or, equivalently, by observing that the current flowing in the series circuit is related to the capacitor voltage by i(t) = Cdv C /dt, and that Eq. (11.55) can be rewritten as (11.57) Note that although different variables appear in the preceding differential equations, both Eqs. (11.55) and (11.57) can be rearranged to appear in the same general form as follows: (11.58) where the general variable y(t) represents either the series current of the circuit of Fig. 11.49 or the capacitor voltage. By analogy with Eq. (11.54), we call Eq. (11.58) a second-order ordinary differential equation with constant coefficients. As the number of energy-storage elements in a circuit increases, one can therefore expect that higher-order differential equations will result. Phasors and Impedance In this section, we introduce an efficient notation to make it possible to represent sinusoidal signals as complex numbers, and to eliminate the need for solving differential equations. Phasors Let us recall that it is possible to express a generalized sinusoid as the real part of a complex vector whose argument, or angle, is given by ( ω t + φ ) and whose length, or magnitude, is equal to the peak amplitude of the sinusoid. The complex phasor corresponding to the sinusoidal signal Acos( ω t + φ ) is therefore defined to be the complex number Ae j φ : (11.59) 1. Any sinusoidal signal may be mathematically represented in one of two ways: a time-domain form and a frequency-domain (or phasor) form 2. A phasor is a complex number, expressed in polar form, consisting of a magnitude equal to the peak amplitude of the sinusoidal signal and a phase angle equal to the phase shift of the sinusoidal signal referenced to a cosine signal. 3. When using phasor notation, it is important to make a note of the specific frequency, ω , of the sinusoidal signal, since this is not explicitly apparent in the phasor expression. Impedance We now analyze the i-v relationship of the three ideal circuit elements in light of the new phasor notation. The result will be a new formulation in which resistors, capacitors, and inductors will be described in the same notation. A direct consequence of this result will be that the circuit theorems of section 11.3 will be extended to AC circuits. In the context of AC circuits, any one of the three ideal circuit elements RC dv C dt LC d 2 v C t() dt 2 v C t()++v S t()= a 2 d 2 yt() dt 2 a 1 dy t() dt a 0 yt()++ Ft()= Ae jf complex phasor notation for A wt f+()cos= vt() A wt f+()cos= V jw() Ae jf = 0066_Frame_C11 Page 32 Wednesday, January 9, 2002 4:14 PM ©2002 CRC Press LLC or, equivalently, by observing that the current flowing in the series circuit is related to the capacitor voltage by i(t) = Cdv C /dt, and that Eq. (11.55) can be rewritten as (11.57) Note that although different variables appear in the preceding differential equations, both Eqs. (11.55) and (11.57) can be rearranged to appear in the same general form as follows: (11.58) where the general variable y(t) represents either the series current of the circuit of Fig. 11.49 or the capacitor voltage. By analogy with Eq. (11.54), we call Eq. (11.58) a second-order ordinary differential equation with constant coefficients. As the number of energy-storage elements in a circuit increases, one can therefore expect that higher-order differential equations will result. Phasors and Impedance In this section, we introduce an efficient notation to make it possible to represent sinusoidal signals as complex numbers, and to eliminate the need for solving differential equations. Phasors Let us recall that it is possible to express a generalized sinusoid as the real part of a complex vector whose argument, or angle, is given by ( ω t + φ ) and whose length, or magnitude, is equal to the peak amplitude of the sinusoid. The complex phasor corresponding to the sinusoidal signal Acos( ω t + φ ) is therefore defined to be the complex number Ae j φ : (11.59) 1. Any sinusoidal signal may be mathematically represented in one of two ways: a time-domain form and a frequency-domain (or phasor) form 2. A phasor is a complex number, expressed in polar form, consisting of a magnitude equal to the peak amplitude of the sinusoidal signal and a phase angle equal to the phase shift of the sinusoidal signal referenced to a cosine signal. 3. When using phasor notation, it is important to make a note of the specific frequency, ω , of the sinusoidal signal, since this is not explicitly apparent in the phasor expression. Impedance We now analyze the i-v relationship of the three ideal circuit elements in light of the new phasor notation. The result will be a new formulation in which resistors, capacitors, and inductors will be described in the same notation. A direct consequence of this result will be that the circuit theorems of section 11.3 will be extended to AC circuits. In the context of AC circuits, any one of the three ideal circuit elements RC dv C dt LC d 2 v C t() dt 2 v C t()++v S t()= a 2 d 2 yt() dt 2 a 1 dy t() dt a 0 yt()++ Ft()= Ae jf complex phasor notation for A wt f+()cos= vt() A wt f+()cos= V jw() Ae jf = 0066_Frame_C11 Page 32 Wednesday, January 9, 2002 4:14 PM ©2002 CRC Press LLC 12 Engineering Thermodynamics 12.1 Fundamentals Basic Concepts and Definitions • Laws of Thermodynamics 12.2 Extensive Property Balances Mass Balance • Energy Balance • Entropy Balance • Control Volumes at Steady State • Exergy Balance 12.3 Property Relations and Data 12.4 Vapor and Gas Power Cycles Although various aspects of what is now known as thermodynamics have been of interest since antiquity, formal study began only in the early nineteenth century through consideration of the motive power of heat: the capacity of hot bodies to produce work. Today the scope is larger, dealing generally with energy and entropy, and with relationships among the properties of matter. Moreover, in the past 25 years engineering thermodynamics has undergone a revolution, both in terms of the presentation of funda- mentals and in the manner that it is applied. In particular, the second law of thermodynamics has emerged as an effective tool for engineering analysis and design. 12.1 Fundamentals Classical thermodynamics is concerned primarily with the macrostructure of matter. It addresses the gross characteristics of large aggregations of molecules and not the behavior of individual molecules. The microstructure of matter is studied in kinetic theory and statistical mechanics (including quantum thermodynamics). In this chapter, the classical approach to thermodynamics is featured. Basic Concepts and Definitions Thermodynamics is both a branch of physics and an engineering science. The scientist is normally interested in gaining a fundamental understanding of the physical and chemical behavior of fixed, quiescent quantities of matter and uses the principles of thermodynamics to relate the properties of matter. Engineers are generally interested in studying systems and how they interact with their surround- ings. To facilitate this, engineers have extended the subject of thermodynamics to the study of systems through which matter flows. System In a thermodynamic analysis, the system is the subject of the investigation. Normally the system is a specified quantity of matter and/or a region that can be separated from everything else by a well-defined surface. The defining surface is known as the control surface or system boundary . The control surface may be movable or fixed. Everything external to the system is the surroundings . A system of fixed mass is Michael J. Moran The Ohio State University ©2002 CRC Press LLC . follows: (11 . 58) where the general variable y(t) represents either the series current of the circuit of Fig. 11 .49 or the capacitor voltage. By analogy with Eq. (11 .54), we call Eq. (11 . 58) a second-order. Ae jf = 0066_Frame_C 11 Page 32 Wednesday, January 9, 2002 4 :14 PM 2002 CRC Press LLC 12 Engineering Thermodynamics 12 .1 Fundamentals Basic Concepts and Definitions • Laws of Thermodynamics 12 .2. Eqs. (11 .55) and (11 .57) can be rearranged to appear in the same general form as follows: (11 . 58) where the general variable y(t) represents either the series current of the circuit of Fig. 11 .49