Conservation of Energy in Fluids

10 Key excerpts on "Conservation of Energy in Fluids"

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  Mechanics of Continuous Media

  • John Botsis et Michel Deville(Author)
  • 2018(Publication Date)
  • PPUR

    (Publisher)

  Chapter 4 Energy 4.1 Introduction Having described the principles of conservation of mass, momentum, and an- gular momentum, we will now introduce the principles related to the thermo- dynamics of continuous media in motion and the conservation of energy. We can recall that all deformations in a material produce a thermal effect in the same way that a thermal effect produces a deformation. This is easily observed by heating a metal bar which lengthens under the action of the heat. In this chapter, we will generally work in the spatial or Eulerian represen- tation. The principle of conservation of total energy is first established. It leads to the principle of conservation of internal energy. Then, we will con- sider the conservation of mechanical energy in the Lagragian representation. Later, we will show that from the principle of conservation of total energy, for which objectivity is imposed, we can infer the other conservation laws. Finally, the chapter ends with the introduction of entropy and the second law of ther- modynamics, which is based on the Clausius–Duhem inequality, a measure of the irreversibility of the phenomena associated with the physics of continuous media. Continuous media thermodynamics is covered in detail by the following authors: [15, 17, 18, 22, 58, 68]. 4.2 Conservation of Energy Let ω(t) be the material volume of a continuous medium at the instant t, such that ω(t) ⊆ R, the deformed configuration of the body B. We generalize the concept of kinetic energy by defining it as the integral over the deformed volume ω(t) of half the density, ρ(x, t), multiplied by the square of the local spatial velocity, v(x, t). The kinetic energy of ω(t), which we denote E k (t), is a scalar given by the relation E k (t) = Z ω(t) ρ(x, t) v(x, t) · v(x, t) 2 dv . (4.1) 142 Energy To simplify, the dependence of ω with respect to time will no longer be explic- itly shown in the following.

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  Thermodynamics

  Concepts and Applications

  • Stephen R. Turns(Author)
  • 2006(Publication Date)
  • Cambridge University Press

    (Publisher)

  We also present and discuss energy conservation expressions that have been commonly adopted for flows where friction is particularly important. We introduce the application of the energy conservation principle to analyses of steady-flow devices. Because of the strong link of this chapter to practical applications, the author highly recommends concurrent study of the indicated sections of Chapter 7 . Chapter 5 Overview 5.1 HISTORICAL CONTEXT The development of an energy conservation principle depended critically on the evolving idea that work was convertible to heat, and vice versa. Benjamin Thompson (1753–1814) (Count Rumford) in 1798 [1], Julius Robert Mayer (1814–1878) in 1842 [2], and James Prescott Joule (1818–1889) in 1849 [3] published results of their research on the mechanical equivalent of heat and presented numerical values. The issue here was to define the specific number of foot-pounds of work that is equivalent to the heat required to raise the page 282 Robert Mayer (1814-1878) Hermann Helmholtz (1821-1894) Rudolf Clausius (1822-1888) CH. 5 CONSERVATION OF ENERGY 283 temperature of 1 lb m of water 1F. (The accepted value today is 778.16 ft-lb f and defines the British thermal unit or Btu.) Mayer was the first person to state a formal conservation of energy principle. In 1842, he wrote [2]: “Forces (energies) are therefore indestructible, convertible, and (in contradistinction to matter) imponderable objects… a force (energy) once in existence, cannot be annihilated.” (Parenthetical material has been added; in the mid 1800s, the word force commonly meant energy [4, 5].) Hermann Helmholtz (1821–1894), who produced an array of scientific achievements, extended the conservation of energy principle in his 1847 paper, “On the Conservation of Force,” to include all known forms of energy: mechanical, thermal, chemical, electrical, and magnetic.

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  The Mechanical Universe

  Mechanics and Heat, Advanced Edition

  • Steven C. Frautschi, Richard P. Olenick, Tom M. Apostol, David L. Goodstein(Authors)
  • 2008(Publication Date)
  • Cambridge University Press

    (Publisher)

