Basic and Applied Thermodynamics Chapter-1 Introduction Prepared By Brij Bhooshan Supported By: Art. Content Page References:

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1 Basic and Applied Thermodynamics Chapter-1 Introduction Prepared By Brij Bhooshan Asst. Professor B. S. A. College of Engg. And Technology Mathura, Uttar Pradesh, (India) Supported By: Purvi Bhooshan In This Chapter We Cover the Following Topics Art. Content Page 1.1 Different Approaches in the Study of Thermodynamics Macroscopic Approach Microscopic Approach Thermodynamic System Thermodynamic Properties, Processes and Cycles Intensive and Extensive Properties Homogeneous and Heterogeneous Systems Thermodynamic Equilibrium Reversibility and Irreversibility Work and Heat Transfer Work Transfer Thermodynamic Definition of Work Path Function and Point Function Flow Work Free Expansion with Zero Work Transfer Heat Transfer Specific Heat and Latent Heat Points to Remember Regarding Heat Transfer and Work Transfer 1.7 State Postulate & Zeroth-Law of Thermodynamics Some Basic Definitions 21 References: 1. M. J. Moran and H. N. Shapiro, Fundamentals of Engineering Thermodynamics, 6e, John Wiley & Sons, Inc., New York, G. J. Van Wylen, R. E. Sonntag, C. Borgnakke, Fundamentals of Thermodynamics, John Wiley & Sons, Inc., New York, J. P. Holman, Thermodynamics, 4e, McGraw-Hill, New York, Copyright by Brij 2013 Page

2 2 Chapter 1: Introduction 4. F. W. Sears, G. L. Salinger, Thermodynamics, Kinetic theory, and Statistical Thermodynamics, 3e, Narosa Publishing House, New Delhi, Y. A. Cengel and M. A. Boles, Thermodynamics: An Engineering Approach, 2e, McGraw-Hill, New York, E. Rathakrishnan, Fundamentals of Engineering Thermodynamics, 2e, PHI Learning Private Limited, New Delhi, P. K. Nag, Basic and Applied Thermodynamics, 1e, McGraw-Hill, New Delhi, V Ganesan, Gas Turbine, 2e, Tata McGraw-Hill, New Delhi, Y. V. C. Rao, An Introduction to Thermodynamics, 1e, New Age International (P) Limited, Publishers, New Delhi, Onkar Singh, Applied Thermodynamics, 2e, New Age International (P) Limited, Publishers, New Delhi, Please welcome for any correction or misprint in the entire manuscript and your valuable suggestions kindly mail us brijrbedu@gmail.com. Copyright by Brij 2013 Page 2

3 Basic and Applied Thermodynamics By Brij Bhooshan 3 The most of general sense of thermodynamics is the study of energy and its relationship to the properties of matter. All activities in nature involve some interaction between energy and matter. The matter is an entity, which can be sensed by touch or can be felt, and occupies space. Thermodynamics is a science that governs the following: Energy and its transformation Feasibility of a process involving transformation of energy Feasibility of a process involving transfer of energy Equilibrium processes More specifically, thermodynamics deals with energy conversion, energy exchange and the direction of exchange. Thermodynamics is the science of energy transfer and its effect on the physical properties of substances. It is based upon observations of common experience which have been formulated into thermodynamic laws. These laws govern the principles of energy conversion. Areas of Application of Thermodynamics All natural processes are governed by the principles of thermodynamics. However, the following engineering devices are typically designed based on the principles of thermodynamics. Automotive engines, Turbines, Compressors, Pumps, Fossil and Nuclear Power Plants, Propulsion systems for the Aircrafts, Separation and Liquefication Plant, Refrigeration, Air-conditioning and Heating Devices. The principles of thermodynamics are summarized in the form of a set of axioms. These axioms are known as four thermodynamic laws: The zeroth law, the first law, the second law and the third law. The Zeroth law The First law Four Laws of thermodynamics The Second law The Third law The Zeroth Law deals with thermal equilibrium and provides a means for measuring temperatures. The First Law of thermodynamics deals with the conservation of energy and introduces the concept of internal energy. The Second Law of thermodynamics provides with the guidelines on the conversion of internal energy of matter into work. It also introduces the concept of entropy. The Third Law of thermodynamics defines the absolute zero of entropy. The entropy of a pure crystalline substance at absolute zero temperature is zero. 1.1 DIFFERENT APPROACHES IN THE STUDY OF THERMODYNAMICS Thermodynamics can be studied through two different approaches: (a) Macroscopic Approach and (b) Microscopic Approach Copyright by Brij 2013 Page 3

