temperature relationship For the initial state 1: 𝑃1 𝑣1 = 𝑚𝑅𝑇1 For the final state 2: 𝑃2 𝑣2 = 𝑚𝑅𝑇2 We know that from general gas equation, 𝑃1 𝑣1 𝑇1 = 𝑃2 𝑣2 𝑇2 Since during the process pressure is constant (P1=P2), so that 𝑣1 𝑇1 = 𝑣2 𝑇2 𝑜𝑟 𝑣 𝑇 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 Mechanical work, 𝑊1−2 = 1 2 𝑃𝑑𝑣 = 𝑃(𝑣2 − 𝑣1) For a non flow process, 𝛿𝑄 = 𝛿𝑊 + 𝛿𝑈 Q1−2 = 𝑃 𝑣2 − 𝑣1 + 𝑈2 − 𝑈1 Q1−2 = 𝑈2 + 𝑃2 𝑣2 − 𝑈1 + 𝑃1 𝑣1
- 29. Q1−2 = (ℎ2 − ℎ1) Since, ℎ = 𝑈 + 𝑃𝑣 ℎ2 − ℎ1 = Q1−2 = 𝑚𝑐 𝑝(𝑇2 − 𝑇1) Where cp= specific heat at constant pressure For unit mass (i.e. m=1) 𝑑ℎ = 𝑐 𝑝 𝑑𝑇 Or 𝑐 𝑝 = 𝑑ℎ 𝑑𝑇 Significance of gas constant R: During constant pressure process, Work done, 𝑊1−2 = 1 2 𝑃𝑑𝑣 = 𝑃(𝑣2 − 𝑣1) Since P1=P2=P; 𝑃1 𝑣1 = 𝑅𝑇1 𝑃2 𝑣2 = 𝑅𝑇2 𝑊1−2 = 𝑃(𝑣2 − 𝑣1) 𝑊1−2 = 𝑅(𝑇2 − 𝑇1) 𝑅 = 𝑊1−2 (𝑇2 − 𝑇1) Thus the gas constant is equal to the work of 1 kg of gas in an isobaric process when the temperature changes by 1 degree.
- 30. Relationship between specific heats (cp and cv) and gas constant R: Let an unit mass of an ideal gas undergo constant volume and constant pressure processes separately through a temperature range from T1 to T2. During isochoric process: 𝑞1−2 = Q1−2 𝑚 = 𝑐 𝑣(𝑇2 − 𝑇1) And 𝑊1−2 = 0 And 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑖𝑛𝑡𝑒𝑟𝑛𝑎𝑙 𝑒𝑛𝑒𝑟𝑔𝑦 𝑈2 − 𝑈1 = 𝑐 𝑣(𝑇2 − 𝑇1) During isobaric process: 𝑞1−2 = Q1−2 𝑚 = 𝑐 𝑝(𝑇2 − 𝑇1) 𝑊1−2 = 𝑃(𝑣2 − 𝑣1) Change in internal energy, Q1−2 = 𝑃 𝑣2 − 𝑣1 + 𝑈2 − 𝑈1 𝑈2 − 𝑈1 = Q1−2 − 𝑃 𝑣2 − 𝑣1 𝑈2 − 𝑈1 = 𝑐 𝑝(𝑇2 − 𝑇1) − 𝑃 𝑣2 − 𝑣1 For an ideal gas internal energy is a function of temperature {U=f(T)}. So that equating the two internal energy equations; 𝑐 𝑝 𝑇2 − 𝑇1 − 𝑃 𝑣2 − 𝑣1 = 𝑐 𝑣(𝑇2 − 𝑇1) 𝑐 𝑝 𝑇2 − 𝑇1 − 𝑅(𝑇2 − 𝑇1) = 𝑐 𝑣(𝑇2 − 𝑇1) {Since, 𝑃1 = 𝑃2 = 𝑃; 𝑃1 𝑣1 = 𝑅𝑇1; 𝑃2 𝑣2 = 𝑅𝑇2} 𝑐 𝑝 − 𝑅 = 𝑐 𝑣 Or 𝒄 𝒑 − 𝒄 𝒗 = 𝑹 The ratio cp/cv is known as isentropic index and is expressed by 𝛾. 𝜸 = 𝒄 𝒑 𝒄 𝒗 We can write from the above to relations, 𝒄 𝒑 = 𝑹 𝜸 𝜸−𝟏 , 𝒄 𝒗 = 𝑹 𝜸−𝟏
