Energy and Power

September 19, 2003 17:5 WSPC/Energy Studies (2nd Edition) bk02-013 CHAPTER 1 ENERGY AND POWER Energy is the capacity or capability to do work. All materials possess energy, because they can all be utilised in some form of energy conversion process. For example, most substances will burn or vaporise, and the consequent heat energy can be harnessed within mechanical energy systems that create motion against some form of mechanical resistance. Energy can take several forms, as classified in Table 1.1. Mass or matter is a form of highly concentrated energy. Some forms of matter can be utilised in nuclear energy applications, as discussed in Chapter 8. Table 1.1. Forms of energy. biofuels (e.g. wood) mass chemical mechanical { kinetic electrical mechanical { potential gravitational nuclear heat (thermal) radiation magnetic sound 1.1. Energy Conversion The many applications of the use of energy usually involve transformations between different forms of energy | a process known as energy conversion. Any conversion between different energy forms is imperfect in that some of the energy has to be used to facilitate the conversion process. The converted energy output is lower than the energy input and this feature is usually described as the conversion efficiency. Figure 1.1 illustrates the large range of variation of energy conversion efficiencies, from very large electricity generators (mechanical to electrical converters) which can operate continuously at about 99% efficiency to the incandescent electric lamp (electrical to radiant converter) which is only a few percent efficient [1]. Some well-known energy conversion processes involve two successive stages. An example 1 September 19, 2003 17:5 WSPC/Energy Studies (2nd Edition) bk02-013 2 Energy Studies Fig. 1.1. Efficiencies of energy converters (based on [1]). September 19, 2003 17:5 WSPC/Energy Studies (2nd Edition) bk02-013 Energy and Power 3 is the motor car engine in which chemical energy in the form of oil or petrol (gaso- line) is converted to heat and then to rotational energy. 1.2. Mechanical Energy The widely used Laws of Motion for bodies of constant mass were developed by the English scientist Isaac Newton in the 17th century. It is now known that in extreme cases Newton's Laws are insufficient | for very small masses quantum mechanics must be employed; with very high speeds, Einstein's theory of special relativity becomes relevant; with very large masses, the concepts of space and time are modified by the theory of general relativity. Nevertheless, for general conduct of life on earth using realistic sizes and time spans, the work of Newton remains valid. 1.2.1. Linear motion When a constant force F is applied to an object and causes it to move through a distance x in the direction of the force, then the work done W is equal to the energy expended: W = F x (1.1) In (1.1), if the force is in newtons (N) and the distance in metres (m), the work or energy W has the unit of joules (J) or newton-metres (Nm). If a body of mass m moves in a straight line with a linear velocity v which is the time rate of change of its position, dx v = for small changes of x dt (1.2) x v = for large changes of x t If a body of mass m moving in a straight line is subjected to changes of velocity, the rate of change of the velocity with time is known as the acceleration a: dv d dx d2x a = = = (1.3) dt dt dt dt2 In S.I. units, mostly used in this book, the velocity is measured in metres/sec (m/s) and the acceleration in metres/sec/sec or metres/sec2 (m/s2). When a force F is applied to a body of constant mass m and causes the linear velocity v to change, the resulting acceleration can be shown experimentally to be proportional to the applied force: dv d2x dv F = ma = m = m = mv (1.4) dt dt2 dx September 19, 2003 17:5 WSPC/Energy Studies (2nd Edition) bk02-013 4 Energy Studies Equation (1.4) is sometimes referred to as Newton's Second Law of Motion. Mass m may be combined with velocity v to define an important physical property known as the momentum: dx Linear Momentum = mv = m (1.5) dt Comparison of (1.4) and (1.5) shows that Force = time rate of change of linear momentum dv d (1.6) F = m = (mv) dt dt Equation (1.6) shows that momentum has the dimension of force time or mass × velocity. × A mass m possesses energy of two kinds, known as potential energy, associated with its position, and kinetic energy, associated with its motion. The gravitational potential energy of a body of mass m, at