4 4 Chapter 1: Introduction Macroscopic Approach: In the macroscopic approach, a certain quantity of matter is considered, without the events occurring at the molecular level being taken into account. Macroscopic thermodynamics is only concerned with the effects of the action of many molecules, and these effects can be perceived by human senses. For example, the macroscopic quantity, pressure, is the average rate of change of momentum due to all the molecular collisions made on a unit area. The effects of pressure can be felt. The macroscopic point of view is not concerned with the action of individual molecules, and the force on a given unit area can be measured by using, e.g., a pressure gauge. These macroscopic observations arc completely independent of the assumptions regarding the nature of matter. Consider a certain amount of gas in a cylindrical container. The volume (V) can be measured by measuring the diameter and the height of the cylinder. The pressure (P) of the gas can be measured by a pressure gauge. The temperature (T) of the gas can be measured using a thermometer. The state of the gas can be specified by the measured P, V and T. The values of these variables are space averaged characteristics of the properties of the gas under consideration. In classical thermodynamics, we often use this macroscopic approach. The macroscopic approach has the following features. The structure of the matter is not considered. A few variables are used to describe the state of the matter under consideration. The values of these variables are measurable following the available techniques of experimental physics. Microscopic Approach: From the microscopic point of view, matter is composed of myriads of molecules. If it is a gas, each molecule at a given instant has a certain position, velocity, and energy, and for each molecule these change very frequently as a result of collisions. The behaviour of the gas is described by summing up the behaviour of each molecule. Such a study is made in microscopic or statistical thermodynamics. On the other hand, the gas can be considered as assemblage of a large number of particles each of which moves randomly with independent velocity. The state of each particle can be specified in terms of position coordinates (xi, yi, zi) and the momentum components (pxi, pyi, pzi). If we consider a gas occupying a volume of 1 cm 3 at ambient temperature and pressure, the number of particles present in it is of the order of The same number of position coordinates and momentum components are needed to specify the state of the gas. The microscopic approach can be summarized as: A knowledge of the molecular structure of matter under consideration is essential. A large number of variables are needed for a complete specification of the state of the matter. All the results of classical or macroscopic thermodynamics can, however, be derived from the microscopic and statistical study of matter. 1.2 THERMODYNAMIC SYSTEM System A thermodynamic system is defined as a definite quantity of matter or a region in space upon which attention is focused in the analysis of a problem. We may want to study a quantity of matter contained within a closed rigid walled chambers, or we may want to consider something such as gas pipeline through which the matter flows. The Copyright by Brij 2013 Page 4

5 Basic and Applied Thermodynamics By Brij Bhooshan 5 composition of the matter inside the system may be fixed or may change through chemical and nuclear reactions. System Boundary Surrounding Diagram 1.1 A system may be arbitrarily defined. It becomes important when exchange of energy between the system and the everything else outside the system is considered. The judgment on the energetic of this exchange is very important. Surroundings and Universe Everything external to the system is called the surroundings or the environment. The system is distinguished from its surroundings by a specified boundary which may be at rest or in motion. The interactions between a system and its surroundings, which take place across the boundary, play an important role in thermodynamics. A system and its surroundings together comprise a universe. The system is separated from the surroundings by the system boundary (Diagram 1.1). The boundary may be either fixed or moving. A system and its surroundings together comprise a universe. Universe = System + Surroundings Types of Systems Two types of systems can be distinguished. These are referred to, respectively, as closed systems and open systems or control volumes. A closed system or a control mass refers to a fixed quantity of matter, whereas a control volume is a region in space through which mass may flow. A special type of closed system that does not interact with its surroundings is called an isolated system. Two types of exchange can occur between the system and its surroundings: 1- Energy exchange (heat or work) and 2- Exchange of matter (movement of molecules across the boundary of the system and surroundings). Based on the types of exchange, one can define (a) isolated systems: no exchange of matter and energy (b) closed systems: no exchange of matter but some exchange of energy (c) open systems: exchange of both matter and energy If the boundary does not allow heat (energy) exchange to take place it is called adiabatic boundary otherwise it is diathermal boundary. Closed System: The closed system (Diagram 1.2a) is a system of fixed mass. There is no mass transfer across the system boundary. There may be energy transfer into or out of the system. A certain quantity of fluid in a cylinder bounded by a piston constitutes a closed system. Examples are Boiling water in a closed pan. Thus, a closed system does not permits any mass transfer across its boundary, but it permits transfer of energy (heat and work). Open System: The open system is one in which matter crosses the boundary of the system (Diagram 1.2b). There may be energy transfer also. Most of the engineering Copyright by Brij 2013 Page 5

6 6 Chapter 1: Introduction devices are generally open systems, e.g., an air compressor in which air enters at low pressure and leaves at high pressure and there are energy transfers across the system boundary. Thus, an open system permits both mass transfer and energy transfer across its boundary, and the mass within the system may not be constant. The properties of the matter occupying the control volume can vary with time. The surface which enclosed a control volume is termed as control surface. Boundary Energy in Mass in System Mass interaction 0 Energy interaction 0 (a) Open system Energy out Surrounding Mass out Boundary Surrounding System Boundary Energy in System Mass interaction = 0 Energy interaction 0 (b) Closed system Energy out Energy Surrounding Mass interaction = 0 Energy interaction = 0 (c) Isolated system Diagram 1.2 Isolated System: Isolated system is one in which there is no interaction between the system and the surrounding (Diagram 1.2c). It is of fixed mass and energy, and there is no mass or energy transfer across the system boundary. An open system with its surroundings is an example of an isolated system. Control Volume For thermodynamic analysis of an open system, such as an air compressor (Diagram 1.3), attention is focussed on a certain volume in space surrounding the compressor, is referred as the control volume, bounded by a surface is termed the control surface. Matter as well as energy crosses the control surface. A closed system is a system closed to matter flow, though its volume can change against a flexible boundary. When there is matter flow, then the system is considered to be a volume of fixed identity, the control volume. There is thus no difference between an open system and a control volume. Control surface Work Motor Air out Heat Control volume Air in Diagram THERMODYNAMIC PROPERTIES, PROCESSES AND CYCLES Every system has certain characteristics by which its physical condition may be described, e.g., volume, temperature, pressure, etc. Such characteristics are called properties of the system. To describe a system and predict its behavior requires a knowledge of its properties and how those properties are related. Properties are macroscopic characteristics of a system such as mass, volume, energy, pressure and Copyright by Brij 2013 Page 6

