On the Calculation of the Effective Polytropic Index in Space Plasmas
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Polytropic Processes
Polytropic Processes a.cyclohexane.molecule November 2015 A polytropic process takes the general form of the common thermodynamic processes, and is defined by the equation n n n pV = C or p1V1 = p2V2 where n is an index we may manipulate to get our common thermodynamic processes. Source: nptel.ac.in Note: as n approaches 1, p has less and less influence; when n = 1, p has no influence and we thus have an isobaric process. Other n values (x in the graph) are possible, but not common. For a physical interpretation of these other values, check the Wikipedia page on polytropic processes. Because polytropic processes are general representations of thermodynamic pro- cesses, any quantities we calculate for a polytropic process will be a general equation for that quantity in all other thermodynamic processes. We proceed to find the work and heat associated with a polytropic process. We start with the definition of work: Z V2 w = − p dV V1 The polytropic equation pV n = C allows us to manipulate our formula for work in the following manner, as long as n 6= 1 V V Z 2 C Z 2 C V C V w = − dV = −C V −n dV = − V 1−n 2 = V 1−n 2 n V1 V1 V1 V V1 1 − n n − 1 Evaluating this expression, C CV 1−n − CV 1−n w = V 1−n − V 1−n = 2 1 n − 1 2 1 n − 1 n n We let the first C be p2V2 and the second C be p1V1 , such that the exponents cancel to obtain p V nV 1−n − p V nV 1−n p V − p V R(T − T ) w = 2 2 2 1 1 1 = 2 2 1 1 = 2 1 n − 1 n − 1 n − 1 where the last equality follows from the ideal gas law pV = RT and our deriva- tion for the work of a polytropic process is complete. -
Nuclear Astrophysics
Nuclear Astrophysics Lecture 3 Thurs. Nov. 3, 2011 Prof. Shawn Bishop, Office 2013, Ex. 12437 www.nucastro.ph.tum.de 1 Summary of Results Thus Far 2 Alternative expressions for Pressures is the number of atoms of atomic species with atomic number “z” in the volume V Mass density of each species is just: where are the atomic mass of species “z” and Avogadro’s number, respectively Mass fraction, in volume V, of species “z” is just And clearly, Collect the algebra to write And so we have for : If species “z” can be ionized, the number of particles can be where is the number of free particles produced by species “z” (nucleus + free electrons). If fully ionized, and 3 The mean molecular weight is defined by the quantity: We can write it out as: is the average of for atomic species Z > 2 For atomic species heavier than helium, average atomic weight is and if fully ionized, Fully ionized gas: Same game can be played for electrons: 4 Temp. vs Density Plane Relativistic - Relativistic Non g cm-3 5 Thermodynamics of the Gas 1st Law of Thermodynamics: Thermal energy of the system (heat) Total energy of the system Assume that , then Substitute into dQ: Heat capacity at constant volume: Heat capacity at constant pressure: We finally have: 6 For an ideal gas: Therefore, And, So, Let’s go back to first law, now, for ideal gas: using For an isentropic change in the gas, dQ = 0 This leads to, after integration of the above with dQ = 0, and 7 First Law for isentropic changes: Take differentials of but Use Finally Because g is constant, we can integrate -
MECH 230 – Thermodynamics 1 Final Workbook Solutions
MECH 230 – Thermodynamics 1 Final Workbook Solutions CREATED BY JUSTIN BONAL MECH 230 Final Exam Workbook Contents 1.0 General Knowledge ........................................................................................................................... 1 1.1 Unit Analysis .................................................................................................................................. 1 1.2 Pressure ......................................................................................................................................... 1 2.0 Energies ............................................................................................................................................ 2 2.1 Work .............................................................................................................................................. 2 2.2 Internal Energy .............................................................................................................................. 3 2.3 Heat ............................................................................................................................................... 3 2.4 First Law of Thermodynamics ....................................................................................................... 3 3.0 Ideal gas ............................................................................................................................................ 4 3.1 Universal Gas Constant ................................................................................................................. -
3. Thermodynamics Thermodynamics
