System boundary List of variablesW. W . mi,vi, hi me, ve, he zi ze m,U . . h z i e m U v m. Q. Q . heat flow Work flow mass flow velocity specific enthalpy elevation denotes inlet variables denotes exit variable mass in system internal energy of system Underlying Physical Principles Chapter 2 Chapter 2 Underlying Physical Principles 11 2.1 Introduction It has been assumed that the student is familiar with: Thermodynamics: • The first and second laws of thermodynamics • the application of these laws to open and closed systems • the ideal gas model. Electrical engineering: • The basic electrical components resistance, capacitance and inductance • the associated relationships between current and voltage over these components • the relations between voltage, current and power in basic electric networks • the difference between DC and AC • the basics of magnetic induction: Faraday and Lorentz. The basic principles of thermodynamics and electrical engineering used in this book are treated in this section. This chapter introduces the terminology and symbols used throughout this book and is to be used as a reference for later chapters. 2.2 Basic thermodynamic principles 2.2.1 First law of thermodynamics Evaluation of energy and energy transfer in thermodynamic processes requires application of the conservation of energy principle. The first law of thermodynamics is based on this principle. The law will first be given in its generic form and will later be applied to closed and open systems. The generic form covers all energetic considerations in this book. General Figure 2.1 is a schematic presentation of a thermodynamic system. The system is separated from the surroundings by the chosen system boundary. Two kinds of thermodynamic systems are used in this book: closed systems (a fixed quantity of mass) and open systems (a fixed control volume). The process is characterised by the heat, work and mass flows that cross the system boundaries and the internal energy and mass of the system. Design of Propulsion and Electric Power Generation Systems 12 System boundary List of variablesW. W . mi,vi, hi me, ve, he zi ze m,U . . h z i e m U v m. Q. Q . heat flow work flow mass flow velocity specific enthalpy elevation denotes inlet variables denotes exit variable mass in system internal energy of system Figure 2.1 A thermodynamic system The first law of thermodynamics states that the change of energy within the system equals the heat flow into the system, minus the work flow delivered by the system, plus the difference in enthalpy H, kinetic energy Ekin and potential energy Epot of the entering and exiting mass flows. The influence of chemical reactions is not included. Equation (2.1) shows the first law in generic terms for a single inlet and a single exit flow. (2.1) 22 ie iiieeedUvvQWmhgzmhgzdt22 §·§·=-+·++·-·++·¨¸¨¸©¹©¹ Additionally the conservation of mass principle must be complied with. The change of mass within the system boundary is the difference between the entering and exiting mass flows. (2.2) iedmmmdt=- Note that equation (2.1) is sometimes given with dE dt instead of dU dt. In that case, energy E equals the sum of the internal, kinetic and potential energy of the system: E = U + Ekin + Epot In most cases, however, kinetic energy Ekin and potential energy Epot of the system do not change and, therefore, can be omitted. Closed system A closed system is a thermodynamic system where no mass flow occurs across the system boundary. So, ?i = ?e = 0, in which case the first law according to equation (2.1) reduces to: (2.3) dUQWdt=- or dUQW=d-d Chapter 2 Underlying Physical Principles 13 After integration for a process between state 1 and state 2 the first law becomes: U2 – U1 = Q1,2 – W1,2 Note the different differential symbols d and d. The internal energy U is a state variable (dU is a total differential). This means that the internal energy of a system is an inherent property of the state of the system; it depends only on its current state and is independent of the way in which the system was brought to that state. However, heat and work are not state variables. The amount of heat or work depends on the path of the process between the beginning and end states of the process. dQ and dW are not total differentials but small amounts of heat and work, also called diminutives. Frequently, closed systems are considered as cycles. A cycle is a sequence of processes of a closed system in which the initial state and the final state of the process are identical. Figure 2.2 shows an example of the Brayton cycle in a p–V diagram. The cycle starts at point 1 and returns to that same point via 2, 3 and 4. The state variables do not change over the cycle: (2.4) m1,start = m1,end ? ?m = 0 (2.5) U1,start = U1,end ? ?U = 0 This means that for a cycle the total heat supplied to the process must be equal to the total work delivered by the process, because equation (2.3) reduces to equation (2.6): (2.6) Qcycle = Wcycle p2 14 V 3 Figure 2.2 The Brayton cycle in p-V diagram Examples of closed systems are a refrigeration system or a steam turbine plant complete with feedwater pump, boiler and condenser. Sometimes it may be useful to consider a diesel engine or a gas turbine as a closed loop thermodynamic cycle. Design of Propulsion and Electric Power Generation Systems 14 Open systems An open system is a thermodynamic system that can be referred to as a volume through which mass may flow. The general form of the first law of thermodynamics is applicable for open systems. For an open system at steady state (an open stationary system), the state variables of the mass in the system do not change with time. Consequently, the internal energy and the mass within the system do not change with time: (2.7) dUdm00dtdt==and The mass flows entering and exiting the system must be equal in order to have no accumulation of mass: ?i = ?e = constant = ? With these assumptions the first law, refer to equation (2.1), for an open system at steady state reduces to: (2.8) ()() 22 ie ieievvWQmhhgzz2 §·-=+·-++·-¨¸©¹ Examples of open systems at steady state Examples of open systems are the compressor (as part of a complete refrigeration system), the turbine (as part of a complete steam turbine plant) or a diesel engine. The following examples of open systems at steady state are often found in marine engineering. Steam turbine A steam turbine generates mechanical