7 Basic and Applied Thermodynamics By Brij Bhooshan 7 temperature to which numerical values can be assigned at a given time without knowledge of the past history of the system. Many other properties are considered during the course of our study. The value of a property of a system is independent of the process or the path followed by the system in reaching a particular state. The change in the value of the property depends only on the initial and the final states. The word state refers to the condition of a system as described by its properties. Mathematically, if P is a property of the system, then the change in the property in going from the initial state 1 to the final state 2 is given by If P = P (x, y) then, where, If then dp is said to be an exact differential, and P is a point function. A thermodynamic property is a point function and not a path function. Pressure, temperature, volume or molar volume are some of the quantities which satisfy these requirements. When all the properties of a system have definite values, the system is said to exist at a definite state. Properties are the coordinates to describe the state of a system. They are the state variables of the system. Any operation in which one or more of the properties of a system changes is called a change of state. The succession of states passed through during a change of state is called the path of the change of state. When the path is completely specified, the change of state is called a process, e.g., a constant pressure process. In other words, a change or a series of changes in the state of a system is termed as process. A thermodynamic cycle is defined as a series of state changes such that the final state is identical with the initial state (Diagram 1.4). thus, if the initial and final states of a system experiencing a series of processes are identical, it is said to have executed a cycle. Properties may be of two types, like intensive properties and extensive properties. Intensive and Extensive Properties There are certain properties which depend on the size or extent of the system, and there are certain properties which are independent of the size or extent of the system. The properties like volume, which depend on the size of the system are called extensive properties. The properties, like temperature and pressure which are independent of the mass of the system are called intensive properties. The test for an intensive property is to observe how it is affected when a given system is combined with some fraction of exact replica of itself to create a new system differing only by size. Intensive properties are those which are unchanged by this process, whereas those properties whose values are increased or decreased in proportion to the enlargement or reduction of the system Copyright by Brij 2013 Page 7

8 8 Chapter 1: Introduction are called extensive properties. Specific extensive properties, i.e. extensive properties per unit mass, are intensive properties. 1 a b A process A cycle Diagram 1.4 Diagram 1.5 Assume two identical systems S1 and S2 as shown in Diagram 1.5. Both the systems are in identical states. Let S3 be the combined system. Is the value of property for S3 same as that for S1 (and S2)? If the answer is yes, then the property is intensive; If the answer is no, then the property is extensive. The ratio of the extensive property to the mass is called the specific value of that property Specific volume, v = V/m = 1/ρ (ρ is the density) Specific internal energy, u = U/m Similarly, the molar properties are defined as the ratios of the properties to the mole number (N) of the substance Molar volume = = V/N Molar internal energy = 2 = U/N Homogeneous and Heterogeneous Systems A quantity of matter homogeneous throughout in chemical composition and physical structure is called a phase. Every substance can exist in any one of the three phases, viz., solid, liquid and gas. A system consisting of a single phase is called a homogeneous system, while a system consisting of more than one phase is known as a heterogeneous system. 1.4 THERMODYNAMIC EQUILIBRIUM Steady state Under the steady state condition, the properties of the system at any location are independent of time. Equilibrium A system which is simultaneously in a state of mechanical equilibrium, thermal equilibrium, and chemical equilibrium is said to be a state of thermodynamic equilibrium. At the state of equilibrium, the properties of the system are uniform and only one value can be assigned to it. A system is said to exist in a state of thermodynamic equilibrium when no change in any macroscopic property is registered, if the system is isolated from its surroundings. An isolated system always reaches in course of time a state of thermodynamic equilibrium and can never depart from it spontaneously. Copyright by Brij 2013 Page 8