Design for Manufacture 3. Thermodynamics Thermodynamics • Thermodynamics – study of heat related matter in motion. • Ma jor deve lopments d uri ng 1650 -1850 • Engineering thermodynamics mainly concerned with work produc ing or utili si ng machi nes such as - Engines, - TbiTurbines and - Compressors together w ith th e worki ng sub stances used i n th e machi nes. • Working substance – fluids: capable of deformation, energy transfer • Air an d s team are common wor king sub st ance Design and Manufacture Pressure F P = A Force per unit area. Unit: Pascal (Pa) N / m 2 1 Bar =105 Pa 1 standard atmosphere = 1.01325 Bar 1 Bar =14.504 Psi (pound force / square inch, 1N= 0.2248 pound force) 1 Psi= 6894.76 Pa Design and Manufacture Phase Nature of substance. Matter can exists in three phases: solid, liquid and gas Cycle If a substance undergoes a series of processes and return to its original state, then it is said to have been taken through a cycle. Design for Manufacture Process A substance is undergone a process if the state is changed by operation carried out on it Isothermal process - Constant temperature process Isobaric process - Constant pressure process Isometric process or isochoric process - Constant volume process Adiabatic Process No heat is transferred, if a process happens so quickly that there is no time to transfer heat, or the system is very well insulated from its surroundings. Polytropic process Occurs with an interchange of both heat and work between the system and its surroundings E.g. Nonadiabatic expansion or compression -
Chapter 7 BOUNDARY WORK
Chapter 7 BOUNDARY WORK Frank is struck - as if for the first time - by how much civilization depends on not seeing certain things and pretending others never occurred. − Francine Prose (Amateur Voodoo) In this chapter we will learn to evaluate Win, which is the work term in the first law of thermodynamics. Work can be of different types, such as boundary work, stirring work, electrical work, magnetic work, work of changing the surface area, and work to overcome friction. In this chapter, we will concentrate on the evaluation of boundary work. 110 Chapter 7 7.1 Boundary Work in Real Life Boundary work is done when the boundary of the system moves, causing either compression or expansion of the system. A real-life application of boundary work, for example, is found in the diesel engine which consists of pistons and cylinders as the prime component of the engine. In a diesel engine, air is fed to the cylinder and is compressed by the upward movement of the piston. The boundary work done by the piston in compressing the air is responsible for the increase in air temperature. Onto the hot air, diesel fuel is sprayed, which leads to spontaneous self ignition of the fuel-air mixture. The chemical energy released during this combustion process would heat up the gases produced during combustion. As the gases expand due to heating, they would push the piston downwards with great force. A major portion of the boundary work done by the gases on the piston gets converted into the energy required to rotate the shaft of the engine. -
Heat Transfer During the Piston-Cylinder Expansion of a Gas
AN ABSTRACT OF THE THESIS OF Matthew Jose Rodrigues for the degree of Master of Science in Mechanical Engineering presented on June 5, 2014. Title: Heat Transfer During the Piston-Cylinder Expansion of a Gas Abstract approved: ______________________________________________________ James A. Liburdy The expansion of a gas within a piston-cylinder arrangement is studied in order to obtain a better understanding of the heat transfer which occurs during this process. While the situation of heat transfer during the expansion of a gas has received considerable attention, the process is still not very well understood. In particular, the time dependence of heat transfer has not been well studied, with many researchers instead focused on average heat transfer rates. Additionally, many of the proposed models are not in agreement with each other and are usually dependent on experimentally determined coefficients which have been found to vary widely between test geometries and operating conditions. Therefore, the expansion process is analyzed in order to determine a model for the time dependence of heat transfer during the expansion. A model is developed for the transient heat transfer by assuming that the expansion behaves in a polytropic manner. This leads to a heat transfer model written in terms of an unknown polytropic exponent, n. By examining the physical significance of this parameter, it is proposed that the polytropic exponent can be related to a ratio of the time scales associated with the expansion process, such as a characteristic Peclet number. Experiments are conducted to test the proposed heat transfer model and to establish a relationship for the polytropic exponent. -
Astronomy 112: the Physics of Stars Class 10 Notes: Applications and Extensions of Polytropes in the Last Class We Saw That Poly