- 31. Constant Temperature process (isothermal process) An isothermal process is one during which temperature T remains constant. Pressure, volume and temperature relationship For the initial state 1: 𝑃1 𝑣1 = 𝑚𝑅𝑇1 For the final state 2: 𝑃2 𝑣2 = 𝑚𝑅𝑇2 We know that from general gas equation, 𝑃1 𝑣1 𝑇1 = 𝑃2 𝑣2 𝑇2 Since during the process temperature is constant (T1=T2), so that 𝑃1 𝑣1 = 𝑃2 𝑣2 𝑜𝑟 𝑃𝑣 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 Work done, 𝑊1−2 = 1 2 𝑃𝑑𝑣 Since, 𝑃𝑣 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 (𝐾) 𝑃 = 𝐾 𝑣
- 32. Substituting the value of P in the equation of work done, 𝑊1−2 = 1 2 𝐾 𝑣 𝑑𝑣 = 𝐾 1 2 1 𝑣 𝑑𝑣 𝑊1−2 = 𝐾 ln(𝑣) 1 2 𝑊1−2 = 𝐾{𝑙𝑛 𝑣2 − 𝑙𝑛 𝑣1 } 𝑾 𝟏−𝟐 = 𝑷 𝟏 𝒗 𝟏 𝒍𝒏 𝒗 𝟐 𝒗 𝟏 = 𝑷 𝟐 𝒗 𝟐 𝒍𝒏 𝒗 𝟐 𝒗 𝟏 Or 𝑾 𝟏−𝟐 = 𝑷 𝟏 𝒗 𝟏 𝒍𝒏 𝑷 𝟏 𝑷 𝟐 = 𝑷 𝟐 𝒗 𝟐 𝒍𝒏 𝑷 𝟏 𝑷 𝟐 {Since, 𝑃1 𝑣1 = 𝑃2 𝑣2}
- 33. Adiabatic process (isentropic process) An adiabatic process is one in which no heat is gained or lost by the system during its expansion or compression. This will happen when the working substance remains thermally insulated, so that no heat enters or leaves it during the process. It may be noted that adiabatic process may be reversible or irreversible. The reversible adiabatic process (frictionless adiabatic process) is known as isentropic process or constant entropy process. But if the friction is involved in the process, then the adiabatic process is irreversible, in this case entropy does not remain constant. Some facts about isentropic process: i. No heat enters or leaves the working substance. ii. The temperature of the gas changes. iii. The change in internal energy is equal to the work done. iv. It is expressed by the relation Pvγ= constant. Where γ is the isentropic index and its valve is 1.4. Also, γ= cp/cv and R= cp-cv Where cp= specific heat at constant pressure and, cv= specific heat at constant volume. R= gas constant.