9 Basic and Applied Thermodynamics By Brij Bhooshan 9 Therefore, there can be no spontaneous change in any macroscopic property if the system exists in an equilibrium state. Thermodynamics studies mainly the properties of physical systems that are found in equilibrium states. In thermodynamics, equilibrium refers to a state of equilibrium with respect to all possible changes, thermal, mechanical and chemical. Mechanical equilibrium: in the absence of any unbalance force within the system itself and also between the system and surrounding, the system is referred as to be in a state of mechanical equilibrium. Pressure should be uniform inside the cylinder and force balance takes place on the piston. Mechanical equilibrium means there is no unbalanced force. In other words, there is no pressure gradient within the system. Thermal equilibrium: Thermal equilibrium is that equilibrium which can be satisfied to be achived in the absence of heat interactions. A state of thermal equilibrium can be described as one in which the temperature of the system is uniform. When a system existing in mechanical and chemical equilibrium is separated from its surroundings by a diathermic wall (diathermic means 'which allows heat to flow') and if there is no spontaneous change in any property of the system, the system is said to exist in a state of thermal equilibrium. When this is not satisfied, the system will undergo a change of state till thermal equilibrium is restored. Chemical equilibrium: If there is no chemical reaction or transfer of matter from one part of the system to another, such as diffusion or solution, the system is said to exist in a state of chemical equilibrium. The criterion for chemical equilibrium is the equality of chemical potential. Superscripts A and B refers to systems and subscript i refers to component If Gibbs function is given by G, G = U + PV TS where ni is the number of moles of substance i. The composition of a system does not undergo any change because of chemical reaction. When the conditions for any one of the three types of equilibrium are not satisfied, a system is said to be in a nonequilibrium state. If the nonequilibrium of the state is due to an unbalanced force in the interior of a system or between the system and the surrounding, the pressure varies from one part of the system to another. There is no single pressure that refers to the system as a whole. Similarly, if the nonequilibrium is because of the temperature of the system being different from that of its surroundings, there is a nonuniform temperature distribution set up within the system and there is no single temperature that stands for the system as a whole. It can thus be inferred that when the conditions for thermodynamic equilibrium are not satisfied, the states passed through by a system cannot be described by thermodynamic properties which represent the system as a whole. Thermodynamic properties are the macroscopic coordinates defined for, and significant to, only thermodynamic equilibrium states. Both classical and statistical thermodynamics study mainly the equilibrium states of a system. Stability of Equilibrium States The equilibrium states of a system can be classified as: Copyright by Brij 2013 Page 9

10 10 Chapter 1: Introduction (a) (b) Stable equilibrium Neutral equilibrium (c) Unstable equilibrium (d) Meta stable equilibrium Considering a mechanical analogy as a example of these equilibrium states are shown in Diagram 1.6. Stable Neutral Unstable Metastable Diagram REVERSIBILITY AND IRREVERSIBILITY In thermodynamics we are mainly concerned with the systems which are in a state of equilibrium. Whenever a system undergoes a change in its condition, from one equilibrium state to another equilibrium state, the system is said to undergo a process. 2 1 Intermediate State Diagram 1.7 Consider a certain amount of gas enclosed in a piston-cylinder assembly as our system. Suppose the piston moves under such a condition that the opposing force is always infinitesimally smaller than the force exerted by the gas. The process followed by the system is reversible. A process is said to be reversible if the system and its surroundings are restored to their respective initial states by reversing the direction of the process. A reversible process has to be quasi-static, but a quasi - static process is not necessarily reversible. The process is irreversible if it does not fulfill the criterion of reversibility. Many processes are characterized by the fact that some property of the system remains constant. Thermodynamic processes may have the change of state occuring in two ways. One is the change of state occuring so that if the system is to restore its original state, it can be had by reversing the factors responsible for occurrence of the process. Other change of state may occur such that the above restoration of original state is not possible. Thermodynamic system that is capable of restoring its original state by reversing the factors responsible for occurrence of the process is called reversible system and the thermodynamic process involved is called reversible process. Thus, upon reversal of a process there shall be no trace of the process being ocurred, i.e. state changes during the forward direction of occurrence of a process are exactly similar to the states passed through by the system during the reversed direction of the process. It is quite obvious that the such reversibility can be realised only if the system maintains its thermodynamic equilibrium throughout the occurrence of process. The irreversibility is the characteristics of the system which forbids system from retracing the same path upon reversal of the factors causing the state change. Thus, Copyright by Brij 2013 Page 10

11 Basic and Applied Thermodynamics By Brij Bhooshan 11 irreversible systems are those which do not maintain equilibrium during the occurrence of a process. Various factors responsible for the nonattainment of equilibrium are generally the reasons responsible for irreversibility. Presence of friction, dissipative effects etc. have been identified as a few of the prominent reasons for irreversibility. The reversible and irreversible processes are shown on p-v diagram in Diagram 1.8 by 1 2 and 2 1 and 3 4 and 4 3 respectively & 2 1 Reversible process following equilibrium states 3 4 & 4 3 Irreversible process following non-equilibrium states Diagram 1.8 Reversible and Irreversible processes 1.6 WORK AND HEAT TRANSFER A closed system and its surroundings can interact in two ways: (a) by work transfer, and (b) by heat transfer. These may be called energy interactions and these bring about changes in the properties of the system. Thermodynamics mainly studies these energy interactions and the associated property changes of the system. Work Transfer Work is one of the basic modes of energy transfer. The work done by a system is a path function, and not a point function. Therefore, work is not a property of the system, and it cannot be said that the work is possessed by the system. It is an interaction across the boundary. What is stored in the system is energy, but not work. A decrease in energy of the system appears as work done. Therefore, work is energy in transit and it can be indentified only when the system undergoes a process. Work must be regarded only as a type of energy in transition across a well defined, zero thickness, boundary of a system. Consequently work, is never a property or any quantity contained within a system. Work is energy driven across by differences in the driving forces on either side of it. Various kinds of work are identified by the kind of driving force involved and the characteristic extensive property change which accompanied it. Work is measured quantitatively in the following way. Any driving force other than temperature, located outside the system on its external boundary, is multiplied by a transported extensive property change within the system which was transferred across the system boundary in response to this force. The result is the numerical value of the work associated with this system and driving force. In static Equilibrium, F = PA (Diagram 1.9). The dx is small so that P does not change. The change in volume of the gas = A.dx. A P F Diagram 1.9 The elemental work, Copyright by Brij 2013 Page 11