height h above a datum plane, is given by WPE = mgh (1.7) where g is the gravitational acceleration constant, of value g = 9:81 m/s2. If the mass m is in kilogrammes and height h is in metres, the potential energy WPE is in joules. While a mass m is in linear motion at a constant velocity v, the kinetic energy WKE associated with the motion is 1 W = mv2 (1.8) KE 2 It can be seen from (1.8) that the derivative of kinetic energy WKE with respect to velocity gives the momentum dW KE = mv (1.9) dv Both kinetic energy and momentum, like mass, satisfy important conservation rules. In this book the most relevant rule is the Principle of Conservation of Energy, which states that \in any physical system the total energy remains constant | energy may be converted to a different form, it may be wasted, but it cannot be destroyed". When a mass m in linear motion is acted upon by a force F , then, in moving between two locations: work done on change of kinetic distance force = or against the = energy between (1.10) × moved mass the two locations September 19, 2003 17:5 WSPC/Energy Studies (2nd Edition) bk02-013 Energy and Power 5 Example 1.1 A mass m initially rests on a ledge at height h metres above ground level, which is the datum plane. Define the conditions of velocity, kinetic energy and potential energy (i) initially, (ii) as the mass falls to ground, (ii) finally after the mass comes to rest. (i) With the mass at rest, its initial velocity vi is zero and therefore so are its initial momentum and kinetic energy. Its total energy is then the potential energy given by (1.7), illustrated in Fig. 1.2. (ii) As the mass falls to the ground it possesses an instantaneous velocity v0, initially zero and increasing uniformly due to gravitational acceleration. Its final velocity becomes zero on impact with the ground. At any arbitrary height h0 during the fall, the mass possesses both potential energy mgh0 and 1 02 kinetic energy 2 mv , which sum to the initial energy mgh. After striking the ground the final velocity vf is zero, the momentum of the motion is transferred to the ground and the kinetic energy is converted to local heat and sound due to impact. Since the ground level is the datum plane, the potential energy after impact is also zero here. m m h v9 h9 m ground (datum!plane) (a) (b) (c) 9 WPE = mgh WPE = mgh WPE = 0 92 WKE = 0 WKE = mv WKE = 0 9 vi = 0 v = v vf = 0 Fig. 1.2. Mass falling freely under gravity. September 19, 2003 17:5 WSPC/Energy Studies (2nd Edition) bk02-013 6 Energy Studies 1.2.2. Rotational motion Most energy conversion processes involving mechanical energy incorporate rotational devices. For example, electromechanical energy converters use rotors that have the form of solid cylinders, Fig. 1.3(a). Petrol engines and diesel engines usually incorporate flywheels, Fig. 1.3(b). The rotor of a water or gas turbine also has the nature of a non-uniform flywheel. To illustrate some of the principles of rotational motion, the example used is that of a concentrated mass m in circular motion at radius r about a fixed cen- tre point, Fig. 1.4. The motion is characterised by the angular velocity ! in radians/sec (rad/s) and the instantaneous tangential velocity v of the mass in metres/sec (m/s), where v = !r (1.11) A centripetal force acting radially inwards is required to keep the mass moving in a circle and is provided along the tie rod. With rotational motion, the externally applied force F acting tangentially on the mass (through a rigid tie-rod), Fig. 1.4, times the radius r is called the torque T , which acts as a rotation producing force: v T = F r = F (1.12) ! Torque is measured in newton-metres (Nm) and is a very important property of rotating energy converters. The tangential or linear acceleration of the mass m is given, from (1.11), by dv d! a = = r (1.13) dt dt Combining (1.12) and (1.13) leads to T = F r = mar dv = m r dt d! ) T = mr2 = mr2α (1.14) dt In (1.14) the term α is the angular acceleration in rad/s2. The quantity mr2 in (1.14) is known as the polar moment of inertia J and is an important physical property in rotational structures, having the dimension kgm2. J = mr2 = [mass] [radius of gyration]2 (1.15) Expression (1.15) is true directly for the flywheel and cylinder of Fig. 1.3. For more complicated structures with distributed, non-uniform mass, the effective radius of gyration is more complicated but the relationship (1.15) is still valid in principle.

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