  CHAPTER ENERGY: CONSERVATION AND CONVERSION . .. You see, therefore, that living force [energy] may be converted into heat, and that heat may be converted into living force, or its equivalent attraction through space. All three, therefore -namely, heat, living force, and attraction through space (to which I might also add light, were it consistent with the scope of the present lecture) - are mutually convertible into one another. In these conversions nothing is ever lost. The same quantity of heat will always be converted into the same quantity of living force. We can therefore express the equivalency in definite language applicable at all times and under all circumstances. James Prescott Joule, On Matter, Living Force, and Heat (1847) 10.1 TOWARD AN IDEA OF ENERGY The law of conservation of energy is one of the most fundamental laws of physics. No matter what you do, energy is always conserved. So why do people tell us to conserve energy? Evidentally the phrase conserve energy has one meaning to a scientist and quite a different meaning to other people, for example, to the president of a utility company or to a politician. What then, exactly, is energy? 219 220 ENERGY: CONSERVATION AND CONVERSION The notion of energy is one of the few elements of mechanics not handed down to us from Isaac Newton. The idea was not dearly grasped until the middle of the nineteenth century. Nevertheless, we can find its germ even earlier than Newton. The essence of the idea of the conservation of energy can be seen in the incredibly fertile experiments that Galileo performed with balls rolling down inclined planes. It is astonishing how many results Gaiileo squeezed out of his simple experiments. Bodies fall much too fast to be timed by the crude water clocks of the seventeenth century, but by slowing down the failing motion with his inclined planes, Galileo showed that uniformly accelerated motion was a part of nature. That alone was an achievement to crown him a genius.

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  Chemical, Biochemical, and Engineering Thermodynamics

  • Stanley I. Sandler(Author)
  • 2017(Publication Date)
  • Wiley

    (Publisher)

  3.1 Conservation of Energy 47 ω I Mass fraction of phase I (quality for steam) ψ Potential energy per unit of mass (J/g) 3.1 CONSERVATION OF ENERGY To derive the energy conservation equation for a single-component system, we again use the black-box system of Figure 2.1-1 and start from the general balance equation, Eq. 2.1-4. Taking θ to be the sum of the internal, kinetic, and potential energy of the system, θ = U + M  v 2 2 + ψ  Here U is the total internal energy, v 2 /2 is the kinetic energy per unit mass (where v is the center of mass velocity), and ψ is the potential energy per unit mass. 1 If gravity is the only force field present, then ψ = gh, where h is the height of the center of mass with respect to some reference, and g is the force of gravity per unit mass. Since energy is a conserved quantity, we can write d dt  U + M  v 2 2 + ψ  =  Rate at which energy enters the system  −  Rate at which energy leaves the system  (3.1-1) To complete the balance it remains only to identify the various mechanisms by which energy can enter and leave the system. These are as follows. Energy flow accompanying mass flow. As a fluid element enters or leaves the sys- tem, it carries its internal, potential, and kinetic energy. This energy flow accompanying the mass flow is simply the product of a mass flow and the energy per unit mass, K  k=1 ˙ M k  ˆ U + v 2 2 + ψ  k (3.1-2) where ˆ U k is the internal energy per unit mass of the kth flow stream, and ˙ M k is its mass flow rate. Heat. We use ˙ Q to denote the total rate of flow of heat into the system, by both conduction and radiation, so that ˙ Q is positive if energy in the form of heat flows into the system and negative if heat flows from the system to its surroundings. If heat flows occur at several different places, the total rate of heat flow into the system is ˙ Q =  ˙ Q j where ˙ Q j is the heat flow at the j th heat flow port.

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  Halliday and Resnick's Principles of Physics

  • David Halliday, Robert Resnick, Jearl Walker(Authors)
  • 2020(Publication Date)
  • Wiley

    (Publisher)

  This law of conservation of energy is not something we have derived from basic physics principles. Rather, it is a law based on countless experiments. The total energy E of a system can change only by amounts of energy that are transferred to or from the system. 168 CHAPTER 8 POTENTIAL ENERGY AND CONSERVATION OF ENERGY Scientists and engineers have never found an exception to it. Energy simply can- not magically appear or disappear. Isolated System If a system is isolated from its environment, there can be no energy transfers to or from it. For that case, the law of conservation of energy states: Many energy transfers may be going on within an isolated system — between, say, kinetic energy and a potential energy or between kinetic energy and ther- mal energy. However, the total of all the types of energy in the system cannot change. Here again, energy cannot magically appear or disappear. We can use the rock climber in Fig. 8-14 as an example, approximating him, his gear, and Earth as an isolated system. As he rappels down the rock face, changing the configuration of the system, he needs to control the transfer of energy from the gravitational potential energy of the system. (That energy cannot just disappear.) Some of it is transferred to his kinetic energy. How- ever, he obviously does not want very much transferred to that type or he will be moving too quickly, so he has wrapped the rope around metal rings to produce friction between the rope and the rings as he moves down. The sliding of the rings on the rope then transfers the gravitational potential energy of the system to thermal energy of the rings and rope in a way that he can control. The total energy of the climber – gear – Earth system (the total of its gravita- tional potential energy, kinetic energy, and thermal energy) does not change during his descent. For an isolated system, the law of conservation of energy can be written in two ways.