- 34. Work done, 𝑊1−2 = 1 2 𝑃𝑑𝑣 Since, 𝑃1 𝑣1 𝛾 = 𝑃2 𝑣2 𝛾 = 𝑃𝑣 𝛾 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 (𝐾) 𝑃 = 𝐾 𝑣 𝛾 Substituting the value of P in the equation of work done, 𝑊1−2 = 1 2 𝐾 𝑣 𝛾 𝑑𝑣 𝑊1−2 = 𝐾 1 2 1 𝑣 𝛾 𝑑𝑣 = 𝐾 1 − 𝛾 (𝑣2 1−𝛾 − 𝑣1 1−𝛾 ) As 𝐾 = 𝑃1 𝑣1 𝛾 = 𝑃2 𝑣2 𝛾 By multiplying the first term inside the bracket by 𝑃2 𝑣2 𝛾 and the second term by 𝑃1 𝑣1 𝛾 , we get, 𝑾 𝟏−𝟐 = 𝑷 𝟐 𝒗 𝟐 − 𝑷 𝟏 𝒗 𝟏 (𝟏 − 𝜸) = 𝑷 𝟏 𝒗 𝟏 − 𝑷 𝟐 𝒗 𝟐 (𝜸 − 𝟏) = 𝑹(𝑻 𝟏 − 𝑻 𝟐) (𝜸 − 𝟏)
- 35. From non-flow process, 𝛿𝑄 = 𝛿𝑊 + 𝛿𝑈 {Since, 𝛿𝑄 = 0} 𝛿𝑊 + 𝛿𝑈 = 0 𝛿𝑊 = −𝛿𝑈 𝛿𝑊 = −(𝑈2 − 𝑈1) Or 𝑼 𝟏 − 𝑼 𝟐 = 𝑷 𝟏 𝒗 𝟏 − 𝑷 𝟐 𝒗 𝟐 (𝜸 − 𝟏) We know that from general gas equation, 𝑃1 𝑣1 𝑇1 = 𝑃2 𝑣2 𝑇2 And as per the isentropic law, 𝑃1 𝑣1 𝛾 = 𝑃2 𝑣2 𝛾 The following relations can be set up, 𝑻 𝟐 𝑻 𝟏 = 𝑷 𝟐 𝑷 𝟏 𝜸−𝟏 𝜸 = 𝒗 𝟏 𝒗 𝟐 𝜸−𝟏
- 36. Polytropic process The polytropic process is also known as general law for the expansion and compression of gases, and it is expressed by the relation: Pvn= constant Where n is a polytropic index. Work done 𝑊1−2 = 1 2 𝑃𝑑𝑣 Since, 𝑃1 𝑣1 𝑛 = 𝑃2 𝑣2 𝑛 = 𝑃𝑣 𝑛 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 (𝐾) 𝑃 = 𝐾 𝑣 𝑛 Substituting the value of P in the equation of work done, 𝑊1−2 = 1 2 𝐾 𝑣 𝑛 𝑑𝑣 𝑊1−2 = 𝐾 1 2 1 𝑣 𝑛 𝑑𝑣 = 𝐾 1 − 𝑛 (𝑣2 1−𝑛 − 𝑣1 1−𝑛 )
- 37. As 𝐾 = 𝑃1 𝑣1 𝑛 = 𝑃2 𝑣2 𝑛 By multiplying the first term inside the bracket by 𝑃2 𝑣2 𝑛 and the second term by 𝑃1 𝑣1 𝑛 , we get, 𝑾 𝟏−𝟐 = 𝑷 𝟐 𝒗 𝟐 − 𝑷 𝟏 𝒗 𝟏 (𝟏 − 𝒏) = 𝑷 𝟏 𝒗 𝟏 − 𝑷 𝟐 𝒗 𝟐 (𝒏 − 𝟏) 𝒘 𝟏−𝟐 = 𝑾 𝟏−𝟐 𝒎 = 𝑷 𝟏 𝒗 𝟏 − 𝑷 𝟐 𝒗 𝟐 (𝒏 − 𝟏) = 𝑹(𝑻 𝟏 − 𝑻 𝟐) (𝒏 − 𝟏) Change in internal energy 𝑼 𝟐 − 𝑼 𝟏 = 𝒎𝒄 𝒗(𝑻 𝟐 − 𝑻 𝟏) 𝒖 𝟐 − 𝒖 𝟏 = 𝒄 𝒗(𝑻 𝟐 − 𝑻 𝟏) Heat interaction 𝑞1−2 = 𝑤1−2 + 𝑢2 − 𝑢1 𝑞1−2 = 𝑃1 𝑣1 − 𝑃2 𝑣2 (𝑛 − 1) + 𝑐 𝑣(𝑇2 − 𝑇1) 𝑞1−2 = 𝑃1 𝑣1 − 𝑃2 𝑣2 (𝑛 − 1) + 𝑅(𝑇1 − 𝑇2) (𝛾 − 1) (Since, 𝑐 𝑣 = 𝑅 𝛾−1 ) 𝑞1−2 = 𝑃1 𝑣1 − 𝑃2 𝑣2 (𝑛 − 1) + 𝑃2 𝑣2 − 𝑃1 𝑣1 (𝛾 − 1) (Since, 𝑃1 𝑣1 − 𝑃2 𝑣2 = 𝑅(𝑇1 − 𝑇2)) 𝑞1−2 = (𝛾 − 𝑛) (𝛾 − 1) × 𝑃1 𝑣1 − 𝑃2 𝑣2 (𝑛 − 1) 𝒒 𝟏−𝟐 = (𝜸 − 𝒏) (𝜸 − 𝟏) × 𝑷𝒐𝒍𝒚𝒕𝒓𝒐𝒑𝒊𝒄 𝒘𝒐𝒓𝒌 The following relations can be set up, We know that from general gas equation, 𝑃1 𝑣1 𝑇1 = 𝑃2 𝑣2 𝑇2 And as per the isentropic law, 𝑃1 𝑣1 𝑛 = 𝑃2 𝑣2 𝑛 𝑻 𝟐 𝑻 𝟏 = 𝑷 𝟐 𝑷 𝟏 𝒏−𝟏 𝒏 = 𝒗 𝟏 𝒗 𝟐 𝒏−𝟏