12 12 Chapter 1: Introduction dw = F.dx = P.A.dx = P.dv [1.4] Thermodynamic Definition of Work In thermodynamics, work done by a system on its surroundings during a process is defined as that interaction whose sole effect, external to the system, could be reduced as the raising of a mass through a distance against gravitational force. Let us consider the raising of mass m from an initial elevation z1 to final elevation z2 against gravitational force. To raise this mass, the force acting on the mass is given by F = mg. The work done on the body is W = mg (z2 z1). System Gas Diagram 1.10 An external agency is needed to act on the system. It can be seen that expansion of the gas gets reduced to raising a mass against gravitational force (Diagram 1.10). dw = F.dx = P.A.dx = P.dv During this expansion process, the external pressure is always infinitesimally smaller than the gas pressure. Switch Switch Resistor Motor System Storage battery Storage battery Diagram 1.11 Compare two systems shown in the Diagram Let the resistor be replaced by a motor drawing the same amount of current as the resistor. The motor can wind a string and thereby raise the mass which is suspended. As far as the battery is concerned, the situations are identical. So, according to thermodynamic definition of work, the interaction of a battery with a resistor is called work. By manipulating the environment, that is external to the battery (system), the effect can be reduced to raising of a mass against the gravitational force and that is the only effect on the surroundings. We can see that the thermodynamic definition of work is more general than that used in mechanics. When work is done by a system, it is arbitrarily taken to be positive, and when work is done on a system, it is taken to be negative (Diagram 1.12). The symbol W is used for work transfer. Copyright by Brij 2013 Page 12

13 Basic and Applied Thermodynamics By Brij Bhooshan 13 System System Surrounding (a) W is positive Surrounding (b) W is negative Diagram 1.12 Work interaction between a system and the surroundings The rate at which work is done by, or upon, the system is termed as power. The unit of power is J/s or watt. Work is one of the forms in which a system and its surroundings can interact with each other. When the piston moves out from position 1 to position 2 with the volume changing from V1 to V2, the amount of work W done by the system will be The magnitude of the work done is given by the area under the path 1-2, as shown in Diagram Since p is at all times a thermodynamic coordinate, all the states passed through by the system as the volume changes from V1 to V2 must be equilibrium states, and the path 1-2 must be quasi-static. The piston moves infinitely slowly so that every state passed through is an equilibrium state. The integration pdv can be performed only on a quasi-static path. 1 Quasi-static process Work transfer Diagram 1.13 Quasi-static p-dv work Situation in which W P.dv. Now suppose the initial volume be v1 and pressure P1, and the final volume be v2 and pressure P2. What should be the work done in this case? Is it equal to P.dv? P.dv = area under the curve indicating the process on P-v diagram. The expansion process may be carried out in steps as shown in Diagram It is possible to draw a smooth curve passing through the points 1bcde2. Does the area under the curve (Diagram 1.15) represent work done by the system? The answer is no, because the process is not reversible. The expansion of the gas is not restrained by an equal and opposing force at the moving boundary. Gas Vacuum 1 Diagram 1.14 Diagram 1.15 Copyright by Brij 2013 Page 13

14 14 Chapter 1: Introduction No external force has moved through any distance in this case, the work done is zero. Therefore, we observe that W = P.dv only for reversible process W P.dv for an irreversible process Gas Paddle System Diagram 1.16 Another exceptional situation The piston is held rigid using latches (Diagram 1.16) dv = 0 Work done on the gas is equal to the decrease in the potential energy of mass m A situation where dv = 0 and yet dw is not zero such work can be done in one direction only. Work is done on the system by the surroundings. Path Function and Point Function With reference to Diagram 1.17, it is possible to take a system from state 1 to state 2 along many quasi-static paths, such as A, B or C. Since the area under each curve represents the work for each process, the amount of work involved in each case is not a function of the end states of the process, and it depends on the path the system follows in going from state 1 to state 2. For this reason, work is called a path function, and dw is an inexact or imperfect differential. 1 A B C 2 Diagram 1.17 Work a path function Thermodynamic properties are point functions, since for a given state, there is a definite value for each property. The change in a thermodynamic property of a system in a change of state is independent of the path the system follows during the change of state, and depends only on the initial and final states of the system. The differentials of point functions are exact or perfect differentials, and the integration is simply The change in volume thus depends only on the end states of the system irrespective of the path the system follows. On the other hand, work done in a quasi-static process between two given states depends on the path followed. Copyright by Brij 2013 Page 14