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  Heat Transfer

  Thermal Management of Electronics

  • Younes Shabany(Author)
  • 2009(Publication Date)
  • CRC Press

    (Publisher)

  29 3 Principle of Conservation of Energy 3.1 FIRST LAW OF THERMODYNAMICS In Chapter 2 it was shown that energy transfers from one system to another as either work or heat. Energy transfer from one system to another is the only way that a system gains or loses energy. The first law of thermodynamics states that energy is not generated or destroyed; it only changes from one form to another or transfers from one system to another. This important physical principle can be expressed by a simple mathematical equation. Consider a system with total initial energy E i as shown in Figure 3.1. The total energy of this system after some time, during which total energy E in enters the system and total energy E out leaves the system, will be E f . The first law of thermodynamics requires that the difference between the total energy entering this system and the total energy leaving it be equal to the difference between its final and initial energies [1–3]: E E E E f i in out − = − . (3.1) The difference between final and initial energies of a system is called change of energy of that system or energy accumulation in that system; ∆ E E E f i system = − . Therefore, the first law of thermodynamics can be expressed by the following equation: E E E in out system − = ∆ . (3.2) This is a simple yet powerful equation that is the basis of all the energy and heat transfer analyses. It is also known as energy balance equation . The energy balance equation, Equation 3.2, deals with initial and final states of a system without being concerned with what happens between these two states or how long it takes to reach from an initial state to a final state. However, these are impor-tant design parameters in most engineering applications. Let’s assume that the state of a system changes during a time interval ∆ t as shown in Figure 3.2. The energy of the system at times t and t + ∆ t are E t and E t + ∆ t , respectively.

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  Exploring Integrated Science

  • Belal E. Baaquie, Frederick H. Willeboordse(Authors)
  • 2009(Publication Date)
  • CRC Press

    (Publisher)

  64 Exploring Integrated Science A powerful idea in physics is that of invariants , that is, things that do not change as a system evolves in time (see also Chapter 2). The existence of invariants and their use allows us to place constraints on the possible dynamics of a system even though we might be ignorant of the details. Denoting total energy by E we have the fundamental (classical) relation Total energy of system D Kinetic Energy C Potential Energy ) E D T C U: (3.40) Suppose the system has energy E 1 at time t 1 and energy E 2 at a later time t 2 , then, the change in energy is E D E 2 NUL E 1 . Conservation of energy implies that E D 0 (3.41) ) T C U D 0: (3.42) Note an important fact that since all we know is that E D 0 , the absolute value of E has not been fixed. Hence, energy is only defined up to a constant, since E and E C constant would both be equally conserved. 3.8 Free energy Energy is everywhere! Einstein’s famous formula E D mc 2 tells us that even the mass of a body is a form of congealed energy (see Chapter 22). All material things are different forms of energy. If energy is indeed everywhere, why is there always a fear that our society is “running out energy”, or that there is a shortage of fuel? Why are we asked to reduce, reuse and recycle? We all intuitively know that energy is precious and that possessing energy is of high value. So we need to wonder: Is all energy equal? Or is there a certain energy that is more desirable than another? Fig. 3.20: Coal provides us with “useful” energy. One of the main developments of science in the nineteenth century was the realization by Sadi Carnot, Rudolf Clausius and others that useful energy — energy that can do mechanical work, energy that can be “controlled” and directed — is a very special kind of energy. This special form of energy is called “free energy” to differentiate it from energy in general. Let us consider the forms of energy that we find useful.

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  Aerodynamics for Engineers

  • John J. Bertin, Russell M. Cummings(Authors)
  • 2021(Publication Date)
  • Cambridge University Press

    (Publisher)

  Internal energy (u e ) : energy associated with the internal fields and the random motion of the molecules So, the energy of the system may be written simply as: e = ke + pe + u e (2.34a) 72 Chap. 2 / Fundamentals of Fluid Mechanics Now we can further examine the terms that comprise the energy of the system. The kinetic energy per unit mass is given by ke = V 2 2 (2.34b) Note that the change in kinetic energy during a process clearly depends only on the ini- tial velocity and final velocity of the system of fluid particles. Assuming that the external force field is that of gravity, the potential energy per unit mass is given by pe = gz (2.34c) Note that the change in the potential energy depends only on the initial and final elevations. Finally, the change in internal energy is also a function of the values at the endpoints only. Substituting equations (2.34) into equation (2.33), we obtain Q # - W # = 0 0t l r a V 2 2 + gz + u e b d(vol) + ∂ r a V 2 2 + gz + u e b V S # n n dA (2.35) You should notice that, while the changes in the energy components are a function of the states, the amount of heat transferred and the amount of work done during a process are path dependent. That is, the changes depend not only on the initial and final states but on the process that takes place between these states. Now we can further consider the term for the rate at which work is done, W # . For convenience, the total work rate is divided into flow work rate (W # f ), viscous work rate (W # v ), and shaft work rate (W # s ), which will be discussed in more detail below. 2.9.3 Flow Work Flow work is the work done by the pressure forces on the surroundings as the fluid moves through space. Consider flow through the streamtube shown in Fig. 2.17. The pressure p 2 acts over the differential area n n 2 dA 2 at the right end (i.e., the down- stream end) of the control volume. Remember that the pressure is a compressive force acting on the system of particles.