- 38. Steady and unsteady flow process: When fluid parameters at any point of the control volume remain constant with respect to time, the flow process is called steady flow process. Let velocity, pressure, temperature etc. Are functions only of location and do not vary with time. If pressure is represented by P then mathematically a steady flow is defined as 𝜕𝑃 𝜕𝑡 = 0, i.e., the rate of change of pressure at a position is zero. Whereas when the fluid parameters vary with respect to time, the flow process is known as unsteady flow process. If pressure is represented by P then mathematically a unsteady flow is defined as 𝜕𝑃 𝜕𝑡 ≠ 0, i.e., the rate of change of pressure at a position is not equal to zero.
- 39. Steady Flow Energy Equation (S.F.E.E.) Assume the flow through a system as shown in figure. During a small time interval dt there occurs a flow of mass and energy into a fixed control volume; entry is at point 1 and exit at point 2. The fluid enters the control volume at point 1 with a average velocity V1, pressure P1, specific volume v1 and internal energy U1.The fluid exit the control volume at point 2 and the corresponding values are V2, P2, v2, U2. During the fluid flow from the two sections, heat Q and mechanical work W may also cross the control surface. The following points are taken into consideration for energy balance equation: (i) Internal energy (ii) Kinetic and potential energies. (iii) Flow work (iv) Heat and mechanical work which cross the control volume.
- 40. From the law of conservation of energy, energy neither be created nor be destroyed we can write, Total energy flow rate into the control volume = Total energy flow rate out of control volume m(energy carried into the system)+m(flow work)+ rate of heat flow= m(energy carried out of the system)+m(flow work)+ rate of work transfer m(I.E.+P.E.+K.E.)1 +m(flow work)1 + 𝑄= m(I.E.+P.E.+K.E.)2 +m(flow work)2 + 𝑊 Where, 𝑄 = 𝑑𝑄 𝑑𝑡 𝑎𝑛𝑑 𝑊 = 𝑑𝑊 𝑑𝑡 𝑚 𝑈1 + 𝑔𝑧1 + 𝑉1 2 2 + 𝑚 𝑃1 𝑣1 + 𝑄 = 𝑚 𝑈2 + 𝑔𝑧2 + 𝑉2 2 2 + 𝑚 𝑃2 𝑣2 + 𝑊
- 41. Arranging the equation, 𝑚 𝑈1 + 𝑃1 𝑣1 + 𝑔𝑧1 + 𝑉1 2 2 + 𝑄 = 𝑚 𝑈2 + 𝑃2 𝑣2 + 𝑔𝑧2 + 𝑉2 2 2 + 𝑊 𝑚 (𝑈1 + 𝑃1 𝑣1) + 𝑔𝑧1 + 𝑉1 2 2 + 𝑄 = 𝑚 (𝑈2 + 𝑃2 𝑣2) + 𝑔𝑧2 + 𝑉2 2 2 + 𝑊 Since ℎ = 𝑈 + 𝑃𝑣 , 𝑠𝑜 𝑡ℎ𝑎𝑡 ℎ1 = 𝑈1 + 𝑃1 𝑣1 𝑎𝑛𝑑 ℎ2 = (𝑈2 + 𝑃2 𝑣2) 𝒎 𝒉 𝟏 + 𝒈𝒛 𝟏 + 𝑽 𝟏 𝟐 𝟐 + 𝑸 = 𝒎 𝒉 𝟐 + 𝒈𝒛 𝟐 + 𝑽 𝟐 𝟐 𝟐 + 𝑾 This equation is known as steady flow energy equation (SFEE). If the mass of fluid is taken as unity then steady flow energy equation is reduces to, ℎ1 + 𝑔𝑧1 + 𝑉1 2 2 + 𝑞1−2 = ℎ2 + 𝑔𝑧2 + 𝑉2 2 2 + 𝑤 All the terms represent energy flow per unit mass of fluid (J/kg).