Astronomy 112: The Physics of Stars Class 10 Notes: Applications and Extensions of Polytropes In the last class we saw that polytropes are a simple approximation to a full solution to the stellar structure equations, which, despite their simplicity, yield important physical insight. This is particularly true for certain types of stars, such as white dwarfs and very low mass stars. In today’s class we will explore further extensions and applications of simple polytropic models, which we can in turn use to generate our first realistic models of typical main sequence stars. I. The Binding Energy of Polytropes We will begin by showing that any realistic model must have n < 5. Of course we already determined that solutions to the Lane-Emden equation reach Θ = 0 at finite ξ only for n < 5, which already hints that n ≥ 5 is a problem. Nonetheless, we have not shown that, as a matter of physical principle, models of this sort are unacceptable for stars. To demonstrate this, we will prove a generally useful result about the energy content of polytropic stars. As a preliminary to this, we will write down the polytropic relation (n+1)/n P = KP ρ in a slightly different form. Consider a polytropic star, and imagine moving down within it to the point where the pressure is larger by a small amount dP . The corre- sponding change in density dρ obeys n + 1 dP = K ρ1/ndρ. P n Similarly, since P = K ρ1/n, ρ P it follows that P ! K 1 dP ! d = P ρ(1−n)/ndρ = ρ n n + 1 ρ We can regard this relation as telling us how much the ratio of pressure to density changes when we move through a star by an amount such that the pressure alone changes by dP . -
Thermodynamic Analysis of an Irreversible Maisotsenko Reciprocating Brayton Cycle
entropy Article Thermodynamic Analysis of an Irreversible Maisotsenko Reciprocating Brayton Cycle Fuli Zhu 1,2,3, Lingen Chen 1,2,3,* and Wenhua Wang 1,2,3 1 Institute of Thermal Science and Power Engineering, Naval University of Engineering, Wuhan 430033, China; [email protected] (F.Z.); [email protected] (W.W.) 2 Military Key Laboratory for Naval Ship Power Engineering, Naval University of Engineering, Wuhan 430033, China 3 College of Power Engineering, Naval University of Engineering, Wuhan 430033, China * Correspondence: [email protected] or [email protected]; Tel.: +86-27-8361-5046; Fax: +86-27-8363-8709 Received: 20 January 2018; Accepted: 2 March 2018; Published: 5 March 2018 Abstract: An irreversible Maisotsenko reciprocating Brayton cycle (MRBC) model is established using the finite time thermodynamic (FTT) theory and taking the heat transfer loss (HTL), piston friction loss (PFL), and internal irreversible losses (IILs) into consideration in this paper. A calculation flowchart of the power output (P) and efficiency (η) of the cycle is provided, and the effects of the mass flow rate (MFR) of the injection of water to the cycle and some other design parameters on the performance of cycle are analyzed by detailed numerical examples. Furthermore, the superiority of irreversible MRBC is verified as the cycle and is compared with the traditional irreversible reciprocating Brayton cycle (RBC). The results can provide certain theoretical guiding significance for the optimal design of practical Maisotsenko reciprocating gas turbine plants. Keywords: finite-time thermodynamics; irreversible Maisotsenko reciprocating Brayton cycle; power output; efficiency 1. Introduction The revolutionary Maisotsenko cycle (M-cycle), utilizing the psychrometric renewable energy from the latent heat of water evaporating, was firstly provided by Maisotsenko in 1976. -
Quantum Mechanics Polytropic Process
Quantum Mechanics_polytropic process A polytropic process is a thermodynamic process that obeys the relation: where p is the pressure, v is specific volume, n, the polytropic index, is any real number, andC is a constant. The polytropic process equation is particularly useful for characterizing expansion and compression processes which include heat transfer. This equation can accurately characterize a very wide range ofthermodynamic processes, that range from n<0 to n= which covers, n=0 (Isobaric), n=1 (Isothermal), n=γ (Isentropic), n= (Isochoric) processes and all values of n in between. Hence the equation is polytropic in the sense that it describes many lines or many processes. In addition to the behavior of gases, it can in some cases represent some liquids and solids. The one restriction is that the process should display an energy transfer ratio of K=δQ/δW=constant during that process. If it deviates from that restriction it suggests the exponent is not a constant. For a particular exponent, other points along the curve can be calculated: Derivation Polytropic processes behave differently with various polytropic indice. Polytropic process can generate other basic thermodynamic processes. The following derivation is taken from Christians.[1] Consider a gas in a closed system undergoing an internally reversible process with negligible changes in kinetic and potential energy. The First Law of Thermodynamics is Define the energy transfer ratio, . For an internally reversible process the only type of work interaction is moving boundary work, given by Pdv. Also assume the gas is calorically perfect (constant specific heat) so du = cvdT. The First Law can then be written (Eq. -