15 Basic and Applied Thermodynamics By Brij Bhooshan 15 Rather, To distinguish an inexact differential dw from an exact differential dv or dp the differential sign is being cut by a line at its top. From Eq. (1.4), Here, 1/p is called the integrating factor. Therefore, an inexact differential dw when multiplied by an integrating factor 1/p becomes an exact differential dv. For a cyclic process, the initial and final states of the system are the same, and hence, the change in any property is zero, i.e. where the symbol denotes the cyclic integral for the closed path. Therefore, the cyclic integral of a property is always zero. 1 Imaginary piston 2 Boundary Diagram Flow work Flow Work The flow work, significant only in a flow process or an open system, represents the energy transferred across the system boundary as a result of the energy imparted to the fluid by a pump, blower or compressor to make the fluid flow across the control volume. Flow work is analogous to displacement work. Let p be the fluid pressure in the plane of the imaginary piston, which acts in a direction normal to it (Diagram 1.18). The work done on this imaginary piston by the external pressure as the piston moves forward is given by dwflow = p.dv [1.9] where dv is the volume of fluid element about to enter the system. dwflow = pv.dm [1.10] where dv= vdm Therefore, flow work at inlet (Diagram 3.16), (dwflow)in = p1 v1 dm1 [1.11] Equation (1.11) can also be derived in a slightly different manner. If the normal pressure p1 is exerted against the area A1, giving a total force (p1a1) against the piston, in time dτ, this force moves a distance V1dτ, where V1 is the velocity of flow (piston). The work in time dτ is p1a1v1dτ, or the work per unit time is p1a1v1. Copyright by Brij 2013 Page 15

16 16 Chapter 1: Introduction Since the flow rate the work done in time dτ becomes (dwflow)in = p1 v1 dm1 Similarly, flow work of the fluid element leaving the system is The flow work per unit mass is thus (dwflow)out = p2 v2 dm2 [1.12] dwflow = p v [1.13] It is the displacement work done at the moving system boundary. Partition Boundary Gas Vacuum Gas Vacuum Partition Insulation Gas Vacuum Diagram 1.19 Free expansion Free Expansion with Zero Work Transfer Work transfer is identified only at the boundaries of a system. It is a boundary phenomenon, and a form of energy in transit crossing the boundary. Let us consider a gas separated from the vacuum by a partition (Diagram 1.19). Let the partition be removed. The gas rushes to fill the entire volume. The expansion of a gas against vacuum is named as free expansion. If we neglect the work associated with the removal of partition, and consider the gas and vacuum together as our system (Diagram 1.19a), there is no work transfer involved here, since no work crosses the system boundary, and hence If only the gas is taken as the system (Diagram 1.19), when the partition is removed there is a change in the volume of the gas, and one is tempted to calculate the work from the expression However, this is not a quasistatic process, although the initial and final end states are in equilibrium. Therefore, the work cannot be calculated from this relation. The two end states can be located on the p-v diagram and these are joined by a dotted line (Diagram 1.19c) to indicate that the process had occurred. However, if the vacuum space is divided into a large number of small volumes by partitions and the partitions are removed one Copyright by Brij 2013 Page 16

17 Basic and Applied Thermodynamics By Brij Bhooshan 17 by one slowly (Diagram 1.19d), then every state passed through by the system is an equilibrium state and the work done can then be estimated from the relation Yet, in free expansion of a gas, there is no resistance to the fluid at the system boundary as the volume of the gas increases to fill up the vacuum space. Work is done by a system to overcome some resistance. Since vacuum does not offer any resistance, there is no work transfer involved in free expansion. Heat Transfer Heat is energy transfer which occurs by virtue of temperature difference across the boundary. Heat like work, is energy in transit. It can be identified only at the boundary of the system. The temperature difference is the 'potential' or 'force' and heat transfer is the 'flux'. The transfer of heat between two bodies in direct contact is called conduction. Heat may be transferred between two bodies separated by empty space or gases by the mechanism of radiation through electromagnetic waves. A third method of heat transfer is convection which refers to the transfer of heat between a wall and a fluid system in motion. Heat is not stored in the body but energy is stored in the body. Heat like work, is not a property of the system and hence its differential is not exact. Heat and work are two different ways of transferring energy across the boundary of the system. The displacement work is given by (Diagram 1.20). It is valid for a quasi-static or reversible process. The work transfer is equal to the integral of the product of the intensive property P and the differential change in the extensive property, dv. Area = Work transfer 1 Quasistatic Work transfer 2 Area = Heat transfer 1 Quasistatic Heat transfer 2 Diagram 1.20 Representation of work transfer and heat transfer in quasi-static processes on p-v and T-x coordinates The direction of heat transfer is taken from the high temperature system to the low temperature system. Heat flow into a system is taken to be positive, and heat flow out of a system is taken as negative (Diagram 1.20). The symbol Q is used for heat transfer, i.e., the quantity of heat transferred within a certain time. Just like displacement work, the heat transfer can also be written as the integral of the product of the intensive property T and the differential change of an extensive property, say X (Diagram 1.20). Copyright by Brij 2013 Page 17