- 42. Applications of Steady flow energy equation Steady flow energy equation is commonly used in flow processes in many engineering plants. Some commonly used engineering systems which works on steady flow energy equation (SFEE) are as follows: (i) Compressor (ii) Condenser (iii) Boiler (iv) Turbine (v) Nozzle and (vi) Pump
- 43. (i) Compressor: Compressor is a device which is used to compress the fluid (may be air) and deliver it at a high pressure and large flow rate. There are two types of compressors as follows: (a) Rotary compressor (b) Reciprocating compressor (a) Rotary compressor: Rotary compressors are the devices which are used to develop high pressure and have a rotor as their primary element. The characteristic features of flow through a rotary compressor are: Work is done on the system so that W is negative. Negligible change in Potential energy. Heat is lost from the system so that Q is negative
- 44. Steady flow energy equation may be written as follows: 𝑚 ℎ1 + 𝑉1 2 2 − 𝑄 = 𝑚 ℎ2 + 𝑉2 2 2 − 𝑊 Or 𝑊 = 𝑚 ℎ2 + 𝑉2 2 2 − 𝑚 ℎ1 + 𝑉1 2 2 + 𝑄 If the change in velocity is negligible and the flow process is assumed as adiabatic (i.e. Q=0) due to very high flow rates, then 𝑊 = 𝑚(ℎ2 − ℎ1) Reciprocating compressor: Reciprocating compressors are the devices which are used to develop high pressure and have a piston cylinder arrangement as their primary element. The characteristic features of flow through a rotary compressor are: Work is done on the system so that W is negative. Negligible change in Potential energy. Heat is lost from the system so that Q is negative
- 45. Steady flow energy equation may be written as follows: 𝑚 ℎ1 + 𝑉1 2 2 − 𝑄 = 𝑚 ℎ2 + 𝑉2 2 2 − 𝑊 Or 𝑊 = 𝑚 ℎ2 + 𝑉2 2 2 − 𝑚 ℎ1 + 𝑉1 2 2 + 𝑄 If the change in velocity is negligible, then 𝑊 = 𝑚(ℎ2 − ℎ1) + 𝑄
- 46. ii. Condenser: Condenser is a type of heat exchanger. It is used to transfer heat from one fluid to another. The characteristic features of a condenser are as follows: No mechanical work (i.e., W=0). No change in kinetic and potential energies. No external heat interaction (Since it is perfectly insulated). Heat is absorbed by the one fluid (Steam) to the another fluid (coolant), so that heat is taken negative. Thus steady flow energy equation reduces to; 𝑚 ℎ1 + 𝑔𝑧1 + 𝑉1 2 2 + 𝑄 = 𝑚 ℎ2 + 𝑔𝑧2 + 𝑉2 2 2 + 𝑊 ℎ1 − 𝑄 = ℎ2 𝑄 = ℎ1 − ℎ2
- 47. (iii) Boiler: Boiler is an equipment used for generation of steam. Thermal energy released by combustion of fuel is transferred to water which vaporizes and gets converted into steam. The characteristic features of a boiler are as follows: No mechanical work (i.e., W=0). No change in kinetic and potential energies Height change between inlet and exit point is negligible. Thus steady flow energy equation reduces to; 𝑚 ℎ1 + 𝑔𝑧1 + 𝑉1 2 2 + 𝑄 = 𝑚 ℎ2 + 𝑔𝑧2 + 𝑉2 2 2 + 𝑊 ℎ1+𝑄 = ℎ2 𝑄 = ℎ2 − ℎ1