Physico-Mathematical Model of a Hot Air Engine Using Heat from Low-Temperature Renewable Sources of Energy
U.P.B. Sci. Bull., Series D, Vol. 78, Iss. D, 2016 ISSN 1223-7027 PHYSICO-MATHEMATICAL MODEL OF A HOT AIR ENGINE USING HEAT FROM LOW-TEMPERATURE RENEWABLE SOURCES OF ENERGY Vlad Mario HOMUTESCU1, Marius-Vasile ATANASIU2, Carmen-Ema PANAITE3, Dan-Teodor BĂLĂNESCU4 The paper provides a novel physico-mathematical model able to estimate the maximum performances of a hot air Ericsson-type engine - which is capable to run on renewable energy, such as solar energy, geothermal energy or biomass. The computing program based on this new model allows to study the influences of the main parameters over the engine performances and to optimize their values. Keywords: Ericsson-type engine, renewable energy, theoretical model, maximum performances. 1. Introduction In the last decades we witnessed an extraordinary development of numerous engines and devices capable of transforming heat into work [1]. The field of the “unconventional” engines has risen again into the attention of the researchers and of the investors. Several former abandoned or marginal technical solutions of hot air engines were reanalyzed by researchers - as the Ericsson-type reciprocating engines [2], Stirling engines or Stirling-related engines [1], [3] and the Vuilleumier machine. The Ericsson-type hot air engine is an external combustion heat engine with pistons (usually, with pistons with reciprocating movement) and with valves. An Ericsson-type engine functional unit comprises two pistons. The compression process takes place inside a compression cylinder, and the expansion process takes place inside an expansion cylinder. Heat is added to the cycle inside a heater. Sometimes a heat recuperator or regenerator is used to recover energy from the exhaust gases, fig. -
20 — Polytropes [Revision : 1.1]
20 — Polytropes [Revision : 1.1] • Mechanical Equations – So far, two differential equations for stellar structure: ∗ hydrostatic equilibrium: dP GM = −ρ r dr r2 ∗ mass-radius relation: dM r = 4πr2ρ dr – Two equations involve three unknowns: pressure P , density ρ, mass variable Mr — cannot solve – Try to relate P and ρ using a gas equation of state — e.g., ideal gas: ρkT P = µ ...but this introduces extra unknown: temperature T – To eliminate T , must consider energy transport • Polytropic Equation-of-State – Alternative to having to do full energy transport – Used historically to create simple stellar models – Assume some process means that pressure and density always related by polytropic equation of state P = Kργ for constant K, γ – Polytropic EOS resembles pressure-density relation for adibatic change; but γ is not necessarily equal to usual ratio of specific heats – Physically, gases that follow polytropic EOS are either ∗ Degenerate — Fermi-Dirac statistics apply (e.g., non-relativistic degenerate gas has P ∝ ρ5/3) ∗ Have some ‘hand-wavy’ energy transport process that somehow maintains a one-to- one pressure-density relation • Polytropes – A polytrope is simplified stellar model constructed using polytropic EOS – To build such a model, first write down hydrostatic equilibrium equation in terms of gradient of graqvitational potential: dP dΦ = −ρ dr dr – Eliminate pressure using polytropic EOS: K dργ dΦ = − ρ dr dr – Rearrange: Kγ dργ−1 dΦ = − γ − 1 dr dr – Solving: K(n + 1)ρ1/n = −Φ where n ≡ 1/(γ − 1) is the polytropic index (do not -
Stellar Structure — Polytrope Models for White Dwarf Density Profiles and Masses
Stellar Structure | Polytrope models for White Dwarf density profiles and masses We assume that the star is spherically symmetric. Then we want to calculate the mass density ρ(r) (units of kg/m3) as a function of distance r from its centre. The mass density will be highest at its centre and as the distance from the centre increases the density will decrease until it becomes very low in the outer atmosphere of the star. At some point ρ(r) will reach some low cutoff value and then we will say that that value of r is the radius of the star. The mass dnesity profile is calculated from two coupled differential equations. The first comes from conservation of mass. It is dm(r) = 4πr2ρ(r); (1) dr where m(r) is the total mass within a sphere of radius r. Note that m(r ! 1) = M, the total mass of the star is that within a sphere large enough of include the whole star. In practice the density profile ρ(r) drops off rapidly in the outer atmosphere of the star and so can obtain M from m(r) at some value of r large enough that the density ρ(r) has become small. Also, of course if r = 0, then m = 0: the mass inside a sphere of 0 radius is 0. Thus the one boundary condition we need for this first-order ODE is m(r = 0) = 0. The second equation comes from a balance of forces on a spherical shell at a radius r. The pull of gravity on the shell must equal the outward pressure at equilibrium; see Ref.