18 18 Chapter 1: Introduction It must also be valid for a quasi-static process only, and the heat transfer involved is represented by the area under the path 1-2 in T-X plot (Diagram 1.20). Heat transfer is, therefore, a path function, i.e., the amount of heat transferred when a system changes from a state 1 to a state 2 depends on the path the system follows (Diagram 1.20). The quantity dq is an inexact differential. dq = T.dX X is an extensive property and dx is an exact differential. The extensive property is yet to be defined. We shall see later that X is nothing but the entropy, S of a system. It is possible to write dx = dq/t or, where 1/T is integrating factor. To make dq integrable, i.e., an exact differential, it must be multiplied by an integrating factor which is, in this case, 1/T. Specific Heat and Latent Heat The specific heat of a substance is defined as the amount of heat required to raise a unit mass of the substance through a unit rise in temperature. The symbol c will be used for specific heat. where Q is the amount of heat transfer (J), m, the mass of the substance (kg), and Δt, the rise in temperature (K). Since heat is not a property, so the specific heat is qualified with the process through which exchange of heat is made. For gases, if the process is at constant pressure, it is cp, and if the process is at constant volume, it is cv. For solids and liquids, however, the specific heat does not depend on the process. The product of mass and specific heat (mc) is named as heat capacity of the substance. The latent heat is the amount of the heat transfer required to cause a phase change in unit mass of a substance at a constant pressure and temperature. There are three phases in which matter can exist: solid, liquid, and vapour or gas. The latent heat of fusion (lfu) is the amount of heat transferred to melt unit mass of solid into liquid, or to freeze unit mass of liquid to solid. The latent heat of vaporization (lvap) is the quantity of heat required to vaporize unit mass of liquid into vapour, or condense unit mass of vapour into liquid. The latent heat of sublimation (lsub) is the amount of heat transferred to convert unit mass of solid into vapour, or vice-versa. The latent heat of fusion (lfu) is not much affected by pressure, whereas latent heat of vaporization (lvap) is highly sensitive to pressure. Points to Remember Regarding Heat Transfer and Work Transfer (a) Heat transfer and work transfer are the energy interactions. A closed system and its surroundings can interact in two ways: by heat transfer and by work transfer. Thermodynamics studies how these interactions bring about property changes in a system. Copyright by Brij 2013 Page 18

19 Basic and Applied Thermodynamics By Brij Bhooshan 19 (b) The same effect in a closed system can be brought about either by heat transfer or by work transfer. Whether heat transfer or work transfer has taken place depends on what constitutes the system. (c) Both heat transfer and work transfer are boundary phenomena. Both are observed at the boundaries of the system, and both represent energy crossing the boundaries of the system. (d) It is wrong to say 'total heat' or 'heat content' of a closed system, because heat or work is not a property of the system. Heat, like work, cannot be stored by the system. Both heat and work are the energy in transit. (e) Heat transfer is the energy interaction due to temperature difference only. All other energy interactions may be termed as work transfer. (f) Both heat and work are path functions and inexact differentials. The magnitude of heat transfer or work transfer depends upon the path the system follows during the change of state. 1.7 STATE POSTULATE & ZEROTH-LAW OF THERMODYNAMICS Introduction to state postulate Since every thermodynamic system contains some matter with energy in its various forms, the system can be completely described by specifying the following variables. The composition of the matter in terms of mole numbers of each constituent. The energy of the system. The volume of the system, and The measurable properties, such as pressure and temperature. By specifying these quantities, the state of the system is defined. Once the system is in a given state, it possesses a unique state of properties like pressure, P, temperature, T, density, etc. All the properties of a system cannot be varied independently since they are interrelated through expressions of the following type f (P, v, T) = 0 For example, the pressure, temperature and molar volume (v) of an ideal gas are related by the expression Pv = RT. Here R is a constant. Only two of the three variables P, v and T can be varied independently. Question is that for a given thermodynamics system, how many variables can be varied independently. The State Postulate As noted earlier, the state of a system is described by its properties. But we know from experience that we do not need to specify all the properties in order to fix a state. Once a sufficient number of properties are specified, the rest of the properties assume certain values automatically. The number of properties required to fix the state of a system is given by the state postulate: The state of a simple compressible system is completely specified by two independent properties. A system is called a simple compressible system in the absence of electrical, magnetic, gravitational, motion, and surface tension effects. These effects are due to external force fields and are negligible for most engineering problems. Otherwise, an additional property needs to be specified for each effect which is significant. If the gravitational Copyright by Brij 2013 Page 19