- 48. (iv) Turbine: Turbine is a device which converts thermal energy into useful work. In turbine fluids expand from high pressure to a low pressure. The work output from the turbine may be used to drive a generator to produce electricity. The characteristic features of a turbine are as follows: Negligible change in velocity so that negligible change in kinetic energy. Negligible change in potential energy. Isentropic expansion takes place since the walls of turbine are thermally insulated. Thus steady flow energy equation reduces to; 𝑚 ℎ1 + 𝑔𝑧1 + 𝑉1 2 2 + 𝑄 = 𝑚 ℎ2 + 𝑔𝑧2 + 𝑉2 2 2 + 𝑊 𝑊 = 𝑚(ℎ2 − ℎ1)
- 49. (v) Nozzle: Nozzle is a device of varying cross-section used for increasing the velocity of a flowing stream at the expense of its pressure drop. In nozzle pressure energy of the fluid is converted into kinetic energy. It is used in turbines, fuel pumps and jet engines etc. The characteristic features of a nozzle are as follows: No mechanical work (i.e. W=0) Flow is isentropic (i.e. Q=0) Change in height between entry and exit is negligible. (i.e. z1=z2) Thus steady flow energy equation reduces to; ℎ1 + 𝑉1 2 2 = ℎ2 + 𝑉2 2 2 Let V1 is known then, 𝑉2 = 2 ℎ1 − ℎ2 + 𝑉1 2
- 50. (vi) Pump: A pump is a device which takes the fluid from a low level and delivers it to a high level. The characteristic features of a pump are as follows: Flow is assumed to be adiabatic (i.e. Q=0) No change in internal energy. Work is done on the system, so that work is taken negative. Thus steady flow energy equation reduces to; 𝑚 ℎ1 + 𝑔𝑧1 + 𝑉1 2 2 + 𝑄 = 𝑚 ℎ2 + 𝑔𝑧2 + 𝑉2 2 2 + 𝑊 𝑚 𝑔𝑧1 + 𝑉1 2 2 = 𝑚 𝑔𝑧2 + 𝑉2 2 2 − 𝑊
- 51. Throttling Process: Throttling is an irreversible expansion process. In this process the expansion of fluid takes place from high pressure to low pressure. This process occurs when the fluid is flowing across a restriction (partially closed valve or a small orifice) placed in the flow passage. This process occurs in a flow through a porous plug as shown in figure. In this process a steady stream of gas at a given pressure (P1) and temperature (T1) flows through a porous plug contained in a thermally insulated horizontal tube. The Fluid exits at a reduced pressure (P2). Throttling process is used for obtaining the dryness fraction of wet steam. The characteristic features of a pump are as follows: Change in kinetic and potential energies are negligible. No mechanical work (i.e. W=0) No heat loss as the tube is thermally insulated (i.e. Q=0)
- 52. Thus steady flow energy equation for unit mass reduces to; ℎ1 + 𝑔𝑧1 + 𝑉1 2 2 + 𝑄 = ℎ2 + 𝑔𝑧2 + 𝑉2 2 2 + 𝑊 ℎ1 = ℎ2 As we know that ℎ = 𝑐 𝑝. 𝑇 Where cp= specific heat at constant pressure So that we can write, 𝑐 𝑝. 𝑇1 = 𝑐 𝑝. 𝑇2 Or 𝑇1 = 𝑇2