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  Transport Processes in Chemically Reacting Flow Systems

  Butterworths Series in Chemical Engineering

  • Daniel E. Rosner, Howard Brenner(Authors)
  • 2013(Publication Date)
  • Butterworth-Heinemann

    (Publisher)

  When would r ^ 4 = 0,rco 2 = 0, etc.? 2.5 For systems that need not be in mechanical, thermal, or chemical equilibrium: a. Would a discontinuity in tangential mass-averaged velocity, v i9 across an interface (e.g., phase boundary) violate any basic conservation principle? b. Would a discontinuity in temperature, T, across an interface violate any conservation principle? c. Would a discontinuity in chemical potential across an interface violate any conservation principle? d. What kind of restrictions do the conservation equations impose in such cases? 2.6 Under what conditions is the notion of specific potential energy (φ) useful? Is the work done on a fluid by the body force due to the earth's gravitation already included in φ, or must it be included separately in the basic energy equation (Eq. (2.5-23))? 2.7 We have stated that a moving fluid element has a kinetic energy equal to v 2 /2 per unit mass. Verify that this is the amount of work that must be done to accelerate a unit mass from rest to velocity v. (Use Newton's second law and the concept that work equals force times displacement (in the direction of the force)). 2.8 Numerically, compute and compare the following energies (after converting all to the same units, say, calories): a. The kinetic energy of a gram of water moving at 1 m/s. b. The potential energy change associated with raising one gram of water Exercises 81 O-m = lg I m/s - z=lm Az=lm through a vertical distance of one meter against gravity (where g = 0.9807 x 10 3 cm/s 2 ). c. The energy required to raise the temperature of one gram of liquid water from 273.2 K to 373.2 K. d. The energy required to melt one gram of ice at 273.2 K. e. The energy released when one gram of H 2 0(g) con-denses at 373 K. f. The energy released when one gram of liquid water is formed from a stoichiometric mixture of hydrogen (H 2 (g)) and oxygen (0 2 (g)) at 273.2 K.

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  Fluid Dynamics via Examples and Solutions

  • Sergey Nazarenko(Author)
  • 2014(Publication Date)
  • CRC Press

    (Publisher)

  Chapter 2 Conservation laws in incompressible fluid flows 2.1 Background theory Many properties of fluid flows can be understood based on very general conservation laws. Some of these laws are universal in physics, such as the energy and the momentum conservation laws. Each of them represents a global conservation property, i.e. the total amount of energy and momentum remain unchanged in the system. The other type of conservation laws in fluids are local or Lagrangian : they refer to conservation of a field along fluid trajectories (e.g. the vorticity in 2D) or conservation over a selected set of moving fluid particles (e.g. the velocity circulation over contours made out of moving fluid particles). 2.1.1 Velocity-vorticity form of the Navier-Stokes equation Let us consider the Navier-Stokes equation for incompressible flow (1.1) under gravity forcing, ( ∂ t + ( u · ∇ )) u = -1 ρ ∇ p -g ˆ z + ν ∇ 2 u . (2.1) Let us define the vorticity ω as ω = ∇ × u (2.2) and use vector identity ( u · ∇ ) u = ω × u + ∇ u 2 2 , which is valid for incompressible flows, i.e. ∇ · u = 0. Then equation (2.1) can be rewritten as ∂ t u + ω × u = -∇ B + ν ∇ 2 u , (2.3) where we have introduced the Bernoulli potential : B = p ρ + u 2 2 + gz. (2.4) 11 12 Fluid Dynamics via Examples and Solutions 2.1.2 Bernoulli theorems Irrotational flows are defined as the flows with zero vorticity field, ω = 0. For the irrotational flows, the velocity field can be represented as a gradient of a velocity potential, u = ∇ φ . For irrotational flow, the second term on the left-hand side of equation (2.3) is zero, and the viscous term is zero too by the incompressibility condition, ∇ 2 φ = ∇ · u = 0. (Another way to see this is to realise that for incompressible fields, ∇ 2 u = -∇ × ω ). Thus, this equation can be integrated once over a path in x . This gives Bernoulli theorem for time-dependent irrotational flow : ∂ t φ + B = C, (2.5) where C is a constant.

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