20 20 Chapter 1: Introduction effects are to be considered, for example, the elevation z needs to be specified in addition to the two properties necessary to fix the state. Nitrogen T=25⁰ C v = 0.9 m 3 /kg Diagram 1.21 The state postulate is also known as the two-property rule. The state postulate requires that the two properties specified be independent to fix the state. Two properties are independent if one property can be varied while the other one is held constant. Temperature and specific volume, for example, are always independent properties, and together they can fix the state of a simple compressible system (Diagram 1.21). For a simple compressible substance (gas), we need the following properties to fix the state of a system: ρv or PT or vt or up or uv. Why not u and T? They are closely related. For ideal gas u = u(t). Temperature and pressure are dependent properties for multi-phase systems. At sea level (P = 1 atm), water boils at 100ºC, but on a mountain-top where the pressure is lower, water boils at a lower temperature. That is, T = f(p) during a phase-change process, thus temperature and pressure are not sufficient to fix the state of a two-phase system. Therefore, by specifying any two properties we can specify all other state properties. Let us choose P and then T = T(P, v), u = u (P, v) etc. Can we say W = W(P, v) No. Work is not a state property, nor is the heat added (Q) to the system. Zeroth Law of Thermodynamics Statement: If a body 1 is in thermal equilibrium with body 2 and body 3, then the body 2 and body 3 are also in thermal equilibrium with each other. System 1 U1, V1, N1 System 2 U2, V2, N2 Rigid and Adiabatic wall Diathermal and rigid wall Diagram 1.22 Two systems 1 and 2 with independent variables (U1, V1, N1) and (U2, V2, N2) are brought into contact with each other through a diathermal wall (Diagram 1.22). Let the system 1 be hot and system 2 be cold. Because of interaction, the energies of both the systems, as well as their independent properties undergo a change. The hot body becomes cold and the cold body becomes hot. After sometime, the states of the two systems do not undergo any further change and they assume fixed values of all thermodynamic properties. These two systems are then said to be in a state of thermal equilibrium with each other. The two bodies which are in thermal equilibrium with each other have a common characteristic called temperature. Therefore temperature is a property which has the same value for all the bodies in thermal equilibrium. Copyright by Brij 2013 Page 20

21 Basic and Applied Thermodynamics By Brij Bhooshan 21 Adiabatic wall Diathermal wall System 1 U1, V1, N1 System 2 U2, V2, N2 Adiabatic wall System 1 U1, V1, N1 System 2 U2, V2, N2 System 3 U3, V3, N3 System 3 U3, V3, N3 Diathermal wall Diagram 1.23 Suppose we have three systems 1, 2 and 3 placed in an adiabatic enclosure as shown in Diagram The systems 1 and 2 do not interact with each other but they interact separately with systems 3 through a diathermal wall. Then system 1 is in thermal equilibrium with system 3 and system 2 is also in thermal equilibrium with system 3. By intuition we can say that though system 1 and 2 are not interacting, they are in thermal equilibrium with each other. Suppose system 1 and 2 are brought into contact with each other by replacing the adiabatic wall by a diathermal wall as shown in Diagram Further they are isolated from system 3 by an adiabatic wall. Then one observes no change in the state of the systems 1 and SOME BASIC DEFINITIONS Temperature Scale Based on zeroth law of thermodynamics, the temperature of a group of bodies can be compared by bringing a particular body (a thermometer) into contact with each of them in turn. To quantify the measurement, the instrument should have thermometric properties. These properties include: The length of a mercury column in a capillary tube, the resistance (electrical) of a wire, the pressure of a gas in a closed vessel, the emf generated at the junction of two dissimilar metal wires etc. are commonly used thermometric properties. To assign numerical values to the thermal state of a system, it is necessary to establish a temperature scale on which temperature of a system can be read. Therefore, the temperature scale is read by assigning numerical values to certain easily reproducible states. For this purpose, it is customary to use. Ice Point: The equilibrium temperature of ice with air saturated water at standard atmospheric pressure which is assigned a value of 0 C. Steam Point: The equilibrium temperature of pure water with its own vapour at standard atmospheric pressure, which is assigned a value of 100 C. This scale is called the Celsius Scale named after Anders Celsius. Perfect Gas Scale An ideal gas obeys the relation where R is the Universal Gas Constant (R = J/mol K). This equation is only an approximation to the actual behavior of the gases. The behavior of all gases approaches the ideal gas limit at sufficiently low pressure (in the limit P 0). The perfect gas temperature scale is based on the observation that the temperature of a gas at constant Copyright by Brij 2013 Page 21

22 22 Chapter 1: Introduction volume increases monotonically with pressure. If the gas pressure is made to approach zero, the gas behavior follows the relation Helium Gas bulb Mercury column Reference mark S Flexible tube Diagram 1.24 a constant volume gas thermometer. The bulb is placed in the system whose temperature is to be measured. The mercury column is so adjusted that the level of mercury stands at the reference mark S. This ensures that the volume of the gas is held at a constant value. Let the pressure of the gas be read as P. Let a similar measurement be made when the gas bulb is maintained at the triple point of water, Ptp. We can obtain triple point by putting water and ice in an insulated chamber and evacuating air (which is then replaced by water vapour). The temperature of the triple point of water has been assigned a value of K. Since for an ideal gas T varies as P, where Ttp is the triple point temperature of water. Suppose a series of measurements with different amounts of gas in the bulb are made. The measured pressures at the triple point as well as at the system temperature change depending on the amount of gas in the bulb. A plot of the temperature Tcal, calculated from the expression T = (P/ Ptp) as a function of the pressure at the triple point, results in a curve as shown in Diagram Tcal Air H2 He Ptp Diagram 1.25 When these curves are extrapolated to zero pressure, all of them yield the same intercept. This behavior can be expected since all gases behave like ideal gas when their pressure approaches zero. The correct temperature of the system can be obtained only when the gas behaves like an ideal gas, and hence the value is to be calculated in limit Ptp 0. Therefore A constant pressure thermometer also can be used to measure the temperature. In that case, Copyright by Brij 2013 Page 22