energy (work) from a steam mass flow. If the turbine is well insulated, the heat flow over the system boundary will be very low and it may be assumed that the process is adiabatic, i.e. Q0= . If the turbine is constructed in such a way that height and velocity differences between intake and exit are also of minor importance, (zi ˜ ze and vi ˜ ve) the work flow can be written as: (2.9) ieWm(hh)=·- Under similar assumptions, this relation is also valid for compressors, pumps and hydraulic motors. Heat exchanger/boiler A heat exchanger or a boiler has no transfer of work across system boundaries, i.e. ? = 0. Using similar assumptions as above it can be shown that the heat transfer is associated with an enthalpy difference: (2.10) eiQm(hh)=·- Chapter 2 Underlying Physical Principles 15 Nozzle/guide vane/diffuser A nozzle, guide vane or diffuser is used to increase or decrease velocity of a fluid. There is no transfer of work and heat. If it is also assumed that height at intake and exit are equal (zi = ze) then: (2.11) 22 eiievv2(hh)-=·- When vi can be considered zero, this results in eiev2(hh)=·- Throttling A throttling process involves a fluid flow over a resistance in a valve or a pipeline. As no work is performed by such a resistance and no heat transfer occurs, the first law of thermodynamics reduces to: (2.12) iehh= This relation holds when changes in velocity or height may be neglected (i.e. zi ˜ ze and vi ˜ve). 2.2.2 Second law of thermodynamics The second law of thermodynamics is used (1) to predict the direction of processes, (2) to establish the conditions of final equilibrium and (3) to determine the best possible theoretical performance of a process. In this section only the aspects of the second law that are necessary for understanding the processes in this book are discussed. Direction of processes and final equilibrium state It can be observed that processes have a natural direction. These spontaneous processes can only be reversed if other processes are added. Two examples are: • If two bodies, with different temperatures are brought into contact, a heat flow will occur from the high temperature body to the low temperature body. This will result in a temperature (internal energy) decrease of the first body and a temperature (internal energy) increase of the second body. The heat flow will cease at the moment that both bodies have the same temperature; this is the final equilibrium state • if two containers filled with air at different pressures are connected, a similar spontaneous process will take place. An air mass flow will occur from the high pressure container to the low pressure container. This will result in a pressure decrease in the first container and a pressure increase in the second container. The airflow will cease at the moment the pressures in the two containers are equal. Reversibility A process of a system is reversible if the system can be restored to the initial state without changing the surroundings. It is essential to include the surroundings in this definition, Design of Propulsion and Electric Power Generation Systems 16 because a system can always be brought back to its initial state by changing the surrounding, such as by increasing the ambient pressure or temperature. The natural processes described above (heat flow between two bodies and air mass flow between two containers) are irreversible, because it is not possible to bring the bodies and the containers back to their original state without a change of state of the surroundings. Entropy The entropy change of a system that is subjected to a reversible process with heat transfer is defined as: (2.13) QdST d= Note that entropy S [J/K], like pressure, temperature, internal energy and enthalpy, is a state variable. The relation states that for a reversible process in which heat transfer takes place, the entropy change is equal to the heat flow divided by temperature. For an irreversible process, the entropy change is greater than the heat flow divided by temperature. Therefore, a way of expressing the second law in general form is: (2.14) QdST d= Or after integration, 2 21 1 QSST d-=? Work on a closed system An example of a closed system that undergoes a reversible process involving volumetric change is a cylinder filled with gas under a piston. Reversibility in this case means that (1) there is no friction between piston and cylinder and (2) that the pressure in the cylinder is equal on every location within the cylinder at every moment. The latter means in practice that compression or expansion should take place slowly. For a volumetric change the work can now be expressed as: (2.15) WpdVd=· Assuming a piston area A, this relation results from the general expression for work as follows: WFdspAdspdVd=·=··=· Chapter 2 Underlying Physical Principles 17 The result is that for a reversible process of a closed system, work can be determined as: (2.16) 2 1,2 1 WpdV=·? Although, compression and expansion are, in practice, never reversible because of friction, the relation found here can frequently be applied to these processes with great accuracy because friction can be considered to be outside the system boundary and then treated afterwards. Heat transfer When heat is transferred in a reversible process, the amount of heat can be determined from: ref. (2.13) QTdSd=· (2.17) 2 1,2 1 QTdS=·? In practice, heat transfer never is a reversible process. However, heat transfer can be imagined as an internally reversible process if the heat flow takes places between two bodies with an infinitesimal small temperature difference dT. The result of this assumption is that the above relation can frequently be applied if the system boundary is suitably chosen. The heating or cooling body, with a distinct temperature difference to the process under consideration, should not be included within the system boundary. Gibbs equations The Gibbs equations are deduced from the first and second laws of thermodynamics. The first law of thermodynamics for a closed system can be expressed as equation (2.3): ref. (2.3) dUQW=d-d The expressions for reversible work and heat, derived in the preceding paragraphs are: ref. (2.15) WpdVd=· ref. (2.13) QTdSd=· The first Gibbs equation is obtained when those expressions are combined: (2.18) dUT.dSpdV=-· The second Gibbs equation is a formulation of enthalpy. By definition enthalpy is: (2.19) defHUpV=+·